diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmjwy" "b/data_all_eng_slimpj/shuffled/split2/finalzzmjwy" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmjwy" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nMagnetic fields are observed on various spatial scales of the\nuniverse: they are detected in planets and stars \\citep{DL09,R12},\nin the interstellar medium \\citep{C12}, and on galactic scales \\citep{B16}.\nAdditionally, observational lower limits on intergalactic\nmagnetic fields have been reported\n\\citep{NV10,DCRFCL11}.\nContrary to the high magnetic field strengths observed on scales below\nthose of galaxy clusters, which can be explained by dynamo amplification\n\\citep[see, e.g.,][]{BS05},\nintergalactic magnetic fields, if confirmed, are most likely of primordial origin.\nBecause of their often large energy densities, magnetic fields can play\nan important role in various astrophysical objects, a prominent\nexample being the $\\alpha\\Omega$ dynamo in solar-like stars that explains\nstellar activity\n\\citep[see, e.g.,][]{P55,P79,M78,KR80,ZRS83,C14}.\n\nWhile there is no doubt about the significant role of magnetic fields in the\ndynamics of the\npresent-day universe, their origin and evolution over cosmic times remain a\nmystery \\citep{R87,GR01,W02,KZ08}.\nNumerous scenarios for the generation of primordial magnetic fields have been\nsuggested\nin the literature. The proposals span from inflation-produced magnetic fields\n\\citep{TW88} to field generation during cosmological phase transitions\n\\citep{SOJ97}.\nEven though strong magnetic fields\ncould be generated shortly after the Big Bang, their strength subsequently\ndecreases in cosmic expansion unless they undergo further amplification.\nBe this as it may, the presence of primordial magnetic fields can affect the\nphysics of the early universe.\nFor example, it has been shown that primordial fields could have\nsignificant effects on the matter power spectrum by suppressing the formation\nof small-scale structures \\citep{KEtAl13,PEtAl15}.\nThis, in turn, could influence cosmological structure formation.\n\nThe theoretical framework for studying the evolution of magnetic fields is\nmagnetohydrodynamics (MHD).\nIn classical plasma physics,\nthe system of equations includes the induction equation,\nwhich is derived from the Maxwell equations and Ohm's law and describes\nthe evolution of magnetic fields,\nthe continuity equation for the fluid density,\nand the Navier-Stokes equation governing the evolution of the\nvelocity field.\n\nAt high energies, for example, in the quark-gluon plasma of the early universe,\nhowever, an additional quantity needs to be taken into account, namely\nthe chiral chemical potential.\nThis quantity is related to an asymmetry between the number densities of\nleft-handed fermions (spin antiparallel to the momentum) and right-handed\nfermions (spin parallel to the momentum). This leads to an additional \ncontribution to the electric current along the magnetic field, known as the\n{\\em chiral magnetic effect} (CME).\nThis phenomenon was discovered by \\cite{Vilenkin:80a} and\nwas later carefully investigated using different\ntheoretical approaches in a number of studies \\citep{RW85,Tsokos:85,\nJS97,AlekseevEtAl1998,Frohlich:2000en,Frohlich:2002fg,\nFukushima:08,Son:2009tf}.\n\nThe CME causes a small-scale dynamo instability \\citep{JS97}, which has also\nbeen revealed from a kinetic description of chiral plasmas \\citep{AY13}.\nThe evolution equation for a nonuniform\nchiral chemical potential has been derived in \\citet{BFR12,BFR15} who used it\nto study the inverse magnetic cascade that results in an\nincrease of the characteristic scale of the magnetic field.\n\\citet{BFR12} have shown that the chiral asymmetry can survive down to energies\nof the order of $10\\,{\\rm MeV}$, due to coupling to an effective axion field.\nThese studies triggered various investigations related to\nchiral MHD turbulence \\citep{Pavlovic:2016gac,Y16}\nand its role in the early universe \\citep{TVV12,DvornikovSemikoz2017},\nas well as in neutron stars \\citep{DvornikovSemikoz2015,SiglLeite2016}.\n\nRecently, a systematic theoretical analysis of the system of \\emph{chiral MHD}\nequations, including the back-reaction of the magnetic field on the chiral\nchemical potential, and the coupling to the plasma velocity field has been\nperformed in \\cite{REtAl17}, referred to here as Paper~I.\nThe main findings of Paper~I include a modification of MHD waves by the\nCME and different kinds of laminar and turbulent dynamos.\nBesides the well-studied laminar chiral dynamo caused by\nthe CME, a chiral--shear dynamo in the presence of a shearing velocity\nwas discussed there.\nIn addition, a mean-field theory of chiral MHD in the presence\nof small-scale nonhelical turbulence was developed in Paper~I,\nwhere a new chiral $\\alpha_\\mu$ effect\nnot related to a kinetic helicity has been found.\nThis effect results from an interaction of chiral magnetic fluctuations\nwith fluctuations of the electric current caused by the tangling magnetic\nfluctuations.\n\nIn the present paper, we report on\nnumerical simulations that confirm and further substantiate the chiral\nlaminar and turbulent dynamos found in Paper~I.\nTo this end, we have implemented the chiral MHD equations in the\n{\\sc Pencil Code}\\footnote{http:\/\/pencil-code.nordita.org\/},\na high-order code suitable for compressible MHD turbulence.\nDifferent situations are considered, from laminar dynamos\nto chiral magnetically driven turbulence\nand large-scale dynamos in externally forced turbulence.\nWith our direct numerical simulations (DNS), we are able to study the dynamical\nevolution of a plasma that includes chiral effects in a large domain of the\nparameter space.\nGiven that the detailed properties of relativistic astrophysical plasmas,\nin particular the initial chiral asymmetry and chiral feedback mechanisms,\nare not well understood at present, a broad analysis of various\nscenarios is essential.\nThe findings from DNS can then be used to explore the possible evolution of \nastrophysical plasmas under different assumptions. \nThese applications should not be regarded as realistic descriptions of \nhigh-energy plasmas; they aim to find out under what conditions the\nCME plays a significant role in the evolution of a\nplasma of relativistic charged fermions (electrons) and to test the\nimportance of chirality flips changing the handedness of the fermions.\nWe are not pretending that the regimes accessible to our simulations\nare truly realistic in the context of the physics of the early universe or in\nneutron stars.\n\nThe outline of the present paper is as follows.\nIn Section~\\ref{sec:simulations} we review the governing equations and\nthe numerical setup,\nand we discuss the physics related to the two main nonlinear effects\nin chiral MHD, which lead to different scenarios of the magnetic\nfield evolution.\nIn Section~\\ref{sec:lamdyn} we present numerical results on laminar chiral\ndynamos.\nIn Section~\\ref{sec_turbdynamo1} we show how magnetic fields, amplified by\nthe CME, produce turbulence (chiral magnetically driven turbulence).\nWe discuss how this gives rise to the chiral $\\alpha_\\mu$ effect.\nWe also study this effect in Section~\\ref{sec_turbdynamo} for a system\nwhere external forcing is employed to produce turbulence.\nAfter a discussion of chiral MHD in astrophysical and cosmological processes in\nSection~\\ref{sec_astro}, we draw conclusions in Section~\\ref{sec_concl}.\n\n\n\n\\section{Chiral MHD in numerical simulations}\n\\label{sec:simulations}\n\n\\subsection{Equations of chiral MHD}\n\\label{sec:eqs-chiral MHD}\n\nThe system of chiral MHD equations\nincludes the induction equation for the magnetic field $\\bm{B}$,\nthe Navier-Stokes equation for the velocity field $\\bm{U}$ of the plasma,\nthe continuity equation for the plasma density $\\rho$,\nand an evolution equation for the normalized chiral chemical potential\n$\\mu$:\n\\begin{eqnarray}\n\\frac{\\partial \\bm{B}}{\\partial t} &=&\n\\bm{\\nabla} \\times \\left[{\\bm{U}} \\times {\\bm{B}}\n- \\eta \\, \\left(\\bm{\\nabla} \\times {\\bm{B}}\n- \\mu {\\bm{B}} \\right) \\right] ,\n\\label{ind-DNS}\\\\\n\\rho{D \\bm{U} \\over D t}&=& (\\bm{\\nabla} \\times {\\bm{B}}) \\times \\bm{B}\n-\\bm{\\nabla} p + \\bm{\\nabla} {\\bm \\cdot} (2\\nu \\rho \\mbox{\\boldmath ${\\sf S}$} {})\n+\\rho \\bm{f} ,\n\\label{UU-DNS}\\\\\n\\frac{D \\rho}{D t} &=& - \\rho \\, \\bm{\\nabla} \\cdot \\bm{U} ,\n\\label{rho-DNS}\\\\\n\\frac{D \\mu}{D t} &=& D_5 \\, \\Delta \\mu\n+ \\lambda \\, \\eta \\, \\left[{\\bm{B}} {\\bm \\cdot} (\\bm{\\nabla} \\times {\\bm{B}})\n- \\mu {\\bm{B}}^2\\right]\n -\\Gamma_{\\rm\\!f}\\mu,\n\\label{mu-DNS}\n\\end{eqnarray}\nwhere $\\bm{B}$ is normalized such that the magnetic energy\ndensity is $\\bm{B}^2\/2$ (without the $4\\pi$ factor), and\n$D\/D t = \\partial\/\\partial t + \\bm{U} \\cdot \\bm{\\nabla}$ is the\nadvective derivative.\nFurther, $\\eta$ is the microscopic magnetic diffusivity,\n$p$ is the fluid pressure,\n${\\sf S}_{ij}={\\textstyle{1\\over2}}(U_{i,j}+U_{j,i})-{\\textstyle{1\\over3}}\\delta_{ij} {\\bm \\nabla}\n{\\bm \\cdot} \\bm{U}$\nare the components of the trace-free strain tensor $\\mbox{\\boldmath ${\\sf S}$} {}$ (commas denote\npartial spatial derivatives),\n$\\nu$ is the kinematic viscosity,\nand $\\bm{f}$ is the turbulent forcing function.\n\nEquation~(\\ref{mu-DNS}) describes the evolution of the chiral\nchemical potential $\\mu_5 \\equiv\\mu_L - \\mu_R$, with $\\mu_L$ and $\\mu_R$ being\nthe chemical potentials of left- and right-handed chiral fermions, which is normalized such that\n$\\mu = (4 \\ensuremath{\\alpha_{\\rm em}} \/\\hbar c) \\mu_5$ has the dimension of an inverse length.\nHere $D_5$ is the diffusion constant of the chiral chemical potential $\\mu$, and\nthe parameter $\\lambda$, referred to in Paper~I as\nthe \\textit{chiral feedback parameter},\ncharacterizes the strength of the\ncoupling of the electromagnetic field to $\\mu$.\nThe expression of the feedback term in Equation~(\\ref{mu-DNS}) was\nderived in Paper~I and is valid for the limit of small magnetic\ndiffusivities.\nFor hot plasmas, when $k_{\\rm B} T \\gg \\max(|\\mu_L|,|\\mu_R|)$,\nthe parameter $\\lambda$ is given\nby\\footnote{The definition of $\\lambda$ in the case of a degenerate Fermi\ngas will be given in Section~\\ref{sec:PNS}.}\n\\begin{eqnarray}\n \\lambda=3 \\hbar c \\left({8 \\ensuremath{\\alpha_{\\rm em}} \\over k_{\\rm B} T} \\right)^2,\n\\label{eq_lambda}\n\\end{eqnarray}\nwhere $\\ensuremath{\\alpha_{\\rm em}} \\approx 1\/137$ is the fine structure constant,\n$T$ is the temperature, $k_{\\rm B}$ is the Boltzmann constant,\n$c$ is the speed of light, and $\\hbar$ is the reduced Planck constant.\nWe note that\n$\\lambda^{-1}$ has the dimension of energy per unit length.\nThe last term in Equation~(\\ref{mu-DNS}), proportional to $\\Gamma_{\\rm\\!f} \\mu$,\ncharacterizes chirality flipping processes due to finite fermion masses.\nThis term is included in a phenomenological way.\nThe detailed\ndependence of $\\Gamma_{\\rm\\!f}$ on the plasma parameters in realistic systems\nis still not fully understood.\nIn most of the runs,\nthe chirality flipping effect is neglected because we concentrate in this paper\non the high-temperature regime, where the other terms in\nEquation~(\\ref{mu-DNS}) dominate.\nHowever, we study its effect on the nonlinear evolution of\n$\\mu$ in Section~\\ref{sec_flip}.\n\nWe stress that the effects related to the chiral anomaly cannot be\nseparated from the rest of the equations. This is one of the\nessential features of the chiral MHD equations that we are studying.\nThe equations interconnect the chiral chemical potential to the\nelectromagnetic field.\nHowever, the chiral anomaly couples the electromagnetic field\nnot directly to the chiral chemical potential but to\nthe chiral charge density, a conjugate variable in the sense of statistical\nmechanics. \nThe parameter $\\lambda$ is nothing but a susceptibility, that is,\na (inverse) proportionality coefficient\nquantifying the response of the\naxial charge to a change in the chiral chemical potential; see Paper~I.\n\nThe system of equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) and their range of\nvalidity have been discussed in detail in Paper~I.\nBelow we present a short summary of the assumptions made in deriving these\nequations.\nWe focus our attention on an isothermal plasma, $T=\\mathrm{const}$.\nThe equilibration rate of the temperature gradients is related to the shortest\ntimescales of the plasma (of the order of the plasma frequency or below) and is\nmuch shorter than the time-scales that we consider in the present study.\nFor an isothermal equation of state, the pressure $p$ is related to\nthe density $\\rho$ via $p=c_{\\rm s}^2\\rho$, where $c_{\\rm s}$ is the sound speed.\nWe apply a one-fluid MHD model that\nfollows from a two-fluid plasma model\n\\citep{Artsimovich-Sagdeev,Melrose-QuantumPlasmadynamicsMagnetized,Biskamp:97}.\nThis implies that we do not consider here kinetic effects and effects related to\nthe two-fluid plasma model.\nWe note that the MHD formalism is valid for scales above the mean free path that can be \napproximated as \\citep{ArnoldEtAl2000}\n\\begin{equation}\n \\ell_\\mathrm{mfp} \\approx \\frac{1}{(4 \\pi \\alpha_\\mathrm{em})^2 \\ln{((4 \\pi \\alpha_\\mathrm{em})^{-1\/2})}} \\frac{\\hbar c}{k_{\\rm B} T}.\n\\label{eq_mfp}\n\\end{equation}\nFurther, we study the nonrelativistic bulk motion of a highly relativistic plasma.\nThe latter leads to a term in the Maxwell equations that destabilizes\nthe nonmagnetic equilibrium and causes an exponential growth of\nthe magnetic field.\nSuch plasmas arise in the description of certain astrophysical systems,\nwhere, for example,\na nonrelativistic plasma interacts with cosmic rays consisting of\nrelativistic particles with small number density;\nsee, for example, \\cite{Schlickeiser-book}.\nWe study the case of small magnetic diffusivity\ntypical of many astrophysical systems with large magnetic Reynolds numbers, so\nwe neglect terms of the order of $\\sim O(\\eta^2)$ in the electric field; see Paper I.\n\n\nA key difference in the induction equations of chiral\nand classical MHD is the last term\n$\\propto \\bm{\\nabla} \\times (\\eta \\, \\mu \\, {\\bm{B}})$ in Equation~(\\ref{ind-DNS}).\nThis is reminiscent of mean-field dynamo theory, where a mean magnetic field $\\overline{\\mbox{\\boldmath $B$}}{}}{$ is\namplified by an $\\alpha$ effect due to a term $\\propto \\bm{\\nabla} \\times (\\alpha \\, {\\overline{\\mbox{\\boldmath $B$}}{}}{})$\nin the mean-field induction equation, which results in an $\\alpha^2$ dynamo.\nIn analogy with mean-field dynamo theory, we use the name\n\\textit{$v_\\mu^2$ dynamo}, introduced in Paper~I, where\n\\begin{equation}\n v_\\mu\\equiv\\eta\\mu_0\n \\label{eq:16}\n\\end{equation}\nplays the role of $\\alpha$ (see Equation~(\\ref{ind-DNS})),\nand $\\mu_0$ is the initial value of the normalized chiral chemical potential.\nThese different notions are motivated by the fact that the $v_\\mu$ effect\nis not related to any turbulence effects; that is, it is not determined by\nthe mean electromotive force, but originates from the\nCME; see Paper~I for details.\nWe will discuss the differences between chiral and classical MHD in more\ndetail in Section~\\ref{sec:difference}.\n\nThe system of Equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) implies a conservation law:\n\\begin{equation}\n\\frac{\\partial }{\\partial t} \\left({\\lambda \\over 2} {\\bm A} {\\bm \\cdot} \\bm{B}\n+ \\mu \\right) + \\bm{\\nabla} {\\bm \\cdot} \\bm{F}_{\\rm tot} = 0,\n\\label{CL}\n\\end{equation}\nwhere\n\\begin{equation}\n\\bm{F}_{\\rm tot} =\n{\\lambda \\over 2} \\left({\\bm \\bm{E}} \\times {\\bm A} + \\bm{B} \\, \\Phi\\right)\n- D_5 \\bm{\\nabla} \\mu\n\\end{equation}\nis the flux of total chirality and\n$\\bm{B} = {\\bm \\nabla} {\\bm \\times} {\\bm A}$,\nwhere ${\\bm A}$ is the vector potential, ${\\bm \\bm{E}}=\n- c^{-1} \\, \\{ {\\bm \\bm{U}} {\\bm \\times} {\\bm{B}} + \\eta \\, \\big(\\mu {\\bm{B}} -\n{\\bm \\nabla} {\\bm \\times} {\\bm{B}} \\big) \\}$ is the electric field, $\\Phi$ is the\nelectrostatic potential, $\\lambda$ is assumed to be constant,\nand the chiral flipping term, $-\\Gamma_{\\rm\\!f}\\mu$,\nin Equation~(\\ref{mu-DNS}) is assumed to be negligibly small.\nThis implies that the total chirality is a conserved quantity:\n\\begin{equation}\n \\label{cons_law}\n\\frac\\lambda 2 \\langle \\bm{A} \\cdot \\bm{B}\\rangle + \\langle \\mu \\rangle = \\mu_0 = \\mathrm{const},\n\\end{equation}\nwhere $\\langle \\mu \\rangle$\nis the value of the chemical potential and\n$ \\langle \\bm{A} \\cdot \\bm{B}\\rangle \\equiv V^{-1}\\int \\bm{A} \\cdot \\bm{B} \\, dV$\nis the mean magnetic helicity density over the volume $V$.\n\n\\subsection{Chiral MHD equations in dimensionless form}\n\nWe study the system of chiral MHD equations~(\\ref{ind-DNS})--(\\ref{mu-DNS})\nin numerical simulations to analyze various laminar and turbulent dynamos,\nas well as the production of turbulence by the CME.\nIt is, therefore, useful to rewrite this system of equations in\ndimensionless form, where velocity is measured in units of the sound speed\n$c_\\mathrm{s}$, length is measured in units of $\\ell_\\mu \\equiv \\mu_0^{-1}$, so time\nis measured in units of $\\ell_\\mu\/c_\\mathrm{s}$,\nthe magnetic field is measured in units of $\\sqrt{\\overline{\\rho}} \\, c_\\mathrm{s}$,\nfluid density is measured in units of $\\overline{\\rho}$\nand the chiral chemical potential is measured in units of $\\ell_\\mu^{-1}$,\nwhere $\\overline{\\rho}$ is the volume-averaged density.\nThus, we introduce the following dimensionless functions, indicated by a\ntilde:\n${\\bm \\bm{B}}=\\sqrt{\\overline{\\rho}} \\, c_\\mathrm{s}\\tilde{\\bm \\bm{B}}$,\n${\\bm \\bm{U}}=c_\\mathrm{s}\\tilde{\\bm \\bm{U}}$,\n$\\mu=\\ell_\\mu^{-1} \\tilde\\mu$\nand $\\rho= \\overline{\\rho} \\, \\tilde\\rho$.\nThe chiral MHD equations in dimensionless form are given by\n\\begin{eqnarray}\n\\frac{\\partial \\tilde\\bm{B}}{\\partial \\tilde t} &=& \\tilde{\\bm \\nabla}\n{\\bm \\times} \\biggl[\\tilde{\\bm \\bm{U}} {\\bm \\times} \\tilde{\\bm{B}} + {\\rm Ma}_\\mu \\,\n\\Big(\\tilde\\mu \\tilde{\\bm{B}} - \\tilde{\\bm \\nabla} {\\bm \\times} {\\tilde\\bm{B}} \\Big)\n\\biggr],\n\\label{ind-NS}\\\\\n\\tilde \\rho {D \\tilde \\bm{U} \\over D \\tilde t}&=& (\\tilde \\bm{\\nabla} \\times \\tilde{\\bm{B}}) \\times\n\\tilde \\bm{B}\n-\\tilde \\bm{\\nabla} \\tilde \\rho + {\\rm Re}_\\mu^{-1} \\tilde\\bm{\\nabla} {\\bm \\cdot} (2\\nu \\tilde\n\\rho \\mbox{\\boldmath ${\\sf S}$} {})\n+ \\tilde \\rho \\bm{f} ,\n\\nonumber\\\\\n\\label{UU-NS}\\\\\n\\frac{D \\tilde\\rho}{D \\tilde t} &=& - \\tilde\\rho \\, \\tilde\\bm{\\nabla} \\cdot \\tilde\\bm{U} ,\n\\label{rho-NS}\\\\\n\\frac{D \\tilde\\mu}{D \\tilde t} &=& D_\\mu \\, \\tilde\\Delta \\tilde\\mu\n+ \\Lambda_\\mu \\, \\Big[{\\tilde\\bm{B}} {\\bm \\cdot} (\\tilde\\bm{\\nabla} {\\bm \\times}\n{\\tilde\\bm{B}})\n- \\tilde\\mu {\\tilde\\bm{B}}^2 \\Big]\n- \\tilde \\Gamma_\\mathrm{f} \\tilde \\mu ,\n\\label{mu-NS}\n\\end{eqnarray}\nwhere we introduce the following dimensionless parameters:\n\\begin{itemize}\n\\item{Chiral Mach number:\n\\begin{eqnarray}\n{\\rm Ma}_\\mu = \\frac{\\eta\\mu_0}{c_\\mathrm{s}} \\equiv \\frac{v_{\\mu}}\n{c_\\mathrm{s}},\n\\label{Ma_mu_def}\n\\end{eqnarray}\n}\n\\item{Magnetic Prandtl number:\n\\begin{eqnarray}\n{\\rm Pr}_{_{\\rm M}} = \\frac{\\nu}{\\eta},\n\\end{eqnarray}\n}\n\\item{Chiral Prandtl number:\n\\begin{eqnarray}\n{\\rm Pr}_\\mu = \\frac{\\nu}{D_5} ,\n\\end{eqnarray}\n}\n\\item{Chiral nonlinearity parameter:\n\\begin{eqnarray}\n \\lambda_\\mu = \\lambda \\eta^2 \\overline{\\rho} ,\n\\label{eq_lambamu}\n\\end{eqnarray}\n}\n\\item{Chiral flipping parameter:\n\\begin{eqnarray}\n \\tilde\\Gamma_\\mathrm{f} = \\frac{\\Gamma_\\mathrm{f}}{\\mu_0 c_\\mathrm{s}} .\n\\end{eqnarray}\n}\n\\end{itemize}\nThen, $D_\\mu={\\rm Ma}_\\mu \\, {\\rm Pr}_{_{\\rm M}} \/ {\\rm Pr}_\\mu$,\n$\\, \\Lambda_\\mu =\\lambda_\\mu\/{\\rm Ma}_\\mu$\nand ${\\rm Re}_\\mu = \\left({\\rm Ma}_\\mu \\, {\\rm Pr}_{_{\\rm M}}\\right)^{-1}$.\n\n\n\\subsection{Physics of different regimes of magnetic field evolution}\n\nThere are two key nonlinear effects that determine the dynamics\nof the magnetic field in chiral MHD.\nThe first nonlinear effect is determined by\nthe conservation law~(\\ref{CL}) for the total chirality,\nwhich follows from the induction equation\nand the equation for the chiral magnetic potential.\nThe second nonlinear effect is determined by the\nLorentz force in the Navier-Stokes equation.\n\nIf the evolution of the magnetic field starts from a very small\nforce-free magnetic field, the\nsecond nonlinear effect, due to the\nLorentz force, vanishes if we assume that\nthe magnetic field remains force-free.\nThe magnetic field is generated by the chiral\nmagnetic dynamo instability\nwith a maximum growth rate $\\gamma^\\mathrm{max}_\\mu=v_\\mu^2\/4 \\eta$\nattained at the wavenumber $k_\\mu=\\mu_0\/2$ \\citep{JS97}.\n\nSince the total chirality is conserved, the increase of\nthe magnetic field in the nonlinear regime results in a decrease of the\nchiral chemical potential, so that the characteristic scale\nat which the growth rate is maximum increases in time.\nThis nonlinear effect has been interpreted in terms of\nan inverse magnetic cascade \\citep{BFR12}.\nThe maximum saturated level of the magnetic field can be\nestimated from the conservation law~(\\ref{CL}):\n$B_{\\rm sat} \\sim (\\mu_0 k_{\\rm M}\/\\lambda)^{1\/2} < \\mu_0\/\\lambda^{1\/2}$.\nHere, $\\mu_{\\rm sat} \\ll \\mu_0$ is the chiral chemical potential at saturation with\nthe characteristic wavenumber $k_{\\rm M} < \\mu_0$, corresponding\nto the maximum of the magnetic energy.\n\nHowever, the growing force-free magnetic field cannot stay\nforce-free in the nonlinear regime of the magnetic field evolution.\nIf the Lorentz force does not vanish, it generates small-scale velocity\nfluctuations.\nThis nonlinear stage begins when the nonlinear term ${\\bm{U}}\\times{\\bm{B}}$ in\nEquation~(\\ref{ind-DNS}) is of the order of the dynamo generating\nterm $v_\\mu {\\bm{B}}$, that is, when the velocity reaches the level\nof $U \\sim v_\\mu$.