A growing dynamo from a saturated Roberts flow dynamo
aa r X i v : . [ a s t r o - ph ] O c t Mon. Not. R. Astron. Soc. , 1–5 (2008) Printed 22 October 2018 (MN L A TEX style file v2.2)
A growing dynamo from a saturated Roberts flow dynamo
Andreas Tilgner and Axel Brandenburg Institute of Geophysics, University of G¨ottingen, Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany NORDITA, Roslagstullsbacken 23, SE - 106 91 Stockholm, Sweden
ABSTRACT
Using direct simulations, weakly nonlinear theory and nonlinear mean-field theory, it isshown that the quenched velocity field of a saturated nonlinear dynamo can itself actas a kinematic dynamo. The flow is driven by a forcing function that would produce aRoberts flow in the absence of a magnetic field. This result confirms an analogous find-ing by F. Cattaneo & S. M. Tobias (arXiv:0809.1801) for the more complicated case ofturbulent convection, suggesting that this may be a common property of nonlinear dy-namos; see also the talk given also online at the Kavli Institute for Theoretical Physics(http://online.kitp.ucsb.edu/online/dynamo c08/cattaneo). It is argued that this property canbe used to test nonlinear mean-field dynamo theories.
Key words: magnetic fields — MHD — hydrodynamics
The magnetic fields of many astrophysical bodies displays orderon scales large compared with the scale of the turbulent fluid mo-tions that are believed to generate these fields via dynamo action. Aleading theory for these types of dynamos is mean-field electrody-namics (Moffatt 1978; Krause & R¨adler 1980), which predicts theevolution of suitably averaged mean magnetic fields. Central to thistheory is the mean electromotive force based on the fluctuations ofvelocity and magnetic fields. This mean electromotive force is thenexpressed in terms of the mean magnetic field and its first derivativewith coefficients α ij and η ijk . The former represents the α effectand the latter the turbulent magnetic diffusivity.Under certain restrictions the coefficients α ij and η ijk canbe calculated using for example the first order smoothing approx-imation, which means that nonlinearities in the evolution equa-tions for the fluctuations are neglected. Whilst this is a valid ap-proach for small magnetic Reynolds numbers or short correla-tion times, it is not well justified in the astrophysically interest-ing case when the magnetic Reynolds number is large and the cor-relation time comparable with the turnover time. However, in re-cent years it has become possible to calculate α ij and η ijk us-ing the so-called test-field method (Schrinner et al. 2005, 2007).For the purpose of this paper we can consider this method essen-tially as a “black box” whose input is the velocity field and its out-put are the coefficients α ij and η ijk . This method has been suc-cessfully applied to the kinematic case of weak magnetic fieldsin the presence of homogeneous turbulence either without shear(Sur et al. 2008; Brandenburg, R¨adler & Schrinner 2008) or withshear (Brandenburg 2005; Brandenburg et al. 2008a).More recently, this method has also been applied to the non-linear case where the velocity field is modified by the Lorentz forceassociated with the dynamo-generated field (Brandenburg et al.2008b) In that case the test-field method consists still of the same black box, whose input is only the velocity field, but now this ve-locity field is based on a solution of the full hydromagnetic equa-tions comprising the continuity, momentum, and induction equa-tions. We emphasize that the magnetic field is quite independent ofthe fields that appear in the test-field method inside the black box.Our present work is stimulated by an interesting and relevantnumerical experiment performed recently by Cattaneo & Tobias(2008). They considered a solution of the full hydromagnetic equa-tions where the magnetic field is generated by turbulent convectivedynamo action and has saturated at a statistically steady value. Theythen used this velocity field and subjected it to an independent in-duction equation, which is equal to the original induction equationexcept that the magnetic field B is now replaced by a passive vec-tor field ˜ B , which does not react back on the momentum equation.Surprisingly, they found that | ˜ B | grows exponentially, even thoughthe velocity field is already quenched by the original