diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhycl" "b/data_all_eng_slimpj/shuffled/split2/finalzzhycl" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhycl" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nLet $X \\subset \\PP^N$ be an $n$-dimensional projective variety over an algebraically closed field $\\Bbbk$ of arbitrary characteristic.\nThe \\emph{Gauss map} $\\gamma$ of $X$\nis defined as a rational map\n\\[\n\\gamma: X \\dashrightarrow \\mathbb{G}(n, \\PP^N),\n\\]\nwhich sends each smooth point $x \\in X$ to the embedded tangent space $\\mathbb{T}_xX$ of $X$ at $x$ in $\\PP^N$.\nThe Gauss map is a classical subject and has been studied by many authors.\nFor example, it is well known\nthat a general fiber of the Gauss map $\\gamma$\nis (an open subset of) a linear subvariety of $\\PP^N$ in characteristic zero\n(P.~Griffiths and J.~Harris \\cite[(2.10)]{GH},\nF.~L.~Zak \\cite[I, 2.3.\\ Theorem (c)]{Zak}).\nThe linearity of general fibers of $\\gamma$ also holds\nin arbitrary characteristic\nif $\\gamma$ is {separable} \\cite[Theorem 1.1]{expshr}.\nWe denote by $\\delta_{\\gamma}(X)$ the dimension of a general fiber of $\\gamma$,\nand call it the \\emph{Gauss defect} of $X$ (see \\cite[2.3.4]{FP}).\nThe Gauss map $\\gamma$ is said to be \\emph{degenerate} if $\\delta_{\\gamma}(X) > 0$.\n\n\\vspace{2mm}\nIn this paper, we investigate the Gauss map of \\emph{toric} $X \\subset \\PP^N$;\nmore precisely, we consider a (not necessarily normal)\ntoric variety $X \\subset \\PP^N$ such that the action of the torus on $X$ extends to the whole space $\\PP^N$.\nIt is known that such $X$ is projectively equivalent to a projective toric variety $X_A$ \nassociated to a finite subset $A$ of a free abelian group $M$ (see \\cite[Ch.~5, Proposition 1.5]{GKZ}).\nThe construction of $X_A$ is as follows.\n\n\nLet $M$ be a free abelian group of rank $n$\nand let $\\Bbbk [M] = \\bigoplus_{u \\in M} \\Bbbk z^u$ be the group ring of $M$ over $\\Bbbk$.\nWe denote by $T_M$ the algebraic torus $\\Spec \\Bbbk[M]$.\nFor a finite subset $A= \\{u_0, \\ldots, u_N\\} \\subset M$,\nwe define the toric variety $X_A$ to be the closure of the image of the morphism\n\\begin{equation}\\label{eq:defphiA}\n \\varphi_A : T_M \\rightarrow \\mathbb{P}^N \\ : \\ t \\mapsto [z^{u_0} (t) : \\cdots: z^{u_N}(t) ].\n\\end{equation}\nWe set $ \\langle A - A \\rangle \\subset M $ (resp.\\ $ \\langle A - A \\rangle_{\\Bbbk} \\subset M_{\\Bbbk}:= M \\otimes_{\\mathbb {Z}} \\Bbbk $)\nto be\nthe subgroup of $M$ (reap.\\ the $\\Bbbk$-vector subspace of $ M_{\\Bbbk}$) generated by $A -A : = \\{ u-u' \\, | \\, u,u' \\in A \\} $.\nThe algebraic torus $T_{ \\langle A - A \\rangle}$ acts on $X_A$,\nand $T_{ \\langle A - A \\rangle}$ is contained in $X_A$ as an open dense orbit.\nIn this paper, for short, we say a projective variety $X \\subset \\PP^N$ is \\emph{projectively embedded toric}\nif $X$ is projectively equivalent to the projective toric variety $X_A$ associated to some finite subset $A$ of a free abelian group $M$ of finite rank.\n\n\n\n\n\n\n\n\n\n\n\nWe denote by $\\Aff(A)$ (resp.\\ $\\Aff_{\\Bbbk}(A)$)\nthe affine sublattice of $M$ (resp.\\ the $\\Bbbk$-affine subspace of $M_{\\Bbbk}$)\nspanned by $A$.\nIn other words,\n$\\Aff(A)$ (resp.\\ $\\Aff_{\\Bbbk}(A)$) is\nthe set of linear combinations $\\sum_i a_i u_i \\in M$ with $a_i \\in \\mathbb{Z}$ (resp.\\ $a_i \\in \\Bbbk$),\n$\\sum_i a_i = 1$, and $u_i \\in A$. We say that $A$ \\emph{spans} the affine lattice $M$\n(resp.\\ the $\\Bbbk$-affine space $M_{\\Bbbk}$)\nif $\\Aff(A) = M$ (resp.\\ $\\Aff_{\\Bbbk}(A) = M_{\\Bbbk}$).\n\n\\vspace{1em}\n\nProjective geometry of toric $X_A$ has been investigated in view of the projective dual\nin many papers (\\cite{CD}, \n\\cite{DDP}, \\cite{DN}, \\cite{DR}, \\cite{GKZ}, \\cite{MT}, etc.).\nOn the other hand, the Gauss map of $X_A$ has not been well studied yet.\nIn the following result, we describe the structure of Gauss maps of toric varieties\nin terms of combinatorics.\n\n\n\\begin{thm}\\label{thm_structure}\n Let $\\Bbbk$ be an algebraically closed field of arbitrary characteristic, and\n let $M$ be a free abelian group of rank $n$.\n For a finite subset $A= \\{u_0, \\ldots, u_N\\} \\subset M$ which spans the affine lattice $M$,\n set\n \\begin{gather*}\n B := \\{ u_{i_0} + u_{i_1} + \\cdots + u_{i_n} \\in M \\, | \\, u_{i_0}, u_{i_1} , \\ldots , u_{i_n} \\text{ span the $\\Bbbk$-affine space } M_{\\Bbbk} \\}\n \\end{gather*}\n and let\n $\\pi: M \\rightarrow M':= M \/ (\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)$ be the natural projection.\n Let $\\gamma: X_A \\dashrightarrow \\mathbb {G}(n, \\mathbb{P}^N)$ be the Gauss map of the toric variety $X_A \\subset \\PP^N$.\n Then the following hold.\n \\begin{enumerate} \\item\\label{thm:item-image}\n The closure $\\overline{\\gamma(X_A)}$ of the image of $\\gamma$,\n which is embedded in a projective space by the Pl\\\"ucker embedding of $ \\mathbb{G} (n, \\mathbb{P}^{N})$,\n is projectively equivalent to the toric variety $ X_{B} $.\n \\item\\label{thm:item-map}\n The restriction of $\\gamma : X_A \\dashrightarrow \\overline{\\gamma (X_A)} \\cong X_{B}$ on $T_M \\subset X_A$\n is the morphism\n \\[\n T_M = \\Spec \\Bbbk[M] \\twoheadrightarrow T_{ \\langle B-B \\rangle} = \\Spec \\Bbbk[ \\langle B-B \\rangle] \\subset X_{B}\n \\]\n induced by the inclusion $ \\langle B-B \\rangle \\subset M$.\n \\item\\label{thm:item-fiber}\n Let $F \\subset T_M$ be an irreducible component of any fiber of $\\gamma|_{T_M}$ with the reduced structure.\n Let $T_{M'} \\hookrightarrow T_M$ be the subtorus induced by $\\pi$.\n Then $F$ is a translation of $T_{M'}$ by an element of $T_M$,\n and \n the closure $\\overline{F} \\subset X_A$\n is projectively equivalent to the toric variety $X_{\\pi(A)}$.\n\n \\end{enumerate}\n In particular,\n we have $ \\delta_{\\gamma}(X_{A}) = \\rk M' = n -\\rank \\langle B-B \\rangle$. \n\\end{thm}\n\n\n\n\n\n\n\n\n\nNote that there is no loss of generality in assuming that $A $ spans the affine lattice $M$ in the above theorem (see \\autoref{thm:span-A-M}).\nWe also study when the Gauss map of toric $X_A$ is degenerate (i.e., $\\rank \\langle B-B \\rangle < n$),\nand give a developability criterion for covering families of $X_A$\n(see \\autoref{sec:criterion-degeneracy}). \n\n\\vspace{2mm}\n\n\nIn the following example, we illustrate the description in \\autoref{thm_structure}.\n\n\n\\begin{ex}\\label{ex_intro}\n Let $M=\\mathbb {Z}^2$ and\n \\[\n A = \\left\\{\n \\begin{bmatrix}\n 0 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 1\n \\end{bmatrix},\n \\begin{bmatrix}\n 1 \\\\ -1\n \\end{bmatrix}\n ,\\begin{bmatrix}\n -1 \\\\ -1\n \\end{bmatrix}\n \\right\\} \\subset \\mathbb {Z}^2,\n \\]\n where we write elements in $\\mathbb {Z}^2$ by column vectors.\n We consider the Gauss map $\\gamma$ of the toric surface $X_A \\subset \\mathbb{P}^3$.\n When $\\chara \\Bbbk \\neq 2$,\n \\[\n B = \\left\\{\n \\begin{bmatrix}\n 0 \\\\ -1\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ -2\n \\end{bmatrix},\n \\begin{bmatrix}\n -1 \\\\ 0\n \\end{bmatrix}\n ,\\begin{bmatrix}\n 1 \\\\ 0\n \\end{bmatrix}\n \\right\\} \\subset \\mathbb {Z}^2.\n \\]\n Hence $ \\langle B-B \\rangle =M =\\mathbb {Z}^2$, and $\\gamma$ is birational due to \\ref{thm:item-map} in \\autoref{thm_structure}.\n On the other hand, when $\\chara \\Bbbk =2$,\n \\begin{align*}\n B &= \\left\\{\n \\begin{bmatrix}\n -1 \\\\ 0\n \\end{bmatrix}\n ,\\begin{bmatrix}\n 1 \\\\ 0\n \\end{bmatrix}\n \\right\\},\\\n \\lin{B-B} = \\left\\langle\n \\begin{bmatrix}\n 2\n \\\\\n 0\n \\end{bmatrix}\n \\right\\rangle,\\\n \\lin{B-B}_{\\mathbb{R}} \\cap M\n = \\left\\langle\n \\begin{bmatrix}\n 1 \\\\ 0\n \\end{bmatrix}\n \\right\\rangle\n \\subset \\mathbb {Z}^2,\n \\end{align*}\n and $\\pi(A) = \\{0, 1,-1\\} \\subset \\mathbb {Z}^2 \/ (\\langle B-B \\rangle_{\\mathbb {R}} \\cap M) = \\mathbb {Z}^1$.\n Thus \\ref{thm:item-image} implies that $\\overline{\\gamma(X_A)} \\cong X_{B} =\\mathbb{P}^1$,\n and \\ref{thm:item-map} implies that\n $\\gamma|_{T_M}: T_M= (\\KK^{\\times})^2 \\twoheadrightarrow T_{\\lin{B-B}}=\\KK^{\\times}$ is given by $(z_1,z_2) \\mapsto z_1^2$.\n From \\ref{thm:item-fiber}, \n a general fiber of $\\gamma$ with the reduced structure is projectively equivalent to the smooth conic $X_{\\pi(A)}$.\n \\begin{figure}[htbp]\n \\[\n \\begin{xy}\n (10,0)=\"A\",(0,10)=\"B\",\n (-10,-10)=\"C\",\n (9.9,0.1)=\"E\",(-5,0.1)=\"F\",\n (9.8,-0.1)=\"I\",(-5,-0.1)=\"J\",\n (-20,0)=\"1\",(20,0)=\"2\",\n (0,-17)=\"3\",(0,18)=\"4\",\n (50,-17)=\"7\",(50,18)=\"8\",\n (0,10)*{\\bullet},(0,0)*{\\bullet},\n (-10,-10)*{\\bullet},(10,-10)*{\\bullet},\n (-10,0)*{\\times},(10,0)*{\\times},\n (50,10)*{\\sqbullet},(50,0)*{\\sqbullet},\n (50,-10)*{\\sqbullet},\n (53,10)*{1},(53,0)*{0},(55,-10)*{-1},\n (16,18)*{\\bullet\\;A}, \n (16,13)*{\\times\\;B},\n (65,16)*{\\sqbullet\\;\\pi(A)},\n (35,3)*{\\pi}\n \\ar \"1\";\"2\"\n \\ar \"3\";\"4\"\n \\ar \"7\";\"8\"\n \\ar (27,0);(43,0)\n \\end{xy}\n \\]\n \\caption{$\\chara \\Bbbk =2$.}\n \\label{figure1}\n \\end{figure}\n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn characteristic zero, it is well known that, if a projective variety\nis the join of some varieties,\nthen its Gauss map is degenerate due to Terracini's lemma (see \\cite[2.2.5]{FP}, \\cite[Ch.\\,II, 1.10.\\,Proposition]{Zak}).\nFor toric varieties in characteristic zero, this is the only case when the Gauss map is degenerate;\nmore precisely, we have:\n\n\n\n\n\n\\begin{cor}\\label{cor_join}\n Let $X \\subset \\mathbb{P}^N$ be a projectively embedded toric variety in $\\chara\\Bbbk = 0$.\n Then there exist disjoint torus invariant closed subvarieties $ X_0, \\ldots,X_{\\delta_{\\gamma}(X)} \\subset X$\n such that $X$ is the join of $ X_0, \\ldots,X_{\\delta_{\\gamma}(X)}$.\n\\end{cor}\n\nIf the Gauss map of $X_A \\subset \\PP^N$ is separable,\n$A$ is written as a \\emph{Cayley sum} of certain finite subsets $A^0, \\dots, A^{\\delta_{\\gamma}(X)}$ in any characteristic\n(see \\autoref{thm:subvar-in-join} for details).\nHowever, the statement of \\autoref{cor_join} does \\emph{not} hold in general in positive characteristic,\neven if the Gauss map is separable\n(see \\autoref{thm:sep-gamma-X-not-join}).\n\n\\vspace{1em}\n\nNext, let us focus on inseparable Gauss maps.\nA.~H.~Wallace \\cite[\\textsection 7]{Wallace} showed that\nthe Gauss map $\\gamma$ of a projective variety can be \\emph{inseparable}\nin positive characteristic.\nIn this case,\nit is possible that\na general fiber of $\\gamma$ is \\emph{not} a linear subvariety of $\\PP^N$;\nthe fiber can be a union of points\n(H.~Kaji \\cite[Example 4.1]{Kaji1986} \\cite{Kaji1989}, J.~Rathmann \\cite[Example~2.13]{Rathmann}, A.~Noma \\cite{Noma2001}),\nand can be a non-linear variety\n(S.~Fukasawa \\cite[\\textsection{}7]{Fukasawa2005}).\nIn fact, Fukasawa \\cite{Fukasawa2006}\nshowed that \\emph{any} projective variety appears as a general fiber of the Gauss map of some projective variety.\n\n\nAs we will see in \\autoref{cor_rk°},\n\\autoref{thm_structure} provides\nseveral computations on the Gauss map $\\gamma$ of toric varieties\n(e.g., the rank, separable degree, inseparable degree). \nWe also obtain the toric version of Fukasawa's result \\cite{Fukasawa2006} as follows:\n\n\n\n\n\n\n\n\n\n\n\n\\begin{thm}[Special case of \\autoref{cor_im&fiber}]\\label{thm_Fukasawa1}\n\n Assume $\\chara \\Bbbk >0$.\n Let $Y \\subset \\mathbb{P}^{N'} $ and $Z \\subset \\mathbb{P}^{N''}$ be projectively embedded toric varieties.\n If $n:=\\dim(Y)+\\dim(Z)$ is greater than or equal to $N'$, then\n there exists an $n$-dimensional projectively embedded toric variety\n $X \\subset \\mathbb{P}^{n+N''}$\n satisfying the following conditions:\n \\begin{enumerate}[\\normalfont (i)]\n \\item\n (The closure of) a general fiber of the Gauss map $\\gamma$ of $X$ with the reduced structure is projectively equivalent to $Y$.\n \\item \n (The closure of) the image of $\\gamma$ is projectively equivalent to $Z$.\n \\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\n\n\nBy \\autoref{thm_Fukasawa1}, any projectively embedded toric variety appears\nas a general fiber and the image of the Gauss map of\na certain projectively embedded \\emph{toric} variety; moreover\nwe can also control the \\emph{rank} of $\\gamma$,\nand the \\emph{number} of the irreducible components of a general fiber of $\\gamma$\n(see \\autoref{sec:posit-char-case}, for details).\n\n\n\\vspace{2mm}\nThis paper is organized as follows.\nIn \\autoref{sec:prel-toric-vari}, we recall some basic properties of toric varieties.\nIn \\autoref{sec:structure-gauss-maps}, we describe the structure of the Gauss maps\nof toric varieties in a combinatorial way, and prove \\autoref{thm_structure}.\nIn \\autoref{sec:degen-gauss-maps}, we investigate when the Gauss maps are degenerate,\nand give a developability criterion.\nAs a result, we show \\autoref{cor_join}.\nIn \\autoref{sec:posit-char-case}, we present two constructions of projectively embedded toric varieties, yielding \\autoref{thm_Fukasawa1}.\n\n\n\\subsection*{Acknowledgments}\n\nThe authors would like to express their gratitude to Professors\nSatoru Fukasawa and Hajime Kaji for their valuable comments and advice.\nIn particular, \\autoref{thm:a-vari-kaji} is due to Professor Kaji.\nThe first author was partially supported by JSPS KAKENHI Grant Number 25800030.\nThe second author was partially supported by JSPS KAKENHI Grant Number 25887010.\n\n\n\n\n\n\\section{Preliminary on toric varieties}\n\\label{sec:prel-toric-vari}\n\n\n\n\n\n\n\nTwo projective varieties $X_1 \\subset \\mathbb{P}^{N_1}$ and $X_2 \\subset \\mathbb{P}^{N_2}$ are said to be\n\\emph{projectively equivalent} if\nthere exist linear embeddings $\\jmath_i: \\mathbb{P}^{N_i} \\hookrightarrow \\PP^N$\n(i.e., $\\jmath_i^*(\\mathscr{O}_{\\PP^N}(1)) = \\mathscr{O}_{\\mathbb{P}^{N_i}}(1)$)\nsuch that $\\jmath_1(X_1) = \\jmath_2(X_2)$.\n(Indeed, we can take $N = \\max \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{N_1,N_2}$.)\n\n\n\\vspace{2mm}\nThe following two lemmas about toric varieties are well known,\nbut we prove them for the convenience of the reader.\n\n\\begin{lem}\\label{lem_lattice_hom}\n Let $\\pi : M \\rightarrow M'$ be a surjective homomorphism between free abelian groups of finite ranks.\n Let $\\iota : T_{M'} \\hookrightarrow T_M$ be the embedding induced by $\\pi$.\n For a finite set $A \\subset M$ with $\\Aff(A)=M$\n the closure of $\\iota(T_{M'})$ in $X_A$ is projectively equivalent to $X_{\\pi(A)}$.\n The translations\n $ \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{t \\cdot \\overline{\\iota(T_{M'})}}_{t \\in T_M}$\n of the closure $\\overline{\\iota(T_{M'})}$\n under the action of $T_M$ on $X_A$ give a covering family of $X_A$, and\n each translation is also projectively equivalent to $X_{\\pi(A)}$.\n\\end{lem}\n\n\n\\begin{proof}\n Let $A=\\{ u_0,\\ldots, u_N\\}$ and $\\pi(A)= \\{u'_0, \\ldots, u'_{N'}\\}$ for $N=\\# A-1$ and $N' = \\# \\pi(A) -1$.\n We define a linear embedding $\\jmath : \\mathbb{P}^{N'} \\rightarrow \\mathbb{P}^N$ by\n \\[\n \\jmath ( [X'_0 : \\cdots: X'_{N'}]) = [X_0 : \\cdots : X_N],\n \\]\n where for each $i$, we set $X_i := X'_j$ for $j$ such that $\\pi(u_i)=u'_j$. \n Then we have the following commutative diagram\n \\[\n \\xymatrix{\n T_{M'} \\ar@{^{(}->}[d]^{\\iota} \\ar[r]^{\\varphi_{\\pi(A)}} \\ar@{}[dr]|\\circlearrowleft & \\mathbb{P}^{N'} \\ar@{^{(}->}[d]^{\\jmath} \\\\\n T_M \\ar[r]^{\\varphi_{A}} & \\mathbb{P}^N.\\\\\n }\n \\]\n Hence\n $\\overline{\\iota(T_{M'})}$ is projectively equivalent to $X_{\\pi(A)}$.\n Since the action of $T_M$ on $X_A$ extends to $\\PP^N$ (see \\cite[Ch.~5, Proposition~1.5]{GKZ}), translations of $\\overline{\\iota(T_{M'})}$ are also\n projectively equivalent to $X_{\\pi(A)}$.\n Since $\\iota(T_{M'})$ is non-empty and contained in $T_M$,\n the translations give a covering family of $X_A$.\n\\end{proof}\n\n\n\nLet $f: X \\dashrightarrow Y$ be a rational map between varieties. For a smooth point $x \\in X$,\nwe denote by $d_xf: t_xX \\rightarrow t_{f(x)}Y$\nthe tangent map\nbetween Zariski tangent spaces at $x$ and $f(x)$.\nThe \\emph{rank} of $f$, denoted by $\\rk (f)$, is defined to be the rank of the $\\Bbbk$-linear map\n$d_xf$ for general $x \\in X$.\nRecall that $f$ is said to be \\emph{separable} if \nthe field extension $K(X)\/K(f(X))$ is separable;\nthis condition is equivalent to $\\rk(f) = \\dim(f(X))$.\n\n\n\n\n\\begin{lem}\\label{lem_rk°}\n Let $M$ be a free abelian group of finite rank.\n Let $M'' $ be a subgroup of $M$ and $g : T_M \\twoheadrightarrow T_{M''} $ be the morphism induced by the inclusion $\\beta : M'' \\hookrightarrow M$.\n \\begin{enumerate}[\\normalfont \\quad (a)]\n \\item\\label{item:lem_rk°:1}\n The inclusions $M'' \\subset M''_{\\mathbb {R}} \\cap M \\subset M$ induce\n a decomposition of $g$\n \\[\n T_M \\stackrel{g_1}{\\rightarrow} T_{M''_{\\mathbb {R}} \\cap M} \\stackrel{g_2}{\\rightarrow} T_{M''},\n \\]\n where\n $g_1$ is a morphism with reduced and irreducible fibers, and $g_2$ is a finite morphism.\n\n \\item\\label{item:lem_rk°:2} The rank of $g$ is equal to the rank of the $\\Bbbk$-linear map $\\beta_{\\Bbbk} : M''_{\\Bbbk} \\rightarrow M_{\\Bbbk} $ obtained by tensoring $\\Bbbk$ with $\\beta : M'' \\hookrightarrow M$.\n In particular, $g$ is separable if and only if\n $\\rk (\\beta_{\\Bbbk}) = \\rank(M'')$.\n \\item\\label{item:lem_rk°:3} Assume $p = \\chara \\Bbbk > 0$.\n Let $a$ be the index $ [M''_{\\mathbb {R}} \\cap M : M'']$ and write $a = p^s b$ with integers $s \\geq 0, b \\geq 1 $ such that $ p \\nmid b$.\n Then the degree, separable degree, inseparable degree of the finite morphism $g_2 $ are $a, b , p^s$ respectively.\n \\end{enumerate}\n\\end{lem}\n\n\n\n\n\\begin{proof}\n Set $n= \\rank M, k= \\rank M''$.\n By the elementary divisors theorem (see \\cite[III, Theorem~7.8]{Lang}, for example),\n there exists a basis $e_1,\\ldots,e_n $ of $M$ such that\n \\[\n M'' = \\mathbb {Z} a_1 e_1 \\oplus \\cdots \\oplus \\mathbb {Z} a_k e_k \\subset \\mathbb {Z} e_1 \\oplus \\cdots \\oplus \\mathbb {Z} e_n =M\n \\]\n for some positive integers $a_i$. Set $e''_i := a_i e_i \\in M''$.\n By the bases $e_1,\\ldots,e_n $ of $M$ and $e''_1,\\ldots, e''_k $ of $M''$,\n we identify $M $ and $ M'' $ with $\\mathbb {Z}^n$ and $ \\mathbb {Z}^k$ respectively.\n Then $g : T_M = (\\KK^{\\times})^n \\rightarrow T_{M''} = (\\KK^{\\times})^k $ is described as\n \\begin{align}\\label{eq_explicit_g}\n (\\KK^{\\times})^n \\rightarrow (\\KK^{\\times})^k \\quad : \\quad (z_1, \\ldots,z_n) \\mapsto (z_1^{a_1}, \\ldots,z_k^{a_k}).\n \\end{align}\n \\vspace{.5mm}\n\n \\noindent\n \\ref{item:lem_rk°:1} \n By \\ref{eq_explicit_g},\n $g$ is decomposed as\n \\[\n (\\KK^{\\times})^n \\rightarrow (\\KK^{\\times})^k \\rightarrow (\\KK^{\\times})^k \\quad : \\quad (z_1, \\ldots,z_n) \\mapsto (z_1, \\ldots,z_k) \\mapsto (z_1^{a_1}, \\ldots,z_k^{a_k}).\n \\]\n Since $M''_{\\mathbb {R}} \\cap M = \\mathbb {Z} e_1 \\oplus \\cdots \\oplus \\mathbb {Z} e_k \\subset M$,\n the assertion of \\ref{item:lem_rk°:1} \n follows.\n\n \\vspace{1ex}\n \\noindent\n \\ref{item:lem_rk°:3} \n Write $a_i = p^{s_i} b_i$ by integers $s_i \\geq 0, b_i \\geq 1 $ such that $ p \\nmid b_i$.\n Since $g_2$ is the morphism\n \\[\n (\\KK^{\\times})^k \\rightarrow (\\KK^{\\times})^k \\quad : \\quad (z_1, \\ldots,z_k) \\mapsto (z_1^{a_1}, \\ldots,z_k^{a_k}) = (z_1^{p^{s_1} b_1}, \\ldots,z_k^{p^{s_k} b_k}),\n \\]\n the degree, separable degree, inseparable degree of $g_2 $ are\n $\\prod_{i=1}^k a_i =a, \\prod_{i=1}^k b_i =b, \\prod_{i=1}^k p^{s_i} = p^s$\n respectively.\n\n\n\n \\vspace{1ex}\n \\noindent\n \\ref{item:lem_rk°:2} \n If $\\chara \\Bbbk=0$, this statement is clear from \\ref{eq_explicit_g}.\n Assume $p= \\chara \\Bbbk >0$ and use the notation $a_i, s_i, b_i$ as above.\n Then the rank of $g$ is equal to\n $\\# \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}*{1 \\leq i \\leq k}{s_i = 0}$.\n On the other hand,\n $\\beta_{\\Bbbk} : M''_{\\Bbbk} = \\Bbbk^{k} \\rightarrow M_{\\Bbbk} = \\Bbbk^n$ is the $\\Bbbk$-linear map defined by $e''_i \\mapsto a_i e_i = p^{s_i} b_i e_i \\in M_{\\Bbbk}$ for $ 1 \\leq i \\leq k$.\n Hence the rank of $\\beta_{\\Bbbk}$ is also equal to $\\# \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}*{1 \\leq i \\leq k}{s_i = 0}$.\n Thus we have $\\rk (g) =\\rk (\\beta_{\\Bbbk})$.\n The last statement follows from $\\dim g(T_M) =\\dim T_{M''} =\\rk (M'')$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Structure of Gauss maps}\n\\label{sec:structure-gauss-maps}\n\nIn this section,\nwe prove Theorem \\ref{thm_structure}\nand describe several invariants (e.g., the rank) of Gauss maps of toric varieties\nby combinatorial data.\n\n\nIn order to investigate a toric variety $X_A$ for $A \\subset M$,\nwe may assume that $A$ spans the affine lattice $M$ due to the following remark.\n\n\\begin{rem}\\label{thm:span-A-M}\n For a finite subset $A \\subset M$,\n let $\\theta : \\mathbb {Z}^m \\rightarrow \\Aff (A) $ be an affine isomorphism for $ m =\\rank \\langle A -A \\rangle $.\n Then $ \\theta^{-1}(A)$ spans $\\mathbb {Z}^m$ as an affine lattice and\n $X_{\\theta^{-1}(A)}$ is naturally identified with $X_A$\n by \\cite[Chapter 5, Proposition 1.2]{GKZ}.\n Hence any projectively embedded toric variety $X$ is projectively equivalent\n to $X_A$ for some $A \\subset M$ with $\\Aff (A)=M$.\n\\end{rem}\n\nLet $A = \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_0, u_1, \\dots, u_N} \\subset M:=\\mathbb{Z}^n$ be a finite subset\nwhich spans the affine lattice $M$.\nWe denote each $u_i$ by a column vector as\n\\[\nu_i =\n\\begin{bmatrix}\n u_{i,1} \\\\ \\vdots \\\\ u_{i,n}\n\\end{bmatrix}.\n\\]\nThen the morphism $\\varphi_A $, defined by \\ref{eq:defphiA} in \\autoref{sec:introduction},\nis described as\n\\[\n\\varphi_A: (\\KK^{\\times})^n \\rightarrow \\PP^N \\quad : \\quad z = (z_1, \\dots, z_n) \\mapsto [z^{u_0} : z^{u_1} : \\dots : z^{u_N}],\n\\]\nwhere\n$z^{u_i} := z_1^{u_{i,1}} z_2^{u_{i,2}}\\cdots z_n^{u_{i,n}}$.\nBy the assumption that $A$ spans $\\mathbb{Z}^n$ as an affine lattice,\n$\\phi_A$ is an isomorphism onto an open subset of $X_A$.\n\n\n\nLet us study the Gauss map\n$\\gamma: X_A \\dashrightarrow \\mathbb{G}(n, \\PP^N)$ of $X_A \\subset \\PP^N$.\n\n\n\\begin{lem}\\label{thm:matrix-Gamma}\n Let $A, \\phi_A$ be as above, and\n let $x \\in (\\KK^{\\times})^n$. Then\n $\\gamma(\\varphi_A(x)) \\in \\mathbb{G}(n, \\PP^N)$\n is expressed by the $\\Bbbk$-valued $(n+1) \\times (N+1)$ matrix $\\Gamma(x)$;\n more precisely, $\\gamma(\\varphi_A(x))$\n corresponds to the $n$-plain (i.e., $n$-dimensional linear subvariety of $\\PP^N$)\n spanned by the $n+1$ points which are given as the row vectors of $\\Gamma(x)$, where\n \\begin{align*}\n \\Gamma &: =\n \\begin{bmatrix}\n z^{u_0} & z^{u_1} & \\cdots & z^{u_N}\n \\\\\n u_{0,1} \\cdot z^{u_0} & u_{1,1} \\cdot z^{u_1} & \\cdots & u_{N,1} \\cdot z^{u_N}\n \\\\\n \\vdots & \\vdots && \\vdots\n \\\\\n u_{0,n} \\cdot z^{u_0} & u_{1,n} \\cdot z^{u_1} & \\cdots & u_{N,n} \\cdot z^{u_N}\n \\end{bmatrix}\n \\\\\n & \\ =\n\\Bigg[ \\,\n z^{u_0} \\cdot \n \\begin{bmatrix}\n 1\n \\\\\n u_0\n \\end{bmatrix}\n \\ \n z^{u_1} \\cdot \n \\begin{bmatrix}\n 1\n \\\\\n u_1\n \\end{bmatrix}\n \\ \n \\cdots\n \\ \n z^{u_N} \\cdot \n \\begin{bmatrix}\n 1\n \\\\\n u_N\n \\end{bmatrix}\n \\,\n \\Bigg]\n\\end{align*}\n\\end{lem}\n\\begin{proof}\n Let $L_x \\subset \\PP^N$ be the $n$-plane spanned by\n the $n+1$ points which are given as the row vectors of\n \\begin{equation}\\label{eq:mat-TTxX-0}\n \\begin{bmatrix}\n z^{u_0} & \\cdots & z^{u_N}\n \\\\\n \\tdiff{(z^{u_0})}{z_1} & \\cdots & \\tdiff{(z^{u_N})}{z_1}\n \\\\\n \\vdots && \\vdots\n \\\\\n \\tdiff{(z^{u_0})}{z_n} & \\cdots & \\tdiff{(z^{u_N})}{z_n}\n \\end{bmatrix}(x).\n \\end{equation}\n Then $L_x$ coincides with the embedded tangent space $\\mathbb{T}_xX$,\n because of the equality $t_{x'}\\hat L_x = t_{x'}\\hat X$ of Zariski tangent spaces in $t_{x'}\\mathbb {A}^{N+1}$ with $x' \\in \\hat x \\setminus \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0}$,\n where $\\hat S \\subset \\mathbb {A}^{N+1}$ means the affine cone of $S \\subset \\PP^N$.\n On the other hand, \\ref{eq:mat-TTxX-0} is calculated as\n \\[\n \\begin{bmatrix}\n z^{u_0} & \\cdots & z^{u_N}\n \\\\\n u_{0,1} \\cdot z^{u_0}\/z_1 & \\cdots & u_{N,1} \\cdot z^{u_N}\/z_1\n \\\\\n \\vdots && \\vdots\n \\\\\n u_{0,n} \\cdot z^{u_0}\/z_n & \\cdots & u_{N,n} \\cdot z^{u_N}\/z_n\n \\end{bmatrix}.\n \\]\n Since each row vector means the homogeneous coordinates of a point of $\\PP^N$,\n by multiplying $z_i$ with the $(i+1)$-th row vector for $1 \\leq i \\leq n$,\n we have the matrix $\\Gamma$ and the assertion.\n\\end{proof}\n\n\nWe interpret \\autoref{thm:matrix-Gamma} by the Pl\\\"ucker embedding.\nWe regard $\\PP^N$ as ${\\PP_{\\!\\! *}}(V) = (V\\setminus \\{0\\}) \/ \\KK^{\\times}$, the projectivization of $V:=\\Bbbk^{N+1}$.\nLet $\\mathbb{G}(n, \\PP^N) \\hookrightarrow {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$ be the Pl\\\"ucker embedding\nand $[p_{i_0,\\ldots,i_n}]_{(i_0 , i_1 , \\ldots , i_n) \\in I}$ be the Pl\\\"ucker coordinates on ${\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$,\nwhere\n\\[\nI := \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{ ( i_0, i_1, \\dots, i_n ) \\in \\mathbb {N}^{n+1} \\, | \\, 0 \\leq i_0 < i_1 < \\cdots < i_n \\leq N }.\n\\]\n\\begin{lem}\\label{lem_plucker}\n The composite morphism $(\\KK^{\\times})^n \\stackrel{\\gamma\\circ \\varphi_A}{\\longrightarrow} \\mathbb{G}(n, \\PP^N) \\hookrightarrow {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$\n maps $ z = (z_1, \\dots, z_n) \\in (\\KK^{\\times})^n $ to \\[\n \\left[\n \\mu_{i_0, i_1, \\dots, i_n}\n \\cdot z^{u_{i_0} + u_{i_1} + \\cdots + u_{i_n}}\n \\right]_{( i_0 ,i_1 , \\ldots , i_n) \\in I} \\in {\\PP_{\\!\\! *}} \\Big( \\bigwedge^{n+1} V \\Big),\n \\]\n where\n \\begin{align*}\n \\mu_{i_0, i_1, \\dots, i_n} &:=\n \\det \\begin{bmatrix}\n 1 & 1 & \\cdots & 1\n \\\\\n u_{i_0} & u_{i_1} & \\cdots & u_{i_n}\n \\end{bmatrix}\n \\in \\Bbbk.\n \\end{align*}\n\\end{lem}\n\\begin{proof}\n This directly follows from \\autoref{thm:matrix-Gamma} and the definition of the Pl\\\"ucker embedding.\n\\end{proof}\n\n\n\n\n\n\nSet\n\\[\nJ := \\{ ( i_0, i_1, \\dots, i_n ) \\in I \\, | \\, \\mu_{i_0, i_1, \\dots, i_n} \\neq 0 \\}.\n\\]\nBy definition, \n$\\mu_{i_0, i_1, \\dots, i_n} \\neq 0$\nin $\\Bbbk$ if and only if $ u_{i_0} , u_{i_1} , \\ldots, u_{i_n}$ span $M_{\\Bbbk}$ as an affine space. Hence\nthe finite set $B$ in \\autoref{thm_structure} is described as\n\\[\nB = \\{ u_{i_0} + u_{i_1} + \\dots + u_{i_n} \\in M \\, | \\, (i_0 , i_1 , \\ldots , i_n) \\in J \\}.\n\\]\nWrite $B = \\{ b_0,b_1,\\cdots,b_{\\# B-1} \\}$ for mutually distinct $b_j \\in M$.\nWe define a linear embedding\n\\begin{equation}\\label{eq:iota-d-hookrar}\n \\begin{aligned}\n \\mathbb{P}^{\\# B -1} &\\hookrightarrow {\\PP_{\\!\\! *}} \\Big(\\bigwedge^{n+1} V\\Big)\n \\\\\n [Y_0 : Y_1 : \\cdots : Y_{\\# B -1}] &\\mapsto [p_{i_0,i_1,\\ldots,i_n}]_{(i_0 , i_1 , \\ldots , i_n) \\in I}\n \\end{aligned}\n\\end{equation}\nas follows. \nWhen $(i_0, i_1, \\dots, i_n) \\in J$,\nthere exists a unique \n$0 \\leq j \\leq \\# B -1$ such that $b_j = u_{i_0} + u_{i_1} + \\dots + u_{i_n}$.\nFor this $j$, we set\n$p_{i_0,i_1,\\ldots,i_n} = \\mu_{i_0, i_1, \\dots, i_n} \\cdot Y_j$.\nWhen $(i_0, i_1, \\dots, i_n) \\not\\in J$,\nwe set\n$p_{i_0,i_1,\\ldots,i_n} = 0$.\n\n\nBy \\autoref{lem_plucker} and the definition of the embedding~\\ref{eq:iota-d-hookrar},\nwe have the following commutative diagram:\n\\begin{equation}\\label{eq:pf-thm1}\n \\begin{split}\n \\xymatrix{\n (\\KK^{\\times})^n \\ar[rrd]^{\\varphi_{B}} \\ar@{^{(}->}[r]^(0.6){\\varphi_A} & X_A \\ar@{-->}[r]^(0.3){\\gamma} & {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V) \\\\\n & & \\mathbb{P}^{\\# B -1} \\ar@{^(->}[u]{}.\n }\n \\end{split}\n\\end{equation}\n\n\\begin{proof}[Proof of \\autoref{thm_structure}]\n By taking a basis of $M$,\n we may assume that $M = \\mathbb {Z}^n$\n and use the notation as above.\n From the above diagram~\\ref{eq:pf-thm1}, the embedding $\\mathbb{P}^{\\# B -1} \\hookrightarrow {\\PP_{\\!\\! *}} (\\bigwedge^{n+1} V)$\n gives an isomorphism between\n \\[\n \\overline{\\gamma(X_A)} = \\overline{\\gamma \\circ \\varphi_A( (\\KK^{\\times})^n)}\n \\textgene{and} X_{B}=\\overline{\\varphi_{B}((\\KK^{\\times})^n )}.\n \\]\n Hence \\ref{thm:item-image} in \\autoref{thm_structure} holds.\n\n\n\n The toric variety $X_{B}=\\overline{\\varphi_{B}((\\KK^{\\times})^n )}$ contains $T_{\\langle B-B \\rangle}$ as an open dense orbit\n and\n the restriction $\\varphi_B |_{T_M}$ is nothing but\n $T_M= (\\KK^{\\times})^n \\twoheadrightarrow T_{\\langle B-B \\rangle}$ induced by the inclusion $\\langle B-B \\rangle \\hookrightarrow M$. Hence \\ref{thm:item-map} in \\autoref{thm_structure} holds by the diagram \\ref{eq:pf-thm1}.\n\n To show \\ref{thm:item-fiber}, we use the following claim.\n\n \\begin{claim}\\label{thm:gammaTM-stein-fact}\n The morphism $ \\gamma|_{T_M} =\\varphi_B$ is decomposed as\n \\begin{align*} T_M \\stackrel{g_1}{\\longrightarrow} T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M} \\stackrel{g_2}{\\longrightarrow} T_{\\langle B-B \\rangle}\n \\end{align*}\n by $ \\langle B-B \\rangle \\subset \\langle B-B \\rangle_{\\mathbb {R}} \\cap M \\subset M$,\n where $g_1$ is a morphism with reduced and irreducible fibers, and $g_2$ is a finite morphism.\n \\end{claim}\n\n \\begin{proof}[Proof of \\autoref{thm:gammaTM-stein-fact}]\n By applying \\ref{item:lem_rk°:1} in \\autoref{lem_rk°} to $ \\langle B-B \\rangle \\subset M$,\n we have the assertion.\n \\end{proof}\n\n\n\n The short exact sequence\n \\[\n 0 \\rightarrow \\langle B-B \\rangle_{\\mathbb {R}} \\cap M \\rightarrow M \\stackrel{\\pi}{\\rightarrow} M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M) \\rightarrow 0\n \\]\n induces a short exact sequence of algebraic tori\n \\begin{align}\\label{eq_alg_gp}\n 1 \\rightarrow T_{M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)} \\rightarrow T_M \\stackrel{g_1}{\\rightarrow} T_{ \\langle B-B \\rangle_{\\mathbb {R}} \\cap M} \\rightarrow 1.\n \\end{align}\n Hence $g_1^{-1}(1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}}) = T_{M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)}$ holds for\n the identity element $1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}} $ of the torus $T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}$.\n Applying \\autoref{lem_lattice_hom} to the surjection $\\pi: M \\rightarrow M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M) $,\n it holds that the closure\n \\[\n \\overline{g_1^{-1}(1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}}) } \\subset X_A\n \\]\n is projectively equivalent to $ X_{\\pi(A)}$.\n\n Let $F $ be an irreducible component of any fiber of $\\gamma |_{T_M}$ with the reduced structure.\n From \\autoref{thm:gammaTM-stein-fact},\n $F$ is a fiber of $ g_1$.\n By \\ref{eq_alg_gp},\n $F$ is the translation of $g_1^{-1}(1_{T_{\\langle B-B \\rangle_{\\mathbb {R}} \\cap M}}) = T_{M \/(\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)} $\n by an element of $ T_M$.\n Hence the closure $ \\overline{F}$ is projectively equivalent to $ X_{\\pi(A)}$\n by \\autoref{lem_lattice_hom}.\n\\end{proof}\n\n\n\n\n\\begin{cor}\\label{cor_rk°}\n Let $A, M, \\gamma, B$ be as in \\autoref{thm_structure}.\n Let $\\gamma |_{T_M} = g_2 \\circ g_1$ be the decomposition of $\\gamma |_{T_M}$ in \\autoref{thm:gammaTM-stein-fact}.\n Then the following hold.\n\n\n \\begin{enumerate}\n \\item\\label{item:rkdeg-rk} The rank of $\\gamma$ is equal to $\\dim (\\Aff_{\\Bbbk} (B))$.\n In particular, $\\gamma $ is separable if and only if $ \\dim (\\Aff_{\\Bbbk} (B)) = \\rank (\\Aff (B))$.\n\n \\item\\label{item:rkdeg-deg} Assume $p = \\chara\\Bbbk > 0$. Then we have\n \\[\n \\deg(g_2) = [\\lin{B-B}_{\\mathbb{R}} \\cap M : \\lin{B-B}].\n \\]\n For the maximum integer $s \\geq 0$ such that $p^s \\mid \\deg(g_2)$,\n the separable degree and the inseparable degree of $g_2 $\n are $\\deg(g_2)\/p^s $ and $ p^s$ respectively.\n Hence the number of the irreducible components of a general fiber of $\\gamma$\n is equal to $\\deg(g_2)\/p^s$, which is coprime to $p$.\n\n \\end{enumerate}\n\\end{cor}\n\nWe note that $\\dim (\\Aff_{\\Bbbk}(B) )= \\dim \\lin{B-B}_{\\Bbbk}$ holds since $ \\Aff_{\\Bbbk}(B)$ is a parallel translation of $\\lin{B-B}_{\\Bbbk} $ in $M_{\\Bbbk}$.\n\n\\begin{proof}[Proof of \\autoref{cor_rk°}]\n We apply \\autoref{lem_rk°} to $ M'' =\\langle B-B \\rangle \\subset M$.\n In this case,\n $g$ in \\autoref{lem_rk°} is $\\gamma |_{T_M}$.\n Since the image of $M''_{\\Bbbk} \\rightarrow M_{\\Bbbk}$ is nothing but $\\langle B-B \\rangle_{\\Bbbk} \\subset M_{\\Bbbk}$,\n it holds that $ \\rk (\\gamma) = \\dim \\langle B-B \\rangle_{\\Bbbk} = \\dim \\Aff_{\\Bbbk} (B)$ by \\ref{item:lem_rk°:2} in \\autoref{lem_rk°}. This implies \\ref{item:rkdeg-rk}.\n On the other hand,\n \\ref{item:rkdeg-deg} follows from \\ref{item:lem_rk°:3} in \\autoref{lem_rk°} directly.\n\\end{proof}\n\nIn the following examples,\nwe denote the separable degree and the inseparable degree of a finite morphism $f$ by $\\deg_s(f)$ and $\\deg_i(f)$ respectively.\n\n\\begin{ex}\\label{ex_intro:rkdeg:2}\n Let $A, B$ be as in \\autoref{ex_intro} and\n assume $\\chara\\Bbbk = 2$.\n Then $\\rk (\\gamma) = \\dim \\lin{B-B}_{\\Bbbk} = 0$.\n From\n \\ref{item:rkdeg-deg} of \\autoref{cor_rk°}, we can calculate $\\deg(g_2) = \\deg_i(g_2) = 2$ and $\\deg_s(g_2) = 1$.\n\\end{ex}\n\n\\begin{ex}\\label{thm:kaji's-ex}\n Kaji's example \\cite[Example 4.1]{Kaji1986} can be interpreted as follows.\n\n Let $A = \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0, 1, cp^m, cp^m+1} \\subset M = \\mathbb{Z}^1$,\n where $c, m$ are positive integers and $p = \\chara\\Bbbk > 0$.\n Assume that $c$ and $p$ are relatively prime.\n Then\n \\[\n B = \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{1, cp^m+1, 2cp^m+1},\\\n \\lin{B-B} = \\lin{cp^m},\\\n \\lin{B-B}_{\\mathbb{R}} \\cap M\n = M.\n \\]\n Therefore $\\deg(\\gamma) = cp^m$, $\\deg_i(\\gamma) = p^m$, $\\deg_s(\\gamma) = c$.\n In particular, a general fiber of $\\gamma$ with the reduced structure is equal to a set of $c$ points.\n \\begin{figure}[htbp]\n \\[\n \\begin{xy}\n (-15,0)=\"1\",(110,0)=\"2\",\n (0,0)*{\\bullet},\n (12,0)*{\\bullet},\n (40,0)*{\\bullet},\n (52,0)*{\\bullet},\n (12,0)*{\\mbox{\\Large$\\times$}},\n (52,0)*{\\mbox{\\Large$\\times$}},\n (92,0)*{\\mbox{\\Large$\\times$}},\n (0,5)*{0},\n (12,5)*{1},\n (40,5)*{cp^m},\n (55,5)*{cp^m+1},\n (92,5)*{2cp^m+1},\n (66,13)*{\\bullet\\;A},\n (79,13)*{\\mbox{\\Large$\\times$}\\,B},\n \\ar \"1\";\"2\"\n \\end{xy}\n \\]\n \\caption{Kaji's example.}\n \\end{figure}\n\n\\end{ex}\n\n\nAs in \\ref{item:rkdeg-deg} of \\autoref{cor_rk°},\nthe number of the irreducible components of a general fiber of $\\gamma$ is coprime to $p = \\chara\\Bbbk$\nin the toric case.\nHowever, the number can be a multiple of $p$ in the non-toric case.\nThe authors learned the following example from H.~Kaji\nin a personal communication.\n\n\\begin{rem}[{A variant of \\cite[Example 4.1]{Kaji1986}}]\n \\label{thm:a-vari-kaji}\n We take $f: \\mathbb{P}^1 \\dashrightarrow \\mathbb{P}^1$ to be a separable rational map whose degree is a multiple of $p$,\n and locally parameterize $g$ by\n $t \\mapsto [1 : f_1(t)]$.\n We set $\\phi: \\mathbb{P}^1 \\dashrightarrow \\mathbb{P}^3$ to be \n the rational map\n which is locally parameterized by\n \\[\n t \\mapsto [1: t: f_1^p: t \\cdot f_1^p]\n \\]\n and set $X \\subset \\mathbb{P}^3$ to be the projective curve $\\overline{\\im(\\phi)}$.\n Then a general fiber of $\\gamma: X \\dashrightarrow \\mathbb{G}(1, \\mathbb{P}^3)$ with the reduced structure\n is equal to a set of $\\deg(f)$ points.\n In order to show this, we may check that\n $\\gamma$ is locally parameterized by\n $t \\mapsto f_1^p(t)$, whose separable degree is equal to $\\deg(f)$.\n We leave the details to the reader.\n\n \n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe following are examples of toric varieties $X_A \\subset \\mathbb{P}^N$ with codimension $1$ or $2$ such that the Gauss maps are birational.\nLater, these examples will be used\nin the proof of \\autoref{cor_rk&fiber}.\n\n\\begin{ex}\\label{prep-cor_rk&fiber}\n Assume $p = \\chara \\Bbbk >0$.\n Let $e_1, \\dots, e_n$ be the standard basis of $M:=\\mathbb {Z}^n$, and set\n \\[\n A = \\{ 0, e_1,\\ldots,e_n, a_1 e_1 + \\ldots + a_n e_n \\} \\subset \\mathbb {Z}^n\n \\]\n for $a_1 , \\ldots, a_n \\in \\mathbb {Z}$ such that\n \\begin{itemize}\n \\item[(i)] $a_1, \\ldots, a_n \\not \\equiv 0 \\mod p$, and \n \\item[(ii)] $a_1 + \\cdots + a_n \\not \\equiv 1 \\mod p$.\n \\end{itemize}\n Then the Gauss map of the toric hypersurface $X_A \\subset \\mathbb{P}^{n+1}$ is birational\n \n as follows.\n\n By condition (i),\n $A \\setminus \\{ e_i\\}$ spans $M_{\\Bbbk}=\\Bbbk^n$ as an affine space for any $1 \\leq i \\leq n$.\n Hence it holds that\n \\begin{equation}\\label{eq:e1+..:1}\n e_1 + \\cdots + e_n + a_1 e_1 + \\ldots + a_n e_n - e_i \\in B.\n \\end{equation}\n By condition (ii),\n $A \\setminus \\{ 0\\}$ spans $M_{\\mathbb {R}}=\\Bbbk^n$ as an affine space.\n Hence \n \\begin{equation}\\label{eq:e1+..:2}\n e_1 + \\cdots + e_n + a_1 e_1 + \\ldots + a_n e_n \\in B.\n \\end{equation}\n Considering the difference between \\ref{eq:e1+..:1} and \\ref{eq:e1+..:2},\n we have $e_i \\in B-B$ for any $i$. Therefore $\\langle B-B \\rangle = M$.\n By \\ref{thm:item-map} in \\autoref{thm_structure},\n the Gauss map of $X_A$ is birational.\n\n\n For example,\n the conditions (i) and (ii) are satisfied for $a_1= \\cdots = a_n=1$ (resp.\\ $a_1= \\cdots = a_n= -1$)\n when $n \\not \\equiv 1 \\mod p$ (resp.\\ $n \\not \\equiv -1 \\mod p$).\n Hence there exists a toric hypersurface in $\\mathbb{P}^{n+1}$ whose Gauss map is birational\n if $\\chara \\Bbbk \\neq 2$ or $n$ is even.\n\\end{ex}\n\n\nAs we will see later in \\autoref{rem_rk_in_char2},\nthe Gauss map of any (not necessarily toric) hypersurface in $\\mathbb{P}^{n+1}$ cannot be birational if $\\chara \\Bbbk =2$ and $n$ is odd.\n\n\n\n\n\n\\begin{ex}\\label{ex_char2}\n Assume $n \\geq 2$.\n Let\n \\[\n A =\\{0, e_1, \\ldots, e_n, e_1 +e_2, e_2+e_3+ \\cdots + e_n \\} \\subset M:=\\mathbb {Z}^n.\n \\]\n \n \n For $S := \\{ 0, e_1, \\ldots, e_n, e_1+e_2\\} \\subset A$, each of\n \\[\n S\\, \\setminus \\{e_1 + e_2\\},\\ S\\, \\setminus \\{e_1\\},\\ S\\, \\setminus \\{e_2\\}\n \\]\n spans the affine space $M_{\\Bbbk}$.\n \n Hence\n it holds that $e_1,e_2 \\in B-B $.\n In addition,\n $\\{0, e_1, \\ldots, e_n, e_2+e_3+ \\cdots + e_n \\} \\setminus \\{e_i\\}$ spans the affine space $M_{\\Bbbk}$ for $2 \\leq i \\leq n$.