\nThe effect described here results in the production of\nchiral magnetically driven turbulence,\nwith the level of turbulent kinetic energy being determined\nby the balance of the nonlinear term, $(\\bm{U} \\cdot \\bm{\\nabla})\\bm{U}$, in\nEquation~(\\ref{UU-DNS}) and the Lorentz force, $(\\bm{\\nabla} \\times {\\bm{B}}) \\times \\bm{B}$,\nso that the turbulent velocity can reach the Alfv\\'en speed\n$v_{\\rm A}=({\\bm{B}^2}\/\\overline{\\rho})^{1\/2}$.\n\nThe chiral magnetically driven turbulence causes\ncomplicated dynamics:\nit produces the mean electromotive force that includes\nthe turbulent magnetic diffusion\nand the chiral $\\alpha_\\mu$ effect that\ngenerates large-scale magnetic fields; see Paper~I.\nThe resulting large-scale magnetic fields are concentrated\nat the wavenumber $k_\\alpha = 2 k_\\mu (\\ln {\\rm Re}_{_\\mathrm{M}})\/(3{\\rm Re}_{_\\mathrm{M}})$,\nfor ${\\rm Re}_{_\\mathrm{M}} \\gg 1$; see Paper~I.\nThe saturated value of the large-scale\nmagnetic field controlled by the conservation law~(\\ref{CL}), is\n$B_{\\rm sat} \\sim (\\mu_0 k_\\alpha\/\\lambda)^{1\/2}$.\nHere, ${\\rm Re}_{_\\mathrm{M}}$ is the magnetic Reynolds number based on the integral\nscale of turbulence and the turbulent velocity at this scale.\n\nDepending on the chiral nonlinearity parameter $\\lambda_\\mu$\n(see Equation~\\ref{eq_lambamu}), there\nare either two or three stages of magnetic field evolution.\nIn particular, when $\\lambda_\\mu$ is very small,\nthere is sufficient time to produce turbulence and excite the large-scale dynamo,\nso that the magnetic field evolution includes three stages:\n\\begin{asparaenum}[\\it (1)]\n\\item{\nthe small-scale chiral dynamo instability,}\n\\item{\nthe production of chiral magnetically driven MHD turbulence\nand the excitation of a large-scale dynamo instability, and}\n\\item{\nthe saturation of magnetic helicity and magnetic field growth\ncontrolled by the conservation law~(\\ref{CL}).}\n\\end{asparaenum}\n\nIf $\\lambda_\\mu$ is not very small, such that the saturated value\nof the magnetic field is not large, there is not enough time to excite\nthe large-scale dynamo instability. In this case,\nthe magnetic field dynamics includes two stages:\n\\begin{asparaenum}[\\it (1)]\n\\item{\nthe chiral dynamo instability, and}\n\\item{\nthe saturation of magnetic helicity and magnetic field growth\ncontrolled by the conservation law~(\\ref{CL}) for the total chirality.}\n\\end{asparaenum}\n\n\\subsection{Characteristic scales of chiral magnetically driven turbulence}\n\\label{sec_chiScales}\n\nIn the nonlinear regime, once turbulence is fully developed,\nsmall-scale magnetic fields can be excited over a broad range of wavenumbers\nup to the diffusion cutoff wavenumber.\nUsing dimensional arguments and numerical simulations,\n\\cite{BSRKBFRK17} found that, for chiral magnetically driven turbulence,\nthe magnetic energy spectrum $E_{\\rm M}(k,t)$ obeys\n\\begin{equation}\n E_{\\rm M}(k,t)=C_\\mu\\,\\overline{\\rho}\\mu_0^3\\eta^2k^{-2},\n\\label{Cmu}\n\\end{equation}\nwhere $C_\\mu\\approx16$ is a chiral magnetic Kolmogorov-type constant.\nHere, $E_{\\rm M}(k,t)$ is normalized such that\n$\\mathcal{E}_{\\rm M} = \\int E_{\\rm M}(k)~\\mathrm{d}k=\\bra{\\bm{B}^2}\/2$\nis the mean magnetic energy density.\nIt was also confirmed numerically in \\cite{BSRKBFRK17}\nthat the magnetic energy spectrum $E_{\\rm M}(k)$ is bound from above\nby $C_\\lambda \\mu_0\/\\lambda$,\nwhere $C_\\lambda\\approx1$ is another empirical constant.\nThis yields a critical minimum wavenumber,\n\\begin{equation}\n k_\\lambda=\\sqrt{\\overline{\\rho}\\lambda \\frac{C_\\mu}{C_\\lambda}}\\,\\mu_0\\eta,\n\\label{klambda}\n\\end{equation}\nbelow which the spectrum will no longer be proportional to $k^{-2}$.\n\nThe spectrum extends to larger wavenumbers up to a diffusive cutoff wavenumber\n$k_{\\rm diff}$.\nThe diffusion scale for magnetically produced turbulence\nis determined by the condition $\\mbox{\\rm Lu}(k_{\\rm diff})=1$,\nwhere $\\mbox{\\rm Lu}(k)=v_{\\rm A}(k)\/\\eta k$ is the scale-dependent Lundquist number,\n$v_{\\rm A}(k)=(\\bra{\\bm{B}^2}_k\/\\overline{\\rho})^{1\/2}$\nis the scale-dependent Alfv\\'en speed,\nand $\\bra{\\bm{B}^2}_k=2\\int_{k_\\lambda}^{k} E_{\\rm M}(k)~\\mathrm{d}k$.\nTo determine the Alfv\\'en speed, $v_{\\rm A}(k)$, we integrate Equation~(\\ref{Cmu}) over $k$\nand obtain\n\\begin{equation}\nv_{\\rm A}(k)=\\eta \\mu_0\\left({2\\,C_\\mu\\,\\mu_0\\over k_\\lambda}\\right)^{1\/2}\n\\left(1 - {k_\\lambda \\over k}\\right)^{1\/2}.\n\\end{equation}\nThe conditions $\\mbox{\\rm Lu}(k_{\\rm diff})=1$ and $k_{\\rm diff} \\gg k_\\lambda$ yield\n\\begin{equation}\n k_{\\rm diff}=\\sqrt{2}\\left(\\frac{C_\\mu C_\\lambda}{\\lambda_\\mu}\\right)^{1\/4}\\mu_0\n \\approx2.8\\,\\lambda_\\mu^{-1\/4}\\,\\mu_0.\n\\label{eq_kdiffchi}\n\\end{equation}\nNumerical simulations reported in \\cite{BSRKBFRK17} have been performed for\n$0.75 \\leq k_{\\rm diff}\/\\mu_0 \\leq 75$.\nIn the present DNS, we use values in the range from 4.5 to 503.\n\n\n\\subsection{Differences between chiral and standard MHD}\n\\label{sec:difference}\n\nThe system of equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) describing chiral\nMHD exhibits the following key differences from standard MHD:\n\\begin{itemize}\n \\item{\n The presence of the term\n ${\\bm \\nabla} \\times (\\eta \\, \\mu \\, {\\bm{B}})$\n in Equation~(\\ref{ind-DNS}) causes a chiral dynamo\n instability and results in chiral magnetically driven turbulence.}\n \\item{\n Because of the finite value of $\\lambda$, the presence of a helical magnetic\n field affects the evolution of $\\mu$; see Equation~(\\ref{mu-DNS}).}\n \\item{\n For $\\Gamma_\\mathrm{f}=0$, the total chirality,\n $\\int ({\\textstyle{1\\over2}} \\lambda {\\bm A} {\\bm \\cdot} \\bm{B} + \\mu) \\, dV$,\n is strictly conserved, and not just in the limit $\\eta\\to 0$.\n This conservation law determines the level of the saturated magnetic field.}\n \\item{\n The excitation of a large-scale magnetic field is caused by\n (i) the combined action of the chiral dynamo instability\n and the inverse magnetic cascade due to the conservation\n of total chirality, as well as by\n (ii) the chiral $\\alpha_\\mu$ effect resulting in\n chiral magnetically driven turbulence.\n This effect is not related to kinetic helicity\n and becomes dominant at large fluid and magnetic\n Reynolds numbers; see Paper~I.}\n\\end{itemize}\n\n\nThe chiral term in Equation~(\\ref{ind-DNS}) and the evolution\nof $\\mu$ governed by Equation~(\\ref{mu-DNS}) are responsible for\ndifferent behaviors in chiral and standard MHD.\nIn particular, in standard MHD,\nthe following phenomena and a conservation law are established:\n\\begin{itemize}\n \\item{\n The magnetic helicity $\\int {\\bm A} {\\bm \\cdot} \\bm{B} \\, dV$\n is only conserved in the limit of $\\eta \\to 0$.}\n \\item{\n Turbulence does not have an intrinsic source.\n Instead, it can be produced externally by a stirring force,\n or due to large-scale shear at large fluid Reynolds numbers,\n the Bell instability in the presence of a cosmic-ray current,\n \\citep{RNBE2012,BL14},\n the magnetorotational instability \\citep{HGB95,BNST95},\n or just an initial irregular magnetic field \\citep{BKT15}.\n }\n \\item{\n A large-scale magnetic field can be generated by:\n (i) helical turbulence with nonzero mean kinetic helicity\n that is produced either by external helical forcing or by\n rotating, density-stratified, or inhomogeneous turbulence\n (so-called mean-field $\\alpha^2$ dynamo);\n (ii) helical turbulence with large-scale shear, which\n results in an additional mechanism of large-scale dynamo action\n referred to as an $\\alpha \\Omega$ or $\\alpha^2 \\Omega$ dynamo\n \\citep{M78,P79,KR80,ZRS83};\n (iii) nonhelical turbulence with large-scale shear, which causes\n a large-scale shear dynamo \\citep{vishniac1997,RK03,RK04,sridhar2010,sridhar2014};\n and\n (iv) in different nonhelical deterministic flows due to negative\n effective magnetic diffusivity \\citep[in Roberts flow IV, see][]{Devlen+13},\n or time delay of an effective pumping velocity of the magnetic field\n associated with the off-diagonal components of the $\\alpha$\n tensor that are either antisymmetric (known as the $\\gamma$\n effect) in Roberts flow III or symmetric in Roberts flow II;\n see \\cite{Rheinhardt+14}.\n All effects in items (i)--(iv) can work in chiral MHD as well.\n However, which one of these effects is dominant depends on the flow\n properties and the\n governing parameters.\n}\n \\end{itemize}\n\n\n\\subsection{DNS with the {\\sc Pencil Code}}\n\\label{sec:Penci-Code}\n\nWe solve Equations~(\\ref{ind-NS})--(\\ref{mu-NS})\nnumerically using the {\\sc Pencil Code}.\nThis code uses sixth-order explicit finite differences in space\nand a third-order accurate time-stepping method \\citep{BD02,Bra03}.\nThe boundary conditions are periodic in all three directions.\nAll simulations presented in Sections~\\ref{sec:lamdyn} and~\\ref{sec_turbdynamo1}\nare performed without external forcing of turbulence.\nIn Section~\\ref{sec_turbdynamo} we apply a turbulent forcing function\n$\\bm{f}$ in the Navier-Stokes equation, which consists of random plane\ntransverse white-in-time, unpolarized waves.\nIn the following, when we discuss numerical simulations,\nall quantities are considered as dimensionless quantities, and we drop the\n``tildes'' in Equations~(\\ref{ind-NS})--(\\ref{mu-NS}) from now on.\nThe wavenumber $k_1=2\\pi\/L$ is based on the size of the box $L=2\\pi$.\nIn all runs, we set $k_1=1$, $c_\\mathrm{s}=1$ and the mean fluid density $\\overline{\\rho}=1$.\n\n\n\n\\begin{table}\n\\centering\n\\caption{\nOverview of Runs for the Laminar $v_\\mu^2$ Dynamos\n(Reference Run in Bold)}\n \\begin{tabular}{l|lllll}\n \\hline\n \\hline\n \\\\\n\tsimulation \t& ${\\rm Pr}_{_{\\rm M}}$ & $\\lambda_\\mu$ & $\\dfrac{{\\rm Ma}_\\mu}{10^{-3}}$ & $\\dfrac{k_\\lambda}{10^{-4}\\mu_0}$ & $\\dfrac{k_\\mathrm{diff}}{\\mu_0}$\t\\\\\t \t\t\\\\\n \\hline\n \tLa2-1B \t& $1.0$\t\t& $1\\times10^{-8}$\t& $2$\t& $4.0$ & $283$ \\\\\n \tLa2-2B \t& $0.5$\t\t& $4\\times10^{-8}$\t& $4$\t& $8.0$ & $200$ \\\\\n \tLa2-3B \t& $0.2$\t\t& $2.5\\times10^{-7}$\t& $10$\t& $20$ & $126$ \\\\\n \tLa2-4B \t& $2.0$\t\t& $2.5\\times10^{-9}$\t& $1$\t& $2.0$ & $400$ \\\\\n \tLa2-5B \t& $1.0$\t\t& $1\\times10^{-9}$\t& $1.5$\t& $1.3$ & $503$ \\\\\n La2-5G \t& $1.0$\t\t& $1\\times10^{-8}$\t& $1.5$\t& $4.0$ & $283$ \\\\\n La2-6G \t& $1.0$\t\t& $1\\times10^{-5}$\t& $2$\t& $130$ & $50$ \\\\\n \tLa2-7B \t& $1.0$\t\t& $1\\times10^{-9}$\t& $3$\t& $4.0$ & $283$ \\\\\n La2-7G \t& $1.0$\t\t& $1\\times10^{-9}$\t& $3$\t& $4.0$ & $283$ \\\\\n \tLa2-8B \t& $1.0$\t\t& $1\\times10^{-9}$\t& $5$\t& $4.0$ & $283$ \\\\\n La2-8G \t& $1.0$\t\t& $1\\times10^{-9}$\t& $5$\t& $4.0$ & $283$ \\\\\n \tLa2-9B \t& $1.0$\t\t& $1\\times10^{-9}$\t& $10$\t& $4.0$ & $283$ \\\\\n La2-9G \t& $1.0$\t\t& $1\\times10^{-9}$\t& $10$\t& $4.0$ & $283$ \\\\\n \tLa2-10B \t& $1.0$\t\t& $1\\times10^{-5}$\t& $20$\t& $130$ & $50$ \\\\\n \tLa2-10Bkmax \t& $1.0$\t\t& $1\\times10^{-5}$\t& $20$\t& $130$ & $50$ \\\\\n \tLa2-10G \t& $1.0$\t\t& $1\\times10^{-5}$\t& $20$\t& $4.0$ & $283$ \\\\\n \tLa2-11B \t& $1.0$\t\t& $1\\times10^{-9}$\t& $50$\t& $1.3$ & $503$ \\\\\n \tLa2-11G \t& $1.0$\t\t& $1\\times10^{-8}$\t& $50$\t& $4.0$ & $283$ \\\\\n \tLa2-12B \t& $1.0$\t\t& $1\\times10^{-9}$\t& $2$\t& $1.3$ & $503$ \\\\\n \tLa2-13B \t& $1.0$\t\t& $1\\times10^{-7}$\t& $2$\t& $13$ & $159$ \\\\\n \tLa2-14B \t& $1.0$\t\t& $3\\times10^{-9}$\t& $2$\t& $2.2$ & $382$ \\\\\n \t\\textbf{La2-15B} & $\\mathbf{1.0}$& $\\mathbf{1\\times10^{-5}}$& $\\mathbf{2}$ & $\\mathbf{130}$ & $\\mathbf{50}$\\\\\n \tLa2-16B \t& $1.0$\t\t& $3\\times10^{-8}$\t& $2$\t& $6.9$ & $215$ \\\\\n \\hline\n \\hline\n \\end{tabular}\n \\label{table_simulations_vmu2}\n\\end{table}\n\\medskip\n\n\n\\section{Laminar chiral dynamos}\n\\label{sec:lamdyn}\n\n\nIn this section, we study numerically laminar chiral dynamos\nin the absence of any turbulence (externally or chiral magnetically driven).\n\n\\subsection{Numerical setup}\n\nParameters and initial conditions for all\nlaminar dynamo simulations are listed in\nTables~\\ref{table_simulations_vmu2} and~\\ref{table_simulations_vmushear}.\nAll of these simulations are two-dimensional and have a resolution of $256^2$.\nRuns with names ending with `B' are with the initial conditions\nfor the magnetic field in the form of a Beltrami magnetic field:\n$B(t=0)=10^{-4}$(0, sin$\\,x$, cos$\\,x$),\nwhile runs with names ending with `G' are initiated with Gaussian noise.\nThe initial conditions for the velocity field for the laminar $v_\\mu^2$ dynamo\nare $U(t=0)=(0,0,0)$, and for the laminar chiral--shear dynamos\n(the $v_\\mu^2$--shear or $v_\\mu$--shear dynamos) are\n$U(t=0)=(0, S_0~\\mathrm{cos}\\,x, 0)$, with\nthe dimensionless shear rate $S_0$ given for all runs.\n\nWe set the chiral Prandtl number $\\Prmu=1$ in all runs.\nIn many runs the magnetic Prandtl number ${\\rm Pr}_{_{\\rm M}}=1$ (except in several\n\nruns for the laminar $v_\\mu^2$ dynamo, see Table~\\ref{table_simulations_vmu2}).\nThe reference runs for the laminar $v_\\mu^2$ dynamo (La2-15B)\nand the chiral--shear dynamos (LaU-4G) are shown in bold in\nTables~\\ref{table_simulations_vmu2} and~\\ref{table_simulations_vmushear}.\nThe results of numerical simulations are compared with\ntheoretical predictions.\n\n\n\\subsection{Laminar $v_\\mu^2$ dynamo}\n\\label{sec_laminara2}\n\nWe start with the situation without an imposed fluid flow,\nwhere the chiral laminar $v_\\mu^2$ dynamo can be excited.\n\n\\subsubsection{Theoretical aspects}\n\nIn this section, we outline the theoretical predictions\nfor a laminar chiral dynamo; for details see Paper~I.\nTo determine the chiral dynamo growth rate, we seek a solution of\nthe linearized Equation~(\\ref{ind-DNS})\nfor small perturbations of the following form:\n$\\bm{B}(t,x,z)=B_y(t,x,z) {\\bm e}_y + \\bm{\\nabla} \\times [A(t,x,z) {\\bm e}_y]$,\nwhere ${\\bm e}_y$ is the unit vector in the $y$ direction.\n\nWe consider the equilibrium configuration:\n$\\mu=\\mu_0={\\rm const}$ and ${\\bm{U}}_0=0$.\nThe functions $B_y(t,x,z)$ and $A(t,x,z)$ are determined by the equations\n\\begin{eqnarray}\n \\frac{\\partial A(t,x,z)}{\\partial t} &=& v_\\mu \\, B_y + \\eta \\Delta A,\n\\label{A-eq} \\\\\n\\frac{\\partial B_y(t,x,z)}{\\partial t}&=& -v_\\mu \\, \\Delta A + \\eta \\Delta B_y ,\n\\label{By-eq}\n\\end{eqnarray}\nwhere $v_\\mu=\\eta \\, \\mu_0$, $\\Delta=\\nabla_x^2 + \\nabla_z^2$,\nand the remaining components of the magnetic field are given by\n$B_x=-\\nabla_z A$ and $B_z=\\nabla_x A$.\nWe seek a solution to Equations~(\\ref{A-eq}) and~(\\ref{By-eq}) of the form\n$A, B_y \\propto \\exp[\\gamma t + i (k_x x + k_z z)]$.\nThe growth rate of the dynamo instability is given by\n\\begin{eqnarray}\n\\gamma = |v_\\mu \\, k| - \\eta k^2 ,\n\\label{gamma}\n\\end{eqnarray}\nwhere $k^2=k_x^2 + k_z^2$.\nThe dynamo instability is excited (i.e., $\\gamma> 0)$ for $k < |\\mu_0|$.\nThe maximum growth rate of the dynamo instability,\n\\begin{eqnarray}\n\\gamma^{\\rm max}_\\mu = \\frac{v_\\mu^2}{4 \\eta},\n\\label{gamma-max}\n\\end{eqnarray}\nis attained at\n\\begin{equation}\nk_\\mu =\\frac12|\\mu_0|.\n\\label{eq_kmax}\n\\end{equation}\n\n\n\\subsubsection{Time evolution}\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=\\columnwidth]{La2_15B__ts}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$ dynamo:}\ntime evolution of $B_\\mathrm{rms}$ (solid black line),\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$ (dashed gray line),\n$\\mu_\\mathrm{rms}$ (multiplied by\n$2\/\\lambda$, dotted blue line), and\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle +2\\mu_\\mathrm{rms}\/\\lambda$\n(dash-dotted red line) for reference run La2-15B\n(see table \\ref{table_simulations_vmu2}).\n}\n\\label{fig__La2_15B__ts}\n\\end{figure}\n\nIn Figure~\\ref{fig__La2_15B__ts} we show the time evolution\nof the rms magnetic field $B_\\mathrm{rms}$,\nthe magnetic helicity $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\nthe chemical potential $\\mu_\\mathrm{rms}$ (multiplied by a factor of\n$2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle\n+2\\mu_\\mathrm{rms}\/\\lambda$ for reference run La2-15B.\nIn simulations, the time is measured in units of diffusion time\n$t_\\eta=(\\eta k_1^2)^{-1}$.\nThe initial conditions for the magnetic field are chosen in the form of\na Beltrami field on $k=k_1=1$.\n\nThe magnetic field is amplified\nexponentially over more than four orders\nof magnitude until it saturates after roughly eight diffusive times.\nWithin the same time, the magnetic helicity\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$ increases\nover more than eight orders of magnitude.\nSince the sum of magnetic helicity and $2 \\mu\/\\lambda$ is conserved,\nthe chemical potential $\\mu$ decreases, in a nonlinear era\nof evolution, from the initial value $\\mu_0=2$ to $\\mu=1$,\nresulting in a saturation of the laminar $v_\\mu^2$ dynamo.\n\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{Gamma_etamu}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$ dynamo:}\ngrowth rates as a function of ${\\rm Ma}_\\mu$,\nfor simulations with $\\mu_0=2$.\nThe black line is the theoretical prediction for the maximum growth rate\n$\\gamma^\\mathrm{max}_\\mu$\n(see Equation~\\ref{gamma-max}) that is attained at\n$k_\\mu=\\mu_0\/2=1$\n(see Equation~\\ref{eq_kmax}).\nThe runs with Gaussian initial fields, shown as red diamonds,\nlie on the theoretically predicted\n$\\gamma^\\mathrm{max}_\\mu$.\nThe dotted line corresponds to the theoretical prediction for the growth rate\n$\\gamma(k=1)$ at the scale of the box.\nThe runs with an initial magnetic Beltrami field on $k=1$,\nshown as blue diamonds, lie on the theoretically predicted dotted\ncurve $\\gamma(k=1)$.\n}\n\\label{fig__Gamma_eta}\n\\end{figure}\n\n\\subsubsection{Dynamo growth rate}\n\nIn Figure~\\ref{fig__Gamma_eta}, we show the growth rate of the magnetic field\nas a function of the chiral Mach number, ${\\rm Ma}_\\mu$.\nThe black solid line\nin this figure shows the theoretical prediction for the maximum\ngrowth rate $\\gamma^\\mathrm{max}_\\mu$ that is attained at $k_\\mu=\\mu_0\/2=1$;\nsee Equations~(\\ref{gamma-max}) and (\\ref{eq_kmax}).\nWhen the initial magnetic field is distributed over all spatial scales,\nlike in the case of initial magnetic Gaussian noise, in which\nthere is a nonvanishing magnetic field at\n$k_\\mu$; that is inside the computational domain,\nthe initial magnetic field is excited with the maximum growth rate\nas observed in the simulations.\nConsequently, the runs with Gaussian initial fields shown as red diamonds in\nFigure~\\ref{fig__Gamma_eta}, lie on the theoretical curve\n$\\gamma^\\mathrm{max}_\\mu$.\nThe dotted line in Figure~\\ref{fig__Gamma_eta} corresponds to the theoretical\nprediction for the growth rate $\\gamma$ at the scale of the box $(k=1)$.\nThe excitation of the magnetic field from an initial Beltrami field on $k=1$\noccurs with growth rates in agreement with the theoretical dotted curve;\nsee blue diamonds in Figure~\\ref{fig__Gamma_eta}.\n\n\n\\subsubsection{Dependence on initial conditions}\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=\\columnwidth]{La2_B_t}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$ dynamo:}\nTime evolution of $B_\\mathrm{rms}$ for two different initial conditions.\nThe black line is for the dynamo instability started from an initial\nBeltrami field at $k=1$ (run La2-10B), while the blue line\nis for an initial Beltrami field with\n$k=10$ (run La2-10Bkmax).\nFits in different regimes are indicated by thin lines.\nBoth runs are for the initial value $\\mu_0=20$, so that $k_\\mu=10$;\nand $\\gamma^\\mathrm{max}_\\mu=0.1$ (see~Equation~\\ref{gamma-max}).\n}\n\\label{fig__La2_B_t}\n\\end{figure}\n\nThe initial conditions for the magnetic field are important mostly at early times.\nIf the magnetic field is initially concentrated on the box scale, we expect to\nobserve a growth rate $\\gamma(k=1)$ as given by Equation~(\\ref{gamma}).\nAt later times, the spectrum of the magnetic field can, however, be changed,\ndue to mode coupling, and be amplified with a larger growth rate.\nThis behavior is observed in Figure~\\ref{fig__La2_B_t}, where\nan initial Beltrami field with $k=10$ is excited with maximum growth rate,\nsince $\\mu_0=20$.\nIn Figure~\\ref{fig__La2_B_t} we also consider another situation\nwhere the dynamo is started from an initial Beltrami field with $k=1$ (La2-10B).\nIn this case, the dynamo starts with a growth rate $\\gamma=0.019$,\nwhich is consistent with the theoretical prediction for $\\gamma(k=1)$.\nLater, after approximately $0.4\\,t_\\eta$, the dynamo growth\nrate increases up to the value $\\gamma=0.07$, which is close to\nthe maximum growth rate $\\gamma^\\mathrm{max}_\\mu=0.1$.\n\n\n\\subsubsection{Saturation}\n\nThe parameter $\\lambda$ in the evolution Equation~(\\ref{mu-DNS}),\nor the corresponding dimensionless parameter $\\lambda_\\mu$\nin Equation~(\\ref{mu-NS}), for the chiral chemical potential determines\nthe nonlinear saturation of the chiral dynamo.\nWe determine the saturation value of the\nmagnetic field $B_\\mathrm{sat}$ numerically for different values of $\\lambda_\\mu$;\nsee Figure~\\ref{fig_Bsat_lambda}.\nWe find that the saturation value of the magnetic field\nincreases with decreasing $\\lambda_\\mu$.\nThis can be expected from the conservation law (\\ref{CL}).\nIf the initial magnetic energy is very small,\nwe find from Equation~(\\ref{CL}) the following estimate\nfor the saturated magnetic field during laminar chiral dynamo action:\n\\begin{eqnarray}\n B_\\mathrm{sat} \\sim \\left[\\frac{\\mu_0(\\mu_0 -\\mu_\\mathrm{sat})}{\\lambda}\\right]^{1\/2},\n\\label{eq_Bsat}\n\\end{eqnarray}\nwhere $\\mu_\\mathrm{sat}$ is the chiral chemical potential at saturation, and\nwe use the estimate $A$ by $2 B \/\\mu_0$.\nInspection of Figure~\\ref{fig_Bsat_lambda} demonstrates\na good agreement between\ntheoretical (solid line) and numerical results (blue diamonds).\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=\\columnwidth]{Bsat_lambda}\n\\end{center}\n\\caption{{\\bf Laminar $v_\\mu^2$ dynamo:}\nthe saturation magnetic field strength for simulations\nwith different $\\lambda_\\mu$.\nDetails for the different runs, given by labeled blue diamonds, can be found in\nTable~\\ref{table_simulations_vmu2}.