Pumping velocity in homogeneous helical turbulence with shear
aa r X i v : . [ a s t r o - ph . S R ] S e p NORDITA-2011-40
Pumping velocity in homogeneous helical turbulence with shear
Igor Rogachevskii, ∗ Nathan Kleeorin, † Petri J. K¨apyl¨a,
2, 3, ‡ and Axel Brandenburg § Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva 84105, Israel Department of Physics, Gustaf H¨allstr¨omin katu 2a (PO Box 64), FI-00064 University of Helsinki, Finland NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden Department of Astronomy, Stockholm University, SE 10691 Stockholm, Sweden (Dated: October 14, 2018, Revision: 1.154 )Using different analytical methods (the quasi-linear approach, the path-integral technique and tau-relaxationapproximation) we develop a comprehensive mean-field theory for a pumping effect of the mean magnetic fieldin homogeneous non-rotating helical turbulence with imposed large-scale shear. The effective pumping velocityis proportional to the product of α effect and large-scale vorticity associated with the shear, and causes a separa-tion of the toroidal and poloidal components of the mean magnetic field along the direction of the mean vorticity.We also perform direct numerical simulations of sheared turbulence in different ranges of hydrodynamic andmagnetic Reynolds numbers and use a kinematic test-field method to determine the effective pumping velocity.The results of the numerical simulations are in agreement with the theoretical predictions. PACS numbers: 47.65.Md
I. INTRODUCTION
The origin of cosmic magnetic fields is one of the funda-mental problems in theoretical physics and astrophysics. It isgenerally believed that solar and galactic magnetic fields arecaused by the combined action of helical turbulent motionsof fluid and differential rotation [1–7]. In most of these stud-ies, differential rotation plays merely the role of enhancing themagnetic field in the toroidal direction. However, in recentyears there has been increased interest in mean-field effectscaused specifically by turbulent shear flows. This interest iscaused by discoveries of the shear dynamo [8, 9] and vortic-ity dynamo [10, 11] in non-helical homogeneous turbulencewith a large-scale shear. In particular, recent numerical ex-periments [12–17] have clearly demonstrated the existence ofa shear dynamo of a large-scale magnetic field in non-helicalturbulence or turbulent convection with superimposed large-scale shear. However, the origin of the shear dynamo is stillsubject of active discussions [8, 9, 15, 18–23].There are three additional phenomena that are also relatedto the presence of shear. One is the vorticity dynamo, which isthe self-excitation of large-scale vorticity in a turbulence withlarge-scale shear. It has been predicted theoretically [10, 11]and detected in recent numerical experiments [13, 14, 24].The vorticity dynamo can also affect the dynamo process ofthe mean magnetic field. Another phenomenon is a non-zero α effect in non-helical turbulence with shear when the systemis inhomogeneous or density stratified. In that case there isan α effect [8, 18] that can lead to an alpha-shear dynamo.Finally, when homogeneous turbulence with shear is heli-cal, there is an effective pumping velocity γ ∝ α W of thelarge-scale magnetic field, where W is the large-scale vortic-ity caused by shear. This effect has so far only been found ∗ [email protected] † [email protected] ‡ petri.kapyla@helsinki.fi § [email protected] in direct numerical simulations (DNS) [25], but there has sofar been no theory for this new effect, nor has there been asystematic survey of DNS for determining the dependence ofpumping on magnetic Reynolds and Prandtl numbers as wellas the turbulent Mach number.The goal of the present study is to develop a comprehensivetheory of mean-field pumping in homogeneous helical turbu-lence with shear and to perform systematic numerical simu-lations designed for detailed comparison with the theoreticalpredictions. It is important to emphasize that the pumping ofthe large-scale magnetic field discussed usually in the litera-ture has always been connected with inhomogeneous turbu-lence [3, 26, 27], but here we study the pumping for homoge-neous, albeit helical turbulence. II. GOVERNING EQUATIONS
We consider homogeneous helical turbulence with a linearshear velocity U = (0 , Sx, . Averaging the induction equa-tion over an ensemble of turbulent velocity field yields themean-field equation: ∂ B ∂t = ∇ × (cid:0) U × B + u × b − η ∇× B (cid:1) , (1)where E i ≡ ( u × b ) i = a ij B j + b ijk ∇ k B j is the meanelectromotive force, u and b are the fluctuations of velocityand magnetic field, overbars denote averaging over an ensem-ble of turbulent velocity fields, B is the mean magnetic field, U is the mean velocity that includes only the imposed large-scale shear, and η is the magnetic diffusion due to electricalconductivity of the fluid. Note that the part a ij B j in the ex-pression for the mean electromotive force determines the ef-fective pumping velocity, γ i = − ǫ ijk a ij , and the α tensor, α ij = ( a ij + a ji ) , i.e., E ( a ) i = α ij B j + ( γ × B ) i , whilethe turbulent magnetic diffusion and the shear-current dynamoeffect are associated with the b ijk term.To determine the turbulent transport coefficients in homo-geneous helical turbulence with mean velocity shear we usethe following equations for fluctuations of velocity and mag-netic field: ∂ u ∂t = − ( U ·∇ ) u − ( u ·∇ ) U − ρ ∇ p + 14 πρ (cid:2) ( b ·∇ ) B +( B ·∇ ) b (cid:3) + ν ∆ u + u N + f ( u ) , (2) ∂ b ∂t = ( B ·∇ ) u − ( u ·∇ ) B + ( b ·∇ ) U − ( U ·∇ ) b + η ∆ b + b N , (3)where ν is the kinematic viscosity, ρ is the mean density of theincompressible fluid flow, p is the fluctuation of total (hydro-dynamic and magnetic) pressure, the magnetic permeabilityof the fluid is included in the definition of the magnetic field, v N and b N are the nonlinear terms, and ρ f ( u ) is the stirringforce for the background velocity fluctuations.We begin by deriving expressions for the pumping effectthat are valid in different regimes, where fluid and magneticReynolds numbers are both small, both are large, or only thefluid Reynolds number is large, but the magnetic Reynoldsnumber is small. These results will then be compared withthose of DNS in the corresponding regimes. A. Small magnetic and hydrodynamic Reynolds numbers
