Radially Dependent Large Scale Dynamos in Global Cylindrical Shear Flows and the Local Cartesian Limit
MMNRAS , 1– ?? (0000) Preprint 6 November 2018 Compiled using MNRAS L A TEX style file v3.0
Radially Dependent Large Scale Dynamos in GlobalCylindrical Shear Flows and the Local Cartesian Limit
F. Ebrahimi , E. G. Blackman , Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08543, USA Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Laboratory for Laser Energetics,University of Rochester, Rochester NY 14623, USA
ABSTRACT
For cylindrical differentially rotating plasmas, we study large-scale magnetic field gen-eration from finite amplitude non-axisymmetric perturbations by comparing numericalsimulations with quasi-linear analytic theory. When initiated with a vertical magneticfield of either zero or finite net flux, our global cylindrical simulations exhibit the mag-netorotational instability (MRI) and large scale dynamo growth of radially alternatingmean fields, averaged over height and azimuth. This dynamo growth is explained byour analytic calculations of a non-axisymmetric fluctuation-induced EMF that is sus-tained by azimuthal shear of the fluctuating fields. The standard “Ω effect” (shear ofthe mean field by differential rotation) is unimportant. For the MRI case, we expressthe large-scale dynamo field as a function of differential rotation. The resulting radiallyalternating large-scale fields may have implications for angular momentum transportin disks and corona. To connect with previous work on large scale dynamos with locallinear shear and identify the minimum conditions needed for large scale field growth,we also solve our equations in local Cartesian coordinates. We find that large scaledynamo growth in a linear shear flow without rotation can be sustained by shear plusnon-axisymmetric fluctuations–even if not helical, a seemingly previously unidentifieddistinction. The linear shear flow dynamo emerges as a more restricted version of ourmore general new global cylindrical calculations.
Astrophysical rotators, such as stars, galaxies, and accretiondisks, commonly show evidence for contemporaneous pres-ence of disordered turbulence and magnetic fields ordered onspatial or temporal scales larger than those of the fluctua-tions. Explaining this circumstance has been a long standingchallenge. In situ amplification of large-scale magnetic fieldsvia some type of large-scale dynamo is likely but how thesedynamos operate and saturate in each context remains anactive subject of research (for reviews see (Brandenburg &Subramanian 2005; Blackman 2015)). How such fields growgiven the presence of fluctuations, what are the best analy-sis methods, and what minimum ingredients for growth areneeded (Vishniac & Brandenburg 1997; Brandenburg et al.2008; Yousef & et al. 2008; Heinemann et al. 2011; Heraultet al. 2011; Squire & Bhattacharjee 2015) are topics of activeinvestigation.Beyond stellar and galactic contexts, evidence for large-scale field growth is seen in magnetically dominated labora-tory plasmas (Ji et al. 1995; Cothran et al. 2009), and in localand global simulations (Brandenburg et al. 1995; Ebrahimiet al. 2009; Lesur & Ogilvie 2010; Davis et al. 2010; Simonet al. 2011; Guan & Gammie 2011; Sorathia et al. 2012;Suzuki & Inutsuka 2014) of the magnetorotational instabil-ity (MRI) (Velikhov 1959; Balbus & Hawley 1991). Large- scale fields in MRI flows have been associated with the suste-nance of MRI turbulence(Lesur & Ogilvie 2008; Davis et al.2010; Simon et al. 2011) are correlated with the convergenceof Maxwell stress (Guan & Gammie 2011; Nauman & Black-man 2014), and can influence corona formation (Blackman& Pessah 2009). For a single MRI mode, large-scale mag-netic fields generated via an EMF can cause MRI satura-tion (Ebrahimi et al. 2009). In short, the large-scale dynamosof MRI-unstable systems are of interest both as phenomenaon their own, and because they may be closely connectedto angular momentum transport in accretion disks by localand nonlocal Maxwell stresses (Blackman & Nauman 2015).In addition to numerical simulations, flow-dominated lab-oratory experiments are also investigating the MRI MHDunstable systems in Taylor-Couette flow geometry (Good-man & Ji 2002; R¨udiger et al. 2003; Kageyama et al. 2004;Noguchi et al. 2002; Sisan et al. 2004; Stefani et al. 2007).The electromotive force (EMF) from correlated veloc-ity and magnetic field fluctuations is important to all largescale dynamo theories (Moffatt 1978). In general, correlatedfluctuations in the EMF facilitate large-scale field amplifica-tion and the form that this takes for cylindrical MHD shearflows is the focus of the present paper.As the role of large-scale dynamos and large scale fieldsfor MRI turbulence and angular momentum has become in-creasingly recognized, there is a need for truly global strat- c (cid:13) a r X i v : . [ a s t r o - ph . H E ] F e b F. Ebrahimi, E. G. Blackman ified simulations in the long run to best compare to realastrophysics disks. But choices must always be made bothdue to limited computational resources and for isolating keyphysical processes that contribute to the global dynamics.The shearing box model has been the workhorse for sim-ulating MRI turbulence for this purpose– but it is a localmodel and has limitations associated with boundary condi-tions, box size and so is a limited model for real astrophysicaldisks (Regev & Umurhan 2008; Bodo et al. 2008; Blackman& Nauman 2015). The cylindrical model used here also hassome complementary limitations due to its boundary con-ditions, but on the other hand provides solutions in a realglobal domain. Moreover, as unstratified shearing simula-tions go, the shearing box in itself might be thought of as amore restrictive approximation of a global unstratified cylin-der of the sort used here.In contrast to previous studies of local Cartesian shear-ing box addressing large scale field growth from nonhelicalturbulence and linear shear, (Brandenburg 2005; Yousef &et al. 2008; Heinemann et al. 2011; Squire & Bhattachar-jee 2015), we focus on the more general large-scale dynamofrom the combination of non-axisymmetric perturbationsand global differentially rotating flows in a cylinder. Weshow how the combination of imposed non-axisymmetricfluctuations and differential rotation, or linear shear of thefluctuating field, is sufficient to source the electromotiveforce and generate a large scale magnetic field in this cylin-drical geometry. We present the complete quasilinear formof the EMF and show that it models favorably the resultsdirect numerical simulations (DNS) of the MRI when themagnitude and growth rate of these initial fluctuations inthe simulations are used as inputs to the quasi-linear sin-gle model analysis. The single mode analysis proves usefulin showing explicitly that mode-mode coupling is not essen-tial for growth, and for identifying which terms in the EMFdominate. We also show that these conditions for large scalefield growth do not depend on whether the shear profile isfavorable or unfavorable to the MRI as long as there is aphysically motivated source of fluctuations.To identify the minimum requirements for large scalegrowth, to connect with previous work, and to compare withthe global cylindrical case, we carry out analogous calcula-tions in local Cartesian coordinates. Comparing cylindricaland local Cartesian models, we find that in each case thefluctuation-induced EMF has separate contributions thatdepend respectively on 1) non-uniformity of the radiallysheared non-axisymmetric perturbations 2) the backgrounddifferential rotation (cylindrical) or linear shear (Cartesian)3) the mean angular velocity. These three vertical EMFterms can separately generate a large-scale magnetic field.We discuss them in the context of our numerical simulationsand the minimum requirements for large scale field growth.We find that non-axisymmetric fluctuations plus linear shearOR uniform rotation provide the two most minimal combi-nations needed for large scale field growth in local Cartesianlimit.In Sec. 2, we present evidence of large-scale toroidalfields from global nonlinear MHD DNS of the MRI in a cylin-drical setup. We derive the general form of the EMF in thequasilinear approximation in cylindrical geometry in Sec. 3.There we also discuss the role of the EMF in field growth(Sec. 3 .2) and consider an example limiting case where ra- dial gradients of the fluctuations are ignored (Sec. 3.3). Wecompare the EMF expression from cylindrical analytics withnumerical calculations in Sec. 4. We find that the terms withdirect dependence on mean differential rotation contributemost to the dynamo seen in the simulations. This is ex-amined for a specific nonaxisymmetric MRI mode and it isshown that a toroidal large-scale field is directly generatedthrough the vertical EMF as the result of the coupling ofa small-scale fluctuations with the differential rotation. Thetraditional “Ω effect” (i.e. growth of mean toroidal field frommean poloidal field by shear)(Moffatt 1978; Parker 1979) isunimportant when the initial mean field is purely vertical.Visualizations of the field lines from our nonlinear MRI sim-ulations of the cylinder are presented in Sec. 5 to highlightwhy non-axisymmetric perturbations are needed for largescale field growth. In Sec. 6 we repeat the quasilinear analy-sis in local Cartesian coordinates and derive general forms ofthe EMF in this geometry. We discuss the associated impli-cations for field growth. By analogy to the specific simplify-ing example in Sec. 3.3, we discuss the limiting case in whichthe x gradients of the fluctuations vanish (Sec. 6.2). Finally,in Sec. 7, we emphasize that large-scale fields can also begenerated even if the rotation profiles would imply stabil-ity to the MRI, as long as there is some external supply ofnon-axisymmetric fluctuations. We conclude in Sec. 8, andalso present a Table summarizing the ingredients needed fordynamo action. We begin with our results from global DNS MHD simula-tions of the MRI in cylindrical (r, φ , z) geometry using theDEBS (Schnack et al. 1987; Ebrahimi et al. 2009) initial-value code to solve the nonlinear, viscous and resistive MHDequations ∂ A ∂t = − E = S V × B − η J (1) ρ ∂ V ∂t = − Sρ V . ∇ V + S J × B + P m ∇ V − S β ∇ P (2) ∂P∂t = − S ∇ · ( P V ) − S (Γ − P ∇ · V (3) ∂ρ∂t = − S ∇ · ( ρ V ) (4) B = ∇ × A (5) J = ∇ × B (6)where the variables, ρ, P, V, B, J, , and Γ are the density,pressure, velocity, magnetic field, current, and ratio of thespecific heats, respectively. We use the same normaliza-tion (Schnack et al. 1987; Ebrahimi et al. 2009; Ebrahimi& Bhattacharjee 2014), where time, radius and velocity arenormalized to the outer radius a , the resistive diffusion time τ R = a /µ η , and the Alfv´en velocity V A = B / √ µ ρ , re-spectively. The dimensionless parameters, S = τ R V A /a and P m , are the Lundquist number and the magnetic Prandtlnumber (the ratio of viscosity to resistivity), respectively.the initial state satisfies the equilibrium force balance con-dition β ∇ p = ρV φ /r , where β ≡ µ P /B is normalizedto the axis value, and the initial pressure and density profiles MNRAS , 1– ????