\n Thus we have $e_i -e_2 \\in B-B $ for $2 \\leq i \\leq n$.\n Hence $\\langle B-B \\rangle = M$ and the Gauss map of $X_A \\subset \\mathbb{P}^{n+2}$ is birational. \n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Degenerate Gauss maps}\n\\label{sec:degen-gauss-maps}\n\n\n\n\n\\subsection{Developability criterion}\n\\label{sec:criterion-degeneracy}\nBy Theorem \\ref{thm_structure},\nthe Gauss map of any given toric variety and\nits general fibers can be explicitly determined\nfrom the computation of $B$ and $\\pi(A)$ as in \\autoref{ex_intro}.\nHowever,\nit is not so clear for what kind of $A$ the Gauss map $\\gamma$ of $X_A \\subset \\PP^N$ is degenerate,\ni.e.,\n$\\rank \\langle B-B \\rangle < n$.\nThe following result tells us\na condition that $\\gamma$ is degenerate.\n\n\\begin{prop}\\label{main_thm}\n Let $M,A,\\pi$ be as in Theorem \\ref{thm_structure}.\n Let $\\tilde\\pi : M \\rightarrow \\tilde{M}'$ be a surjective homomorphism of free abelian groups.\n Then $\\tilde\\pi$ \n factors through $\\pi: M \\rightarrow M'$,\n i.e., $\\langle B-B \\rangle \\subset \\ker \\tilde \\pi$ \n if and only if\n \\[\n \\sum_{j=0}^{\\tilde{N}'} \\dim \\Aff_{\\Bbbk}(A_j) =n- \\tilde{N}',\n \\]\n where $\\tilde{N}' := \\# \\tilde\\pi(A) -1$,\n $\\tilde\\pi(A ) = \\{ \\tilde u'_0,\\ldots, \\tilde u'_{\\tilde{N}'}\\}$, and $A_j= \\tilde\\pi^{-1}(\\tilde u'_j) \\cap A$.\n\\end{prop}\n\nFor a surjective homomorphism $\\tilde\\pi: M \\rightarrow \\tilde M'$\nof free abelian groups,\nthe toric variety\n$X_A$ is covered by the translations of\n$\\overline{T_{\\tilde M'}} = X_{\\tilde\\pi(A)}$ by elements of $T_M$\ndue to \\autoref{lem_lattice_hom}.\nThe homomorphism $\\tilde\\pi$ factors through $\\pi$\nif and only if\n$\\overline{T_{\\tilde M'}}$ (or equivalently,\nthe translation of $\\overline{T_{\\tilde M'}}$ by any element of $T_M$)\nis contracted to one point by $\\gamma$.\n\n\n\n\nIn general, a covering family $ \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{F_{\\alpha}}$ of a projective variety $X \\subset \\PP^N$ by subvarieties $F_{\\alpha} \\subset X$ is said to be \\emph{developable} if\n$F_{\\alpha}$ is contracted to one point by the Gauss map of $X$\n(i.e., $\\mathbb{T}_xX$ is constant on general $x \\in F_{\\alpha}$) for general $\\alpha$.\n\\autoref{main_thm} is regarded as\na toric version of the developability criterion\n(cf.\\ \\cite[2.2.4]{FP}, \\cite{Fukasawa2005}, \\cite[\\textsection{}4]{expshr}; see also \\autoref{sec:separable-gauss-maps} for the separable case).\n\n\n\nBefore the proof,\nwe illustrate \\autoref{main_thm} by an example.\n\n\n\n\\begin{ex}\\label{ex_intro2}\n Let $A \\subset \\mathbb {Z}^2$ be as in \\autoref{ex_intro}.\n For the projection $\\tilde \\pi : \\mathbb {Z}^2 \\rightarrow \\mathbb {Z}^1$ to the second factor,\n $\\tilde \\pi(A) = \\{0,1, -1\\}$. Then\n $A_0 =\\tilde \\pi^{-1}(0) \\cap A,\\, A_1 =\\tilde \\pi^{-1}(1) \\cap A,\\, A_2 =\\tilde \\pi^{-1}(-1) \\cap A$\n are given by\n \\[\n A_0 = \\left\\{\n \\begin{bmatrix}\n 0 \\\\ 0\n \\end{bmatrix}\n \\right\\},\\,\n A_1 = \\left\\{\n \\begin{bmatrix}\n 0 \\\\ 1\n \\end{bmatrix}\n \\right\\},\\,\n A_2 = \\left\\{\n \\begin{bmatrix}\n 1 \\\\ -1\n \\end{bmatrix},\n \\begin{bmatrix}\n -1 \\\\ -1\n \\end{bmatrix}\n \\right\\}.\n \\]\n Thus\n \\[\n \\sum_{j=0}^{\\tilde{N}'} \\dim \\Aff_{\\Bbbk}(A_j) = \\left\\{\n \\begin{array}{cl} \n 1 & \\text{when } \\chara \\Bbbk \\neq2,\\\\ 0 & \\text{when } \\chara \\Bbbk=2.\\\\\n \\end{array} \\right.\n \\]\n On the other hand,\n $n-\\tilde{N}'=2-2=0$ holds.\n Hence the equality in \\autoref{main_thm} holds if and only if $\\chara \\Bbbk =2$.\n Note that, in this example, the above\n $\\tilde\\pi$ can be identified with the natural projection\n $\\pi: M \\rightarrow M'$ in \\autoref{thm_structure} when $\\chara \\Bbbk=2$.\n\\end{ex}\n\n\nTo prove \\autoref{main_thm}, we need the following lemma.\n\n\n\n\\begin{lem}\\label{thm:AffAj-eq}\n In the setting of \\autoref{main_thm},\n the homomorphism $\\tilde \\pi$ factors through $\\pi$ if and only if\n \\[\n \\Aff_{\\Bbbk}(A_j) = \\Aff_{\\Bbbk}( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j)\n \\quad (\\RNj)\n \\]\n for any $u_{i_0}, \\dots, u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n\n\\end{lem}\n\n\\begin{proof}\n First, we show the ``only if'' part.\n Assume that $\\tilde \\pi$ factors through $\\pi$.\n The inclusion ``$\\supset$'' always holds.\n We show ``$\\subset$''.\n Let $u \\in A_j$.\n Since $u_{i_0}, \\dots, u_{i_n}$ span the affine space $M_{\\Bbbk}$,\n we can write $u = \\sum_{k=0}^n c_{i_k} u_{i_k}$\n with $\\sum_{k=0}^n c_{i_k} = 1$ and $c_{i_k} \\in \\Bbbk$.\n For any $k$ with $c_{i_{k}} \\neq 0$, we have $u_{i_{k}} \\in A_j$ as follows.\n\n If $c_{i_{k}} \\neq 0$, we find that $\\{u\\} \\cup \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_{k'}}}_{0 \\leq k' \\leq n, k' \\neq k}$ span\n the affine space $M_{\\Bbbk}$. Thus $u + \\sum_{0 \\leq k' \\leq n, k' \\neq k} u_{i_{k'}} \\in B$, \n and then $u - u_{i_{k}} \\in B-B$.\n Since $\\tilde \\pi$ factors through $\\pi$, we have $ \\tilde\\pi(u_{i_{k}}) = \\tilde\\pi(u) = \\tilde{u}'_j$ by $u \\in A_j = \\tilde{\\pi}^{-1}(\\tilde{u}'_j) \\cap A$;\n hence $u_{i_{k}} \\in A_j$.\n \n Thus we have $u = \\sum_{u_{i_k} \\in A_j} c_{i_k} u_{i_k}$ with $\\sum_{u_{i_k} \\in A_j} c_{i_k} =1 $,\n i.e.,\n $u $ is contained in $ \\Aff_{\\Bbbk}( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j)$.\n This implies the assertion.\n\n \\vspace{2mm}\n Next, we show the ``if'' part.\n For any $b \\in B$,\n we can write $b= u_{i_0} + \\cdots + u_{i_n}$ with $u_{i_0} , \\ldots , u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n Since the $n$-dimensional affine space $M_{\\Bbbk} $ is spanned by $n+1$ elements $u_{i_0} , \\ldots , u_{i_n} \\in A$,\n we have\n \\[\n \\# ( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, \\dots, u_{i_n} }\\cap A_j) = \\dim \\Aff_{\\Bbbk}( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j) +1.\n \\] \n Hence $ \\# ( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, \\dots, u_{i_n} }\\cap A_j) = \\dim \\Aff_{\\Bbbk}(A_j) + 1 $ holds for each $0 \\leq j \\leq \\tilde{N}'$ by assumption.\n In particular,\n it holds that\n \\[\n \\tilde \\pi(b)= \\tilde \\pi (u_{i_0} + \\cdots + u_{i_n}) = \\sum_{0 \\leq j \\leq \\tilde{N}'} (\\dim \\Aff_{\\Bbbk}(A_j) + 1) \\cdot u'_j ,\n \\]\n which does not depend on $b \\in B$.\n Thus we have $B-B \\subset \\ker \\tilde \\pi$,\n i.e.,\n $\\tilde \\pi $ factors through $\\pi$.\n\\end{proof}\n\n\\begin{rem}\\label{thm:B-B-wr-Aj-Aj}\n Assume that $\\tilde\\pi$ factors through $\\pi$.\n From the ``if'' part in the above proof,\n $\\# ( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, \\dots, u_{i_n} }\\cap A_j) $ does not depend on $u_{i_0}, \\dots, u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n Thus each element of $B -B$ is written as a linear combination\n of elements of $\\bigcup_{\\RNj} (A_j-A_j)$.\n\\end{rem}\n\n\n\n\n\n\n\n\n\\begin{proof}[Proof of \\autoref{main_thm}]\n Let us take $u_{i_0}, \\dots, u_{i_n} \\in A$ which span the affine space $M_{\\Bbbk}$.\n Then the latter condition of \\autoref{thm:AffAj-eq} holds\n if and only if\n \\begin{equation}\\label{eq:sub-AffAj}\n \\dim \\Aff_{\\Bbbk}(A_j) = \\#( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j) - 1\n \\end{equation}\n for any $\\RNj$ (we note that ``$\\geq$'' always holds).\n On the other hand, since $A = A_0 \\sqcup A_1 \\sqcup \\dots \\sqcup A_{\\tilde{N}'}$, we have\n \\[\n \\sum_{\\RNj} (\\#( \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} \\cap A_j) -1)\n = \\# \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_{i_0}, u_{i_1}, \\dots, u_{i_n}} - (\\tilde{N}' + 1) = n-\\tilde{N}'.\n \\]\n Hence $\\sum_{\\RNj[\\tilde{N}']} \\dim \\Aff_{\\Bbbk}(A_j) =n- \\tilde{N}'$ holds\n if and only if the equality~\\ref{eq:sub-AffAj} holds for any $\\RNj$.\n Therefore this proposition follows from \\autoref{thm:AffAj-eq}.\n\\end{proof}\n\n\\begin{cor}\\label{cor_fiber_separable}\n Let $A, \\pi: M \\rightarrow M'$ be as in \\autoref{thm_structure}.\n Then it holds that $\\rk (\\gamma) \\leq n-(\\#\\pi(A)-1)$. Moreover, if $\\gamma$ is separable,\n then we have $\\# \\pi(A) = \\rank M' +1$, which means $ X_{\\pi(A)}$ is a linear projective space of dimension $\\rank M'$.\n\\end{cor}\n\n\\begin{proof}\n We apply \\autoref{main_thm} to the homomorphism $\\pi$. Then\n it holds that $\\sum_{j} \\dim \\lin{A_j-A_j}_{\\Bbbk} =n- (\\#\\pi(A)-1)$.\n From \\autoref{cor_rk°}, we have $\\rk (\\gamma) = \\dim \\Aff_{\\Bbbk} (B) = \\dim \\lin{B-B}_{\\Bbbk}$.\n By \\autoref{thm:B-B-wr-Aj-Aj},\n $\\lin{B-B}$ is contained in the space $\\lin{ \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{A_j-A_j}_j}$.\n Thus $\\rk (\\gamma) \\leq n- (\\#\\pi(A)-1)$ holds.\n\n If $\\gamma$ is separable, $\\rk (\\gamma) = \\rank \\langle B-B \\rangle = n - \\rank (M')$ holds by \\autoref{cor_rk°}.\n Hence $\\rank(M') \\geq \\#\\pi(A)-1$ holds by the above inequality.\n The converse inequality ``$\\leq$'' always holds since $\\pi(A)$ spans the affine lattice $M'$.\n Since $\\pi(A)$ spans the affine lattice $ M'$,\n the equality $\\# \\pi(A) = \\rank M' +1$ means $ X_{\\pi(A)}$ is a linear projective space of dimension $\\rank M'$.\n\\end{proof}\n\n\n\\begin{rem}\n The equality ``$\\rk (\\gamma) = n-(\\#\\pi(A)-1)$'' does \\emph{not} hold in general.\n For example, set $A=\\{0,1,p\\} \\subset M = \\mathbb {Z}^1$ with $p=\\chara \\Bbbk >0$.\n Then we have\n \\[\n B=\\{1, p+1\\}, \\quad \\lin{B-B} = \\lin{p} , \\quad \\lin{B-B}_{\\mathbb{R}} \\cap M =M.\n \\]\n Thus $\\pi : M=\\mathbb {Z}^1 \\rightarrow M\/(\\lin{B-B}_{\\mathbb{R}} \\cap M) =\\{0 \\}$ is the zero map.\n Here we have $\\rk (\\gamma) = \\dim \\lin{B-B}_{\\Bbbk} = 0$\n and \n $n-(\\#\\pi(A) - 1) = 1-(1-1) = 1$.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\\subsection{Separable Gauss maps, Cayley sums, and joins}\n\\label{sec:separable-gauss-maps}\n\nIn this subsection,\nwe study the case when the Gauss map is separable,\nand prove Corollary \\ref{cor_join} in the characteristic zero case.\n\n\n\n\n\n\\begin{defn}\\label{def_cayley}\n Let $l \\leq n$ be non-negative integers. Let $e_1,\\ldots,e_l$ be the standard basis of $ \\mathbb {Z}^l$.\n For finite sets $A^0,\\ldots,A^l \\subset \\mathbb {Z}^{n-l}$,\n the Cayley sum $A^0 * \\cdots * A^l $ of $A^0,\\ldots,A^l$ is defined to be\n \\[\n A^0 * \\cdots * A^l := (A^0 \\times \\{0\\}) \\cup (A^1 \\times \\{e_1 \\}) \\cup \\cdots (A^l \\times \\{e_l\\}) \\subset \\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l.\n \\]\n\\end{defn}\n\nLet $A$ be the Cayley sum of $A^0,\\ldots,A^l \\subset \\mathbb {Z}^{n-l}$,\nand assume that $A$ spans the affine lattice $ \\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l$.\nFor the projection $\\tilde{\\pi} : \\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l \\rightarrow \\mathbb {Z}^l $\nto the second factor, $X_{\\tilde{\\pi}(A)}$ is an $l$-plane\nsince $\\tilde\\pi(A) = \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0, e_1, \\dots, e_l}$.\nBy \\autoref{lem_lattice_hom}, $X_A$ is covered by $l$-planes,\nwhich are translations of $X_{\\tilde{\\pi}(A)} = \\overline{T_{\\mathbb {Z}^l}}$.\nFor this $\\tilde\\pi$,\n$\\tilde N'$ in \\autoref{main_thm} is equal to $l$.\nThus,\nthe subtorus $ T_{\\mathbb {Z}^l} \\subset T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^l}$ is contracted to one point by the Gauss map of $X_A$\nif and only if\n\\begin{align}\\label{eq_condition_for_sum}\n \\sum_{j=0}^{l} \\dim \\Aff_{\\Bbbk}(A^j) =n- l.\n\\end{align}\nIn other words,\n\\ref{eq_condition_for_sum} is the condition for the developability\nof the covering family obtained by translations of $\\overline{T_{\\mathbb {Z}^l}}$.\nIn fact,\nany toric variety with separable Gauss map is described by a Cayley sum with the condition~\\ref{eq_condition_for_sum}, as follows.\n\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{thm:subvar-in-join}\n Let $A, M$ be as in \\autoref{thm_structure}.\n Assume that the Gauss map $\\gamma$ of $X_A \\subset \\PP^N$ is separable, and set $l = \\delta_{\\gamma}(X)$.\n Then there exist finite subsets $A^0, \\dots, A^{l} \\subset \\mathbb{Z}^{n-l}$\n with $\\sum_{j=0}^{l} \\dim \\Aff_{\\Bbbk}(A^j) =n- l$\n such that $A$ is identified with the Cayley sum of $A^0, \\dots, A^{l} \\subset \\mathbb{Z}^{n-l}$ under some affine isomorphism $M \\simeq \\mathbb{Z}^{n-l} \\times \\mathbb{Z}^{l}$.\n\\end{thm}\n\n\n\n\n\n\\begin{proof}\n Let $A = \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{u_0, u_1, \\dots, u_N}$ and $\\pi : M \\rightarrow M'$ be\n as in \\autoref{thm_structure},\n where we may assume $u_0=0 \\in M$.\n From \\autoref{thm_structure} and \\autoref{cor_fiber_separable}, it follows that\n $\\rank M' = l$ and $\\# \\pi (A) = l+1$ since $\\gamma$ is separable by assumption.\n Set $\\pi (A)=\\{u'_0, \\ldots,u'_l \\} $.\n Without loss of generality,\n we may assume that $u'_0 =\\pi(u_0)=0 \\in M'$.\n Then $ u'_1, \\ldots,u'_l $ form a basis of $M'$\n since $\\pi(A) $ spans the affine lattice $M' \\simeq \\mathbb {Z}^l$.\n Set $A_j = \\pi^{-1}(u'_j) \\cap A$.\n\n\n\n Fix a splitting $s : M' \\rightarrow M$ of the short exact sequence $0 \\rightarrow \\ker \\pi \\rightarrow M \\stackrel{\\pi}{\\rightarrow} M' \\rightarrow 0$.\n Then the induced isomorphism\n \\[\n M \\stackrel{\\sim}{\\rightarrow} \\ker \\pi \\times M' \\quad : \\quad u \\mapsto (u - s ( \\pi(u)), \\pi(u))\n \\]\n gives an identification of $A \\subset M$ with\n \\begin{equation}\\label{eq:expressA-Cayley}\n \\bigcup_{j=0}^l \\left( A^j \\times \\{u'_j \\} \\right) \\subset \\ker \\pi \\times M' ,\n \\end{equation}\n where $A^j := A_j -s(u'_j) \\subset \\ker \\pi$ is the parallel translation of $A_j$ by $s(u'_j)$.\n Since\n \n $ u'_1, \\ldots,u'_l $ form a basis of $M'$, $u_0' = 0$,\n and $\\ker \\pi \\simeq \\mathbb{Z}^{n-l}$,\n this theorem follows.\n\\end{proof}\n\n\nIn order to prove \\autoref{cor_join}, we consider a relation between\nCayley sums and joins.\nFor projective varieties $X_1, \\dots, X_m \\subset \\PP^N$,\nwe define the \\emph{join} of $X_1, \\dots, X_m$ to be the closure of\n$\\bigcup_{x_1 \\in X_1, \\dots, x_m \\in X_m} \\Lambda_{x_1, \\dots, x_m} \\subset \\PP^N$,\nwhere $\\Lambda_{x_1, \\dots, x_m}$ is the linear variety spanned by the points $x_1, \\dots, x_m$.\n\n\n\\begin{lem}\\label{lem_torus_inv_sub}\n Let $A \\subset \\mathbb{Z}^{n-l} \\times \\mathbb{Z}^l$ be the Cayley sum of $A^0, \\dots, A^l \\subset \\mathbb{Z}^{n-l}$ with $\\Aff(A) = \\mathbb{Z}^{n-l} \\times \\mathbb{Z}^l$.\n Then the following hold.\n \\begin{enumerate}[\\quad \\normalfont (a)]\n \\item \\label{lem_torus_inv_sub:1}\n $X_{A^0}, \\ldots,X_{A^l}$ are embedded into $X_A$ as torus invariant subvarieties,\n and they are mutually disjoint.\n\n \\item \\label{lem_torus_inv_sub:2}\n $X_{A}$ is contained in the join of $X_{A^0}, \\ldots,X_{A^l} \\subset \\PP^N$,\n and the codimension of $X_{A}$ in the join is\n $l - n + \\sum_{j=0}^l \\rank \\Aff(A^j)$.\n\n \\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n \\begin{inparaenum}[(a)]\n \\item \n Write $A^j = \\{u^j_{0}, \\ldots,u^j_{N_j}\\} \\subset \\mathbb {Z}^{n-l}$ for $ N_j= \\# A^j -1$.\n Set $N=\\# A -1 = \\sum_{j=0}^l (N_j+1) -1$\n and let $\\{ X^j_{i}\\}_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l}$ be the homogeneous coordinates on $ \\mathbb{P}^{N}$.\n By the definition of the Cayley sum $A$,\n it holds that\n \\begin{equation}\\label{eq:phiAyz}\n \\begin{aligned}\n \\varphi_{A }(z, w) &= [w_j z^{u_i^{j}}]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l} \\\\\n &= [ w_0z^{u^0_{0}} : \\cdots : w_0 z^{u^0_{N_0}} :\n w_1 z^{u^1_{0}} : \\cdots : w_1 z^{u^1_{N_1}} :\n \\cdots: w_l z^{u^l_{0}} : \\cdots : w_l z^{u^l_{N_l}}]\n \\end{aligned}\n \\end{equation}\n in $\\mathbb{P}^N$\n for $(z, w)=(z_1,\\ldots,z_{n-l}, w_1,\\ldots,w_l)\n \\in (\\KK^{\\times})^{n-l} \\times (\\KK^{\\times})^{l} = T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}}$,\n where we set $w_0 = 1$.\n For fixed $0 \\leq j \\leq l$ and $z \\in (\\KK^{\\times})^{n-l} = T_{\\mathbb {Z}^{n-l}}$,\n $\\phi_A(z, w)$ converges to\n \\begin{equation}\\label{eq:phiAjz}\n [0:\\cdots : 0 : z^{u^j_{0}} : \\cdots : z^{u^j_{N_j}} : 0:\\cdots : 0] \\in \\mathbb{P}^N\n \\end{equation}\n when $w_k\/w_j \\rightarrow 0$ for $0 \\leq k \\neq j \\leq l$.\n Thus, the point \\ref{eq:phiAjz} is\n contained in the closure $\\overline{\\varphi_{A }(T_{\\mathbb {Z}^l \\times \\mathbb {Z}^{n-l}} )}=X_A$.\n In other words, $\\varphi_{A^j}(z) = [z^{u^j_{0}} : \\cdots : z^{u^j_{N_j}} ] \\in \\mathbb{P}^{N^j} $ is contained in $X_A$ for any $z \\in (\\KK^{\\times})^{n-l} = T_{\\mathbb {Z}^{n-l}}$\n by embedding $\\mathbb{P}^{N_j}$ into $\\mathbb{P}^N$ as\n \\begin{equation}\\label{eq:PNj=Xji}\n \\mathbb{P}^{N_j} = (X^{j'}_{i}=0)_{j' \\neq j, 0 \\leq i \\leq N_{j'}} \\subset \\mathbb{P}^{N} .\n \\end{equation}\n Since $X_{A^j}$ is the closure of $\\varphi_{A^j}(T_{\\mathbb {Z}^{n-l}})$,\n $X_{A^j} \\subset \\mathbb{P}^{N_j}$ is contained in $X_A$.\n\n\n The action of $T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}} $ on $X_A$ is described as\n \\[\n (z,w) \\cdot [X^j_i]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l} = [ w_j z^{u^j_i} X^j_i]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l}\n \\]\n for $ (z,w ) = (z_1,\\ldots,z_{n-l}, w_1,\\ldots,w_l) \\in T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}} $ and $ [X^j_i]_{0 \\leq i \\leq N_j, 0 \\leq j \\leq l} \\in X_A$,\n where $w_0 = 1$ as before.\n Therefore $X_{A^j} \\subset X_A$ is a torus invariant subvariety.\n Since $\\mathbb{P}^{N_0}, \\ldots, \\mathbb{P}^{N_l} \\subset \\mathbb{P}^N$ are mutually disjoint by \\ref{eq:PNj=Xji},\n so are $X_{A^0}, \\ldots, X_{A^l} \\subset X_A$.\n\n\n \\vspace{1ex}\n \\noindent\n \\item\n For $(z,w ) \\in T_{\\mathbb {Z}^{n-l} \\times \\mathbb {Z}^{l}}$,\n the image $\\varphi_{A}(z,w) \\in X_A \\subset \\PP^N$ is described by \\ref{eq:phiAyz}, and\n $\\varphi_{A^j}(z) \\in X_{A^j} \\subset \\PP^N$ is described by \\ref{eq:phiAjz}\n for each $0 \\leq j \\leq l$.\n Hence $\\varphi_{A}(z,w) $ is contained in\n the $l$-plane spanned by $\\varphi_{A^0}(z), \\varphi_{A^1}(z), \\cdots, \\varphi_{A^l}(z) $.\n Thus $X_{A}$ is contained in the join of $X_{A^0}, \\ldots,X_{A^l}$.\n From \\ref{eq:PNj=Xji},\n the dimension of the join of $X_{A^0}, \\ldots,X_{A^l}$ is\n $l+ \\sum_{j=0}^l \\dim X_{A^j}$,\n which is equal to $l + \\sum_{j=0}^l \\rank \\Aff(A^j)$.\n Since $\\dim(X_A) = n$, the assertion about the codimension follows.\n \\end{inparaenum}\n\\end{proof}\n\n\n\n\n\\begin{ex}\n Let $A \\subset \\mathbb {Z}^2 \\times \\mathbb {Z}^1$ be the Cayley sum of\n \\[\n A^0 = \\Set{\\begin{bmatrix}\n 0 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 1 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 2 \\\\ 0\n \\end{bmatrix}\n }, \\\n A^1 = \\Set{\\begin{bmatrix}\n 0 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 1\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 2\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 3\n \\end{bmatrix}\n }\n \\subset \\mathbb {Z}^2.\n \\] \n Then it holds that\n \\[\n l - n + \\sum_{j=0}^l \\rank \\Aff(A^j) = 1 - 3 + (1+1) =0.\n \\] \n Hence $X_A$ is the join of $X_{A^0}$ and $X_{A^1}$.\n In fact,\n \\[\n X_A = \\overline{ \\big\\{ [1: x: x^2 : w : w y : w y^2 : w y^3] \\, | \\, (x,y,w) \\in (\\KK^{\\times})^3= T_{\\mathbb {Z}^2 \\times \\mathbb {Z}^1} \\big\\} } \\subset \\mathbb{P}^6,\n \\]\n and the conic $X_{A^0} \\subset \\mathbb{P}^2$ and the twisted cubic $X_{A^1} \\subset \\mathbb{P}^3$ are embedded into $X_A$ as\n \\begin{align*}\n X_{A^0} &=\\overline{ \\big\\{ [1: x: x^2 : 0 : 0 : 0 : 0] \\, | \\, x \\in \\KK^{\\times} \\big\\} } \\hspace{1mm} \\subset X_A, \\\\\n X_{A^1} &=\\overline{ \\big\\{ [0:0:0 : 1 : y : y^2 : y^3] \\, | \\, y \\in \\KK^{\\times} \\big\\} } \\subset X_A.\n \\end{align*}\n\\end{ex}\n\n\\begin{rem}\\label{thm:subvar-in-join:rem}\n In \\autoref{thm:subvar-in-join},\n the codimension of $X_A$ in the join of $X_{A^0}, \\dots, X_{A^l}$\n is\n $\\sum_{j=0}^l (\\rank \\Aff(A^j) - \\dim \\Aff_{\\Bbbk}(A^j))$\n by \\autoref{lem_torus_inv_sub}.\n\\end{rem}\n\n\n\n\n\\vspace{1mm}\nNow we can prove \\autoref{cor_join} immediately.\n\n\\begin{proof}[Proof of Corollary \\ref{cor_join}]\n We may assume that\n $X=X_A$ for some finite set $A \\subset M$ with $\\Aff (A)=M$.\n Then the assertion follows from\n \\autoref{thm:subvar-in-join} and \\autoref{thm:subvar-in-join:rem} since\n the equality $\\rank \\Aff(A^j) = \\dim \\Aff_{\\Bbbk}(A^j)$ holds in $\\chara\\Bbbk = 0$.\n\\end{proof}\n\n\\begin{cor}\n Assume $\\chara\\Bbbk=0$.\n If a toric variety $X_A \\subset \\PP^N$ is the join of some projective varieties,\n then\n $X_A$ is the join of some toric varieties $X_{A^0}, X_{A^1}, \\dots, X_{A^{l}}$ for some $l >0$.\n\\end{cor}\n\\begin{proof}\n Since $X_A$ is the join in $\\chara\\Bbbk=0$,\n the Gauss defect $\\delta_{\\gamma}(X_A)$ is positive (due to Terracini's lemma). Hence this corollary follows from \\autoref{cor_join}\n for $l = \\delta_{\\gamma}(X)$.\n\\end{proof}\n\n\n\n\n\n\nThe assumption $\\chara\\Bbbk = 0$ is crucial in the above proof of \\autoref{cor_join}.\nIn positive characteristic,\neven if the Gauss map $\\gamma$ of toric $X_A$ is separable (equivalently, \na general fiber of $\\gamma$ is scheme-theoretically an open subset of a linear variety of $\\PP^N$),\nit is possible that $\\gamma$ is degenerate but $X_A$ is \\emph{not} the join of any varieties, as follows.\n\n\\begin{ex}\\label{thm:sep-gamma-X-not-join}\n Let $p = \\chara\\Bbbk \\geq 3$.\n Set\n \\[\n A^0=\\{0,1,-1\\}, A^1 =\\{ 0,p\\} \\subset \\mathbb {Z}^1\n \\]\n and let $A \\subset \\mathbb {Z}^1 \\times \\mathbb {Z}^1$ be the Cayley sum of $A^0,A^1$.\n Then\n \\[\n \\lin{B-B} = \\mathbb {Z}^1 \\times \\{0\\} \\subset \\mathbb {Z}^1 \\times \\mathbb {Z}^1.\n \\]\n Hence $\\pi: M \\rightarrow M\/ (\\lin{B-B}_{\\mathbb{R}} \\cap M)$ coincides with\n the projection $\\mathbb {Z}^1 \\times \\mathbb {Z}^1 \\rightarrow \\mathbb{Z}^1$ to the second factor.\n In this setting, the following hold.\n\n \\begin{enumerate}\n \\item The Gauss map $\\gamma$ of the surface $X_A \\subset \\mathbb{P}^4$ is separable.\n A general fiber of $\\gamma$ is a line; in particular,\n $\\gamma$ is degenerate.\n\n \\item The conic $X_{A^0}$ and the line $X_{A^1}$ are embedded into $X_A$.\n\n \\item $X_A$ is of codimension one in the join of $X_{A^0}$ and $X_{A^1}$.\n On the other hand, $X_A$ itself is not the join of any varieties.\n \\end{enumerate}\n\n The reason is as follows.\n \\begin{inparaenum}\n \\item The separability of $\\gamma$ follows from \\autoref{cor_rk°}.\n A general fiber of $\\gamma$ is projectively equivalent to $X_{\\pi(A)}$, which is a line.\n \\item The embedding of $X_{A^j}$ is given as in \n \\autoref{lem_torus_inv_sub}.\n \\item It follows from \\autoref{thm:subvar-in-join} that\n $X_A$ is contained in the join of $X_{A^0}$ and $X_{A^1}$.\n Since the join is of dimension $3$,\n the codimension of $X_A$ in the join is equal to $1$. By \\autoref{thm:subvar-in-join:rem}, the codimension is also calculated from\n \\begin{gather*}\n \\rk \\Aff (A^0) - \\dim \\Aff_{\\Bbbk} (A^0) = 1-1 = 0,\n \\\\\n \\rk \\Aff (A^1) - \\dim \\Aff_{\\Bbbk} (A^1) = 1-0 = 1.\n \\end{gather*}\n On the other hand, $X_A$ is not the join of any varieties;\n this is because,\n a projective surface $X \\subset \\PP^N$ is the join of some varieties\n if and only if $X$ is the cone of a curve with a vertex.\n \\end{inparaenum}\n\n\n \\begin{figure}[htbp]\n \\[\n \\begin{xy}\n (10,0)=\"A\",(0,10)=\"B\",\n (-10,-10)=\"C\",\n (9.9,0.1)=\"E\",(-5,0.1)=\"F\",\n (9.8,-0.1)=\"I\",(-5,-0.1)=\"J\",\n (-20,0)=\"1\",(65,0)=\"2\",\n (0,-10)=\"3\",(0,18)=\"4\",\n (95,-10)=\"7\",(95,18)=\"8\",\n (10,-4)*{1},\n (-10,-4)*{-1},\n (45,0)*{\\mbox{\\scriptsize$\\mid$}},\n (45,-4)*{p},\n (0.1,10)*{\\mbox{\\Large$\\times$}},\n (-10,10)*{\\mbox{\\Large$\\times$}},\n (10,10)*{\\mbox{\\Large$\\times$}},\n (35,10)*{\\mbox{\\Large$\\times$}},\n (45,10)*{\\mbox{\\Large$\\times$}},\n (55,10)*{\\mbox{\\Large$\\times$}},\n (45,10)*{\\mbox{\\large$\\circ$}},\n (0,10)*{\\mbox{\\large$\\circ$}},\n (0,0)*{\\bullet},\n (-10,0)*{\\bullet},(10,0)*{\\bullet},\n (95,10)*{\\sqbullet},(95,0)*{\\sqbullet},\n (98,10)*{1},(98,0)*{0},\n (45,-14)*{\n \\bullet\\;A^0 \\times \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0}\\quad \n \\mbox{\\large$\\circ$}\\;A^1 \\times \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{1}\\quad \n \\mbox{\\Large$\\times$}\\;B\n }, \n (110,-14)*{\\sqbullet\\;\\pi(A)},\n (80,3)*{\\pi}\n \\ar \"1\";\"2\"\n \\ar \"3\";\"4\"\n \\ar \"7\";\"8\"\n \\ar (72,0);(88,0)\n \\end{xy}\n \\]\n \\caption{Cayley sum: $A = A^0 * A^1$ \\ ($p \\geq 3$).}\n \\label{figure3}\n \\end{figure}\n\n\n\n\n\n\n\n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Constructions in positive characteristic}\n\\label{sec:posit-char-case}\n\n\nThis section presents two constructions of projectively embedded toric varieties\nin positive characteristic.\nWe consider whether it is possible to find a toric variety\nwhose Gauss map $\\gamma$ has given data about\n\\begin{enumerate}\n\\item[{(F)}] each irreducible component of a general fiber of $\\gamma$;\n\\item[{(I)}] the image of $\\gamma$;\n\\item[{(c)}]\n the number of the irreducible components of a general fiber of $\\gamma$;\n\\item[{(r)}]\n the rank of $\\gamma$.\n\\end{enumerate}\nThe statement of \\autoref{thm_Fukasawa1}\nmeans that, a projectively embedded toric variety $X$ is constructed for given (F) and (I).\nIn fact,\nin the construction of \\autoref{cor_im&fiber}, we can control (F), (I), and (c),\nbut not (r) (indeed, $\\rk (\\gamma) = 0$ for $X$ in our proof).\nOn the other hand, in the construction of \\autoref{cor_rk&fiber},\nwe can control (F), (r), and (c).\nHereafter we assume that $p= \\chara \\Bbbk$ is positive.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{cor_im&fiber}\n Assume $p = \\chara \\Bbbk >0$.\n Let $A'$ and $A''$ be finite subsets of free abelian groups $M'$ and $M''$\n respectively\n such that $\\Aff (A') = M'$ and $ \\Aff (A'')=M''$.\n Let $c > 0$ be an integer coprime to $p$.\n Assume $n:= \\rk(M') + \\rk(M'') \\geq \\#A'-1$ and $\\rk(M'') \\geq 1$.\n Then there exists a finite subset $A \\subset M := \\mathbb {Z}^n$ with $\\# A = n + \\# A''$ and $\\Aff (A)= M$\n such that the Gauss map $\\gamma$ of $X_A \\subset \\mathbb{P}^{\\#A-1}$ satisfies the following conditions:\n \\begin{enumerate}[\\normalfont(i)]\n \\item (The closure of) each irreducible component of a general fiber of $\\gamma$\n is projectively equivalent to $X_{A'}$.\n \\item (The closure of) the image $\\gamma (X_A)$ is projectively equivalent to $X_{A''}$.\n \\item The number of the irreducible components of a general fiber of $\\gamma$ is equal to $c$.\n \\end{enumerate}\n\\end{thm}\n\n\\newcommand{N'}\\newcommand{\\sAddm}{N''}{N'}\\newcommand{\\sAddm}{N''}\n\n\\begin{proof}We set $N'}\\newcommand{\\sAddm}{N'' = \\# A'-1$\n and $A'=\\{ u'_0,\\ldots, u'_{N'}\\newcommand{\\sAddm}{N''}\\}$,\n and let $e_1, \\ldots, e_{n}$\n be the standard basis of $M = \\mathbb {Z}^{n}$.\n Without loss of generality,\n we may assume that $u'_0=0$.\n We define a group homomorphism $\\pi$ by\n \\begin{align*}\n \\pi : M \\rightarrow M' \\quad: \\quad e_i \\mapsto \\left\\{ \n \\begin{array}{cl} \n u'_i & \\text{for } 1 \\leq i \\leq N'}\\newcommand{\\sAddm}{N'' ,\\\\ \n 0 & \\text{for } N'}\\newcommand{\\sAddm}{N'' +1 \\leq i \\leq n .\\\\\n \\end{array} \\right.\n \\end{align*}\n We note that $n \\geq N'}\\newcommand{\\sAddm}{N''$ holds by assumption.\n Since $\\Aff (A') = M'$ and $u'_0=0 \\in M'$,\n $\\pi$ is surjective.\n Hence $\\ker \\pi$ is a free abelian group whose rank is $\\rk (M'')$.\n Since $\\rk(M'') \\geq 1$, we can take and\n fix an injective group homomorphism\n \\[\n M'' \\hookrightarrow \\ker \\pi\n \\]\n whose cokernel is isomorphic to $\\mathbb {Z} \/ \\lin{c}$. \n Let $A''=\\{ f_0,\\ldots, f_{\\sAddm}\\} \\subset M'' \\subset \\ker \\pi$ for $\\sAddm = \\# A'' -1$.\n Without loss of generality,\n we may assume that $f_0=0 \\in M''$.\n Set\n \\[\n A =\\{ e_1,\\ldots, e_{n}, p f_0, \\ldots, p f_{\\sAddm} \\} \\subset M.\n \\]\n Since $e_1, \\ldots, e_{n} $ is a basis of $M$ and $ p f_0 =0 \\in M$,\n $A$ spans the affine lattice $M$.\n\n Let $B$ be as in the statement of Theorem \\ref{thm_structure} for the above $A$.\n Choose $n+1$ elements $u_{i_0}, u_{i_1} , \\ldots , u_{i_{n}} \\in A$ which span the affine space $M_{\\Bbbk}$.\n Since $p f_s=0 $ in $M_{\\Bbbk} $ for $0 \\leq s \\leq \\sAddm$,\n at most one element of $\\{p f_0, p f_1,\\ldots, p f_{\\sAddm} \\}$ is contained in $\\{u_{i_0}, u_{i_1} , \\ldots , u_{i_{n}}\\}$.\n Hence $\\{u_{i_0}, u_{i_1} , \\ldots , u_{i_{n}}\\} =\\{ p f_s, e_1,\\ldots, e_{n} \\}$ holds for some $0 \\leq s \\leq \\sAddm $.\n Thus we have\n \\[\n B = \\{ p f_0 + e_1+ \\cdots +e_{n} ,\\ldots, p f_{\\sAddm} +e_1+ \\cdots +e_{n} \\},\n \\]\n that is, $B $ is the parallel translation of $p \\cdot A'' $ by $e_1+ \\cdots +e_{n}$.\n Hence we have $X_B = X_{p \\cdot A''} = X_{A''}$.\n Since the closure of the image $\\gamma(X_A)$ is projectively equivalent to $X_B$ by \\ref{thm:item-image} in Theorem \\ref{thm_structure},\n the condition~(ii) in this theorem holds.\n\n\n\n Since $A''= \\{f_0, \\ldots, f_{\\sAddm} \\}$ spans the affine lattice $M''$,\n it holds that\n \\[\n \\langle B -B \\rangle = p \\cdot \\langle A''-A'' \\rangle =p \\cdot M'' \\subset M'' \\subset \\ker \\pi.\n \\]\n Therefore $\\langle B-B \\rangle_{\\mathbb {R}} \\cap M= \\ker \\pi$\n and the natural projection $M \\rightarrow M \/ (\\langle B-B \\rangle_{\\mathbb {R}} \\cap M)$ coincides with $\\pi : M \\rightarrow M'$.\n Since $\\pi(A) =A'$ by the definition of $\\pi$ and $A$,\n the condition~(i) in this theorem follows from \\ref{thm:item-fiber} in Theorem \\ref{thm_structure}.\n\n Since $ \\ker \\pi \/ M'' \\simeq \\mathbb {Z} \/ \\lin{c}$,\n the order of the finite group\n \\[\n (\\langle B-B \\rangle_{\\mathbb {R}} \\cap \\mathbb {Z}^{n}) \/ \\langle B-B \\rangle = \\ker \\pi \/ (p \\cdot M'')\n \\]\n is $p^{\\rk(M'')} c$.\n Hence (iii) in this theorem follows from (2) in \\autoref{cor_rk°}.\n\\end{proof}\n\nNote that, in the above construction, $\\rk (\\gamma) = \\dim\\lin{B-B}_{\\Bbbk} = 0$.\n\n\n\\begin{ex}\n\n\n\n\n\n\n\n In this example,\n we illustrate \\autoref{cor_im&fiber} for \n \\[\n A' = \\Set{\n \\begin{bmatrix}\n 0 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 1 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 1\n \\end{bmatrix},\n \\begin{bmatrix}\n 1 \\\\ 1\n \\end{bmatrix}\n } \\subset \\mathbb{Z}^2,\\\n A'' = \\@ifstar{\\@setstar}{\\@set}}\\newcommand{\\@setstar}[2]{\\{\\, #1 \\mid #2 \\,\\}{0, 1, 2, 3} \\subset \\mathbb{Z}^1 \n \\]\n and an integer $c >0$ coprime to $p = \\chara \\Bbbk$.\n In this case, $X_{A'} = \\mathbb{P}^1 \\times \\mathbb{P}^1 \\subset \\mathbb{P}^3$ is a smooth quadric surface\n and $X_{A''} \\subset \\mathbb{P}^3$ is a twisted cubic curve. \n Since $n= 2 +1 \\geq \\# A' -1 =4-1$,\n we can apply \\autoref{cor_im&fiber}.\n\n \n \n \n We use the notations in the proof of \\autoref{cor_im&fiber}.\n Since $\\pi : M = \\mathbb {Z}^3 \\rightarrow M' = \\mathbb {Z}^2$ is defined by\n \\[\n \\pi(e_1)=\n \\begin{bmatrix}\n 1 \\\\ 0\n \\end{bmatrix}, \\quad\n \\pi(e_2)=\n \\begin{bmatrix}\n 0 \\\\ 1\n \\end{bmatrix}, \\quad\n \\pi(e_3)=\n \\begin{bmatrix}\n 1 \\\\ 1\n \\end{bmatrix}\n \\]\n for the standard basis $e_1,e_2,e_3$ of $\\mathbb {Z}^3$,\n $\\ker \\pi $ is generated by $e_1 +e_2 -e_3$.\n Hence an injection $M'' =\\mathbb {Z}^1 \\hookrightarrow \\ker \\pi$ with cokernel $\\mathbb {Z} \/ \\lin{c}$ is given by mapping $1 \\in \\mathbb {Z}^1$ to $ c (e_1 +e_2 -e_3)$.\n Thus\n $A$ in the proof of \\autoref{cor_im&fiber} is \n \\[\n A = \\Set{\n \\begin{bmatrix}\n 1 \\\\ 0 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 1 \\\\ 0\n \\end{bmatrix},\n \\begin{bmatrix}\n 0 \\\\ 0 \\\\ 1\n \\end{bmatrix},\n p \\cdot 0 ,\n p \\cdot f,\n p \\cdot 2 f,\n p \\cdot 3 f\n }\n \\text{ for }\n f= \\begin{bmatrix}\n c \\\\ c \\\\ -c\n \\end{bmatrix}.\n \\]\n\n \\vspace{2mm}\n We can see directly that (i) - (iii) in \\autoref{cor_im&fiber} hold for this $A$ as follows:\n In this case,\n $X_A$ is the image of \n $\\phi_A: (\\KK^{\\times})^3 \\hookrightarrow \\mathbb{P}^{6}$ defined by \n \\begin{align}\\label{eq_varphi}\n (x,y,z) \\mapsto [x : y : z : 1 : (xyz^{-1})^{pc} : (xyz^{-1})^{2pc} : (xyz^{-1})^{3pc}].\n \\end{align}\n We embed $\\mathbb{P}^3 $ into $ \\mathbb{G} (3,\\mathbb{P}^6)$ by mapping $[X:Y:Z:W] \\in \\mathbb{P}^3$ to the $3$-plane in $\\mathbb{P}^6$ spanned by the $4$ points which are given as the row vectors of\n \\begin{align*}\n \\begin{bmatrix}\n 0 & 0 & 0 & X & Y & Z & W \n \\\\\n 1 & 0 & 0 & 0 & 0 & 0 & 0 \n \\\\\n 0 & 1 & 0 & 0 & 0 & 0 & 0 \n \\\\\n 0 & 0 & 1 & 0 & 0 & 0 & 0 \n \\end{bmatrix}.\n \\end{align*}\n Then the image of $\\varphi_A(x,y,z)$ by the Gauss map $\\gamma$ of $X_A$ is\n \\begin{align}\\label{eq_image}\n [1 : (xyz^{-1})^{pc} : (xyz^{-1})^{2pc} : (xyz^{-1})^{3pc}] \\in \\mathbb{P}^3 \\subset \\mathbb{G}(3, \\mathbb{P}^6)\n \\end{align}\n from \\autoref{thm:matrix-Gamma}.\n Hence the closure $\\overline{\\gamma (X_A)}$ is the twisted cubic curve $X_{A''} \\subset \\mathbb{P}^3$.\n Thus (ii) holds.\n\n From \\ref{eq_varphi} and \\ref{eq_image},\n the fiber of $\\gamma |_{T_M}$ over $[1:1:1:1] \\in \\overline{\\gamma(X_A)}= X_{A''} \\subset \\mathbb{P}^3$ is\n \\[\n \\left\\{ [x : y : z : 1 : 1 : 1 : 1] \\in X_A \\subset \\mathbb{P}^6 \\, | \\, (x,y,z ) \\in (\\KK^{\\times})^3, (xyz^{-1})^{pc} =1 \\right\\}.\n \\]\n As a set,\n this is the disjoint union of\n \\begin{align}\\label{eq_fib_comp}\n \\left\\{ [s : t : \\zeta^k st : 1 : 1 : 1 : 1] \\in X_A \\subset \\mathbb{P}^6 \\, | \\, (s,t ) \\in (\\KK^{\\times})^2 \\right\\}\n \\end{align}\n for $0 \\leq k \\leq c-1$, where $\\zeta \\in \\KK^{\\times}$ is a primitive $c$-th root of unity.\n Since the closure of each \\ref{eq_fib_comp} is projectively equivalent to $X_{A'}= \\mathbb{P}^1 \\times \\mathbb{P}^1 \\subset \\mathbb{P}^3$,\n (i) and (iii) are satisfied.\n\\end{ex}\n\n\n\n\n\n\n\n\n\n\n\n\n\nNext, we consider how to construct $X$ for a given integer $r > 0$ such that the rank of the Gauss map of $X$\nis equal to $r$.\nFrom the following remark, we need to assume $r \\neq 1$ if $\\chara\\Bbbk = 2$.\n\n\\begin{rem}\\label{rem_rk_in_char2}\n In characteristic $2$, it is known that\n the rank of the Gauss map of any projective variety $X \\subset \\PP^N$ \\emph{cannot} be equal to $1$.\n In addition, if $X$ is a hypersurface, then the rank of the Gauss map is \\emph{even}.\n The reason is as follows.\n \n Let $X \\subset \\PP^N$ be a projective variety in $\\chara\\Bbbk = 2$,\n and let $x \\in X$ be a general point.\n As in \\cite[\\textsection{}2]{fukaji2010},\n choosing homogeneous coordinates on $\\PP^N$, we may assume that\n $X$ is locally parameterized at $x = [1:0:\\dots:0]$\n by $[1:z_1: \\dots: z_n: f_{n+1}: \\dots: f_{N}]$,\n where $z_1, \\dots, z_n$ form a regular system of parameters of $\\mathscr{O}_{X,x}$, and\n $f_{n+1}, \\dots, f_N \\in \\mathscr{O}_{X, x}$.\n Then\n $\\rk d_{x}\\gamma$\n is equal to the rank of the $n \\times (n(N-n))$ matrix\n \\begin{equation}\\label{eq:Hessians}\n \\begin{bmatrix}\n H(f_{n+1}) & H(f_{n+2}) & \\cdots & H(f_{N})\n \\end{bmatrix},\n \\end{equation}\n where\n $H(f) :=\n [\\partial^2 f \/ \\partial z_i \\partial z_j]_{1 \\leq i,j \\leq n}\n $\n is the Hessian matrix of a function $f$.\n Assume that $\\rk (\\gamma)$ ($= \\rk d_x \\gamma$) is nonzero.\n Then the matrix \\ref{eq:Hessians} is nonzero; in particular,\n one of the Hessian matrix $H(f_{k})$ is nonzero.\n Since $\\chara\\Bbbk = 2$, we have $\\partial^2 f_k \/ \\partial z_i\\partial z_i = 0$,\n i.e., the diagonal entries of $H(f_{k})$ are zero. Hence \n some $\\partial^2 f_k \/ \\partial z_i\\partial z_j$ with $i \\neq j$ must be nonzero.\n Therefore $H(f_{k})$ has $2\\times 2$ submatrix\n \\[\n \\begin{bmatrix}\n \\partial^2 f_k \/ \\partial z_i\\partial z_i & \\partial^2 f_k \/ \\partial z_j\\partial z_i\n \\\\\n \\partial^2 f_k \/ \\partial z_i \\partial z_j & \\partial^2 f_k \/ \\partial z_j\\partial z_j\n \\end{bmatrix}\n =\n \\begin{bmatrix}\n 0 & \\partial^2 f_k \/ \\partial z_i\\partial z_j\n \\\\\n \\partial^2 f_k \/ \\partial z_i \\partial z_j & 0\n \\end{bmatrix},\n \\]\n whose determinant is nonzero. This implies that $\\rk (\\gamma) \\geq 2$.\n\n\n Now assume that $X \\subset \\PP^N$ is a hypersurface.\n Then $X$ is locally parametrized by $[1:z_1:\\dots:z_n:f_{n+1}]$,\n and hence $\\rk d_x \\gamma = \\rk H(f_{n+1})$.\n Since $\\chara\\Bbbk = 2$, the symmetric matrix $H(f_{n+1})$ is skew-symmetric,\n whose rank is even\n (for example, see \\cite[\\textsection{}5, n$^\\text{o}$1, Corollaire 3]{bourbaki}).\n\n\\end{rem}\n\n\n\\begin{thm}\\label{cor_rk&fiber}\n Assume $p = \\chara \\Bbbk > 0$.\n Let $A'$ be a finite subset of a free abelian group $M'$ with $\\Aff (A') = M'$.\n Let $ r,c > 0$ be positive integers such that $(p,r) \\neq(2,1)$ and $c$ is coprime to $p$.\n Assume that positive integers $n, N$ satisfy\n \\[\n n \\geq \\max \\{ (\\# A' -1) +r , \\rk (M') + r +1\\}\n \\]\n and\n \\begin{align}\\label{cond_N}\n N \\geq \\left\\{ \n \\begin{array}{cl} \n 2n-\\rk(M')-r+1 & \\text{if } p \\geq 3, \\text{ or } \\ p=2, r : \\text{even} ,\\\\ \n 2n-\\rk(M')-r+2 & \\text{if } p=2, r : \\text{odd} .\\\\ \n \\end{array} \\right.\n \\end{align}\n Then there exists a finite subset $A \\subset M:= \\mathbb {Z}^{n}$\n with $\\Aff (A)= M$ and $\\# A =N+1$\n such that the Gauss map $\\gamma$ of $X_A \\subset \\PP^N$ satisfies the following conditions:\n \\begin{enumerate}[\\normalfont(i)]\n \\item (The closure of) each irreducible component of a general fiber of $\\gamma$\n is projectively equivalent to $X_{A'}$.\n \\item The rank of $\\gamma$ is equal to $r$.\n \\item The number of the irreducible components of a general fiber of $\\gamma$ is equal to $c$.\n \\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n We set $n' = \\rk(M')$ and $N'}\\newcommand{\\sAddm}{N'' = \\# A'-1$,\n and $A'=\\{ u'_0,\\ldots, u'_{N'}\\newcommand{\\sAddm}{N''}\\}$.\n Let $e_1, \\ldots, e_{n}$ be the standard basis of $M=\\mathbb {Z}^{n}$.\n Without loss of generality,\n we may assume that $u'_0=0$.\n As in the proof of \\autoref{cor_im&fiber},\n we define a surjective group homomorphism $\\pi : \\mathbb {Z}^n \\rightarrow M' $\n by $\\pi(e_i) = u'_i$ for $1 \\leq i \\leq N'}\\newcommand{\\sAddm}{N''$ and $\\pi(e_i)=0$ for $N'}\\newcommand{\\sAddm}{N''+1 \\leq i \\leq n$.