We use the quasi-linear or second order correlation ap-proximation (SOCA) applied to shear flow turbulence (see[18, 20]). This approach is valid for small magnetic and hy-drodynamic Reynolds numbers. To exclude the pressure termfrom the equation of motion (2) we calculate ∇× ( ∇× u ) ,then we rewrite the obtained equation and Eq. (3) in Fourierspace, apply the two-scale approach (i.e., we use large-scale and small-scale variables), neglect nonlinear terms inEqs. (2)–(3), but retain molecular dissipative terms in theseequations. We seek a solution for fluctuations of velocity andmagnetic fields as an expansion for weak velocity shear: u = u (0) + u (1) + ... , (4) b = b (0) + b (1) + ... , (5)where b (0) i ( k , ω ) = G η ( k, ω ) (cid:20) i ( k · B ) δ ij − (cid:16) δ ij k m ∂∂k n + δ im δ jn (cid:17) ( ∇ n B m ) (cid:21) u (0) j ( k , ω ) , (6) u (1) i ( k , ω ) = G ν ( k, ω ) (cid:18) k iq δ jp + δ ij k q ∂∂k p − δ iq δ jp (cid:19) × ( ∇ p U q ) u (0) j ( k , ω ) , (7) b (1) i ( k , ω ) = G η ( k, ω ) (cid:26)h i ( k · B ) δ ij − (cid:16) δ ij k m ∂∂k n + δ im δ jn (cid:17) ( ∇ n B m ) i u (1) j ( k , ω ) + h δ ij k q ∂∂k p + δ iq δ jp i b (0) j ( k , ω ) ( ∇ p U q ) (cid:27) . (8) Here G ν ( k, ω ) = ( νk − iω ) − , G η ( k, ω ) = ( ηk − iω ) − ,and δ ij is the Kronecker tensor. The statistical propertiesof the background velocity fluctuations with a zero large-scale shear, u (0) , are assumed to be given. For derivation ofEqs. (6)–(8) we use the identity Z U q ( Q ) b n ( k − Q ) d Q = i ( ∇ p U q ) ∂b n ∂k p , which is valid in the framework of the mean-field approach,i.e., it is assumed that there is scale separation. Equations (6)–(8) coincide with those derived by [18], and they allow usto determine the cross-helicity tensor g (1) ij = h u (0) i b (1) j i + h u (1) i b (0) j i . This procedure yields the contributions E ( S ) m = ε mij R g (1) ij ( k , ω ) d k dω to the mean electromotive forcecaused by sheared helical turbulence. We are interested firstof all in the contributions to the mean electromotive forcewhich are proportional to the mean magnetic field, i.e., E ( a ) i = α ij B j + ( γ × B ) i . For the integration in ω -space and in k -space we have to specify a model for the background shear-free helical turbulence (with B = 0) , which is determined byequation: h u i ( k , ω ) u j ( − k , − ω ) i (0) = E ( k ) Φ( ω )8 π k h(cid:16) δ ij − k i k j k (cid:17) ×h u i (0) − ik ε ijl k l h u · ( ∇ × u ) i (0) i , (9)where E ( k ) is the energy spectrum (e.g., a power-law spec-trum, E ( k ) ∝ ( k/k f ) − q with the exponent < q < for the wavenumbers k f ≤ k ≤ k d , where k f and k d arethe forcing and dissipation wavenumbers), and ε ijk is thefully antisymmetric Levi-Civit`a tensor. We consider the fre-quency function Φ( ω ) in the form of the Lorentz profile: Φ( ω ) = νk / [ π ( ω + ν k )] . This model for the frequencyfunction corresponds to the correlation function h u i ( t ) u j ( t + τ ) i ∝ exp( − τ νk ) . (10)In that case, and under the assumption of small magnetic andhydrodynamic Reynolds numbers, the effective pumping ve-locity, γ , and the off-diagonal components of the tensor α ij are given by γ = C ( q )2 (cid:18) Pm1 + Pm (cid:19) Re τ f α ∗ W , (11) α ij = C ( q )5 (2Pm + 1) Pm(1 + Pm) Re τ f α ∗ ( ∂U ) ij , (12) C ( q ) = Z k d k f E ( k ) (cid:18) kk f (cid:19) − dk = (cid:18) q − q + 3 (cid:19) (cid:20) − ( k f /k d ) q +3 − ( k f /k d ) q − (cid:21) , (13)where α ∗ = − ( τ f / h u · ( ∇ × u ) i (0) , Pm = ν/η is themagnetic Prandtl number,
Re = τ f h u i (0) /ν is the hydro-dynamic Reynolds number, Rm = Re Pm is the magneticReynolds number, τ f = ℓ f /u rms is the turnover time, where ℓ f = 1 /k f is the energy-containing (forcing) scale of a ran-dom velocity field, and u rms = p h u i (0) . For the integrationin ω -space we use the integrals I n ( k ) given in Appendix A.For linear shear velocity, U = (0 , Sx, , the mean vorticityis W = ∇× U = (0 , , S ) , and the mean symmetric tensor ( ∂U ) ij = ( ∇ i U j + ∇ j U i ) / has only two nonzero compo-nents: ( ∂U ) = ( ∂U ) = S/ . Therefore, α ij has twonon-zero off-diagonal components caused by both, shear andhelical turbulence α = α , while the effective pumping ve-locity, γ , has only one component directed along the verticalaxis, γ = (0 , , γ ) : γ = C ( q )2 (cid:18) Pm1 + Pm (cid:19) Re α ∗ Sh , (14) α = α = C ( q )10 (2Pm + 1) Pm(1 + Pm) Re α ∗ Sh , (15)where Sh = τ f S is the shear parameter. As follows fromEqs. (14) and (15), γ ∝ Pm and α ∝ Pm for Pm ≪ ,while for Pm ≫ the effective pumping velocity γ and α are independent of Pm . For all values of the magnetic Prandtlnumbers, γ and α are positive. This asymptotic behaviorwhich is valid for Re ≪ , is in agreement with Figs. 1 and 2(see Sect. III). Note that the diagonal components of the ten-sor α ij in this case are α = − C ( q )3 (cid:18) Rm1 + Pm (cid:19) τ f h u · ( ∇ × u ) i (0) , (16) C ( q ) = Z k d k f E ( k ) (cid:18) kk f (cid:19) − dk = (cid:18) q − q + 1 (cid:19) (cid:20) − ( k f /k d ) q +1 − ( k f /k d ) q − (cid:21) . (17) B. Large magnetic and hydrodynamic Reynolds numbers