magnetic field.One might have expected that, because the velocity is modi-fied such that it produces a statistically steady solution to the origi-nal induction equation, ˜ B should decay or also display statisticallysteady behavior. The argument sounds particularly convincing fortime-independent flows because, if a growing ˜ B were to exist, onewould expect this alternative field to grow and replace the initialfield. This view is supported by recent simulations in which theflow field from a geodynamo simulation in a spherical shell wasused as velocity field in kinematic dynamo computations, and nogrowing ˜ B was found (Tilgner 2008). However, it turns out thatthis reasoning is not correct in general. One finds counterexampleseven within the confines of mean field MHD using analytical tools.The existence of a growing ˜ B thus is not tied to chaotic flows orfluctuating small-scale dynamos.This finding of Cattaneo & Tobias (2008) is interesting inview of the applicability of the test-field method to the nonlinearcase. Of course, the equations used in the test-field method are dif- c (cid:13) A. Tilgner & A. Brandenburg ferent from the original induction equation. (The equations used inthe test-field method include an inhomogeneous term and the meanelectromotive force is subtracted, but they are otherwise similar tothe original induction equation.) Given the seemingly unphysicalbehaviour of the induction equation in the presence of a vector fielddifferent from the actual magnetic field, it would be tempting to ar-gue that one should choose test fields whose shape is rather closeto that of the actual magnetic field (Cattaneo & Hughes 2008). Onthe other hand, the α ij and η ijk tensors should give the correct re-sponse to all possible fields, not just the B field that grew out of aparticular initial condition, but also the passive ˜ B field that obeys aseparate induction equation. It is therefore important to choose a setof test fields that are orthogonal to each other, even if none of thefields are solutions of the induction equation. One goal of this paperis to show that the α ij and η ijk tensors obtained in this way pro-vide not only interesting diagnostics of the flow, but they are alsoable to explain the surprising result of Cattaneo & Tobias (2008) inthe context of a simpler example. However, let us begin by repeat-ing the numerical experiment of Cattaneo & Tobias (2008) usingthe simpler case of a Roberts flow. Next, we consider a weaklynonlinear analysis of this problem and turn then to its mean-fielddescription. In order to examine the possibility of a growing passive vector field,we first considered the case of a driven ABC flow. Such a flow isnon-integrable and has chaotic streamlines. Growing passive vec-tor fields were found. To simplify matters even further, we considernow the case of a Roberts flow, which is integrable, has non-chaoticstreamlines, and the dynamo can only be a slow one, i.e. the growthrate goes to zero in the limit of large magnetic Reynolds number.This is however not an issue here, because we will only be consid-ering finite values of the magnetic Reynolds number.In the following we consider both incompressible and isother-mal cases. The governing equations for any externally driven ve-locity field (turbulence, ABC flow, or Roberts flow) are then givenby ∂ U ∂t = − U · ∇ U − ∇ H + 1 ρ J × B + f + F visc , (1) ∂ B ∂t = ∇ × ( U × B ) + η ∇ B , (2)where U is the velocity, B is the magnetic field, ρ is the density, H is the specific enthalpy, J = ∇ × B /µ is the current density, µ is the vacuum permeability, f is the forcing function, F visc isthe viscous force per unit mass, and η = const is the magneticdiffusivity. In the incompressible case, ∇ · U = 0 , we have H = p/ρ , where p is the pressure and ρ = const . The viscous force isthen given by F visc = ν ∇ U . In the isothermal case, the densityobeys the usual continuity equation ∂ρ∂t = − ∇ · ( ρ U ) , (3)but now p = c ρ , where c s is the isothermal sound speed. In thatcase H = c ln ρ and the viscous force is given by F visc = ν ∇ U + ν ∇∇ · U + 2 ν S ∇ ln( ρν ) , (4)where S ij = ( U i,j + U j,i ) − δ ij ∇ · U is the traceless rate ofstrain matrix.In order to compute the evolution of an additional passive vec-tor field ˜ B we also solve the equation ∂ ˜ B ∂t = ∇ × ( U × ˜ B ) + η ∇ ˜ B . (5)In the case of the