}\n\\label{fig_Bsat_lambda}\n\\end{figure}\n\n\n\\subsubsection{Effect of a nonvanishing flipping rate}\n\\label{sec_flip}\n\nIn this section, we consider the influence of a nonvanishing chiral\nflipping rate on the $v_\\mu^2$ dynamo.\nA large flipping rate $\\Gamma_\\mathrm{f}$ decreases the chiral chemical\npotential $\\mu$; see Equation~(\\ref{mu-DNS}). It can stop\nthe growth of the magnetic field caused by the chiral dynamo instability.\n\nQuantitatively, the influence of the flipping term can be estimated by\ncomparing the last two terms of Equation~(\\ref{mu-DNS}).\nThe ratio of these terms is\n\\begin{eqnarray}\n f_\\mu \\equiv \\frac{\\Gamma_\\mathrm{f}\\mu_0}{\\lambda\\eta\\mu_0 B_\\mathrm{sat}^2}\n = \\frac{\\Gamma_\\mathrm{f}}{\\eta\\mu_0^2},\n\\label{eq_fmu}\n\\end{eqnarray}\nwhere we have used Equation~(\\ref{eq_Bsat}) with $\\mu_\\mathrm{sat} \\ll \\mu_0$\nfor the saturation value of the magnetic field strength.\nIn Figure~\\ref{fig_ts_flip} we present the time evolution of\n$B_\\mathrm{rms}$ and $\\mu_\\mathrm{rms}$\nfor different values of $f_\\mu$.\nThe reference run La2-15B, with zero flipping rate ($f_\\mu=0$),\nhas been repeated with a finite flipping term.\nAs a result, the magnetic field grows more\nslowly in the nonlinear era, due to the flipping effect, and it\ndecreases the saturation level of the magnetic field; see Figure~\\ref{fig_ts_flip}.\nFor larger values of $f_\\mu$, the chiral chemical potential\n$\\mu$ decreases quickly, leading to strong quenching of the $v_\\mu^2$ dynamo;\nsee the blue lines in Figure~\\ref{fig_ts_flip}.\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[width=\\columnwidth]{ts_flip.ps}\n\\end{center}\n\\caption{{\\bf Laminar $v_\\mu^2$ dynamo:}\ntime evolution of the chiral chemical potential $\\mu_{\\rm rms}$ (black lines) and\nthe magnetic field $B_{\\rm rms}$ (blue lines)\nfor $f_\\mu=0$ (solid), $f_\\mu = 0.0025$ (dashed), and $f_\\mu = 0.01$ (dotted).\n}\n\\label{fig_ts_flip}\n\\end{figure}\n\n\n\\subsection{Laminar chiral--shear dynamos}\n\\label{sec_laminarashear}\n\n\nIn this section, we consider laminar chiral dynamos\nin the presence of an imposed shearing velocity.\nSuch a nonuniform velocity profile can be created\nin different astrophysical flows.\n\n\\subsubsection{Theoretical aspects}\n\nWe start by outlining the theoretical predictions\nfor laminar chiral dynamos in the presence\nof an imposed shearing velocity; for details see Paper~I.\nWe consider the equilibrium configuration specified by the shear velocity\n${\\bm{U}}_{\\rm eq}=(0,S\\, x,0)$,\nand $\\mu=\\mu_0=$ const.\nThis implies that the fluid has nonzero vorticity\n${\\bm W} = (0,0,S)$ similar to differential (nonuniform) rotation.\nThe functions $B_y(t,x,z)$ and $A(t,x,z)$ are determined by\n\\begin{eqnarray}\n&&\\frac{\\partial A(t,x,z)}{\\partial t} = v_\\mu \\, B_y + \\eta \\Delta A,\n\\label{A1-eq}\\\\\n&&\\frac{\\partial B_y(t,x,z)}{\\partial t} = - S\\nabla_z A - v_\\mu \\, \\Delta A\n+ \\eta \\Delta B_y .\n\\label{By1-eq}\n\\end{eqnarray}\nWe look for a solution to Equations~(\\ref{A1-eq})\nand~(\\ref{By1-eq}) of the form $A, B_y \\propto \\exp[\\gamma t + i (k_x x + k_z z-\n\\omega t)]$.\nThe growth rate of the dynamo instability and the frequency of the dynamo waves\nare given by\n\\begin{eqnarray}\n \\gamma = {|v_\\mu \\, k| \\over \\sqrt{2}} \\,\n\\left\\{1 + \\left[1 + \\left({S k_z \\over v_\\mu \\, k^2}\\right)^2 \\right]^{1\\over 2}\n\\right\\}^{1\\over 2} - \\eta k^2\n\\label{eq_gamma_aS}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n \\omega= {\\rm sgn} \\left(\\mu_0 k_z\\right) \\, {S k_z \\over \\sqrt{2} k}\n\\,\n\\left\\{1 + \\left[1 + \\left({S k_z \\over v_\\mu \\, k^2}\\right)^2\\right]^{1\\over 2}\n\\right\\}^{-{1\\over 2}} ,\n\\nonumber\\\\\n\\label{omega10}\n\\end{eqnarray}\nrespectively.\nThis solution describes a laminar $v_\\mu^2$--shear dynamo\nfor arbitrary values of the shear rate $S$.\n\n\nNext, we consider a situation where the shear term on the\nright side of Equation~(\\ref{By1-eq}) dominates,\nthat is, where $|S \\nabla_z A| \\gg |v_\\mu \\, \\Delta A|$.\nThe growth rate of the dynamo instability\nand the frequency of the dynamo waves are then given by\n\\begin{eqnarray}\n&& \\gamma = \\left({ |v_\\mu \\, S \\, k_z| \\over 2}\\right)^{1\/2} - \\eta k^2 ,\n\\label{gamma1}\\\\\n&& \\omega= {\\rm sgn} \\left(\\mu_0 k_z\\right) \\, \\left({|v_\\mu \\, S \\,\nk_z| \\over 2}\\right)^{1\/2} .\n\\label{omega}\n\\end{eqnarray}\nThe dynamo is excited for $k < |v_\\mu \\, S \\, k_z \/2\\eta^2|^{1\/4}$.\nThe maximum growth rate of the dynamo instability\nand the frequency $\\omega=\\omega (k=k_z^\\mu)$\nof the dynamo waves are attained at\n\\begin{equation}\n k_z^\\mu ={1 \\over 4} \\left({2|S \\, v_\\mu| \\over \\eta^2} \\right)^{1\/3},\n\\label{kz-max}\n\\end{equation}\nand are given by\n\\begin{eqnarray}\n&& \\gamma^{\\rm max}_\\mu\n = {3 \\over 8} \\left({S^2 \\, v_\\mu^2 \\over 2\\eta}\\right)^{1\/3}\n - \\eta k_x^2,\n\\label{gam-max}\\\\\n&& \\omega(k=k_z^\\mu)\n = {{\\rm sgn} \\left(v_\\mu \\, k_z\\right) \\over 2\\eta} \\,\n \\left({S^2 \\, v_\\mu^2 \\over 2 \\eta}\\right)^{1\/3} .\n\\label{omega-max}\n\\end{eqnarray}\nThis solution describes the laminar $v_\\mu$--shear dynamo.\n\n\n\\subsubsection{Simulations of the laminar $v_\\mu$--shear dynamo}\n\n\n\\begin{table}\n\\centering\n\\caption{\nOverview of Runs for the Chiral--Shear Dynamos\n(Reference Run in Bold)}\n \\begin{tabular}{l|llllll}\n \\hline\n \\hline\n \\\\\n\tsimulation \t& $\\lambda_\\mu$ \t& $\\dfrac{{\\rm Ma}_\\mu}{10^{-3}}$ & $u_S$\t& $\\dfrac{k_\\lambda}{10^{-4}\\mu_0}$ & $\\dfrac{k_\\mathrm{diff}}{\\mu_0}$\t\\\\\\\\\t \t\t\\\\\n \\hline\n \tLaU-1B\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.01\t& 1.3 & 503 \\\\\n \tLaU-1G\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.01\t& 1.3 & 503 \\\\\n \tLaU-2B\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.02\t& 1.3 & 503 \\\\\n \tLaU-2G\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.02\t& 1.3 & 503 \\\\\n \tLaU-3B\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.05\t& 1.3 & 503 \\\\\n \tLaU-3G\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.05\t& 1.3 & 503 \\\\\n \tLaU-4B\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.10\t& 1.3 & 503 \\\\\n \t\\textbf{LaU-4G} & $\\mathbf{1\\times10^{-5}}$\t& $\\mathbf{2.0}$ & $\\mathbf{0.10}$ & $\\mathbf{126}$ & $\\mathbf{50}$ \\\\\n \tLaU-5B\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.20 \t& 1.3 & 503 \\\\\n \tLaU-5G\t \t& $1\\times10^{-9}$\t& $2.0$\t& 0.20 & 1.3 & 503 \\\\\n \tLaU-6B\t \t& $1\\times10^{-9}$\t& $2.0$ & 0.50\t& 1.3 & 503 \\\\\n \tLaU-6G\t \t& $1\\times10^{-9}$\t& $2.0$ & 0.50\t& 1.3 & 503 \\\\\n \tLaU-7G\t \t& $1\\times10^{-8}$\t& $10$ & 0.01\t& 4.0 & 283 \\\\\n \tLaU-8G\t \t& $1\\times10^{-8}$\t& $10$ & 0.05\t& 4.0 & 283 \\\\\n \tLaU-9G\t \t& $1\\times10^{-8}$\t& $10$ & 0.10\t& 4.0 & 283 \\\\\n \tLaU-10G\t\t& $1\\times10^{-8}$\t& $10$\t& 0.50\t& 4.0 & 283 \\\\\n \\hline\n \\hline\n \\end{tabular}\n \\label{table_simulations_vmushear}\n\\end{table}\n\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=\\columnwidth]{LaU_9G__ts}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu$--shear dynamo:}\ntime evolution of the magnetic field\n$B_\\mathrm{rms}$, the velocity $u_\\mathrm{rms}$, the magnetic helicity\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\nthe chemical potential $\\mu_\\mathrm{rms}$ (multiplied by a factor\nof $2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B}\n\\rangle +2\\mu_\\mathrm{rms}\/\\lambda$\n(run LaU-4G).}\n\\label{fig_AlphaShear_t}\n\\end{figure}\n\nSince our simulations have periodic boundary conditions, we model shear\nvelocities as $U_S=(0, u_S \\cos x, 0)$.\nThe mean shear velocity $\\overline{u}_S$ over half the box is\n$\\overline{u}_S = (2\/\\pi) u_S$.\nIn Figure~\\ref{fig_AlphaShear_t} we show the time evolution of the magnetic field\n(which starts to be excited from a Gaussian initial field),\nthe velocity $u_\\mathrm{rms}$, the magnetic helicity\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\nthe chemical potential $\\mu_\\mathrm{rms}$ (multiplied by a factor\nof $2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B}\n\\rangle +2\\mu_\\mathrm{rms}\/\\lambda$ for run LaU-4G.\nThe growth rate for the chiral--shear dynamo (the $v_\\mu^2$--shear dynamo)\nis larger than that for the laminar chiral dynamo (the $v_\\mu^2$--dynamo).\nAfter a time of roughly $0.03~t_\\eta$, the system enters a nonlinear\nphase, in which the velocity field is affected by the magnetic field,\nbut the magnetic field can still increase slowly.\nSaturation of the dynamo occurs after approximately $0.1~t_\\eta$.\n\nFor Gaussian initial fields, we have observed a short delay in the\ngrowth of the magnetic field.\nIn both cases, the dynamo growth rate increases with increasing shear.\nAs for the chiral $v_\\mu^2$ dynamo, we observe perfect\nconservation of the quantity $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle +\n2\\mu_\\mathrm{rms}\/\\lambda$ in the simulations of the laminar $v_\\mu$--shear\ndynamo.\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=\\columnwidth]{Beltrami_GammaOmega_Us}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu$--shear dynamo:}\ngrowth rate (top panel) and dynamo frequency (bottom panel) as a function\nof the mean shear $\\overline{u}_S$ for the Beltrami initial field\n(runs LaU-$n$B with $n=1$--$6$; see Table~\\ref{table_simulations_vmushear}).}\n\\label{Gamma_Us_Beltrami}\n\\end{figure}\n\nIn Figure~\\ref{Gamma_Us_Beltrami} we show\nthe theoretical dependence of the growth rate $\\gamma$ and the dynamo frequency\n$\\omega$ on the shear velocity $\\overline{u}_S$\nfor Beltrami initial conditions at different wavenumbers;\nsee Equations~(\\ref{gamma1}) and~(\\ref{gam-max}).\nThe dynamo growth rate is estimated from an exponential fit.\nThe result of the fit depends slightly on the fitting regime, leading\nto an error of the order of 10\\%.\nThe dynamo frequency is determined afterward by dividing the magnetic field\nstrength by $\\mathrm{exp}(\\gamma t)$ and fitting a sine function.\nDue to the small amplitude and a limited number of periods of dynamo waves,\nthe result is sensitive to the fit regime considered.\nHence we assume a conservative error of 50\\% for the dynamo frequency.\nThe blue diamonds correspond to the numerical results.\nWithin the error bars, the theoretical and numerical results are in agreement.\n\n\\subsubsection{Simulations of the laminar $v_\\mu^2$--shear dynamo}\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=\\columnwidth]{a2shear_Gamma_Us}\n\\end{center}\n\\caption[]{\\textbf{Laminar $v_\\mu^2$--shear dynamo:}\ngrowth rate $\\gamma$ as a function of\nmean shear $\\overline{u}_S$.\nFor comparison, we plot the maximum growth rate of $v_\\mu^2$ dynamo\n(\\ref{gamma-max}) and of the $v_\\mu$--shear dynamo (\\ref{gam-max}).\nThe solid black line is the theoretically predicted maximum growth rate\n(see Equation~\\ref{eq_gamma_aS}).\n}\\label{fig_Gamma_Us_aShear}\n\\end{figure}\n\n\nThe growth rate of chiral--shear dynamos versus mean shear in\nthe range between $u_S=0.01$\nand $0.5$ is shown in Figure~\\ref{fig_Gamma_Us_aShear}.\nWe choose a large initial value of the chemical potential, i.e.\\\n$\\mu_0=10$, to ensure that $k_\\mathrm{max}$ is\ninside the box for all values of $\\overline{u}_S$.\nWe overplot the growth rates found from the simulations with the maximum growth\nrate given by Equation (\\ref{eq_gamma_aS}).\nIn addition, we show the theoretical predictions for the limiting cases\nof the $v_\\mu^2$ and\n$v_\\mu$--shear dynamos; see Equations~(\\ref{gamma-max}) and (\\ref{gam-max}).\nInspection of Figure~\\ref{fig_Gamma_Us_aShear} shows that the results\nobtained from the simulations agree with theoretical predictions.\n\n\n\n\n\\section{Chiral magnetically driven turbulence}\n\\label{sec_turbdynamo1}\n\n\nIn this section we show that the CME\ncan drive turbulence via the Lorentz force in the Navier-Stokes equation.\nWhen the magnetic field increases exponentially, due to the\nsmall-scale chiral magnetic dynamo with growth rate $\\gamma$,\nthe Lorentz force,\n$({\\bm \\nabla} {\\bm \\times} {\\bm{B}}) {\\bm \\times} \\bm{B}$,\nincreases at the rate $2\\gamma$.\nThe laminar dynamo occurs only up to the first nonlinear phase,\nwhen the Lorentz force starts to produce turbulence\n(referred to as chiral magnetically driven turbulence).\nWe will also demonstrate here that, during the second nonlinear phase,\na large-scale dynamo is excited by the chiral $\\alpha_\\mu$ effect\narising in chiral magnetically driven turbulence.\nThe chiral $\\alpha_\\mu$ effect was studied using different\nanalytical approaches in Paper~I.\nThis effect is caused by an interaction of the CME\nand fluctuations of the small-scale\ncurrent produced by tangling magnetic fluctuations.\nThese fluctuations are generated by tangling of the large-scale\nmagnetic field through sheared velocity fluctuations.\nOnce the large-scale magnetic field becomes strong enough,\nthe chiral chemical potential decreases, resulting in the saturation\nof the large-scale dynamo instability.\n\nThis situation is similar to that of\ndriving small-scale turbulence via the Bell instability\nin a system with an external cosmic-ray current\n\\citep{B04,BL14}, and the generation of a\nlarge-scale magnetic field by the Bell turbulence; see \\citet{RNBE2012}\nfor details.\n\n\\subsection{Mean-field theory for large-scale dynamos}\n\\label{sec_meanfieldMHD}\n\nIn this section, we outline the theoretical predictions\nfor large-scale dynamos based on mean-field theory;\nsee Paper~I for details.\nThe mean induction equation is given by\n\\begin{eqnarray}\n\\frac{\\partial \\overline{\\mbox{\\boldmath $B$}}{}}{}{\\partial t} &=&\n\\bm{\\nabla} \\times \\left[\\overline{\\bm{U}} \\times \\overline{\\mbox{\\boldmath $B$}}{}}{\n+ (\\overline{v}_\\mu + \\alpha_\\mu) \\overline{\\mbox{\\boldmath $B$}}{}}{\n- (\\eta+ \\, \\eta_{_{T}})\\bm{\\nabla} \\times \\overline{\\mbox{\\boldmath $B$}}{}}{\\right], \\nonumber\\\\\n\\label{ind4-eq}\n\\end{eqnarray}\nwhere $\\overline{v}_\\mu = \\eta \\overline{\\mu}_{0}$, and we consider the following equilibrium\nstate: $\\overline{\\mu}_{\\rm eq}=\\overline{\\mu}_{0}={\\rm const}$ and ${\\bm \\overline{\\bm{U}}}_{\\rm eq}=0$.\nThis mean-field equation contains additional terms that are\nrelated to the chiral $\\alpha_\\mu$ effect and the turbulent magnetic diffusivity\n$\\eta_{_{T}}$.\nIn the mean-field equation, the chiral $v_\\mu$ effect is replaced\nby the mean chiral $\\overline{v}_\\mu$ effect.\nNote, however, that at large fluid and magnetic Reynolds numbers, the $\\alpha_\\mu$ effect\ndominates the $\\overline{v}_\\mu$ effect.\n\nTo study the large-scale dynamo,\nwe seek a solution to Equation~(\\ref{ind4-eq}), for small perturbations in\nthe form\n$\\overline{\\mbox{\\boldmath $B$}}{}}{(t,x,z)=\\overline{B}_y(t,x,z) {\\bm e}_y + \\bm{\\nabla} \\times\n[\\overline{A}(t,x,z) {\\bm e}_y]$,\nwhere ${\\bm e}_y$ is the unit vector directed along the $y$ axis.\nThe functions $\\overline{B}_y(t,x,z)$ and $\\overline{A}(t,x,z)$ are determined by\n\\begin{multline}\n \\frac{\\partial \\overline{A}(t,x,z)}{\\partial t}\n =(\\overline{v}_\\mu + \\alpha_\\mu)\\, \\overline{B}_y\n + (\\eta+ \\, \\eta_{_{T}}) \\, \\Delta \\overline{A},\n \\label{me-A-eq}\n\\end{multline}\n\\begin{multline}\n \\frac{\\partial \\overline{B}_y(t,x,z)}{\\partial t}\n =-(\\overline{v}_\\mu + \\alpha_\\mu) \\, \\Delta \\overline{A}\n + (\\eta+ \\, \\eta_{_{T}}) \\, \\Delta \\overline{B}_y ,\n\\label{me-By-eq}\n\\end{multline}\nwhere $\\Delta=\\nabla_x^2 + \\nabla_z^2$, and the other components of the magnetic\nfield are $\\overline{B}_x=-\\nabla_z \\overline{A}$ and $\\overline{B}_z=\\nabla_x \\overline{A}$.\n\nWe look for a solution of the mean-field equations~(\\ref{me-A-eq})\nand~(\\ref{me-By-eq}) in the form\n\\begin{eqnarray}\n \\overline{A}, \\overline{B}_y \\propto \\exp[\\gamma t + i (k_x x + k_z z)],\n\\end{eqnarray}\nwhere the growth rate of the large-scale dynamo instability is given by\n\\begin{eqnarray}\n\\gamma = |(\\overline{v}_\\mu + \\alpha_\\mu)\\, k| - (\\eta+ \\, \\eta_{_{T}}) \\, k^2,\n\\label{gamma_turb}\n\\end{eqnarray}\nwith $k^2=k_x^2 + k_z^2$.\nThe maximum growth rate of the large-scale dynamo instability, attained at\nthe wavenumber\n\\begin{equation}\n k \\equiv k_\\alpha\n ={|\\overline{v}_\\mu + \\alpha_\\mu|\n \\over 2(\\eta+ \\, \\eta_{_{T}})},\n\\label{kmax_turb}\n\\end{equation}\nis given by\n\\begin{eqnarray}\n\\gamma^{\\rm max}_\\alpha\n= {(\\overline{v}_\\mu + \\alpha_\\mu)^2\\over 4 (\\eta+ \\, \\eta_{_{T}})}\n= {(\\overline{v}_\\mu + \\alpha_\\mu)^2\\over 4 \\eta \\, (1 + \\, {\\rm Re}_{_\\mathrm{M}}\/3)}.\n\\label{gammamax_turb}\n\\end{eqnarray}\nFor small magnetic Reynolds numbers,\n${\\rm Re}_{_\\mathrm{M}}=u_0 \\ell_0\/\\eta = 3 \\eta_{_{T}}\/\\eta$, this equation\nyields the correct result for the laminar $v_\\mu^2$ dynamo;\nsee Equation~(\\ref{gamma-max}).\n\nAs was shown in Paper~I, the CME\nin the presence of turbulence gives rise to the chiral $\\alpha_\\mu$ effect.\nThe expression for $\\alpha_\\mu$\nfound for large Reynolds numbers and a weak\nmean magnetic field is\n\\begin{eqnarray}\n \\alpha_\\mu = - {2 \\over 3} \\overline{v}_\\mu \\ln {\\rm Re}_{_\\mathrm{M}}.\n\\label{alphamu}\n\\end{eqnarray}\nSince the $\\alpha_\\mu$ effect in homogeneous turbulence\nis always negative, while the $\\overline{v}_\\mu$ effect is positive,\nthe chiral $\\alpha_\\mu$ effect decreases the $\\overline{v}_\\mu$ effect.\nBoth effects compensate each other at ${\\rm Re}_{_\\mathrm{M}}=4.5$ (see Paper~I).\nHowever, for large fluid and magnetic Reynolds numbers, $\\overline{v}_\\mu \\ll\n|\\alpha_\\mu|$, and we can neglect $\\overline{v}_\\mu$ in these equations.\nThis regime corresponds to the large-scale $\\alpha_\\mu^2$ dynamo.\n\n\n\\subsection{DNS of chiral magnetically driven turbulence}\n\nWe have performed a higher resolution $(576^3)$ three-dimensional\nnumerical simulation to study chiral magnetically driven turbulence.\nThe chiral Mach number of this simulation is ${\\rm Ma}_\\mu=2\\times10^{-3}$,\nthe chiral nonlinearity parameter is $\\lambda_\\mu=2\\times10^{-7}$, and the\nmagnetic and the chiral Prandtl numbers are unity.\nThe velocity field is initially zero, and the magnetic field is Gaussian noise,\nwith $B=10^{-6}$.\n\n\\begin{figure}[t]\n\\centering\n \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_ts_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nTime evolution for different quantities.\n}\n \\label{fig_LTts}\n\\end{figure}\n\nThe time evolution of $B_\\mathrm{rms}$, $u_\\mathrm{rms}$,\n$\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle$,\n$\\mu_\\mathrm{rms}$ (multiplied by\n$2\/\\lambda$), and $\\langle \\mathbf{A} \\cdot \\mathbf{B}\n\\rangle +2\\mu_\\mathrm{rms}\/\\lambda$\nof chiral magnetically driven turbulence\nis shown in the top panel of Figure~\\ref{fig_LTts}.\nFour phases can be distinguished:\n\\begin{asparaenum}[\\it (1)]\n\\item{The kinematic phase of small-scale chiral dynamo instability\nresulting in exponential growth of small-scale magnetic field\ndue to the CME.\nThis phase ends approximately at $t=0.05 t_\\eta$.\n}\n\\item{The first nonlinear phase resulting in production\nof chiral magnetically driven turbulence.\nIn this phase, $u_\\mathrm{rms}$ grows from very weak noise\nover seven orders of magnitude up to nearly the equipartition value\nbetween turbulent kinetic and magnetic energies,\ndue to the Lorentz force in the Navier-Stokes equation.\n}\n\\item{The second nonlinear phase resulting in large-scale dynamos.\nIn particular, the evolution of $B_\\mathrm{rms}$ for $t > 0.12 t_\\eta$\nis affected by the velocity field.\nDuring this phase, the velocity stays approximately constant,\nwhile the magnetic field continues to increase\nat a reduced growth rate in comparison with that of the\nsmall-scale chiral dynamo instability.\nIn this phase, we also observe the formation of inverse energy transfer\nwith a $k^{-2}$ magnetic energy spectrum that was previously found\nand comprehensively analyzed by \\cite{BSRKBFRK17} in DNS of chiral MHD\nwith different parameters.\n}\n\\item{The third nonlinear phase resulting in saturation of the large-scale\ndynamos, which ends at $\\approx 0.45 t_\\eta$ when the\nlarge-scale magnetic field reaches the maximum value.\nThe conserved quantity $\\langle \\mathbf{A} \\cdot \\mathbf{B} \\rangle\n+2\\mu_\\mathrm{rms}\/\\lambda$ stays constant over all four phases.\nSaturation is caused by the $\\lambda$ term in the evolution equation of the chiral\nchemical potential, which leads to a decrease of $\\mu$ from its initial value to 1.\n}\n\\end{asparaenum}\n\nThe middle panel of Figure~\\ref{fig_LTts} shows the measured growth rate of\n$B_\\mathrm{rms}$ as a function of time.\nIn the kinematic phase, $\\gamma$ agrees\nwith the theoretical prediction for the\nlaminar chiral dynamo instability; see Equation~(\\ref{gamma-max}),\nwhich is indicated by the dashed red horizontal line in the middle panel of\nFigure~\\ref{fig_LTts}.\nDuring this phase, the growth rate\nof the velocity field, given by the dotted gray line in Figure~\\ref{fig_LTts}, is\nlarger by roughly a factor of two than that of the magnetic field.\nThis is expected when turbulence is driven via the Lorentz force, which is\nquadratic in the magnetic field.\n\nOnce the kinetic energy is of the same order\nas the magnetic energy, the growth rate of\nthe magnetic field decreases abruptly by a factor of more than five.\nThis is expected in the presence of turbulence, because\nthe energy dissipation of the magnetic field is increased by turbulence due to\nturbulent magnetic diffusion.\nAdditionally, however, a positive contribution to the\ngrowth rate comes from the chiral $\\alpha_\\mu$ effect\nthat causes large-scale dynamo instability.\n\nThe time evolution of the ratio of the mean magnetic field to the total\nfield, $\\overline{B}\/B_\\mathrm{rms}$, is presented in the bottom panel\nof Figure~\\ref{fig_LTts}.\nThe mean magnetic field grows faster than the rms of the total magnetic\nfield in the time interval between 0.14 and 0.2 $t_\\eta$.\nDuring this time, the large-scale (mean-field) dynamo operates,\nso magnetic energy is transferred to larger spatial scales.\nWe now determine, directly from DNS,\nthe growth rate of the large-scale dynamo using\nEquation~(\\ref{gamma_turb}).\nTo this end, we determine the Reynolds number and the strength of the $\\alpha_\\mu$\neffect using the data from our DNS.