Astrophysical rotators, such as stars, galaxies, and accretiondisks, commonly show evidence for contemporaneous pres-ence of disordered turbulence and magnetic fields ordered onspatial or temporal scales larger than those of the fluctua-tions. Explaining this circumstance has been a long standingchallenge. In situ amplification of large-scale magnetic fieldsvia some type of large-scale dynamo is likely but how thesedynamos operate and saturate in each context remains anactive subject of research (for reviews see (Brandenburg &Subramanian 2005; Blackman 2015)). How such fields growgiven the presence of fluctuations, what are the best analy-sis methods, and what minimum ingredients for growth areneeded (Vishniac & Brandenburg 1997; Brandenburg et al.2008; Yousef & et al. 2008; Heinemann et al. 2011; Heraultet al. 2011; Squire & Bhattacharjee 2015) are topics of activeinvestigation.Beyond stellar and galactic contexts, evidence for large-scale field growth is seen in magnetically dominated labora-tory plasmas (Ji et al. 1995; Cothran et al. 2009), and in localand global simulations (Brandenburg et al. 1995; Ebrahimiet al. 2009; Lesur & Ogilvie 2010; Davis et al. 2010; Simonet al. 2011; Guan & Gammie 2011; Sorathia et al. 2012;Suzuki & Inutsuka 2014) of the magnetorotational instabil-ity (MRI) (Velikhov 1959; Balbus & Hawley 1991). Large- scale fields in MRI flows have been associated with the suste-nance of MRI turbulence(Lesur & Ogilvie 2008; Davis et al.2010; Simon et al. 2011) are correlated with the convergenceof Maxwell stress (Guan & Gammie 2011; Nauman & Black-man 2014), and can influence corona formation (Blackman& Pessah 2009). For a single MRI mode, large-scale mag-netic fields generated via an EMF can cause MRI satura-tion (Ebrahimi et al. 2009). In short, the large-scale dynamosof MRI-unstable systems are of interest both as phenomenaon their own, and because they may be closely connectedto angular momentum transport in accretion disks by localand nonlocal Maxwell stresses (Blackman & Nauman 2015).In addition to numerical simulations, flow-dominated lab-oratory experiments are also investigating the MRI MHDunstable systems in Taylor-Couette flow geometry (Good-man & Ji 2002; R¨udiger et al. 2003; Kageyama et al. 2004;Noguchi et al. 2002; Sisan et al. 2004; Stefani et al. 2007).The electromotive force (EMF) from correlated veloc-ity and magnetic field fluctuations is important to all largescale dynamo theories (Moffatt 1978). In general, correlatedfluctuations in the EMF facilitate large-scale field amplifica-tion and the form that this takes for cylindrical MHD shearflows is the focus of the present paper.As the role of large-scale dynamos and large scale fieldsfor MRI turbulence and angular momentum has become in-creasingly recognized, there is a need for truly global strat- c (cid:13) a r X i v : . [ a s t r o - ph . H E ] F e b F. Ebrahimi, E. G. Blackman ified simulations in the long run to best compare to realastrophysics disks. But choices must always be made bothdue to limited computational resources and for isolating keyphysical processes that contribute to the global dynamics.The shearing box model has been the workhorse for sim-ulating MRI turbulence for this purpose– but it is a localmodel and has limitations associated with boundary condi-tions, box size and so is a limited model for real astrophysicaldisks (Regev & Umurhan 2008; Bodo et al. 2008; Blackman& Nauman 2015). The cylindrical model used here also hassome complementary limitations due to its boundary con-ditions, but on the other hand provides solutions in a realglobal domain. Moreover, as unstratified shearing simula-tions go, the shearing box in itself might be thought of as amore restrictive approximation of a global unstratified cylin-der of the sort used here.In contrast to previous studies of local Cartesian shear-ing box addressing large scale field growth from nonhelicalturbulence and linear shear, (Brandenburg 2005; Yousef &et al. 2008; Heinemann et al. 2011; Squire & Bhattachar-jee 2015), we focus on the more general large-scale dynamofrom the combination of non-axisymmetric perturbationsand global differentially rotating flows in a cylinder. Weshow how the combination of imposed non-axisymmetricfluctuations and differential rotation, or linear shear of thefluctuating field, is sufficient to source the electromotiveforce and generate a large scale magnetic field in this cylin-drical geometry. We present the complete quasilinear formof the EMF and show that it models favorably the resultsdirect numerical simulations (DNS) of the MRI when themagnitude and growth rate of these initial fluctuations inthe simulations are used as inputs to the quasi-linear sin-gle model analysis. The single mode analysis proves usefulin showing explicitly that mode-mode coupling is not essen-tial for growth, and for identifying which terms in the EMFdominate. We also show that these conditions for large scalefield growth do not depend on whether the shear profile isfavorable or unfavorable to the MRI as long as there is aphysically motivated source of fluctuations.To identify the minimum requirements for large scalegrowth, to connect with previous work, and to compare withthe global cylindrical case, we carry out analogous calcula-tions in local Cartesian coordinates. Comparing cylindricaland local Cartesian models, we find that in each case thefluctuation-induced EMF has separate contributions thatdepend respectively on 1) non-uniformity of the radiallysheared non-axisymmetric perturbations 2) the backgrounddifferential rotation (cylindrical) or linear shear (Cartesian)3) the mean angular velocity. These three vertical EMFterms can separately generate a large-scale magnetic field.We discuss them in the context of our numerical simulationsand the minimum requirements for large scale field growth.We find that non-axisymmetric fluctuations plus linear shearOR uniform rotation provide the two most minimal combi-nations needed for large scale field growth in local Cartesianlimit.In Sec. 2, we present evidence of large-scale toroidalfields from global nonlinear MHD DNS of the MRI in a cylin-drical setup. We derive the general form of the EMF in thequasilinear approximation in cylindrical geometry in Sec. 3.There we also discuss the role of the EMF in field growth(Sec. 3 .2) and consider an example limiting case where ra- dial gradients of the fluctuations are ignored (Sec. 3.3). Wecompare the EMF expression from cylindrical analytics withnumerical calculations in Sec. 4. We find that the terms withdirect dependence on mean differential rotation contributemost to the dynamo seen in the simulations. This is ex-amined for a specific nonaxisymmetric MRI mode and it isshown that a toroidal large-scale field is directly generatedthrough the vertical EMF as the result of the coupling ofa small-scale fluctuations with the differential rotation. Thetraditional “Ω effect” (i.e. growth of mean toroidal field frommean poloidal field by shear)(Moffatt 1978; Parker 1979) isunimportant when the initial mean field is purely vertical.Visualizations of the field lines from our nonlinear MRI sim-ulations of the cylinder are presented in Sec. 5 to highlightwhy non-axisymmetric perturbations are needed for largescale field growth. In Sec. 6 we repeat the quasilinear analy-sis in local Cartesian coordinates and derive general forms ofthe EMF in this geometry. We discuss the associated impli-cations for field growth. By analogy to the specific simplify-ing example in Sec. 3.3, we discuss the limiting case in whichthe x gradients of the fluctuations vanish (Sec. 6.2). Finally,in Sec. 7, we emphasize that large-scale fields can also begenerated even if the rotation profiles would imply stabil-ity to the MRI, as long as there is some external supply ofnon-axisymmetric fluctuations. We conclude in Sec. 8, andalso present a Table summarizing the ingredients needed fordynamo action. We begin with our results from global DNS MHD simula-tions of the MRI in cylindrical (r, φ , z) geometry using theDEBS (Schnack et al. 1987; Ebrahimi et al. 2009) initial-value code to solve the nonlinear, viscous and resistive MHDequations ∂ A ∂t = − E = S V × B − η J (1) ρ ∂ V ∂t = − Sρ V . ∇ V + S J × B + P m ∇ V − S β ∇ P (2) ∂P∂t = − S ∇ · ( P V ) − S (Γ − P ∇ · V (3) ∂ρ∂t = − S ∇ · ( ρ V ) (4) B = ∇ × A (5) J = ∇ × B (6)where the variables, ρ, P, V, B, J, , and Γ are the density,pressure, velocity, magnetic field, current, and ratio of thespecific heats, respectively. We use the same normaliza-tion (Schnack et al. 1987; Ebrahimi et al. 2009; Ebrahimi& Bhattacharjee 2014), where time, radius and velocity arenormalized to the outer radius a , the resistive diffusion time τ R = a /µ η , and the Alfv´en velocity V A = B / √ µ ρ , re-spectively. The dimensionless parameters, S = τ R V A /a and P m , are the Lundquist number and the magnetic Prandtlnumber (the ratio of viscosity to resistivity), respectively.the initial state satisfies the equilibrium force balance con-dition β ∇ p = ρV φ /r , where β ≡ µ P /B is normalizedto the axis value, and the initial pressure and density profiles MNRAS , 1– ???? (0000) arge Scale Dynamos in Cylinders Figure 1.