Roberts flow we use the forcing function f = νk U Rob , (6)where U Rob = k f ψ ˆ z − ˆ z × ∇ ψ (7)with ψ = ( u /k ) cos k x cos k y (8)and k f = √ k . We consider a domain of size L x × L y × L z . In allcases we consider L x = L y = L z = 2 π/k . Our model is charac-terized by the choice of fluid and magnetic Reynolds numbers thatare here based on the inverse wavenumber k ,Re = u /νk , R m = u /ηk . (9) We solve equations (1)–(5) for the isothermal case using the P EN - CIL C ODE , which is a high-order public domain code (sixthorder in space and third order in time) for solving partial dif-ferential equations. Equation (5) is solved using the test-fieldmodule with the input parameters lignore uxbtestm=T , and itestfield=’B=0’ , which means that the inhomogeneousterm of the test-field equation is set to zero and the subtractionof the mean electromotive force has been disabled. In this waywe solve equation (5), instead of the original test-field equation.We focus on the case of small fluid Reynolds number, Re = 0 . .The initial conditions for B and ˜ B are Beltrami fields, (cos( k z + ϕ ) , sin( k z + ϕ ) , , with an arbitrarily chosen phase ϕ = 0 . , butfor ˜ B we ϕ = 0 . In Fig. 1 we plot the evolution of the rms val-ues of B and ˜ B for a weakly supercritical case with R m = 6 . .(In our case with Re = 0 . the critical value for dynamo action is R m ≈ . ; for Re → the critical value would be R m ≈ . .)Both B and ˜ B grow at first exponentially at the same rate. How-ever, when B reaches saturation, the growth of ˜ B slows down tem-porarily, but then resumes to nearly its original value. This confirmsthe result of Cattaneo & Tobias (2008) for the much simpler caseof a Roberts flow.In Fig. 2 we compare horizontal cross-sections of the twofields. Note that the two are phase shifted in the z direction by aquarter wavelength. The short interval in Fig. 1 during which thegrowth of ˜ B slowed down temporarily is therefore related to thefact that the solution needed to “adjust” to this particular form. A weakly nonlinear analysis of the Roberts flow is presented inTilgner & Busse (2001). Two nonlinear terms enter the full dynamoproblem. In order to make the calculation analytically tractable, itis assumed that the fluid has infinite magnetic Prandtl number sothat the inertial terms (and hence the advection term) drop from theNavier-Stokes equation. The second nonlinear term, the Lorentzforce, is assumed to be small compared with the driving force f and is treated perturbatively. The linear induction equation is trans-formed into a mean field equation assuming separation of length c (cid:13) , 1–5 growing dynamo from a saturated Roberts flow dynamo Figure 1.
Evolution of the rms values of B and ˜ B for R m = 6 . . Thegrowth rate of ˜ B is . u k in the kinematic phase and . u k inthe nonlinear phase. scales and small magnetic Reynolds numbers. One can under theseassumptions compute the modifications of the velocity field U Rob due to the presence of a mean field B , which we define here as B ( z, t ) = Z L y Z L x B ( x, y, z, t ) d x d y/L x L y . (10)The magnetically modified velocity field then becomes U = (1 − γ ) U Rob + 2 γ B x B y B x + B y ˆ U (11)with ˆ U = u sin k x cos k y − cos k x sin k y √ k x sin k y ! (12)and γ = ( B x + B y ) / (2 ηνk ρµ ) . The original flow U Rob isreduced in magnitude and another 2D periodic flow component isadded. The mean-field induction equation with flow U given by(11), written for a passive vector field ˜ B , is: ∂ ˜ B ∂t + ∇ × A ˜ B x ˜ B y ! − ∇ × CB x B y ˜ B y ˜ B x ! = η ∇ ˜ B (13)with A = ˜ R m v and C = ˜ R P m v ρµ , (14)where ˜ R m = v ηk f , P m = νη , and v = u (1 − γ ) (15)have been introduced. This equation corresponds to equation (10)of Tilgner & Busse (2001). Apart from a change of notation, thedistinction between the passive vector field ˜ B and the field distort-ing the Roberts flow, B , has been made. In addition, equation (10)of Tilgner & Busse (2001) was intended as a model of the Karls-ruhe dynamo and used u as control parameter, whereas here, weconsider f as given. For this reason, the two equations are identicalonly to first order in γ .Equation (13) reduces to the usual dynamo problem for ˜ B = B and leads to the kinematic dynamo problem if it is further-more linearized in B , which corresponds to dropping the third Figure 2.