\nWhereas the rms velocity is a direct output of the simulation, the turbulent\nforcing scale can be found from analysis of the energy spectra.\nThe theoretical value based on these estimates at the time $0.2\\,t_\\eta$ is\nindicated as the solid red horizontal line in the middle\npanel of Figure~\\ref{fig_LTts}.\n\n\\begin{figure}[t]\n\\centering\n \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_spec_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nMagnetic (blue lines) and kinetic (black lines) energy\nspectra are calculated at equal time differences,\nand the very last spectra are shown as solid lines.}\n \\label{fig_LTspec}\n\\end{figure}\n\n\nThe evolution of kinetic and magnetic energy spectra is\nshown in Figure~\\ref{fig_LTspec}.\nWe use equal time steps between the different spectra, covering the whole\nsimulation time.\nThe magnetic energy, indicated by blue\nlines, increases initially at $k=\\mu_0\/2=10$, which agrees with the theoretical\nprediction for the chiral laminar dynamo.\nThe magnetic field drives a turbulent spectrum of the kinetic energy, as can\nclearly be seen in\nFigure~\\ref{fig_LTspec} (indicated by black lines in Figure~\\ref{fig_LTspec}).\nThe final spectral slope of the kinetic energy is roughly $-5\/3$.\nThe magnetic field continues to grow at small wavenumbers,\nproducing a peak at $k=1$ in the final stage of the time evolution.\n\n\\begin{figure}[t]\n\\centering\n \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_kmax_t_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nThe black solid line shows the inverse correlation length, $k_{\\rm M}$,\nof the magnetic energy, defined by Equation (\\ref{eq_kcorr}), as a\nfunction of time $t$.\nUsing this wavenumber and the rms velocity, the fluid and magnetic\nReynolds numbers are estimated (see Equation~\\ref{eq_Rmkcorr}), that is\nshown by the dashed blue line.}\n \\label{fig_LTkmax}\n\\end{figure}\n\n\nWe determine the correlation length of the magnetic field from\nthe magnetic energy spectrum via\n\\begin{equation}\n\\xi_{\\rm M}(t)\\equiv\n k_{\\rm M}^{-1}(t) = \\frac{1}{\\mathcal{E}_{\\rm M}(t)} \\int k^{-1}\n E_{\\rm M}(k,t)~\\mathrm{d}k .\n\\label{eq_kcorr}\n\\end{equation}\nThe wavenumber $k_{\\rm M}$ so defined coincides (up to a numerical factor of order unity) with\nthe so-called tracking solution, $\\Delta \\mu_{\\rm tr}$ in \\citet{BFR12}.\nThere it was demonstrated that, in the course of evolution, the chiral\nchemical potential follows $k_{\\rm M}(t)$.\nAnd, indeed, the evolution of $k_{\\rm M}$, shown in Figure~\\ref{fig_LTkmax},\nstarts at around 10 (the value of $\\mu_0\/2$ in this simulation) and then decreases to\n$k_{\\rm M} = k_1$ (corresponding to the simulation box size) at $t\\approx0.18~t_\\eta$.\nInterestingly, the chemical potential is affected by\nmagnetic helicity only at much later times, as can be seen in\nFigure~\\ref{fig_LTkmax}.\nBased on the wavenumber, $k_{\\rm M}$, we estimate\nthe Reynolds numbers as\n\\begin{equation}\n {\\rm Re}_{_\\mathrm{M}} = \\mathrm{Re} = \\frac{u_\\mathrm{rms}}{\\nu k_{\\rm M}} .\n\\label{eq_Rmkcorr}\n\\end{equation}\nFigure~\\ref{fig_LTkmax} shows that the Reynolds number increases exponentially,\nmostly due to the fast increase of $u_\\mathrm{rms}$, and saturates\nlater at ${\\rm Re}_{_\\mathrm{M}}\\approx 10^2$.\nSimilarly, the turbulent diffusivity can be estimated as\n\\begin{equation}\n \\eta_{_{\\rm T}} = \\frac{u_\\mathrm{rms}}{3~k_{\\rm M}} .\n\\label{eq_etaTkcorr}\n\\end{equation}\nDuring the operation of the mean-field large-scale dynamo, we find\nthat $\\eta_{_{\\rm T}}\\approx2.4\\times10^{-3}$,\nwhich is about 24 times larger than the molecular diffusivity $\\eta$.\nUsing these estimates, we determine\nthe chiral magnetic $\\alpha_\\mu$ effect from Equation~(\\ref{alphamu}).\nThe large-scale dynamo growth rate (\\ref{gamma_turb}) is\nshown as the solid red horizontal line in the middle panel of Figure~\\ref{fig_LTts}\nand is in agreement with the DNS results shown as the black solid line.\n\n\n\\begin{figure}[t]\n\\centering\n \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_ts4_mu20}}\n \\subfigure{\\includegraphics[width=\\columnwidth]{LamTurb_gammak_mu20}}\n\\caption{\n{\\bf Chiral magnetically driven turbulence.}\nThe evolution of the magnetic energy $E_{\\rm M}$ on different wavenumbers $k$\n(top panel).\nThe growth rate as a function of $k$ in different time intervals as given in the\nplot legend. The black line corresponds to a fit, while the theoretical\nexpectations are given as a red\nline.}\n \\label{fig_LT_gammak}\n\\end{figure}\n\nFurther analysis of the evolution of the magnetic field at different wavenumbers\nis presented in Figure~\\ref{fig_LT_gammak}.\nIn the top panel, we display the magnetic energy at various\nwavenumbers as a function of time.\nIn the kinematic phase, for $t<0.1~t_\\eta$, the fastest amplification\noccurs at $k=10$, as can also be seen in the energy spectra.\nAt wavenumbers $k1$ needs\nto be fulfilled.\nAs shown in Equation (\\ref{kmax_turb}), $k_\\mathrm{max}$ is proportional\nto $\\eta\/\\eta_{_{\\rm T}}$, which is inversely proportional to the magnetic\nReynolds number ${\\rm Re}_{_\\mathrm{M}}$.\nAs a result, the chemical potential needs to be sufficiently large for\n$k_\\mathrm{max}>1$.}\n\\item{\nDue to nonlocal effects, the turbulent diffusivity $\\eta_{_{\\rm T}}$\nis generally scale-dependent and\ndecreases above $k_\\mathrm{f}$ \\citep{BRS08}.\nFor comparison with mean-field theory, the chiral dynamo\ninstability has to occur on scales $k < k_\\mathrm{f}$, where\n$\\eta_{_{\\rm T}} \\approx u_\\mathrm{rms}\/(3k_\\mathrm{f})$.\nNote, however, that\nthe presence of a mean kinetic helicity in the system\ncaused by the CME (see Paper~I)\ncan increase the turbulent diffusivity $\\eta_{_{\\rm T}}$\nfor moderate magnetic Reynolds numbers by up to 50\\% \\citep{BSR17}.}\n\\item{\nTo simplify the system, we\navoid classical small-scale dynamo action, which\noccurs at magnetic Reynolds numbers larger than\n${\\rm Re}_{_{\\mathrm{M,crit}}}\\approx50$.}\n\\end{itemize}\n\n\n\n\\begin{table}[t]\n\\caption{\nOverview of Runs With Externally Forced Turbulence\n(Reference Run in Bold)}\n\\centering\n \\begin{tabular}{l|lllllll}\n \\hline\n \\hline\n \\\\\n\t~ \t & $\\mu_0$ \t& $\\dfrac{{\\rm Ma}_\\mu}{10^{-3}}$ \t& $\\dfrac{\\lambda_\\mu}{10^{-6}}$ \t& $\\dfrac{k_\\lambda}{10^{-3} \\mu_0}$ & $\\dfrac{k_\\mathrm{diff}}{\\mu_0}$ & $k_\\mathrm{f}$ \t& ${\\rm Re}_{_\\mathrm{M}}$\t\t\\\\\t\n\t~ \t & \t\t& ~\t\t \t& ~\t\t \t& ~\t\t& ~\t\t& ~\t \t& (early $\\rightarrow$ late)\t\t\\\\\t \\hline\n \tTa2-1 \t& $20$\t\t& $8$\t& $16$ \t& 160\t\t \t& 4.5\t& $10$ \t\t& $24 \\rightarrow 19$ \\\\\n \tTa2-2 \t& $20$\t\t& $4$\t& $4.0$ & 80\t\t \t& 63\t& $10$ \t\t& $36 \\rightarrow 28$ \t\\\\\n \tTa2-3 \t& $20$\t\t& $8$\t& $16$ \t& 160\t\t \t& 45\t& $10$ \t\t& $16 \\rightarrow 14$ \t\\\\\n \tTa2-4 \t& $20$\t\t& $4$\t& $4.0$ & 80\t\t \t& 63\t& $10$ \t\t& $4 \\rightarrow 13$ \t\\\\\n \t\\textbf{Ta2-5} & $\\mathbf{20}$\t& $\\mathbf{8}$ & $\\mathbf{160}$ & $\\mathbf{51}$ \t& $\\mathbf{80}$\t& $\\mathbf{10}$ \t& $\\mathbf{24} \\rightarrow\\mathbf{18}$ \t\\\\\n \tTa2-6 \t& $20$\t\t& $8$\t& $1.6$ & 51\t\t \t& 80\t& $10$ \t\t& $16 \\rightarrow 14$ \t\\\\\n \tTa2-7 \t& $30$\t\t& $12$\t& $32$ \t& 230\t\t \t& 38\t& $4$ \t\t& $42 \\rightarrow 58$ \t\\\\\n \tTa2-8 \t& $30$\t\t& $9$\t& $18$ \t& 160\t\t \t& 43\t& $4$ \t\t& $58 \\rightarrow 65$ \t\\\\\n \tTa2-9 \t& $30$\t\t& $9$\t& $13.5$ & 150\t\t \t& 47\t& $4$ \t\t& $82 \\rightarrow 74$ \t\\\\\n \tTa2-10 & $40$\t\t& $8$\t& $16$ \t& 160\t\t \t& 45\t& $4$ \t\t& $119 \\rightarrow 107$ \\\\\n \\hline\n \\hline\n \\end{tabular}\n \\label{table_simulations_forced}\n\\end{table}\n\n\n\\subsection{DNS of chiral dynamos in forced turbulence}\n\\label{sec:forced-chiral-dynamos}\n\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=\\columnwidth]{ts__TurbRef}\n\\caption{\n{\\bf Externally forced turbulence.}\nTime evolution of the magnetic field, the velocity field, and\nthe chemical potential, as well as the mean value of the magnetic\nhelicity (top panel).\nThe middle panel shows the growth rate of $B_\\mathrm{rms}$ as a function of\ntime (solid black line).\nThe red lines are theoretical expectations in different dynamo phases.\nIn the bottom panel, the ratio of the mean magnetic field to the total field\n$B_\\mathrm{rms}$ is presented.}\n\\label{fig_alphamu_ts}\n\\end{figure}\n\nThe time evolution of different quantities in our reference run is presented in\nFigure~\\ref{fig_alphamu_ts}.\nThe magnetic field first increases first exponentially,\nwith a growth rate $\\gamma \\approx 60~t_\\eta^{-1}$, which is\nabout a factor of 1.6 lower than that expected for the laminar $v_\\mu^2$ dynamo;\nsee the middle panel of Figure~\\ref{fig_alphamu_ts}.\nThis difference seems to be caused by the presence of random forcing;\nsee discussion below.\nAt approximately 0.2 $t_\\eta$, the growth rate decreases to a value\nof $\\gamma\\approx 15~t_\\eta^{-1}$ consistent with that of the mean-field\nchiral $\\alpha_\\mu^2$ dynamo, before saturation occurs at $0.4\\,t_\\eta$.\nThe evolution of $B_\\mathrm{rms}$ is comparable qualitatively\nin chiral magnetically produced turbulence; see Figure~\\ref{fig_LTts}.\nAn additional difference from the latter\nis the value of $u_\\mathrm{rms} \\approx 0.1$ for\nexternally forced turbulence, which is controlled by the intensity\nof the forcing function.\nAn indication of the presence of a mean-field dynamo is the evolution of\n$\\overline{B}\/B_\\mathrm{rms}$ in the bottom panel of\nFigure~\\ref{fig_alphamu_ts}, which reaches a value of unity at $0.3\\,t_\\eta$.\n\nThe energy spectra presented in Figure~\\ref{fig_alphamu_spec} support the\nlarge-scale dynamo scenario.\nFirst, the magnetic energy increases at all scales,\nand, at later times, the maximum of the magnetic energy\nis shifted to smaller wavenumbers, finally producing a\npeak at $k=1$, i.e., the smallest possible wavenumber in our periodic domain.\n\nA detailed analysis of the growth of magnetic energy is presented in\nFigure~\\ref{fig_alphamu_specana}.\nIn the first phase, the growth rate of the magnetic field is\nindependent of the wavenumber $k$ (see top panel), due to a coupling\nbetween different modes.\nThe growth rate measured in this phase is less than that\nin the laminar case (see middle panel), due to a scale-dependent\nturbulent diffusion caused by the random forcing.\n\nWithin the time interval $(0.22$--$0.28)\\,t_\\eta$, only the magnetic field at\n$k=1$ increases.\nThis is clearly seen in the bottom panel of Figure~\\ref{fig_alphamu_specana},\nwhere we show the evolution of the magnetic energy at different wavenumbers $k$.\nThe growth rate of the mean-field dynamo, which is determined\nat $k=1$, agrees with the result from\nmean-field theory, given by Equation~(\\ref{gammamax_turbRm}).\nThere is a small dependence of the resulting mean-field growth rate on the\nexact fitting regime.\nIf the phase of the\nmean-field dynamo is very short, changing the fitting range can affect the\nresult by a factor up to 30 \\%.\nWe use the latter value as an estimate of the uncertainty in the growth rate, and,\nin addition, indicate an error of 20 \\% in determining the Reynolds number,\nwhich is caused by the temporal variations of $u_\\mathrm{rms}$.\n\n\\begin{figure}[t]\n\\centering\n \\subfigure{\\includegraphics[width=\\columnwidth]{spec__TurbRef}}\n\\caption{\n{\\bf Externally forced turbulence.}\nEvolution of kinetic (black lines) and magnetic energy spectra (blue lines)\nfor the reference run Ta2-5.\nThe ratio $\\mu_0\/\\lambda$ is indicated by the horizontal dashed line.}\n\\label{fig_alphamu_spec}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n \\subfigure{\\includegraphics[width=\\columnwidth]{Emag_t__TurbRef}}\n \\subfigure{\\includegraphics[width=\\columnwidth]{gamma_k__TurbRef}}\n\\caption{\n{\\bf Externally forced turbulence.}\nTime evolution of the magnetic energy at different wavenumbers $k$\n(top panel).\nThe remaining panels show the growth rates as a function of scale in\ndifferent fit intervals.}\n\\label{fig_alphamu_specana}\n\\end{figure}\n\n\n\\subsection{Dependence on the magnetic Reynolds number}\n\nBased on the mean-field theory developed in Paper~I, we expect\nthe following.\nUsing the expression for the $\\alpha_\\mu$ effect\ngiven by\nEquation (\\ref{alphamu}), the maximum growth rate (\\ref{gammamax_turb})\nfor the mean-field dynamo can be rewritten as a function of the magnetic\nReynolds number:\n\\begin{eqnarray}\n\\gamma_{\\rm max} ({\\rm Re}_{_\\mathrm{M}}) = \\frac{\\overline{v}_\\mu^2 (1 - 2\/3~\\ln{\\rm Re}_{_\\mathrm{M}} )^2}{4 \\eta \\,\n(1 + \\, {\\rm Re}_{_\\mathrm{M}}\/3)} ,\n\\label{gammamax_turbRm}\n\\end{eqnarray}\nwhere the ratio $\\eta_{_{\\rm T}}\/\\eta = {\\rm Re}_{_\\mathrm{M}}\/3$.\n\nWe perform DNS with different Reynolds numbers to\ntest the scaling of $\\gamma_{\\rm max}({\\rm Re}_{_\\mathrm{M}})$ given by\nEquation~(\\ref{gammamax_turbRm}).\nThe parameters of the runs with externally forced\nturbulence are summarized in Table \\ref{table_simulations_forced}.\nWe vary $\\nu \\, (=\\eta$), the forcing wavenumber $k_\\mathrm{f}$, as well as the\namplitude of the forcing,\nto determine the function $\\gamma_{\\rm max}({\\rm Re}_{_\\mathrm{M}})$.\nIn the initial phase, $u_\\mathrm{rms}$ is constant in time.\nOnce large-scale turbulent dynamo action occurs,\nthere are additional minor\nvariations in $u_\\mathrm{rms}$, because the system is already\nin the nonlinear phase.\nThe nonlinear terms in the Navier-Stokes equation lead to a modification of\nthe velocity field at small spatial scales, which affects the value of\n$u_\\mathrm{rms}$ and results in\nthe small difference between\nthe initial and final values of the Reynolds numbers\n(see Table \\ref{table_simulations_forced}).\n\nAccording to Equation~(\\ref{kmax_turb}),\nthe wavenumber associated with the maximum growth rate\nof the large-scale turbulent\ndynamo instability decreases with increasing ${\\rm Re}_{_\\mathrm{M}}$.\nIn order to keep this mode inside the computational domain and hence to\ncompare the measured growth rate with the maximum one given by\nEquation~(\\ref{gammamax_turbRm}),\nwe vary the value of $\\mu_0$ in our simulations.\nThe variation of $\\mu_0$ and the additional variation of $\\eta$ for scanning\nthrough the ${\\rm Re}_{_\\mathrm{M}}$ parameter space, implies that ${\\rm Ma}_\\mu$ changes\ncorrespondingly.\n\nThe values of the nonlinear parameter $\\lambda$ should be within a certain range.\nIndeed, the saturation value of the magnetic field, given by\nEquation~(\\ref{eq_Bsat}), is proportional to $\\lambda^{-1\/2}$.\nIn order for the Alfv\\'en velocity not to exceed the sound speed,\nwhich would result in a very small time step in DNS,\n$\\lambda$ should not be below a certain value.\nOn the other hand, $\\lambda$ should not be too large,\nas in this case the dynamo would saturate quickly, and there is only a very short\ntime interval of the large-scale dynamo.\nIn this case, determining the growth rate of the mean-field dynamo,\nand hence comparing with the mean-field theory, is difficult.\n\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=\\columnwidth]{gamma_Rm}\n\\caption{\n{\\bf Externally forced turbulence and chiral magnetically driven turbulence.}\nThe normalized growth rate $\\gamma~\\eta\/v_\\mu^2$ of the magnetic\nfield as a function of the magnetic Reynolds number ${\\rm Re}_{_\\mathrm{M}}$.\nThe gray data points show the\ngrowth rate in the initial, purely kinematic phase of the simulations.\nThe blue data points show the measured growth rate of the magnetic field\non $k=1$, when the large-scale dynamo occurs.\nThe diamond-shaped data points represent simulations of forced turbulence, while\nthe dot-shaped data points refer to the case of\nchiral magnetically driven turbulence.\nThe growth rate observed in the initial laminar phase\nfor the case of chiral magnetically driven turbulence\nis shown at ${\\rm Re}_{_\\mathrm{M}}=2$, with the left arrow indicating that the actual ${\\rm Re}_{_\\mathrm{M}}$ is much\nlower and out of the plot range at this time; see Figure~\\ref{fig_LTkmax}.}\n \\label{fig_gamma_Re}\n\\end{figure}\n\nIn Figure~\\ref{fig_gamma_Re} we show the normalized growth\nrate $\\gamma~\\eta\/v_\\mu^2$ of the magnetic\nfield as a function of the magnetic Reynolds number ${\\rm Re}_{_\\mathrm{M}}$.\nThe gray data points show the growth rate in the initial,\npurely kinematic phase of the simulations.\nThe blue data points show the measured growth rate of the magnetic field\non $k=1$, when the large-scale dynamo occurs.\nFor comparison of the results with externally forced turbulence\n(indicated as diamond-shaped data points), we show\nin Figure~\\ref{fig_gamma_Re} also the results obtained for the\ndynamo in chiral magnetically driven turbulence, which are indicated as dots.\n\nIn DNS with externally forced turbulence, we see in all cases a\nreduced growth rate due to mode coupling.\nContrary to the case with externally forced turbulence,\nin DNS with the chiral magnetically driven turbulence,\nwe do initially observe the purely laminar dynamo\nwith the growth rate given by Equation~(\\ref{gamma-max}),\nbecause there is no mode coupling in the initial phase\nof the magnetic field evolution in this case.\nOn the other hand, the measured growth rates of the mean-field dynamo\nin both cases agree (within the error bars) with the growth rates\nobtained from the mean-field theory.\n\n\n\n\n\n\\section{Chiral MHD dynamos in astrophysical relativistic plasmas}\n\\label{sec_astro}\n\n\nIn this section, the results for the\nnonlinear evolution of the chiral chemical potential,\nthe magnetic field, and the turbulent state of the plasma found in this paper\nare applied to astrophysical relativistic plasmas.\nWe begin by discussing the role of chiral dynamos in the early universe and\nidentify conditions under which the CME affects the generation and evolution of\ncosmic magnetic fields.\nFinally, in Section~\\ref{sec:PNS}, we examine the importance of the CME\nin proto-neutron stars (PNEs).\n\n\n\\subsection{Early Universe}\n\\label{sec:early-universe}\n\nIn spite of many possible mechanisms that can produce magnetic fields in the\nearly universe\n\\citep[see, e.g.,][for reviews]{W02,WEtAl12,DN13,Giovannini:2003yn,S16},\nunderstanding the origin of cosmic magnetic fields remains an open problem.\nTheir generation is often associated with nonequilibrium events in the universe\n(e.g., inflation or phase transitions).\nA period of particular interest is the electroweak (EW) epoch, characterized\nby temperatures of $\\unit[10^{15}]{K}$ ($k_{\\rm B} T \\sim \\unit[100]{GeV}$).\nSeveral important events take place around this time: the electroweak\nsymmetry gets broken, photons appear while intermediate vector bosons\nbecome massive, and the asymmetry between matter and antimatter appears\nin the electroweak baryogenesis scenario \\citep{KRS85};\nsee, for example, the review by \\citet{MRM12}.\nMagnetic fields of appreciable strength can be generated as a consequence of\nthese events \\citep{V91,Olesen:1992np,Enqvist:1993np,Enqvist:1994dq,\nVachaspati:1994ng,GGV95, 1996PhLB..380..253D,BBM96,V01,Semikoz:2010zua}.\nTheir typical correlation length $\\xi_{\\rm M}^{(\\rm ew)} \\sim (\\ensuremath{\\alpha_{\\rm em}} T)^{-1}$\ncorresponds to only a few centimeters today -- much less than the observed correlation\nscales of magnetic fields in galaxies or galaxy clusters.\nTherefore, in the absence of mechanisms that can increase\nthe comoving scale of the magnetic field beyond\n$\\xi_{\\rm M}^{(\\rm ew)}$, such fields were deemed to be irrelevant\nto the problem of cosmic magnetic fields \\citep[for discussion,\nsee, e.g.,][]{Durrer:2003ja,Caprini:2009pr,Saveliev:2012ea,KTBN13}.\n\nThe situation may change if (i) the magnetic fields are helical and\n(ii) the plasma is turbulent.\nIn this case, an inverse transfer of magnetic energy may develop,\nwhich leads to a shift of the typical scale of the magnetic field to\nprogressively larger scales \\citep{BEO96, CHB01, BJ04, KTBN13}.\nThe origin of such turbulence has been unknown.\nAn often considered paradigm is that a random magnetic field, generated at small\nscales, produces turbulent motions via the Lorentz force.\nHowever, continuous energy input is required.\nIf this is not the case, the magnetic field decays:\n$\\langle \\bm{B}^2 \\rangle \\sim t^{-2\/3}$ as the correlation scale grows\n\\citep{BM99PhRvL, KTBN13}, so that $\\langle \\bm{B}^2 \\rangle \\xi_{\\rm M} = {\\rm const}$.\n\nIn the present work, we demonstrated that the presence of a finite chiral\ncharge in the plasma at the EW epoch is sufficient to satisfy the above\nrequirements (i) and (ii).\nAs a result,\n\\begin{asparaenum}[\\it (1)]\n\\item helical magnetic fields are excited,\n\\item turbulence with large ${\\rm Re}_{_\\mathrm{M}}$ is produced, and\n\\item the comoving correlation scale increases.\n\\end{asparaenum}\nWe discuss each of these phases in detail below.\n\n\n\\subsubsection{Generation and evolution of cosmic magnetic fields in the\npresence of a chiral chemical potential}\n\\label{subsec_cosmologybounds}\n\nAlthough it is not possible to perform numerical simulations with\nparameters matching those of the early universe, the results of the\npresent paper allow us to make qualitative predictions about the fate\nof cosmological magnetic fields generated at the EW epoch in the\npresence of a chiral chemical potential.\n\nAll of the main stages of the magnetic field evolution,\nsummarized in Section~\\ref{sec:stages}, can occur in the early universe\n(a sketch of the main phases is provided in \\Fig{fig:phases}).\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\columnwidth]{sketch_phases_2}\n \\caption{\n \\textbf{Chiral MHD dynamos in the early universe.}\nSketch of the different phases of the chiral mean-field dynamo.\nFrom left to right:\nsmall-scale chiral dynamo (phase 1), large-scale turbulent dynamo (phase 2), and\nsaturation (phase 3).\nAfter saturation of the dynamo, the magnetic magnetic field dissipates.\nThe upper horizontal dotted line shows the initial value of $\\mu$ and the lower\none the ``saturation limit,'' given by Eq.~\\protect\\eqref{eq:2}.}\n \\label{fig:phases}\n\\end{figure}\n\nPhase 1.