\n Since $\\ker \\pi \\simeq \\mathbb {Z}^{n-n'}$ and $e_{N'}\\newcommand{\\sAddm}{N''+1} , \\ldots, e_{N' + r} \\in \\ker \\pi$\n (note that $N'}\\newcommand{\\sAddm}{N'' +r = (\\# A' -1) +r \\leq n$ holds by assumption),\n there exist\n \\[\n f_1, \\ldots, f_{n-n'-r} \\in \\ker \\pi\n \\]\n such that $e_{N'+1} , \\ldots, e_{N'+r}, f_1, \\ldots, f_{n-n'-r} $\n form a basis of $\\ker \\pi$.\n By assumption,\n $n-n' -r = n - \\rk(M') -r \\geq 1$ holds.\n\n\n\n\n\n\n \\vspace{1mm}\n First,\n we consider the case\n when $N$ is equal to the right hand side of \\ref{cond_N}.\n Set\n \\[\n A = C \\cup D \\subset M,\n \\]\n where\n \\begin{align*}\n C &= \\{ e_1,\\ldots,e_n, 0, c pf_1, p f_2, \\ldots, p f_{n-n'-r}\\},\n \\\\\n D &= \\left\\{ \n \\begin{array}{cl} \n \\{e_{N'}\\newcommand{\\sAddm}{N''+1} + \\cdots + e_{N'}\\newcommand{\\sAddm}{N''+r} \\} & \\text{for } r \\not \\equiv 1 \\text{ mod } p ,\\\\ \n \\{- e_{N'}\\newcommand{\\sAddm}{N''+1} - \\cdots - e_{N'}\\newcommand{\\sAddm}{N''+r} \\} & \\text{for } r \\equiv 1 , r \\not \\equiv - 1 \\text{ mod } p ,\\\\ \n \\{e_{N'}\\newcommand{\\sAddm}{N''+1} + e_{N'}\\newcommand{\\sAddm}{N''+2}, e_{N'}\\newcommand{\\sAddm}{N''+2} + \\cdots + e_{N'}\\newcommand{\\sAddm}{N''+r} \\} & \\text{for } p=2, r : \\text{odd}, r \\geq 3.\n \\end{array} \\right.\n \\end{align*}\n By a similar argument in the proof of \\autoref{cor_im&fiber} and by Examples \\ref{prep-cor_rk&fiber}, \\ref{ex_char2},\n we have\n \\begin{align}\\label{eq_B-B}\n \\langle B-B \\rangle = \\bigoplus_{i=N'}\\newcommand{\\sAddm}{N''+1}^{N'}\\newcommand{\\sAddm}{N''+r} \\mathbb {Z} e_{i} \\oplus \\mathbb {Z} cp f_1 \\oplus \\bigoplus_{j=2}^{n-n'-r} \\mathbb {Z} p f_j .\n \\end{align}\n Hence \n $\\langle B-B \\rangle_{\\mathbb {R}} \\cap M= \\ker \\pi$.\n Since $A' = \\pi (A)$,\n (i) and (iii) in this corollary follows as in \\autoref{cor_im&fiber}.\n\n Since $e_{N'+1} , \\ldots, e_{N'+r}, f_1, \\ldots, f_{n-n'-r} $ form a basis of $\\ker \\pi$,\n we have $\\langle B-B \\rangle_{\\Bbbk} =\\bigoplus_{i=N'}\\newcommand{\\sAddm}{N''+1}^{N'}\\newcommand{\\sAddm}{N''+r} \\Bbbk e_{j} $.\n Thus the rank of $\\gamma $ is $\\dim \\langle B-B \\rangle_{\\Bbbk} = r$ by \\autoref{cor_rk°}.\n\n \\vspace{3mm}\n Next, we consider any integer $N$ satisfying the inequality~\\ref{cond_N}.\n We take\n a finite subset $E$ of the right hand side of \\ref{eq_B-B} such that\n $N = \\# (C \\cup D \\cup E) -1$\n for the above $C$ and $D$.\n Set\n $A := C \\cup D \\cup E \\subset M$.\n Since\n $E$ is contained in the right hand side of \\ref{eq_B-B}, the subgroup\n $\\langle B-B \\rangle $ for this $A$ is the same as \\ref{eq_B-B}.\n Hence the assertion follows. \\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe Muon Ionization Cooling Experiment~\\cite{MICE} is under construction at the UK's Rutherford Appleton Laboratory (RAL) in order to demonstrate for the first time the feasibility and efficacy of ionization cooling of muons~\\cite{cooling1,cooling2}. The MICE apparatus (Fig.~\\ref{fig:MICE}) comprises one cooling lattice cell based on a design from Neutrino Factory Feasibility Study II (FS-II)~\\cite{FSII} sandwiched by the input and output spectrometers and particle identification and timing detectors that will be used to characterize the ionization-cooling process experimentally. Since an affordable cooling section cools by only $\\sim$\\,10\\% (see Fig.~\\ref{fig:MICE}), too small an effect to measure reliably using standard beam instrumentation, MICE employs a low-intensity muon beam~\\cite{beam} and measures each muon individually. Once completed, it will thereby demonstrate that the ionization-cooling process is understood in detail in both its physics and engineering aspects, and that it works as simulated. In order to afford a thorough validation of the codes used to design ionization-cooling channels, MICE will be operated in various modes and optics configurations. Full results are expected by about 2020, with analyses of some configurations available up to five years earlier. Early results will include important validations of the models used in ionization-cooling simulation codes, as well as the first experimental test of muon transverse--longitudinal emittance exchange (needed for six-dimensional cooling, e.g., for a muon collider) in a wedge absorber.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=.62\\linewidth, trim=450 0 300 0 mm,clip]{MICE-3D-cutaway-new}\\hfill\\includegraphics[width=.37\\linewidth]{MICE-cool-new}}\n\\vspace{-.1in}\n\\caption{Left: cutaway rendering of MICE (``Step VI\") apparatus: two precision scintillating-fiber solenoidal spectrometers surround a cooling cell based on the FS-II ``SFOFO\" design~\\cite{FSII} (particle-ID detectors not shown). Right: normalized-emittance change vs input emittance in the nominal MICE optics configuration, showing 2.5$\\pi$\\,mm$\\cdot$rad equilibrium transverse emittance.\\\\[-.15in]\n}\\label{fig:MICE}\n\\end{figure}\n\nIonization cooling is inherently a transverse effect. A simple and intuitive way to see this is to consider that ionization energy loss exerts a braking effect on each muon, reducing both the transverse and longitudinal momentum components, while reacceleration in RF cavities restores only the longitudinal momentum component. With repeated braking and reacceleration the divergence of the beam is progressively reduced. Alternating-gradient focusing brings a concomitant reduction of the beam's cross-sectional area. The process can beneficially continue until an equilibrium is reached between the cooling effect of ionization and the heating effect of multiple scattering, i.e., until \n\\begin{eqnarray} \n\\frac{d\\epsilon_n}{ds}\\approx\n-\\frac{1}{\\beta^2} \\left\\langle\\!\\frac{dE_{\\mu}}{ds}\\!\\!\\right\\rangle\\frac{\\epsilon_n}{E_{\\mu}}\n +\n\\frac{1}{\\beta^3} \\frac{\\beta_\\perp\n(0.014\\,{\\rm GeV})^2}{2E_{\\mu}m_{\\mu}L_R}\\to 0 \\,,\n\\label{eq:cool} \n\\end{eqnarray} \n $d\\epsilon_n\/ds$ being the rate of normalized-emittance change within the absorber; $\\beta c$, $E_\\mu$, and $m_\\mu$\nthe muon velocity, energy, and mass; $\\beta_\\perp$ the lattice betatron function at the absorber; and $L_R$ the radiation length of the absorber material~\\cite{cooling2}. Ionization cooling works optimally for momenta near the ionization minimum, hence MICE is designed for the momentum range 140 to 240\\,MeV\/$c$.\n\n\n\\section{MICE Steps}\n\nThe MICE precision goal, 0.1\\% emittance resolution, allows even the small emittance decrements (increments) when operating just above (below) the cooling cell's equilibrium emittance to be well measured. This places a premium on careful spectrometer characterization and calibration. This goal and the anticipated funding schedule led to a planned MICE buildup in a series of six ``Steps,\" but exigencies of complex apparatus construction have resulted in the abbreviated sequence of Fig.~\\ref{fig:Steps}. Step I has already taken place, and the results presented~\\cite{beam,PID}, with the exception of the Oct.\\ 2013 EMR (``electron--muon ranger\" total-absorption calorimeter) run. \n\\begin{figure}\n\\begin{minipage}[b]{20pc}\n\\includegraphics[width=1.13\\linewidth\n]{mice-steps-I}\\\\\n\\includegraphics[width=1.03\\linewidth,\n]{mice-steps-IV-VI}\n\\caption{The Steps of MICE}\\label{fig:Steps}\n\\end{minipage}\\hfill\n\\begin{minipage}[b]{16pc}\n\\centerline{\\includegraphics[width=\\linewidth]{SS}}\n\\caption{MICE spectrometer solenoids at vendor}\\label{fig:SS}\n\\end{minipage}\n\n\\end{figure}\n\n\\section{Step IV}\n\nMICE Step IV will represent a first study of muon ionization cooling, with results expected starting in 2015. \nIn addition to the Step I components (pion-production target inserted periodically into the ISIS synchrotron beam, pion-to-muon beamline, particle-ID and muon-timing detectors), Step IV requires the two trackers with their spectrometer solenoids (SS) and one absorber--focus-coil (AFC) module. The trackers are in hand and have been tested with cosmic rays~\\cite{tracker}. Each SS comprises five superconducting coils: three to provide a 4\\,T field uniform to better than 1\\% over the 1-m-long, 40-cm-diameter tracking volume, and two to match the beam into or out of the cooling cell. The spectrometer solenoids (Fig.~\\ref{fig:SS}) have been built by Wang NMR in Livermore, CA, USA, and the focus coils by Tesla Engineering Ltd., UK. As of this writing, after some retrofitting, the first SS has been successfully trained, field-mapped, and delivered to RAL, and training of the second has commenced. The first AFC has been trained to just above the baseline current in ``flip\" mode (the two coils powered in opposite polarities) and to the full design current in ``solenoid\" mode (same polarities). The training of a second AFC is commencing and the performance of the first will be assessed in the light of that experience.\n\nStep IV will include tests of LH$_2$, LiH, and plastic or other low-$Z$ absorbers. A demonstration of emittance exchange is also planned. Muon colliders require muon cooling in all six phase-space dimensions, and rely on emittance exchange in order to couple the transverse ionization cooling effect into the longitudinal phase plane. This is accomplished via a suitable correlation between muon momentum and path length in absorbers, and thus requires lattices with dispersion. While there is naturally some dispersion in the MICE beamline, since in MICE each muon is measured individually, the desired range of dispersion will be created via off-line selection of muons.\n\n\\section{Toward Step VI}\n\nMICE Step IV will allow the detailed characterization of ionization energy loss, multiple Coulomb scattering, and their effect on the ionization cooling process, and will thus validate the models used in muon ionization-cooling simulations. However, it constitutes a ``non-sustainable\" cooling configuration, since the energy lost in the absorber is not replenished. The first engineering demonstration of ``sustainable\" cooling will thus be Step VI (depicted in Fig.~\\ref{fig:MICE}), with a possible ``stopover\" in Step V, depending on component availability. By allowing a thorough exploration of the optics of ionization-cooling lattices, including those with periodic field flips and those with a modulated solenoid field, Step VI will furnish a complete validation of the simulations.\n\nStep VI requires (besides the components of Step IV) two additional AFC modules and two RF-cavity--coupling-coil (RFCC) modules, each containing four 201\\,MHz RF cavities and one large-diameter coupling-coil (CC) solenoid. All eight RF cavities have been fabricated and the first (Fig.~\\ref{fig:201MHz}) is being readied for testing in the Fermilab MuCool Test Area (MTA). At the FS-II design gradient these cavities each require 4 MW of input power, provided through two coaxial couplers. Prototype tests at the MTA revealed RF-coupler breakdown as an issue. Couplers of a revised design are now in fabrication and will soon be tested in the MTA. \n\nA full test requires in addition a coupling coil, so that the behavior of the cavity in a multi-tesla solenoid field can be explored. The first CC cold mass (Fig.~\\ref{fig:CC}), built for Harbin Institute of Technology by Qi Huan Co., Beijing, China, and planned for the MTA rather than MICE, is now under test at Fermilab. Once it is trained, three more (including one spare) will be fabricated. Fabrication of the needed cryostats and vacuum vessels \n{(at LBNL, based on initial designs developed by SINAP)}, and of the cavity tuners, is also in progress, as is assembly of the RF control and power distribution systems by LBNL and Daresbury Lab.\n\n\\begin{figure}\n\\begin{minipage}[b]{17pc}\n\\centerline{\\includegraphics[width=\\linewidth,trim=40 0 40 0 mm,clip]{201MHz}}\n\\caption{First MICE RF cavity being fitted with tuners at Fermilab.}\\label{fig:201MHz}\n\\end{minipage}\\hfill\n\\begin{minipage}[b]{17pc}\n\\centerline{\\includegraphics[width=\\linewidth,trim=0 0 0 10 0mm,clip]{CC}}\\vspace{-.1in}\n\\caption{First CC cold mass fitted out at LBNL prior to shipment to Fermilab.}\\label{fig:CC}\n\\end{minipage}\n\\end{figure}\n\nThe FS-II design~\\cite{FSII} calls for 32\\,MW of RF power per cell in order to allow off-crest operation. The MICE Step VI goal is on-crest acceleration, with power provided by four 2\\,MW supplies from CERN and LBNL refurbished by Daresbury Lab. The first supply has recently reached full power, with a single-supply test at MICE scheduled for later this year. \n\n\\section{Conclusions}\nMICE is progressing towards the first experimental study of muon ionization cooling. Step IV is planned for 2015 and the concluding Step VI of MICE by the end of this decade. Once ionization cooling is established, construction of a neutrino factory with cooling could commence. \n\n\\section*{Acknowledgments}\nThe author thanks his MICE collaborators \nfor many stimulating interchanges on these topics. Work supported by the U.S.\\ Dept.\\ of Energy (through the Muon Accelerator Program) and National Science Foundation.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:intro}Introduction}\nNumerous geophysical and astrophysical flows present a two-layer configuration, with a turbulent convective layer standing above or below a stably stratified one. Examples include planetary atmospheres, stars interiors, and possibly the outermost layer of the Earth liquid core \\cite{hirose_composition_2013}. The dynamics of coupled stratified and convective layers are quite complex. Due to the convective motions, internal gravity waves (IGWs) are generated at the interface between the two layers, and propagate in the stratified one. IGWs transport energy and momentum \\cite{rogers_internal_2012,bretherton_momentum_1969} from where they are generated to where they are damped. Thanks to their transport properties and non-linear interactions, IGWs are able to generate and sustain large-scale horizontal flows \\cite{plumb_interaction_1977,rogers_internal_2012}. Examples of such large-scale flows driven by IGWs are the oscillations of equatorial zonal winds observed in some planets' atmosphere \\cite{fouchet_equatorial_2008,leovy_quasiquadrennial_1991}, including the Earth where it is called the Quasi-Biennial Oscillation (QBO) \\cite{baldwin_quasi-biennial_2001}.\\\\\n\nThe IGWs generation by turbulent dynamics has been studied in various experiments. The generation by a single buoyant plume was experimentally studied by Ansong \\& Sutherland \\cite{ansong_internal_2010}. The penetration of the plume within the stratified layer and the spectral characteristics of the generated IGWs were studied. They found that the peak frequency of the generated IGWs lies in a range close to $0.7N$, where $N$ is the Brunt-V\\\"ais\\\"al\\\"a (or buoyancy) frequency, and that the radial wavenumber is set by the plume cap and not by the width of the plume at the interface.\n\nDeardoff et al. \\cite{deardorff_laboratory_1969} and later Michaelian et al. \\cite{michaelian_coupling_2002} studied the effect of penetrative convection in a stratified layer in a transient, Rayleigh-B\\'enard type experiment. Stratification was initially set up thermally from the top to the bottom of the tank. Then, the fluid was suddenly warmed up at the bottom, triggering Rayleigh-B\\'enard convection. IGWs were measured transiently \\cite{michaelian_coupling_2002} while the stratified (resp. convective) layer was decreasing (resp. increasing) in size. Eventually, there was no more stratified layer to sustain the propagation of IGWs. \n\nTownsend \\cite{townsend_natural_1964} introduced an original set-up to study the quasi-steady generation of IGWs by Rayleigh-B\\'enard convection. Using the fact that water maximum density is around $4^{\\circ}$C, a two-layer system is spontaneously generated by cooling the bottom of a tank at $0^{\\circ}$C and heating its top above $4^{\\circ}$C. The density gradient is unstable at temperature below 4 $^{\\circ}$C and stable at temperature above. This creates a self-organising system, with a turbulent convective layer adjacent to a stratified layer. With dye visualisation and temperature measurements, he observed IGWs propagating close to the interface between the two layers. The $4^\\circ$C convection was also studied experimentally by Le Gal \\cite{legal_penetrative_1997} in a laminar flow, at low Rayleigh number. Convection displayed an hexagonal pattern and viscous entrainment of the fluid above the convective cells was observed. Perrard et al. \\cite{perrard_experimental_2013} and Le Bars et al. \\cite{bars_experimental_2015} re-investigated this setup in a quasi two-dimensional tank using Particle Image Velocimetry (PIV) and temperature measurements to obtain detailed data of IGWs generated by the convection. They observed a wide spectrum of waves generated at different frequencies. Favoured frequencies were related to the differential attenuation length of the waves depending on frequency, in good agreement with linear theory. No large-scale flow in the stratified layer was observed in this 2D geometry. Numerical simulations of the same configuration were performed by Lecoanet et al. \\cite{lecoanet_numerical_2015}. They demonstrated that IGWs are mainly generated by the Reynolds stresses in the bulk of the convection below the interface, rather than by the displacement of the interface between the two layers. Numerical studies by Couston et al. \\cite{couston_dynamics_2017,couston_order_2018,couston_energy_2018} extended these results by considering a generic non-linear equation of state (piecewise linear around an inversion temperature, with adjustable slopes), both in 2D and 3D horizontally periodic geometries. Various flow regimes and the energy transport by IGWs were quantitatively studied. Interestingly, long simulations -- accessible in 2D studies only -- showed, for low Prandtl numbers ($Pr < 1$), a large-scale horizontal flow with reversals in the stratified layer, similar to the QBO phenomenon introduced above \\cite{couston_order_2018}. \n\\\\\n\nSeveral experiments took interest in the generation and reversal of a large-scale horizontal mean flow in a stratified domain, driven by IGWs. The well-known experiment designed by Plumb and McEwan \\cite{plumb_instability_1978}, later reproduced and extended by Semin et al. \\cite{semin_generation_2016,semin_nonlinear_2018}, is capable of driving a QBO from the mechanical forcing of a standing wave pattern (\\textit{i.e.} two waves with the same frequency and opposite phase speed) in a salty water stratification. With this system, Plumb and McEwan managed to observe few oscillations of the driven large-scale flow before the stratification was destroyed by mixing. The experiment gave results in good agreement with the theory \\cite{richard_s._lindzen_theory_1968,lindzen_updated_1972,plumb_interaction_1977}, notably with reversals starting away from the forcing and propagating towards it.\nSemin et al. \\cite{semin_generation_2016,semin_nonlinear_2018} improved the system by constantly injecting fluid to rebuild the stratification, while removing mixed fluid close to the forcing. This method allowed to run the experiment longer and to study the nature of the bifurcation in the Plumb model which can be either supercritical or subcritical, depending on the dominant dissipative process. In those experimental realisations of the QBO mechanism, the wave forcing remains monochromatic, as opposed to the natural mechanism where it is due to chaotic tropical storms \\cite{baldwin_quasi-biennial_2001}. The forcing is driven by interface displacements, as opposed to the observations of \\cite{lecoanet_numerical_2015}. Besides, only the stratified layer is modelled. It thus remains a challenge to observe experimentally a large-scale, reversing flow from a turbulent source and a wide range of naturally excited IGWs.\n\\\\\n\nIn the present study, we extend the work of Townsend \\cite{townsend_natural_1964,perrard_experimental_2013,bars_experimental_2015} in a cylindrical, 3D geometry reminiscent of Plumb and McEwan's set-up \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}. Our purpose is threefold: to characterise the generation of IGWs in such a self-organising two-layer system, to quantify the coupling between the layers, and to investigate the possible generation of large-scale horizontal flows. Our experiments are complemented by direct numerical simulations of the same configuration. The experimental setup and numerical model are presented in section \\ref{sec:methods}, results are analysed in section \\ref{sec:results}, and conclusions and future works are discussed in section \\ref{sec:discussion}. \n\n\n\n\\section{\\label{sec:methods}Methods}\n\\subsection{Experimental set-up}\nThe set-up consists in a cubic tank whose lateral boundaries are made of 2 cm thick acrylic walls. The bottom boundary is a chrome plated copper plate in which refrigerated fluid is forced to circulate. The top boundary is a commercial, transparent electric heater. The tank inner dimensions are $32 \\times32$ cm for the horizontal section and $H=20$ cm in height. Preliminary experiments were conducted in this cubic geometry. Eventually, a cylinder of outer diameter $D = 29$ cm and thickness $e = 0.4 $ cm was added inside the cubic tank, to reproduce the axisymmetric geometry of \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}, which seems prone to the development of large-scale flows. We are interested in the flow within the cylinder: the fluid in the gap between the cylinder and the cubic tank follows a similar dynamics and thus imposes to the working fluid a (close to) no-flux lateral boundary condition.\n\nThe temperature of the bottom boundary is controlled by water circulating in the copper plate. Water is cooled down by a thermal bath with a fixed temperature set at $-1.25^{\\circ}$C. Due to thermal losses in the pipes alimenting the copper plate, bottom tank temperature is $0.2 \\pm 0.05^\\circ$C. Plate thickness and circulation circuit were optimised so as to ensure a uniform temperature over the whole plate. At the top boundary, the heater is set at a temperature of $35^{\\circ}$C. Its temperature control is custom made: a PT100 probe measures the heater temperature in real time, driving through a feed-back loop the input power injected in the heater. This is a very simple and inexpensive system to impose a temperature while having a transparent top boundary, allowing visualisation and velocity measurements by PIV. Nonetheless, it is necessary to point out that the temperature over the heater area is not perfectly homogeneous. Temperature is maximal at the centre where $T \\sim 38^{\\circ}$C, while the edges are indeed at the requested $T = 35\\pm 0.1^\\circ$C. This inhomogeneity of the top temperature by $\\delta T = 3^{\\circ}$C induces slow convective motions below the heater, in a $\\sim 2$ cm high layer. By performing an experiment where the whole fluid is stably stratified with an overall temperature gradient similar to the one in the stratified layer studied here, but above the density reversal at $4^{\\circ}$C (i.e. bottom boundary set at 10$^{\\circ}$C and top boundary at 70$^{\\circ}$C), we have checked that those top convective motions have no significant impact on the dynamics of the two-layer system. It is also important to say that despite the thick acrylic wall and the intermediate fluid layer between the cylinder and the tank, the working region is not fully thermally insulated on the sides. Nevertheless, our fully stratified, test experiment has shown no motion within the fluid driven by these lateral losses. \n\\\\\n\nThe equation of state of water is non-linear with a maximum density close to 4$^{\\circ}$C (International Equation of State of Seawater, 1980):\n\\begin{equation}\n \\begin{split} \n \\rho(T)= 999.842594+6.793952.10^{-2}T-9.095290.10^{-3}T^2+1.001685.10^{-4}T^3 \\\\\n -1.120083.10^{-6}T^4+6.536332.10^{-9}T^5.\\\\\n \\end{split}\n \\label{eq:eos_eau}\n\\end{equation}\nThus, due to the imposed temperature at top and bottom boundaries, the bottom part of the tank, between 0$^{\\circ}$C and 4$^{\\circ}$C, is convectively unstable (see figure \\ref{fig:setup}). Cold plumes detach from the bottom plate and rise in the tank due to buoyancy. Reciprocally, ``hot'' fluid sinks from the 4$^{\\circ}$C interface due to gravity. While convective motion takes place in the lower layer, the upper part of the tank, between 4$^{\\circ}$C and 35$^{\\circ}$C, is stably stratified, with an assumed linear temperature profile at equilibrium. The temperature is indeed linear for an ideal case without thermal losses, assuming that the stratified layer has a bottom boundary at fixed temperature $4^\\circ$C and top boundary at $35^\\circ$C (\\textit{i.e.} constant diffusive flux through the whole layer). However, the density profile is not linear, due to the non-linear equation of state of water. Stratification is characterised by the Brunt-V\\\"ais\\\"al\\\"a frequency $N^* = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\partial \\rho}{\\partial z}}$. Because of the non-linear equation of state, $N^*$ is not constant with depth, as shown in figure \\ref{fig:setup}. For simplicity, we also use below the global buoyancy frequency defined as $N = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\Delta \\rho}{\\Delta z}}$, where ${\\Delta \\rho}$ is the global density contrast within the stratified layer of depth ${\\Delta z}$.\n\\\\\n\n\\begin{figure}[h]\n\n \\includegraphics[scale=.3]{schema_cuve_profilN_NB_v3.png}\n \\caption{2D sketch of the tank. A cylinder (light grey shaded area) is placed in a larger cubic tank. The system is cooled down at the bottom at 0$^{\\circ}$C and heated up at the top at 35$^{\\circ}$C. The bottom half is convective with an almost constant density, apart from the bottom boundary layer. The upper half is stably stratified and waves are generated due to the fluid motions in the convective layer. The graph on the right shows the theoretical profile for the buoyancy frequency $N^*$. It is computed considering a linear temperature profile and the equation of state of water (\\ref{eq:eos_eau}). The dashed line is the global buoyancy frequency $N$ calculated on the stratified layer. The various length scales are the cylinder diameter $D$, the vertical extent of the tank $H$ and the vertical extent of the convective layer $h$. $\\delta$ is the minimal width between the outer square tank and the inner cylinder. The $x$, $y$ and $z$-velocity components are noted $u$, $v$ and $w$ respectively.}\n \\label{fig:setup}\n\\end{figure}\n\nBefore starting the experiment, the bath and the heater are allowed to reach their assigned temperature. Then, the upper half of the tank is filled with water stratified in temperature from 4$^{\\circ}$C to 35$^{\\circ}$C, using the double bucket technique \\cite{oster_density_1965}. The bottom half is filled with water with a temperature close to 4$^{\\circ}$C. This filling process is used to avoid tremendously long transient before reaching steady state by thermal diffusion. Typically, we fill the tank at the end of a day and start the measurements the next day in order to let the system reach its equilibrium state over night. Each experiment then lasts four days, with no change in the location of the interface. Note that this steady interface position is the result of the heat budget equilibrium between the convective heat flux extracted from the bottom plate, the diffusive heat flux through the stratified layer, and the lateral heat losses.\n\nTo perform PIV measurements, particles are chosen as small as possible and with a density as close as possible to water density in order to avoid their sedimentation in the stratified layer over the long duration of the experiment. We use fluorescent orange polyethylene micro-spheres whose size ranges from $10 \\, \\mathrm{\\mu m}$ to $20 \\, \\mathrm{\\mu m}$ and density is $1.00 \\pm 0.01$~g\/cc. The fluorescent property allows us, with a high pass filter on the camera, to remove any laser reflection, significantly enhancing the images quality. The tank is illuminated with a green laser $532$~nm. Power is set at $1$~W. \nWe perform side view PIV to measure convection and IGWs spectral characteristics, and top view PIV to observe the large scale flow and its fluctuations over time. The camera used for the side view PIV is a HiSense Zyla $2560 \\times 2160$ pixels recorded on 12 bits. Acquisition rate is $2$~Hz with $100$~ms exposure time. Typical acquisition time for spectral characteristics is $50$~min.\nFor the top view, we use a Point grey camera $1920\\times1080$ pixels on 8~bits. Exposure time is $1$~s, acquisition rate $0.1$~Hz and acquisition time $8$~hours. Captured movies are processed either by the DantecDynamics software DynamicStudio for the side view or by DPIVSoft \\cite{meunier2003analysis} for the top view. Both are resolved into $32\\times32$ pixels boxes with 50\\% overlapping. \n\nSide view PIVs are performed in the middle of the tank at $y=16$~cm in a laser sheet crossing the cylinder along its diameter. This is the case for all figures shown in the $(x,z)$ plane and thus not mentioned in the results section. The vertical fields (see an example in figure \\ref{fig:LSCexpe}) do not show the whole $(x,z)$ plane (where the origin $(0,0)$ is located in the bottom left corner of the cubic tank): it was indeed chosen to zoom in, in order to have the best resolution for the very weak motions in the stratified layer. The interface between the layers is localised between $11 \\mathrm{~cm} \\leqslant z \\leqslant 12$~cm. Typical Rayleigh number for the convection based on this depth is $Ra = 7 \\times 10^6$, and the global Brunt-V\\\"ais\\\"al\\\"a frequency is $N = 1.35 \\times 10^{-1}$~Hz.\n\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[scale = .45]{champ_int_730_colored.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .45]{LSC_0510_1800_colored.pdf}\n \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=150$ mm, and transported by the large-scale circulation. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes signal. The large-scale circulation is a counter clockwise cell. No inversion of the circulation has been seen in our experiment. The velocities under $z=10$ mm are noisy and thus not shown here. Maximum instantaneous velocities are $2.5$ times bigger than the maximum averaged velocities. The left edge of the cylinder is located at $x=19$~mm and the centre of the cylinder is at $x=160$~mm. Approximately $40$~mm of the right side of the cylinder is not captured with our PIV measurments.}\n \\label{fig:LSCexpe}\n\\end{figure*}\n\n\nTo observe the large-scale flow, horizontal views at different heights are performed. A linear translation axis platform from Igus, driven by homemade software, is used to automate the process. With two mirrors, one fixed at the exit of the laser head and the other fixed on the translating platform, it is possible to sweep along one direction with the laser sheet. We typically make measurements during 15~s in each of 11 successive planes separated by 0.5~cm. The full scan duration is about 3~min, and is repeated during at least 8~hours.\n\\\\\n\nThe cylindrical geometry described here differs from the cylindrical shell geometry in \\cite{plumb_instability_1978,semin_generation_2016, semin_nonlinear_2018}. We first tried to work in that annular geometry by adding a second, smaller cylinder in our cubic tank. Three different gap sizes were tested but none showed any interesting dynamics. Indeed, the convection was too confined in the shell to provide an efficient chaotic excitation, and IGWs did not propagate well within the stratification, attenuated quite fast by wall friction. During these tests, we observed the most interesting dynamics within the innermost cylinder, so we decided to use that geometry. The point of the cylindrical shell geometry is that it is a close analogue to the equatorial band of the stratosphere where QBO takes place. By working in a cylinder, the geometry analogy is lost but the physics of the problem remains the same, still a priori allowing for large-scale, reversing, axisymmetric horizontal flows.\n\n\\clearpage\n\\subsection{Numerical model}\\label{sec:methods_num}\n\nTo complement the experiments, we also performed Direct Numerical Simulations (DNS) of the same configuration. We solve the Navier Stokes equations using a non-Oberbeck Boussinesq model. The density variations are consider small compared to the reference density $\\frac{\\Delta \\rho }{\\rho_o} \\ll 1$. Therefore, density fluctuations only appear in the buoyancy force. However, temperature variations affect the value of the thermal expansion coefficient to account for the non-linear equation of state of water. Variations of the thermal diffusivity $\\kappa$ and kinematic viscosity $\\nu$ are neglected. Governing equations are given in dimensionless form by:\n\\begin{align}\n\\label{eq:momentum}\n\\frac{\\partial\\bm{u}}{\\partial t}+\\bm{u}\\cdot\\nabla\\bm{u} = & -\\nabla P+Pr\\nabla^2\\bm{u}-Pr Ra \\ \\theta^2 \\bm{e}_z \\\\\n\\label{eq:heat}\n\\frac{\\partial \\theta}{\\partial t}+\\bm{u}\\cdot\\nabla\\theta = & \\ \\nabla^2 \\theta \\\\\n\\label{eq:mass}\n\\nabla\\cdot\\bm{u} = & \\ 0\n\\end{align}\nwhere we used the depth $H$ of the container as a unit of length, the thermal diffusive timescale $H^2\/\\kappa$ as a unit of time and the difference between the bottom and the inversion temperature $T_0 - T_i$ as a temperature scale. These equations are characterised by the Prandtl number $Pr=\\nu\/\\kappa$, the global Rayleigh number $Ra=\\alpha g (T_0-T_i)^2 H^3\/(\\nu\\kappa)$ and the imposed dimensionless top temperature $\\theta_{top}=(T_{top}-T_i)\/(T_0-T_i)$.\nNote that the quadratic temperature term in the momentum equation is a direct consequence of the nonlinear equation of state of water given by equation~\\eqref{eq:eos_eau}, which is approximed in our model by the quadratic equation $\\rho(T) \\approx \\rho_0 (1-\\alpha(T-T_i)^2)$.\nThe coefficient $\\alpha$ is not the usual thermal expansion coefficient but has a unit of $(\\degree \\mathrm{C})^{-2}$ and is given by $\\alpha\\approx8.1\\times10^{-6}(\\degree \\mathrm{C})^{-2}$ (see also \\cite{lecoanet_numerical_2015}).\n\nWe consider a cylindrical fluid cavity of diameter $D=3H\/2$ as in the experiments.\nBoth horizontal plates are assumed to be no-slip and with fixed temperature.\nThe side wall is assumed to be no-slip and perfectly insulating.\nThis is of course not the case in the experiment, for which lateral heat losses are inevitable and top temperature is not exactly constant, but the objective is to check whether the conclusions drawn from the experimental results are robust and do not depend on these effects.\nSince the experiment runs with water, we use $Pr=7$.\nThe Rayleigh number of the experiment is $Ra=7 \\times 10^7$ while its dimensionless top temperature is $\\theta_{top}=-7.75$. If we were to run the simulation with these parameters, the interface will be located very close to the top boundary. It is not the case in the experiment because of the lateral heat losses, which tend to reduce the effective Rayleigh number. For that reason, and instead of taking into account these losses as in \\cite{lecoanet_numerical_2015}, we kept the insulating lateral boundaries and use the slightly adjusted parameters $Ra=10^7$ and $\\theta_{top} = -11$ instead, which leads to an interface located approximately at $z\\approx120$~mm, as in the experiment. The Rayleigh number could not be lowered under $10^7$ in order to keep the convective flow turbulent enough, thus we had to increase the top temperature to have the interface located at $z\\approx120$~mm.\n\nWe perform DNS of equations~\\eqref{eq:momentum}-\\eqref{eq:mass} using the spectral element solver Nek5000 \\citep{Nek5000}.\nThe global geometry is decomposed into hexahedral elements, with vertical refinement close to the horizontal boundaries and around the mid-plane where the inversion isotherm is expected to be located.\nVelocity, buoyancy and pressure variables are represented as tensor product Lagrange polynomials of order $N$ and $N-2$ based on Gauss or Gauss-Lobatto quadrature points.\nThe total number of grid points is given by $\\mathcal{E}N^3$ where $\\mathcal{E}$ is the number\nof elements.\nFor all the results discussed in this paper, the number of elements is $\\mathcal{E}=8960$ and we use a polynomial order of $N=11$. Numerical convergence was checked by increasing the polynomial order $N$.\nTime integration is performed with a third-order mixed implicit-explicit scheme.\nThe simulations are initialised with a small amplitude buoyancy perturbation and a temperature profile varying linearly between the top and bottom boundaries.\nSimulations are run until a quasi-steady state is reached, after which data is accumulated to compute statistics.\n\n\\section{\\label{sec:results}Results}\n\\subsection{\\label{sec:results_exp}Experiments}\n\\subsubsection{\\label{sec:results_conv}Convection}\nPIV side view is used to quantify horizontal and vertical velocities in the convection zone. Examples of vertical velocities measured at one point in a given location are shown in figure \\ref{fig:panache}, for both ascending cold and descending hot structures. Measurements are consistent with the numerical simulations \\cite{lecoanet_numerical_2015,couston_dynamics_2017} showing intense, localised, cold rising plumes and more diffusive descending plumes. Moreover, these structures are advected by a large-scale circulation encompassing the whole convective layer, as shown in figure \\ref{fig:LSCexpe}.\n\n\\begin{figure}[h]\n \\centering\n {\\label{fig:panacheup}\\includegraphics[scale=0.43]{sonde_panache_up.pdf}}\n \\hspace{.2pt}\n {\\label{fig:panachedown}\\includegraphics[scale=0.43]{sonde_panache_down.pdf}}\n \\caption{Time evolution of the vertical velocity $w$ within: (Left) upward plumes at $x = 200$ mm, $z = 45$ mm, (Right) downward structures at $x= 100$ mm, $z = 95$ mm. }\n \\label{fig:panache}\n\\end{figure}\n\nSpectral analysis is performed to extract power spectral density (PSD) from the velocity signals. Figure \\ref{fig:spectreconv} shows the PSD of the convection vertical velocity $w$. For the two panels, the spectrum is flat with a lot of energy for low frequencies, then the energy drops above some cut-off frequency. Left panel of figure \\ref{fig:spectreconv} shows the vertical velocity PSD at a single point close to the top of the convective layer. A small peak can be seen close to $f = 10^{-2}$~Hz. This is the quasi-periodic signal of the plumes dropping from the top thermal boundary layer. The theoretical characteristic time of convection can be computed from \\cite{gortler_convection_1966}:\n\\begin{equation}\n \\tau = \\frac{h^2}{\\pi \\kappa}\\left(\\frac{\\Delta T}{\\Delta T_{local}} \\times \\frac{Ra_c}{Ra}\\right)^{2\/3}\n\\end{equation}\n\n\\begin{figure*}[b]\n \\centering\n \\includegraphics[scale = .43]{spectre_1_point_cd67_58_vf.