Grey-scale representations of horizontal cross-sections of B and ˜ B for R m = 6 . . Here, k z ≈ − . . Both fields are scaled symmetri-cally around zero (grey shade) with dark shades indicating negative valuesand light shades indicating positive vales, as indicated on the greyscale bar. term in Eq. (13). For u > η k f k , the equation has growing so-lutions of the form (cos kz, sin kz, . Weakly nonlinear analysisdetermines the amplitude B of a saturated solution of the form B = B (cos kz, sin kz, by inserting this ansatz for B and ˜ B = B into equation (13) and by projecting equation (13) onto (cos kz, sin kz, and integrating spatially over a periodicity cell(Tilgner & Busse 2001).We proceed similarly to find growing solutions of equa-tion (13). Assume B = B (cos kz, sin kz, . Obvious candidatesfor passive vector fields growing at rate p are ˜ B ∝ e pt (cos( kz + ϕ ) , sin( kz + ϕ ) , . Since the third term in equation (13) is a per-turbation, a growing solution must be of a form such that the otherterms maximize the growth rate. These other terms are identical tothe kinematic problem, so that ˜ B must have the same general formas the kinematic dynamo field except for the phase shift ϕ which c (cid:13) , 1–5 A. Tilgner & A. Brandenburg measures the phase angle between the saturated field B and thepassive vector field ˜ B . The velocity field U Rob is independent of z ,so that any solution of the kinematic dynamo problem remains a so-lution after translation along z . However, neither the Lorentz forcedue to B nor the flow modified by that Lorentz force are indepen-dent of z , so that the phase angle ϕ matters for ˜ B . The above ansatzfor ˜ B will not be an exact solution of equation (13) because of the z -dependence of B x B y , but it represents the leading Fourier com-ponent. In order to determine the optimal ϕ , we insert this ansatzinto equation (13), project onto (cos( kz + ϕ ) , sin( kz + ϕ ) , andintegrate over a periodicity cell to find p = − ηk + Ak − CB k
14 cos 2 ϕ. (16)The saturation amplitude B is determined from this equation bysetting p = 0 and ϕ = 0 . For any given B , the maximum of p is obtained for ϕ = π/ . The fastest growing mean passive vectorfield is thus expected to have the same form as the mean dynamofield except for a translation by a quarter wavelength along the z -axis. This is in agreement with the simulation results of Sect. 3.The weakly nonlinear analysis in summary delineates the fol-lowing physical picture: As detailed in Tilgner & Busse (2001), thesaturated dynamo field modifies the flow in two different ways.Firstly, it reduces the amplitude of U Rob by the factor − γ , andsecondly, it introduces a new set of vortices which lead to the thirdterm in equation (13). The reduction of the amplitude of the Robertsflow affects all magnetic mean fields with a spatial dependence in (cos( kz + ϕ ) , sin( kz + ϕ ) , in the same manner, independentlyof ϕ . The additional vortices, however, have a quenching effect onthe field that created them (e.g. ϕ = 0 ) but are amplifying for afield shifted by ϕ = π/ with respect to the saturated field.We were able to find a simple growing passive vector fieldthanks to the periodic boundary conditions in z . The same construc-tion is impossible for vacuum boundaries at z = 0 and z = 2 π/k .Numerical simulations of the Roberts dynamo with vacuum bound-aries, not reported in detail here, have revealed that growing ˜ B ex-ist in this geometry nonetheless, but they bear a more complicatedrelation with B than a simple translation. At present, the flow ofthe convection driven dynamo in a spherical shell used in Tilgner(2008) seems to be the only known example of a dynamo whichdoes not allow for growing ˜ B . In mean-field dynamo theory for a flow such as the Roberts flowone solves an equation for the horizontally averaged mean field, asdefined in equation (10). The mean electromotive force is definedas E = u × b , where u = U − U and b = B − B are thefluctuating components of magnetic and velocity fields. The meanelectromotive force can be expressed in terms of the mean fields as E i = α ij B j − η ij J j , (17)where we have used the fact that for mean fields that depend onlyon one spatial coordinate one can express all first derivatives of thecomponents of B in terms of those of J alone.The tensorial forms of α ij and η ij are ignored in many mean-field dynamo applications, but here their tensorial forms turn out tobe of crucial importance. Of course, there is always the anisotropywith respect to the z direction, but this is unimportant in our one-dimensional mean field problem, because of solenoidality of B and J and suitable initial conditions on B such that B z = J z = 0 . However, the dynamo-generated magnetic fields will introduce ananisotropy in the x and y directions. If B is the only vector givinga preferred direction to the system, the α ij and η ij tensors must beof the form α ij = α ( B ) δ ij + α ( B ) ˆ B i ˆ B j + ..., (18) η ij = η ( B ) δ ij + η ( B ) ˆ B i ˆ B j + ..., (19)where ˆ B = B / | B | is the unit vector of the dynamo-generatedmean magnetic field, and dots indicate the presence of terms relatedto the anisotropy in the z direction inherent to the Roberts flow. Asindicated above, equation (18) is correct without these terms onlyin the ( x, y ) plane. However, the terms represented by the dots donot enter the considerations below because we are only interestedin fields with B z = 0 .In order to predict the evolution of ˜ B in the saturated state,we need to know the effect on α ij (and in principle also on η ij ,but η is small; see below). Thus, we now need to know B . Themean magnetic field generated by the Roberts flow is a force-freeBeltrami field of the form B = (cos k z, sin k z, , (20)so ˆ B i ˆ B j = cos k z cos k z sin k z k z sin k z sin k z