The generation of large-scale toroidal magnetic field(averaged over vertical and toroidal directions) in the zero-netflux simulations in the r-t plane, (a) when all the Fourier modesare included and (b) only one non-axisymmetric mode m=1 isevolved (Rm=3100, Pm=1, β = 10 , V /V A = 31). are assumed to be radially uniform and unstratified. Pres-sure and density are evolved, however, they remain fairlyuniform during the computations. A no-slip boundary con-dition is used for the poloidal flow and flow fluctuations. Theinner and outer radial boundaries are perfectly conductingso that the tangential electric field, the normal component ofthe magnetic field, and the normal component of the velocityvanish. The tangential component of the velocity is the ro-tational velocity of the wall. The azimuthal ( φ ) and axial ( z )boundaries are periodic. We assume a radial pressure gradi-ent balances the centrifugal force in equilibrium, but radialgravity and a radial pressure force are interchangeable forour incompressible, unstratified circumstance. The pressuregradient, rather than gravity, is what balances the centrifu-gal force in cylindrical laboratory experiments designed totest the MRI (Goodman & Ji (2002)).All variables are decomposed as f ( r, φ, z, t ) = (cid:80) ( m,k ) (cid:98) f m,k ( r, t ) e i ( − mφ + kz ) = (cid:104) f ( r, t ) (cid:105) + (cid:101) f ( r, φ z, t ), where (cid:104) f (cid:105) is the mean ( m = k = 0) component, and (cid:101) f is thefluctuating component. Mean quantities (indicated bybrackets ( (cid:104)(cid:105) ) or overbars) are azimuthally and axiallyaveraged, but remain dependent on radius (r). Equations(1-6) are then integrated forward in time using the DEBScode. The DEBS code uses a finite difference method with astaggered grid for radial discretization and pseudospectralmethod for azimuthal and vertical coordinates. In thisdecomposition, each mode satisfies a separate equationof the form ∂ ˜ f m,k /∂t = L m,k ˜ f m,k + (cid:80) ( m (cid:48) ,k (cid:48) ) N m,k,m (cid:48) ,k (cid:48) ,where L m,k is a linear operator that depends on ˜ f , ( r, t ),and N m,k,m (cid:48) ,k (cid:48) is a nonlinear term that represents thecoupling of the mode ( m, k ) to all other modes ( m (cid:48) , k (cid:48) ). ( Figure 2.
Total magnetic energies, W φ = 1 / (cid:82) B φ dr and W r =1 / (cid:82) B r dr , and large-scale toroidal magnetic energy, (cid:104) W φ (cid:105) =1 / (cid:82) (cid:104) B φ (cid:105) dr , vs. time for net-zero flux 3-D MRI computations. This latter term is evaluated pseudospectrally.) The timeadvance is a combination of the leapfrog and semi-implicitmethods (Schnack et al. 1987).We initiate simulations with a Keplerian flow (cid:104) V φ ( r ) (cid:105) = V r − / and uniform magnetic field B = B ˆ z (with non-zeroinitial net-flux) or B = B sin(2 π ( r − r ) / ( r − r )) /r ˆ z , (withzero-net-flux) where r , r are the inner and outer radii.Fully nonlinear simulations with all Fourier modes included(with radial, azimuthal and axial resolutions of n r =220,0 < m <
43 and − < n <
43) show that large-scalemagnetic fields are generated [Figure 1(a)]. In all of oursimulations, the initially weak vertical magnetic field, getsredistributed, amplified at inner radii and reduced at outerradii. Initially B φ =0, but a toroidal large-scale (averaged in φ and z ) field grows via the correlation of non-axisymmetricMRI-induced fluctuations. The sustenance of total toroidaland radial magnetic field energies, as well as the large-scaletoroidal magnetic energy during the computation fornet-zero flux are shown in Fig. 2. The radially dependentlarge-scale field [Figure 1(a)] is also sustained in time.The code can also be used to compute the nonlin-ear evolution of a single mode evolution for which theinitial conditions consist of an equilibrium (cid:104) f ( r ) (cid:105) plus asingle mode ˜ f m,k ( r, ( imφ + ikz ) perturbation. The initialamplitude ˜ f m,k ( r,
0) is a polynomial in r that satisfiesthe boundary conditions at r = r and r = r . Theinitial amplitudes of all other modes are set to zero. Onlythe mode ( m, k ) is then evolved; however, the m = 0, k = 0 component (the background) is allowed to evolveself-consistently. The evolution of the background profile˜ f , can affect the evolution of the mode ( m, k ) and causethe mode to saturate. In a fully nonlinear computation,all modes are initialized with small random amplitudeand are evolved in time, including the full nonlinear term( N m,k,m (cid:48) ,k (cid:48) ).To facilitate the analytic investigation of the large-scalefield generation, we carried out nonlinear single-mode(i.e. single value of m and k ) simulation of m=1 non-axisymmetric MRI (in which only one fluctuation mode andthe mean fields self-consistently grow) for zero-net-flux andnet-flux configurations. As seen in Fig. 1(b) at t (cid:39) .
02, asthe m=1 MRI mode amplitude approaches saturation, alarge-scale toroidal field is also generated.