To determine the the effective pumping velocity and thetensor α ij in homogeneous helical turbulence with mean ve-locity shear for large magnetic and hydrodynamic Reynoldsnumbers we use the procedure which is similar to that appliedin [9] in earlier investigations of shear flow turbulence. Letus derive equations for the second moments. We apply thetwo-scale approach, e.g., we use large scale R = ( x + y ) / , K = k + k and small scale r = x − y , k = ( k − k ) / variables (see, e.g., [28]). We derive equations for the follow-ing correlation functions: f ij ( k ) = ˆ L ( u i ; u j ) , h ij ( k ) = ˆ L ( b i ; b j ) ,g ij ( k ) = (4 πρ ) − ˆ L ( b i ; u j ) , where ˆ L ( a ; c ) = Z h a ( k + K / c ( − k + K / i exp ( i K · R ) d K , and h ... i denotes averaging over ensemble of turbulent veloc-ity field. The equations for these correlation functions aregiven by (see [9]) ∂f ij ( k ) ∂t = i ( k · B )Φ ij + I fij + I Sijmn ( U ) f mn + F ij + ˆ N f ij , ∂h ij ( k ) ∂t = − i ( k · B )Φ ij + I hij + E Sijmn ( U ) h mn + ˆ N h ij ,∂g ij ( k ) ∂t = i ( k · B )[ f ij ( k ) − h ij ( k ) − h ( H ) ij ] + I gij + J Sijmn ( U ) g mn + ˆ N g ij , (18)where hereafter we omit the arguments t and R in the correla-tion functions and neglect small terms ∼ O ( ∇ ) . Here F ij isrelated to the forcing term and ∇ = ∂/∂ R . In Eqs. (18), Φ ij ( k ) = (4 πρ ) − [ g ij ( k ) − g ji ( − k )] , and ˆ N f ij , ˆ N h ij , ˆ N g ij , are the third-order moments appearing due to the non-linear terms which include also molecular dissipation terms.The tensors I Sijmn ( U ) , E Sijmn ( U ) and J Sijmn ( U ) are givenby I Sijmn ( U ) = (cid:18) k iq δ mp δ jn + 2 k jq δ im δ pn − δ im δ jq δ pn − δ iq δ jn δ pm + δ im δ jn k q ∂∂k p (cid:19) ∇ p U q ,E Sijmn ( U ) = (cid:18) δ im δ jq δ pn + δ jm δ iq δ pn + δ im δ jn k q ∂∂k p (cid:19) ∇ p U q ,J Sijmn ( U ) = (cid:18) k jq δ im δ pn − δ im δ pn δ jq + δ jn δ pm δ iq + δ im δ jn k q ∂∂k p (cid:19) ∇ p U q , where k ij = k i k j /k . The source terms I fij , I hij , and I gij whichcontain the large-scale spatial derivatives of the magnetic field B , are given in [9]. Next, in Eqs. (18) we split the tensor formagnetic fluctuations into nonhelical, h ij , and helical, h ( H ) ij , parts. The helical part of the tensor of magnetic fluctuations h ( H ) ij depends on the magnetic helicity and it follows frommagnetic helicity conservation arguments (see, e.g., [29–32]and [7] for a review).The second-moment equations include the first-order spa-tial differential operators ˆ N applied to the third-order mo-ments M (III) . A problem arises how to close the system,i.e., how to express the set of the third-order terms ˆ N M (III) through the lower moments M (II) (see, e.g., [33–35]). We usethe spectral τ -closure-approximation which postulates that thedeviations of the third-moment terms, ˆ N M (III) ( k ) , from thecontributions to these terms due to the background turbulence, ˆ N M (III , ( k ) , are expressed through the similar deviations ofthe second moments, M (II) ( k ) − M (II , ( k ) : ˆ N M (III) ( k ) − ˆ N M (III , ( k )= − τ r ( k ) h M (II) ( k ) − M (II , ( k ) i , (19)(see, e.g., [33, 36, 37]), where τ r ( k ) is the scale-dependentrelaxation time, which can be identified with the correlationtime, τ f , of the turbulent velocity field for large hydrodynamicand magnetic Reynolds numbers. The quantities with the su-perscript (0) correspond to the background shear-free turbu-lence with a zero mean magnetic field. We apply the spectral τ approximation only for the nonhelical part h ij of the tensorof magnetic fluctuations. Note that a justification of the τ ap-proximation for different situations has been performed in anumber of numerical simulations and analytical studies (see,e.g., [7, 38–45]).We take into account that the characteristic time of variationof the magnetic field B is substantially longer than the corre-lation time τ f . This allows us to obtain a stationary solutionfor Eqs. (18) for the second-order moments, M (II) ( k ) , whichare the sums of contributions caused by shear-free and shearedturbulence. The contributions to the mean electromotive forcecaused by a shear-free turbulence and sheared non-helical tur-bulence are given in [9]. In particular, the contributions tothe electromotive force caused by the sheared turbulence read: E ( S ) m = ε mji R g ( S ) ij ( k ) d k , where the corresponding contri-butions to the cross-helicity tensor g ( S ) ij in the kinematic ap-proximation, are given by g ( S ) ij ( k ) = iτ r ( k ) h J Sijmn τ r ( k ) ( k · B )+ τ r ( k ) ( k · B ) I Sijmn i f (0) mn , (20)and we use the following model for the background shear-freehelical turbulence (with B = 0) : f (0) ij = h u i ( k ) u j ( − k , ) i (0) = h(cid:16) δ ij − k i k j k (cid:17) h u i (0) − ik ε ijl k l h u · ( ∇× u ) i (0) i E ( k )8 π k , (21)where the energy spectrum is E ( k ) = ( q −
1) ( k/k f ) − q ,k f = 1 /ℓ f and the length ℓ f is the maximum scale of tur-bulent motions. The turbulent correlation time is τ r ( k ) =2 τ f ( k/k f ) − q . Therefore, for large magnetic and hydrody-namic Reynolds number the effective pumping velocity, γ ,and the off-diagonal components of the tensor α ij caused bysheared helical turbulence are given by γ = 23 τ f α ∗ W , (22) α ij = −
45 (5 − q ) τ f α ∗ ( ∂U ) ij . (23)Since the mean symmetric tensor ( ∂U ) ij has only twononzero components: ( ∂U ) = ( ∂U ) = S/ , the tensor α ij has only two non-zero off-diagonal components, α = α . In particular, γ = 23 α ∗ Sh , (24) α = α = −