00 0 0 ! . (21)The coefficients α , α , η , and η have previously been deter-mined for the case of homogeneous turbulence (Brandenburg et al.2008b) and it turned out at α and α have opposite sign, and that η is negligible. This is also true in the present case, for which wehave determined α /u = − . , α /u = +0 . , η k /u =0 . , and η k /u = 0 . . The microscopic value of η is 0.160,so the steady state condition, α + α + ( η + η + η ) k = 0 ,is obeyed. In the kinematic regime we have α /u = − . , η k /u = 0 . , with α = η = 0 , resulting in a positivegrowth rate of . u k . Thus, even though α increases in thiscase, the sum α + α is being quenched. This, together with theincrease of η + η , leads to saturation of B .Returning now to the mean-field problem for ˜ B , this too willbe governed by the same α ij and η ij tensors, but now the tensor ˆ B i ˆ B j is fixed and independent of ˜ B . The solution for ˜ B will be onethat maximizes the growth, so it must experience minimal quench-ing. Such a solution is given by that eigenvector of ˆ B i ˆ B j that min-imizes the quenching of ˜ B . In the case of our Beltrami field (18),the minimizing eigenvector is given by ˜ B = (sin k z, − cos k z, , (22)which satisfies ˆ B i ˆ B j ˜ B j = 0 . This is indeed the same result thatwe found both numerically and using weakly nonlinear theory. Thegrowth rate of ˜ B is then expected to be | α | k − ( η + η ) k =0 . u k , which is indeed positive, but it is somewhat bigger thanthe one seen in Fig. 1.Let us emphasize once more that by determining the full α ij and η ij tensors in the nonlinear case, we have been able to predictthe behavior of the passive vector field as well. This adds to the We note that the sign of α is opposite to the sign of the kinetic helicity,but since the Roberts flow has positive helicity, α must be negative, whichis indeed the case. c (cid:13) , 1–5 growing dynamo from a saturated Roberts flow dynamo credence of the test-field method in the nonlinear case, and con-firms that the test-fields can well be very different from the actualsolution.The considerations above suggest that solutions to the passivevector equation equation (5) can be used to provide an independenttest of proposed forms of α quenching. Isotropic formulations of α quenching would not reproduce the growth of a passive vector field,and so such quenching expressions can be ruled out, even thoughthe resulting electromotive force for B would be the same. We sug-gest therefore that the eigenvalues and eigenvectors of equation (5)with a velocity field from a saturated dynamo can be used to charac-terize the quenching of dynamo parameters ( α effect and turbulentdiffusivity) and thereby to test proposed forms of α quenching. The most fundamental question of dynamo theory beyond kine-matic dynamo theory is “how do magnetic fields saturate?”. In thesimplest picture, the velocity field reorganizes in response to theLorentz force such that all magnetic fields decay except one whichhas zero growth rate and which is the one we observe. This pic-ture is already questionable for chaotic dynamos. In a chaotic sys-tem, nearby initial conditions lead to exponentially separating timeevolutions. If one takes a magnetic field B with (on time average)zero growth rate which is the saturated solution of a chaotic dy-namo, and solves the kinematic dynamo problem for a passive vec-tor field ˜ B with initial conditions different from B , one is preparedto find growing ˜ B . Examples for growing ˜ B in chaotic dynamoshave been given by Cattaneo & Tobias (2008).For a time-independent saturated dynamo, on the other hand,the simple picture seems to be adequate at first. However, wehave shown in this paper that growing ˜ B also exist in the time-independent Roberts dynamo. The origin of the growing ˜ B can inthis case be understood with the help of weakly nonlinear theory.The growing ˜ B has the same shape as the saturated dynamo fieldbut is translated in space.What was wrong with the naive intuition invoked above? Itwas based on a stability argument (the equilibrated magnetic fieldshould be replaced by ˜ B if there is a growing ˜ B ). However, thelinear stability problem for a solution of the full dynamo equationsis different from the kinematic dynamo problem for ˜ B , because inthe latter, the velocity field is fixed. Both problems are closely re-lated eigenvalue problems, but standard mathematical theorems donot provide us with a relation between the spectra of both prob-lems. The numerical computation above gives an example of a sta-ble Roberts dynamo, showing that the linear stability problem forthe solution found there has only negative eigenvalues. Solving thesame eigenvalue problem with velocity fixed, which is the kine-matic problem for ˜ B , can very well lead to positive eigenvalues,and indeed, it does. The dynamo is therefore only stable becausethe velocity field is able to adjust to perturbations in the magneticfield. The magnetic field on its own is unstable. ACKNOWLEDGMENTS
We acknowledge the Kavli Institute for Theoretical Physics for pro-viding a stimulating atmosphere during the program on dynamotheory. This work has been initiated through discussions with SteveM. Tobias and Fausto Cattaneo. We thank Eric G. Blackman, K.-H. R¨adler, and Kandaswamy Subramanian for comments on the pa-per. This research was supported in part by the National ScienceFoundation under grant PHY05-51164.
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