MNRAS , 1– ?? (0000) F. Ebrahimi, E. G. Blackman
To identify the origin of large-scale magnetic field growth inthe DNS simulations, we employ quasilinear analytical cal-culations for the single-mode case. Given initial fluctuationswe calculate the fluctuation-induced EMF E ≡ (cid:104) (cid:101) V × (cid:101) B (cid:105) fromlinearized eigenfunctions. The EMF is the source of largescale field growth. All averaged correlations are presentedin terms of the radial Lagrangian displacement ξ r of a fluidplasma element, (Frieman & Rotenberg 1960), and the meanquantities. We assume perturbed quantities of the form ξξξ ( r, φ, z, t ) = [ ξ r ( r ) , ξ φ ( r ) , ξ z ( r )] exp ( γ c t − imφ + ik z z ) fora cylinder of outer radius a and height L = 2 π/k . In thepresence of an equilibrium mean flow, self-adjointness of thelinear stability problem is lost (Frieman & Rotenberg 1960).Thus, for nonaxisymmetric modes (nonzero m ), the eigen-values γ c = γ + iω r and the eigenvectors ξ ( r ) can be com-plex, where γ and ω r are the growth rate and the oscillationfrequency of the mode, respectively. To isolate the role ofshear flow on the dynamo effect in the quasilinear analyticalcalculations below, we impose an initial, uniform B .For a single MRI Fourier mode, the cylindrical coor-dinate components of the linearized momentum equationin terms of the Lagrangian displacement vector ξ (Chan-drasekar 1961) are[¯ γ + ω A + 2 r Ω( r )Ω (cid:48) ( r )] ξ r − γ Ω( r ) ξ φ = − ∂X∂r (7)(¯ γ + ω A ) ξ φ + 2¯ γ Ω( r ) ξ r = imXr (8)(¯ γ + ω A ) ξ z = − ik z X, (9)where ¯ γ = γ + iω r − im Ω( r ), X = (cid:101) P + (cid:101) B · B , ω A = k z B /ρ and Ω( r ) = V φ ( r ) /r is the angular velocity. By includingthis imposed mean flow in the definition of the Lagrangiandisplacement vector, the velocity fluctuations in an Eulerianframe are given by (cid:101) V = ∂ ξ /∂t + ∇ × ( ξ × V ), with compo-nents (cid:101) V r = ¯ γξ r , (cid:101) V φ = ¯ γξ φ − [ ∂V φ ∂r − V φ r ] ξ r and (cid:101) V z = ¯ γξ z .For small resistivity, the magnetic field perturbationscan be directly related to the displacement via (cid:101) B = ik z B ξ .Using this along with incompressibility and Eqs.( 7-9), wecan eliminate X and the azimuthal and vertical displace-ments can be written in terms of ξ r as ξ φ = 11 + m / ( r k z ) (cid:20) − r )¯ γ (¯ γ + ω A ) ξ r − imr k z ( rξ r ) (cid:48) (cid:21) (10)and ξ z = mk z r ( r k z + m ) (cid:18) − r )¯ γ ¯ γ + ω A (cid:19) ξ r + irk z (cid:18) − m r k z + m (cid:19) ( rξ r ) (cid:48) , (11)where the primes indicate radial derivatives. The quasilinearEMF components E z = (cid:104) (cid:101) V × (cid:101) B (cid:105) z = Re ( (cid:101) V ∗ r (cid:101) B φ − (cid:101) V ∗ φ (cid:101) B r ) and E φ = (cid:104) (cid:101) V × (cid:101) B (cid:105) φ = Re ( (cid:101) V ∗ z (cid:101) B r − (cid:101) V ∗ r (cid:101) B z ) can now be writtenin terms of the radial velocity fluctuations (cid:101) V r = ¯ γξ r , E Z = (cid:104) (cid:101) V × (cid:101) B (cid:105) z = E ZV r (cid:48) + E Z Ω (cid:48) + E Z Ω , (12) where E ZV r (cid:48) = mγk z B ( m + k z r ) (cid:34) r (cid:102) V r (cid:102) V ∗ r (cid:48) γ + Ω ( r ) (cid:35) . (13) E Z Ω (cid:48) = − m γk z B ( m + k z r ) (cid:34) Ω (cid:48) ( r )Ω( r ) r | (cid:102) V r | ( γ + Ω ( r )) (cid:35) , (14) E Z Ω = 2 γk z B Ω( r ) r Ω G ( γ + Ω − ω A ) | (cid:102) V r | + mγk z B ( m + k z r ) (cid:34) | (cid:102) V r | γ + Ω ( r ) (cid:35) , (15)and E φ = (cid:104) (cid:101) V × (cid:101) B (cid:105) φ = E φV r (cid:48) + E φ Ω (cid:48) + E φ Ω , (16)where E φV r (cid:48) = (cid:20) γB r − m γB r ( m + k z r ) (cid:21) (cid:34) ( r (cid:102) V r (cid:102) V ∗ r (cid:48) ) γ + Ω ( r ) (cid:35) , (17) E φ Ω (cid:48) = − (cid:20) γB r − m γB r ( m + k z r ) (cid:21) (cid:34) Ω (cid:48) ( r )Ω( r ) r | (cid:102) V r | ( γ + Ω ( r )) (cid:35) , (18) E φ Ω = 2 γk z rB Ω m Ω( r ) G ( ω A − γ − Ω ) | (cid:102) V r | + (cid:20) γB r − m γB r ( m + k z r ) (cid:21) (cid:34) | (cid:102) V r | γ + Ω ( r ) (cid:35) , (19)where G = ( m + k z r )[ γ +Ω ( r )][4 γ Ω ( r )+( γ +Ω ( r )+ ω A ) ](20)and Ω( r ) = m Ω( r ) − ω r . In an ideal MHD cylindrical plasma,Eq. (12) provides the complete quasilinear form of the ver-tical fluctuation-induced EMF in terms of radial perturba-tions. The first term on the RHS, E ZV r (cid:48) , depends on thenon-uniformity of the radial displacement of the mode. Thesecond term E Z Ω (cid:48) , which depends on the differential rotationΩ (cid:48) ( r ) is sufficient to directly produce a nonzero fluctuation-induced dynamo term. The free energy source d Ω dlnr appearsin this term. The third term, E Z Ω , shows the dependence ofthe vertical EMF on angular velocity.The linearized cylindrical solutions ( γ and ξ r ) for non-axisymmetric flow-driven and MRI modes have been pre-viously examined (Bondeson et al. 1987; Ogilvie & Pringle1996; Keppens et al. 2002). Here we do not solve the eigen-value problem to find the ξ r for nonaxisymmetric modesbut (in Sec. 4,) extract the linearized solutions directly fromDNS for a single mode and verify the quasilinear forms ofEMFs.The above quasilinear EMF terms allow us to identifythe source of large scale magnetic field growth in a rotatingplasma with or without radially sheared non-axisymmetricperturbations. The dominance of the fluctuation-induced quasilinear E Z in the generation of the large-scale toroidal magnetic field MNRAS , 1– ????
To identify the origin of large-scale magnetic field growth inthe DNS simulations, we employ quasilinear analytical cal-culations for the single-mode case. Given initial fluctuationswe calculate the fluctuation-induced EMF E ≡ (cid:104) (cid:101) V × (cid:101) B (cid:105) fromlinearized eigenfunctions. The EMF is the source of largescale field growth. All averaged correlations are presentedin terms of the radial Lagrangian displacement ξ r of a fluidplasma element, (Frieman & Rotenberg 1960), and the meanquantities. We assume perturbed quantities of the form ξξξ ( r, φ, z, t ) = [ ξ r ( r ) , ξ φ ( r ) , ξ z ( r )] exp ( γ c t − imφ + ik z z ) fora cylinder of outer radius a and height L = 2 π/k . In thepresence of an equilibrium mean flow, self-adjointness of thelinear stability problem is lost (Frieman & Rotenberg 1960).Thus, for nonaxisymmetric modes (nonzero m ), the eigen-values γ c = γ + iω r and the eigenvectors ξ ( r ) can be com-plex, where γ and ω r are the growth rate and the oscillationfrequency of the mode, respectively. To isolate the role ofshear flow on the dynamo effect in the quasilinear analyticalcalculations below, we impose an initial, uniform B .For a single MRI Fourier mode, the cylindrical coor-dinate components of the linearized momentum equationin terms of the Lagrangian displacement vector ξ (Chan-drasekar 1961) are[¯ γ + ω A + 2 r Ω( r )Ω (cid:48) ( r )] ξ r − γ Ω( r ) ξ φ = − ∂X∂r (7)(¯ γ + ω A ) ξ φ + 2¯ γ Ω( r ) ξ r = imXr (8)(¯ γ + ω A ) ξ z = − ik z X, (9)where ¯ γ = γ + iω r − im Ω( r ), X = (cid:101) P + (cid:101) B · B , ω A = k z B /ρ and Ω( r ) = V φ ( r ) /r is the angular velocity. By includingthis imposed mean flow in the definition of the Lagrangiandisplacement vector, the velocity fluctuations in an Eulerianframe are given by (cid:101) V = ∂ ξ /∂t + ∇ × ( ξ × V ), with compo-nents (cid:101) V r = ¯ γξ r , (cid:101) V φ = ¯ γξ φ − [ ∂V φ ∂r − V φ r ] ξ r and (cid:101) V z = ¯ γξ z .For small resistivity, the magnetic field perturbationscan be directly related to the displacement via (cid:101) B = ik z B ξ .Using this along with incompressibility and Eqs.