25 (5 − q ) α ∗ Sh = − α ∗ Sh , (25)where we have used the Kolmogorov kinetic energy spectrumexponent q = 5 / in Eq. (25). The diagonal components ofthe tensor α ij in this case are α = α ∗ (see, e.g., [1, 3]). Theseresults for large magnetic and hydrodynamic Reynolds num-ber are in qualitative agreement with DNS performed in [25]. C. Large magnetic Reynolds numbers and smallhydrodynamic Reynolds numbers
To develop a mean-field theory for large magnetic Reynoldsnumbers and small hydrodynamic Reynolds numbers we usestochastic calculus for a random velocity field. To derive anequation for the mean magnetic field we use an exact solu-tion of the induction equation for the total field B (which isthe sum of the mean B and fluctuating b parts) with an initialcondition B ( t = t , x ) = B ( t , x ) in the form of a func-tional integral: B i ( t, x ) = h G ij ( t, t , ξ ) exp( ˆ ξ · ∇ ) B j ( t , x ) i w , (26)(see, e.g., [46, 47]), where the operator exp( ˆ ξ · ∇ ) is deter-mined by exp( ˆ ξ · ∇ ) = ∞ X k =0 k ! ( ˆ ξ · ∇ ) k , (27) ˆ ξ = ξ − x (see Appendix B). The Wiener trajectory ξ ( t, t , x ) is determined by ξ ( t, t , x ) = x − Z t − t v ( t σ , ξ ) dσ + (2 η ) / w ( t − t ) , (28)where t σ = t − σ , and the velocity field v is the sum ofthe mean shear velocity U and fluctuating u parts. We con-sider large magnetic Reynolds number, but take into accountsmall yet finite magnetic diffusion η . The magnetic diffu-sion can be described by a random Wiener process w ( t ) thatis defined by the following properties: h w i ( t ) i w = 0 and h w i ( t + τ ) w j ( t ) i w = τ δ ij , where h·i w denotes the averagingover the statistics of the Wiener random process. The function G ij ( t, s, ξ ) is determined by equation: dG ij ( t, s, ξ ) ds = N ik G kj ( t, s, ξ ) , (29)with the initial condition G ij ( t = s ) = δ ij and N ij = ∇ j v i .The form of the exact solution (26) allows us to separate theaveraging over random Brownian motion of particles (i.e., theaveraging over a random Wiener process w ( t ) ) and a randomvelocity u .We consider a random flow with a small yet finite Strouhalnumber (that is the ratio the correlation time of a random fluidflow to the turnover time ℓ f /u rms ). A random velocity fieldwith a small Strouhal number can be modelled by a randomvelocity field with a constant renewal time τ . Assume that inthe intervals . . . ( − τ, , τ ); ( τ, τ ); . . . the velocity fieldsare statistically independent and have the same statistics. Thisimplies that the velocity field looses memory at the prescribedinstants t = mτ , where m = 0 , ± , ± , . . . . This velocityfield cannot be considered as a stationary velocity field forsmall times ∼ τ , however, it behaves like a stationary field for t ≫ τ . Averaging Eq. (26) over the random velocity field wearrive at the equation for the mean magnetic field, B : ∂B i ∂t = (cid:2) ∇ × ( U × B ) (cid:3) i + A ijm ∇ m B j + D ijmn ∇ m ∇ n B j , (30)(see Appendix B), where A ijm = 1 τ hh ˆ ξ m G ij ii w , (31) D ijmn = 12 τ hh ˆ ξ m ˆ ξ n G ij ii w , (32)the angular brackets h·i denote an ensemble average over therandom velocity field. Therefore, the mean magnetic field isdetermined by double averaging over two independent ran-dom processes, i.e., by the ensemble average over the randomvelocity field and by the average over Wiener random process w ( t ) .We are interested in the lowest-order contributions to themean electromotive force which are proportional to the meanmagnetic field, E ( a ) i = a ij B j , where a ij = (1 / ε inm A njm and the tensor A ijm reads: A ijm = − τ Z τ dt Z τ dt ′ D(cid:2) v m ( t, ξ ) (cid:3) x (cid:2) ∇ j v i ( t ′ , ξ ) (cid:3) y E , (33)where x → y and (cid:2) v m ( t, ξ ) (cid:3) x denotes the Eulerian velocitydetermined at the Wiener trajectory ξ that passes through thepoint x at instant t . Hereafter the angular brackets denotedouble averaging over a random velocity field and over thestatistics of the Wiener process.For small hydrodynamic Reynolds numbers we seek the so-lutions of the linearized Navier-Stokes equation (2) for incom-pressible velocity field u as superpositions of the Orr-Kelvinrandom shearing waves u ( t, r ) = R u ( t, k ) exp[ i k ( t ) · r ] d k , where k = ( k x , k y , k z ) , k ( t ) = ( k x − Sk y t, k y , k z ) (see, e.g., [23, 48–50]). Therefore, the effec-tive pumping velocity, γ , and the off-diagonal components ofthe tensor α ij are given by γ n = 12 ε nji a ij = 14 A kmm = − i τ Z τ dt Z τ dt ′ × k m ( t ′ ) h v m ( t, k ) v ∗ n ( t ′ , k ) i , (34) α ij = 12 ( a ij + a ji ) = 14 ( ε inm A njm + ε jnm A nim )= − i τ Z τ dt Z τ dt ′ ( ε inm k j ( t ′ ) + ε jnm k i ( t ′ )) ×h v m ( t, k ) v ∗ n ( t ′ , k ) i . (35)Using these equations and Eqs. (C13)–(C10) in Appendix Cwe obtain the effective pumping velocity, γ = (0 , , γ ) , andthe off-diagonal components α = α of the tensor α ij forlarge magnetic Reynolds numbers and small hydrodynamicReynolds numbers: γ = C ( q )3 α ∗ Sh Re , (36) α = α = (cid:18) C ( q ) Re ττ f − C ( q )2 Re (cid:19) α ∗ Sh , (37)where Re ≪ τ /τ f < . The diagonal components of thetensor α ij in this case obtained using path-integral approach are α = − (1 / h τ u · ( ∇ × u ) i (0) (see, e.g., [46, 51]). In thenext section we discuss comparison with new systematic DNSdesigned for comparison with our theoretical predictions. III. COMPARISON WITH DNSA. Numerical model