\nAt this initial stage, the small-scale chiral dynamo instability develops at\nscales around $\\xi_\\mu$, where\n\\begin{equation}\n \\xi_\\mu \\equiv \\frac{2}{|\\mu_0|} ,\n \\label{eq_mu-1}\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:1}\n \\mu_0 \\approx 4\\ensuremath{\\alpha_{\\rm em}}\\frac{\\mu_5}{\\hbar c}\\approx\n\\unit[1.5\\times10^{14}]{cm^{-1}}\\frac{\\mu_5}{\\unit[100]{GeV}}.\n\\end{equation}\nThe chemical potential $\\mu_5$ can be approximated by the thermal energy\n$k_{\\rm B} T$ for order-of-magnitude estimates.\nIn what follows, we provide numerical estimates for $\\mu_5 = 100\\,{\\rm GeV}$ which\ncorresponds to the typical thermal energy of relativistic particles\nat the EW epoch.\nThe characteristic energy at the quantum chromodynamics phase transition\nis $\\approx 100\\,{\\rm MeV}$ where the quark--gluon plasma turns into hadrons.\nWe stress, however, that the MHD formalism is only valid if the scales\nconsidered are larger than the mean free path given by Equation~(\\ref{eq_mfp}).\nComparing the chiral instability scale $k_\\mu^{-1}$ with $\\ell_\\mathrm{mfp}$ \nresults in the condition \n$\\mu_5 \\ll k_{\\rm B} T\\, 4\\pi^2 \\ensuremath{\\alpha_{\\rm em}} \\ln{((4\\pi \\ensuremath{\\alpha_{\\rm em}})^{-1\/2})}$. \nStrictly speaking, modeling a system that does not fulfill this condition \nrequires full kinetic theory as described, for example, in \\citet{CPWW13} or\nin \\citet{AY13}.\n\nThe growth rate of an initially weak magnetic field in the linear\nstage of the chiral dynamo instability is given by \\Eq{gamma-max}:\n\\begin{equation}\n\\label{eq:9}\n \\gamma^{\\rm max}_\\mu = \\frac{\\mu_0^2\\eta}4 \\approx 2.4\\times 10^{19}T_{100}^{-1}\\,{\\rm s}^{-1}.\n\\end{equation}\nFor the value of the magnetic diffusivity $\\eta = c^2\/(4 \\pi \\sigma)$\nin the early universe, we adopted the conductivity $\\sigma$ from\nEquation~(1.11) of \\cite{ArnoldEtAl2000}.\nNumerically,\n\\begin{equation}\n\\eta(T)={7.3\\times 10^{-4}}\\,{\\hbar c^2\\overk_{\\rm B} T}\\approx\n{4.3\\times10^{-9}}T_{100}^{-1}\\,{\\rm cm}^2\\,{\\rm s}^{-1} ,\n\\label{eq_rDiffvA}\n\\end{equation}\nwhere $T_{100}=1.2\\times10^{15}\\,{\\rm K}$ (so that $k_{\\rm B} T_{100} =\\unit[100]{GeV}$).\nAs a result, the number of $e$-foldings over one Hubble time $t_H$ is\n$$\\gamma^{\\rm max}_\\mu t_H \\gg 1,$$ where\n\\begin{equation}\n \\label{eq:tH}\n t_H = H^{-1}(T) \\approx 4.8\\times10^{-11}\\,g_{100}^{-1\/2}T_{100}^{-2}\\,\\,{\\rm s}\n\\end{equation}\n(here $g_*$ is the number of relativistic degrees of freedom and $g_{100}=g_*\/100$).\nWe should stress that this picture has been known before and was described\nin many previous works \\citep{JS97,Frohlich:2000en,Frohlich:2002fg,BFR12}.\n\nWe note that a nonzero chiral flipping rate\n$\\Gamma_\\mathrm{f}$ has been discussed in the literature\n\\citep{CDEO92,BFR12,DS15,BFR15,SiglLeite2016}.\nIn Section~\\ref{sec_flip}, we have found\nin numerical simulations that the flipping term affects the evolution\nof the magnetic field only\nfor large values of $f_\\mu$, when the flipping term is of the order of or\nlarger than the $\\lambda_\\mu$ term in Equation~(\\ref{mu-NS});\nsee also Equation~(\\ref{eq_fmu}) and Figure~\\ref{fig_ts_flip}.\nWhen adopting the estimate in \\citet{BSRKBFRK17} of $f_\\mu\\approx 1.6\\times10^{-7}$,\nchirality flipping is not likely to play a significant role for\nthe laminar $v_\\mu^2$ dynamo in the early universe at very high temperatures\nof the order of $100 \\,{\\rm GeV}$.\nHowever, $\\Gamma_\\mathrm{f}$ depends on the ratio $m_e c^2\/(k_{\\rm B} T)$ and\nthus suppresses all chiral effects once the universe has cooled down to\n$k_{\\rm B} T \\approx m_e c^2$ \\citep{BFR12}.\nAt this point, we stress again that the true value of $\\mu_0$ is unknown\nand has here been set to the thermal energy in Equation~(\\ref{eq:1}).\nIf it turns out that the initial value of the chiral chemical potential\nis much smaller than the thermal energy, $f_\\mu$ becomes larger, and\nthe flipping rate can play a more important role already during the initial\nphases of the chiral instability in the early universe.\nThis scenario is not considered in the following discussion.\n\nIn the regime of the laminar $v_\\mu^2$ dynamo, one could reach\n$\\mathcal{O}(10^9)$ $e$-folds over the Hubble time $t_{\\rm H}$; see lower panel\nof Figure~\\ref{fig:chiral_turbulence}.\nHowever, as shown in this work, already after a few hundred $e$-foldings,\nthe magnetic field starts to excite turbulence via the Lorentz force.\nThis happens once the magnetic field is no longer force-free.\nOnce the flow velocities reach the level $v_\\mu = \\mu_0\\eta$, nonlinear\nterms are no longer small, small-scale turbulence is produced, and\nthe next phase begins.\n\n\nPhase 2.\nThe subsequent evolution of the magnetic field depends on the strength\nof the chiral magnetically excited turbulence.\nThis has been shown in the mean-field analysis of \\citet{REtAl17} and\nis confirmed by the present work; see, for example, Figure~\\ref{fig_gamma_Re}.\nThe growth rate and instability scale depend on the magnetic Reynolds\nnumber; see Equations~(\\ref{gamma_turb})--(\\ref{gammamax_turb}).\nThe maximum growth rate for ${\\rm Re}_{_\\mathrm{M}} \\gg 1$ is given by\n\\begin{equation}\n \\label{eq:10}\n \\gamma^{\\rm max}_{\\alpha} = \\gamma_\\mu^{\\rm max} \\frac{4}3 \\frac{(\\ln {\\rm Re}_{_\\mathrm{M}})^2}{{\\rm Re}_{_\\mathrm{M}}},\n\\end{equation}\nwhere $\\gamma_\\mu^{\\rm max}$ is given by \\Eq{eq:9}.\nFor the early universe, it is impossible to determine the exact value of the\nmagnetic Reynolds number from the numerical simulations, but one\nexpects ${\\rm Re}_{_\\mathrm{M}} \\gg 1$ and we show in \\Fig{fig:chiral_turbulence}\nthat, in a wide range of magnetic Reynolds numbers,\n$1 \\ll {\\rm Re}_{_\\mathrm{M}} \\ll 6\\times 10^{12}$, the number of $e$-foldings during one\nHubble time is much larger than $1$.\nThe turbulence efficiently excites magnetic fields at scales much larger than\n$\\xi_\\mu$ (Figure~\\ref{fig:chiral_turbulence}, top panel).\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\columnwidth]{TurbDyn_EU}\n \\caption[Turbulence-driven instability in the early universe]\n {\\textbf{Chiral MHD dynamos in the early universe. }\n The ratios between $\\xi_\\alpha$ of the turbulence-driven dynamo (Eq.~(\\ref{kmax_turb}))\n and scale $\\xi_\\mu$ (Equation~\\protect\\eqref{eq_mu-1}), as well as the ratio\n between $\\xi_\\mu$ and the Hubble radius at different temperatures.\n In the top panel, furthermore, the ratio $\\xi_\\mu\/\\xi_\\lambda$ is presented.\n Maximum growth rates over the Hubble time for laminar ($\\gamma_\\mu^{\\rm max}$)\n and turbulent ($\\gamma_\\alpha^{\\rm max}$) regimes are shown in the bottom panel.\n }\\label{fig:chiral_turbulence}\n\\end{figure}\n\nUsing dimensional analysis and DNS, \\citet{BSRKBFRK17} demonstrated\nthat the resulting spectrum of the magnetic fields behaves as\n$E_{\\rm M}\\propto k^{-2}$ between $k_\\mu$ and $k_\\lambda$, given by\nEquation~(\\ref{klambda}).\nThe wavenumber $k_\\lambda$ depends on the nonlinearity parameter $\\lambda$,\ndefined by Equation~(\\ref{eq_lambda}), which, in the early universe, is given by\n\\begin{equation}\n \\lambda=3 \\hbar c\\,\\left({8\\ensuremath{\\alpha_{\\rm em}}\\overk_{\\rm B} T}\\right)^2\\approx1.3\\times10^{-17}\\,T_{100}^{-2}\\,\n\\,{\\rm cm}\\,{\\rm erg}^{-1}.\n \\label{eq_lambda_1}\n\\end{equation}\nWe note that this expression is, strictly speaking, only valid when\n$k_{\\rm B} T \\gg \\max(|\\mu_L|,|\\mu_R|)$ and modifications might be expected \noutside of this regime.\nFurther, the mean density of the plasma\n\\begin{equation}\n \\overline{\\rho}=\\frac{\\pi^2}{30}\\,g_*\\frac{(k_{\\rm B} T)^4}{\\hbar^3c^5}\n \\approx 7.6\\times10^{26}g_{100}T_{100}^4\\,{\\rm g}\\,{\\rm cm}^{-3}.\n\\end{equation}\nThe ratio $\\xi_\\lambda\/\\xi_\\mu = k_\\mu\/k_\\lambda$ is presented in the top\npanel of \\Fig{fig:chiral_turbulence}, but we note that the exact numerical\ncoefficient in the condition $k_\\mu\/k_\\lambda \\gg 1$ might depend on ${\\rm Re}_{_\\mathrm{M}}$.\n\nPhase 3.\nThe stage of large-scale turbulent dynamo action ends with the\n\\emph{saturation phase} (see Section~\\ref{sec:stages} and\n\\Fig{fig:phases}).\nAt this stage, the total chiral charge (determined by the initial\nconditions) gets transferred to magnetic helicity.\nAs shown in \\citet{BFR12} (see also~\\citet{JS97} for earlier work, as well\nas \\cite{TVV12} and \\cite{Hirono:2015rla} for more discussion), and confirmed by\nnumerical simulations in \\citet{BSRKBFRK17} and in the present work,\nthe chiral chemical potential $\\mu$ follows $k_{\\rm M}$ at this stage\nand thus decreases with time.\nTherefore, most of the chiral charge will be transferred with time into magnetic helicity,\n\\begin{equation}\n \\langle \\bm{A} \\cdot \\bm{B}\\rangle \\simeq \\xi_{\\rm M} \\langle \\bm{B}^2 \\rangle \\to \\frac{2\\mu_0}\\lambda ,\n \\label{eq:2}\n\\end{equation}\nswitching off the CME (the end of Phase 3 in \\Fig{fig:phases}).\n\n\\subsubsection{Chiral MHD and cosmic magnetic fields}\n\\label{sec:end_chiral_MHD}\n\nMagnetic fields produced by chiral dynamos are fully helical.\nOnce the CME has become negligible, the subsequent phase of decaying\nhelical turbulence begins and the\nmagnetic energy decreases, while the magnetic correlation length increases\nin such a way that the magnetic helicity~\\eqref{eq:2} is conserved\nfor very small magnetic diffusivity \\citep{BM99PhRvL, KTBN13}.\n\nBased on \\Eq{eq:2}, one can estimate the magnetic helicity \\emph{today}; see also \\cite{BSRKBFRK17}.\nTaking as an estimate for the chiral chemical potential $\\mu_5 \\sim k_{\\rm B} T$ (this means that the density of the chiral charge is of the order of the\nnumber density of photons), one finds\n\\begin{multline}\n \\label{eq:Br17:3}\n \\bra{\\bm{B}^2}\\xi_{\\rm M} \\simeq\n \\frac{\\hbar c}{4\\ensuremath{\\alpha_{\\rm em}}} \\frac{g_{0}}{g_\\ast} n_\\gamma^{(0)}\n \\simeq 6\\times 10^{-38}\\,{\\rm G}^2\\,{\\rm Mpc} .\n\\end{multline}\nHere, the present number density of photons is $n_\\gamma^{(0)} = 411\\,{\\rm cm}^{-3}$,\nand the ratio $g_{0}\/g_\\ast\\approx3.36\/106.75$ of the effective\nrelativistic degrees of freedom today and at the EW epoch appears,\nbecause the photon number density dilutes as $T^3$ while the magnetic helicity\ndilutes as $a^{-3}$.\nWe recall that, to arrive at the numerical value in $\\,{\\rm G}^2\\,{\\rm Mpc}$\ngiven in Equation~(\\ref{eq:Br17:3}), an additional $4\\pi$ factor\nwas applied to convert to Gaussian units.\n\nUnder the assumption that the spectrum of the cosmic magnetic field is sharply\npeaked at some scale\n$\\xi_0$ (as is the case in all of the simulations presented here),\nthe lower bounds on magnetic fields, inferred\nfrom the nonobservation of GeV cascades from TeV sources\n\\citep{NV10,Tavecchio:10,Dolag:10} can be directly translated into\na bound on magnetic helicity today. The observational bound scales\nas $|\\bm{B}| \\propto \\xi_0^{-1\/2}$ for $\\xi_0 < 1\\,{\\rm Mpc}$ \\citep{NV10} and\ntherefore $\\bra{\\bm{B}^2}\\xi_0 = {\\rm const} > 8\\times 10^{-38}\\,{\\rm G}^2\\,{\\rm Mpc}$.\nThe numerical value is obtained using the most conservative bound\n$|\\bm{B}| \\ge 10^{-18}\\,{\\rm G}$ at $1\\,{\\rm Mpc}$ (\\citealt{DCRFCL11},\nsee also~\\citealt{DN13}).\nThese observational constraints for intergalactic magnetic fields are\ncompared to the magnetic field produced in chiral MHD for different values\nof the initial chiral chemical potential in Figure~\\ref{fig_B_xi__comov}.\n\nThe limit given by Equation~(\\ref{eq:Br17:3}) is quite general.\nIt does not rely on chiral MHD or the CME, but simply\nreinterprets the bounds of \\cite{NV10}, \\cite{Tavecchio:10}, \\cite{Dolag:10},\nand \\cite{DCRFCL11} as bounds on magnetic helicity.\nGiven such an interpretation, we conclude that \\emph{if cosmic magnetic fields\nare helical and have a cosmological origin, then at some moment in the history\nof the universe the density of chiral charge was much larger\nthan $n_\\gamma(T)$}.\nThis chiral charge can be, for example, in the form of magnetic helicity or of\nchiral asymmetry of fermions, or both.\nTo generate such a charge density, some new physics beyond the Standard Model of\nelementary particles is required.\nBelow we list several possible mechanisms that can generate large initial\nchiral charge density:\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\columnwidth]{B_xi__comov}\n \\caption{\n {\\bf Chiral MHD dynamos in the early universe.}\nThe magnetic field strength resulting from a chiral dynamo as a function of correlation length in comoving\nunits and comparison with observational constraints.\nThe differently colored lines show the chiral magnetically produced magnetic field\nstrength in the range between the injection length $\\mu^{-1}$ and the\nsaturation length $k_\\lambda^{-1}$; see Equations~(\\ref{eq_mu-1}) and\n(\\ref{klambda}), respectively.\nThe colors indicate different values of the chiral chemical potential:\nRed refers to the value of $\\mu_0$ given in Equation~(\\ref{eq:1}), blue to\n$10^{-2}\\mu_0$, and purple to $10^{2}\\mu_0$.\nThe dashed gray line is an upper limit on the intergalactic magnetic field from\nZeeman splitting.\nSolid gray lines,\nrefer to the lower limits reported by \\citet{NV10} (``NV10'') and \\citet{DCRFCL11} (``D+11''), respectively.\nThe vertical dotted gray lines show the\nhorizon at $k_{\\rm B} T = 100\\,{\\rm GeV}$ and $100\\,{\\rm MeV}$ correspondingly.\nThe thin colored arrows refer to the nonlinear evolution of magnetic fields in\nan inverse cascade in helical turbulence up to the final value as given in\n\\citet{BJ04}\n(line ``BJ04'').\n}\n \\label{fig_B_xi__comov}\n\\end{figure}\n\n\\begin{asparaenum}[\\it (1)]\n\\item The upper bound in \\Eq{eq:Br17:3} assumes that only one fermion of the\nStandard Model developed a chiral asymmetry $\\sim n_\\gamma$.\n Many fermionic species are present in the plasma at the electroweak epoch.\n They all can have a left--right asymmetric population of comparable size,\n increasing the total chirality by a factor $\\mathcal{O}(10)$, which\n makes the estimate~(\\ref{eq:Br17:3}) consistent with the lower bound\n from \\citet{DCRFCL11}.\n One should check, of course, whether for more massive fermions the\n chirality flipping rate is much slower than the dynamo growth rate\n determined by Equation~(\\ref{eq:9}).\n\\item The estimate~(\\ref{eq:Br17:3}) assumed that left--right asymmetry was\ncreated via thermal processes. Of course, new physics at the EW epoch can result in\nnonthermal production of chiral asymmetry (e.g.\\ via decays of some long-lived\nparticles), thus leading to $n_5 \\gg n_\\gamma$ and so increasing the limit\n(\\ref{eq:Br17:3}).\n\\item\n The left--right asymmetry may be produced as a consequence of the decay\n of helical \\emph{hypermagnetic} fields prior to the EW epoch.\n Such a scenario, relating hypermagnetic helicity to the\n chiral asymmetry, has been discussed previously, such as in\n \\cite{Giovannini:1997eg} and \\cite{Semikoz:2012ka}.\n A conservation law similar to that of~(\\ref{cons_law}) exists also for\n hypermagnetic fields, and the decay of the latter may cause asymmetric\n populations of left and right states.\n\\item\nIn our analysis, we have not taken into account the chiral vortical effect\n\\citep{Vilenkin:79}.\nFor nonvanishing chemical potential, it leads to an additional current\nalong the direction of vorticity \\citep[see, e.g.,][]{TVV12}.\n\\end{asparaenum}\n\nFrom the point of view of chiral MHD, the value of $\\mu_0$ (to which\nthis bound is proportional) is just an initial condition and therefore\ncan take arbitrary values.\nOnce an initial condition with a large value of $\\mu_0$ has been\ngenerated, the subsequent evolution (as described above) does not require\nany new physics.\n\nMoreover, the coupled evolution of magnetic helicity and chiral chemical\npotential is \\emph{unavoidable} in the relativistic plasma and should be an\nintegral part of relativistic MHD (as was discussed in Paper~I).\n\n\n\n\n\\subsection{PNSs and the CME}\n\\label{sec:PNS}\n\n\nIn this section, we explore whether the CME and chiral dynamos can play a\nrole in the development of strong magnetic fields in neutron stars.\nA PNS is a stage of stellar evolution after\nthe supernova core collapse and before the cold and dense neutron star is\nformed \\citep[see, e.g.,][]{PRPLM99}.\nPNSs are characterized by high temperatures\n(typically $k_{\\rm B} T \\sim \\unit[\\mathcal{O}(10)]{MeV}\\gg m_e c^2$),\nlarge lepton number density (electrons Fermi energy $\\mu_e \\sim$ a few\nhundreds of MeV), the presence of turbulent flows in the interior, and quickly\nchanging environments.\nOnce the formation of a neutron star is completed, its magnetic\nfield can be extremely large.\nNeutron stars that exceed the quantum electrodynamic limit\n$B_\\mathrm{QED}\\equiv m_e^2c^3\/(e \\hbar)\\approx4.4\\times10^{13}~\\,{\\rm G}$ are known as\n``magnetars'' \\citep[see, e.g.,][for recent reviews]{MPM15,TZ15,KB17}.\nThe origin of such strong magnetic fields remains unknown, although\nmany explanations have been proposed; see, for example, \\citet{DT92}, \n\\cite{AWML03}, and \\cite{FW06}.\n\nThe role of the CME in the physics of (proto)neutron stars and\ntheir contribution to the generation of strong magnetic fields have been\ndiscussed in a number of works\n\\citep{Charbonneau:2009ax,Ohnishi:2014uea,Dvornikov:2015lea,DS15,\nGKR15,Dvornikov:2016cmz,SiglLeite2016,Yamamoto:2015gzz}.\n\n\n\\subsubsection{Chiral MHD in PNSs}\n\\label{sec:chiral-mhd-PNS}\n\nDuring the formation of a PNS, electrons and protons are converted into neutrons,\nleaving behind left-handed neutrinos.\nThis is known as the Urca process\n\\citep[$e + p \\to n + \\nu_e$;][]{Haensel:95}.\nIf the chirality-flipping timescale, determined by the electron's\nmass, is longer than the instability scale, the net chiral\nasymmetry in the PNS can lead to the generation\nof magnetic fields. This scenario has been discussed previously\n\\citep{Ohnishi:2014uea,SiglLeite2016,GKR15}.\nThe chiral turbulent dynamos discussed in this work can be relevant for\nthe physics of PNSs and can affect our conclusions\nabout the importance of the CME.\nHowever, to make a detailed quantitative analysis, a number of factors\nshould be taken into account:\n\\begin{asparaenum}[\\it (1)]\n\\item The rate of the Urca process is strongly temperature dependent\n \\citep{LPPH91,Haensel:95}.\n The temperatures inside PNSs are only known with large uncertainties, and\n the cooling occurs on a scale of seconds \\citep[see, e.g.,][]{PRPLM99},\n making estimates of the Urca rates uncertain by orders of magnitude.\n\\item The chirality flipping rate that aims to restore the depleted population of\n left-chiral electrons is also expected to be temperature dependent\n \\citep[see, e.g.,][]{GKR15,SiglLeite2016}.\n\\item The neutrinos produced via the Urca process are trapped in the\n interior of a PNS and can release the chiral asymmetry back into the plasma\n via the $n + \\nu_e \\to e + p$ process.\n Therefore, only when the star becomes transparent to neutrinos\n (as the temperature drops to a few MeV) does the creation of chiral asymmetry\n can become significant.\n\\end{asparaenum}\n\n\nModeling the details of PNS cooling and neutrino propagation is\nbeyond the scope of this paper.\nBelow we perform the estimates that demonstrate that chiral MHD can\nsignificantly change the picture of the evolution of a PNS.\n\n\n\\subsubsection{Estimates of the relevant parameters}\n\nAn upper limit of the chiral chemical potential can be estimated by\nassuming that\n$n_\\mathrm{L}=0$ and $n_\\mathrm{R}=n_e$ (all left-chiral electrons have been\nconverted into neutrinos, and the rate of chirality flipping is much slower than\nother relevant processes).\nThis leads to the estimate $\\mu_5 \\simeq \\mu_e$ and correspondingly\n\\begin{equation}\n \\mu_{\\rm max} = 4\\ensuremath{\\alpha_{\\rm em}} \\frac{\\mu_e}{\\hbar c} \\approx \\unit[4\\times 10^{11} ]{cm}^{-1}\\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right),\n \\label{eq_muupper_NS}\n\\end{equation}\nwhere we have used a typical value of the electron's Fermi energy $\\mu_e$\n\\citep{PRPLM99}.\nFor an ultrarelativistic \\emph{degenerate} electron gas (i.e., when\n$\\mu_e \\gg k_{\\rm B} T \\gg m_e c^2$), the relation between the number density\nof electrons, $n_e$, and their Fermi energy, $\\mu_e$, is\n\\begin{equation}\n \\label{eq:5}\n \\mu_e = \\hbar c(2\\pi^2 n_e)^{1\/3}\\approx 250\\,{\\rm MeV} \\left(\\frac{n_e}{\\unit[10^{38}]{cm^{-3}}}\\right)^{1\/3}.\n\\end{equation}\nThe interior of neutron stars is a conducting medium whose\nconductivity is estimated to be \\citep{BPP69,Kelly1973}:\n\\begin{eqnarray}\n \\sigma(T) &=& \\sqrt 3\\left(\\frac4\\pi\\right)^{3\/2}\\frac{\\hbar^4 c^2}{e\\,m_p^{3\/2}}\\frac{n_e^{3\/2}}{k_{\\rm B}^2 T^2}\\\\\n &\\approx& 1\\times 10^{27}\\left(\\frac{1~\\mathrm{MeV}}{k_{\\rm B} T}\\right)^{2}\\left(\\frac{n_e}{\\unit[10^{38}]{cm^{-3}}}\\right)^{3\/2}\\,{\\rm s}^{-1}\n\\label{eq_sigmaNS}\n\\end{eqnarray}\n(there is actually a difference in the numerical coefficient $\\mathcal{O}(1)$\nbetween the results of \\citet{BPP69} and \\citet{Kelly1973}).\nUsing Equation~(\\ref{eq_sigmaNS}), we find the magnetic diffusion coefficient to be\n\\begin{equation}\n \\label{eq:3}\n \\eta(T)\n \\approx 7\\times 10^{-8}\\,{\\rm cm}^2\\,{\\rm s}^{-1} \\left(\\frac{k_{\\rm B} T}{1~\\mathrm{MeV}}\\right)^{2}\\left(\\frac{\\unit[10^{38}]{cm^{-3}}}{n_e}\\right)^{3\/2}.\n\\end{equation}\nTherefore, we can determine the\nthe maximum growth rate of the small-scale\nchiral instability (\\ref{gamma-max}) as\n\\begin{equation}\n \\label{eq:4}\n \\gamma^{\\rm max}_\\mu = \\frac{\\mu_{\\rm max}^2\\eta}4 \\approx 2\\times 10^{15}\\,{\\rm s}^{-1} \\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right)^2 \\left(\\frac{k_{\\rm B} T}{1~\\mathrm{MeV}}\\right)^{2}.\n\\end{equation}\nWe see that over a characteristic time $\\tau_\\mathrm{cool}\\sim 1 \\,{\\rm s}$\n(the typical cooling time), the magnetic field would increase by many\n$e$-foldings.\nIn fact, using a flipping rate of $\\Gamma_\\mathrm{f}=10^{14} \\,{\\rm s}$, as\nsuggested in \\citet{GKR15} for $\\mu_\\mathrm{e}= 100 \\,{\\rm MeV}$ and $k_{\\rm B} T=30 \\,{\\rm MeV}$,\nwe find that $f_\\mu$ ranges from\n$\\approx 9\\times 10^{-3}$ down to $\\approx 9\\times 10^{-7}$ for the range\nbetween $k_{\\rm B} T = 1 \\,{\\rm MeV}$ and $k_{\\rm B} T = 100 \\,{\\rm MeV}$.