( 7-9), wecan eliminate X and the azimuthal and vertical displace-ments can be written in terms of ξ r as ξ φ = 11 + m / ( r k z ) (cid:20) − r )¯ γ (¯ γ + ω A ) ξ r − imr k z ( rξ r ) (cid:48) (cid:21) (10)and ξ z = mk z r ( r k z + m ) (cid:18) − r )¯ γ ¯ γ + ω A (cid:19) ξ r + irk z (cid:18) − m r k z + m (cid:19) ( rξ r ) (cid:48) , (11)where the primes indicate radial derivatives. The quasilinearEMF components E z = (cid:104) (cid:101) V × (cid:101) B (cid:105) z = Re ( (cid:101) V ∗ r (cid:101) B φ − (cid:101) V ∗ φ (cid:101) B r ) and E φ = (cid:104) (cid:101) V × (cid:101) B (cid:105) φ = Re ( (cid:101) V ∗ z (cid:101) B r − (cid:101) V ∗ r (cid:101) B z ) can now be writtenin terms of the radial velocity fluctuations (cid:101) V r = ¯ γξ r , E Z = (cid:104) (cid:101) V × (cid:101) B (cid:105) z = E ZV r (cid:48) + E Z Ω (cid:48) + E Z Ω , (12) where E ZV r (cid:48) = mγk z B ( m + k z r ) (cid:34) r (cid:102) V r (cid:102) V ∗ r (cid:48) γ + Ω ( r ) (cid:35) . (13) E Z Ω (cid:48) = − m γk z B ( m + k z r ) (cid:34) Ω (cid:48) ( r )Ω( r ) r | (cid:102) V r | ( γ + Ω ( r )) (cid:35) , (14) E Z Ω = 2 γk z B Ω( r ) r Ω G ( γ + Ω − ω A ) | (cid:102) V r | + mγk z B ( m + k z r ) (cid:34) | (cid:102) V r | γ + Ω ( r ) (cid:35) , (15)and E φ = (cid:104) (cid:101) V × (cid:101) B (cid:105) φ = E φV r (cid:48) + E φ Ω (cid:48) + E φ Ω , (16)where E φV r (cid:48) = (cid:20) γB r − m γB r ( m + k z r ) (cid:21) (cid:34) ( r (cid:102) V r (cid:102) V ∗ r (cid:48) ) γ + Ω ( r ) (cid:35) , (17) E φ Ω (cid:48) = − (cid:20) γB r − m γB r ( m + k z r ) (cid:21) (cid:34) Ω (cid:48) ( r )Ω( r ) r | (cid:102) V r | ( γ + Ω ( r )) (cid:35) , (18) E φ Ω = 2 γk z rB Ω m Ω( r ) G ( ω A − γ − Ω ) | (cid:102) V r | + (cid:20) γB r − m γB r ( m + k z r ) (cid:21) (cid:34) | (cid:102) V r | γ + Ω ( r ) (cid:35) , (19)where G = ( m + k z r )[ γ +Ω ( r )][4 γ Ω ( r )+( γ +Ω ( r )+ ω A ) ](20)and Ω( r ) = m Ω( r ) − ω r . In an ideal MHD cylindrical plasma,Eq. (12) provides the complete quasilinear form of the ver-tical fluctuation-induced EMF in terms of radial perturba-tions. The first term on the RHS, E ZV r (cid:48) , depends on thenon-uniformity of the radial displacement of the mode. Thesecond term E Z Ω (cid:48) , which depends on the differential rotationΩ (cid:48) ( r ) is sufficient to directly produce a nonzero fluctuation-induced dynamo term. The free energy source d Ω dlnr appearsin this term. The third term, E Z Ω , shows the dependence ofthe vertical EMF on angular velocity.The linearized cylindrical solutions ( γ and ξ r ) for non-axisymmetric flow-driven and MRI modes have been pre-viously examined (Bondeson et al. 1987; Ogilvie & Pringle1996; Keppens et al. 2002). Here we do not solve the eigen-value problem to find the ξ r for nonaxisymmetric modesbut (in Sec. 4,) extract the linearized solutions directly fromDNS for a single mode and verify the quasilinear forms ofEMFs.The above quasilinear EMF terms allow us to identifythe source of large scale magnetic field growth in a rotatingplasma with or without radially sheared non-axisymmetricperturbations. The dominance of the fluctuation-induced quasilinear E Z in the generation of the large-scale toroidal magnetic field MNRAS , 1– ???? (0000) arge Scale Dynamos in Cylinders can be seen by examining the mean (averaged in φ and z )toroidal component of the induction equation, ignoring re-sistivity. This equation is ∂ B φ ∂t = − ∂ E z ∂r + ( B · ∇ ) V | φ − ( V · ∇ ) B | φ . (21)Since there is neither a mean radial magnetic field B r , norvelocity field V r , so the second and third terms on the rightof Eq. (21) vanish. Note that the second term on the right ofEq. (21) is the traditional ”Ω effect” which thus vanishes forour setup and averaging procedure. The shear (differentialrotation) does enter through E z , and the first term on theright of Eq. (21) is the dominant term.Keeping only the first term on the right of Eq. (21)( ∂ B φ ∂t = − ∂ E z ∂r ) we then see that the three terms on theright of Eq. (12) provide distinct paths for large scale fieldsto grow: (1) Radially sheared non-axisymmetric perturba-tions i. e., the first term in Eq. (12), E ZV r (cid:48) , proportional tothe non-uniformity of the radial displacement of the nonax-isymmetric perturbation (2) Uniform non-axisymmetric per-turbations (stable or unstable) but with background shearin the angular flow i. e. the last two terms in Eq. (12).We emphasize that all three terms on the right of E Z (Eq.12) vanish explicitly for axisymmetric modes (m=0 modes,Ω = 0), as axisymmetric modes are purely growing or de-caying (i.e. ω r = 0) (Chandrasekar 1961).The induction equation for the vertical mean field is ∂ B z ∂t = ∂ E φ ∂r + ( B · ∇ ) V | z − ( V · ∇ ) B | z . (22)Here again, as in equation (21), the last two terms on theright vanish and the mean field evolves only through theEMF term ∂ E φ /dr . For this equation, axisymmetric modescan contribute through the radial variations of (cid:101) V r in E φV r (cid:48) (Eq. 16) and the last term in Eq. (19) to give E φ ( m =0) = (cid:18) B γ (cid:19) [ (cid:102) V r (cid:102) V ∗ r (cid:48) + | (cid:102) V r | /r ] , E Z ( m =0) = 0 (23)but only non-axisymmetric modes allow evolution of both B z and B φ through the EMF terms. k r = 0To simplify pinpointing the dominant contributions to theEMF from the quasi-linear theory appropriate for MRIdriven fluctuations for an initially vertical field, we assume (cid:102) V (cid:48) r = 0 (in the limit of k r = 0). This is justified sinceFig. 3(a) from DNS in the following section shows that theterm arising from this radial gradient is subdominant. Wecan now solve the quasilinear equations for E z without know-ing the exact form of the global eigenfunctions. For largegrowth rates (i.e. γ , Ω > ω A ), Eq. (12) then reduces to E Z ( r ) ∼ γkB ( m + k r ) (cid:34) m − m Ω (cid:48) ( r )Ω( r ) rγ + Ω + 2 k V φ r Ω γ + Ω (cid:35) × ( | (cid:102) V r | γ + Ω ( r ) ) . (24) Figure 3.
The profiles of (a) (in black) direct numerical calcu-lations of total vertical EMF term, LHS of Eq. (12); (in red)RHS side of Eq. (12) based on the quasilinear calculations;(in blue) the first term in Eq. (12). Linear growth rate of themode at the early phase γτ orbit ∼ . ω r τ orbit ∼ . (cid:104) B φ (cid:105) and − γ ∂ E z ∂r at t/τ orbit = 16 and t/τ orbit = 10 during the growth of m=1 MRImode from DNS with non-zero net flux (solid lines) and zero netflux (dashed lines), respectively. The dimensionless magnetic andvelocity fluctuations at this time of DNS are | (cid:102) B φ /B | ∼ . | (cid:102) B r /B | = k z a | (cid:102) V r /V A | k γτ A ∼ . γτ A ∼ | (cid:102) V r /V A | ∼ . V /V A = 16). This vertical EMF for a single nonaxisymmetic mode andthe mean flow (e.g. Keplerian) with γ ∼ Ω (where Ω isthe angular frequency at the inner radial boundary) in the k z r > m and γ > Ω limits are then related by E z ∼− m B k z r γ d Ω( r ) dlnr | (cid:102) V r | +2 mkB r )Ω( r ) γ | (cid:102) V r | ≡ Q ( r ) (cid:102) V r | . Usingthis equation and B φ ∼ − γ ∂ E z ∂r in the limit of constant (cid:101) V r and constant mean vertical magnetic field B , the large-scaletoroidal magnetic field can be written as B φ ( r ) ∼ − m B k z γ (cid:20) r d Ω( r ) dlnr (cid:21) (cid:48) | (cid:102) V r | + 2 mkB γ (cid:2) Ω( r )Ω( r ) (cid:3) (cid:48) | (cid:102) V r | , (25)showing a direct relationship between differential rotationand the generation of large-scale magnetic field. From DNS of a nonaxisymmetric mode m = 1, k z a =12 incylindrical model (Sec. 2), we evaluate (cid:104) (cid:101) V × (cid:101) B (cid:105) z and com-pare it to quasilinear calculations of the right side (RHS) ofEq. (12) in terms of radial velocity fluctuations (cid:101) V r = ¯ γξ r .For the RHS of Eq. (12), the radial velocity fluctuationsand the eigenvalues from DNS are inserted into the analyt-ical forms. Fig. 3(a) shows good agreement between these MNRAS , 1– ?? (0000) F. Ebrahimi, E. G. Blackman (a)(b)
Figure 4.