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .43]{spectre_W_50toend_10to57_vf.pdf}\n \\caption{PSD for the vertical velocity fluctuations. (Left): PSD computed at a single point $x=100$~mm, $z=95$~mm (signal shown in figure \\ref{fig:panache} right). The plume forcing frequency can be seen around $f = 10^{-2}$~Hz (red dashed line). (Right): PSD spatially averaged over the whole convective cell in the measured $(x,z)$ plane (all points above $z=10$~mm and below $z=110$~mm). }\n \\label{fig:spectreconv}\n\\end{figure*}\n\nwith $h$ the height of the convective layer, $\\kappa$ the thermal diffusivity, $\\Delta T$ the temperature difference between the top and bottom of the convective domain, $\\Delta T_{local}$ the temperature difference between the top and bottom of the thermal boundary layer, and $Ra_c$ the critical Rayleigh number. The critical Rayleigh number in the presence of free and solid interfaces and for fixed temperature is $Ra_c = 1100.65$. For our experiment, the characteristic time is $\\tau = 96$~s, thus characteristic frequency is $1\/ \\tau \\sim 10^{-2}$~Hz, which is close to the observed peak in the left panel of figure \\ref{fig:spectreconv}. At frequencies lower than this characteristic frequency, the spectrum is flat. This is explained by the combined effect of the randomness of the plumes (see figure \\ref{fig:panache}) and of the slow fluctuations of the large-scale circulation. Right panel of figure \\ref{fig:spectreconv} shows the PSD of vertical velocities, averaged over the whole convective cell in the $(x,z)$ plane. It shows a similar trend, with a lower cut-off frequency compared to right panel spectrum. Actually, the plumes signal is more localised and less intense on average than the large-scale circulation signal, which hence dominates the space-averaged PSD.\n\nThe probability density function (PDF) of the vertical velocities in the whole convective layer $\\mathrm{P}(w)$ is computed and shown in figure \\ref{fig:pdf}. It is normalised such that $\\int \\mathrm{P}(w)\\mathrm{d}w = 1$. The PDF describes important features of the convection: it is skewed towards positive values, with positive velocities reaching higher magnitude than negative velocities, \\textit{i.e.} the ascending plumes are stronger than the descending structures. However, the central part of the PDF is close to gaussian profile. The distribution obtained here is in good agreement with the probability density function computed in an idealised 2D numerical model by Couston et al. \\cite{couston_dynamics_2017}. Note that this asymmetry is specific to our model, for which the usual upside-down symmetry in Boussinesq Rayleigh-B\\'enard convection is broken.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale = .6]{pdf_conv.pdf}\n \\caption{Probability density function of the vertical velocities in the convective layer. All PIV points under $z=110$~mm have been used to compute the PDF.}\n \\label{fig:pdf}\n\\end{figure}\n\n\\subsubsection{\\label{sec:results:buffer}Buffer layer}\nAn intermediate layer (we name it the buffer layer in the following) is present between the convective layer and the stratified layer. It was first reported in the quasi-2D 4$^{\\circ}$C convection experiment by Perrard et al. \\cite{perrard_experimental_2013}. Their temperature measurements showed that the buffer layer corresponds to the area where the temperature goes from 4$^{\\circ}$C to 8$^{\\circ}$C. This actually corresponds to the overshooting region for rising cold plumes (note that this type of convection is called \"penetrative convection\" because of this effect). Indeed, since the density of water is close to quadratic around $4^{\\circ}$C, densities at e.g. $0^{\\circ}$C and $8^{\\circ}$C are the same, and the 8$^{\\circ}$C isotherm is the theoretical maximum height reachable by an ascending cold plume at $0^{\\circ}$C in the absence of thermal diffusion. Simultaneously, the overall density profile between $4^{\\circ}$C and $8^{\\circ}$C is stable, as in the stratified layer above $8^{\\circ}$C. The buffer layer is thus a very specific location supporting simultaneously convective overshooting motions and IGWs, as observed with PIV measurements \\cite{perrard_experimental_2013}. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = .6]{buffer_U-xavg.pdf}\n \\caption{Time evolution of the horizontal average of the horizontal velocity, noted $u_X$. Red (resp. blue) regions correspond to mean flow going towards the right (resp. left).}\n \\label{fig:bufferlayer}\n\\end{figure}\n\nTo complete the description of this buffer layer now using velocity measurements, we plot in figure \\ref{fig:bufferlayer} the spatio-temporal graph of the horizontal average of the horizontal velocity $u$. The graph exhibits a strong shear around $z=120$ mm. Since the fluid is going in opposite directions above and below $z=120$ mm with a sharp interface, viscous entrainment by the convective layer is excluded. A special kind of thermal coupling might explain the observed dynamics, as sketched in figure \\ref{fig:schema_couplage}. Indeed, when a cold ascending plume from the convection zone reaches the interface and overshoots in the buffer region, its associated velocity perturbation dissipates more rapidly than its temperature perturbation. Due to gravity, the distorted part of the buffer region wants to sink back to its initial state (pictured by the green arrows), while the fluid above the buffer layer moves towards the impact point of the plume to take the place of the falling water (pictured by the red arrows). The buffer layer then needs some compensating fluid from the convective layer. This mechanism works when the velocity perturbation of the plume at the interface dissipates more rapidly than the thermal perturbation, hence for a Prandtl number $Pr \\geq 1$. One might expect the shearing zone to decrease in size and amplitude when thermal diffusion increases (i.e. when the Prandtl number decreases), since the overshooting rising cold plume will then equilibrate thermally during its ascent more rapidly. This may explain why no interfacial shear was reported in the systematic numerical study of \\cite{couston_dynamics_2017,couston_order_2018} where $Pr \\leq 1$. Global temperature field measurements (using e.g. Temperature Dependent, Laser Induced Fluorescence) are now required to confirm or infirm the proposed model, but those are beyond the scope of the present paper. Note that by extension, we call \"buffer layer\" in the following the region including the $T=4^\\circ$C to $T=8^\\circ$C overshooting region and the shear region. In the experiment, the shear region extends from $z=120$~mm to $z \\approx135$~mm.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[scale = .75]{schema_couplage_final2.pdf}\n \\caption{Sketch of the thermal coupling between the convective and buffer layers. On the left, a cold plume moves upwards towards the interface between the two layers. On the right, isotherms are deflected due to the impact.}\n \\label{fig:schema_couplage}\n\\end{figure}\n\n\\clearpage\n\\subsubsection{\\label{sec:results_waves}Internal gravity waves}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = .83]{visu_onde_colormap_col_02to03_axisequal_schema.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .53]{panache_v1.pdf\n \\caption{Velocity fields showing IGWs propagating. (Top) Velocities in $(x,z)$ plane. The signal is frequency-filtered to enhance the visualisation of oscillatory motions: only frequencies between $0.02$ and $0.03$ Hz are shown, propagating at an angle of roughly $75^\\circ$ with the vertical. The angle of propagation is the angle between the constant phase line and the vertical. (Bottom) Velocities in $(x,y)$ plane located at $z \\approx 125$~mm. In the $(x,y)$ plane, IGWs take the form of oscillating rings. Note that this figure is from a previous experiment without any internal cylinder and is therefore only displayed here as an illustration of the IGWs seen from above.}\n \\label{fig:ondeN}\n\\end{figure}\n\n The convective motions induce a Reynolds stress at and below the interface which generates IGWs propagating in the stratified area \\cite{lecoanet_numerical_2015}. An example is shown in figure \\ref{fig:ondeN}. The vector field has been frequency filtered in the band $[0.02-0.03]$ Hz to isolate a single propagating wave train. We can measure an angle close to $\\theta \\simeq 75^{\\circ}$ between contant phase lines and the vertical. This observation is in good agreement with the inviscid dispersion relation $\\omega = \\pm N \\mathrm{cos}(\\theta)$, which relates the frequency and the propagation angle of IGWs. Indeed, at $z=120$~mm, $N \\sim 0.1$~Hz, thus $\\theta = \\mathrm{cos^{-1}}\\left(\\frac{\\omega}{N}\\right) = 78.5^\\circ$. The motion within the stratified area is a superimposition of many such IGWs oscillating at different frequencies.\n\nTo further investigate the waves signal, waves spectra are plotted in figure \\ref{fig:spectro}, showing the power spectral density of oscillatory motions within the stratified layer at every height, averaged horizontally for each height. The grey line is the theoretically computed buoyancy frequency profile. Figure \\ref{fig:spectro} shows that energy is present in a wide frequency band, from the lowest measured frequencies to the buoyancy frequency $N$. Low frequency motions $f < 4\\times10^{-3}$~Hz are very intense and propagate high in the stratified layer. Motions with frequency ranging from $4 \\times 10^{-3}$~Hz to $N$ are less intense, but still propagate into the stratified layer. Motions propagating at frequencies higher than the buoyancy frequency $N$ are greatly attenuated after a centimetre as IGWs of frequency larger than $N$ are evanescent. The weak signal at low frequencies above $z=180$ mm comes from the convective motions due to the non-homogeneous heating at the top. These motions are confined at the very top of the experimental container. \n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale=0.55]{propagation_nonorm_120_vff.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale =.55]{plot_attenuation_norm_v2.pdf}\n \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ above the convective layer. The grey curve shows the theoretical buoyancy frequency profile, assuming a linear profile for the temperature, from $4^\\circ$C to $35^\\circ$C. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=120$~mm (noted $\\mathrm{PSD_o}$). The PIV measurements are performed for 50 minutes and results (using the pwelch Matlab function) are horizontally averaged at each height to obtain the averaged power spectral density.}\n \\label{fig:spectro}\n\\end{figure*}\n\nRight panel of figure \\ref{fig:spectro} shows two vertical profiles of the PSD re-scaled by the PSD at the top of the convective layer ($z=120$~mm), taken at two different frequencies. The energy decrease is quite similar between $z=120$~mm and $z=140$~mm for both frequencies. However, for $z>140$~mm, the energy for the higher frequency decreases slower than the energy for the lower frequency. This dependence of the attenuation length regarding the frequency of the signal is a characteristic of IGWs. Indeed, the dispersion relation of IGWs relates the frequency and the wave vector direction. Moreover, energy propagates perpendicularly to the wave vector for IGWs (i.e. group and phase velocities are perpendicular). The closer to $N$ the wave frequency $\\omega$ is, the more horizontal the phase propagates, hence the more vertical energy propagates. High frequency waves are thus capable of transporting energy to high altitudes before being damped. On the opposite, waves with low frequency compared to $N$ propagate energy almost horizontally, and are thus attenuated before reaching high altitudes. At frequencies $f<4\\times 10^{-3}$~Hz, a lot of energy is seen and the attenuation length does not depend on the frequency. There is no reason why IGWs should disappear below a certain frequency, but we would expect to see the attenuation length to keep decreasing with decreasing frequency. We thus deduce that IGWs at frequencies $f \\leqslant 4\\times 10^{-3}$~Hz are hidden in the energy spectrum by some very energetic large-scale slowly-varying flow, which we will describe below.\n\n\nMore than one order of magnitude separates the buoyancy frequency and the fastest large-scale flow fluctuations. The large-scale flow penetrates deep into the stratified layer. It globally decreases in amplitude with height, but with some local increases at $z\\sim 125$ mm (i.e. close to the interface between convective and buffer layers) and $z \\sim 145$ mm.\nThe IGWs signal can be seen between $f = 4 \\times 10^{-3}$ Hz and the buoyancy frequency. A peak that reaches the top of the stratified layer is seen around $f= 1.2 \\times 10^{-2}$ Hz, \\textit{i.e.} the same frequency as the convective forcing discussed in section \\ref{sec:results_conv}. It corresponds to the strong excitation provided by the cold rising and hot sinking turbulent plumes.\nHowever, top panel of figure \\ref{fig:spectro} also shows a sudden drop of the energy at frequencies $f>1.2 \\times 10^{-2}$~Hz. Indeed, wave attenuation is strong at these frequencies, even if they are close to (but below) the buoyancy frequency $N$. Actually, energy dissipation also depends on the norm of the wave vector squared. There is no reason that all excited waves have the same wave vector norm; one could even expect that fastest waves are excited by fastest, hence smallest convective patterns, and are thus also at smallest scale: they then dissipate more rapidly.\n\n\n\\subsubsection{\\label{sec:results_lsf}Large-scale flow in the stratified layer}\nFigure \\ref{fig:spectro} shows an important amount of energy at low frequencies which has been interpreted as the signature of a large-scale slowly-varying flow in the stratified layer. We will now investigate the nature of these fluctuations to see if they relate to reversals similar to the QBO.\n\nFigure \\ref{fig:meanflow} shows horizontal vector fields at the same depth at different times. In figure \\ref{fig:meanflow}(a), the flow goes counter-clockwise inside the cylinder. Figure \\ref{fig:meanflow}(b) shows that two contra-rotating vortices with a smaller amplitude typical velocities have appeared. Figure \\ref{fig:meanflow}(c) shows a mostly clockwise rotating flow, where one of the preceding eddy pairs has nearly disappeared. The large-scale flow thus evolves drastically over time. A criterion is computed to extract a typical mean velocity from those fields that accounts for the ``direction'' of the large-scale flow: as illustrated in figure \\ref{fig:critere}, we compute a mean azimuthal velocity, taken along a ring centred in the cylinder. Other criteria to extract a representative value for the large scale flow direction have also been tested, including: the mean vorticity over the cylinder area, the average of the azimuthal velocity over several rings with different radii, and the azimuthal velocity averaged over thick rings. They all give similar results for the large-scale flow measurement. \n\\\\\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale = .8]{evolution_mf_vff.pdf}\n \\caption{Horizontal velocity vector fields in the stratified layer at different times. The laser sheet is located at $z=150$~mm. The large-scale flow reverses from (a) to (c). Time between (a) and (c) is approximately half an hour. Maximum velocities are $0.1$ mm\/s.}\n \\label{fig:meanflow}\n\\end{figure*}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale = .45]{critere_mf.pdf}\n \\caption{Criterion used to extract a significant value for the large scale flow and its direction: the azimuthal velocity is averaged over the ring shown in red.}\n \\label{fig:critere}\n\\end{figure}\n\nIn order to investigate the vertical phase propagation of the reversals, and thus, to compare the reversal dynamics observed to a QBO-like phenomenon, the setup has been equipped with a linear translating platform that allows us to perform horizontal laser sheet sweeping along the vertical. Horizontal velocities are measured in an horizontal plane, every $5$~mm from the top of the convective layer $z=110$~mm to the middle of the stratified layer $z=160$~mm. Any trace of downward phase propagation of the reversals, as observed on the QBO on Earth \\cite{baldwin_quasi-biennial_2001} and on the historical Plumb experiment \\cite{plumb_interaction_1977, semin_nonlinear_2018}, would be a significant evidence for QBO-like phenomenon in the experiment. \nIndeed, the phase propagation of the reversals due to IGWs non-linear interactions is theorised as follows: an IGW propagating in a stratified layer with an horizontal phase velocity in the same direction as the existing base flow propagates upward until reaching a critical height $z_c$, where it deposits all its energy locally. At $z=z_c$, the flow accelerates. Thus the critical height where the flow is intense enough to damp the wave is lowered. As time goes on, this critical height moves towards the location where the waves are emitted. Here, the waves are emitted at the bottom of the stratified layer. We would expect a downward phase propagation if the reversals are driven by IGWs non-linear interactions. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[scale=0.7]{QBOfinal_avecpentediffusionv2.pdf}\n \\caption{Reversals of the large-scale flow. $z=115 - 120$ mm is the convective \/ buffer layers interface. Ascending plumes often perturb the buffer layer flow. The velocity was measured at 11 heights, marked by each tick in the vertical axis of the figure, and interpolated in between. The slope of the black dot lines represent the viscous coupling phase velocity.} \\label{fig:QBO}\n\\end{figure*}\n\n\nWe performed long time experiments (around 8 hours). Typical results extracted from the criterion described above are shown in figure \\ref{fig:QBO}. Blue patches (resp. red patches) represent large scale flow going counter clockwise (resp. clockwise). The present measurements mainly confirm the interpretation of figure \\ref{fig:spectro} for the lowest frequencies: the large-scale flow is horizontal and extends over the whole depth of the stratified layer with an amplitude attenuated with height, and exhibits slow reversals. Additionally, some intense events at $z = 110$ mm are directly related to penetrative plumes from the convection. Reversals times range from $400$s to $1800$s. However, no downward phase propagation of the reversals is observed. On the contrary, the reversals seem to occur along the whole stratification height at the same time, or even with a rapid upward phase propagation. Since the phase propagation is not towards the location where the waves are emitted, the reversals are unlikely driven by the non-linear interactions of IGWs. However, as seen in section \\ref{sec:results_waves}, IGWs propagate in the stratified layer and carry energy. Therefore, they give energy to the large-scale flow through non-linear interactions. Yet, the process is not dominant in the reversals dynamics.\n\\\\\nSince, the reversals observed in figure \\ref{fig:QBO} do not have a downward phase propagation, we look for other mechanisms than the QBO mechanism to explain the reversals. Two other mechanisms can be investigated. The first one relies on a specific convective dynamics within the overall stratified layer driven by horizontal gradients related to imperfect top and side boundary conditions. The second mechanism relies on viscous coupling with the underlying convective and buffer layers. \n\nOur fully stratified reference experiment described in section \\ref{sec:methods} precludes the first scenario. Indeed, setting the bottom boundary at 10$^{\\circ}$C and the top boundary at 70$^{\\circ}$C, no motion is observed for the bottom $3\/4$ of the tank. In this test-experiment, the top $1\/4$ of the tank is animated by convective motions due to the non-homogeneous top heat source (in the standard $4^\\circ$C experiment, where $T_{top} = 35\\degree$C, only $\\sim 2$ cm are affected by the convection at the top of the tank, because the non-homogeneity of the heat source is less important for lower temperature, thus the horizontal convection is weaker). However, these are inefficient to generate waves below and to drive any large-scale flow observable away from the top region.\n\n\n\\begin{figure}[h]\n\\includegraphics[scale = 0.55]{balayage_corr_vf.pdf}\n \\caption{Velocity vector fields in the horizontal plane. Different columns represent different sweeping cycles $t^*$ (one sweeping cycle corresponds to the 11 steps needed to go from the lowest position $z=110$~mm to the highest position $z=160$~mm). Different rows represent different heights within the same sweeping cycle: first row is the top of the convection $z=110$~mm, second row is in the buffer layer $z=120$~mm and third row is in the stratified layer $z=145$~mm. Convective plumes are easily noticeable on the first row fields.}\n \\label{fig:champ_correle}\n\\end{figure}\n\n\\begin{figure}[h]\n \\includegraphics[scale = 0.55]{balayage_NOcorr_vf.pdf}\n \\caption{Same as figure \\ref{fig:champ_correle} but for different sweeping cycles. Note that in this set of velocity fields, the buffer and stratified layers are less correlated than they are in figure \\ref{fig:champ_correle}.}\n \\label{fig:champ_pascorrele}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale=0.6]{correProdscalaire_BW_vf.pdf}\n \n \\caption{Velocity correlation between the three layers. Dashed black lines are the $-0.5$ and $0.5$ values. Correlation coefficient are window-averaged over $10$~min to smooth the curve.}\n \\label{fig:correlation}\n\\end{figure}\n\nThis leaves viscous entrainment as a possible driving mechanism. The dotted lines on figure \\ref{fig:QBO} show a theoretical viscous time, computed from the time for viscous entrainment to drive 20\\% of the horizontal velocity at $z = 115$ mm to $z=160$ mm, starting from a base state flow at rest. The 20\\% value corresponds to the measured value of the large scale flow at $z=160$ mm compared to the value at $z = 115$ mm (noted $u_b$). The theoretical viscous time entrainment is given by $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. The reversals occur in a time scale comparable with this theoretical viscous time. The similarity between the slope of the dashed lines and the slope of the upward phases suggests that reversals are driven viscously.\n\nHowever, the existence of the buffer layer and its associated intense shear, with opposite horizontal velocities with the convective layer below (see figure \\ref{fig:bufferlayer}) precludes direct viscous coupling between the convective and stratified layers. Besides, no reversal has been observed in the convective region. We thus propose a thermal coupling between the convective and buffer layers as seen in section \\ref{sec:results:buffer}, associated to a viscous coupling between the buffer and stratified ones. To further quantify this possibility, figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele} show horizontal velocity fields at different heights and at different times. For each of the columns shown, first row is the mean flow in the convective layer at depth $z=110$~mm, second row is the mean flow in the buffer layer at depth $z=120$~mm and third row is the mean flow in the stratified layer at depth $z=145$~mm. The correlation coefficients through time between (i) the convective and buffer layers, (ii) the buffer and stratified layers, and (iii) the convective and stratified layers have been computed. It consists in a scalar product of the velocity vector for each position at two different heights rescaled by the product of the norm of the velocity vector at the two heights, \\textit{i.e.}: \n\n\\begin{equation}\nR_{ij} = R(x_i,y_j) = \\frac{u(x_i,y_j,z_1) \\times u(x_i,y_j,z_2) + v(x_i,y_j,z_1) \\times v(x_i,y_j,z_2)}{\\left(u(x_i,y_j,z_1)^2+v(x_i,y_j,z_1)^2\\right)^{1\/2} \\times \\left(u(x_i,y_j,z_2)^2 + v(x_i,y_j,z_2)^2 \\right)^{1\/2}}\n\\end{equation}\nThis gives a correlation coefficient $R_{ij}$ for each PIV position in the horizontal plane. The global correlation coefficient $R$ is computed by spatially averaging the local correlation coefficients. \n\nResults are shown in figure \\ref{fig:correlation}. The convective and buffer layers are negatively correlated: the correlation coefficient is most of the time close to $R=-0.5$. This can also be seen at all times in figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele}, where horizontal velocities in the convective and buffer layers have opposite direction. A diverging flow coming from an impinging plume in the convective zone corresponds to a converging flow in the buffer layer towards the impact zone, hence confirming the thermal coupling mechanism described in section \\ref{sec:results:buffer}. This converging flow may lead either to a clockwise or anticlockwise azimuthal mean flow, depending on the details of the chaotic excitation from the convective plumes. The correlation coefficient between the convective and stratified layers can be positive or negative, and is anyway most of the time less than 0.2, in absolute value. The correlation coefficient between the buffer and stratified layers shows a lot of temporal variations. However, it remains always positive. At a given time, the large-scale flow in the stratified layer may switch between a regime strongly dominated by the buffer layer (see also figure \\ref{fig:champ_correle}), and a second regime where the flow in the stratified layer is quite different from the flow in the buffer layer (see also figure \\ref{fig:champ_pascorrele}). \n\nWe thus conclude that the stratified layer is globally viscously driven by the buffer layer. However, the stratified layer exhibits additional complexities. These might be due to IGWs interacting with the large-scale flow. The results from Couston et al. \\cite{couston_order_2018} show that the lower the Prandtl number, the more regular the QBO. In the experiment, the Prandtl number is close to $Pr = 7$: the typical associated QBO-type flow is irregular, with low amplitude. We thus propose that large-scale flow driven by IGWs non-linear interaction superimposes on the viscously driven flow, but remains secondary. We do not know at this point how to disentangle those two potential contributions from the available data. \n\n\n\\clearpage\n\\subsection{\\label{sec:results_num} Numerical simulations}\nThe experimental results are not fully sufficient to explain, with complete certainty, the origin of the buffer layer and of the large-scale flow observed in the stratified layer. In addition, the effects of the lateral heat losses and top temperature heterogeneity are difficult to distinguish. To answer these questions, 3D DNS of a configuration similar to our experiments are performed, reproducing the 4$^{\\circ}$C convection but with idealised boundary conditions (i.e. no flux on the sides, and fixed temperature at the top and bottom). As mentioned in section \\ref{sec:methods_num}, the Rayleigh number $Ra$ and $T_{top}$ are tuned so that the interface depth in the experiment and the numerical simulation are similar. We have $Ra=10^7$ and $T_{top}=48^\\circ$C. All the numerical simulations are run dimensionless, but results are shown in dimensional values. The length scale is $H = 200$~mm, the vertical extent of the whole domain (hence diameter is $D=300$~mm), the timescale is the thermal diffusive time $\\tau = \\frac{H^2}{\\kappa} = \\frac{0.2^2}{1.5 \\, 10^{-7}} = 2.67 \\times 10^{5}$~s, and the temperature is given by the dimensionless temperature $\\theta = \\frac{T - T_i}{T_0 - T_i}$, where $T, T_i, T_0$ are respectively the dimensional temperature, the inversion temperature of the equation of state (i.e. $4^\\circ$C), and the bottom temperature (i.e. $0^\\circ$C). Results for sections \\ref{sec:results_num_conv} - \\ref{sec:results_num_igw} are computed from a $(x,z)$ vertical plane located along a cylinder diameter.\n\n\\subsubsection{\\label{sec:results_num_conv}Large-scale circulation in the convection zone and buffer layer}\nFigure \\ref{fig:LSC_num} shows that a large-scale circulation takes place in the convective layer. It consists of a cell filling the whole convective layer, and exhibits no reversal over the whole course of the simulation. The fluid rotates counter clockwise in the vertical plane. This is qualitatively consistent with the mean flow observed in the experiment and shown in the right panel of figure \\ref{fig:LSCexpe}. As in the experiments, a counter current exists on the top of the convective layer at $z= 120$~mm, creating a strong shear and demonstrating the existence of a buffer layer in the numerical simulation as well. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = .47]{quiver_inst_color.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .47]{quiver_moyen_color.pdf}\n \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=230$ mm. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes recording. The large-scale circulation is a counter clockwise cell. Maximum instantaneous velocities are $3$ times bigger than the maximum averaged velocities.}\n \\label{fig:LSC_num}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale = 0.5]{Spatio_temp_Ux_v3_dim.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = 0.5]{profil_T_all.pdf}\n \\caption{(Left) Horizontal average of the horizontal velocity $u$ over a vertical cross-section in the middle of the tank. The buffer layer can be seen above $z=120$~mm. A stationary large-scale circulation is present in the convective layer, even if it appears quite perturbed at the end of the signal. (Right) Temperature profiles along the $z$-axis.}\n \\label{fig:num_buffer}\n\\end{figure*}\n\nThe space-time diagram of the mean horizontal flow shown in figure \\ref{fig:num_buffer} confirms it. Observing the buffer layer in the absence of side thermal losses and top temperature heterogeneity is an additional argument accounting for the fact that it is not an artefact driven by imperfect experimental conditions.\nWe also observe that the flow within the convection stays positive through time at the bottom and negative at the top. This is evidence of the steady large scale circulation taking place in the convective layer. Some events appear at $t > 1.42 \\times 10^{5}$~s and are interpreted as quasi-reversal of the large-scale circulation. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=.5]{quiverIsotherme.pdf}\n \\caption{Velocity field and temperature isotherms at the end of an upward plume impact on the interface.}\n \\label{fig:num_quiverisoT}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=.6]{corre_WetU.pdf}\n \\caption{(Left) Spatio-temporal diagram of the horizontal velocity $u$ at $z=128$~mm. (Right) Spatio-temporal diagram of the vertical velocity $w$ at $z=108$~mm. The event at $t \\approx 1.052 \\times 10^5$~s is shown in figure \\ref{fig:num_quiverisoT}.}\n \\label{fig:num_correWU}\n\\end{figure}\n\nThe temperature profile along the $z$ axis is also plotted on the right panel of figure \\ref{fig:num_buffer}. The figure shows a temporal and horizontal average of the temperature field (black thick curve), two temporal averages at two different positions $x=14$~mm (left side of the tank - dashed grey) and $x=277$~mm (right side - dotted grey) and an instantaneous profile at $x=145$~mm (middle of the tank - thick grey with crosses). The thermal boundary layer can be seen, between $z=0$~mm and $z=10$~mm. Then, between $z=10$~mm and $z=100$~mm lies a layer of constant temperature $T \\sim 2.8^\\circ$C. Between $100 \\mathrm{~mm} \\leqslant z \\leqslant 115 \\mathrm{~mm}$, the temperature profile evolves from constant to linear for $z > 115$ ~mm. The $T=4^\\circ$C (respectively $T=8^\\circ$C) isotherm is located at $z = 110$~mm (resp. $z = 120$~mm). Note that the temporal average of the temperature profiles are different on the left and right sides of the tank. Indeed, the constant temperature height goes to $z=90$~mm for the left side whereas it goes to $z=115$~mm for the right side. This suggests that the convective \/ buffer layer interface does not lies at one height over the whole tank but is a function of time and space. This is very likely due to the large-scale circulation. Thus, the thermal coupling described in \\ref{sec:results:buffer} will likely occur at different heights, depending on time and horizontal position.\n\n\nThe thermal coupling as schematised in figure \\ref{fig:schema_couplage} can be found in the numerical simulation. This is represented in figure \\ref{fig:num_quiverisoT}. An upward plume impacting the convective \/ buffer layer interface is seen. The isotherms ranging from $T=4^\\circ$C to $T=11^\\circ$C are deflected upward, due to the plume bringing cold fluid upward. On the contrary, the isotherms $T = 12 -14^\\circ$C are deflected downward by the converging flow. Isotherms at $T \\geqslant 15^\\circ$C remain horizontal. After the impact on the interface, the plume is deflected outwards. One could expect the fluid above the impact to be viscously entrained by this outward deflection. However, as observed in figure \\ref{fig:num_quiverisoT} for the simulation and figures \\ref{fig:champ_correle}-\\ref{fig:champ_pascorrele} for the experiment, the fluid above the interface is going towards the plume, \\textit{i.e.} in the opposite direction of the fluid below, hence explaining the observed shear (see figures \\ref{fig:num_quiverisoT} and \\ref{fig:schema_couplage}). The time evolution of these dynamics is shown in figure \\ref{fig:num_correWU}.\n\n\nFigure \\ref{fig:num_correWU} shows the time evolution of the horizontal velocity $u$ in the shear layer at $z=128$~mm and the time evolution of the vertical velocity $w$ in the convective layer at $z=108$~mm. Comparing the two panels of figure \\ref{fig:num_correWU} shows that upward plumes are concomitant with converging horizontal velocities towards the plume impact. Indeed, the spatio-temporal diagram of $w$ exhibits local strong upward plumes. These plumes, as suggested by the dashed black lines, are correlated in time and space with converging horizontal velocities. For instance, an upward plume is seen at $x\\approx220$~mm and $t\\approx1.043 \\times 10^5$~s. At the same horizontal position and time, the positive horizontal velocity becomes stronger and the negative horizontal velocity patch increases in size to reach $x\\approx220$~mm. The converging horizontal velocities event occurs a short time after the impact of the plumes. Thus, it can be concluded that the plume induces the converging flow, as suggested by our explanation in section \\ref{sec:results:buffer}.\n\n\\clearpage\n\n\\subsubsection{\\label{sec:results_num_igw}Internal gravity waves}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = 0.55]{spectro_vfinale.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .55]{plot_attenuation_norm_num_v2.pdf}\n \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ in the buffer and stratified layers. The grey curve shows the buoyancy frequency profile computed from the spatial and temporal average of the temperature field. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=118$~mm.}\n \\label{fig:num_spectro}\n\\end{figure}\n\nPSDs are computed in the stratified and buffer layers and are plotted in figure \\ref{fig:num_spectro}. As for the experiment (figure \\ref{fig:spectro}), numerical results show oscillatory motions at different frequencies attenuated with height. Experimental results (figure \\ref{fig:spectro}) and numerical results (figure \\ref{fig:num_spectro}) show strongly similar dynamics: most of the energy is present at low frequencies ($f<3 \\times 10^{-3}$~Hz). The motion with frequencies ranging from $3 \\times 10^{-3}$~Hz to $N$ are less intense, and almost no energy is seen at frequencies $f>N$. \n\nRight panel of figure \\ref{fig:num_spectro} shows two selected vertical profiles (shown by the white dashed line on the left panel figure) of the the PSD re-scaled by the PSD at $z=118$~mm. The energy for the higher frequency ($f=1.4\\times10^{-2}$~Hz) decreases slower than the energy for the lower frequency ($f=5.0\\times10^{-3}$~Hz). This is, as experimental results, in agreement with the dispersion relation of IGWs.\nThe overall behaviour of waves spectra is similar in experiment and numerical simulation, with an attenuation length independent of the frequency in the low-frequency signal thus confirming a viscous coupling origin of the large-scale flow, and increasing when the frequency goes towards $N$ in the wave domain.\n\n\\subsubsection{Large-scale flow within the stratified layer}\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_square_dim.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_zoom_dim.pdf}\n \\caption{Spatio-temporal diagrams of the azimuthal averaging of the azimuthal velocity inside a virtual cylinder of radius $r = 140~$mm. The bottom figure is a zoom on the stratified zone, delimited in the top figure by the black square. The slope of the black lines show the theoretical viscous diffusive time.