Scale dependence of alpha effect and turbulent diffusivity
aa r X i v : . [ a s t r o - ph ] J a n Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018October 22, 2018
Scale dependence of alpha effect and turbulent diffusivity
A. Brandenburg , K.-H. R¨adler , and M. Schrinner NORDITA, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden Astrophysical Institute Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany Max-Planck-Institut f¨ur Sonnensystemforschung, 37191 Katlenburg-Lindau, GermanyOctober 22, 2018
ABSTRACT
Aims.
To determine alpha e ff ect and turbulent magnetic di ff usivity for mean magnetic fields with profiles of di ff erent length scalefrom simulations of isotropic turbulence, and to relate these results to nonlocal formulations in which alpha and the turbulent magneticdi ff usivity correspond to integral kernels. Methods.
A set of evolution equations for magnetic fields is solved which gives the response to imposed test fields, that is, meanmagnetic fields with various wavenumbers. Both an imposed fully helical steady flow consisting of a pattern of screw-like mo-tions (Roberts flow) and time-dependent statistically steady isotropic turbulence are considered. In the latter case the aforementionedevolution equations are solved simultaneously with the momentum and continuity equations. The corresponding results for the elec-tromotive force are used to calculate alpha and magnetic di ff usivity tensors. Results.
For both the Roberts flow under the second–order correlation approximation and isotropic turbulence alpha and turbulentmagnetic di ff usivity are largest at large scales and these values diminish toward smaller scales. In both cases the alpha e ff ect and tur-bulent di ff usion kernels are well approximated by exponentials, corresponding to Lorentzian profiles in Fourier space. For isotropicturbulence the turbulent di ff usion kernel is half as wide as the alpha e ff ect kernel. For the Roberts flow beyond the second–ordercorrelation approximation the turbulent di ff usion kernel becomes negative at large scales. Key words.
Magnetohydrodynamics (MHD) – Turbulence
1. Introduction
Stars and galaxies harbour magnetic fields whose scales are largecompared with the scale of the underlying turbulence. This phe-nomenon is successfully explained in terms of mean–field dy-namo theory discussed in detail in a number of text books and re-views (e.g. Mo ff att 1978, Krause & R¨adler 1980, Brandenburg &Subramanian 2005a). In this context velocity and magnetic fieldsare split into large–scale and small–scale components, U = U + u and B = B + b , respectively. The crucial quantity of the the-ory is the mean electromotive force caused by the small–scalefields, E = u × b . In many representations it is discussed un-der strongly simplifying assumptions. Often the relationship be-tween the mean electromotive force and the mean magnetic fieldis tacitly taken as (almost) local and as instantaneous, that is, E in a given point in space and time is considered as determinedby B and its first spatial derivatives in this point only, and thepossibility of a small–scale dynamo is ignored. Then the meanelectromotive force is given by E i = α i j B j + η i jk ∂ B j /∂ x k (1)with two tensors α i j and η i jk . If the turbulence is isotropic thetwo tensors are isotropic, too, that is α i j = αδ i j and η i jk = η t ǫ i jk with two scalar coe ffi cients α and η t . Then the expression (1)simplifies to E = α B − η t J , (2)where we have denoted ∇ × B simply by J (so that J is µ timesthe electric current density, where µ is the magnetic permeabil-ity of free space). The coe ffi cient α is, unlike η t , only non–zero if the turbulence lacks mirror–symmetry. The coe ffi cient η t is re-ferred to as the turbulent magnetic di ff usivity.In general, the mean electromotive force has the form E = E + K ◦ B , (3)where E stands for a part of E that is independent of B , and K ◦ B denotes a convolution in space and time of a kernel K with B (see, e.g., Krause & R¨adler 1980, R¨adler 2000, R¨adler &Rheinhardt 2007). Due to this convolution, E in a given point inspace and time depends on B in a certain neighborhood of thispoint, with the exception of future times. This corresponds to amodification of (1) such that also higher spatial and also timederivatives of B occur.In this paper we ignore the possibility of coherent e ff ectsresulting from small–scale dynamo action and put therefore E equal to zero. For the sake of simplicity we assume further theconnection between E and B to be instantaneous so that the con-volution K ◦ B refers to space coordinates only. The memorye ff ect, which we so ignore, has been studied previously by solv-ing an evolution equation for E (Blackman & Field 2002).For homogeneous isotropic turbulence we may then write,analogously to (2), E = ˆ α ◦ B − ˆ η t ◦ J , (4)or, in more explicit form, E ( x ) = Z h ˆ α ( ξ ) B ( x − ξ ) − ˆ η t ( ξ ) J ( x − ξ ) i d ξ (5) A. Brandenburg et al.: Scale dependence of alpha e ff ect with two functions ˆ α and ˆ η t of ξ = | ξ | that vanish for large ξ . Theintegration is in general over all ξ –space. Although E and B aswell as ˆ α and ˆ η t may depend on time the argument t is droppedeverywhere. For a detailed justification of the relations (4) and(5) we refer to Appendix A. In the limit