Radial profiles of time-averaged (b) saturated large-scale fields (cid:104) B z (cid:105) , (cid:104) B φ (cid:105) , (b) E · B ( = S (cid:104) (cid:101) V × (cid:101) B (cid:105) ·(cid:104) B (cid:105) from DNSof Eq. (1-6) during nonlinear evolution of m=1 MRI. two calculations. Fig. 3(a) also shows that the first term onthe RHS of Eq. (12) is subdominant to the last two termswhich depend on the mean flow.We have verified the dominance of the first term on theright of Eq. (21) from DNS of a single-mode m=1 MRI. Thelarge-scale B φ starts to grow, even when initially zero, as theinstability develops. Figure 3(b) shows (cid:104) B φ (cid:105) as computedfrom the DNS during the linear phase of single-mode simula-tions with non-zero net flux and the first term on the RHS ofEq. (21), right before the saturation, as also measured fromthe DNS. As seen, the mean toroidal field is correlated with,and directly generated by the vertical EMF. Similarly, themean B φ generated in the net-zero flux simulations shownin Fig. 1 is also correlated with the vertical EMF. Fig 3(a)shows that the main contribution to the EMF comes fromthe last two terms of Eq. (12). Thus B φ ∼ − γ ∂ (cid:104) ( (cid:101) V × (cid:101) B ) z (cid:105) ∂r is directly dependent on the shear-flow ( V (cid:48) φ ) or differentialrotation ( d Ω /dr ) in the presence of a finite amplitude fluc-tuation.In addition to the EMF components themselves, themagnetic field-aligned EMF plays important role in thequasi-linear regime as the mode starts to saturate. Evo-lution of both components (toroidal and vertical) of thelarge-scale magnetic field is only possible for nonaxisymmet-ric fluctuations because only in this case are both E z and E φ (Eq. 12, 16) non-zero. The generation of the large-scaletoroidal field is due to the vertical EMF and the redistri-bution (and amplification around r=0.2-0.4) of B z is dueto the nonzero toroidal EMF. Figure 4 shows the profilesof time-averaged saturated large-scale toroidal and verticalfields as well as the EMF parallel to B .Simulations with axisymmetric fluctuations also showthe amplification of vertical large-scale field ( B z ) but with-out a generation of B φ . The amplification of B z from ax-isymmetric modes (Eq. 23), which may contribute to the Figure 5.
Field line visualizations during nonlinear MRI simula-tions in a cylinder (a) 2-D (b) 3-D when m=1 mode perturbationis dominant (c) toroidal top view in 3-D shows the twisting of thefield lines. mode saturation in a cylinder (Ebrahimi et al. 2009) resultsfrom the curvature terms and is absent for channel modesin a local Cartesian model, as discussed below.
The physical picture of generating (cid:104) B φ (cid:105) can further be exam-ined through comparing visualizations of the field lines with-out and with nonaxisymmetric MRI perturbations in 2-Dand 3-D as shown in Fig. 5. For toroidal m=0 perturbations,weak vertical magnetic field lines are toroidally stretched(Fig. 5(a)) according to the second term in Eq. (21). How-ever due to toroidal symmetry, (cid:104) B φ (cid:105) = 0 as the positiveand negative contributions from the perturbations remainon the same vertical surface, and only the mean verticalfield is amplified through ∂ E φ /dr . (Ebrahimi et al. 2009).In the presence of nonaxisymmetric perturbations in 3-Dnonlinear simulations however, the field lines are stretchedand twisted (Fig. 5 b,c). As a consequence, (cid:104) B φ (cid:105) (cid:54) = 0 sincenow the oppositely signed toroidal field contributions fromperturbations are displaced radially from one another. Since MNRAS , 1– ????
The physical picture of generating (cid:104) B φ (cid:105) can further be exam-ined through comparing visualizations of the field lines with-out and with nonaxisymmetric MRI perturbations in 2-Dand 3-D as shown in Fig. 5. For toroidal m=0 perturbations,weak vertical magnetic field lines are toroidally stretched(Fig. 5(a)) according to the second term in Eq. (21). How-ever due to toroidal symmetry, (cid:104) B φ (cid:105) = 0 as the positiveand negative contributions from the perturbations remainon the same vertical surface, and only the mean verticalfield is amplified through ∂ E φ /dr . (Ebrahimi et al. 2009).In the presence of nonaxisymmetric perturbations in 3-Dnonlinear simulations however, the field lines are stretchedand twisted (Fig. 5 b,c). As a consequence, (cid:104) B φ (cid:105) (cid:54) = 0 sincenow the oppositely signed toroidal field contributions fromperturbations are displaced radially from one another. Since MNRAS , 1– ???? (0000) arge Scale Dynamos in Cylinders (cid:104) B r (cid:105) is zero, the standard “Ω effect” [ (cid:104) B r (cid:105) V (cid:48) φ ] contributionin Eq. (21) is zero. Here, we present the analog to Eqs. (7-12) in local Carte-sian coordinates ( x, y, z ) in a frame rotating with fixed an-gular velocity Ω = Ω e z and a linear shear velocity of V = V y ( x ) e y . We again assume a vertical field B ˆ z but nowassume perturbed velocity and magnetic field of the form ξξξ ( x, y, z, t ) = [ ξ x ( x ) , ξ y ( x ) , ξ z ( x )] exp ( γ c t − ik y y + ik z z ),where, γ c = γ + iω r . In this rotating unstratified system, weinclude the Coriolis force and the centrifugal force, and againassume that in equilibrium, the latter is canceled by the ra-dial pressure gradient. (As for the cylindrical case, the roleof gravity vs. radial pressure gradient are interchangeablefor our incompressible, unstratified case. ) The momentumequation ρ ∂ V ∂t = − ρ V . ∇ V +2( Ω × V )+ Ω × Ω × r + J × B −∇ P and the induction equation ∂ t B = ∇× ( V × B ) are linearizedin the incompressible limit to give:¯ γ (cid:101) V x − (cid:101) V y = ik z B (cid:101) B x /ρ − ∂X∂x ¯ γ (cid:101) V y + V (cid:48) y ( x ) (cid:101) V x + 2Ω (cid:101) V x = ik z B (cid:101) B y /ρ + ik y X ¯ γ (cid:101) V z = ik z B (cid:101) B z /ρ − ik z X (26)and¯ γ (cid:101) B x = ik z B (cid:101) V x ¯ γ (cid:101) B y = V y ( x ) (cid:48) (cid:101) B x + ik z B (cid:101) V y ¯ γ (cid:101) B z = ik z B (cid:101) V z , (27)where primes denote variation in x direction, ( e.g. ∂/∂x )Using (cid:101) V x = ¯ γξ x , (cid:101) V y = ¯ γξ y − V (cid:48) y ( x ) ξ x and (cid:101) V z = ¯ γξ z , Eqs. (26)and (27) can be written in terms of the displacement vectoras [¯ γ + ω A + 2 r Ω V (cid:48) y ( x )] ξ x − γ Ω ξ y = − ∂X∂x (28)(¯ γ + ω A ) ξ y + 2¯ γ Ω ξ x = ik y X (29)(¯ γ + ω A ) ξ z = − ik z X. (30)Analogous to the cylindrical case (Sec. 3), from thesesets of equations, the quasilinear vertical EMF in terms ofthe Eulerian velocity fluctuations ( (cid:101) V = ∂ ξ /∂t + V · ∇ ξ − ξ ·∇ V ), (cid:101) V x = ¯ γξ x is reduced to: E z = (cid:104) (cid:101) V × (cid:101) B (cid:105) z = E ZV r (cid:48) + E Z ( shear ) + E Z Ω , (31)where E ZV x (cid:48) = k y γk z B ( k y + k z ) (cid:34) (cid:102) V x (cid:102) V ∗ x (cid:48) γ + ϑ ( x ) (cid:35) , (32) E Z ( shear ) = − k y γk z B ( k y + k z ) (cid:20) k y V y ( x ) (cid:48) ( k y V y ( x ) − ω r )( γ + ϑ ( x )) (cid:21) | (cid:102) V x | , (33)and E Z Ω0 = 2 γk z B Ω ϑ ( x ) G ( γ + ϑ ( x ) − ω A ) | (cid:102) V x | , (34) where ϑ ( x ) = k y V y ( x ) − ω r , ¯ γ = γ − iϑ ( x ), and G = ( k y + k z )( γ + ϑ ( x ))[4 γ ϑ ( x ) + ( γ + ϑ ( x ) + ω A ) ]. As seen inthese equations, in the local Cartesian model, the rotationand shear are independent.The first contribution E ZV x (cid:48) shows how nonaxisym-metric perturbations with radial shear, even without anyexplicit mean shear flow can source a vertical EMF. Inthe absence of rotation, the second contribution, E Z ( shear ) ,shows a direct dependence of vertical EMF on the lin-ear shear. A mean shear flow, V y ( x ) (cid:48) combined with aradially uniform non-axisymmertic ( k y (cid:54) = 0) perturbationis