}\n \\label{fig:num_vthetacyl}\n\\end{figure*}\nSimilarly to what has been done for the experimental data, figure \\ref{fig:num_vthetacyl} shows the mean azimuthal velocity over the whole height of a virtual cylinder of radius $r = 140$ mm. We observe reversals within the convective layer ($z<120$ mm), which are not systematically correlated with the signal in the stratified layer. The mean velocity in the stratified layer also exhibits reversals. They are characterised by an upward phase propagation from the buffer zone at $z=120$ mm, as shown in the zoom (bottom panel of figure \\ref{fig:num_vthetacyl}). The phase velocity seen in figure \\ref{fig:num_vthetacyl} is in good agreement with the theoretical time for viscous propagation $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. This corroborates the fact that the reversals observed within the stratified layer are viscously driven from the dynamics occurring in the buffer layer, as it has been seen for the experiment. Reversals time ranges from $300$~s to $1500$~s. Those reversals times are similar to the experimental ones, though slightly shorter (numerical reversals are $\\sim20$\\% faster than experimental reversals). \n\n\\clearpage\n\\section{Conclusion}\\label{sec:discussion}\nThe 4$^{\\circ}$C convection experiment, originally performed by Townsend \\cite{townsend_natural_1964}, has been re-investigated using long-term PIV measurements in a vertical cross-section, and in several horizontal cross-sections within the stratified layer. This last type of measurements has allowed to investigate for the first time the long-term horizontal mean flow in the stratified layer. Experiments have been complemented by direct numerical simulations. The first result of this paper is the confirmation, in 3D and with ideal boundary conditions, of the presence of a buffer layer, including an overshooting region as first observed by Perrard \\cite{perrard_experimental_2013}, and a shear region. We have argued that the buffer layer is driven by thermal coupling with the convection, due to the non-linear equation of state of water, and that this mechanism is a priori related to a Prandtl number larger than one. The second result is that the buffer layer viscously drives slow reversals of the horizontal large-scale flow within the stratified layer. \n\n\nAdditionally, IGWs at different frequencies propagate in the stratified layer. They likely interact with the horizontal large-scale flow, and probably also produce a reversing flow, which superimposes to the viscously driven one. From Couston et al. \\cite{couston_order_2018}, we know that the Prandtl number has a strong influence on this QBO-like mechanism: the lower the Prandtl number, the stronger the amplitude of the QBO. In water, $Pr \\sim 7$, and the expected amplitude of the large-scale QBO flow is weak, hence dominated by the viscous driving. Further experimental studies at lower Prandtl number should allow deciphering the two contributions. One could for instance suggest using gas as a working fluid; however, the absence of density reversal around a given temperature will necessitate to consider either transient experiments like \\cite{deardorff_laboratory_1969, michaelian_coupling_2002}, or two-gas experiments which might then be prone to double diffusive instabilities. Experimentally, the question also remains to understand why the only successful QBO experiment has been performed in salty water, hence with a Schmidt number (equivalent to Pr) of 700.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale=.6]{U_avgX.pdf}\n \\caption{Horizontal average of the horizontal velocity $u$ over a vertical cross section in the middle of the numerical domain for $Pr = 0.1$.}\n \\label{fig:num_pr01}\n\\end{figure}\n\nIn the meantime, it is straightforward to change the Prandtl number in the numerical simulation of our set-up. We have thus run a second simulation with the same Rayleigh number $Ra = 10^7$ and top temperature $\\theta_{top} = 11$ but with $Pr = 0.1$. In this simulation, as shown in figure \\ref{fig:num_pr01}, no buffer layer is observed, but strong signatures of a QBO like mechanism are visible, marked by downward phase propagation of the reversals of the large-scale flow. This configuration thus deserves a more systematic study in the future.\n\n\\section*{Acknowledgements} \nThe authors acknowledge funding by the European Research Council under the European Union's Horizon 2020 research and innovation program through Grant No. 681835-FLUDYCO-ERC-2015-CoG. This work was granted access to the HPC resources of Aix-Marseille Universit\\'e financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program \"Investissements d'Avenir\" supervised by the Agence Nationale de la Recherche. Computations were also conducted with the support of the HPC resources of GENCI-IDRIS (Grant No.A0060407543).\n\n\n\\section{\\label{sec:intro}Introduction}\nNumerous geophysical and astrophysical flows present a two-layer configuration, with a turbulent convective layer standing above or below a stably stratified one. Examples include planetary atmospheres, stars interiors, and possibly the outermost layer of the Earth liquid core \\cite{hirose_composition_2013}. The dynamics of coupled stratified and convective layers are quite complex. Due to the convective motions, internal gravity waves (IGWs) are generated at the interface between the two layers, and propagate in the stratified one. IGWs transport energy and momentum \\cite{rogers_internal_2012,bretherton_momentum_1969} from where they are generated to where they are damped. Thanks to their transport properties and non-linear interactions, IGWs are able to generate and sustain large-scale horizontal flows \\cite{plumb_interaction_1977,rogers_internal_2012}. Examples of such large-scale flows driven by IGWs are the oscillations of equatorial zonal winds observed in some planets' atmosphere \\cite{fouchet_equatorial_2008,leovy_quasiquadrennial_1991}, including the Earth where it is called the Quasi-Biennial Oscillation (QBO) \\cite{baldwin_quasi-biennial_2001}.\\\\\n\nThe IGWs generation by turbulent dynamics has been studied in various experiments. The generation by a single buoyant plume was experimentally studied by Ansong \\& Sutherland \\cite{ansong_internal_2010}. The penetration of the plume within the stratified layer and the spectral characteristics of the generated IGWs were studied. They found that the peak frequency of the generated IGWs lies in a range close to $0.7N$, where $N$ is the Brunt-V\\\"ais\\\"al\\\"a (or buoyancy) frequency, and that the radial wavenumber is set by the plume cap and not by the width of the plume at the interface.\n\nDeardoff et al. \\cite{deardorff_laboratory_1969} and later Michaelian et al. \\cite{michaelian_coupling_2002} studied the effect of penetrative convection in a stratified layer in a transient, Rayleigh-B\\'enard type experiment. Stratification was initially set up thermally from the top to the bottom of the tank. Then, the fluid was suddenly warmed up at the bottom, triggering Rayleigh-B\\'enard convection. IGWs were measured transiently \\cite{michaelian_coupling_2002} while the stratified (resp. convective) layer was decreasing (resp. increasing) in size. Eventually, there was no more stratified layer to sustain the propagation of IGWs. \n\nTownsend \\cite{townsend_natural_1964} introduced an original set-up to study the quasi-steady generation of IGWs by Rayleigh-B\\'enard convection. Using the fact that water maximum density is around $4^{\\circ}$C, a two-layer system is spontaneously generated by cooling the bottom of a tank at $0^{\\circ}$C and heating its top above $4^{\\circ}$C. The density gradient is unstable at temperature below 4 $^{\\circ}$C and stable at temperature above. This creates a self-organising system, with a turbulent convective layer adjacent to a stratified layer. With dye visualisation and temperature measurements, he observed IGWs propagating close to the interface between the two layers. The $4^\\circ$C convection was also studied experimentally by Le Gal \\cite{legal_penetrative_1997} in a laminar flow, at low Rayleigh number. Convection displayed an hexagonal pattern and viscous entrainment of the fluid above the convective cells was observed. Perrard et al. \\cite{perrard_experimental_2013} and Le Bars et al. \\cite{bars_experimental_2015} re-investigated this setup in a quasi two-dimensional tank using Particle Image Velocimetry (PIV) and temperature measurements to obtain detailed data of IGWs generated by the convection. They observed a wide spectrum of waves generated at different frequencies. Favoured frequencies were related to the differential attenuation length of the waves depending on frequency, in good agreement with linear theory. No large-scale flow in the stratified layer was observed in this 2D geometry. Numerical simulations of the same configuration were performed by Lecoanet et al. \\cite{lecoanet_numerical_2015}. They demonstrated that IGWs are mainly generated by the Reynolds stresses in the bulk of the convection below the interface, rather than by the displacement of the interface between the two layers. Numerical studies by Couston et al. \\cite{couston_dynamics_2017,couston_order_2018,couston_energy_2018} extended these results by considering a generic non-linear equation of state (piecewise linear around an inversion temperature, with adjustable slopes), both in 2D and 3D horizontally periodic geometries. Various flow regimes and the energy transport by IGWs were quantitatively studied. Interestingly, long simulations -- accessible in 2D studies only -- showed, for low Prandtl numbers ($Pr < 1$), a large-scale horizontal flow with reversals in the stratified layer, similar to the QBO phenomenon introduced above \\cite{couston_order_2018}. \n\\\\\n\nSeveral experiments took interest in the generation and reversal of a large-scale horizontal mean flow in a stratified domain, driven by IGWs. The well-known experiment designed by Plumb and McEwan \\cite{plumb_instability_1978}, later reproduced and extended by Semin et al. \\cite{semin_generation_2016,semin_nonlinear_2018}, is capable of driving a QBO from the mechanical forcing of a standing wave pattern (\\textit{i.e.} two waves with the same frequency and opposite phase speed) in a salty water stratification. With this system, Plumb and McEwan managed to observe few oscillations of the driven large-scale flow before the stratification was destroyed by mixing. The experiment gave results in good agreement with the theory \\cite{richard_s._lindzen_theory_1968,lindzen_updated_1972,plumb_interaction_1977}, notably with reversals starting away from the forcing and propagating towards it.\nSemin et al. \\cite{semin_generation_2016,semin_nonlinear_2018} improved the system by constantly injecting fluid to rebuild the stratification, while removing mixed fluid close to the forcing. This method allowed to run the experiment longer and to study the nature of the bifurcation in the Plumb model which can be either supercritical or subcritical, depending on the dominant dissipative process. In those experimental realisations of the QBO mechanism, the wave forcing remains monochromatic, as opposed to the natural mechanism where it is due to chaotic tropical storms \\cite{baldwin_quasi-biennial_2001}. The forcing is driven by interface displacements, as opposed to the observations of \\cite{lecoanet_numerical_2015}. Besides, only the stratified layer is modelled. It thus remains a challenge to observe experimentally a large-scale, reversing flow from a turbulent source and a wide range of naturally excited IGWs.\n\\\\\n\nIn the present study, we extend the work of Townsend \\cite{townsend_natural_1964,perrard_experimental_2013,bars_experimental_2015} in a cylindrical, 3D geometry reminiscent of Plumb and McEwan's set-up \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}. Our purpose is threefold: to characterise the generation of IGWs in such a self-organising two-layer system, to quantify the coupling between the layers, and to investigate the possible generation of large-scale horizontal flows. Our experiments are complemented by direct numerical simulations of the same configuration. The experimental setup and numerical model are presented in section \\ref{sec:methods}, results are analysed in section \\ref{sec:results}, and conclusions and future works are discussed in section \\ref{sec:discussion}. \n\n\n\n\\section{\\label{sec:methods}Methods}\n\\subsection{Experimental set-up}\nThe set-up consists in a cubic tank whose lateral boundaries are made of 2 cm thick acrylic walls. The bottom boundary is a chrome plated copper plate in which refrigerated fluid is forced to circulate. The top boundary is a commercial, transparent electric heater. The tank inner dimensions are $32 \\times32$ cm for the horizontal section and $H=20$ cm in height. Preliminary experiments were conducted in this cubic geometry. Eventually, a cylinder of outer diameter $D = 29$ cm and thickness $e = 0.4 $ cm was added inside the cubic tank, to reproduce the axisymmetric geometry of \\cite{plumb_instability_1978,semin_generation_2016,semin_nonlinear_2018}, which seems prone to the development of large-scale flows. We are interested in the flow within the cylinder: the fluid in the gap between the cylinder and the cubic tank follows a similar dynamics and thus imposes to the working fluid a (close to) no-flux lateral boundary condition.\n\nThe temperature of the bottom boundary is controlled by water circulating in the copper plate. Water is cooled down by a thermal bath with a fixed temperature set at $-1.25^{\\circ}$C. Due to thermal losses in the pipes alimenting the copper plate, bottom tank temperature is $0.2 \\pm 0.05^\\circ$C. Plate thickness and circulation circuit were optimised so as to ensure a uniform temperature over the whole plate. At the top boundary, the heater is set at a temperature of $35^{\\circ}$C. Its temperature control is custom made: a PT100 probe measures the heater temperature in real time, driving through a feed-back loop the input power injected in the heater. This is a very simple and inexpensive system to impose a temperature while having a transparent top boundary, allowing visualisation and velocity measurements by PIV. Nonetheless, it is necessary to point out that the temperature over the heater area is not perfectly homogeneous. Temperature is maximal at the centre where $T \\sim 38^{\\circ}$C, while the edges are indeed at the requested $T = 35\\pm 0.1^\\circ$C. This inhomogeneity of the top temperature by $\\delta T = 3^{\\circ}$C induces slow convective motions below the heater, in a $\\sim 2$ cm high layer. By performing an experiment where the whole fluid is stably stratified with an overall temperature gradient similar to the one in the stratified layer studied here, but above the density reversal at $4^{\\circ}$C (i.e. bottom boundary set at 10$^{\\circ}$C and top boundary at 70$^{\\circ}$C), we have checked that those top convective motions have no significant impact on the dynamics of the two-layer system. It is also important to say that despite the thick acrylic wall and the intermediate fluid layer between the cylinder and the tank, the working region is not fully thermally insulated on the sides. Nevertheless, our fully stratified, test experiment has shown no motion within the fluid driven by these lateral losses. \n\\\\\n\nThe equation of state of water is non-linear with a maximum density close to 4$^{\\circ}$C (International Equation of State of Seawater, 1980):\n\\begin{equation}\n \\begin{split} \n \\rho(T)= 999.842594+6.793952.10^{-2}T-9.095290.10^{-3}T^2+1.001685.10^{-4}T^3 \\\\\n -1.120083.10^{-6}T^4+6.536332.10^{-9}T^5.\\\\\n \\end{split}\n \\label{eq:eos_eau}\n\\end{equation}\nThus, due to the imposed temperature at top and bottom boundaries, the bottom part of the tank, between 0$^{\\circ}$C and 4$^{\\circ}$C, is convectively unstable (see figure \\ref{fig:setup}). Cold plumes detach from the bottom plate and rise in the tank due to buoyancy. Reciprocally, ``hot'' fluid sinks from the 4$^{\\circ}$C interface due to gravity. While convective motion takes place in the lower layer, the upper part of the tank, between 4$^{\\circ}$C and 35$^{\\circ}$C, is stably stratified, with an assumed linear temperature profile at equilibrium. The temperature is indeed linear for an ideal case without thermal losses, assuming that the stratified layer has a bottom boundary at fixed temperature $4^\\circ$C and top boundary at $35^\\circ$C (\\textit{i.e.} constant diffusive flux through the whole layer). However, the density profile is not linear, due to the non-linear equation of state of water. Stratification is characterised by the Brunt-V\\\"ais\\\"al\\\"a frequency $N^* = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\partial \\rho}{\\partial z}}$. Because of the non-linear equation of state, $N^*$ is not constant with depth, as shown in figure \\ref{fig:setup}. For simplicity, we also use below the global buoyancy frequency defined as $N = \\frac{1}{2 \\pi} \\sqrt{-\\frac{g}{\\rho_0} \\frac{\\Delta \\rho}{\\Delta z}}$, where ${\\Delta \\rho}$ is the global density contrast within the stratified layer of depth ${\\Delta z}$.\n\\\\\n\n\\begin{figure}[h]\n\n \\includegraphics[scale=.3]{schema_cuve_profilN_NB_v3.png}\n \\caption{2D sketch of the tank. A cylinder (light grey shaded area) is placed in a larger cubic tank. The system is cooled down at the bottom at 0$^{\\circ}$C and heated up at the top at 35$^{\\circ}$C. The bottom half is convective with an almost constant density, apart from the bottom boundary layer. The upper half is stably stratified and waves are generated due to the fluid motions in the convective layer. The graph on the right shows the theoretical profile for the buoyancy frequency $N^*$. It is computed considering a linear temperature profile and the equation of state of water (\\ref{eq:eos_eau}). The dashed line is the global buoyancy frequency $N$ calculated on the stratified layer. The various length scales are the cylinder diameter $D$, the vertical extent of the tank $H$ and the vertical extent of the convective layer $h$. $\\delta$ is the minimal width between the outer square tank and the inner cylinder. The $x$, $y$ and $z$-velocity components are noted $u$, $v$ and $w$ respectively.}\n \\label{fig:setup}\n\\end{figure}\n\nBefore starting the experiment, the bath and the heater are allowed to reach their assigned temperature. Then, the upper half of the tank is filled with water stratified in temperature from 4$^{\\circ}$C to 35$^{\\circ}$C, using the double bucket technique \\cite{oster_density_1965}. The bottom half is filled with water with a temperature close to 4$^{\\circ}$C. This filling process is used to avoid tremendously long transient before reaching steady state by thermal diffusion. Typically, we fill the tank at the end of a day and start the measurements the next day in order to let the system reach its equilibrium state over night. Each experiment then lasts four days, with no change in the location of the interface. Note that this steady interface position is the result of the heat budget equilibrium between the convective heat flux extracted from the bottom plate, the diffusive heat flux through the stratified layer, and the lateral heat losses.\n\nTo perform PIV measurements, particles are chosen as small as possible and with a density as close as possible to water density in order to avoid their sedimentation in the stratified layer over the long duration of the experiment. We use fluorescent orange polyethylene micro-spheres whose size ranges from $10 \\, \\mathrm{\\mu m}$ to $20 \\, \\mathrm{\\mu m}$ and density is $1.00 \\pm 0.01$~g\/cc. The fluorescent property allows us, with a high pass filter on the camera, to remove any laser reflection, significantly enhancing the images quality. The tank is illuminated with a green laser $532$~nm. Power is set at $1$~W. \nWe perform side view PIV to measure convection and IGWs spectral characteristics, and top view PIV to observe the large scale flow and its fluctuations over time. The camera used for the side view PIV is a HiSense Zyla $2560 \\times 2160$ pixels recorded on 12 bits. Acquisition rate is $2$~Hz with $100$~ms exposure time. Typical acquisition time for spectral characteristics is $50$~min.\nFor the top view, we use a Point grey camera $1920\\times1080$ pixels on 8~bits. Exposure time is $1$~s, acquisition rate $0.1$~Hz and acquisition time $8$~hours. Captured movies are processed either by the DantecDynamics software DynamicStudio for the side view or by DPIVSoft \\cite{meunier2003analysis} for the top view. Both are resolved into $32\\times32$ pixels boxes with 50\\% overlapping. \n\nSide view PIVs are performed in the middle of the tank at $y=16$~cm in a laser sheet crossing the cylinder along its diameter. This is the case for all figures shown in the $(x,z)$ plane and thus not mentioned in the results section. The vertical fields (see an example in figure \\ref{fig:LSCexpe}) do not show the whole $(x,z)$ plane (where the origin $(0,0)$ is located in the bottom left corner of the cubic tank): it was indeed chosen to zoom in, in order to have the best resolution for the very weak motions in the stratified layer. The interface between the layers is localised between $11 \\mathrm{~cm} \\leqslant z \\leqslant 12$~cm. Typical Rayleigh number for the convection based on this depth is $Ra = 7 \\times 10^6$, and the global Brunt-V\\\"ais\\\"al\\\"a frequency is $N = 1.35 \\times 10^{-1}$~Hz.\n\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[scale = .45]{champ_int_730_colored.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .45]{LSC_0510_1800_colored.pdf}\n \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=150$ mm, and transported by the large-scale circulation. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes signal. The large-scale circulation is a counter clockwise cell. No inversion of the circulation has been seen in our experiment. The velocities under $z=10$ mm are noisy and thus not shown here. Maximum instantaneous velocities are $2.5$ times bigger than the maximum averaged velocities. The left edge of the cylinder is located at $x=19$~mm and the centre of the cylinder is at $x=160$~mm. Approximately $40$~mm of the right side of the cylinder is not captured with our PIV measurments.}\n \\label{fig:LSCexpe}\n\\end{figure*}\n\n\nTo observe the large-scale flow, horizontal views at different heights are performed. A linear translation axis platform from Igus, driven by homemade software, is used to automate the process. With two mirrors, one fixed at the exit of the laser head and the other fixed on the translating platform, it is possible to sweep along one direction with the laser sheet. We typically make measurements during 15~s in each of 11 successive planes separated by 0.5~cm. The full scan duration is about 3~min, and is repeated during at least 8~hours.\n\\\\\n\nThe cylindrical geometry described here differs from the cylindrical shell geometry in \\cite{plumb_instability_1978,semin_generation_2016, semin_nonlinear_2018}. We first tried to work in that annular geometry by adding a second, smaller cylinder in our cubic tank. Three different gap sizes were tested but none showed any interesting dynamics. Indeed, the convection was too confined in the shell to provide an efficient chaotic excitation, and IGWs did not propagate well within the stratification, attenuated quite fast by wall friction. During these tests, we observed the most interesting dynamics within the innermost cylinder, so we decided to use that geometry. The point of the cylindrical shell geometry is that it is a close analogue to the equatorial band of the stratosphere where QBO takes place. By working in a cylinder, the geometry analogy is lost but the physics of the problem remains the same, still a priori allowing for large-scale, reversing, axisymmetric horizontal flows.\n\n\\clearpage\n\\subsection{Numerical model}\\label{sec:methods_num}\n\nTo complement the experiments, we also performed Direct Numerical Simulations (DNS) of the same configuration. We solve the Navier Stokes equations using a non-Oberbeck Boussinesq model. The density variations are consider small compared to the reference density $\\frac{\\Delta \\rho }{\\rho_o} \\ll 1$. Therefore, density fluctuations only appear in the buoyancy force. However, temperature variations affect the value of the thermal expansion coefficient to account for the non-linear equation of state of water. Variations of the thermal diffusivity $\\kappa$ and kinematic viscosity $\\nu$ are neglected. Governing equations are given in dimensionless form by:\n\\begin{align}\n\\label{eq:momentum}\n\\frac{\\partial\\bm{u}}{\\partial t}+\\bm{u}\\cdot\\nabla\\bm{u} = & -\\nabla P+Pr\\nabla^2\\bm{u}-Pr Ra \\ \\theta^2 \\bm{e}_z \\\\\n\\label{eq:heat}\n\\frac{\\partial \\theta}{\\partial t}+\\bm{u}\\cdot\\nabla\\theta = & \\ \\nabla^2 \\theta \\\\\n\\label{eq:mass}\n\\nabla\\cdot\\bm{u} = & \\ 0\n\\end{align}\nwhere we used the depth $H$ of the container as a unit of length, the thermal diffusive timescale $H^2\/\\kappa$ as a unit of time and the difference between the bottom and the inversion temperature $T_0 - T_i$ as a temperature scale. These equations are characterised by the Prandtl number $Pr=\\nu\/\\kappa$, the global Rayleigh number $Ra=\\alpha g (T_0-T_i)^2 H^3\/(\\nu\\kappa)$ and the imposed dimensionless top temperature $\\theta_{top}=(T_{top}-T_i)\/(T_0-T_i)$.\nNote that the quadratic temperature term in the momentum equation is a direct consequence of the nonlinear equation of state of water given by equation~\\eqref{eq:eos_eau}, which is approximed in our model by the quadratic equation $\\rho(T) \\approx \\rho_0 (1-\\alpha(T-T_i)^2)$.\nThe coefficient $\\alpha$ is not the usual thermal expansion coefficient but has a unit of $(\\degree \\mathrm{C})^{-2}$ and is given by $\\alpha\\approx8.1\\times10^{-6}(\\degree \\mathrm{C})^{-2}$ (see also \\cite{lecoanet_numerical_2015}).\n\nWe consider a cylindrical fluid cavity of diameter $D=3H\/2$ as in the experiments.\nBoth horizontal plates are assumed to be no-slip and with fixed temperature.\nThe side wall is assumed to be no-slip and perfectly insulating.\nThis is of course not the case in the experiment, for which lateral heat losses are inevitable and top temperature is not exactly constant, but the objective is to check whether the conclusions drawn from the experimental results are robust and do not depend on these effects.\nSince the experiment runs with water, we use $Pr=7$.\nThe Rayleigh number of the experiment is $Ra=7 \\times 10^7$ while its dimensionless top temperature is $\\theta_{top}=-7.75$. If we were to run the simulation with these parameters, the interface will be located very close to the top boundary. It is not the case in the experiment because of the lateral heat losses, which tend to reduce the effective Rayleigh number. For that reason, and instead of taking into account these losses as in \\cite{lecoanet_numerical_2015}, we kept the insulating lateral boundaries and use the slightly adjusted parameters $Ra=10^7$ and $\\theta_{top} = -11$ instead, which leads to an interface located approximately at $z\\approx120$~mm, as in the experiment. The Rayleigh number could not be lowered under $10^7$ in order to keep the convective flow turbulent enough, thus we had to increase the top temperature to have the interface located at $z\\approx120$~mm.\n\nWe perform DNS of equations~\\eqref{eq:momentum}-\\eqref{eq:mass} using the spectral element solver Nek5000 \\citep{Nek5000}.\nThe global geometry is decomposed into hexahedral elements, with vertical refinement close to the horizontal boundaries and around the mid-plane where the inversion isotherm is expected to be located.\nVelocity, buoyancy and pressure variables are represented as tensor product Lagrange polynomials of order $N$ and $N-2$ based on Gauss or Gauss-Lobatto quadrature points.\nThe total number of grid points is given by $\\mathcal{E}N^3$ where $\\mathcal{E}$ is the number\nof elements.\nFor all the results discussed in this paper, the number of elements is $\\mathcal{E}=8960$ and we use a polynomial order of $N=11$. Numerical convergence was checked by increasing the polynomial order $N$.\nTime integration is performed with a third-order mixed implicit-explicit scheme.\nThe simulations are initialised with a small amplitude buoyancy perturbation and a temperature profile varying linearly between the top and bottom boundaries.\nSimulations are run until a quasi-steady state is reached, after which data is accumulated to compute statistics.\n\n\\section{\\label{sec:results}Results}\n\\subsection{\\label{sec:results_exp}Experiments}\n\\subsubsection{\\label{sec:results_conv}Convection}\nPIV side view is used to quantify horizontal and vertical velocities in the convection zone. Examples of vertical velocities measured at one point in a given location are shown in figure \\ref{fig:panache}, for both ascending cold and descending hot structures. Measurements are consistent with the numerical simulations \\cite{lecoanet_numerical_2015,couston_dynamics_2017} showing intense, localised, cold rising plumes and more diffusive descending plumes. Moreover, these structures are advected by a large-scale circulation encompassing the whole convective layer, as shown in figure \\ref{fig:LSCexpe}.\n\n\\begin{figure}[h]\n \\centering\n {\\label{fig:panacheup}\\includegraphics[scale=0.43]{sonde_panache_up.pdf}}\n \\hspace{.2pt}\n {\\label{fig:panachedown}\\includegraphics[scale=0.43]{sonde_panache_down.pdf}}\n \\caption{Time evolution of the vertical velocity $w$ within: (Left) upward plumes at $x = 200$ mm, $z = 45$ mm, (Right) downward structures at $x= 100$ mm, $z = 95$ mm. }\n \\label{fig:panache}\n\\end{figure}\n\nSpectral analysis is performed to extract power spectral density (PSD) from the velocity signals. Figure \\ref{fig:spectreconv} shows the PSD of the convection vertical velocity $w$. For the two panels, the spectrum is flat with a lot of energy for low frequencies, then the energy drops above some cut-off frequency. Left panel of figure \\ref{fig:spectreconv} shows the vertical velocity PSD at a single point close to the top of the convective layer. A small peak can be seen close to $f = 10^{-2}$~Hz. This is the quasi-periodic signal of the plumes dropping from the top thermal boundary layer. The theoretical characteristic time of convection can be computed from \\cite{gortler_convection_1966}:\n\\begin{equation}\n \\tau = \\frac{h^2}{\\pi \\kappa}\\left(\\frac{\\Delta T}{\\Delta T_{local}} \\times \\frac{Ra_c}{Ra}\\right)^{2\/3}\n\\end{equation}\n\n\\begin{figure*}[b]\n \\centering\n \\includegraphics[scale = .43]{spectre_1_point_cd67_58_vf.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .43]{spectre_W_50toend_10to57_vf.pdf}\n \\caption{PSD for the vertical velocity fluctuations. (Left): PSD computed at a single point $x=100$~mm, $z=95$~mm (signal shown in figure \\ref{fig:panache} right). The plume forcing frequency can be seen around $f = 10^{-2}$~Hz (red dashed line). (Right): PSD spatially averaged over the whole convective cell in the measured $(x,z)$ plane (all points above $z=10$~mm and below $z=110$~mm). }\n \\label{fig:spectreconv}\n\\end{figure*}\n\nwith $h$ the height of the convective layer, $\\kappa$ the thermal diffusivity, $\\Delta T$ the temperature difference between the top and bottom of the convective domain, $\\Delta T_{local}$ the temperature difference between the top and bottom of the thermal boundary layer, and $Ra_c$ the critical Rayleigh number. The critical Rayleigh number in the presence of free and solid interfaces and for fixed temperature is $Ra_c = 1100.65$. For our experiment, the characteristic time is $\\tau = 96$~s, thus characteristic frequency is $1\/ \\tau \\sim 10^{-2}$~Hz, which is close to the observed peak in the left panel of figure \\ref{fig:spectreconv}. At frequencies lower than this characteristic frequency, the spectrum is flat. This is explained by the combined effect of the randomness of the plumes (see figure \\ref{fig:panache}) and of the slow fluctuations of the large-scale circulation. Right panel of figure \\ref{fig:spectreconv} shows the PSD of vertical velocities, averaged over the whole convective cell in the $(x,z)$ plane. It shows a similar trend, with a lower cut-off frequency compared to right panel spectrum. Actually, the plumes signal is more localised and less intense on average than the large-scale circulation signal, which hence dominates the space-averaged PSD.\n\nThe probability density function (PDF) of the vertical velocities in the whole convective layer $\\mathrm{P}(w)$ is computed and shown in figure \\ref{fig:pdf}. It is normalised such that $\\int \\mathrm{P}(w)\\mathrm{d}w = 1$. The PDF describes important features of the convection: it is skewed towards positive values, with positive velocities reaching higher magnitude than negative velocities, \\textit{i.e.} the ascending plumes are stronger than the descending structures. However, the central part of the PDF is close to gaussian profile. The distribution obtained here is in good agreement with the probability density function computed in an idealised 2D numerical model by Couston et al. \\cite{couston_dynamics_2017}. Note that this asymmetry is specific to our model, for which the usual upside-down symmetry in Boussinesq Rayleigh-B\\'enard convection is broken.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale = .6]{pdf_conv.pdf}\n \\caption{Probability density function of the vertical velocities in the convective layer. All PIV points under $z=110$~mm have been used to compute the PDF.}\n \\label{fig:pdf}\n\\end{figure}\n\n\\subsubsection{\\label{sec:results:buffer}Buffer layer}\nAn intermediate layer (we name it the buffer layer in the following) is present between the convective layer and the stratified layer. It was first reported in the quasi-2D 4$^{\\circ}$C convection experiment by Perrard et al. \\cite{perrard_experimental_2013}. Their temperature measurements showed that the buffer layer corresponds to the area where the temperature goes from 4$^{\\circ}$C to 8$^{\\circ}$C. This actually corresponds to the overshooting region for rising cold plumes (note that this type of convection is called \"penetrative convection\" because of this effect). Indeed, since the density of water is close to quadratic around $4^{\\circ}$C, densities at e.g. $0^{\\circ}$C and $8^{\\circ}$C are the same, and the 8$^{\\circ}$C isotherm is the theoretical maximum height reachable by an ascending cold plume at $0^{\\circ}$C in the absence of thermal diffusion. Simultaneously, the overall density profile between $4^{\\circ}$C and $8^{\\circ}$C is stable, as in the stratified layer above $8^{\\circ}$C. The buffer layer is thus a very specific location supporting simultaneously convective overshooting motions and IGWs, as observed with PIV measurements \\cite{perrard_experimental_2013}. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = .6]{buffer_U-xavg.pdf}\n \\caption{Time evolution of the horizontal average of the horizontal velocity, noted $u_X$. Red (resp. blue) regions correspond to mean flow going towards the right (resp. left).}\n \\label{fig:bufferlayer}\n\\end{figure}\n\nTo complete the description of this buffer layer now using velocity measurements, we plot in figure \\ref{fig:bufferlayer} the spatio-temporal graph of the horizontal average of the horizontal velocity $u$. The graph exhibits a strong shear around $z=120$ mm. Since the fluid is going in opposite directions above and below $z=120$ mm with a sharp interface, viscous entrainment by the convective layer is excluded. A special kind of thermal coupling might explain the observed dynamics, as sketched in figure \\ref{fig:schema_couplage}. Indeed, when a cold ascending plume from the convection zone reaches the interface and overshoots in the buffer region, its associated velocity perturbation dissipates more rapidly than its temperature perturbation. Due to gravity, the distorted part of the buffer region wants to sink back to its initial state (pictured by the green arrows), while the fluid above the buffer layer moves towards the impact point of the plume to take the place of the falling water (pictured by the red arrows). The buffer layer then needs some compensating fluid from the convective layer. This mechanism works when the velocity perturbation of the plume at the interface dissipates more rapidly than the thermal perturbation, hence for a Prandtl number $Pr \\geq 1$. One might expect the shearing zone to decrease in size and amplitude when thermal diffusion increases (i.e. when the Prandtl number decreases), since the overshooting rising cold plume will then equilibrate thermally during its ascent more rapidly. This may explain why no interfacial shear was reported in the systematic numerical study of \\cite{couston_dynamics_2017,couston_order_2018} where $Pr \\leq 1$. Global temperature field measurements (using e.g. Temperature Dependent, Laser Induced Fluorescence) are now required to confirm or infirm the proposed model, but those are beyond the scope of the present paper. Note that by extension, we call \"buffer layer\" in the following the region including the $T=4^\\circ$C to $T=8^\\circ$C overshooting region and the shear region. In the experiment, the shear region extends from $z=120$~mm to $z \\approx135$~mm.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[scale = .75]{schema_couplage_final2.pdf}\n \\caption{Sketch of the thermal coupling between the convective and buffer layers. On the left, a cold plume moves upwards towards the interface between the two layers. On the right, isotherms are deflected due to the impact.}\n \\label{fig:schema_couplage}\n\\end{figure}\n\n\\clearpage\n\\subsubsection{\\label{sec:results_waves}Internal gravity waves}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = .83]{visu_onde_colormap_col_02to03_axisequal_schema.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .53]{panache_v1.pdf\n \\caption{Velocity fields showing IGWs propagating. (Top) Velocities in $(x,z)$ plane. The signal is frequency-filtered to enhance the visualisation of oscillatory motions: only frequencies between $0.02$ and $0.03$ Hz are shown, propagating at an angle of roughly $75^\\circ$ with the vertical. The angle of propagation is the angle between the constant phase line and the vertical. (Bottom) Velocities in $(x,y)$ plane located at $z \\approx 125$~mm. In the $(x,y)$ plane, IGWs take the form of oscillating rings. Note that this figure is from a previous experiment without any internal cylinder and is therefore only displayed here as an illustration of the IGWs seen from above.}\n \\label{fig:ondeN}\n\\end{figure}\n\n The convective motions induce a Reynolds stress at and below the interface which generates IGWs propagating in the stratified area \\cite{lecoanet_numerical_2015}. An example is shown in figure \\ref{fig:ondeN}. The vector field has been frequency filtered in the band $[0.02-0.03]$ Hz to isolate a single propagating wave train. We can measure an angle close to $\\theta \\simeq 75^{\\circ}$ between contant phase lines and the vertical. This observation is in good agreement with the inviscid dispersion relation $\\omega = \\pm N \\mathrm{cos}(\\theta)$, which relates the frequency and the propagation angle of IGWs. Indeed, at $z=120$~mm, $N \\sim 0.1$~Hz, thus $\\theta = \\mathrm{cos^{-1}}\\left(\\frac{\\omega}{N}\\right) = 78.5^\\circ$. The motion within the stratified area is a superimposition of many such IGWs oscillating at different frequencies.