\nHence the evolution of the chemical potential\nand the chiral dynamo\nis weakly affected by flipping reactions.\n\nAs in Section~\\ref{subsec_cosmologybounds}, the phase of the small-scale\ninstability ends when turbulence is excited.\nIt should be stressed, however, that unlike the early universe, the\ninteriors of PNSs are expected to be turbulent with high ${\\rm Re}_{_\\mathrm{M}}$ even in\nthe absence of chiral effects\n(with ${\\rm Re}_{_\\mathrm{M}}$ as large as $10^{17}$); see \\cite{TD93}.\nTherefore, the system may find itself in the forced turbulence regime of Section~\\ref{sec:forced-chiral-dynamos}.\nFigure~\\ref{fig:turbulence_PNS} shows that in a wide range of magnetic\nReynolds numbers, one can have many $e$-foldings over a typical timescale of\nthe PNS and that the scale of the magnetic field can reach macroscopic size.\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\columnwidth]{scales_NS}\n \\caption[Laminar and turbulent scales in PNS]{\n \n \\textbf{Chiral MHD dynamos in PNSs.}\nLaminar and turbulent growth rate multiplied by the\ncooling timescale (top panel) and the characteristic scales of chiral MHD\nnormalized by the typical radius of the PNS $r_{\\rm NS} \\sim 10\\,{\\rm km}$\n(bottom panel).\nThe estimates are presented as a function of ${\\rm Re}_{_\\mathrm{M}}$.\nThe initial value of the chiral chemical potential is assumed at the\nlevel~(\\protect\\ref{eq_muupper_NS}) and the we use $\\mu_e =\\unit[250]{MeV}$.\nSince the conductivity is temperature dependent, the ratios including $\\eta$ are\npresented for both $k_{\\rm B} T = 1~\\mathrm{MeV}$ and $k_{\\rm B} T = 10~\\mathrm{MeV}$.\n }\n \\label{fig:turbulence_PNS}\n\\end{figure}\n\n\\subsubsection{Estimate of magnetic field strengths}\n\nA dedicated analysis, taking into account temperature and density evolution of\nthe PNS as well as its turbulent regimes, is needed to make detailed predictions.\nHere we will make the estimates of the strength of the magnetic field,\nsimilar to Section~\\ref{sec:early-universe} above.\nTo this end, we use the conservation law~(\\ref{cons_law}), assuming\n$\\mu_0 = \\mu_{\\rm max}$.\nIn the PNS case, the plasma is degenerate, and therefore the relation between\n$n_5$ and $\\mu_5$ is given by\n\\begin{equation}\n n_5 = \\frac{\\mu_5}{3\\pi^2}(3\\mu_e^2 + \\pi^2 T^2)\n \\label{eq:7}\n\\end{equation}\n(in the limit $\\mu_5 \\ll T$).\n{\nAs a result, the chiral feedback parameter\n$\\lambda$ is\n\\begin{equation}\n \\label{lambda_PNS}\n \\lambda_{\\rm PNS} = \\frac{\\hbar c\\pi^2}{2}\\left(\\frac{8\\ensuremath{\\alpha_{\\rm em}}}{\\mu_e}\\right)^2,\n\\end{equation}\nwhich determines the wavenumber $k_\\lambda$; see Equation~(\\ref{klambda}).\nThe corresponding length scale $\\xi_\\lambda = k_\\lambda^{-1}$ is presented in\nthe top panel of Figure~\\ref{fig:turbulence_PNS}, where we assume a\nmean density of the PNS of $\\overline{\\rho}_{\\rm PNS} = 2.8\\times10^{14}~\\mathrm{g}\\mathrm{cm}^{-3}$.\n\nUsing \\Eqs{eq_muupper_NS}{lambda_PNS}, we find\n\\begin{eqnarray}\n \\label{eq:8}\n (B^2\\xi)_{\\max}\n &=&\\frac{4\\pi\\mu_{\\rm max}}{\\lambda_{\\rm PNS}} = \\frac{\\mu_e^3}{2(\\hbar c)^2\\ensuremath{\\alpha_{\\rm em}}}\\\\\n &\\approx& 1.4\\times 10^{24}\\,{\\rm G}^2\\,{\\rm cm}\\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right)^3.\n\\end{eqnarray}\nAssuming for the maximum correlation scale $\\xi_{\\rm PNS} \\sim 1\\,{\\rm cm}$\n(see \\Fig{fig:turbulence_PNS}), we find that\nmagnetic field strength is of the order of\n\\begin{equation}\n \\label{eq:6}\n B_{\\rm max} \\approx 1.2\\times 10^{12}\\,{\\rm G} \\left(\\frac{\\mu_e}{\\unit[250]{MeV}}\\right)^{3\/2}\n \\left(\\frac {1\\,{\\rm cm}}{\\xi_{\\rm M}}\\right)^{1\/2} .\n\\end{equation}\nNotice that the estimate~(\\ref{eq:8}) is independent of $T$\n(but depends strongly on the assumed value of $\\mu_e$).\n\n\nOur estimates have demonstrated that the chiral MHD could be capable of generating\nstrong small-scale magnetic fields. \\emph{Therefore, chiral effects should be\nincluded in the modeling of evolution of PNSs.}}\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec_concl}\n\nIn this work, we have presented results from numerical simulations of chiral MHD\nthat include the temporal and spatial evolution of magnetic fields,\nplasma motions, and the chiral chemical potential.\nThe latter, characterizing the asymmetry between left- and right-handed fermions,\ngives rise to the CME, which\nresults in the excitation of a small-scale chiral dynamo instability.\n\nOur numerical simulations are performed for the system of\nchiral MHD equations~(\\ref{ind-DNS})--(\\ref{mu-DNS}) that was derived in\nPaper~I.\nThis system of equations is valid for plasmas with high electric conductivity,\nthat is, in the limit of high and moderately high Reynolds numbers.\nChiral flipping reactions are neglected in most of the simulations.\nIn the majority of the runs, the initial conditions are a very weak magnetic seed\nfield and a high chiral chemical potential.\nBoth initially force-free systems and systems with external forcing of\nturbulence are considered.\nWith our numerical simulations, we confirm\nvarious theoretical predictions of the chiral laminar\nand turbulent large-scale dynamos discussed in Paper~I.\n\nOur findings from DNS can be summarized as follows:\n\\begin{itemize}\n\\item{\nThe evolution of magnetic fields studied here in DNS agrees with\nthe predictions made in Paper~I for all types of laminar dynamos.\nIn particular, the scalings of\nthe maximum growth rate of the chiral dynamo instability\n$\\gamma_\\mu\\propto v_\\mu^2$ for the\n$v_\\mu^2$ dynamo (see Figure~\\ref{fig__Gamma_eta}) and\n$\\gamma_\\mu\\propto (S v_\\mu)^{2\/3}$\nfor the $v_\\mu$--shear dynamo (see Figure~\\ref{Gamma_Us_Beltrami}) have been\nconfirmed.\nAdditionally, the transitional regime of an $v_\\mu^2$--shear dynamo, where the\ncontributions from the $v_\\mu^2$- and shear terms\nare comparable, agrees with theoretical predictions, as can be seen in\nFigure~\\ref{fig_Gamma_Us_aShear}.\nIn our DNS, the scale-dependent amplification of the magnetic field\nin the laminar chiral dynamo is observed in the energy spectra; see, for example,\nFigure~\\ref{fig_LTspec} where the maximum\ngrowth rate of the $v_\\mu^2$ dynamo instability is attained\nat wavenumber $k_\\mu=\\mu_0\/2$.\n}\n\\item{\nThe conservation law~(\\ref{CL}) for total chirality\nimplies a maximum magnetic field strength\nof the order of $B_\\mathrm{sat}\\approx(\\mu_0 \\xi_\\mathrm{M}\/\\lambda)^{1\/2}$.\nThis dependence of $B_\\mathrm{sat}$ on the chiral nonlinearity parameter\n$\\lambda$ has been confirmed numerically and is presented in\nFigure~\\ref{fig_Bsat_lambda}.\n}\n\\item{\nThe CME can drive turbulence efficiently via the Lorentz force,\nwhich has been demonstrated in our numerical simulations\nthrough the measured growth rate of the turbulent velocity, which is\nlarger by approximately a factor of two than that of the magnetic field;\nsee, for example, the middle panel of Figure~\\ref{fig_LTts}.\n}\n\\item{\nIn the presence of small-scale turbulence, the large-scale dynamo\noperates due to the chiral $\\alpha_\\mu$ effect,\nwhich is not related to the kinetic helicity;\nsee Equation~(\\ref{alphamu}).\nIn the limit of large magnetic Reynolds numbers,\nthe maximum growth rate of the large-scale dynamo instability\nis reduced by a factor of\n$(4\/3)(\\ln{\\rm Re}_{_\\mathrm{M}})^2\/{\\rm Re}_{_\\mathrm{M}}$ as\ncompared to the laminar case; see Equation~(\\ref{gammamax_turbRm}).\nThe dynamo growth rate is close to this prediction\nof mean-field chiral MHD for both\nchiral magnetically produced turbulence and\nfor externally driven turbulence; see\nsee Figure~\\ref{fig_gamma_Re}.\n}\n\\item{\nUsing DNS, we found a new scenario of the magnetic field evolution\nconsisting of three phases\n(see also the schematic overview in Figure~\\ref{fig:phases}):\n\\begin{asparaenum}[\\it (1)]\n\\item small-scale chiral dynamo instability;\n\\item production of small-scale turbulence, inverse transfer of magnetic energy,\nand generation of a large-scale magnetic field by the chiral $\\alpha_\\mu$ effect;\n\\item\nsaturation of the large-scale chiral dynamo\nby a decrease of the CME\ncontrolled by the conservation law for\nthe total chirality:\n$\\lambda \\, \\bra{{\\bm A} {\\bm \\cdot} \\bm{B}}\/2 + \\bra{\\mu} = \\mu_0$.\n\\end{asparaenum}\nThe previously discussed scenario of magnetic field\nevolution caused by the CME \\citep{BFR12}\ndid not include the second phase.}\n\\end{itemize}\n\nWhile the results summarized above have been obtained\nin simulations of well-resolved periodic domains,\nastrophysical parameters are beyond the regime accessible to DNS.\nHence we can only estimate the effects of the chiral anomaly in relativistic\nastrophysical plasmas, like in the early universe or in neutron stars.\nThe main conclusions from the astrophysical applications are the following:\n\\begin{itemize}\n\\item{The chiral MHD scenario found in DNS may help to explain the origin of\nthe magnetic field observed in the interstellar\nmedium. The chiral dynamo instability produces helical magnetic\nfields initially at\nsmall spatial scales and simultaneously drives turbulence,\nwhich generates a magnetic field on large scales.\nAfter the chiral chemical potential has been transformed into magnetic\nhelicity during the dynamo saturation phase,\nthe magnetic field cascades to\nlarger spatial scales according to the phenomenology of decaying MHD\nturbulence.\nWe have estimated the values of $\\mu_0$ and $\\lambda$ for the early universe.\nThese parameters determine the time and spatial scales associated with\nthe chiral dynamo instability\n(see Figure~\\ref{fig:chiral_turbulence}) and the maximum magnetic\nhelicity (see Equation~\\ref{eq:Br17:3}).\nOur estimates for magnetic fields produced by chiral dynamos in the\nearly universe are consistent with the observational lower limits\nfound by \\citet{DCRFCL11} (see Figure~\\ref{fig_B_xi__comov})\nif we assume that the initial chiral chemical potential is of the order of\nthe thermal energy density.\n}\n\\item{In PNSs, chiral dynamos\noperating in the first tens of seconds after the supernova explosion\ncan produce magnetic fields of approximately $10^{12}\\,{\\rm G}$ at a magnetic\ncorrelation length of $1\\,{\\rm cm}$; see Equation~(\\ref{eq:6}).\nHowever, we stress that many questions remain open, especially regarding the\ngeneration of a chiral asymmetry and the role of the chiral flipping term\nin PNEs.\n}\n\\end{itemize}\n\nFinally, we stress again that the parameters and the initial conditions,\nincluding the initial chiral asymmetry, are unknown in the astrophysical \nsystems discussed in this paper. \nHence, the purpose of our applications should be classified as a study\nof the conditions under which the CME plays a significant \nrole in the evolution of a plasma of relativistic charged fermions. \nWith the regimes accessible to our simulations not being truly realistic in the \ncontext of the physics of the early universe and \nin neutron stars, our applications have a rather exploratory nature. \nIn this sense, our results from DNS can be used to answer\nthe question in which area of plasma physics -- the physics of the early\nuniverse, the physics of neutron stars, or the physics of heavy ion\ncollisions---the CME is important and can modify the\nevolution of magnetic fields.\n\n\n\\begin{acknowledgements}\nWe thank the anonymous referee for constructive criticism that\nimproved our manuscript.\nWe acknowledge support from Nordita, which is funded by the\nNordic Council of Ministers, the Swedish Research Council, and the two host\nuniversities, the Royal Institute of Technology (KTH) and\nStockholm University.\nThis project has received funding from the\nEuropean Union's Horizon 2020 research and\ninnovation program under the Marie Sk{\\l}odowska-Curie grant\nNo.\\ 665667.\nWe also acknowledge the University of Colorado's support through\nthe George Ellery Hale visiting faculty appointment.\nSupport through the NSF Astrophysics and Astronomy Grant Program (grant 1615100),\nthe Research Council of Norway (FRINATEK grant 231444),\nand the European Research Council (grant number 694896) are\ngratefully acknowledged. Simulations presented in this work have been performed\nwith computing resources\nprovided by the Swedish National Allocations Committee at the Center for\nParallel Computers at the Royal Institute of Technology in Stockholm.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nThe celebrated laws of black hole thermodynamics~\\cite{Bek72,Bek74,Haw74,Haw75} ascribe physical properties to the event horizon of a black hole. However, the event horizon is defined globally, as the boundary of the past of future infinity. Thus, the location of the thermodynamic object depends on the future history of the spacetime. For example, an observer in a perfectly flat spacetime region might already be inside a black hole, if a null shell is collapsing outside their past light-cone. By causality, Hawking radiation and the first and second law of black hole thermodynamics should have no manifestation for such an observer. Conversely, once a black hole has formed, its thermodynamic properties should be observable at finite distance, regardless of whether the collapsed region already coincides with the true event horizon, or is headed for substantial growth in the distant future.\n\nHere we consider the problem of finding a geometric object that is locally defined, and which obeys a classical law analogous to one of the laws of thermodynamics. We will focus on the second law, whose manifestation in classical General Relativity is the statement that the area of certain surfaces cannot decrease. For the cross-sections of an event horizon this was proven by Hawking in 1971~\\cite{Haw71}, but as noted above the event horizon is not locally defined.\n\nWe will formulate and prove a new area theorem. It is obeyed by what we shall call a {\\em future (or past) holographic screen}, $H$. $H$ is a hypersurface foliated by marginally (anti-)trapped surfaces, which are called {\\em leaves}. This definition is local, unlike that of an event horizon. It requires knowledge only of an infinitesimal neighborhood of each leaf.\n\\begin{figure*}[ht]\n\\subfigure[]{\n\\includegraphics[width=0.45 \\textwidth]{1aPRD.pdf}\n}\n\\subfigure[]{\n \\includegraphics[width=0.45 \\textwidth]{1bPRD.pdf}\n}\n\\caption{Penrose diagrams showing examples of holographic screens. The green diagonal lines show a null slicing of the spacetime; green dots mark the maximal area sphere on each slice. These surfaces combine to form a holographic screen (blue lines); we prove that their area increases monotonically in a uniform direction on the screen (blue triangles). (a) A black hole is formed by collapse of a star (inner shaded region); later another massive shell collapses onto the black hole (outer shaded region). At all other times an arbitrarily small amount of matter accretes (white regions); this suffices to satisfy our generic conditions. The black hole interior contains a future holographic screen that begins at the singularity and asymptotes to the event horizon. It is timelike in the dense regions and spacelike in the dilute regions. (b) In a closed universe filled with dust, marginally antitrapped spheres form a past holographic screen in the expanding region; its area increases towards the future. Marginally trapped spheres form a future holographic screen in the collapsing region; its area increases towards the past. The equator of the three-sphere at the turnaround time (black circle) belongs to neither the past nor the future screen; it is extremal in the sense of Ref.~\\cite{HubRan07}.}\n\\label{fig-examplesPRD}\n\\end{figure*} \nA future holographic screen exists (nonuniquely) in generic spacetimes that have a future event horizon. It is disjoint from the event horizon but it may asymptote to it; see Fig.~\\ref{fig-examplesPRD}a. Past holographic screens exist in expanding universes such as ours, regardless of whether they have a past event horizon. Because $H$ is not defined in terms of distant regions, past and future holographic screens can exist in spacetimes with no distant boundary at all, such as a recollapsing closed universe; see Fig.~\\ref{fig-examplesPRD}b. Our area law applies to all future and past holographic screens.\n\n\\noindent{\\bf Relation to previous work} The notion of future or past holographic screen has roots in two distinct bodies of research, which had not been connected until now. It can be regarded as a refinement of the notion of ``preferred holographic screen hypersurface''~\\cite{CEB2}, which need not have monotonic area. Alternatively, it can be viewed as a generalization of the notion of ``dynamical horizon'', which obeys a straightforward area law but is not known to exist in many realistic solutions. We will now discuss these two connections for context and attribution; see also~\\cite{BouEng15a}. We stress, however, that our theorem and proof are self-contained. They rely only on classical General Relativity, and not, for example, on any conjecture about semiclassical or quantum gravity.\n\nFirst, let us discuss the relation to the holographic principle. (See~\\cite{Tho93,Sus95,FisSus98} for earlier work and Ref.~\\cite{RMP} for a review.) To an arbitrary codimension 2 spatial surface $B$, one can associate a light-sheet~\\cite{CEB1}: a null hypersurface orthogonal to $B$ with everywhere nonpositive expansion (i.e., locally nonincreasing area), in the direction away from $B$. The covariant entropy bound (Bousso bound)~\\cite{CEB1} is the conjecture that the entropy of the matter on the light-sheet cannot exceed the area of $B$, in Planck units. The conjecture has broad support; it has been proven in certain limiting regimes~\\cite{FlaMar99,BouFla03,StrTho03,BouCas14a,BouCas14b}.\n\nThere are four null directions orthogonal to any surface. In each direction, the orthogonal null congruence generates a null hypersurface with boundary $B$. The expansion in opposing directions, such as future-outward and past-inward, differs only by a sign. In typical settings, therefore, there will be two directions with initially negative expansion, each of which gives rise to a light-sheet. For example, a sphere in Minkowski space admits light-sheets in the future and past inward directions, but not in the outward directions. A large enough sphere near the big bang is anti-trapped: it admits light-sheets in the past inward and outward directions. Spheres near the singularity of a black hole are trapped: the light-sheets point in the future inward and outward directions.\n\nHowever, it is possible to find surfaces that are {\\em marginal}: they have vanishing expansion in one opposing pair of null directions. Hence they admit a pair of light-sheets whose union forms an entire null slice of the spacetime~\\cite{CEB1}. In fact, in strongly gravitating regions one can readily construct a continuous family of marginal surfaces, which foliate a hypersurface called ``preferred holographic screen hypersurface''. The opposing pairs of light-sheets attached to each leaf foliate the spacetime. The Bousso bound is particularly powerful when applied to these light-sheets. It constrains the entropy of the entire spacetime, slice by slice, in terms of the area of the leaves. All quantum information in the spacetime can be stored on the leaves, at no more than about one qubit per Planck area. In this sense the world is a hologram.\n\nFor event horizons, a classical area theorem~\\cite{Haw71} preceded the interpretation of area as physical entropy~\\cite{Bek72}. For holographic screens, the present work belatedly supplies a classical area law for an object whose relevance to geometric entropy had long been conjectured~\\cite{CEB2}. What took so long?\n\nIn fact, the notion of ``preferred holographic screen hypersurface'' lacked a key refinement, without which our theorem would not hold: the distinction between past and future holographic screens. The leaves of a ``preferred holographic screen hypersurface'' are {\\em marginal}, that is, one orthogonal null congruence has vanishing expansion. However, they were not required to be either marginally trapped, or marginally anti-trapped. That is, no definite sign was imposed on the expansion of the second, independent orthogonal null congruence. Fig.~\\ref{fig-examplesPRD}b shows a spacetime in which a ``preferred holographic screen hypersurface'' fails to obey an area law. Once we distinguish between marginally trapped and anti-trapped surfaces, however, we recognize that there are in fact two disconnected objects: a past and a future holographic screen. Each obeys an area law, as our proof guarantees, but in different directions of evolution. This is analogous to the distinction between past and future event horizons. From this perspective, it is not surprising that ``preferred holographic screen hypersurfaces'' fail to satisfy an area law without the refinement we introduce here.\n\nThis brings us to the second body of research to which the present work owes debt. Previous attempts to find a quasi-local alternative to the event horizon culminated in the elegant notions of a future outer trapping horizon (FOTH)~\\cite{Hay93,Hay97,HayMuk98} or dynamical horizon~\\cite{AshKri02,AshKri03,AshGal} (see~\\cite{AshKri04,Booth05} for reviews). In a generic, classical setting their definitions are equivalent: a dynamical horizon is a spacelike hypersurface foliated by marginally trapped surfaces. \n\n``Preferred holographic screen hypersurface'' was a weaker notion than future holographic screen; ``dynamical horizon'' is a stronger notion. It adds not only the crucial refinement from marginal to marginally trapped, but also the requirement that the hypersurface be spacelike. This immediately implies that the area increases in the outward direction~\\cite{Hay93,AshKri02}. (Note the brevity of the proof of Theorem~\\ref{thm-area} below, which alone would imply an area law without the need for any of the previous theorems, if a spacelike assumption is imposed.) \n\nHowever, our present work shows that the spacelike requirement is not needed for an area theorem. This is important, because the spacelike requirement is forbiddingly restrictive~\\cite{BoothBrits}: no dynamical horizons are known to exist in simple, realistic systems such as a collapsing star or an expanding universe dominated by matter, radiation, and\/or vacuum energy. \n\nThus, the notion of a dynamical horizon (or of a FOTH) appears to be inapplicable in a large class of realistic regions in which gravity dominates the dynamics. We are not aware of a proof of nonexistence. But we show here that an area theorem holds for the more general notion of future holographic screen, whose existence is obvious and whose construction is straightforward in the same settings. Thus we see little reason for retaining the additional restriction to hypersurfaces of spacelike signature, at least in the context of the second law.