sufficient to produce E Z ( shear ) . The last contribution E Z Ω , which vanishes in the absence of angular velocity,shows that a contribution to the vertical EMF can resultfrom a finite angular velocity (Ω ) for non-axisymmetricperturbations.Similarly the azimuthal EMF is given by E y = (cid:104) (cid:101) V × (cid:101) B (cid:105) y = E yV r (cid:48) + E y ( shear ) + E y Ω , (35)where E yV x (cid:48) = γB (cid:20) − k y ( k y + k z ) (cid:21) (cid:34) ( (cid:102) V x (cid:102) V ∗ x (cid:48) )( γ + ϑ ( x )) (cid:35) , (36) E y ( shear ) = − γB (cid:20) − k y ( k y + k z ) (cid:21) (cid:34) k y V y ( x ) (cid:48) ( k y V y ( x ) − ω r ) | (cid:102) V x | ( γ + ϑ ( x )) (cid:35) , (37)and E y Ω0 = 2 γk z B Ω k y ( k y V y ( x ) − ω r ) G ( ω A − γ − ϑ ( x )) | (cid:102) V x | . (38)Spatial derivatives of Eq. (35) could, in principle, alsogenerate and redistribute the vertical field B z due to non-axisymmetric modes ( k y (cid:54) = 0). However, the fastest growingaxisymmetric modes–the channel modes ( k y = k x = 0)—do NOT contribute in either the vertical or azimuthal EMF(Eq. 31 and Eq. 35) obtained above. In contrast, for theglobal cylindrical model, even for radially uniform axisym-metric modes ( k r = m = 0), the last term in Eq. (23) DOEScontribute in the amplification of vertical field. This dis-tinction highlights at least one circumstance in which theabsence of curvature in the Cartesian model removes a con-tribution that could be present in the global rotator. k x = 0A large-scale magnetic field can be generated by the EMF ofEq. (31) from any of the independent contributions in (32-34). In the absence of rotation, a large-scale magnetic field, B y ∼ − γ ∂ E z ∂x , can directly be generated via a linear flow-shear and a radially uniform non-axisymmetric ( k y , k z (cid:54) = 0, k x = 0) perturbation, B y ( x ) = k y k z B ( k y + k z ) (cid:20) k y V y ( x ) (cid:48) ( k y V y ( x ) − ω r )( γ + ϑ ( x )) (cid:21) (cid:48) | (cid:102) V x | , (39) MNRAS , 1– ?? (0000) F. Ebrahimi, E. G. Blackman
Figure 6. B φ generated with positive shear flow ( V φ ( r ) /V A =80( r/a ) ) and large initial amplitude forcing (solid line); and onlywith forcing amplitude varying with radius (dashed line). This is an exact analytical equation for a large-scale az-imuthal magnetic field generated via a linear mean shear-flow and any perturbations with nonzero k y and k z .The large-scale field given in Eq. (39) is consistent withprevious studies of large scale field growth from the combina-tion of linear shear with randomly forced turbulence (Vish-niac & Brandenburg 1997; Yousef & et al. 2008; Heinemannet al. 2011; Mitra & Brandenburg 2012; Sridhar & Singh2014). However, our calculations explicitly reveal the mostminimalist conditions needed for growth in the absence ofrotation: a background linear shear and an imposed non-axisymmetric perturbation with nonzero k y , k z . Helical ve-locity perturbations are not required.Generation of B y in this case of mean shear can be vi-sualized by considering an perturbation in the x directionand then considering why both z and y variations are neededto produce a net field in vertically averaged planes. If therewere no z variation in the perturbation then the mean shearwould produce no toroidal field even before vertically aver-aging. And if there were a vertical variation but no y vari-ation, then the the mean shear would produce B y ( x ) = 0from vertical averaging. Our quasi-linear theory imposes fluctuations and back-ground shear as a starting point whereas in DNS, the fluc-tuations can directly result from the MRI. The quasilineartheory shows that the growth of B φ in Eq. (21) via ∂ r E z does not requires a shear profile favorable to the MRI, just asource of fluctuations and differential rotation of either sign.To show this, we numerically computed the large-scale fieldgrowth from quasilinear theory by initializing single modefluctuations ( ˜ f m,k ( r,
0) with a polynomial dependence on r )in the simulations and forcing amplitudes of 10% on top ofa stable equilibrium flow d Ω /dr > B φ grows via E z from(Eq. 12). Figure 6 shows the large-scale toroidal field gen-erated using V φ ( r ) /V A = 80( r/a ) . The profiles are time-averaged during the decay phase. In Figure 6 we have alsoshown the case when B φ is generated by forcing only withthe same radially dependent fluctuations but in the absenceof mean shear. For this latter case, E z in Eq. (12) is thendominated by the first term on the right. Comparing the two cases, we see that for small r/a the case with onlyradius dependent fluctuations (dashed line in Fig. 6) alsocaptures the growth of B φ as for the case with both fluctua-tions and shear. But for radii of large shear r/a > .
75, theflow-dependent EMF terms (the last two terms of Eq. 12)dominate.
In summary, we have shown from both numerical simula-tions and semi-analytic quasi-linear theory how radially al-ternating large-scale toroidal fields averaged vertically andazimuthally can be generated from MHD flow-driven fluctu-ations. These fields are found in MHD DNS for both zero-net-flux and non-zero-net-flux initial configurations in boththe quasi-linear regime and the fully saturated non-linearregime.Given nonaxisymmetric fluctuations (with nonzero ver-tical and azimuthal perturbations), we calculated the con-tributions to the quasilinear fluctuation-induced EMFs inboth cylindrical and Cartesian coordinates. We have sepa-rated the derivation of the global and local models so thatthe reader can study them separately.We have not presented physical interpretations of all cir-cumstances that can lead to growth from these equations,but have provided the general forms of the EMFs and iden-tified the minimum requirements for growth. Table 1 sum-marizes these requirements for a nonzero EMF and large-scale field growth in both cylindrical and Cartesian models.In general, we find a direct relationship between dynamogenerating EMFs and differential rotation in the cylindri-cal model, or linear shear in the local Cartesian model. Thevertical EMF associated with fluctuations in the presence ofan initial vertical field is sufficient to generate an azimuthallarge-scale field for non-zero differential rotation (in rotat-ing system) but requires non-zero flow shear in the localCartesian model for a non-rotating system.Table 1 also highlights that due to the absence of cur-vature terms, the local Cartesian model is more restrictivefor field growth than global cylindrical model. According toour Cartesian EMF calculations, the fastest growing channelmodes (with k y = k x = 0) (Goodman & Xu 1994) found inshearing box simulations do NOT contribute to the EMFsin the local approximation (and thus the saturation of thesemodes) but the analogous modes can amplify large-scalefields and contribute to MRI saturation in global cylindricalsimulations (Ebrahimi et al. 2009).In the case of a large scale flow-driven instability, thefree energy source from the large scale motion can be thesource of the needed fluctuations. For the global cylinder,we have indeed found explicit dynamo generation of (cid:104) B φ (cid:105) from DNS where the MRI produces a fluctuation-inducedvertical EMF E z . The DNS provide properties of the fluctu-ations that we use as inputs to a quasi-linear calculation ofthe dynamo growth for a single mode. The DNS large scalefield growth and the associated quasi-linear dynamo calcu-lations are in reasonable agreement. Our study of the singlemode evolution its correspondence with DNS highlights that”turbulence” (defined as non-linear mode coupling) is notactually essential for the large-scale field growth and thatinsight is gained even from single mode analyses. MNRAS , 1– ????