\n\nTo further investigate the waves signal, waves spectra are plotted in figure \\ref{fig:spectro}, showing the power spectral density of oscillatory motions within the stratified layer at every height, averaged horizontally for each height. The grey line is the theoretically computed buoyancy frequency profile. Figure \\ref{fig:spectro} shows that energy is present in a wide frequency band, from the lowest measured frequencies to the buoyancy frequency $N$. Low frequency motions $f < 4\\times10^{-3}$~Hz are very intense and propagate high in the stratified layer. Motions with frequency ranging from $4 \\times 10^{-3}$~Hz to $N$ are less intense, but still propagate into the stratified layer. Motions propagating at frequencies higher than the buoyancy frequency $N$ are greatly attenuated after a centimetre as IGWs of frequency larger than $N$ are evanescent. The weak signal at low frequencies above $z=180$ mm comes from the convective motions due to the non-homogeneous heating at the top. These motions are confined at the very top of the experimental container. \n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale=0.55]{propagation_nonorm_120_vff.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale =.55]{plot_attenuation_norm_v2.pdf}\n \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ above the convective layer. The grey curve shows the theoretical buoyancy frequency profile, assuming a linear profile for the temperature, from $4^\\circ$C to $35^\\circ$C. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=120$~mm (noted $\\mathrm{PSD_o}$). The PIV measurements are performed for 50 minutes and results (using the pwelch Matlab function) are horizontally averaged at each height to obtain the averaged power spectral density.}\n \\label{fig:spectro}\n\\end{figure*}\n\nRight panel of figure \\ref{fig:spectro} shows two vertical profiles of the PSD re-scaled by the PSD at the top of the convective layer ($z=120$~mm), taken at two different frequencies. The energy decrease is quite similar between $z=120$~mm and $z=140$~mm for both frequencies. However, for $z>140$~mm, the energy for the higher frequency decreases slower than the energy for the lower frequency. This dependence of the attenuation length regarding the frequency of the signal is a characteristic of IGWs. Indeed, the dispersion relation of IGWs relates the frequency and the wave vector direction. Moreover, energy propagates perpendicularly to the wave vector for IGWs (i.e. group and phase velocities are perpendicular). The closer to $N$ the wave frequency $\\omega$ is, the more horizontal the phase propagates, hence the more vertical energy propagates. High frequency waves are thus capable of transporting energy to high altitudes before being damped. On the opposite, waves with low frequency compared to $N$ propagate energy almost horizontally, and are thus attenuated before reaching high altitudes. At frequencies $f<4\\times 10^{-3}$~Hz, a lot of energy is seen and the attenuation length does not depend on the frequency. There is no reason why IGWs should disappear below a certain frequency, but we would expect to see the attenuation length to keep decreasing with decreasing frequency. We thus deduce that IGWs at frequencies $f \\leqslant 4\\times 10^{-3}$~Hz are hidden in the energy spectrum by some very energetic large-scale slowly-varying flow, which we will describe below.\n\n\nMore than one order of magnitude separates the buoyancy frequency and the fastest large-scale flow fluctuations. The large-scale flow penetrates deep into the stratified layer. It globally decreases in amplitude with height, but with some local increases at $z\\sim 125$ mm (i.e. close to the interface between convective and buffer layers) and $z \\sim 145$ mm.\nThe IGWs signal can be seen between $f = 4 \\times 10^{-3}$ Hz and the buoyancy frequency. A peak that reaches the top of the stratified layer is seen around $f= 1.2 \\times 10^{-2}$ Hz, \\textit{i.e.} the same frequency as the convective forcing discussed in section \\ref{sec:results_conv}. It corresponds to the strong excitation provided by the cold rising and hot sinking turbulent plumes.\nHowever, top panel of figure \\ref{fig:spectro} also shows a sudden drop of the energy at frequencies $f>1.2 \\times 10^{-2}$~Hz. Indeed, wave attenuation is strong at these frequencies, even if they are close to (but below) the buoyancy frequency $N$. Actually, energy dissipation also depends on the norm of the wave vector squared. There is no reason that all excited waves have the same wave vector norm; one could even expect that fastest waves are excited by fastest, hence smallest convective patterns, and are thus also at smallest scale: they then dissipate more rapidly.\n\n\n\\subsubsection{\\label{sec:results_lsf}Large-scale flow in the stratified layer}\nFigure \\ref{fig:spectro} shows an important amount of energy at low frequencies which has been interpreted as the signature of a large-scale slowly-varying flow in the stratified layer. We will now investigate the nature of these fluctuations to see if they relate to reversals similar to the QBO.\n\nFigure \\ref{fig:meanflow} shows horizontal vector fields at the same depth at different times. In figure \\ref{fig:meanflow}(a), the flow goes counter-clockwise inside the cylinder. Figure \\ref{fig:meanflow}(b) shows that two contra-rotating vortices with a smaller amplitude typical velocities have appeared. Figure \\ref{fig:meanflow}(c) shows a mostly clockwise rotating flow, where one of the preceding eddy pairs has nearly disappeared. The large-scale flow thus evolves drastically over time. A criterion is computed to extract a typical mean velocity from those fields that accounts for the ``direction'' of the large-scale flow: as illustrated in figure \\ref{fig:critere}, we compute a mean azimuthal velocity, taken along a ring centred in the cylinder. Other criteria to extract a representative value for the large scale flow direction have also been tested, including: the mean vorticity over the cylinder area, the average of the azimuthal velocity over several rings with different radii, and the azimuthal velocity averaged over thick rings. They all give similar results for the large-scale flow measurement. \n\\\\\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale = .8]{evolution_mf_vff.pdf}\n \\caption{Horizontal velocity vector fields in the stratified layer at different times. The laser sheet is located at $z=150$~mm. The large-scale flow reverses from (a) to (c). Time between (a) and (c) is approximately half an hour. Maximum velocities are $0.1$ mm\/s.}\n \\label{fig:meanflow}\n\\end{figure*}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale = .45]{critere_mf.pdf}\n \\caption{Criterion used to extract a significant value for the large scale flow and its direction: the azimuthal velocity is averaged over the ring shown in red.}\n \\label{fig:critere}\n\\end{figure}\n\nIn order to investigate the vertical phase propagation of the reversals, and thus, to compare the reversal dynamics observed to a QBO-like phenomenon, the setup has been equipped with a linear translating platform that allows us to perform horizontal laser sheet sweeping along the vertical. Horizontal velocities are measured in an horizontal plane, every $5$~mm from the top of the convective layer $z=110$~mm to the middle of the stratified layer $z=160$~mm. Any trace of downward phase propagation of the reversals, as observed on the QBO on Earth \\cite{baldwin_quasi-biennial_2001} and on the historical Plumb experiment \\cite{plumb_interaction_1977, semin_nonlinear_2018}, would be a significant evidence for QBO-like phenomenon in the experiment. \nIndeed, the phase propagation of the reversals due to IGWs non-linear interactions is theorised as follows: an IGW propagating in a stratified layer with an horizontal phase velocity in the same direction as the existing base flow propagates upward until reaching a critical height $z_c$, where it deposits all its energy locally. At $z=z_c$, the flow accelerates. Thus the critical height where the flow is intense enough to damp the wave is lowered. As time goes on, this critical height moves towards the location where the waves are emitted. Here, the waves are emitted at the bottom of the stratified layer. We would expect a downward phase propagation if the reversals are driven by IGWs non-linear interactions. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[scale=0.7]{QBOfinal_avecpentediffusionv2.pdf}\n \\caption{Reversals of the large-scale flow. $z=115 - 120$ mm is the convective \/ buffer layers interface. Ascending plumes often perturb the buffer layer flow. The velocity was measured at 11 heights, marked by each tick in the vertical axis of the figure, and interpolated in between. The slope of the black dot lines represent the viscous coupling phase velocity.} \\label{fig:QBO}\n\\end{figure*}\n\n\nWe performed long time experiments (around 8 hours). Typical results extracted from the criterion described above are shown in figure \\ref{fig:QBO}. Blue patches (resp. red patches) represent large scale flow going counter clockwise (resp. clockwise). The present measurements mainly confirm the interpretation of figure \\ref{fig:spectro} for the lowest frequencies: the large-scale flow is horizontal and extends over the whole depth of the stratified layer with an amplitude attenuated with height, and exhibits slow reversals. Additionally, some intense events at $z = 110$ mm are directly related to penetrative plumes from the convection. Reversals times range from $400$s to $1800$s. However, no downward phase propagation of the reversals is observed. On the contrary, the reversals seem to occur along the whole stratification height at the same time, or even with a rapid upward phase propagation. Since the phase propagation is not towards the location where the waves are emitted, the reversals are unlikely driven by the non-linear interactions of IGWs. However, as seen in section \\ref{sec:results_waves}, IGWs propagate in the stratified layer and carry energy. Therefore, they give energy to the large-scale flow through non-linear interactions. Yet, the process is not dominant in the reversals dynamics.\n\\\\\nSince, the reversals observed in figure \\ref{fig:QBO} do not have a downward phase propagation, we look for other mechanisms than the QBO mechanism to explain the reversals. Two other mechanisms can be investigated. The first one relies on a specific convective dynamics within the overall stratified layer driven by horizontal gradients related to imperfect top and side boundary conditions. The second mechanism relies on viscous coupling with the underlying convective and buffer layers. \n\nOur fully stratified reference experiment described in section \\ref{sec:methods} precludes the first scenario. Indeed, setting the bottom boundary at 10$^{\\circ}$C and the top boundary at 70$^{\\circ}$C, no motion is observed for the bottom $3\/4$ of the tank. In this test-experiment, the top $1\/4$ of the tank is animated by convective motions due to the non-homogeneous top heat source (in the standard $4^\\circ$C experiment, where $T_{top} = 35\\degree$C, only $\\sim 2$ cm are affected by the convection at the top of the tank, because the non-homogeneity of the heat source is less important for lower temperature, thus the horizontal convection is weaker). However, these are inefficient to generate waves below and to drive any large-scale flow observable away from the top region.\n\n\n\\begin{figure}[h]\n\\includegraphics[scale = 0.55]{balayage_corr_vf.pdf}\n \\caption{Velocity vector fields in the horizontal plane. Different columns represent different sweeping cycles $t^*$ (one sweeping cycle corresponds to the 11 steps needed to go from the lowest position $z=110$~mm to the highest position $z=160$~mm). Different rows represent different heights within the same sweeping cycle: first row is the top of the convection $z=110$~mm, second row is in the buffer layer $z=120$~mm and third row is in the stratified layer $z=145$~mm. Convective plumes are easily noticeable on the first row fields.}\n \\label{fig:champ_correle}\n\\end{figure}\n\n\\begin{figure}[h]\n \\includegraphics[scale = 0.55]{balayage_NOcorr_vf.pdf}\n \\caption{Same as figure \\ref{fig:champ_correle} but for different sweeping cycles. Note that in this set of velocity fields, the buffer and stratified layers are less correlated than they are in figure \\ref{fig:champ_correle}.}\n \\label{fig:champ_pascorrele}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale=0.6]{correProdscalaire_BW_vf.pdf}\n \n \\caption{Velocity correlation between the three layers. Dashed black lines are the $-0.5$ and $0.5$ values. Correlation coefficient are window-averaged over $10$~min to smooth the curve.}\n \\label{fig:correlation}\n\\end{figure}\n\nThis leaves viscous entrainment as a possible driving mechanism. The dotted lines on figure \\ref{fig:QBO} show a theoretical viscous time, computed from the time for viscous entrainment to drive 20\\% of the horizontal velocity at $z = 115$ mm to $z=160$ mm, starting from a base state flow at rest. The 20\\% value corresponds to the measured value of the large scale flow at $z=160$ mm compared to the value at $z = 115$ mm (noted $u_b$). The theoretical viscous time entrainment is given by $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. The reversals occur in a time scale comparable with this theoretical viscous time. The similarity between the slope of the dashed lines and the slope of the upward phases suggests that reversals are driven viscously.\n\nHowever, the existence of the buffer layer and its associated intense shear, with opposite horizontal velocities with the convective layer below (see figure \\ref{fig:bufferlayer}) precludes direct viscous coupling between the convective and stratified layers. Besides, no reversal has been observed in the convective region. We thus propose a thermal coupling between the convective and buffer layers as seen in section \\ref{sec:results:buffer}, associated to a viscous coupling between the buffer and stratified ones. To further quantify this possibility, figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele} show horizontal velocity fields at different heights and at different times. For each of the columns shown, first row is the mean flow in the convective layer at depth $z=110$~mm, second row is the mean flow in the buffer layer at depth $z=120$~mm and third row is the mean flow in the stratified layer at depth $z=145$~mm. The correlation coefficients through time between (i) the convective and buffer layers, (ii) the buffer and stratified layers, and (iii) the convective and stratified layers have been computed. It consists in a scalar product of the velocity vector for each position at two different heights rescaled by the product of the norm of the velocity vector at the two heights, \\textit{i.e.}: \n\n\\begin{equation}\nR_{ij} = R(x_i,y_j) = \\frac{u(x_i,y_j,z_1) \\times u(x_i,y_j,z_2) + v(x_i,y_j,z_1) \\times v(x_i,y_j,z_2)}{\\left(u(x_i,y_j,z_1)^2+v(x_i,y_j,z_1)^2\\right)^{1\/2} \\times \\left(u(x_i,y_j,z_2)^2 + v(x_i,y_j,z_2)^2 \\right)^{1\/2}}\n\\end{equation}\nThis gives a correlation coefficient $R_{ij}$ for each PIV position in the horizontal plane. The global correlation coefficient $R$ is computed by spatially averaging the local correlation coefficients. \n\nResults are shown in figure \\ref{fig:correlation}. The convective and buffer layers are negatively correlated: the correlation coefficient is most of the time close to $R=-0.5$. This can also be seen at all times in figures \\ref{fig:champ_correle} and \\ref{fig:champ_pascorrele}, where horizontal velocities in the convective and buffer layers have opposite direction. A diverging flow coming from an impinging plume in the convective zone corresponds to a converging flow in the buffer layer towards the impact zone, hence confirming the thermal coupling mechanism described in section \\ref{sec:results:buffer}. This converging flow may lead either to a clockwise or anticlockwise azimuthal mean flow, depending on the details of the chaotic excitation from the convective plumes. The correlation coefficient between the convective and stratified layers can be positive or negative, and is anyway most of the time less than 0.2, in absolute value. The correlation coefficient between the buffer and stratified layers shows a lot of temporal variations. However, it remains always positive. At a given time, the large-scale flow in the stratified layer may switch between a regime strongly dominated by the buffer layer (see also figure \\ref{fig:champ_correle}), and a second regime where the flow in the stratified layer is quite different from the flow in the buffer layer (see also figure \\ref{fig:champ_pascorrele}). \n\nWe thus conclude that the stratified layer is globally viscously driven by the buffer layer. However, the stratified layer exhibits additional complexities. These might be due to IGWs interacting with the large-scale flow. The results from Couston et al. \\cite{couston_order_2018} show that the lower the Prandtl number, the more regular the QBO. In the experiment, the Prandtl number is close to $Pr = 7$: the typical associated QBO-type flow is irregular, with low amplitude. We thus propose that large-scale flow driven by IGWs non-linear interaction superimposes on the viscously driven flow, but remains secondary. We do not know at this point how to disentangle those two potential contributions from the available data. \n\n\n\\clearpage\n\\subsection{\\label{sec:results_num} Numerical simulations}\nThe experimental results are not fully sufficient to explain, with complete certainty, the origin of the buffer layer and of the large-scale flow observed in the stratified layer. In addition, the effects of the lateral heat losses and top temperature heterogeneity are difficult to distinguish. To answer these questions, 3D DNS of a configuration similar to our experiments are performed, reproducing the 4$^{\\circ}$C convection but with idealised boundary conditions (i.e. no flux on the sides, and fixed temperature at the top and bottom). As mentioned in section \\ref{sec:methods_num}, the Rayleigh number $Ra$ and $T_{top}$ are tuned so that the interface depth in the experiment and the numerical simulation are similar. We have $Ra=10^7$ and $T_{top}=48^\\circ$C. All the numerical simulations are run dimensionless, but results are shown in dimensional values. The length scale is $H = 200$~mm, the vertical extent of the whole domain (hence diameter is $D=300$~mm), the timescale is the thermal diffusive time $\\tau = \\frac{H^2}{\\kappa} = \\frac{0.2^2}{1.5 \\, 10^{-7}} = 2.67 \\times 10^{5}$~s, and the temperature is given by the dimensionless temperature $\\theta = \\frac{T - T_i}{T_0 - T_i}$, where $T, T_i, T_0$ are respectively the dimensional temperature, the inversion temperature of the equation of state (i.e. $4^\\circ$C), and the bottom temperature (i.e. $0^\\circ$C). Results for sections \\ref{sec:results_num_conv} - \\ref{sec:results_num_igw} are computed from a $(x,z)$ vertical plane located along a cylinder diameter.\n\n\\subsubsection{\\label{sec:results_num_conv}Large-scale circulation in the convection zone and buffer layer}\nFigure \\ref{fig:LSC_num} shows that a large-scale circulation takes place in the convective layer. It consists of a cell filling the whole convective layer, and exhibits no reversal over the whole course of the simulation. The fluid rotates counter clockwise in the vertical plane. This is qualitatively consistent with the mean flow observed in the experiment and shown in the right panel of figure \\ref{fig:LSCexpe}. As in the experiments, a counter current exists on the top of the convective layer at $z= 120$~mm, creating a strong shear and demonstrating the existence of a buffer layer in the numerical simulation as well. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = .47]{quiver_inst_color.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .47]{quiver_moyen_color.pdf}\n \\caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=230$ mm. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes recording. The large-scale circulation is a counter clockwise cell. Maximum instantaneous velocities are $3$ times bigger than the maximum averaged velocities.}\n \\label{fig:LSC_num}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale = 0.5]{Spatio_temp_Ux_v3_dim.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = 0.5]{profil_T_all.pdf}\n \\caption{(Left) Horizontal average of the horizontal velocity $u$ over a vertical cross-section in the middle of the tank. The buffer layer can be seen above $z=120$~mm. A stationary large-scale circulation is present in the convective layer, even if it appears quite perturbed at the end of the signal. (Right) Temperature profiles along the $z$-axis.}\n \\label{fig:num_buffer}\n\\end{figure*}\n\nThe space-time diagram of the mean horizontal flow shown in figure \\ref{fig:num_buffer} confirms it. Observing the buffer layer in the absence of side thermal losses and top temperature heterogeneity is an additional argument accounting for the fact that it is not an artefact driven by imperfect experimental conditions.\nWe also observe that the flow within the convection stays positive through time at the bottom and negative at the top. This is evidence of the steady large scale circulation taking place in the convective layer. Some events appear at $t > 1.42 \\times 10^{5}$~s and are interpreted as quasi-reversal of the large-scale circulation. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=.5]{quiverIsotherme.pdf}\n \\caption{Velocity field and temperature isotherms at the end of an upward plume impact on the interface.}\n \\label{fig:num_quiverisoT}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=.6]{corre_WetU.pdf}\n \\caption{(Left) Spatio-temporal diagram of the horizontal velocity $u$ at $z=128$~mm. (Right) Spatio-temporal diagram of the vertical velocity $w$ at $z=108$~mm. The event at $t \\approx 1.052 \\times 10^5$~s is shown in figure \\ref{fig:num_quiverisoT}.}\n \\label{fig:num_correWU}\n\\end{figure}\n\nThe temperature profile along the $z$ axis is also plotted on the right panel of figure \\ref{fig:num_buffer}. The figure shows a temporal and horizontal average of the temperature field (black thick curve), two temporal averages at two different positions $x=14$~mm (left side of the tank - dashed grey) and $x=277$~mm (right side - dotted grey) and an instantaneous profile at $x=145$~mm (middle of the tank - thick grey with crosses). The thermal boundary layer can be seen, between $z=0$~mm and $z=10$~mm. Then, between $z=10$~mm and $z=100$~mm lies a layer of constant temperature $T \\sim 2.8^\\circ$C. Between $100 \\mathrm{~mm} \\leqslant z \\leqslant 115 \\mathrm{~mm}$, the temperature profile evolves from constant to linear for $z > 115$ ~mm. The $T=4^\\circ$C (respectively $T=8^\\circ$C) isotherm is located at $z = 110$~mm (resp. $z = 120$~mm). Note that the temporal average of the temperature profiles are different on the left and right sides of the tank. Indeed, the constant temperature height goes to $z=90$~mm for the left side whereas it goes to $z=115$~mm for the right side. This suggests that the convective \/ buffer layer interface does not lies at one height over the whole tank but is a function of time and space. This is very likely due to the large-scale circulation. Thus, the thermal coupling described in \\ref{sec:results:buffer} will likely occur at different heights, depending on time and horizontal position.\n\n\nThe thermal coupling as schematised in figure \\ref{fig:schema_couplage} can be found in the numerical simulation. This is represented in figure \\ref{fig:num_quiverisoT}. An upward plume impacting the convective \/ buffer layer interface is seen. The isotherms ranging from $T=4^\\circ$C to $T=11^\\circ$C are deflected upward, due to the plume bringing cold fluid upward. On the contrary, the isotherms $T = 12 -14^\\circ$C are deflected downward by the converging flow. Isotherms at $T \\geqslant 15^\\circ$C remain horizontal. After the impact on the interface, the plume is deflected outwards. One could expect the fluid above the impact to be viscously entrained by this outward deflection. However, as observed in figure \\ref{fig:num_quiverisoT} for the simulation and figures \\ref{fig:champ_correle}-\\ref{fig:champ_pascorrele} for the experiment, the fluid above the interface is going towards the plume, \\textit{i.e.} in the opposite direction of the fluid below, hence explaining the observed shear (see figures \\ref{fig:num_quiverisoT} and \\ref{fig:schema_couplage}). The time evolution of these dynamics is shown in figure \\ref{fig:num_correWU}.\n\n\nFigure \\ref{fig:num_correWU} shows the time evolution of the horizontal velocity $u$ in the shear layer at $z=128$~mm and the time evolution of the vertical velocity $w$ in the convective layer at $z=108$~mm. Comparing the two panels of figure \\ref{fig:num_correWU} shows that upward plumes are concomitant with converging horizontal velocities towards the plume impact. Indeed, the spatio-temporal diagram of $w$ exhibits local strong upward plumes. These plumes, as suggested by the dashed black lines, are correlated in time and space with converging horizontal velocities. For instance, an upward plume is seen at $x\\approx220$~mm and $t\\approx1.043 \\times 10^5$~s. At the same horizontal position and time, the positive horizontal velocity becomes stronger and the negative horizontal velocity patch increases in size to reach $x\\approx220$~mm. The converging horizontal velocities event occurs a short time after the impact of the plumes. Thus, it can be concluded that the plume induces the converging flow, as suggested by our explanation in section \\ref{sec:results:buffer}.\n\n\\clearpage\n\n\\subsubsection{\\label{sec:results_num_igw}Internal gravity waves}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale = 0.55]{spectro_vfinale.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = .55]{plot_attenuation_norm_num_v2.pdf}\n \\caption{(Left) Power spectral density of the absolute velocity $\\sqrt{u^2+w^2}$ in the buffer and stratified layers. The grey curve shows the buoyancy frequency profile computed from the spatial and temporal average of the temperature field. (Right) Two selected profiles (taken at frequencies shown by dashed lines on the left graph) of the re-scaled PSDs by the PSD at the top of the convective layer, \\textit{i.e.} $z=118$~mm.}\n \\label{fig:num_spectro}\n\\end{figure}\n\nPSDs are computed in the stratified and buffer layers and are plotted in figure \\ref{fig:num_spectro}. As for the experiment (figure \\ref{fig:spectro}), numerical results show oscillatory motions at different frequencies attenuated with height. Experimental results (figure \\ref{fig:spectro}) and numerical results (figure \\ref{fig:num_spectro}) show strongly similar dynamics: most of the energy is present at low frequencies ($f<3 \\times 10^{-3}$~Hz). The motion with frequencies ranging from $3 \\times 10^{-3}$~Hz to $N$ are less intense, and almost no energy is seen at frequencies $f>N$. \n\nRight panel of figure \\ref{fig:num_spectro} shows two selected vertical profiles (shown by the white dashed line on the left panel figure) of the the PSD re-scaled by the PSD at $z=118$~mm. The energy for the higher frequency ($f=1.4\\times10^{-2}$~Hz) decreases slower than the energy for the lower frequency ($f=5.0\\times10^{-3}$~Hz). This is, as experimental results, in agreement with the dispersion relation of IGWs.\nThe overall behaviour of waves spectra is similar in experiment and numerical simulation, with an attenuation length independent of the frequency in the low-frequency signal thus confirming a viscous coupling origin of the large-scale flow, and increasing when the frequency goes towards $N$ in the wave domain.\n\n\\subsubsection{Large-scale flow within the stratified layer}\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_square_dim.pdf}\n \\hspace{.1cm}\n \\includegraphics[scale = 0.6]{Spatiotemp_Vtheta_cyl_v2_zoom_dim.pdf}\n \\caption{Spatio-temporal diagrams of the azimuthal averaging of the azimuthal velocity inside a virtual cylinder of radius $r = 140~$mm. The bottom figure is a zoom on the stratified zone, delimited in the top figure by the black square. The slope of the black lines show the theoretical viscous diffusive time.}\n \\label{fig:num_vthetacyl}\n\\end{figure*}\nSimilarly to what has been done for the experimental data, figure \\ref{fig:num_vthetacyl} shows the mean azimuthal velocity over the whole height of a virtual cylinder of radius $r = 140$ mm. We observe reversals within the convective layer ($z<120$ mm), which are not systematically correlated with the signal in the stratified layer. The mean velocity in the stratified layer also exhibits reversals. They are characterised by an upward phase propagation from the buffer zone at $z=120$ mm, as shown in the zoom (bottom panel of figure \\ref{fig:num_vthetacyl}). The phase velocity seen in figure \\ref{fig:num_vthetacyl} is in good agreement with the theoretical time for viscous propagation $t = \\frac{z^2}{4\\,\\nu\\, \\mathrm{erf}^{-2}\\left(\\left(\\frac{u}{u_b}-1\\right)\\right)}$. This corroborates the fact that the reversals observed within the stratified layer are viscously driven from the dynamics occurring in the buffer layer, as it has been seen for the experiment. Reversals time ranges from $300$~s to $1500$~s. Those reversals times are similar to the experimental ones, though slightly shorter (numerical reversals are $\\sim20$\\% faster than experimental reversals). \n\n\\clearpage\n\\section{Conclusion}\\label{sec:discussion}\nThe 4$^{\\circ}$C convection experiment, originally performed by Townsend \\cite{townsend_natural_1964}, has been re-investigated using long-term PIV measurements in a vertical cross-section, and in several horizontal cross-sections within the stratified layer. This last type of measurements has allowed to investigate for the first time the long-term horizontal mean flow in the stratified layer. Experiments have been complemented by direct numerical simulations. The first result of this paper is the confirmation, in 3D and with ideal boundary conditions, of the presence of a buffer layer, including an overshooting region as first observed by Perrard \\cite{perrard_experimental_2013}, and a shear region. We have argued that the buffer layer is driven by thermal coupling with the convection, due to the non-linear equation of state of water, and that this mechanism is a priori related to a Prandtl number larger than one. The second result is that the buffer layer viscously drives slow reversals of the horizontal large-scale flow within the stratified layer. \n\n\nAdditionally, IGWs at different frequencies propagate in the stratified layer. They likely interact with the horizontal large-scale flow, and probably also produce a reversing flow, which superimposes to the viscously driven one. From Couston et al. \\cite{couston_order_2018}, we know that the Prandtl number has a strong influence on this QBO-like mechanism: the lower the Prandtl number, the stronger the amplitude of the QBO. In water, $Pr \\sim 7$, and the expected amplitude of the large-scale QBO flow is weak, hence dominated by the viscous driving. Further experimental studies at lower Prandtl number should allow deciphering the two contributions. One could for instance suggest using gas as a working fluid; however, the absence of density reversal around a given temperature will necessitate to consider either transient experiments like \\cite{deardorff_laboratory_1969, michaelian_coupling_2002}, or two-gas experiments which might then be prone to double diffusive instabilities. Experimentally, the question also remains to understand why the only successful QBO experiment has been performed in salty water, hence with a Schmidt number (equivalent to Pr) of 700.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale=.6]{U_avgX.pdf}\n \\caption{Horizontal average of the horizontal velocity $u$ over a vertical cross section in the middle of the numerical domain for $Pr = 0.1$.}\n \\label{fig:num_pr01}\n\\end{figure}\n\nIn the meantime, it is straightforward to change the Prandtl number in the numerical simulation of our set-up. We have thus run a second simulation with the same Rayleigh number $Ra = 10^7$ and top temperature $\\theta_{top} = 11$ but with $Pr = 0.1$. In this simulation, as shown in figure \\ref{fig:num_pr01}, no buffer layer is observed, but strong signatures of a QBO like mechanism are visible, marked by downward phase propagation of the reversals of the large-scale flow. This configuration thus deserves a more systematic study in the future.\n\n\\section*{Acknowledgements} \nThe authors acknowledge funding by the European Research Council under the European Union's Horizon 2020 research and innovation program through Grant No. 681835-FLUDYCO-ERC-2015-CoG. This work was granted access to the HPC resources of Aix-Marseille Universit\\'e financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program \"Investissements d'Avenir\" supervised by the Agence Nationale de la Recherche. Computations were also conducted with the support of the HPC resources of GENCI-IDRIS (Grant No.A0060407543).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe study of universal scaling behaviors associated with the nonequilibrium critical phenomena is an attractive and fascinating field of statistical physics that has attracted a considerable attention in recent years \\cite{kardar1998nonequilibrium,fisher1998collective,amaral1995scaling,kawamura2012statistical}. \n Indeed, it is expected that a wide range of models at the critical point could well be characterized by the same universal parameters, as is well known from equilibrium critical phenomena \\cite{odor2004universality}. \nIs it possible to derive these parameters to determine the universality of critical phase transitions in out-of-equilibrium models? Recent studies have focused on the dynamical characteristics of a vast range of problems, including fracture propagation in solids \\cite{gao1989first,schmittbuhl1995interfacial,tanguy1998individual,alava2006statistical,kawamura2012statistical}, charge density waves in anisotropic\nconductors \\cite{fisher1985sliding,gruner1988dynamics}, vortices in type-II superconductors \\cite{blatter1994vortices}, domain walls\nin ferromagnetic \\cite{lemerle1998domain} or ferroelectric \\cite{tybell2002domain} systems, \nthe contact line of a fluid drop on a disordered substrate \\cite{cieplak1988dynamical,moulinet2004width,alava2004imbibition},\nthe deformation of crystals \\cite{alava2006statistical}, crackling noise in a wide range of physical systems from magnetic materials to paper crumpling\n \\cite{sethna2001crackling,bonamy2008crackling}, friction and lubrication \\cite{cule1996tribology,kawamura2012statistical,vanossi2011modeling}, \nthe motion of geological faults \\cite{fisher1997statistics,kawamura2012statistical}, tumour growth \\cite{bru2004pinning}, and many others. This diverse set of processes may be described as an extended elastic manifold driven over quenched disorder, which has a complicated dynamics that includes non-equilibrium phase transitions.\n\nThe competition between the deformation induced by quenched disorder (induced by the presence of impurities in the host environment) and the elastic material's response to an applied driving force is the key factor determining their dynamical behaviour in all of these complex non-linear systems.\n\nThe \"depinning transition\" phenomenon is a significant result of this competition \\cite{surface2}. \nAt zero temperature and in the absence of an external driving force $F$, \nthe system is disordered but it does not move and remains \\textit{pinned}\nby the quench disorder. When the external force is increased from zero, \nthe elastic object \\textit{unpins} and reaches a finite steady-state velocity \\cite{fisher1998collective}.\nThis describes the critical phase transition of the elastic interface \nat the critical force $F = F_c$, where the driving force $F$ plays the\nrole of the control parameter and the mean velocity $v$ is the\norder parameter \\cite{kardar1998nonequilibrium}. Note that the critical value of the external force $F_c$ is not\nuniversal and its value depends on the details of the model. \nThe steady-state average velocity follows a power-law characteristic as $v\\sim (F-F_c)^\\theta$ while approaching the critical point from above, where $theta$ is a universal parameter. Other measures, such as the local width, the correlation functions, the correlation length, and the structure factor, may be used to extract the exponents associated to the criticality of the elastic interface. These techniques have been extensively used to investigate the self-affine surface structure's scaling properties \\cite{surface2,mckane2013scale,krug1997origins}.\n\nConsider a single-valued function $u(\\textbf{x},t)$ that describes an elastic interface. \nThe global surface width $W= \\sqrt{\\langle (u(\\textbf{x},t) - \\langle u \\rangle_{\\textbf{x}})^2 \\rangle_{\\textbf{x}}}$, is the simplest quantity used to characterize the scaling characteristics of elastic interfaces near the critical point, which is defined as the standard deviation around the mean position. \nFor a finite system of size $L$, the roughening of $u$ from a \nflat initial condition, scales as: \n\\begin{eqnarray}\nW(L,t) \\sim t^\\beta f(L\/t^{1\/\\nu}),\n\\end{eqnarray}\nwhere the exponents $\\beta$ and $\\nu$ are called the growth and the dynamical exponent. The scaling \nfunction $f(x)$ is such that $f(x)\\sim \\textrm{const}$ for $x\\gg 1$, and $f(x) \\sim x^{\\zeta_{g}}$ \nfor $x\\ll 1$ (the exponent $\\zeta_g$ is known as the global roughness exponent). Finite size effects, as expected, occur when $t^{\\times}_W \\sim L^\\nu$. The self-affine\nscaling relates now $\\zeta_g$, $\\beta$, and the dynamical exponent $\\nu$ through $\\nu = \\zeta_g\/\\beta$ \\cite{surface2}.\n\nThe average velocity, which corresponds to the order parameter of the pinning-depinning transition of a driven interface, may be assumed to be a homogeneous function of time $t$ and $|F-F_c|$, similar to critical phenomena as:\n\\begin{eqnarray}\nv(t,F) \\sim t^{-\\sigma} g(|F-F_c|t^{\\sigma\/\\theta}),\n\\end{eqnarray}\nwhere $\\sigma$ is a universal scaling exponent. For $F > F_c$ there is a crossover time-scale, $t^{\\times}_v\\sim |F-F_c|^{\\theta\/\\sigma}$ between two regimes: $g(x) \\rightarrow \\textrm{const}$ for $t\\ll t^{\\times}_v$ and $g(x)\\sim x^\\theta$ for $t \\gg t^{\\times}_v$. \n\nThe equilibrium configuration of an elastic rough interface in the critical point is expected to be self-affine, and the tow-point correlation function is supposed to obey the scaling form\n\\begin{eqnarray}\nC(r)=\\langle [ u(\\textbf{x})- u(\\textbf{x}^\\prime) ] ^2\\rangle \\sim |\\textbf{x}-\\textbf{x}^\\prime |^{2\\zeta_l},\n\\end{eqnarray}\nwhere $\\zeta_l$ is the local roughness exponent. \n\nVarious experimental, analytical, and numerical works have been proposed to compute the critical exponents $theta$, $beta$, $zeta_g$, $zeta_l$, and $sigma$ characterizing the ``pinning-depinning'' phase transition, in a similar fashion to the equilibrium critical phenomena.