Our DNS model is identical to that used in [25]. We be-gin by testing the analytical results numerically using three-dimensional simulations of isotropically forced turbulencein a fully periodic cube of size (2 π ) . The uniform shear U = (0 , Sx, is imposed using the shearing box method andthe gas obeys an isothermal equation of state characterized bythe constant speed of sound c s . We solve the continuity andNavier–Stokes equations in the form D ln ρ D t = − U · ∇ ln ρ − ∇ · U , (38) D U D t = − U · ∇ U − SU x ˆ y − c ∇ ln ρ + f + F visc , (39)where the imposed shear is subsumed in the advective deriva-tive DD t ≡ ∂∂t + Sx ∂∂y . (40)Here ρ is the density, U is the velocity, f describes the forc-ing, and F visc = ρ − ∇ · (2 ρν S ) is the viscous force, where ν is the kinematic viscosity, and S ij = ( U i,j + U j,i ) − ∇ · U (41)is the traceless rate of strain tensor. The forcing function f isgiven in [52]: f ( x , t ) = Re { N f k ( t ) exp[i k ( t ) · x + i φ ( t )] } , (42)where x is the position vector. The wavevector k ( t ) and therandom phase − π < φ ( t ) ≤ π change at every time step, so f ( x , t ) is δ -correlated in time. The normalization factor N ischosen on dimensional grounds to be N = f c s ( | k | c s /δt ) / ,where f is a nondimensional forcing amplitude. At eachtimestep we select randomly one of many possible wavevec-tors in a certain range around a given forcing wavenumber.The average wavenumber is referred to as k f . In the presentstudy we always use k f /k = 5 . We force the system withtransverse helical waves [53], f k = R · f (nohel) k with R ij = δ ij − i σǫ ijk ˆ k k √ σ , (43)where σ = 1 for the fully helical case with positive helicity ofthe forcing function, f (nohel) k = ( k × ˆ e ) / p k − ( k · ˆ e ) , (44)is a non-helical forcing function, and ˆ e is an arbitrary unitvector not aligned with k ; note that | f k | = 1 . We use fullyhelical forcing, i.e. σ = 1 , in all of our runs.The boundary conditions in the y and z directions are peri-odic, whereas shearing-periodic conditions are used in the x direction. The simulations are governed by the fluid and mag-netic Reynolds numbers, the magnetic Prandtl number, andthe shear and Mach numbers:Re = u rms νk f , Rm = u rms ηk f , Pm = νη , (45)Sh = Su rms k f , Ma = u rms c s . Here u rms is the root mean square velocity of turbulent mo-tions and η is the magnetic diffusivity. We use the P ENCIL C ODE to perform the simulations. B. Test field method
We apply the kinematic test-field method (see, e.g., [15,54, 55]) to compute the effective pumping velocity, γ , and allcomponents of the tensor α ij . The essence of this method isthat a set of prescribed test fields B ( p,q ) and the flow from theDNS are used to evolve separate realizations of small-scalefields b ( p,q ) . Neither the test fields B ( p,q ) nor the small-scalefields b ( p,q ) act back on the flow. These small-scale fieldsare then used to compute the electromotive force E ( p,q ) cor-responding to the test field B ( p,q ) . The number and form ofthe test fields used depends on the problem at hand. For thepurposes of the present study we use uniform horizontal testfields B (1) = ( B , , and B (2) = (0 , B , , in which casethe series expansion of the electromotive force contains onlya single term E ( a ) i = a ij B j . (46)We present the results using the quantities: α = ( a + a ) , (47) α = α = ( a + a ) , (48) γ = ( a − a ) . (49)We use α = u rms as a normalization factor when present-ing numerical results. Errors are estimated by dividing thetime series into three equally long parts and computing timeaverages for each of them. The largest departure from thetime average computed over the entire time series representsthe error. This definition of the error bar gives an indicationabout the mean value that one would obtain for shorter partsof the time series. With this definition, the error bars do nor-mally become shorter for longer runs, provided the time seriesis stationary. This would not be the case for the rms value ofthe deviations, which might sometimes also be of interest. http://pencil-code.googlecode.com/ FIG. 1. Pumping coefficient γ = ( a − a ) normalized by α = u rms as a function of Pm for two values of Re (Sets A1 and A2).The shear parameter Sh = − . ( − . ) for Re = 0 . ( . ).Analytical results according to Eq. (14) are overplotted with dottedlines. The values of C ( q ) are used as fit parameters and indicated inthe legends. C. Results
We perform several sets of simulations where we vary theparameters Pm , Rm, Sh, and Ma individually to study theanalytical results derived in Sect. II; see Table I. The setupused here is prone to exhibit the so-called vorticity dynamo[10, 24], due to which large-scale vorticity is generated, andcomplications can arise in the interpretation of the simulationdata. Here we restrict the studied parameter range so that thevalues of Re and Sh are subcritical for the vorticity dynamo.In our runs where the Reynolds numbers are of the order ofunity or less, a low grid resolution of is often sufficient.Indeed, in Table II we show the results obtained for differ-ent resolutions ranging from to for Rm around 1,which demonstrates good convergence of the results withinerror bars.
1. Dependence on Pm Figure 1 shows our results for γ as a function of magneticPrandtl number Pm . We find that the numerical results coin-cide with the analytical formula, Eq. (14). Values of the orderof C ( q ) ≈ fit the DNS results within the error estimates.Figure 2 shows the results for α as a function of Pm fortwo values of Re. The data for α shows significantly largerfluctuations than the corresponding results for γ . However, theDNS results seem to fall in line with the analytical expression,Eq. (15), although the value of C ( q ) needed to fit the data isan order of magnitude larger than in the case of γ . This canbe explained by comparing Eqs. (36) and (37), which showthat γ ∝ Re , while α ∝ Re ( τ /τ f ) , where τ is the flow TABLE I. Summary of the runs.Set Re Pm Sh Ma gridA1 0.04 0.05. . . 25 − .
20 0 .
010 32 . . . A2 0.16 0.02. . . 20 − .
13 0 .
016 32 . . . B1 0.08. . . 81 1 − .
025 0 .