\n\nIn the early literature on FOTHs\/dynamical horizons, future holographic screens were already defined and discussed, under the name ``marginally trapped tube''~\\cite{AshKri04}.\\footnote{The definition of ``trapping horizon''~\\cite{Hay93} excludes the junctions between inner and outer trapping horizons and thus precludes the consideration of such objects as a single hypersurface.} Ultimately, two separate area laws were proven, one for the spacelike and one for the timelike portions of the future holographic screens. These follow readily from the definitions. The first, for FOTHs\/dynamical horizons, was mentioned above. The second states that the area decreases toward the future along any single timelike portion (known as ``future inner trapping horizons''~\\cite{Hay93} or ``timelike membranes''~\\cite{AshKri04}). \n\nIn these pioneering works, no unified area law was proposed for ``marginally trapped tubes''\/future holographic screens. Perhaps this is because it was natural to think of their timelike portions as future directed and thus area-decreasing. Moreover, the close relation to ``preferred holographic screen hypersurfaces''~\\cite{CEB2} was not recognized, so the area of leaves lacked a natural interpretation in terms of entropy.\\footnote{It is crucial that the entropy associated with the area of leaves on a future holographic screen $H$ is taken to reside on the light-sheets of the leaves, as we assert, and not on $H$ itself. The latter choice---called a ``covariant bound'' in Refs.~\\cite{He:2007qd, He:2007dh, He:2007xd, He:2008em} but related to~\\cite{BakRey99} and distinct from~\\cite{CEB1}---is excluded by a counterexample~\\cite{KalLin99} and would not lead to a valid Generalized Second Law.} And finally, it is not immediately obvious that an area law can hold once timelike and spacelike portions are considered together. Indeed, the central difficulty in the proof we present here is our demonstration that such portions can only meet in ways that uphold area monotonicity for the entire future holographic screen under continuous flow. A key element of our proof builds on relatively recent work~\\cite{Wal10QST}.\n\nThere is an intriguing shift of perspective in a brief remark in later work by Booth {\\em et al.}~\\cite{BoothBrits}. After explicitly finding a ``marginally trapped tube'' (i.e., what we call a future holographic screen) in a number of spherically symmetric collapse solutions, the authors point out that it could be considered as a single object, rather than a collection of dynamical horizon\/``timelike membrane'' pairs. They note that with this viewpoint the area increases monotonically in the examples considered. Our present work proves that this behavior is indeed general.\n\nAnalogues of a first law of thermodynamics have been formulated for dynamical horizons and trapping horizons. We expect that this can be extended to future holographic screens. However, here we shall focus on the second law and its classical manifestation as an area theorem.\n\n\\noindent{\\bf Outline \\ } In Sec.~\\ref{sec-screens}, we give a precise definition of future and past holographic screens, and we establish notation and nomenclature. We also describe a crucial mathematical structure derived from the foliation of $H$ by marginally (anti-)trapped leaves $\\sigma(r)$: there exists a vector field $h^a$ tangent to $H$ and normal to its leaves, which can be written as a linear combination of the orthogonal null vector fields $k^a$ and $l^a$. Its integral curves are called fibers of $H$. \n\nIt is relatively easy to see that the area of leaves is monotonic if $h^a l_a$ has definite sign, i.e., if $H$ evolves towards the past or exterior of each leaf. The difficulty lies in showing that it does so everywhere.\n\nOur proof is lengthy and involves nontrivial intermediate results. Given an arbitrary two-surface $\\sigma$ that splits a Cauchy surface into complementary spatial regions, we show in Sec.~\\ref{sec-csss} that a null hypersurface $N(\\sigma)\\supset \\sigma$ partitions the entire spacetime into two complementary spacetime regions: $K^+(\\sigma)$, the future-and-interior of $\\sigma$; and $K^-(\\sigma)$, the past-and-exterior of $\\sigma$.\n\nIn Sec.~\\ref{sec-ssmono}, we consider a hypersurface foliated by Cauchy-splitting surfaces $\\sigma(r)$. We prove that $K^+(r)$ grows monotonically under inclusion, if the surfaces $\\sigma(r)$ evolve towards their own past-and-exterior. This puts on a rigorous footing the equivalence (implicit in the constructions of~\\cite{CEB2}) between foliations of $H$ and null foliations of spacetime regions. The proofs in Sec.~\\ref{sec-monosplit} do not use all of the properties of $H$; in particular they do not use the marginally trapped property of its leaves. Thus our results up to this point apply to more general classes of hypersurfaces.\n\nIn Sec.~\\ref{sec-arealaw}, we do use the assumption that the leaves of $H$ are marginally trapped, and we combine it with the monotonicity of $K^+(r)$ that we established for past-and-exterior evolution. This allows us to show that the evolution of leaves $\\sigma(r)$ on a future holographic screen $H$ must be {\\em everywhere} to the past or exterior (assuming the null energy condition and certain generic conditions). This is the core of our proof. We then demonstrate that such evolution implies that the area $A(r)$ of $\\sigma(r)$ increases strictly monotonically with $r$. \n\nWe close Sec.~\\ref{sec-arealaw} with a theorem establishing the uniqueness of the foliation of a given holographic screen. The holographic screens themselves are highly nonunique. For example, one can associate a past (future) holographic screen with any observer, by finding the maximal area surfaces on the past (future) light-cones of each point on the observer's worldline.\n\n\n\n\n\\section{Holographic Screens}\n\\label{sec-screens}\n\nWe assume throughout this paper that the spacetime is globally hyperbolic (with an appropriate generalization for asymptotically AdS geometries~\\cite{Wal10QST,EngWal14}). We assume the null curvature condition (NCC): $R_{ab}k^{a}k^{b}\\geq 0$ where $k^{a}$ is any null vector. In a spacetime with matter satisfying Einstein's equations this is equivalent to the null energy condition: $T_{ab}k^{a}k^{b}\\geq 0$.\n\n\\begin{defn}\nA {\\em future holographic screen}~\\cite{CEB2} (or {\\em marginally trapped tube}~\\cite{AshGal, AshKri04}) $H$ is a smooth\nhypersurface admitting a foliation by marginally trapped surfaces called {\\em leaves}.\n\nA {\\em past holographic screen} is defined similarly but in terms of marginally anti-trapped surfaces. Without loss of generality, we will consider future holographic screens in general discussions and proofs.\n\nBy {\\em foliation} we mean that every point $p\\in H$ lies on exactly one leaf. A {\\em marginally trapped surface} is a codimension 2 compact spatial surface $\\sigma$ whose two future-directed orthogonal null geodesic congruences satisfy\n\\begin{eqnarray}\n\\theta_{k} &=& 0 \\label{marginal}~,\\\\\n\\theta_{l} &<& 0 \\label{trapped}~.\n\\end{eqnarray} \nThe opposite inequality defines ``marginally anti-trapped'', and thus, past holographic screens. Here $\\theta_{k}= \\hat{\\nabla}_a k^a$ and $\\theta_{l} = \\hat{\\nabla}_a l^a$ are the null expansions~\\cite{Wald} (where $\\hat{\\nabla}_{a}$ is computed with respect to the induced metric on $\\sigma$), and $k^{a}$ and $l^{a}$ are the two future directed null vector fields orthogonal to $\\sigma$.\n\nWe will refer to the $k^a$ direction as {\\em outward} and to the $l^a$ direction as {\\em inward}. For screens in asymptotically flat or AdS spacetimes, these notions agree with the intuitive ones. Furthermore, in such spacetimes any marginally trapped surface, and hence any holographic screen, lies behind an event horizon. However, holographic screens may exist in cosmological spacetimes where an independent notion of outward, such as conformal infinity, need not exist (e.g., a closed FRW universe). In this case the definition of $H$ requires only that there exist some continuous assignment of $k^a$ and $l^a$ on $H$ such that all leaves are marginally trapped. See Fig.~\\ref{fig-examplesPRD} for examples of holographic screens.\n\\end{defn}\n\n\\begin{defn}\nThe defining foliation of $H$ into leaves $\\sigma$ determines a $(D-2)$-parameter family of leaf-orthogonal curves $\\gamma$, such that every point $p\\in H$ lies on exactly one curve that is orthogonal to $\\sigma(p)$. We will refer to this set of curves as the {\\em fibration of $H$}, and to any element as a {\\em fiber} of $H$.\n\\end{defn}\n\n\\begin{conv} \\label{conv-rh}\nThus it is possible to choose a (non-unique) {\\em evolution parameter} $r$ along the screen $H$ such that $r$ is constant on any leaf and increases monotonically along the fibers $\\gamma$. We will label leaves by this parameter: $\\sigma(r)$. \n\nThe tangent vectors to the fibers define a vector field $h^a$ on $H$. For any choice of evolution parameter the normalization of this vector field can be fixed by requiring that the function $r$ increases at unit rate along $h^a$: $h(r)=h^a\\, (dr)_a=1$. (Since $H$ can change signature, unit normalization of $h^a$ would be possible only piecewise, and hence would not be compatible with the desired smoothness of $h^a$.)\n\\end{conv}\n\n\\begin{rem}\nSince fibers are orthogonal to leaves, a tangent vector field $h^a$ can be written as a (unique) linear combination of the two null vector fields orthogonal to each leaf:\n\\begin{equation}\nh^{a} = \\alpha l^{a} + \\beta k^{a}\n\\label{eq-hlk}\n\\end{equation}\nMoreover, the foliation structure guarantees that $h^a$ vanishes nowhere: it is impossible to have $\\alpha=\\beta=0$ anywhere on $H$. (These remarks hold independently of the requirement that each leaf be marginally trapped.) \\label{rem-h}\n\\end{rem}\n\\begin{figure}[ht]\n\n\\includegraphics[width=2in]{AlphaBetaDiagram.pdf}\n\\caption{The null vectors $l^{a}$ and $k^{a}$ orthogonal to a leaf $\\sigma$ of the foliation of $H$ at some point. The evolution of $H$ is characterized by vector $h^a$ normal to the leaves and tangent to $H$. Depending on the quadrant $h^a$ points to, $H$ evolves locally to the future, exterior, past, or interior (clockwise from top).}\n\\label{fig-st}\n\n\\end{figure}\n\n\\begin{conv} \\label{conv-stnotation}\nAs shown in Fig.~\\ref{fig-st}, $h^a$ is spacelike and outward-directed if $\\alpha<0, \\beta>0$; timelike and past-directed if $\\alpha<0, \\beta<0$; spacelike and inward-directed if $\\alpha>0, \\beta<0$; and finally, timelike and future-directed if $\\alpha>0, \\beta>0$. We denote such regions, in this order (and somewhat redundantly): $S_{-+},T_{--},S_{+-},T_{++}$. \n\\end{conv}\n\\begin{rem} \\label{rem-result}\nOur key technical result below will be to demonstrate that $\\alpha$ cannot change sign on $H$. Thus on a given screen $H$, either only the first two, or only the second two possibilities are realized. (The latter case can be reduced to the former by taking $r\\to -r$.)\n\\end{rem}\n\\begin{rem} \\label{rem-borders}\nBecause $\\alpha$ and $\\beta$ cannot simultaneously vanish, $S_{+-}$ and $S_{--}$ regions cannot share a boundary or be separated by a null region; they must be separated by a timelike region. Similarly $T_{++}$ and $T_{--}$ regions must be separated by a spacelike region.\n\\end{rem}\n\nBelow we will consider only holographic screens that satisfy additional technical assumptions:\n\\begin{defn} \\label{def-technical}\nA holographic screen $H$ is {\\em regular} if\n\\begin{enumerate}[(a)]\n\\item \\label{def-technical1} the {\\em first generic condition} is met, that $R_{ab} k^a k^b + \\varsigma_{ab}\\varsigma^{ab}>0$ everywhere on $H$, where $\\varsigma_{ab}$ is the shear of the null congruence in the $k^a$-direction;\n\\item \\label{def-technical2} the {\\em second generic condition} is met: let $H_+$, $H_-$, $H_0$ be the set of points in $H$ with, respectively, $\\alpha>0$, $\\alpha<0$, and $\\alpha=0$. Then $H_0= \\dot H_- = \\dot H_+$. \n\\item \\label{def-technical3} every inextendible portion $H_i\\subset H$ with definite sign of $\\alpha$ either contains a complete leaf, or is entirely timelike.\n\\item \\label{def-technical4} every leaf $\\sigma$ splits a Cauchy surface $\\Sigma$ into two disjoint portions $\\Sigma^\\pm$.\n\\end{enumerate} \n\\end{defn}\nAnalogous assumptions have been used in the more restricted context of dynamical horizons. The first generic condition is identical to the regularity condition of~\\cite{AshGal}. Together with the null curvature condition, $R_{ab}k^a k^b\\geq 0$, it ensures that the expansion of the $k^a$-congruence becomes negative away from each leaf. The second generic condition excludes the degenerate case where $\\alpha$ vanishes along $H$ without changing sign. Either condition excludes the existence of an open neighborhood in $H$ with $\\alpha=0$. Both are aptly called ``generic'' since they can fail only in situations of infinitely fine-tuned geometric symmetry and matter distributions.\nThe third assumption is substantially weaker than the definition of a dynamical horizon, since we do not require global spacelike signature of $H$. The fourth assumption will play a role analogous to the assumption of achronality of the dynamical horizon. It holds in typical spacetimes of interest (including settings with nontrivial spatial topology, such as $S^1\\times S^2$, as long as the holographic screen is sufficiently localized on the sphere). We leave the question of relaxing some or all of these assumptions to future work.\n\n\\begin{rem} \\label{rem-s0exists}\nAssumption~\\ref{def-technical}.c and Remark~\\ref{rem-borders} imply that $H$ contains at least one complete leaf with definite sign of $\\alpha$.\n\\end{rem}\n\\begin{conv}\n\\label{conv-orient}\nLet $\\sigma(0)\\subset H$ be an arbitrary leaf with definite sign of $\\alpha$. We will take the parameter $r$ to be oriented so that $\\alpha< 0$ on $\\sigma(0)$, and we take $r=0$ on $\\sigma(0)$. By convention \\ref{conv-rh} this also determines the global orientation of the vector field $h^a$. For past holographic screens, it is convenient to choose the opposite convention, $\\alpha>0$ on $\\sigma(0)$.\n\\end{conv}\n\n\n\\section{Leaves Induce a Monotonic Spacetime Splitting}\n\\label{sec-monosplit}\n\nIn this section, we will use only a subset of the defining properties of a holographic screen. In Sec.~\\ref{sec-csss}, we examine the implications of Assumption~\\ref{def-technical}.d, that each leaf split a Cauchy surface. We show that a null surface orthogonal to such a leaf splits the entire spacetime into two disconnected regions $K^\\pm(\\sigma)$.\n\nIn Sec.~\\ref{sec-ssmono}, we use the foliation property of the holographic screen. (However, nowhere in this section do we use the condition that each leaf be marginally trapped, or Assumptions~\\ref{def-technical}.a-c.) We show that in portions of $H$ where $\\alpha$ is of constant sign, the sets $K^\\pm(\\sigma(r))$ satisfy inclusion relations that are monotonic in the evolution parameter $r$. \n\nTogether these results imply that an $\\alpha<0$ foliation of any hypersurface $H$ into Cauchy-splitting surfaces $\\sigma$ induces a null foliation of the spacetime, such that each null hypersurface $N(\\sigma)$ splits the entire spacetime into disconnected regions $K^\\pm(\\sigma)$. \n\nIn the following section, we will add the marginally trapped condition and the remaining technical assumptions, to show that on a future holographic screen, $\\alpha$ {\\em must}\\\/ have constant sign.\n\n\\subsection{From Cauchy Splitting to Spacetime Splitting}\n\\label{sec-csss}\n\nBy Assumption~\\ref{def-technical}.d, every leaf $\\sigma$ splits a Cauchy surface $\\Sigma$ into two disconnected portions $\\Sigma^+$ and $\\Sigma^-$:\n\\begin{equation}\n\\Sigma = \\Sigma^+ \\cup \\sigma \\cup \\Sigma^-~,~~\\sigma = \\dot\\Sigma^\\pm~.\n\\end{equation}\nWe take $\\Sigma^\\pm$ to be open in the induced topology on $\\Sigma$, so that $\\Sigma^\\pm\\cap\\sigma=\\varnothing$.\nWe consider the following sets shown in Fig.~\\ref{fig-sets}a:\n\\begin{itemize} \n\\item $I^+(\\Sigma^+)$, the chronological future of $\\Sigma^+$: this is the set of points that lie on a timelike future-directed curve starting at $\\Sigma^+$. (Note that this set does not include $\\Sigma^+$.)\n\\item $D^-(\\Sigma^+)$, the past domain of dependence of $\\Sigma^+$: this is the set of points $p$ such that every future-directed causal curve through $p$ must intersect $\\Sigma^+$. (This set does include $\\Sigma^+$.)\n\\item Similarly, we consider $I^-(\\Sigma^-)$ and $D^+(\\Sigma^-)$.\n\\end{itemize} \n\n\\begin{figure*}[ht]\n\n\\subfigure[ ]{\n\n\\includegraphics[height=0.4 \\textwidth]{SigmaDecompositionv2.pdf}\n\\label{fig-sets-a}}\n\\qquad\n\\subfigure[]{\n\\includegraphics[height=0.4\\textwidth]{Ksv2.pdf}\n\\label{fig-sets-b}}\n\\caption{(a) Each leaf $\\sigma$ splits a Cauchy surface. This defines a partition of the entire spacetime into four regions, given by the past or future domains of dependence and the chronological future or past of the two partial Cauchy surfaces. (b) The pairwise unions $K^\\pm$ depend only on $\\sigma$, not on the choice of Cauchy surface. They can be thought as past and future in a null foliation defined by the lightsheets $N$.}\n\\label{fig-sets}\n\\end{figure*}\n\n\n\\begin{defn}\\label{def-sets}\nFrom the Cauchy-splitting property of $\\sigma$, it follows\\footnote{The proofs of the following statements are straightforward and use only well-known properties of $I^\\pm$ and $D^\\pm$.} that the four sets defined above have no mutual overlap. However they share null boundaries:\n\\begin{align} \nN^+(\\sigma) & \\equiv & \\dot I^+(\\Sigma^+) -\\Sigma^+ = \\dot D^+(\\Sigma^-) - I^-(D^+(\\Sigma^-)) \\\\\nN^-(\\sigma) & \\equiv & \\dot I^-(\\Sigma^-) - \\Sigma^- = \\dot D^-(\\Sigma^+) - I^+(D^-(\\Sigma^+))\n\\end{align} \nNote that $N^+(\\sigma)\\cap N^-(\\sigma)=\\sigma$. We define\n\\begin{eqnarray} \nK^+(\\sigma) & \\equiv & I^+(\\Sigma^+) \\cup D^-(\\Sigma^+)-N^+(\\sigma)~; \\\\\nK^-(\\sigma) & \\equiv & D^+(\\Sigma^-) \\cup I^-(\\Sigma^-)-N^-(\\sigma)~; \\\\\nN(\\sigma) &\\equiv & N^+(\\sigma) \\cup N^-(\\sigma)\n\\end{eqnarray}\nThus\n\\begin{equation}\nN(\\sigma) = \\dot K^+(\\sigma) = \\dot K^-(\\sigma)~;\n\\end{equation}\nand the sets $N$, $K^+$, and $K^-$ provide a partition of the spacetime (Fig.~\\ref{fig-sets}b). \n\\end{defn}\n\n\\begin{lem}\\label{lem-secondN}\nThere exists an independent characterization of $N^+$, $N^-$, and thus of $N$:\n$N^+(\\sigma)$ is generated by the future-directed null geodesic congruence orthogonal to $\\sigma$ in the $\\Sigma^-$ direction up to intersections: $p\\in N^+(\\sigma)$ if and only if no conjugate point or nonlocal intersection with any other geodesic in the congruence lies between $\\sigma$ and $p$.\n\\end{lem}\nThis follows from a significantly strengthened version of Theorem 9.3.11 in Ref.~\\cite{Wald}, a proof of which will appear elsewhere. Similarly $N^-$ is generated by the past-directed $\\sigma$-orthogonal null congruence towards $\\Sigma^+$. (Hence if $\\sigma$ is marginally trapped then $N^\\pm$ both are light-sheets of $\\sigma$~\\cite{CEB1}.)\n\n\\begin{cor}\\label{cor-onlysigma}\nLemma~\\ref{lem-secondN} implies that $N$ depends only on $\\sigma$, not on the Cauchy surface $\\Sigma$. Moreover, the sets $K^+$ and $K^-$ are then uniquely fixed by the fact that $N$ splits the spacetime: $K^+$ is the largest connected set that contains $I^+(N)$ but not $N$.\n\\end{cor}\nThus our use of $\\sigma$ (as opposed to $\\Sigma^+$ and\/or $\\Sigma^-$) as the argument of the sets $K^\\pm$, $N^\\pm$ is appropriate. \n\n\\subsection{Monotonicity of the Spacetime Splitting}\n\\label{sec-ssmono}\n\nUntil now, we have only used the Cauchy-splitting property of $\\sigma$. We will now consider a family of such leaves, $\\sigma(r)$, that foliate a hypersurface ${\\cal H}$. (We use this notation instead of $H$, in order to emphasize that ${\\cal H}$ need not satisfy the additional assumptions defining a future holographic screen.) A tangent vector field $h^a$ can be defined as described in Remark~\\ref{rem-h}, with decomposition $h^a = \\alpha l^a + \\beta k^a$ into the null vectors orthogonal to each leaf. We take $\\Sigma^+$ to be the side towards which the vector $l^a$ points. (This convention anticipates Sec.~\\ref{sec-arealaw}. In the current section, $k^a$ and $l^a$ need not be distinguished by conditions on the corresponding expansions.) To simplify notation, we denote $K^+(\\sigma(r))$ as $K^+(r)$, etc.\n\n\\begin{thm}\\label{thm-kmono}\nConsider a foliated hypersurface ${\\cal H}$ with tangent vector field $h^a$ defined as above. Suppose that $\\alpha<0$ on all leaves $\\sigma(r)$ in some open interval, $r\\in I$. Then\n\\begin{equation}\n\\bar K^+(r_1)\\subset K^+(r_2)~,\n\\label{eq-kmono}\n\\end{equation}\nor equivalently $K^-(r_1)\\supset \\bar K^-(r_2)$, for all $r_1, r_2 \\in I$ with $r_10$ between $\\sigma(r)$ and $\\sigma(r+\\delta r)$, i.e., if $\\delta {\\cal H}$ is spacelike. Both the future of a set, and the past domain of dependence of a set cannot become smaller when the set is enlarged; hence,\n\\begin{eqnarray} \nI^+(X) & \\supset & I^+(\\Sigma^+(r))~,\\nonumber \\\\\nD^-(X) & \\supset & D^-(\\Sigma^+(r))~,\n\\label{eq-idsuper}\n\\end{eqnarray} \nand so the infinitesimal version of Eq.~(\\ref{eq-kmono}) follows trivially from the definition of $K^+$.\n\nNow consider the general case, with no restriction on the sign of $\\beta$. Thus, $\\delta {\\cal H}$ may be spacelike, timelike ($\\beta<0$), or null ($\\beta=0$); indeed, it may be spacelike at some portion of $\\sigma(r)$ and timelike at another. One can still define the submanifold $X$ as the extension of $\\Sigma^+(r)$ by $\\delta {\\cal H}$, as in Eq.