In summary, we have shown from both numerical simula-tions and semi-analytic quasi-linear theory how radially al-ternating large-scale toroidal fields averaged vertically andazimuthally can be generated from MHD flow-driven fluctu-ations. These fields are found in MHD DNS for both zero-net-flux and non-zero-net-flux initial configurations in boththe quasi-linear regime and the fully saturated non-linearregime.Given nonaxisymmetric fluctuations (with nonzero ver-tical and azimuthal perturbations), we calculated the con-tributions to the quasilinear fluctuation-induced EMFs inboth cylindrical and Cartesian coordinates. We have sepa-rated the derivation of the global and local models so thatthe reader can study them separately.We have not presented physical interpretations of all cir-cumstances that can lead to growth from these equations,but have provided the general forms of the EMFs and iden-tified the minimum requirements for growth. Table 1 sum-marizes these requirements for a nonzero EMF and large-scale field growth in both cylindrical and Cartesian models.In general, we find a direct relationship between dynamogenerating EMFs and differential rotation in the cylindri-cal model, or linear shear in the local Cartesian model. Thevertical EMF associated with fluctuations in the presence ofan initial vertical field is sufficient to generate an azimuthallarge-scale field for non-zero differential rotation (in rotat-ing system) but requires non-zero flow shear in the localCartesian model for a non-rotating system.Table 1 also highlights that due to the absence of cur-vature terms, the local Cartesian model is more restrictivefor field growth than global cylindrical model. According toour Cartesian EMF calculations, the fastest growing channelmodes (with k y = k x = 0) (Goodman & Xu 1994) found inshearing box simulations do NOT contribute to the EMFsin the local approximation (and thus the saturation of thesemodes) but the analogous modes can amplify large-scalefields and contribute to MRI saturation in global cylindricalsimulations (Ebrahimi et al. 2009).In the case of a large scale flow-driven instability, thefree energy source from the large scale motion can be thesource of the needed fluctuations. For the global cylinder,we have indeed found explicit dynamo generation of (cid:104) B φ (cid:105) from DNS where the MRI produces a fluctuation-inducedvertical EMF E z . The DNS provide properties of the fluctu-ations that we use as inputs to a quasi-linear calculation ofthe dynamo growth for a single mode. The DNS large scalefield growth and the associated quasi-linear dynamo calcu-lations are in reasonable agreement. Our study of the singlemode evolution its correspondence with DNS highlights that”turbulence” (defined as non-linear mode coupling) is notactually essential for the large-scale field growth and thatinsight is gained even from single mode analyses. MNRAS , 1– ???? (0000) arge Scale Dynamos in Cylinders EMF radial derivative Restriction Maintains Finiteness? global cylinder (with Ω >
0; Ω (cid:48) (cid:54) = 0; k (cid:54) = 0; k r (cid:54) = 0; m > k r = 0 Ω (cid:48) ( r ) = 0 m = 0 E (cid:48) Z ( r ) (cid:54) = 0 yes yes no E (cid:48) φ ( r ) (cid:54) = 0 yes yes yes E (cid:48) φ ( r ) and E (cid:48) Z ( r ) (cid:54) = 0 (dynamo) yes yes nolocal Cartesian (with Ω > V (cid:48) y (cid:54) = 0; k (cid:54) = 0; k x (cid:54) = 0; k y > k x = 0 V (cid:48) y ( x ) = 0 k y = 0 E (cid:48) Z ( x ) (cid:54) = 0 yes no no E (cid:48) y ( x ) (cid:54) = 0 yes no no E (cid:48) y ( x ) and E (cid:48) Z ( x ) (cid:54) = 0 (dynamo) yes no nolocal Cartesian (with Ω = 0; V (cid:48) y (cid:54) = 0; k (cid:54) = 0; k x (cid:54) = 0; k y > k x = 0 V (cid:48) y ( x ) = 0 k y = 0 E (cid:48) Z ( x ) (cid:54) = 0 yes no no E (cid:48) y ( x ) (cid:54) = 0 yes no no E (cid:48) y ( x ) and E (cid:48) Z ( x ) (cid:54) = 0 (dynamo) yes no no Table 1.
Table summarizing the minimal ingredients needed to maintain the finiteness of the two components of the EMFs separatelyand together. The first column indicates three general cases: the global cylinder and two Cartesian cases. To read the table for each ofthese cases, consider the global cylinder case as an example: the second row of the first column indicates general ingredients that ourglobal cylinder could have (rotation Ω >
0, differential rotation Ω (cid:48) (cid:54) = 0, and general perturbations with k (cid:54) = 0 , k r (cid:54) = 0, and m > > = 0) listed, a first and second row provide the information onthe finiteness of each component of the and the third row provides the information on the finiteness of both components together. Thislatter circumstances is needed to supply the large scale dynamo. Note that if either of the two separate components has a ”no” entrythen that also implies that both together cannot be finite. The conditions for finiteness (and thus dynamo action) are different in thetwo geometries because curvature terms are absent in the Cartesian approximation. Our results also show that the traditional “Ω effect” ofshear on the mean field is absent when the initial mean fieldis vertical and the averaging is over vertical surfaces. Insteadthe essential shear operates on the fluctuations. The fieldgrowth can be entirely described by working with the EMFdirectly, non-axisymmetric (though not necessarily helical)velocity perturbations are essential for large scale growth asevidenced from direct visualization of the field lines in DNSand from the quasi-linear theory. We should also note thatin much of the MRI dynamo literature, large-scale fields inshearing boxes are computed via planar averages (and av-eraged over the direction of the nonuniformaty of the meanflow) leaving mean fields as a function of z direction. There,because of the averaging and boundary conditions for theshear box “Ω effect” can still survive. Here, our averaging isover vertical surfaces, and not along the direction of meanflow variation. An important lesson is that the averagingprocedure and boundary conditions have important impli-cations for the dominant contributions to the EMF.By calculating the complete form of EMF for bothglobal cylindrical and Cartesian cases, we have demon-strated the minimum ingredients for large scale field growthin both of these two models. Our results suggest that thequasilinear and nonlinear fluctuation-induced EMF mayprovide fundamental insight into the growth and sustenanceof large-scale dynamo in these flow-driven systems. The cal-culations herein provide a more general approach to iden-tifying the origin and minimal ingredients needed for largescale dynamo growth in unstratified rotating and differen-tially rotating systems or linearly sheared systems.Although we leave a detailed analysis making explicitconnections to previous approaches of incoherent alpha(Vishniac & Brandenburg 1997; Brandenburg 2005; Bran-denburg et al. 2008; Mitra & Brandenburg 2012; Sridhar &Singh 2014) and or shear current effects as an opportunity for further work, we emphasize two points in this context.First we have intentionally avoided using the α formalismand worked directly with only the EMF. Second, we findthat the absolute minimum conditions for radially depen-dent large scale field growth are non-axisymmetric velocityfluctuations plus linear shear. The velocity fluctuations donot need to be helical at any time. In this way our globaland local calculations provide a more minimalist set of con-ditions for growth than the that of a fluctuating kinetic helic-ity (Vishniac & Brandenburg 1997). We note however, thatin the quasi-linear regime, the large scale magnetic field does(as a function of radius) (see Fig.4) develop a field alignedEMF, which is a source term for sum of the time derivativeof large scale magnetic helicity and divergence of large scalehelicity flux, as previously confirmed for the global cylin-drical case (Ebrahimi & Bhattacharjee 2014). Here we havenot studied the non-linear/saturating effects of the growthof small scale magnetic helicity helical fluctuations, nor theEMF and mean magnetic field correspondence during thenonlinear saturation. More detailed calculations for the non-linear phase of DNS (by P. Bhat et.al in preparation) doshow a direct correlation of large-scale field with the EMFterms in the nonlinear regime that we have presently com-puted only in the quasli-linear approximation.Finally, we note that our large scale fields show radialreversals and these would be sites of current sheets. If wethink toward generalizations to stratified rotators that formcoronae, only magnetic structures of large enough scale sur-vive buoyant rise into coronae where they can dissipate andtransport angular momentum non-locally (Blackman & Pes-sah 2009). If our present toroidal field structures and rever-sal scales survive stratified generalizations, they provide ascale for coronal structures and current sheets that link thelarge scale field directly to structures associated with coro-nal transport and dissipation. MNRAS , 1– ?? (0000) F. Ebrahimi, E. G. Blackman
ACKNOWLEDGMENTS
We thank H. Ji for useful discussions, and thank AxelBrandenburg for useful comments. FE acknowledges grantsupport from DOE, DE-FG02-12ER55142 and NSF PHY-0821899 CMSO. This work was also facilitated by theMPPC. EB acknowledges support from NSF-AST-1109285,HST-AR-13916.002, a Simons Fellowship, and the IBM-Einstein Fellowship Fund at the Institute for AdvancedStudy during part of this work.
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