080 32 . . . B2 0.08. . . 83 1 − .
075 0 .
080 32 . . . B3 0.08. . . 3.5 1 − .
25 0 .
080 32 B4 0.08. . . 0.4 1 − . .
080 32 C1 0.04 1 − . . . . − .
19 0 .
010 32 C2 0.16 1 − . . . . − .
12 0 .
016 32 C3 0.45 1 − . . . . − .
09 0 .
023 32 C4 1.3 1 − . . . . − .
07 0 .
032 32 D1 0.08 1 − .
010 0 . . . . .
41 32 TABLE II. Convergence study of γ and α for Rm = 1 . andSh = − . from simulations with different grid sizes.Run γ/α [10 − ] α /α [10 − ] gridE1 . ± .
12 0 . ± .
25 16 E2 . ± .
07 0 . ± .
20 32 E3 . ± .
06 0 . ± .
55 64 E4 . ± .
18 0 . ± .
23 128 FIG. 2. Symmetric part of a ij , α = ( a + a ) normalized by α = u rms as a function of Pm for the same runs as in Fig. 1. Thedotted lines show the analytical result according to Eq. (15), with thevalues of C ( q ) indicated in the legends. renovating time, and τ f = ℓ f /u rms is the turnover time ofturbulent eddies. Note that Eqs. (36) and (37) are obtained forlarge magnetic Reynolds numbers, while Re ≪ τ /τ f < .This implies that for these conditions α ≫ γ . The latter is FIG. 3. α -effect as a function of Pm normalized by α = u rms forthe same runs as in Fig. 1. Analytical results according to Eq. (16)are overplotted with dotted lines. The values of C ( q ) are used as fitparameters and indicated in the legends. in agreement with DNS results (see Figs. 1 and 2).In Fig. 3 we show α -effect (the diagonal elements In Fig. 3we show the α -effect (the diagonal elements of the α ij tensor)as a function of the magnetic Prandtl number Pm . These re-sults are in a good agreement with the analytical results (16).
2. Dependence on Rm Our results for γ as a function of Rm are shown in Fig. 4.We find that for Rm smaller than roughly two, γ is well de-scribed by the analytical result, Eq. (14) obtained for Rm ≪ and Re ≪ . For greater Rm, γ is consistent with a constantvalue as a function of Rm, and is in accordance with Eq. (24)derived for Rm ≫ and Re ≫ . Note also that for the largest FIG. 4. γ as a function of Rm for Pm = 1 and for four values ofSh ( − . , − . , − . , and − . ; see Sets B1 to B4). Thelines show the analytical results according to Eqs. (14) and (24) with C ( q ) = 1 , for Sets B1 (dotted lines), B2 (dashed), B3 (dot-dashed),and B4 (triple-dot dashed), respectively.FIG. 5. Symmetric contribution α as a function of Rm for Pm = 1 and four values of shear as indicated by the legend (Sets B1 to B4). values of the shear parameter, Sh = − . ( − . ), there is avorticity dynamo for Rm > (Rm > ), so no points areplotted in those cases.The off-diagonal component α , shown in Fig. 5, is pro-portional to Re for small Rm, while the analytical expres-sion (15) yields α ∝ Re . A sign change occurs forRm ≈ , and the values of α are consistently negative inthis regime in agreement with Eq. (25) derived for Rm ≫ and Re ≫ . The data is noisy but suggest that α could beindependent of Rm at high Rm in an agreement with the ana-lytical result (25). Furthermore, for small Rm the dependenceon shear is weak, although a clearer dependence on shear isseen for Rm greater than around 10. FIG. 6. α -effect as a function of Rm normalized by α = u rms fortwo values of Re (Sets B1 and B2). The dotted line is proportionalto Rm. The inset shows the normalized kinetic helicity of the flow.FIG. 7. Pumping velocity γ = ( a − a ) normalized by α as afunction of Sh for Pm = 1 and different values of Rm as indicatedin the legend (Sets C1–C4). Analytical results according to Eqs. (14)with C ( q ) = 1 , and (24) are overplotted with dotted and dashedlines, respectively. In Fig. 6 we show α as a function of Rm. We find that α is proportional to Rm for small magnetic Reynolds numbersin agreement with Eq. (16). For Rm greater than roughly five, α decreases slightly, while the theory suggests that α is inde-pendent of Rm for Rm ≫ . This inconsistency can be un-derstood in terms of the relative kinetic helicity H / ( k f u ) ,where H = ω · u , which decreases by about 20 per cent be-tween Rm 8 and 83 (see the inset in Fig. 6). Since α ∝ H ,this explains the decrease of α with Rm for Rm ≫ . FIG. 8. Symmetric contribution α normalized by α as a functionof Sh for Pm = 1 and different values of Rm as indicated in thelegend (Sets C1–C4). Runs with Rm = 16 are shown with asterisksand connected by a dashed line. For these runs α < so the plotshows − α .FIG. 9. Pumping coefficient γ = ( a − a ) as a function of theMach number for Pm = 1 (Set D1). The normalization factor is α = u rms , and Sh = − . .