\n\nThe purpose of this research is to describe and investigate the statics and dynamics of a generalized model for the investigation of a range of other reported phenomena in which the pinning-deppinig phase transition may occur.\nThe paper is organized as follows. Section \\ref{Definition sec} introduces the model. \nSection\n\\ref{Numerical method sec} describes the numerical formalism. \nIn Sec. \\ref{Numerical results sec} we discuss our findings. In the final section, we summarize the obtained results and our conclusions.\n\n\n\n\\section{Definition of the model}\\label{Definition sec}\nDespite the significant variations in theoretical models, many of the computations were performed using the linear assumption of the elasticity $u(\\textbf{x},t)$. The following equation can be used to explain the motion of an interface in an isotropic disordered material at this level of precision \\cite{fisher1998collective}:\n\\begin{eqnarray}\n\\label{fisher_formula}\n\\frac{\\partial u(\\textbf{x},t)}{\\partial t} = F + f_p(\\mathbf{x},u(\\mathbf{x},t)) -\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right]~,\n\\end{eqnarray} \nwhere $F$ is a uniform external force which is also the control parameter and $f_p$ representing\nthe ``non-thermal'' quenched random forces due to the randomness and impurities of the heterogeneous medium. \nThe quenched random noise $f_p(\\mathbf{x},u(\\mathbf{x},t))$, can be taken to have zero mean satisfying the relation \n$\\langle f_p(\\mathbf{x},u) f_p(\\mathbf{x}^\\prime,u^\\prime) \\rangle = 2D \\delta (\\mathbf{x}-\\mathbf{x}^\\prime) \\mathcal{R} (u-u^\\prime)$, where $\\mathcal{R}(u-u^\\prime)$ assumed to decay rapidly for large values of its argument. The final term $\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right]$ in Eq. (\\ref{fisher_formula}) describes the elastic forces between different parts. It has the form\n\\begin{eqnarray}\n\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right] = \\int d^D\\textbf{x}^\\prime \\int dt^\\prime \\mathcal{J}(\\textbf{x}-\\textbf{x}^\\prime,t-t^\\prime)\\nonumber \\\\\n\\times \\left[ u(\\textbf{x}^\\prime,t^\\prime) - u(\\textbf{x},t) \\right],\n\\end{eqnarray}\nwhere $D$ is the space dimension and $\\mathcal{J}(\\textbf{x}-\\textbf{x}^\\prime,t-t^\\prime)$ is defined as the propagation kernel to transmit the stress on the interface from its elasticity. Moreover, systems\nwith short range elasticity of the interface are\ncharacterized by $\\mathcal{J}(\\textbf{x},t)\\propto \\delta (t) \\nabla^2 \\delta (\\textbf{x})$ \\cite{fisher1998collective}.\n\nTheoretical studies on quenched disordered systems, such as a contact line of a liquid meniscus on a disordered substrate \\cite{rosso2002roughness,moulinet2004width}, \ncrack propagation \\cite{rosso2002roughness,laurson2010avalanches} \nand solid friction \\cite{moretti2004depinning}, have shown that it is possible to express\nthe kernel $\\mathcal{K}[u]$ in a long-range form \n\\begin{eqnarray}\n\\label{long range kernel}\n\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right]\\propto \\int d^D\\textbf{x}^\\prime \\frac{u(\\textbf{x},t)-u(\\textbf{x}^\\prime,t)}{|\\textbf{x}-\\textbf{x}^\\prime|^{D+z}},\n\\end{eqnarray}\nwhere the exponent $z$ is a variable that depends on the model chosen to represent the elastic interface \\cite{tanguy1998individual}. The most important aspect of the singular integration Eq. (\\ref{long range kernel}) is that it may be used to rewrite the elastic force $\\mathcal{K}[u]$ as \n\\begin{eqnarray}\\label{fractional fisher}\n\\mathcal{K}\\left[ u(\\mathbf{x},t) \\right] = \\left( - \\Laplace \\right) ^{z\/2} \nu(\\mathbf{x},t),\n\\end{eqnarray}\nwhere $\\left( - \\Laplace \\right) ^{z\/2}$ is the fractional Laplacian \ndefined by its Fourier transform \n$\\widehat{\\left( - \\Laplace \\right)} ^{z\/2} \\Phi (\\textbf{k}) = \n|\\textbf{k}|^z \\widehat{\\Phi }(\\textbf{k})$ \\cite{fractiobaloperators}.\nAccording to Eqs. (\\ref{fisher_formula}) and (\\ref{fractional fisher}),\none can rewrite the Eq. (\\ref{fisher_formula}) as follows:\n\\begin{eqnarray}\n\\label{generalized_fisher_formula}\n\\frac{\\partial u(\\mathbf{x},t)}{\\partial t} =F +f_p(\\mathbf{x},u(\\mathbf{x},t))-\n\\left( - \\Laplace \\right) ^{z\/2} u(\\mathbf{x},t) \n~.\n\\end{eqnarray}\n\n\nIt is indeed worth mentioning that the dynamics given by Eq. (\\ref{generalized_fisher_formula} are essentially generalizations of the quenched Edwards-Wilkinson (qEW) and quenched Mullins-Herring (qMH) equations, which are the simplest and most often used equations to explain the interface pinning-depinning transition in quenched random media, with $z=2 \\textrm{ and } 4$, respectively.\n\nMany researches have been carried on the qEW and qMH equations, as well as the related models. Early studies investigated numerically the crucial characteristics of the qEW equation \\cite{leschhorn1993interface},\nand it has been the subject of many\ntheoretical and numerical studies in recent years \n\\cite{ramasco2000generic,rosso2001origin,lacombe2001force,rosso2003depinning,kolton2006short,kolton2009universal}.\nRecently, a novel and very efficient approach investigated the qEW equation's depinning threshold and critical exponents \\cite{ferrero2013numerical,ferrero2013nonsteady}.\nThe scaling properties of the qMH equation at the critical point of the pinning-depinning transition have been quantitatively explored \\cite{lee2000growth,lee2006depinning,boltz2014depinning}.\nIt is worth noting that, for the \nso-called space-fractional quenched equation Eq. (\\ref{generalized_fisher_formula}), the scaling hypothesis has been established in Ref. \\cite{xia2012depinning} \n(The fractional power $z$ is expected to be in the range $1.5 \\leqslant z\\leqslant 2$). The Grunwald-Letnikov form of a fractional derivative has been used to discretize the space-fractional quenched equation, which is essentially an integro-differential equation, as noted in Ref.\\cite{xia2012depinning}.\n\nDespite the success of\nEqs. (\\ref{fisher_formula}) and (\\ref{generalized_fisher_formula}) \nin describing the dynamics of elastic interfaces driven through a disordered medium, this toy model had one weakness: hydrodynamic interactions were not included. This is the case, for instance, of polymers \\cite{haidara2008competitiv,d2010single},\nmembranes \\cite{nissen2001interface,verma2014rough}, the dynamics of colloid suspensions, macromolecular solutions and multicomponent systems \\cite{clague1996hindered,miguel2003deblocking,cui2004anomalous,Sbragaglia2014,stannard2011dewetting}. \nBecause of the long-range hydrodynamic interaction, the dynamical behavior of these systems is correlated via flows.\n\n\nThe generalized elastic model (GEM), proposed in Ref. \\cite{Taloniprl}, is a suitable linear model that may capture the essence of criticality and phase transition (see \\cite{Talonipre,Taloniepl,Taloniperturb,Talonirev} for more details). In this case, we used this model in the presence of a quenched disorder. The quenched form of the generalized elastic model (qGEM) is represented by the stochastic linear integrodifferential equation shown below\n\\begin{eqnarray}\n\\label{qGEM}\n\\frac{\\partial u(\\mathbf{x},t)}{\\partial t} =F +\n\\int d^d x^\\prime \\Lambda(\\vert \\mathbf{x}-\\mathbf{x}^\\prime\\vert)\\frac{\\partial^z}\n{\\partial |\\mathbf{x}^\\prime | ^z}u(\\mathbf{x}^\\prime,t)\\nonumber \\\\\n+f_p(\\mathbf{x},u(\\mathbf{x},t))\n~,\n\\end{eqnarray}\nwhere the dynamical variables of the system $u(\\mathbf{x},t)$ describes \nan elastic interface driven through a disordered media. \n$F$ is the driving force on the interface and\n$f_p$ represents the quenched pinning forces\nwhich its distribution can be chosen Gaussian with the\nfirst two moments, $\\langle f_p (\\mathbf{x},u)\\rangle = 0$ and\n$\\langle f_p(\\mathbf{x},u) f_p(\\mathbf{x}^\\prime,u^\\prime)\\rangle \\propto \\delta (\\mathbf{x}-\\mathbf{x}^\\prime)\\delta(u-u^\\prime) $.\nThe hydrodynamic interaction term $\\Lambda(\\vert \\mathbf{x}-\n\\mathbf{x}^\\prime\\vert)$, corresponds to the non-local coupling of \ndifferent sites $\\mathbf{x}$ and $\\mathbf{x}'$.\nHere, $\\partial^z\/\\partial|\\mathbf{x}|^z$ is the multidimensional\nRiesz-Feller fractional derivative operator, which is \ndefined via its\nFourier transform \n$\\mathcal{F}\\left\\{\\frac{\\partial^z}{\\partial|\\mathbf{x}|^z}\\Phi (\\textbf{x})\\right\\}\n\\equiv-|\\mathbf{k}|^z \\Phi (\\textbf{k})$, \nimmediately implies that the Riesz-Feller fractional derivative has \nthe same meaning as the \nfractional Laplacian operator $\\partial^z\/\\partial|\\mathbf{x}|^z:=-\\left( - \n\\Laplace \\right) ^{z\/2}$ \\cite{fractiobaloperators}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=7cm,clip]{vel_z_1_alpha_05.eps}\n\\includegraphics[width=7cm,clip]{vel_z_3_alpha_05.eps}\n\\includegraphics[width=7cm,clip]{vel_z_4_alpha_05.eps}\n\\end{center}\n\\caption{(Color online)The numerical evaluation for the average velocity $v(t,F) = \\frac{d}{dt} \\langle \\int u(x,t)dx \\rangle$ for the generalized elastic model with quenched disorder which corresponds to the order parameter of the pinning-depinning transition of the visco-elastic interface driven through a disordered media. The behavior of the order parameter is strongly influenced by the hydrodynamic interaction parameter $\\alpha$ and the fractional power $z$. Top: The order parameter $v(t)$ as a function of time for three different values of the external force $F$. It goes to zero, when $t\\to \\infty$. The saturation values of the order parameter $v(F)$ is shown in the inset. Note that there is no phase transition for the values $\\alpha = 0.5$ and $z=1.0$ in Eq. (\\ref{qGEM}). Middle: The order parameter as a function of time and force for the so called $1$st order phase transition for the values $\\alpha = 0.5$ and $z=3.0$ Bottom: The same analysis for the values $\\alpha = 0.5$ and $z=4.0$ shows an ordinary pinning-depinnig phase transition. Solid red line corresponds to the scaling relation $v(t) \\sim t^{-\\sigma}$ for the critical point $F=F_c$. \n}\n\\label{Figure:1}\n\\end{figure} \n\n\nAt this point a specification\nof the hydrodynamic interaction kernel $\\Lambda (\\vec{r})$ is called for. \nFor no fluid-mediated interactions, one may suppose that the friction kernel is local, $\\Lambda (\\vec{r})=\\delta (|\\vec{r}|)$ (examples). \nFor systems having non-local interactions, such as membranes, polymers, or viscoelastic surfaces, where the hydrodynamic interactions take on a long-range power-law form, a different scenario \n\\begin{eqnarray}\\label{non local hydrodynamics kernel}\n\\Lambda (\\vec{r}) \\sim \\frac{1}{|\\vec{r}|^{\\alpha}},\n\\end{eqnarray}\noccurs (where $\\frac{D-1}{2}<\\alpha\\alpha\/2$, where $C_{\\alpha,\\frac{\\alpha}{2}+n}$ are binomial coefficients \\cite{zoia2007fractional}. \n \nBringing together Eqs. (\\ref{time finite difference}) and (\\ref{HOLR})\nand substitution into the Eq. (\\ref{qGEM}) leads to the discrete version of \nthe qGEM.\nWe employ the finite difference method to investigate the numerical \ndiscretization of Eq.(\\ref{qGEM}), in the form\n\\begin{eqnarray}\\label{discretqGEM}\nu_i^{n+1} =u_i^n &+ \\Delta t \\lbrace \\frac{1}{(\\Delta x)^z} \\sum_{j= 0}^{L}\\sum_{k= 0}^{L} \\Lambda (\\vert i-j\\vert)\\mathbb{K}_{j,k}u_k^n \\nonumber \\\\\n& +F + f_p(x_i,u_i^n)\\rbrace\n ~, \n\\end{eqnarray}\nwhere $u_i^n$ approximates the interface profile $u(x_i , t_n )$ at the $i$th lattice point and the $n$th time step. The lattice constant $\\Delta x$ has been set equal \nto one and the grid steps $\\Delta t$ in time was chosen small enough to \navoid numerical instabilities. \n\nIn order to numerically generate quenched random field \n $f_p(x_i,u_i^n)$, without loss of generality we assumed the continuous stochastic variables $u(x_i , t_n )$ are discretized into a finite numbers of integer values $[u_i^n\/\\epsilon]$ where $\\epsilon \\ll 1$ is an arbitrary small parameter and $[\\dots]$ represents the bracket notation for the integer part of a given continuous variable.\nThen the quenched random field $f_p$ is defined on a square array where each cell $[i,h]$ ($1\\leqslant i \\leqslant L$ and $h =[u_i^n\/\\epsilon ]$) is assigned an identically distributed random variables $\\eta(i,h)$ with normal Gaussian\ndistribution with zero mean and unit variance. The random disorder $f_p(x_i,u_i^n)$ is obtained by the linear interpolation of the random\nforce between two random variables $\\eta(i,h)$ and $\\eta(i,h+1)$ where $h = [u_i^n\/\\epsilon]$.\n\nThe numerical investigation of the scaling characteristics and critical exponents of the quenched generalized elastic model for different values of the fractional order $z$ and the non-local hydrodynamic interaction power $\\alpha$ is presented in detail in the next section.\n\n\n\n\\section{Numerical results}\\label{Numerical results sec}\nTo determine the time evolution of the interface specified by $u(x,t)$\nand to obtain the critical properties of the qGEM, \nthe simulation is started with initial condition $u(x,0) = 0$ , and boundary condition $u(x,t) = u(x+L,t)$.\nWe simulated this model on a lattice of size $L = 256$. In addition, we carefully\nchoose the time increment $\\Delta t$ small enough to\nensure the stability of the numerical algorithm.\n\nIn order to determine the criticality of the qGEM (\\ref{qGEM}) and (\\ref{discretqGEM}) for various parameter\nvalues of the fractional order $z$ and the hydrodynamic interaction parameter $\\alpha$, we first compute the average velocity\n$v(t,F) = \\frac{d}{dt} \\langle \\int u(x,t)dx \\rangle$ as a\nfunction of time for various values of the external homogeneous force $F$. \n\n\nSurprisingly,\nour simulations indicate that, the qGEM model in the limit $t\\rightarrow \\infty$ exhibits three quite different behaviours depending on the values of $z$ and $\\alpha$. \nWhen hydrodynamic interactions is strongly long-range $\\alpha \\ll 1$ and the fractional power $z\\ll 4$, there exists\nno phase transition between a pinned phase and a moving phase. In this regime $\\lim _{t\\rightarrow\\infty} v(t,F)=0$ for an arbitrary external driving force $F$. Such a behaviour is shown on the top panel of Fig. (\\ref{Figure:1}) for $\\alpha =0.5$ and $z = 1.0$. \nIn the opposite limit when the parameters $\\alpha \\leq 1 $ and $z \\gg 1 $ the velocity of the interface remains zero (\\textit{pinned} phase) up to a critical force $F_c$ and above $F_c$ \nthe velocity $v(t)$ decreases\nas a power-law at the beginning and then becomes constant at all later time \\textit{i.e.} $\\lim _{t\\rightarrow\\infty} \\frac{d}{dt} v(t,F)=0$ (\\textit{moving} phase). \nAs indicated in the bottom panel Fig. (\\ref{Figure:1}), $v(F)$ is a continuous function of $F$. Thus the transition,\nlooks similar to the continuous phase transition in the context of the critical phenomena. \nAnother surprising features of the qGEM model is the anomalous pinning-depinning transition for some specific values of the parameters $\\alpha$ and $z$ in the ($\\alpha$,$z$) plane. In the anomalous regime, an elastic interface which exhibits non-trivial phase transition behavior is pinned when $F F_c$ we observe a jump in the average velocity as a function of $F$ (see Fig. (\\ref{Figure:1})), may lead to a first order\nphase transition in which the order parameter of the system changes discontinualy from zero to a finite value. Note that above $F_c$ the average velocity varies with time $\\lim _{t\\rightarrow\\infty} \\frac{d}{dt} v(t,F)\\neq 0$, which is noticeably\n different from a standard pinning-depinning phase transition appears in\n the elastic interface models. In Fig. (\\ref{Figure:2}) one may see a so-called phase diagram calculated\nfor the generalized elastic model with quenched disorder. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8cm,clip]{phase_diagram_1.eps}\n\\end{center}\n\\caption{(Color online)The phase diagram of the generalized elastic model with \nquenched disorder. There are three different regimes, depending on the values of\nthe parameters $z$ and $\\alpha$ in Eq. (\\ref{qGEM}):\nThe first regime is when $z\\ll 4$ and $\\alpha<1$ where there is no phase\ntransition between pinned and moving phases. The second regime is when \n$z < 4$ and $\\alpha \\gg 0$ where the order parameter of the system $v(F)$ \nas a function of the control parameter $F$ changes\ncontinuously from zero to the non-zero values ($2nd$ order phase transition).\nIn the third regime the mean velocity $v(F)$ \nas a function of $F$ changes\ndiscontinuously from zero to the non-zero values ($1st$ order phase transition). \n}\n\\label{Figure:2}\n\\end{figure} \n \n \nWe here focus on one aspect of the\nproblem namely the scaling\nbehavior with characteristic critical exponents of the qGEM close to the depinning critical point (This part is incomplete and we will put our results here very soon). \n\n\n\\section{Conclusions}\\label{Conclusion sec}\nIn this paper we have studied the the depinning transition of the elastic interface with non-local hydrodynamic interactions. As we mentioned earlier this model is called generalized elastic model in the presence of quenched disorder. We numerically studied different aspects of this model for different values of the fractional order $z$ and the non-local hydrodynamic interaction power $\\alpha$.\nWe found that the behaviour of order parameter $v(F)$ as function of the external force $F$ highly depends on the values of $z$ and $\\alpha$. There are three distinct phases in the phase space $(z-\\alpha)$. In the small values of $z$ and $alpha$ the order parameter vanishes and in the thermodynamic limit the steady-state order parameter approaches zero.\nIn opposite limit, where $\\alpha \\sim 1$ and $z>>1$, the model exhibits second-order phase transition and the order parameter $v(F)$ continuously changes from zero to none-zero values. And finally there is an additional phase with the order parameter changes discontinuously changes from zero to non-zero values, which is characterized by a first-order phase transition. We have analysed in detail the steady state of the model\nin the second-order phase transition regime. Our model displays naturally scaling features near critical point $F_c$. We measured different scaling exponents as functions of $z$ and $\\alpha$. Our results are in a good agreements with the well-known models. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe Sun is powered by nuclear fusion reactions occurring in its core~\\cite{Bethe:1939bt,Bahcall:2004pz,Bahcall:2000nu,Antonelli:2012qu,Davis:2016hil}. This involves hydrogen and helium predominantly, but also heavier elements such as lithium, carbon, nitrogen, oxygen, beryllium and boron. Each heavier element is produced through nuclear fusion from the lighter ones in a chain-reaction, with some steps in the chain releasing neutrinos of a characteristic energy spectrum. There are two main chains which convert hydrogen to helium in stars: the proton-proton chain and the carbon-nitrogen-oxygen (CNO) cycle~\\cite{Bethe:1939bt}, both of which lead to neutrino production.\nFor the proton-proton chain there are five neutrino components (pp, $^7$Be, $^8$B, pep and hep) each with a different spectrum, while for the CNO cycle there are three ($^{13}$N, $^{15}$O and $^{17}$F), and we refer to the sum of the latter as the CNO neutrinos.\n\nTheoretical predictions for the fluxes of these components depend on solar models, which themselves depend on various inputs such as the abundance of heavier ``metal'' elements (specifically all elements heavier than $^4$He, for example $^{12}$C, $^{13}$N and $^{15}$O) i.e. the metallicity. Solar models based on abundances that were inferred from earlier observations and modelling of the photosphere were also in excellent agreement with helioseismological observations (e.g. GS98 \\cite{Basu1997,Basu2004}). However, advances in photosphere and line-formation modelling led to a downward revision of most of the abundances of elements heavier than helium, leading to the more recent ``low-metallicitly'' models~\\cite{Asplund:2009fu}, now incompatible with helioseismological data~\\cite{2009ASPC..416..193B,Vinyoles:2017bqj}. This disagreement is known as the solar metallicity, or solar abundance problem \\cite{Bahcall:2004yr, Bahcall06, Yang07, Basu08, Serenelli:2009yc}, and its solution will require additional, independent data (it may also be a sign of new physics, see e.g. ref.~\\cite{Frandsen:2010yj,Vincent:2014jia}). In particular, the CNO neutrino flux is very sensitive to the metallicity of the solar core, as can be seen in Table~\\ref{cno_flux_table}, and hence a precise measurement of this flux would help in improving solar models, by providing another metallicity measurement against which they can be tested~\\cite{Song:2017kvf,Bergstrom:2016cbh}. \n\n\nWhile the components of the proton-proton chain have all been measured (pp~\\cite{Bellini:2014uqa}, pep~\\cite{Collaboration:2011nga}, $^7$Be~\\cite{Bellini:2013lnn}, $^8$B~\\cite{Agostini:2017cav,Aharmim:2011vm,Abe:2016nxk}, hep at $1\\sigma$~\\cite{Bergstrom:2016cbh,Aharmim:2006wq}), a measurement of the CNO neutrino flux has not yet been achieved~\\cite{Bergstrom:2016cbh}. This is primarily because CNO neutrinos have neither a high energy, like $^8$B neutrinos, nor a huge flux, like pp neutrinos, but instead form a sub-dominant component of the total solar neutrino spectrum at energies below approximately $1.5$~MeV.\nFurthermore, due to effects from the finite energy-resolution and the adopted detection reaction of experiments, the observed spectrum in a detector resulting from CNO neutrinos is expected to resemble strongly that from pep neutrinos, leading to systematic uncertainties in determining the CNO flux even for a background-free experiment.\n\nMany current and future experiments will attempt to measure the CNO flux by looking for neutrinos scattering on electrons, such as Borexino~\\cite{Bellini:2013lnn}, SNO+~\\cite{Andringa:2015tza}, the Jinping Neutrino Experiment~\\cite{JinpingNeutrinoExperimentgroup:2016nol} and liquid argon experiments such as DarkSide and Argo~\\cite{Aalseth:2017fik,Franco:2015pha}. It should also be possible to look for CNO neutrinos through scattering with nuclei, in a similar manner to direct searches for dark matter. However this would likely require new technologies to achieve the required low energy threshold~\\cite{Strigari:2016ztv}, for example refs.~\\cite{Strauss:2017cam,Angloher:2017sxg,Maris:2017xvi,Petricca:2017zdp,Agnese:2017jvy,Schutz:2016tid,Budnik:2017sbu,Aguilar-Arevalo:2016ndq,Arnaud:2017bjh,Agnese:2016cpb,Yang:2017yaw,Arnaud:2017usi}.\n\nIn this work we determine the accuracy to which these future experiments can measure the CNO neutrino flux, and whether this is enough to distinguish between the two solar metallicity scenarios.\nFor experiments looking for electron-recoils from solar neutrinos (e.g. Borexino, Argo and SNO+), some individual projections have been carried out by the experimental collaborations themselves for their respective experiments~\\cite{Bellini:2013lnn,Andringa:2015tza,Franco:2015pha}. \nThe purpose of this work is not to refute these estimates, but rather to gather the predictions in the same place and to compare as closely as possible the discovery possibilities from a theory perspective. \n\nOur main aim is to obtain a time-scale for when the CNO flux will be measured. Furthermore, as we will see, it is difficult but not impossible for nuclear recoil experiments to see this flux. By establishing a time-frame we hope to provide crucial input for experimentalists who plan to run a low-threshold nuclear-recoil experiment, which may also be designed to search for light dark matter.\n\n\n\n\n\n\n\\begin{table}[bt]\n\\centering\n \\renewcommand{\\arraystretch}{1.4}\n\\begin{tabular}{ p{3cm} || c | c | c }\n\\multirow{2}{3cm}{\\textbf{Solar Metallicity}} & \\multicolumn{3}{c }{\\textbf{CNO Neutrino Flux [cm$^{-2}$ s$^{-1}$] }} \\\\ \n\\cline{2-4}\n& $^{13}$N [$10^8$] & $^{15}$O [$10^8$] & $^{17}$F [$10^6$] \\\\ \\hline\nHigh & 2.78 $\\pm$ 0.42 & 2.05 $\\pm$ 0.35 & 5.92 $\\pm$ 1.06 \\\\ \\hline\nLow & 2.04 $\\pm$ 0.29 & 1.44 $\\pm$ 0.23 & 3.26 $\\pm$ 0.59 \\\\\n\\hline\n\\end{tabular}\n\\caption{Predicted fluxes of each component of the total neutrino emission from the CNO cycle, used in this work, for the high and low solar metallicity models (GS98 and AGSS09met respectively)~\\cite{Vinyoles:2017bqj,Basu1997,Basu2004}.}\n\\label{cno_flux_table}\n\\end{table}\n\n\n\n\\section{Electron-recoil experiments searching for CNO neutrinos}\nExperiments searching for electronic recoils induced by interactions with neutrinos will be sensitive to CNO neutrinos, provided their low-energy threshold is below approximately $1.5$~MeV. Here we consider only elastic scattering i.e. $\\nu_e + e^- \\rightarrow \\nu_e + e^-$, and not inelastic scattering as will be seen for example in the DUNE experiment~\\cite{Acciarri:2015uup}.\n\nFor these experiments there exist various backgrounds whose spectra are similar to that expected from CNO neutrino-induced electronic recoils, and which are often specific to the detector or target. In some cases, the rates of these backgrounds can be determined externally, for example through measurements of the decay rate of daughter nuclei in a radioactive decay chain. It is vital that not only should these background rates be kept as low as possible, to reduce statistical uncertainties, but also ideally that they are known \\emph{a priori} to keep systematic uncertainties small.\n\n\nThe aim of our analysis is to quantify the precision with which the CNO flux can be measured for up-coming experimental runs, given the uncertainties on the other solar neutrino fluxes and backgrounds in the energy region of interest. Of particular importance are the systematic uncertainties on determining the CNO neutrino flux, which arise through degeneracies between the CNO flux and both the other neutrino fluxes and the detector-specific background rates. Motivated by this fact, we perform a Markov Chain Monte Carlo (MCMC) parameter scan over all of the neutrino fluxes and background rates, by comparing the spectra of these sources to simulated data with a Poisson likelihood.\nEach relevant background rate and neutrino flux has one parameter which determines its total energy-integrated rate, and the spectra are kept fixed up to this normalisation i.e. we fit spectra, but vary only the total rates for backgrounds or fluxes for solar neutrinos.\n\n\n\n\nThe MCMC analysis requires that each parameter has with it an associated prior distribution, which reflects the amount of knowledge we have about this parameter before the analysis is performed. For each solar neutrino flux we assume that we have no knowledge and therefore assume a uniform (or flat) prior. In practice all uniform priors are constant between zero and a value much greater than the fiducial value used to generate the simulated data. We will also use a Gaussian prior for some backgrounds when their rates are known to a given precision.\n\nAfter the analysis is complete the MCMC returns sampled values of each of the parameters, whose histogram is the posterior distribution, which tells us which values of the various parameters fit best to the simulated data. The posterior has a number of dimensions equal to the number of free parameters, and so in order to find the precision with which the CNO flux can be measured we marginalise the posterior over all other parameters.\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hist_fiducial_Borexino.pdf}\n\\caption{Differential electron recoil event rate from CNO neutrinos (solid black) compared with various backgrounds for Borexino~\\cite{Agostini:2017aaa,Collaboration:2011nga,Bellini:2013lnn}. }\n\\label{fig:spec_borexino}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_borexino_ny_10_0_lab_gran_sassofixed_C11_flag_1_0_v3.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_borexino_ny_10_0_lab_gran_sassovery_fixed_C11_flag_1_0_v3.pdf}\n\\caption{Two-dimensional marginalised posterior distributions for Borexino running for 10~years with the pessimistic prior set, assuming $10\\%$ uncertainty on the $^{210}$Bi rate (top) or the optimistic prior set assuming $1\\%$ uncertainty on the $^{210}$Bi rate (bottom).The dashed black lines show the fiducial values used to generate simulated data. In each panel the CNO flux is compared to different background rates and other solar neutrino fluxes, where the red contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$.}\n\\label{fig:borexino_mcmc}\n\\end{figure*}\n\nIn the rest of this section we detail, for each experiment, the detector-specific backgrounds and their priors, and the results of our MCMC analysis on the precision with which the CNO neutrino flux can be measured. We consider Borexino, SNO+ and dual-phase liquid argon experiments such as DarkSide or Argo. We do not perform a dedicated analysis for the Jinping Neutrino Experiment. However its projected sensitivity should be similar to that of SNO+, corrected for the different target mass~\\cite{JinpingNeutrinoExperimentgroup:2016nol}.\n\n\n\\subsection{Borexino}\n\nBorexino is a liquid scintillator experiment situated in the Gran Sasso laboratory, with a target mass of 278 tons and a fiducial mass in the most recent run of 71.3 tons~\\cite{Bellini:2013lnn,Agostini:2017ixy}. Neutrinos are detected by observing scintillation light from the electrons, which have been given MeV-levels of energy though elastic scattering.\nIn their previous Phase-I run, the Borexino collaboration did not observe CNO neutrinos and so set a limit on the CNO interaction rate of $R_{\\mathrm{CNO}} < 7.9~[100 \\, \\mathrm{ton} \\, \\mathrm{day}]^{-1}$ at $95\\%$ confidence~\\cite{Bellini:2013lnn}, with a similar limit set in Phase-II~\\cite{Agostini:2017ixy}. Here we consider a future run of Borexino, taking into account various improvements as detailed below.\nAs the only running experiment considered in this work, it will have the most realistic projections for measuring the CNO flux, since for example its backgrounds and energy-resolution are well-understood already.\n\n\nAs shown in figure~\\ref{fig:spec_borexino}, apart from the other solar neutrino fluxes ($^7$Be and pep) the largest backgrounds in the region of interest for a CNO neutrino search by Borexino arise from $^{210}$Bi beta-decays (originating from the slow decay of $^{210}$Pb) and $^{11}$C decay to positrons~\\cite{Agostini:2017ixy,Agostini:2017aaa,Collaboration:2011nga}. The former is particularly troublesome as its spectrum follows closely that expected from the CNO neutrinos, and so any measurement of the CNO neutrino rate will be partially degenerate with the rate of $^{210}$Bi beta-decay electrons. \nFor Borexino, it has been suggested that the magnitude of this $^{210}$Bi background can be inferred to an accuracy of $10\\%$ (i.e. around $6.4$ counts per ton per year) or better through measurements of the alpha-decay of its daughter nucleus $^{210}$Po~\\cite{Villante:2011zh}. Recent upgrades to the Borexino experiment, to establish secular equilibrium in the $^{210}$Pb decay chain, mean that such a measurement is now possible~\\cite{calaprice_talk_borexino}.\nThe $^{11}$C positron background arises from cosmogenic muons interacting with $^{12}$C nuclei. Its contamination in the region of interest for a CNO search is reduced by an order of magnitude using ``three-fold'' coincidence cuts on the cosmogenic muons and neutrons which are commonly produced with the $^{11}$C nuclei~\\cite{Agostini:2017ixy,Agostini:2017aaa,Collaboration:2011nga}. \n\n\n\n\\begin{table}[tb]\n\\centering\n \\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ p{2.5cm} || c | c | c | c }\n\\multirow{3}{3cm}{\\textbf{Background}} & \\multicolumn{2}{c |}{\\textbf{Optimistic}} & \\multicolumn{2}{c}{\\textbf{Pessimistic}} \\\\ \n\\cline{2-5}\n& Value & Error & Value & Error \\\\ \n& [(ton yr)$^{-1}$] & [1$\\sigma$] & [(ton yr)$^{-1}$] & [1$\\sigma$] \\\\ \\hline\n$^{210}$Bi& 64 & $1\\%$ & 64 & $10\\%$ \\\\\n\\hline\n$^{210}$Po& 950 & Free & 950 & Free \\\\ \\hline\n$^{11}$C& 9.49 & Free & 9.49 & Free \\\\ \n\\hline\n\\end{tabular}\n\\caption{Fiducial values and relative uncertainties for the backgrounds in Borexino based on Phase-II data~\\cite{Agostini:2017ixy}. We provide either the one-sigma error on the \nGaussian prior, with central value equal to the fiducial value, or allow ``Free'' errors, meaning the prior distribution is uniform and unconstrained.}\n\\label{borexino_priors}\n\\end{table}\n\nIn order to understand the projected sensitivity of Borexino to CNO neutrinos in the future, and the effect of this technique to measure the $^{210}$Bi background, we perform two MCMC runs each with a different prior distribution for the $^{210}$Bi rate. In the first case, we assume that the method proposed in ref.~\\cite{Villante:2011zh} allows the $^{210}$Bi background rate to be measured to a precision of $10\\%$ of the total rate (i.e. around $6.4$ counts per ton per year), and so the $^{210}$Bi rate is given a Gaussian prior with this value as its one-sigma standard deviation. While in the second case we assume an optimistic scenario in which the $^{210}$Bi rate will be measured to $1\\%$ accuracy (i.e. around $0.64$ counts per ton per year). Our different priors and background rates are summarized in Table~\\ref{borexino_priors}, based on the Phase-II\\footnote{We use Phase-II as this is the most recent published Borexino data, though note that the backgrounds in the next phase of Borexino will likely be smaller due to the slow decay of $^{210}$Pb and its daughters.} run in ref.~\\cite{Agostini:2017ixy}, which we also use to obtain the spectra of the background components. \n\n\n\n\n\n\n\nFigure~\\ref{fig:borexino_mcmc} shows a comparison of the posterior distributions from our MCMC projection for Borexino with either the optimistic or pessimistic prior sets, assuming $10$~years of data-taking. Each panel is a two-dimensional projection of the total posterior distribution i.e. the posterior distribution summed over all free parameters except those on the $x$ and $y$ axes. This allows us to see degeneracies between the CNO flux and the different backgrounds and other solar neutrino spectra. For example, the left-most panel on each plot shows that the CNO flux measurement is strongly degenerate with the $^{210}$Bi background, as expected since they have similar spectra. The precision on the CNO flux measurement is then obtained by marginalising over the other parameters, for example by summer over the $x$-axis in any of the two-dimensional panels, and so projecting onto the $y$-axis.\n\nFor the pessimistic priors there is a wide range of pairs of values for the $^{210}$Bi rate and CNO flux which provide a good fit to the data, which means that we have only weak constraints on the CNO flux, since it can be compensated by changing the $^{210}$Bi rate. However, with the optimistic prior set values of the $^{210}$Bi rate close to its fiducial value are strongly preferred, since we know those values further away can not be physical as we assume the $^{210}$Bi rate has been measured to $1\\%$ accuracy. Hence the degeneracy is broken and the allowed range of CNO flux values is much smaller. Importantly for Borexino this degeneracy is very strong, which is why the effect of the optimistic priors is so clear, indicating that the ability of Borexino to measure the CNO flux will be extremely sensitive to the precision with which the $^{210}$Bi rate is measured, as well as to the total rate of this background.\n\nSimilarly, in the right-most panel we show the degeneracy between the CNO and pep neutrinos fluxes, resulting from their similar elastic-scattering spectra. This significantly degrades the ability of Borexino, and indeed all neutrino detectors, to measure the CNO flux. For example, if we knew the pep neutrino flux precisely then the CNO flux could be constrained to be within the region where the vertical dashed line in the right-most panel intersects with the shaded regions, projected onto the $y$-axis, but unfortunately disentangling these two neutrinos fluxes is not possible. \n\n\\subsection{SNO+}\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hist_fiducial_SNO.pdf}\n\\caption{The spectrum of electronic recoil events from CNO neutrinos (solid black) compared with various backgrounds expected for SNO+~\\cite{Andringa:2015tza}. }\n\\label{fig:spec_snoplus}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_sno_plus_ny_0_5_lab_sno_lab_priors_fixed_realistic.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_sno_plus_ny_3_0_lab_sno_lab_priors_fixed_realistic.pdf}\n\\caption{Marginalised posteriors for SNO+ with 6 months (top) or 3 years (bottom) of data and pessimistic priors. A diagonal shaped distribution means the parameters have some degeneracy between them. The dashed black lines show the fiducial values used to generate simulated data. In each panel the CNO flux is compared to different background rates and other solar neutrino fluxes, where the red contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$. Note that all total rates are calculated above an energy of $0.3$~MeV.}\n\\label{fig:sno_plus}\n\\end{figure*}\n\nSNO+ is an upcoming liquid scintillator experiment situated at SNOLAB, with a 780 ton target mass~\\cite{Andringa:2015tza}. \nThe expected neutrino signals and backgrounds are shown in figure~\\ref{fig:spec_snoplus}, where following ref.~\\cite{Andringa:2015tza} the background rates are based on those already achieved in Borexino. Due to its location within a deeper site compared with Borexino, its cosmogenic backgrounds such as $^{11}$C are smaller. The primary purpose of the SNO+ experiment is to measure neutrino-less double-beta decay using $^{130}$Te loaded into the liquid scintillator~\\cite{Andringa:2015tza}. Due to this, the SNO+ detector may only be sensitive to CNO neutrinos for a fraction of its total running time, since this $^{130}$Te beta-decay leads to a large background in the CNO neutrino energy range. Hence in this work we consider only SNO+ in its pure scintillator mode i.e. without $^{130}$Te doping.\n\nAs with Borexino, and shown in figure~\\ref{fig:spec_snoplus}, the SNO+ detector will have a difficult background from $^{210}$Bi, although it should have better energy-resolution making this background easier to separate from the CNO neutrino spectrum~\\cite{Andringa:2015tza}. It is also expected to have backgrounds originating from the thorium and uranium decay chains and $^{85}$Kr, $^{210}$Po and $^{11}$C. As detailed in ref.