~(\\ref{eq-x}); see Fig.~\\ref{fig-deltah}. Again, this extension cannot decrease the future of the set, nor its past domain of dependence,\\footnote{The future of a set is defined for arbitrary sets. The domain of dependence is usually defined only for certain sets, for example for closed achronal sets in Ref.~\\cite{Wald}. Here we extend the usual definition to the more general set $X$: $p\\in D^-(X)$ iff every future-inextendible causal curve through $p$ intersects $X$. This is useful for our purposes; however, we caution that certain theorems involving $D^\\pm$ need not hold with this broader definition.} as described in Eq.~(\\ref{eq-idsuper}).\n\nHowever, $X$ need not be achronal and hence, it need not lie on any Cauchy surface. In this case, we consider a new Cauchy surface that contains $\\sigma(r+\\delta r)$. Because $\\alpha<0$, this surface can be chosen so that $\\Sigma^+(r+\\delta r)$ is nowhere to the future of $X$; see Fig.~\\ref{fig-deltah}. Since $X$ and $\\Sigma^+(r+\\delta r)$ share the same boundary $\\sigma(r+\\delta r)$, $\\alpha>0$ then implies that $X$ is entirely in the future of $\\Sigma^+(r+\\delta r)$: \n\\begin{equation}\nX\\subset I^+(\\Sigma^+(r+\\delta r))\n\\label{eq-xfut}\n\\end{equation}\nMoreover, the set $X$ together with $\\bar\\Sigma^+(r+\\delta r)$ forms a ``box'' that bounds an open spacetime region $Y$, such that \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.0in]{NestedK2.pdf}\n\\caption{Proof that $K^+(r)$ grows monotonically under inclusion, for any foliation $\\sigma(r)$ of a hypersurface $\\cal H$ with $\\alpha<0$. See the main text for details and definitions.}\n\\label{fig-deltah}\n\\end{center}\n\\end{figure}\n\\begin{equation}\nY\\subset I^+(\\Sigma^+(r+\\delta r))~.\n\\label{eq-ysubi}\n\\end{equation}\nAll future-directed timelike curves that pass through $\\Sigma^+(r+\\delta r)$ enter $Y$ and then can exit $Y$ only through $X$. Hence $D^-(X) \\subset Y \\cup D^-(\\Sigma^+(r+\\delta r))$. Since $\\alpha<0$, for all points outside of $Y \\cup D^-(\\Sigma^+(r+\\delta r))$ there exist future-directed timelike curves that evade $X$. Hence equality holds:\n\\begin{equation}\nD^-(X) = Y \\cup D^-(\\Sigma^+(r+\\delta r))~.\n\\label{eq-dmx}\n\\end{equation}\nTo obtain the infinitesimal inclusion relation, \n\\begin{equation}\nK^+(r+\\delta r)\\supset K^+(r)~,\n\\label{eq-minimono}\n\\end{equation}\nby Eq.~(\\ref{eq-idsuper}) it suffices to show that $K^+(r+\\delta r)\\supset I^+(X)\\cup D^-(X)$. Indeed if $p\\in I^+(X)$ by Eq.~(\\ref{eq-xfut}) $p\\in I^+(\\Sigma^+(r+\\delta r))\\subset K^+(r+\\delta r)$. And if $p\\in D^-(X)$ then by Eqs.~(\\ref{eq-dmx}) and (\\ref{eq-ysubi}) we again have $p\\in K^+(r+\\delta r)$.\n\nTo obtain the stricter relation\n\\begin{equation}\nK^+(r+\\delta r)\\supset \\bar K^+(r)~,\n\\label{eq-minimonos}\n\\end{equation}\nwe note that $\\sigma(r)\\subset X$; hence by Eq.~(\\ref{eq-xfut}), for every point $p\\in\\sigma(r)$ there exists a timelike curve from $\\Sigma^+(r+\\delta r)$ to $p$. This curve can be continued along the null generator of $N^+(r)$ starting at $p$ to a point $q\\in N^+(r)$, and then slightly deformed into a timelike curve connecting $p$ to $q$. By Lemma~\\ref{lem-secondN}, every point in $N^+(r)$ lies on a generator starting at $\\sigma(r)$. Hence, $N^+(r)\\subset K^+(r+\\delta r)$. A similar argument yields $N^-(r)\\subset K^+(r+\\delta r)$. Since $N(r)=N^+(r)\\cup N^-(r)$ and $\\bar K^+(r) = K^+(r)\\cup N(r)$, Eq.~(\\ref{eq-minimonos}) follows. \n\nTo extend Eq.~(\\ref{eq-minimonos}) to Eq.~(\\ref{eq-kmono}), one may iterate the above infinitesimal construction. The only way this could fail is if the iteration gets stuck because the steps $\\delta r$ have to be taken ever smaller to keep Eq.~(\\ref{eq-rdr}) satisfied. Suppose therefore that the iteration can only reach an open set $(r,r_*)$ but no leaves in the set $(r_*,r_2)$. But this contradicts the assumption that $\\alpha<0$ at $r_*$.\n\\end{proof}\n\n\\section{Area Law}\n\\label{sec-arealaw}\n\nIn this section, we prove our main result: that the area of the holographic screen is monotonic. The most difficult part of this task is proving that $\\alpha$ cannot change sign on $H$, Theorem~\\ref{thm-alpha}).\nWe then prove our Area Theorem~\\ref{thm-area}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.35 \\textwidth]{lemma3D.pdf}\n\\caption{An example illustrating Lemma~\\ref{monotonicity}: in Minkowski space, the spatial sphere $\\chi$ is tangent to the null plane $N$ at $p$ and lies outside the past of $N$ near $p$. It is easy to see that this implies that $\\chi$ is a cross-section of a future light-cone that shares one null generator with $N$. In this example it is obvious that $\\chi$ expands faster than $N$ at $p$, as claimed in Lemma~\\ref{monotonicity}.}\n\\label{fig-lemma}\n\\end{figure}\nWe begin by stating a useful Lemma.\n\\begin{lem} \\label{monotonicity}\nLet $N$ be a null hypersurface and let $\\chi$ be a spacelike surface tangent to $N$ at a point $p$. That is, we assume that one of the two future-directed null vectors orthogonal to $\\chi$, $\\kappa^a$, is also orthogonal to $N$ at $p$. We may normalize the (null) normal vector field to $N$ so that it coincides with $\\kappa^a$ at $p$. Let $\\theta^{(\\chi)}$ be the null expansion of the congruence orthogonal to $\\chi$ in the $\\kappa^a$ direction, and let $\\theta^{(N)}$ be the null expansion of the generators of $N$. Then:\n\\begin{itemize} \n\\item If there exists an open neighborhood $O(p)\\cap \\chi$ that lies entirely outside the past of $N$,\\footnote{I.e., there exists no past directed causal curve from any point on $N$ to any point in $O(p)\\cap \\chi$.} then $\\theta^{(\\chi)}\\geq \\theta^{(N)}$ at $p$.\n\\item If there exists an open neighborhood $O(p)\\cap \\chi$ that lies entirely outside the future of $N$, then $\\theta^{(\\chi)}\\leq \\theta^{(N)}$ at $p$.\n\\end{itemize} \n\\end{lem}\n\\begin{proof}\nSee Lemma A in Ref.~\\cite{Wal10QST}. Our Lemma is stronger but the proof is the same; so instead of reproducing it here, we offer Fig.~\\ref{fig-lemma} to illustrate the result geometrically. It generalizes to null hypersurfaces an obvious relation in Riemannian space, between the extrinsic curvature scalars of two codimension 1 surfaces that are tangent at a point in a Riemannian space but do not cross near that point.\n\\end{proof}\n\n\n\n\\begin{thm} Let $H$ be a regular future holographic screen with leaf-orthogonal tangent vector field $h^{a}=\\alpha l^{a} + \\beta k^{a}$, whose orientation is chosen so that $\\alpha<0$ at the leaf $\\sigma(0)$. Then $\\alpha< 0$ everywhere on $H$. \\label{thm-alpha}\n\\end{thm}\n\n\\begin{proof}\nBy contradiction: suppose that $H$ contains a point with $\\alpha\\geq 0$. It immediately follows that the subset $H_+\\subset H$ of points with $\\alpha> 0$ is nonempty, since by Assumption~\\ref{def-technical}.b, $\\alpha=0$ can occur only as a transition between $\\alpha<0$ and $\\alpha>0$ regions. Let $\\sigma(0)$ be the complete leaf that exists by Remark~\\ref{rem-s0exists} and has $r=0$, $\\alpha<0$, by Convention~\\ref{conv-orient}. By continuity of $\\alpha$, there exists an open neighborhood of $\\sigma(0)$ where $\\alpha<0$. \n\n\\begin{figure}[t]\n\\includegraphics[height=0.3 \\textwidth]{Forbidden.pdf}\n\\caption{The four types of spacelike-timelike transitions on a future holographic screen that would violate the monotonicity of the area, and which our proof in Sec.~\\ref{sec-arealaw} will exclude. Near $\\sigma(0)$, the area increases in the direction of the arrow. On the far side of the ``bend'' the area would decrease, in the same direction. There are other types of spacelike-timelike transitions which preserve area monotonicity under uniform flow; these do arise generically (see Fig.~\\ref{fig-examplesPRD}a).}\n\\label{fig-sphsym}\n\\end{figure}\nWe first consider the case where $H_+$ has a component in the $r>0$ part of $H$ (cases 1 and 2 in Fig.~\\ref{fig-sphsym}). Let $\\sigma(1)$ be the ``last slice'' on which $\\alpha\\leq 0$, i.e., we use our freedom to rescale $r$ to set\n\\begin{equation}\n1=\\inf\\{r:r>0,\\sigma(r)\\cap H_+ \\neq \\varnothing\\}\n\\end{equation}\nBy the second generic condition~\\ref{def-technical}.b, $\\alpha<0$ for all leaves $\\sigma(r)$ with $00$ in $O(p)$, so that the assumed sign change from $\\alpha<0$ to $\\alpha>0$ corresponds to a transition of $h^a$ from spacelike-outward ($S_{-+}$) to timelike-future-directed ($T_{++}$). The following construction is illustrated in Fig. \\ref{fig-Case1LR}.\n\nLet $\\sigma^+(1+\\epsilon)$ be the set of points with $\\alpha>0$ on the leaf $\\sigma(1+\\epsilon)$. If there is more than one connected component, we choose $\\sigma^+(1+\\epsilon)$ to be the component at least one of whose fibers intersects $p$. By choosing $\\epsilon$ sufficiently small, we can ensure that $\\sigma^+(1+\\epsilon) \\subset O(p)$. Let $\\Gamma$ be the set of fibers that pass through $\\sigma^+(1+\\epsilon)$. \n\nBecause $\\alpha>0$, all fibers in $\\Gamma$ enter $K^-(1+\\epsilon)$ as we trace them back to smaller values of $r$. But $\\sigma(0)$ is entirely outside of this set: by definition, $\\sigma(0)\\cap K^-(0)=\\varnothing$, so Eq.~(\\ref{eq-nocon}) implies $\\sigma(0)\\cap K^-(1+\\epsilon)=\\varnothing$. Hence, all fibers in $\\Gamma$ also intersect $N(1+\\epsilon)$, at some positive value of $r< 1+\\epsilon$. Because $\\beta>0$ in $O(p)$, this intersection will be with $N^-(1+\\epsilon)$. By smoothness and the second generic assumption, the intersection will consist of one point per fiber. (Otherwise a fiber would coincide with a null generator of $N^{-}(1+\\epsilon)$ in a closed interval.) \nThe set of all such intersection points, one for each fiber in $\\Gamma$, defines a surface $\\phi$, and the fibers define a continuous, one-to-one map $\\sigma^+(1+\\epsilon)$ to $\\phi$. Similarly, the closures of both sets, $\\bar\\sigma^+(1+\\epsilon)$ and $\\bar\\phi$ are related by such a map. Note that these two sets share the same boundary at $r=1+\\epsilon$.\n\nLet $R$ be the minimum value of $r$ on the intersection: $R\\equiv\\inf\\{r(q): q\\in \\bar\\phi\\}$. Since $\\bar\\sigma^+(1+\\epsilon)$ is a closed subset of a compact set, it is compact; and by the fiber map, $\\bar\\phi$ is also compact. Therefore $R$ is attained on one or more points in $\\bar\\phi$. Let $Q$ be such a point. Since $R<1$ but $\\dot\\phi\\subset \\sigma(1+\\epsilon)$, $Q\\notin\\dot\\phi$, and hence $Q$ represents a local minimum of $r$. Hence the leaf $\\sigma(R)$ is tangent to the null hypersurface $N^{-}(1+\\epsilon)$ at $Q$.\n\nSince $Q$ achieves a global minimum of $r$ on $\\bar\\phi$, $\\sigma(R)$ lies nowhere in the past of $N^{-}(1+\\epsilon)$ in a sufficiently small open neighborhood of $Q$. For suppose there existed no such neighborhood. Then fibers arbitrarily close to the one containing $Q$ (and hence connected to $\\sigma^+(1+\\epsilon)$ would still be inside $K^-(1+\\epsilon)$ at $R$. Hence we could find a value $r0$ at $Q$. But this contradicts the defining property of holographic screens, that all leaves are marginally trapped ($\\theta_k^{\\sigma(r)}=0$ for all $r$). \n\n\\vskip .6cm\n\\noindent {\\bf Case 2 \\ } Next we consider the case where $\\beta<0$ in the neighborhood of the assumed transition from $\\alpha<0$ to $\\alpha>0$ that begins at $r=1$ (see Fig.~\\ref{fig-sphsym}). This corresponds to the appearance of a spacelike-inward-directed region within a timelike-past-directed region: $T_{--}\\to S_{+-}$.\n\nWe note that the direct analogue of the above proof by contradiction fails: tracing back the generators from $\\sigma^+(1+\\epsilon)$ to $\\sigma(0)$, one finds that they pass through $N^+(1+\\epsilon)$, rather than $N^-(1+\\epsilon)$. But $N^+$ has negative expansion by the first generic condition, whereas $N^-$ had positive expansion. There is no compensating sign change elsewhere in the argument; in particular, the tangent leaf $\\sigma(R)$ with vanishing expansion again lies nowhere in the past of $N^+$ in a neighborhood of the tangent point $Q$. Thus no contradiction arises with Lemma~\\ref{monotonicity}.\n\nInstead, we show that every case 2 transition implies the existence of a case 1 transition at a {\\em different}\\\/ point on $H$, under the reverse flow $r\\to c-r$. Since we have already shown that case 1 transitions are impossible, this implies that case 2 transitions also cannot occur. \n\nLet us first illustrate this argument in the simple case where the transition occurs entirely on a single leaf: $\\alpha<0$ for $0\\leq r<1$, $\\alpha=0$ at $r=1$, and $\\alpha>0$ for $10$ region contains a complete leaf $\\sigma(2+\\epsilon)$. In the text we show that the complete-leaf region begins at some leaf $\\sigma(2)$ where a $T_{--}\\to S_{+-}$ boundary comes to an end: either the original one (a), or a different one containing a $T_{--}$ region with no complete leaf (b). The endpoint (green dot) becomes the starting point of a case 1 transition ($S_{-+}\\to T_{++}$) under reversal of the flow direction; but this case has already been ruled out.}\n\\label{fig-case2proof}\n\\end{figure*}\n\n\nIn general, the case 2 transition need not occur on a single leaf, so we shall assume for contradiction only that $\\alpha$ first becomes positive at some point on or subset of $\\sigma(1)$, as in the case 1 proof, and that $\\beta<0$ in a neighborhood of this set. Let $\\tilde H_+$ denote the connected region with $\\alpha>0$ that begins at this transition. Since the transition is $T_{--}\\to S_{+-}$, $\\tilde H_+$ contains some spacelike points; and hence by Def.~\\ref{def-technical}.c, $\\tilde H_+$ contains a complete leaf with $\\alpha>0$. We use our freedom to rescale $r$ to set\n\\begin{equation}\n2=\\inf\\{r:r>0,\\sigma(r)\\subset \\tilde H_+\\}\n\\label{eq-leaf2}\n\\end{equation}\nBy the second generic assumption, Def.~\\ref{def-technical}.b, this choice implies the existence of an open interval $(2,2+\\epsilon)$ such that every leaf in this interval is a complete leaf with $\\alpha>0$. Let us call this intermediate result (*); see Fig.~\\ref{fig-case2proof} which also illustrates the remaining arguments.\n\nWe now consider the boundary $B$ that separates the $\\alpha<0$ from the $\\alpha>0$ region, i.e., the connected set of points with $\\alpha=0$ that begins at $r=1$. Because $\\alpha$ and $\\beta$ cannot simultaneously vanish, we have $\\beta<0$ in an open neighborhood of all of $B$. Thus, $B$ separates a $T_{--}$ region at smaller $r$ from a $S_{+-}$ region at larger $r$. We note that $B$ must intersect every fiber, or else $H_+$ would not contain a complete leaf. Moreover, $B$ must end at some $r_*\\leq 2$, or else there would be points with $\\alpha<0$ in the interval $(2,2+\\epsilon)$, in contradiction with (*). \n\nIf $r_*=2$ then under the reverse flow starting from the complete leaf at $r=2+\\epsilon$ there is a case 1 transition at $r=2$ from $S_{-+}$ to $T_{++}$, and we are done. This is shown in Fig.~\\ref{fig-case2proof}a. \n\nThe only remaining possibility is that $B$ ends at some $r_*\\in (1,2)$; this is shown in Fig.~\\ref{fig-case2proof}b. Then every leaf with $r\\in (r_*,2)$ must contain points with $\\alpha<0$, or else there would be a complete leaf with $\\alpha>0$ at some $r<2$, in contradiction with Eq.~(\\ref{eq-leaf2}). Therefore each leaf with $r\\in (r_*,2)$ must intersect one or more $\\alpha<0$ regions $\\tilde H_-^{(i)}$ that are disconnected from the $T_{--}$ region bounded by $B$. None of these regions $\\tilde H_-^{(i)}$ can contain a complete $\\alpha<0$ leaf, because this would imply that $\\tilde H_+$ does not contain a complete $\\alpha>0$ leaf. From Def.~\\ref{def-technical}.c it follows that each region $\\tilde H_-^{(i)}$ is everywhere timelike, i.e., of type $T_{--}$. But this implies that a $T_{--}$ region ends at $r=2$ where $\\alpha$ becomes positive. Moreover, the $S_{+-}$ region in which the $T_{--}$ region ends has complete leaves in some open interval $(2,2+\\epsilon)$ by our result (*). Thus we find again that under the reverse flow starting from the complete leaf at $r=2+\\epsilon$ there is a case 1 transition at $r=2$ from $S_{-+}$ to $T_{++}$.\n\nWe have thus established that a case 2 transition at $r=1$ implies a case 1 transition at the same or a larger value of $r$, after reversal of the direction of flow. Since case 1 transitions are impossible, we conclude that case 2 transitions are also impossible.\n\\begin{figure*}[ht]\n\n\\includegraphics[width= 0.85\\textwidth]{Panelv3.pdf}\n\\caption{(a) Case 3 is ruled out analogously to case 1, by contradiction. (b) Case 4 is analogous to case 2: the transition is impossible because it would imply a case 3 transition elsewhere on $H$, under reversal of the flow direction.}\n\\label{fig-cases}\n\\end{figure*}\n\n\n\\vskip .6cm\n\\noindent {\\bf Cases 3 and 4} \nOur consideration of cases 1 and 2 has ruled out the possibility of points with $\\alpha>0$ at any $r>0$. (Recall that $r=0$ corresponds to a complete leaf with $\\alpha<0$.) We must now also rule out the possibility that $\\alpha$ might be positive in the region $r<0$; this corresponds to cases 3 and 4 in Fig.~\\ref{fig-sphsym}. Again, assume for contradiction that such a transition occurs, and focus on the transition nearest to $r=0$. We may rescale $r$ so that this transition ends at $r=-1$. That is, $\\alpha<0$ for all $r\\in (-1,0)$, but all leaves in some interval $(-(1+\\epsilon),-1)$ contain points with $\\alpha>0$. Again, a further case distinction arises depending on the sign of $\\beta$ at this transition.\n \nThe proof of case 3 (Fig.~\\ref{fig-cases}a), where $\\beta>0$ at the transition, proceeds exactly analogous to that of case 1. Fibers that connect the offending region to $r=0$ must cross the null hypersurface $N^+(-(1+\\epsilon))$, implying the existence of a leaf $\\sigma(R)$, $-10~.\n\\label{eq-area}\n\\end{equation}\n\\end{thm}\n\\begin{proof} \nBy Theorem~\\ref{thm-alpha}, $\\alpha<0$ everywhere on $H$. In regions where $\\beta$ is of definite sign, the result would then follow from the analysis of Hayward~\\cite{Hay93} (using a 2+2 lightlike formalism) or that of Ashtekar and Krishnan~\\cite{AshKri02} who used a standard 3+1 decomposition. It should be straightforward to generalize their proofs to the case where $\\beta$ may not have definite sign on some or all leaves. However, since this would necessitate the introduction of additional formalism, we will give here a simple, geometrically intuitive proof. Our construction is shown in Fig.~\\ref{fig-zigzag}. \n\nConsider two infinitesimally nearby leaves at $r$ and $r+dr$, $dr>0$. Construct the null hypersurface $N(r)$ in a neighborhood of $\\sigma(r)$. Also, construct the null hypersurface $L^+(r+dr)$ generated by the future-directed null geodesics with tangent vector $l^a$, in a neighborhood of $\\sigma(r+dr)$. By Theorem~\\ref{thm-alpha}, for sufficiently small $dr$ these null hypersurfaces intersect on a two-dimensional surface $\\hat\\sigma(r,r+dr)$, such that every generator of each congruence lies on a unique point in $\\hat\\sigma(r,r+dr)$. \n\nNote that in regions where $H$ is spacelike, $\\beta>0$, the intersection will lie in $N^+(r)$; if $H$ is timelike, $\\beta<0$, the intersection will lie in $N^-(r)$; but this makes no difference to the remainder of the argument. Crucially, Theorem~\\ref{thm-alpha} guarantees that the intersection always lies in $L^+(r+dr)$, and never on $L^-(r+dr)$, the null hypersurface generated by the past-directed null geodesics with tangent vector $-l^a$.\nWe now exploit the defining property of $H$, that each leaf is marginally trapped ($\\theta^{\\sigma(r)}_k=0$). This implies\n\\begin{eqnarray} \nA[\\hat\\sigma]-A[\\sigma(r)] & = & O(dr^2)~;\\\\\nA[\\sigma(r+dr)]-A[\\hat\\sigma] & = & O(dr)>0~.\n\\end{eqnarray} \nHence, the area increases linearly in $dr$ between any two nearby leaves $\\sigma(r)$, $\\sigma(r+dr)$. This implies that the area increases strictly monotonically with $r$.\n\n\\end{proof}\n\n\\begin{cor}\\label{cor-quant}\nThe above construction implies, more specifically, that the area of leaves increases at the rate\n\\begin{equation}\n\\frac{dA}{dr} = \\int_{\\sigma(r)} \\sqrt{h^{\\sigma(r)}}~ \\alpha \\theta^{\\sigma(r)}_l~.\n\\label{eq-specarea}\n\\end{equation}\nwhere $h_{ab}^{\\sigma(r)}$ is the induced metric on the leaf $\\sigma(r)$ and $h^{\\sigma(r)}$ is its determinant. Note that the integrand is positive definite since $\\alpha<0$ and all leaves are marginally trapped; in this sense the area theorem is local. However, the theorem applies to complete leaves only, not to arbitrary deformations of leaves.\n\\end{cor}\n\n\\begin{cor}\\label{cor-past}\nFor past holographic screens, we recall the contrasting convention that $\\alpha>0$ on $\\sigma(0)$. The above arguments then establish that $\\alpha>0$ everywhere on $H$. Eqs.~(\\ref{eq-area}) and (\\ref{eq-specarea}) hold as an area theorem. \n\\end{cor}\n\n\\begin{rem}\\label{rem-arrow}\nWe note that the area increases in the outside or future direction along a past holographic screen. With an interpretation of area as entropy, the holographic screens of an expanding universe thus have a standard arrow of time. \n\\end{rem}\n\\begin{rem}\\label{rem-backarrow}\nBy contrast, the area increases in the outside or {\\em past} direction along a future holographic screen. Thus, {\\em the arrow of time runs backwards on the holographic screens inside black holes, and near a big crunch}. Perhaps this intriguing result is related to the difficulty of reconciling unitary quantum mechanics with the equivalence principle~\\cite{Haw76,AMPS,Bou12c,AMPSS,Bou13,MarPol13,Bou13a,Bou13b}.\n\\end{rem}\n\nWe close with a final theorem that establishes the uniqueness of the foliation of $H$:\n\\begin{thm}\nLet $H$ be a regular future holographic screen with foliation $\\{\\sigma(r)\\}$. Every marginally trapped surface $s\\subset H$ is one of the leaves $\\sigma(r)$. \n\\end{thm}\n\\begin{proof}\nBy contradiction: suppose that $s$ is marginally trapped and distinct from any $\\sigma(r)$. Thus $s$ intersects the original foliation in a nontrivial closed interval $[r_1,r_2]$ and is tangent to $\\sigma(r_1)$ and $\\sigma(r_2)$. The $\\theta=0$ null vector field orthogonal to $s$ must coincide with $k^a$ at the tangent point with $\\sigma(r_2)$. Since $r_1