3. Dependence on shear
Figure 7 shows the pumping velocity γ normalized by α as a function of the shear number, Sh, for Pm = 1 and differ-ent values of Rm. Linear dependence of γ on shear is clearlyseen in Fig. 7. This is in agreement with the analytical re-sult of Eq. (14). Rather surprisingly, the data for α suggestthat there is no dependence on shear (Fig. 8), in contradictionwith the analytical result of Eq. (15) that was derived for smallshear, Sτ f ≪ .Note that our theory has been developed for incompressible flow since the DNS results are nearly independent of Machnumber for Ma < . . This is shown in Fig. 9, where we no-tice a sharp decline of γ for larger values of the Mach number.We are not aware of similar findings for mean-field transportcoefficients as a function of Mach number. IV. DISCUSSION AND CONCLUSIONS
To clarify the physical effect related to the pumping ve-locity, γ , and the off-diagonal components of the tensor α ij we rewrite the contributions to the mean electromotive forcewhich are proportional to the mean magnetic field in the fol-lowing form: E ( S ) i = α ij B j + ( γ × B ) i = h γ ( P ) × B ( P ) + γ ( T ) × B ( T ) i i , (50)where B ( T ) is the toroidal mean magnetic field directed alongthe mean shear velocity U (along the y axis), B ( P ) is thepoloidal mean magnetic field directed perpendicular to both,the mean shear velocity U and the mean vorticity (along the x axis), while the pumping velocities, γ ( T ) and γ ( P ) , of thetoroidal and poloidal components of the mean magnetic fieldare given by: γ ( P ) = ˆ z ( α + γ ) , (51) γ ( T ) = − ˆ z ( α − γ ) . (52)Here we take into account the following identities for the off-diagonal components of the tensor α ij = (ˆ x i ˆ y j + ˆ x j ˆ y i ) α and α ij B j = α ˆ z × ( B ( P ) − B ( T ) ) , where α = α and ˆ x , ˆ y , ˆ z are the unit vectors directed along x , y and z axes,respectively.It follows from these equations that, when α > γ > ,the effective pumping velocity of the poloidal mean magneticfield is directed upward (along the z axis), while the effec-tive pumping velocity of the toroidal mean magnetic field isdirected downward. When α < , but | α | > γ , the sit-uation is opposite, i.e., the effective pumping velocity of thetoroidal mean magnetic field is directed upward, while the ef-fective pumping velocity of the poloidal mean magnetic fieldis directed downward. Therefore, the effective pumping ve-locity, γ , as well as the off-diagonal components of the tensor α ij , result in a separation of toroidal and poloidal componentsof the mean magnetic field. This effect is very important forlarge-scale dynamo action in shear flow turbulence.Another reason for the different pumping velocity oftoroidal and poloidal components of the mean magnetic fieldis a combination of the effects of rotation and stratification onsmall-scale turbulence. The effect of the separation of toroidaland poloidal components of the mean magnetic field was earlyidentified in analytic calculations of rotating stratified turbu-lence in [26, 56], confirmed in DNS of rotating stratified con-vection [57, 58], and included in numerical mean-field mod-eling of the solar dynamo in [59]. Note also that a nonlinearfeedback of the mean magnetic field to turbulent fluid flowcauses a different pumping velocity of toroidal and poloidal0components of the mean magnetic field [9]. The latter effectwas included in numerical mean-field modeling of the solardynamo in [60].In summary, we have developed a mean-field theory for apumping effect of the mean magnetic field in homogeneoushelical turbulence with imposed large-scale shear. In our anal-ysis we use the quasi-linear approach, the path-integral tech-nique and tau-relaxation approximation, which allow us to de-termine all components of the α tensor in different ranges ofhydrodynamic and magnetic Reynolds numbers. The pump-ing effect depends on the α effect and on shear. Using DNSand the kinematic test-field method we were able to determineall components of the α tensor from numerical simulationsof sheared helical turbulence. The major part of the numer-ical results for the effective pumping velocity, the diagonaland off-diagonal components of the α tensor are in a goodagreement with the theoretical results. However, the numeri-cal results for α suggest that there is no dependence of theoff-diagonal component on shear in contradiction with the an-alytical result. In addition, according to the numerical results α ( Re ) is proportional to Re for small Rm, while the theoryyields α ∝ Re . On the other hand, the change of the signof α from positive for small Rm to negative for large Rmobserved in DNS is in agreement with the theoretical predic-tions. ACKNOWLEDGMENTS
Numerous illuminating discussions with AlexanderSchekochihin on the shearing waves approach are kindlyacknowledged. The numerical simulations were performedwith the supercomputers hosted by CSC – IT Center for Sci-ence in Espoo, Finland, who are administered by the FinnishMinistry of Education. Financial support from the Academyof Finland grant Nos. 136189, 140970, the Swedish ResearchCouncil grant 621-2007-4064, COST Action MP0806, andthe European Research Council under the AstroDyn ResearchProject 227952 are acknowledged. The authors acknowledgethe hospitality of NORDITA.
Appendix A: The integrals of the Green functions