~\\cite{Andringa:2015tza}, each of these backgrounds can be measured to some extent using coincidence decays and tagging of daughter nuclei. For $^{210}$Po, $\\alpha$-tagging will be used to reduce this background by $95\\%$. Hence we adopt Gaussian priors on these background rates which reflect the expected precision to which these backgrounds will be determined, as shown in table~\\ref{sno_plus_priors}. We perform two runs, an optimistic and pessimistic scenario, where in the former case the $^{210}$Bi background can be measured to an accuracy of $1\\%$, while in the latter case it will be measured to $10\\%$ precision, both employing the method proposed in ref.~\\cite{Villante:2011zh}.\nIt remains to be seen\nwhether the SNO+ background projections\nare actually achievable in practice.\n\n\\begin{table}[t]\n\\centering\n \\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ p{3cm} || c | c | c | c }\n\\multirow{3}{3cm}{\\textbf{Background}} & \\multicolumn{2}{c |}{\\textbf{Optimistic}} & \\multicolumn{2}{c}{\\textbf{Pessimistic}} \\\\ \n\\cline{2-5}\n& Value & Error & Value & Error \\\\ \n& [(ton yr)$^{-1}$] &[1$\\sigma$] & [(ton yr)$^{-1}$] & [1$\\sigma$] \\\\ \\hline\n$^{210}$Bi& 45.4 & $1\\%$ & 45.4 & $10\\%$ \\\\\n\\hline\n$^{210}$Po& 1530 & $20\\%$ & 1530 & $20\\%$ \\\\ \\hline\n$^{11}$C& 1.74 & Free & 1.74 & Free \\\\ \\hline\n$^{85}$Kr& 96.4 & $50\\%$ & 96.4 & $50\\%$ \\\\ \\hline\nU chain& 74.4 & $7\\%$ & 74.4 & $7\\%$ \\\\ \\hline\nTh chain& 11.1 & $25\\%$ & 11.1 & $25\\%$ \\\\ \n\\hline\n\\end{tabular}\n\\caption{Fiducial values and relative uncertainties for the backgrounds in SNO+ above a threshold energy of $0.3$~MeV (without $^{130}$Te doping)~\\cite{Andringa:2015tza}. A percentage uncertainty means the one-sigma error on the \nGaussian prior, with central value equal to the fiducial value. Columns with ``Free'' errors mean the prior distribution is uniform\ni.e. unconstrained.}\n\\label{sno_plus_priors}\n\\end{table}\n\n\n\n\nFigure~\\ref{fig:sno_plus} shows the results of our MCMC analysis for SNO+, assuming a $50\\%$ fiducial mass cut. There is a significant systematic uncertainty on the CNO flux arising from the $^{210}$Bi background, as can be seen by the degeneracy between these two parameters in the left-most panel. The uncertainty on the CNO flux alone is then obtained by marginalising over the $^{210}$Bi rate, and all of the other parameters. We have made the assumption that the $^{210}$Bi background rate can be measured to an accuracy of $10\\%$ i.e. the pessimistic priors, which makes the CNO-$^{210}$Bi degeneracy clear. With six months of data, the degeneracy is such that a zero CNO flux can not be excluded with high significance. \n\n\nThe improvement on the CNO flux measurement going from 6 months to 3 years of SNO+ data is large. The reason for this is subtle, as more data also helps relieve the partial degeneracy between both $^{210}$Bi and pep and CNO, especially when the tails of their spectra can be measured precisely. This can be seen in the left-most panel of each plot, where the best-fit region between CNO and $^{210}$Bi rotates towards the horizontal, meaning the degeneracy between these two parameters has been reduced. This is because the increased statistics means that the spectra of the CNO and $^{210}$Bi components, as shown in figure~\\ref{fig:spec_snoplus}, can be more easily distinguished. Hence an increased amount of data has not only reduced statistical uncertainties, but also systematics .\nThere is little degeneracy between the CNO flux and the $^{11}$C or U-chain backgrounds, since their spectra are not expected to be similar to that from CNO. though there is a small amount of degeneracy for the 3~year run between the Th-chain and CNO.\n\n\n\n\\subsection{Liquid argon TPCs}\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hist_fiducial_argon.pdf}\n\\caption{The spectrum of electronic recoil events from CNO neutrinos (solid black) compared with various backgrounds expected for a liquid argon experiment operating in the Gran Sasso lab~\\cite{Franco:2015pha}. The $^{222}$Rn contamination is assumed to be $10$~$\\mu$Bq per $100$~ton. }\n\\label{fig:spec_argon_gs}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_gran_sasso_priors_free.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_gran_sasso_priors_fixed_radon.pdf} \\\\\n\\caption{Contours showing the degeneracy between the CNO flux and various backgrounds and other solar neutrino fluxes for a liquid argon electronic-recoil experiment running in the Gran Sasso lab, with an exposure of 1000 ton-years with an unknown radon background (top) or a radon background known to a precision of $10\\%$ at one-sigma (bottom). The dashed black lines show the fiducial values used to generate simulated data. In each panel the red contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$.}\n\\label{fig:argon_GS_1000tonyr}\n\\end{figure*}\n\n\n\nIn addition to the currently-running DEAP-3600~\\cite{Amaudruz:2017ekt} there exist several upcoming or proposed experiments, such as DarkSide-20k (with a fiducial mass of 20~tons) and Argo (with a fiducial mass of 100~tons)~\\cite{Aalseth:2017fik,Franco:2015pha}, based on a dual-phase argon\\footnote{We do not consider liquid xenon experiments for an electronic recoil CNO neutrino search due to their large backgrounds, especially from $^{136}$Xe double-beta decay~\\cite{Baudis:2013qla}.} time-projection chamber (TPC) set-up which will look for both dark matter, primarily through nuclear-recoils, and neutrinos~\\cite{Aalseth:2017fik}. Here we take a more general view and will not look at a specific experiment in detail, with the motivation of determining the best experimental set-up to maximize sensitivity to CNO neutrinos for an argon experiment. Additionally, although we know that DarkSide-20k will be housed in the Gran Sasso lab~\\cite{Aalseth:2017fik}, we do not know if this will be the case for Argo, and so we will consider also the case where Argo is housed in SNOLAB or Jinping~\\cite{JinpingNeutrinoExperimentgroup:2016nol}. \n\nThe analysis of ref.~\\cite{Franco:2015pha} identified two major sources of backgrounds relevant to a search for CNO neutrinos using electronic recoils in a liquid argon experiment: cosmogenic backgrounds and $^{222}$Rn from the detector environment. The background from $^{222}$Rn comes from $\\beta$-decay electrons emitted by $^{214}$Pb and $^{214}$Bi, two of the daughters of $^{222}$Rn $\\alpha$-decay. In this work we use the background spectra calculated in ref.~\\cite{Franco:2015pha}, which we show in figure~\\ref{fig:spec_argon_gs}. \nAs can be seen from\nthe figure, the excellent energy resolution\nmakes the pep spectrum easy to distinguish\nfrom CNO.\n\nThe size of the background from the $^{222}$Rn decay chain depends on the specific detector set-up of the argon experiment, and needs to be smaller than $\\sim 100$~$\\mu$Bq per $100$~ton to make a measurement of the CNO flux achievable~\\cite{Franco:2015pha}. The DEAP-3600 experiment had a $^{222}$Rn contamination level of $(1.8 \\pm 0.2) \\cdot 10^4$~$\\mu$Bq per $100$~ton in its most recent run~\\cite{Amaudruz:2017ekt}, however a larger experiment such as DarkSide-20k or Argo should have a smaller contamination by volume, as the $^{222}$Rn enters through the walls of the argon tank and scales less strongly with increasing experiment size than the total target mass. Throughout this work we assume a $^{222}$Rn contamination of $10$~$\\mu$Bq per $100$~ton.\nThe magnitude of the radon background can be measured through observations of the delayed coincidence between $^{214}$Bi $\\beta$-decay and $^{214}$Po $\\alpha$-decay, another daughter of $^{222}$Rn $\\alpha$-decay~\\cite{Franco:2015pha,Amaudruz:2017ekt}. Hence in order to understand the effect of this measurement on the CNO sensitivity, we perform analyses with the $^{222}$Rn freely-varying and with it fixed \\emph{a priori} to a given precision. All of the fiducial values and priors we use in our analysis are given in table~\\ref{argon_priors}. \n\n\n\n\n\n\\begin{table}[b]\n\\centering\n \\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ p{3.5cm} || c | c | c | c }\n\\multirow{3}{3.5cm}{\\textbf{Background}} & \\multicolumn{2}{c |}{\\textbf{Optimistic}} & \\multicolumn{2}{c}{\\textbf{Pessimistic}} \\\\ \n\\cline{2-5}\n& Value & Error & Value & Error \\\\ \n& [(ton yr)$^{-1}$] & & [(ton yr)$^{-1}$] & \\\\ \\hline\n$^{222}$Rn daughters& 0.84 & $10\\%$ & 0.84 & Free \\\\\n\\hline\n$^{32}$P& 0.78 & Free & 0.78 & Free \\\\ \\hline\nOther cosmogenics& 5.4 & Free & 5.4 & Free \\\\ \n\\hline\n\\end{tabular}\n\\caption{Fiducial values and uncertainties for the backgrounds in a liquid argon experiment, based in Gran Sasso~\\cite{Franco:2015pha}, where ``Other cosmogenics\" refers to all cosmogenic backgrounds except for $^{32}$P. A percentage uncertainty means the one-sigma error on the \nGaussian prior, with central value equal to the fiducial value. Columns with ``Free'' errors mean the prior distribution is uniform\ni.e. unconstrained.}\n\\label{argon_priors}\n\\end{table}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_sno_lab_priors_free.pdf} \\\\\n\\includegraphics[width=0.99\\textwidth]{corner_plot_only_bottom_line_colour_argon_ny_50_0_lab_sno_lab_priors_fixed_radon.pdf} \n\\caption{Contours showing the degeneracy between the CNO flux and various backgrounds and other solar neutrino fluxes for a liquid argon electronic-recoil experiment running in SNOLAB or Jinping, with an exposure of 1000 ton-years with an unknown radon background (top) or a radon background known to a precision of $10\\%$ at one-sigma (bottom). The dashed black lines show the fiducial values used to generate simulated data. In each panel the red contour bounds $68\\%$ of the posterior distribution, while yellow bounds $95\\%$ and blue $99\\%$.}\n\\label{fig:argon_SNOlab_1000tonyr}\n\\end{figure*}\n\n\nThe result of our MCMC scan for a liquid argon experiment based in Gran Sasso lab with a $1000$~ton-year exposure is shown in figure~\\ref{fig:argon_GS_1000tonyr}. It is clear that the CNO flux is degenerate with measurements of the radon background and the background from $^{32}$P decays, as expected from their similar spectra at lower energies seen in figure~\\ref{fig:spec_argon_gs}. Indeed, as can be seen in the second panel from the left in each plot, knowledge of the radon background vastly improves the precision to which the CNO flux can be measured, by eliminating regions of parameter space where a larger radon background can compensate for a smaller CNO flux, and vice versa. For the pessimistic case, the $99\\%$ region extends to values where the CNO flux is zero, and the radon background is around $5$ counts per day per ton, and so when marginalising over the radon background SNO+ can not exclude the possibility of a zero CNO flux at $99\\%$ confidence. By contrast, when we fix the radon background for the optimistic case, a zero CNO flux will be strongly disfavoured by the experiment. Hence in order to measure the CNO neutrino flux at high-precision with an argon-based experiment, it is crucial that the radon background is kept as small as possible and is measured.\n\n\n\n\n\nIn the case where the liquid argon experiment, such as Argo, is based in SNOLAB or Jinping, we assume that the $^{32}$P and ``Other cosmogenics'' background rates in Table~\\ref{argon_priors} are 100 times lower owing to the smaller atmospheric muon flux~\\cite{JinpingNeutrinoExperimentgroup:2016nol}. As shown in figure~\\ref{fig:argon_SNOlab_1000tonyr}, the improvement gained by this reduction in the cosmogenic background is larger than one might expect, and is due to two effects: the first is the reduction in statistical uncertainty from the lower total background. The second is a partial breaking of the degeneracy between the CNO and $^{222}$Rn spectra, as the radon background spectrum has a predominant tail which will be difficult to measure above the cosmogenic background in Gran Sasso (see figure~\\ref{fig:spec_argon_gs}), but is easily visible for Argo, or a similar experiment, based in SNOLAB or Jinping. This also means that an external measurement of the $^{222}$Rn background through coincidence decays is less important for an experiment based in SNOLAB or Jinping, since its rate can be determined well from the beta-spectrum alone. Indeed as can be seen in the second-from-left panel in figure~\\ref{fig:argon_SNOlab_1000tonyr} there is still degeneracy between the CNO flux and the $^{222}$Rn rate in the pessimistic case, but it no longer extends to values of zero for the CNO flux. This is due to the fact that a zero CNO flux would need to be compensated by a larger radon background which would also introduce too many events at higher energies above the cut-off of the CNO spectrum, while in the Gran Sasso case these extra events would not be easily visible above the large cosmogenic background. \n\n\n\n\\subsection{Comparison of potential CNO flux measurements}\n\n\n\n\nFigure~\\ref{fig:ER_comparison} shows the expected precision with which our various projected experimental runs can measure the CNO neutrino flux at $3 \\sigma$ confidence. For each run, the error bars bracket the region where the value of the CNO neutrino flux provides a fit to the simulated data which deviates by $3 \\sigma$ or less from the best-fit value, marginalising over the other parameters in the MCMC fit. We have chosen $3 \\sigma$ since it is generally considered to be a sufficient level of confidence for discovery in astrophysics. \n\nA liquid argon electronic recoil experiment based in Gran Sasso lab will obtain the required precision with a $1000$~ton~year exposure and a radon background which has been measured \\emph{a priori} to a precision of $10\\%$ at $1 \\sigma$ (or better). This exposure could be obtained by the proposed Argo experiment in just over $3$~years if it has a mass of $300$~tons, but will likely take up to $10$~years since the plan is to use $100$~tons as the fiducial mass~\\cite{Davini:2016vpd,Aalseth:2017fik,Franco:2015pha}. With such an exposure Argo in Gran Sasso will be able to distinguish between the low and high metallicity scenarios for the CNO flux at $3 \\sigma$ confidence, though as expected from figure~\\ref{fig:argon_GS_1000tonyr} this is not the case if the size of the radon background is unknown or poorly measured. Unfortunately for a lower exposure the precision degrades significantly, and so there is no prospect for measuring the CNO flux with DarkSide-20k. \nIn all such instances the argon experiments\ngain a significant advantage from their excellent energy resolution.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.98\\textwidth]{CNO_categories_all_v5_metallicity_low_nsigma_3_v2_best_fit_and_err.pdf}\n\\caption{Comparison of experiments searching for CNO neutrinos using electronic recoils, with either exposure or running-time labelled. The error-bars show the projected $3 \\sigma$ precision with which each experimental run will be able to measure the flux of CNO neutrinos.\nThe optimistic and pessimistic scenarios differ by how accurately key backgrounds will be measured, and are detailed in Table~\\ref{argon_priors} for liquid argon experiments (e.g. DarkSide-20k or Argo)~\\cite{Aalseth:2017fik,Franco:2015pha}, Table~\\ref{sno_plus_priors} for SNO+~\\cite{Andringa:2015tza} and Table~\\ref{borexino_priors} for Borexino~\\cite{Bellini:2013lnn,Agostini:2017ixy}. The horizontal lines labelled ``High\" and ``Low\" refer to the high and low metallicity scenarios respectively~\\cite{Vinyoles:2017bqj,Basu1997,Basu2004}. Each column and colour represents a different experiment or lab combination.}\n\\label{fig:ER_comparison}\n\\end{figure*}\n\nFor Argo, or a similar experiment, based in SNOLAB or Jinping the prospects are even better. This is mainly because the $^{222}$Rn background is much easier to distinguish from the CNO spectrum, compared with Gran Sasso where the larger cosmogenic background makes it difficult to precisely measure the radon from its beta-spectrum alone. In this case a precise determination of the CNO flux can be made in approximately half the time, with a $500$~ton~year exposure, even if the radon background is not measured using observations of coincidence decays. This would take Argo (in SNOLAB or Jinping) $5$~years with a $100$~ton fiducial mass. In both cases this requires a radon background at the level of $10 \\mu$Bq per 100 tons contamination.\n\n\n\n\n\nSNO+ in its pure liquid-scintillator mode (i.e. without $^{130}$Te added) should achieve a good measurement of the CNO flux with three years of running, and with five years of data will be able to distinguish the two scenarios for the solar metallicity using CNO neutrinos at $3 \\sigma$, provided that the $^{210}$Bi background can be measured to an accuracy of $1\\%$. As can be seen in figure~\\ref{fig:ER_comparison}, without this constraint on the $^{210}$Bi background rate, the measurement of the CNO flux will be much less precise due to the large degeneracy between the CNO flux and the $^{210}$Bi rate. Hence an accurate determination of the $^{210}$Bi rate in SNO+ is crucial for a CNO neutrino search.\nA SNO+ run of six months should be able to detect CNO neutrinos with 99\\% confidence, but will lead to only a modest constraint on the absolute flux, which is unlikely to be better than the limit already set by Borexino~\\cite{Bellini:2013lnn,Agostini:2017ixy}. Hence SNO+ would need to run for between $3$ and $5$ years to be confident of measuring the CNO flux to enough precision to solve the solar metallicity problem, provided that its background levels are as low as those already measured in Borexino.\n\nAfter ten years of running, Borexino could measure the CNO flux with enough precision to separate the two solar models, provided that the $^{210}$Bi background is measured to a precision of $1\\%$ i.e. the optimistic prior scenario.\nHowever, this projection is extremely sensitive to the level of precision to which the backgrounds, especially $^{210}$Bi can be measured to i.e. for Borexino the difference in CNO flux precision between the optimistic and pessimistic cases is particularly large.\nIf the future run of Borexino has larger background rates than we have assumed or if these are not measured to the precision assumed in our optimistic prior case, then Borexino is unlikely to measure the CNO flux even after ten years, as is clear from the large error bars for the pessimistic case. \n\n\nNote also that we have only fit energy spectra in our analysis of each experiment, while the experimental collaborations will have access to additional information. Hence our projections should be considered as conservative estimates. \nFor example, the Borexino collaboration will have access to more data such as the spatial position of each interaction, information on coincidences between different detected events and the pulse-shape of each event, which may improve their sensitivity to CNO neutrinos~\\cite{Agostini:2017ixy,Agostini:2017aaa,Collaboration:2011nga}. The SNO+ and Argo groups would likely have access to similar information, though in all cases the improvement is not likely to be large, considering that we have made efforts to implement the effects of these cuts where possible.\n\nWhich of these experimental runs gets to the CNO flux measurement first, especially with enough precision to solve the solar metallicity problem, depends on both background control and on the amount of running time dedicated to a CNO search. If the SNO+ collaboration commit most of their experimental run-time to a search for neutrino-less double-beta decay~\\cite{Andringa:2015tza} then it is possible (though perhaps unlikely) they will be overtaken by Argo, or potentially the Jinping Neutrino Experiment~\\cite{JinpingNeutrinoExperimentgroup:2016nol}, despite the fact that SNO+ is in a more advanced stage of development. As an estimate, if SNO+ finishes its neutrino-less double-beta decay search in 2022, then a CNO flux measurement may be possible by around 2025 to 2027, but possibly later. This relies crucially on the estimates of the $^{210}$Bi rate in ref.~\\cite{Andringa:2015tza} being correct. The Borexino experiment has the advantage that it is already running, and may be the first to exclude a zero CNO flux, provided its backgrounds are kept under control. \nImportantly, this also means that its background rates will be the most realistic of\nthe experiments considered in this work, in\ncontrast to the projections of SNO+ and\nArgo, and so for a fair comparison this fact\nmust be taken into account.\n\n\nAs a final point, we note that it is possible that new technologies may allow the CNO flux to be measured by electron-recoil experiments sooner, in particular the development of experiments which can detect both scintillation and Cherenkov light, such as THEIA~\\cite{Gann:2015fba,Alonso:2014fwf,Caravaca:2016fjg}. This would mean that the direction of the recoiling electrons could be measured in addition to their energies, which would break the degeneracy between solar neutrinos and background such as $^{210}$Bi.\n\n\\section{Nuclear-recoil experiments searching for CNO neutrinos}\nIn the previous section we considered experiments looking for CNO neutrinos through their scattering with electrons, now we turn to nuclear-recoil searches. For the case of CNO neutrinos scattering with nuclei the energy range of interest is below a few hundred eV, with the exact value depending on the particular target nucleus~\\cite{Strigari:2016ztv} i.e. targets with heavier nuclei require lower thresholds to observe CNO neutrinos, as shown in figure~\\ref{fig:CNO_NR}. Such low-energy recoils are difficult to observe, limiting the sensitivity of these searches. However nuclear-recoil searches have the advantage of lower backgrounds, and a larger cross section of interaction between neutrinos and nuclei (compared with electrons), arising from coherent enhancement by approximately the square of the number of neutrons in the target nucleus. This also means that target exposures (effective mass times experiment live time) can in principle be much lower. Hence, in contrast to the previous section, our focus here will not be on background control, but instead we are predominantly concerned with finding the required combination of target nucleus, low-energy threshold and exposure in order to measure the CNO neutrino flux with precision.\n\n\\begin{figure}[bt]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{CNO_NR.pdf}\n\\caption{The expected spectrum of nuclear-recoils from elastic scattering by CNO neutrinos in the high-metallicity case, for various different target nuclei.}\n\\label{fig:CNO_NR}\n\\end{figure}\n\nCurrent nuclear-recoil experiments focus on searching for dark matter, but they also work well as neutrino detectors~\\cite{Cerdeno:2016sfi,Lang:2016zhv,Strigari:2016ztv,Billard:2013qya,Billard:2014yka}. \nThe energy thresholds of the most successful dark matter direct detection experiments are currently too high for a CNO neutrino search, for example the liquid-xenon-based LZ and XENON1T experiments have thresholds around a keV, meaning they will not see any CNO neutrinos~\\cite{Akerib:2016vxi,Aprile:2017iyp,Mount:2017qzi}. However there has been much recent progress on the development of experiments with lower thresholds. For example, the CRESST-III experiment which looks for small temperature changes of a cryogenically-cooled CaWO$_4$ crystal, had a low-energy threshold conservatively set\nat 100 eV for their most recent analysis with $2.39$~kg-days of data~\\cite{Petricca:2017zdp}. In addition, the same collaboration has developed the $\\nu$-cleus detector with a $20$~eV threshold using a $0.5$g Al$_2$O$_3$ crystal target~\\cite{Angloher:2017sxg}. There has also been progress using germanium-based experiments such as SuperCDMS, which has achieved energy thresholds as small as 75eV and 56eV in their most recent runs with an exposure around 70kg-days~\\cite{Agnese:2017jvy}. This has been possible thanks to a special (high-voltage) operation mode and new efforts in understanding and reducing the overall background rate.\n\nThe next stage in this experiment, SuperCDMS in SNOLAB, could go as low as a 40eV threshold with an exposure in high-voltage detectors around $44$kg-yr for germanium and $10$~kg-yr for silicon~\\cite{Agnese:2016cpb}. In addition, the next phase of the EDELWEISS experiment, which also uses a germanium target, could have a mass as large as 100kg~\\cite{Arnaud:2017usi}. Finally, the NEWS-G experiment, based on the technology of gaseous spherical detectors, had a 720eV threshold in their recent run with a 9.7kg-days exposure using neon and CH$_4$ targets~\\cite{Arnaud:2017bjh}. In this kind of detector, the low energy threshold is limited only by the mean ionization energy of the gas mixture, which, depending on the specific target, can be as low as a few eV \\cite{Bougamont:2011zz,Savvidis:2016wei}.\n\nIn the rest of this section we will focus on making projections for future versions of such searches, since at present their low target masses, much smaller than for the electron-recoil experiments, will not provide enough data for a precise measurement of the CNO flux. We will determine exactly what exposures will be needed, for a given low-energy threshold and target nucleus, for a positive detection of the CNO flux.\n\n\n\n\\subsection{Statistical procedure}\n\n\n\n\nIn order to do so, we test the ability of an experiment with a given target nucleus, recoil detection threshold and exposure to discriminate the CNO flux from the pp, pep, $^8$B and $^7$Be neutrino fluxes originating in the pp chain. Thus, we\n\\begin{enumerate}\n\\item randomly generate events based on the total expected spectrum, from both the proton-proton-chain and CNO. For each pseudoexperiment, the normalizations of the proton-proton-chain fluxes are randomly obtained, within measured uncertainties \\cite{Bergstrom:2016cbh}\\footnote{11\\%, 8.5\\%,15\\% and 3\\% for pp, $^8$B, pep and $^7$Be, respectively.}. The CNO flux is fixed to the sum of the $^{13}$N and $^{15}$O fluxes. \n\\item We construct an extended unbinned likelihood: \n\\begin{equation}\nL = e^{-N_{\\mathrm{exp}}}\\prod_{i}^{N_{\\mathrm{obs}}}\\left[\\sum_c \\phi_{0,c}\\frac{dR}{dE_R}(E_i)\\right] P(\\phi_{0,c}),\n\\label{eq:LLNR}\n\\end{equation}\nwhere $\\frac{dR}{dE_R}$ is the expected differential neutrino-nucleus scattering rate, $N_{\\mathrm{obs}}$ is the number of ``observed'' events in the sample, and $P(\\phi_{0,c})$ is a Gaussian prior on the neutrino fluxes, with mean and width based on the currently measured fluxes. The index c runs over the components $c = \\{$pp, pep, $^8$B, $^7$Be, ($^{13}$N + $^{15}$O) $\\}$. The total solar fluxes $\\phi_{0,c}$ are normalized such that the total number of expected events is:\n\\begin{equation}\nN_{\\mathrm{exp}} = \\sum_c N_{\\mathrm{exp},c}= \\sum_c \\phi_{0,c'}\\int_{E_{th}}^{E_{\\mathrm{max}}}\\frac{dR}{dE_R}dE_R\n\\end{equation}\nWe do not consider background events, which implies a certain level of discrimination between electronic and nuclear recoils.\n\nThis is because solar neutrinos from other populations, especially $pp$, will induce electron recoils in the energy range relevant for a nuclear-recoil search for CNO neutrinos. Despite $pp$ neutrinos being much more abundant, coherent scattering with nuclei benefits from a larger cross section than for electronic scattering, including a factor of the nucleon number squared. \nWith this consideration, the rate expected from $pp$ neutrino-induced electron scattering is smaller than the one for CNO neutrino nuclear recoils, by approximately a factor 10 in xenon for example, in the region-of-interest~\\cite{Cerdeno:2016sfi,Akerib:2018lyp}. \nThus, although a certain level of discrimination between electron and nuclear recoils would be desirable, it is only really important for the lightest nuclei, where one would preferably have around $90\\%$ electron-recoil rejection or better.\n\n\\item The likelihood in equation \\eqref{eq:LLNR} is then maximised setting $\\phi_{0,\\mathrm{CNO}} = 0$, then again for allowing the total CNO flux to vary freely. The quantity $ts = 2(\\log(L_{\\mathrm{CNO}}) - \\log(L_{\\mathrm{CNO}=0}))$ is constructed. We count an experiment as successful when $ts > 3.84$, corresponding to a 95\\% CL detection ($ts$ is indeed distributed as a chi-squared with one degree of freedom). \n\\item Steps 1-3 are repeated 5000 times for each point in exposure-threshold parameter space. \n\\end{enumerate}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{cleanedNR}\n\\caption{Required exposure versus threshold to discover the CNO neutrino flux via coherent neutrino-nucleus interaction at 95\\% CL, in the high metallicity scenario. For each target element all parameters above the lines will lead to a CNO detection. In the low-metallicity case, we are unable to significantly distinguish the CNO flux from the other solar neutrino components.}\n\\label{fig:NR_results_high}\n\\end{figure}\n\n\\subsection{Results}\n\nFig.~\\ref{fig:NR_results_high} shows the result of this procedure, for nuclear recoil experiments using helium, oxygen, neon, silicon, germanium and xenon in the high-metallicity scenario. The contours represent the line above which 90\\% of the pseudoexperiments are able to see the CNO flux at 95\\% CL or more. \nAs expected, lighter targets are able to measure the CNO flux with a higher threshold, as larger momenta can be transferred from the neutrinos. The drawback is a suppression in event rate due to the smaller $N^2$ enhancement, where $N$ is the number of neutrons. \n\n\n\n\n\n\n\n\nFigure~\\ref{fig:NR_results_high} allows us to assess the suitability of the different experimental techniques to look for CNO neutrinos. For example, despite their large projected exposures, liquid xenon detectors would require a threshold energy smaller than 5eV, which is three orders of magnitude smaller than their current values. It is not clear if this is physically possible, see e.g. ref. \\cite{Baldini:2004td}.\nConversely, the threshold needed for gaseous spherical detectors with light targets seems within the reach of future experiments, however, the required exposures are still very far from those projected in future detectors, and it is not clear whether discrimination from electron recoils is possible. The situation is similar for experiments with oxygen-based crystal targets, such as CRESST and $\\nu$-cleus~\\cite{Angloher:2017sxg}.\n\n\nOf all the nuclear-recoil technologies, low-temperature solid state detectors appear to be best suited for the observation of CNO neutrinos. Indeed, the exposure needed for germanium is just a factor of three larger than what SuperCDMS SNOLAB will achieve, although the threshold would need to be reduced to $\\sim 10$eV. Likewise, silicon would require a threshold of approximately 30eV, smaller than the projected 70eV of SuperCDMS SNOLAB, and the exposure needed is approximately ten times larger.\nIn principle, these thresholds could be reduced since the required energy to produce a single electron-hole ($e-h$) pair in germanium and silicon is of the order of 3eV and 3.6eV, respectively. Although this is not within the reach of current technology, future improvements could make this possible. \n\n\nFor detection of CNO neutrinos in the low-metallicity scenario, the prospects of direct detection experiments are less promising. The $^{13}$N and $^{15}$O fluxes decrease substantially and the resulting spectrum blends in to that of the pep neutrinos. As a result, much larger exposures are needed, as well as much lower thresholds. As an example, a silicon experiment would require a threshold of approx 10~eV and a minimum exposure of $0.5$~ton-yr to start observing these events, whereas a germanium-based experiment would require a threshold close to the minimum energy to excite an $e-h$ pair.\n\n\n\n\\section{Conclusion}\nThere is no consistent model of the Sun which explains both helioseismological data and measurements of the solar metallicity using spectroscopic data, leading to the so-called ``solar metallicity problem''~\\cite{Asplund:2009fu,2009ASPC..416..193B}.\nThe aim of this paper has been to work out when experiments will measure the CNO neutrino flux with enough precision to help solve this problem, and the challenges this involves. We have studied both nuclear recoil experiments, which need to be sensitive to very low-energy recoils but have small backgrounds, and electronic recoil experiments, which generally have larger target masses but also higher background rates. We determine whether such nuclear-recoil technologies, which are important to the search for light dark matter, can catch up with the electronic-recoil experiments.\n\nFor experiments searching for CNO neutrinos using electron-recoils, we have made projections with an MCMC analysis, to compare how precisely they will be able to measure the CNO flux, given various assumptions about the backgrounds and how well their rates will be measured. Figure~\\ref{fig:borexino_mcmc} shows our results for Borexino, figure~\\ref{fig:sno_plus} for SNO+ and figures~\\ref{fig:argon_GS_1000tonyr} and \\ref{fig:argon_SNOlab_1000tonyr} for a liquid argon experiment, such as Argo~\\cite{Aalseth:2017fik,Franco:2015pha}, in Gran Sasso and SNOLAB respectively. Each two-dimensional panel shows the range of parameters which provide a good fit to simulated data, highlighting the degeneracies between parameters which lead to systematic uncertainties. For Borexino and SNO+ the main degeneracy is between the CNO flux and the $^{210}$Bi background, while for argon experiments the CNO flux is degenerate with the background from the decay of the $^{222}$Rn daughters. Any technology which could break this degeneracy would significantly improve the accuracy of a CNO flux measurement (see e.g. refs.~\\cite{Gann:2015fba,Alonso:2014fwf,Caravaca:2016fjg}). In all cases the CNO flux is strongly degenerate with the pep neutrino flux.\n\nSince we have performed our own analysis, we are able to compare our projected sensitivities for each experiment. Our comparison is shown in figure~\\ref{fig:ER_comparison}. For each experimental run we have chosen an optimistic and pessimistic prior set, with the former imposing strong constraints on the sizes of key backgrounds and thereby reducing systematic uncertainties, while in the latter case we use only weak constraints. The comparison between these different assumptions makes it clear that controlling key systematics is just as vital as obtaining more data.\n\n\n\nFor experiments looking for CNO neutrinos through their scattering with nuclei we focused on future searches, and determined the required low-energy threshold and exposure for different target nuclei, in order to measure the CNO neutrino flux precisely. Our results for nuclear-recoils are shown in figure~\\ref{fig:NR_results_high}.\n\n\n\n\nOur analysis highlights several ways in which the CNO flux may be measured to the precision needed to separate the two solar metallicity scenarios, which we list below:\n\\begin{itemize}\n\\item If Argo is built in Gran Sasso lab with a $100$~ton fiducial mass then it will measure the CNO flux to the required precision after ten years, provided that the $^{222}$Rn-chain background is measured to a accuracy of $10\\%$ or better.\n\\item Building Argo in SNOLAB or Jinping, where the cosmogenic backgrounds will be smaller, should lead to an accurate CNO flux measurement in only $5$~years, assuming a $100$~ton fiducial mass. In all cases the argon experiments rely on their extremely good projected energy resolution.\n\\item SNO+ will measure the CNO flux in $5$~years provided that the $^{210}$Bi rate is known to at least $1\\%$ accuracy. However this is only if it is kept running in its liquid scintillator mode, and not while doped with $^{130}$Te for the neutrino-less double-beta decay search. It is likely the best candidate to see CNO neutrinos, with a potential for detection between 2025 to 2027 if its double-beta search ends in 2022, but only if the $^{210}$Bi rate meets projections to be at least as low as that already achieved in Borexino~\\cite{Andringa:2015tza}.\n\\item Borexino will set strong limits on the CNO neutrino flux, and could obtain a good measurement after ten years, provided it can measure the $^{210}$Bi rate to an accuracy of better than $0.5$~counts per day. It is the most sensitive of the experiments to systematics from backgrounds, and so an accurate measurement of $^{210}$Bi is particularly important. Crucially though, it is the only experiment we have considered which is currently running, and so has the most realistic background assumptions.\n\\item Nuclear recoils of CNO neutrinos could also be within the reach of dark matter experiments, although these would require very low-thresholds and large exposures. Low-temperature solid state detectors seem the best alternative to confirm or rule-out the high-metallicity scenario, but the technology will have to improve to be able to lower the experimental threshold down to 10eV for germanium or 30eV for silicon. Probing the low-metallicity scenario is much more challenging, as lower thresholds and larger exposures are needed. Such an experiment would also revolutionize constraints on light dark matter by many orders of magnitude~\\cite{Agnese:2017jvy}.\n\\end{itemize}\n\nIt is clear is that, despite the challenges, neutrino experiments will be able to contribute significantly to the solar metallicity problem in the near future. A full solution will likely also need us to understand why helioseismology disagrees with the standard solar model. Which experiment gets there first will depend on the amount of time dedicated to a CNO neutrino search and crucially, how well the backgrounds can be controlled for electron-recoil searches, and how low the energy threshold will be for nuclear-recoil searches.\n\n\\acknowledgments\nWe thank Davide Franco, Gabriel Orebi Gann, Gilles Gervais, El\\'ias L\\'opez-Asamar, Valentina Lozza, Ryan Martin and Oleg Smirnov.\nDGC acknowledges support from STFC.\nJD and MF are funded by the European Research Council through the project DARKHORIZONS under the European Union's Horizon 2020 program (ERC Grant Agreement no.648680). The work of MF was also supported partly by the STFC Grant ST\/L000326\/1 and this project has benefited from a Durham IPPP Fellowship. \n\n\\bibliographystyle{JHEP}\n\n\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}