For the integration in ω -space in the case of small magneticand hydrodynamic Reynolds numbers we used the followingintegrals in Eqs. (11) and (12): I ( k ) = Z G η G ν G ∗ ν dω = πν ( ν + η ) k ,I ( k ) = Z G η G ν G ∗ ν dω = π ν ( ν + η ) k ,I ( k ) = Z G η G ν ( G ∗ ν ) dω = π (5 ν + η )2 ν ( ν + η ) k ,I ( k ) = Z G η G ν ( G ∗ ν ) dω = π ν ( ν + η ) k × [2 ν ( ν + η ) + ( ν + η ) + 4 ν ] , I ( k ) = Z G η G ν ( G ∗ ν ) dω = π (3 ν + η )2 ν ( ν + η ) k ,I ( k ) = Z G η G ν G ∗ ν dω = π ν ( ν + η ) k ,I ( k ) = Z G η G ν G ∗ ν dω = π ν ( ν + η ) k ,I ( k ) = Z G η G ν G ∗ ν dω = πν ( ν + η ) k ,I ( k ) = Z G η G ν G ∗ ν dω = πν ( ν + η ) k . Appendix B: Derivation of Eqs. (26) and (30) in path-integralapproach
To derive Eq. (26) we use an exact solution of the inductionequation with an initial condition B ( t = t , x ) = B ( t , x ) in the form of the Feynman-Kac formula: B i ( t, x ) = h G ij ( t, t , ξ ) B j ( t , ξ ) i w , (B1)and assume that B j ( t , ξ ) = Z exp( i ξ · q ) B j ( t , q ) d q . (B2)Substituting Eq. (B2) into Eq. (B1) we obtain B i ( t, x ) = Z h G ij ( t, t , ξ ) exp[ i ˆ ξ · q ] i w × B j ( t , q ) exp( i q · x ) d q , (B3)where ˆ ξ = ξ − x . In Eq. (B3) we expand the function exp[ i ˆ ξ · q ] in Taylor series at q = 0 : exp( i ˆ ξ · q ) = ∞ X k =0 k ! ( i ˆ ξ · q ) k , and use the identity: ∇ k exp( i x · q ) = ( i q ) k exp( i x · q ) . This allows us to rewrite Eq. (B3) as follows: B i ( t, x ) = h G ij ( t, t , ξ ) h ∞ X k =0 k ! ( ˆ ξ · ∇ ) k i i w × Z B j ( t , q ) exp( i q · x ) d q . (B4)After the inverse Fourier transformation, B j ( t , x ) = R B j ( t , q ) exp( i q · x ) d q , in Eq. (B4) we obtain Eq. (26).Equation (B2) can be formally considered as an inverseFourier transformation of the function B j ( t , ξ ) . Equa-tion (26) has been also derived by a rigorous method, using theFeynman-Kac formula and Cameron-Martin-Girsanov theo-rem (see [47]).Averaging Eq. (26) over the random velocity field yields theequation for the mean magnetic field B i (( m + 1) τ, x ) = hh G ij ( t, s, ξ ) exp( ˆ ξ · ∇ ) ii w × B j ( mτ, x ) , (B5)1where the angular brackets h·i denote the ensemble averageover the random velocity field. Now we use the identity B i ( t + τ, x ) = exp (cid:18) τ ∂∂t (cid:19) B i ( t, x ) , (B6)which follows from the Taylor expansion f ( t + τ ) = ∞ X m =1 (cid:18) τ ∂∂t (cid:19) m f ( t ) = exp (cid:18) τ ∂∂t (cid:19) f ( t ) m ! . Therefore, Eqs. (B5)–(B6) yield exp (cid:18) τ ∂∂t (cid:19) B i ( t, x ) = ( G ij + G ij ξ m ∇ m + A ijm ∇ m + C ijmn ∇ m ∇ n ) B j ≡ exp( τ ˆ L ) B , (B7)where G ij = hh G ij ii w = δ ij + U i,j τ + O [( Sτ ) ] , ξ i = hh ˆ ξ i ii w = − U i τ + O [( Sτ ) ] , A ijm = hh ˆ ξ m G ij ii w ,C ijmn = hh ˆ ξ m ˆ ξ n G ij ii w , and we introduced the operator ˆ L, which allows us to reduce the integral equation (B5) to a par-tial differential equation. Indeed, Eq. (B7), which is rewrittenin the form exp (cid:20) τ (cid:18) ˆ L − ∂∂t (cid:19)(cid:21) B = B , (B8)reduces to ∂ B ∂t = ˆ L B . (B9)Taylor expansion of the function exp( τ ˆ L ) reads exp( τ ˆ L ) = ˆ E + τ ˆ L + ( τ ˆ L ) / ..., (B10)where ˆ E is the unit operator. Thus, Eqs. (B7) and (B10) yield ˆ L ≡ L ij = 1 τ ( G ij − δ ij + ξ m G ij ∇ m + A ijm ∇ m )+ D ijmn ∇ m ∇ n + O ( ∇ ) , (B11)where D ijmn = ( C ijmn − A ikm A kjn ) / τ . This yields Eq.(30). Appendix C: Orr-Kelvin random shearing waves for smallhydrodynamic Reynolds numbers
We explain here the details that led to the derivation ofEqs. (36) and (37). We seek the solutions of the linearizedEq. (2) for incompressible velocity field u as superpositionsof the Orr-Kelvin shearing waves: u ( t, r ) = Z u ( t, k ) exp[ i k ( t ) · r ] d k , (C1)(see, e.g., [23, 48–50]), where k = ( k x , k y , k z ) , k ( t ) =( k x − Sk y t, k y , k z ) and we neglected weak Lorentz force.The amplitudes of the shearing waves satisfy the followingequations: ∂u x ( t, k ) ∂t = (cid:20) S k y k x ( t ) k ( t ) − νk ( t ) (cid:21) u x ( t, k ) + f x , (C2) ∂u z ( t, k ) ∂t = 2 S k y k z k ( t ) u x ( t, k ) − νk ( t ) u z ( t, k ) + f z . (C3) These equations were obtained by taking twice curl of Eq. (2).Equations (C2) and (C3) have explicit solutions: u x ( t, k ) = 1 k ( t ) Z t dt ′ ˜ G ν ( t, t ′ ) k ( t ′ ) f x ( t ′ , k ) , (C4) u z ( t, k ) = u (1) z ( t, k ) + u (2) z ( t, k ) , (C5) u y ( t, k ) = − k y [ k x ( t ) u x ( t, k ) + k z u z ( t, k )] , (C6) u (1) z ( t, k ) = Z t dt ′ ˜ G ν ( t, t ′ ) f z ( t ′ , k ) , (C7) u (2) z ( t, k ) = 2 S k y k z Z t dt ′ ˜ G ν ( t, t ′ ) k ( t ′ ) u x ( t ′ , k ) , (C8)where ˜ G ν ( t, t ′ ) = exp h − ν R tt ′ dt ′′ k ( t ′′ ) i . Equations (C4)–(C8) for a white-in-time forcing yield the following formulasfor non-instantaneous two-point correlation functions: h u x ( t, k ) u ∗ (1) z ( t ′ , k ) i = ˜ G ν ( t, t ′ ) k ( t ′ ) k ( t ) × h u x ( t ′ , k ) u ∗ (1) z ( t ′ , k ) i , (C9) h u (1) z ( t, k ) u ∗ x ( t ′ , k ) i = ˜ G ν ( t, t ′ ) × h u (1) z ( t ′ , k ) u ∗ x ( t ′ , k ) i , (C10) h u x ( t, k ) u ∗ (2) z ( t ′ , k ) i = 2 S k y k z Z t ′ dt ′′ ˜ G ν ( t ′ , t ′′ ) k ( t ′′ ) × h u x ( t, k ) u ∗ x ( t ′′ , k ) i , (C11) h u (2) z ( t, k ) u ∗ x ( t ′ , k ) i = 2 S k y k z Z t dt ′′ ˜ G ν ( t, t ′′ ) k ( t ′′ ) × h u x ( t ′′ , k ) u ∗ x ( t ′ , k ) i , (C12)where for t ′′ < t ′ h u x ( t ′′ , k ) u ∗ x ( t ′ , k ) i = ˜ G ν ( t ′ , t ′′ ) k ( t ′′ ) k ( t ′ ) × h u x ( t ′′ , k ) u ∗ x ( t ′′ , k ) i , (C13)and for t ′′ > t ′ h u x ( t ′′ , k ) u ∗ x ( t ′ , k ) i = ˜ G ν ( t ′′ , t ′ ) k ( t ′ ) k ( t ′′ ) × h u x ( t ′ , k ) u ∗ x ( t ′ , k ) i . (C14)2 [1] H. K. Moffatt, Magnetic Field Generation in Electrically Con-ducting Fluids (Cambridge University Press, New York, 1978).[2] E. Parker,
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