of a weak dependence of B and J on space coordinates, i.e. when the variations of B ( x − ξ )and J ( x − ξ ) with ξ are small in the range of ξ where ˆ α ( ξ ) andˆ η t ( ξ ) are markedly di ff erent from zero, the relations (4) or (5)turn into (2), and we see that α = R ˆ α ( ξ ) d ξ and η t = R ˆ η t ( ξ ) d ξ .At first glance the representations (4) and (5) of E look ratherdi ff erent from (3). Considering J = ∇ × B and carrying out anintegration by parts we may however easily rewrite (5) into E i ( x ) = Z K i j ( ξ ) B j ( x − ξ ) d ξ (6)with K i j ( ξ ) = ˆ α ( ξ ) δ i j − ξ ∂ ˆ η t ( ξ ) ∂ξ ǫ i jk ξ k . (7)We further note that due to the symmetry of ˆ α ( ξ ) in ξ only thepart of B ( x − ξ ) that is symmetric in ξ , i.e. the part that can bedescribed by B ( x ) and its derivatives of even order, contributesto the ˆ α terms in (5) or in (6) and (7). The symmetry of ˆ η t ( ξ )implies that only the part of B ( x − ξ ) antisymmetric in ξ , whichcorresponds to the derivatives of B ( x ) of odd order, contributesto the ˆ η t terms.Finally, referring to a Cartesian coordinate system ( x , y , z )we define mean fields by averaging over all x and y so that inparticular E and B depend only on z and on time. Then (5) turnsinto E ( z ) = Z h ˆ α ( ζ ) B ( z − ζ ) − ˆ η t ( ζ ) J ( z − ζ ) i d ζ . (8)The functions ˆ α ( ζ ) and ˆ η t ( ζ ) are just averages of ˆ α ( ξ ) and ˆ η t ( ξ )over all ξ x and ξ y . They are therefore real and symmetric in ξ z ≡ ζ . The integration in (8) is in general over all ζ . The remark onthe limit of weak dependences of B and J on space coordinatesmade above in connection with (4) and (5) applies analogouslyto (8). We have now α = R ˆ α ( ζ ) d ζ and η t = R ˆ η t ( ζ ) d ζ .Relation (8) can also be brought in a form analogous to (6)and (7), E i ( z ) = Z K i j ( ζ ) B j ( z − ζ ) d ζ (9)with K i j ( ζ ) = ˆ α ( ζ ) δ i j − ∂ ˆ η t ( ζ ) ∂ζ ǫ i j . (10)The remarks made under (6) and (7) apply, now due to the sym-metries of ˆ α ( ζ ) and ˆ η t ( ζ ) in ζ , analogously to (8), (9) and (10).It is useful to consider in addition to (8) also the correspond-ing Fourier representation. We define the Fourier transformationin this paper by Q ( z ) = R ˜ Q ( k ) exp(i kz ) d( k / π ). Then this repre-sentation reads˜ E ( k ) = ˜ α ( k ) ˜ B ( k ) − ˜ η t ( k ) ˜ J ( k ) . (11)Both ˜ α ( k ) and ˜ η t ( k ) are real quantities, and they are symmetric in k . The limit of weak dependences of B and J on z correspondshere to k →
0, and we have α = ˜ α (0) and η t = ˜ η t (0). Detailedanalytic expressions for ˆ α ( ζ ) and ˆ η t ( ζ ), or ˜ α ( k ) and ˜ η t ( k ), can be derived, e.g., from results presented in Krause & R¨adler (1980).A numerical determination of quantities corresponding to ˆ α ( ζ )and ˆ η t ( ζ ) has been attempted by Brandenburg & Sokolo ff (2002)for shear flow turbulence.In this paper two specifications of the velocity field u will beconsidered. In the first case u is chosen such that it correspondsto a steady Roberts flow, which is periodic in x and y and inde-pendent of z . A mean–field theory of a magnetic field in fluidflows of this type, that are of course di ff erent from genuine tur-bulence, has been developed in the context of the Karlsruhe dy-namo experiment (R¨adler et al. 2002a,b, R¨adler & Brandenburg2003). It turned out that the mean electromotive force E , exceptits z component, satisfies relation (2) if any nonlocality in theabove sense is ignored (see also Appendix B). Several analyticaland numerical results are available for comparison with those ofthis paper. In the second case u is understood as homogeneous,isotropic, statistically steady turbulence, for which the above ex-planations apply immediately. Employing the method developedby Schrinner et al. (2005, 2007) we will in both cases numeri-cally calculate the functions ˜ α ( k ) and ˜ η t ( k ) as well as ˆ α ( ζ ) andˆ η t ( ζ ).
2. The method
We first relax the assumption of isotropic turbulence used in theSect. 1 (but will later return to it). We remain however with thedefinition of mean fields by averaging over all x and y . Then, asalready roughly indicated above, B x and B y may only depend on z and time but B z , because of ∇ · B =
0, must be independentof z . Furthermore all first–order spatial derivatives of B can beexpressed by the components of ∇ × B , that is, of J , where J z =
0. Instead of (8) we have then E i ( z ) = Z h ˆ α i j ( ζ ) B j ( z − ζ ) − ˆ η i j ( ζ ) J j ( z − ζ ) i d ζ (12)and instead of (11)˜ E i ( k ) = ˜ α i j ( k ) ˜ B j ( k ) − ˜ η i j ( k ) ˜ J j ( k ) , (13)with real ˆ α i j ( ζ ) and ˆ η i j ( ζ ), which are even in ζ , and real ˜ α i j ( k )and ˜ η i j ( k ), which are even in k . A justification of these relationsis given in Appendix A. We have further˜ α i j ( k ) = Z ˆ α i j ( ζ ) cos k ζ d ζ , ˜ η i j ( k ) = Z ˆ η i j ( ζ ) cos k ζ d ζ . (14)Since J = η i , as well as the ˜ η i , are of no interest.In the following we restrict attention to E x and E y and assumethat B z is equal to zero. We note that E z and the contributions of B z to E x and E y are anyway without interest for the mean–fieldinduction equation, which contains E only in the form ∇ × E ,that is, they do not a ff ect the evolution of B . We may formulatethe above restriction in a slightly di ff erent way by saying that weconsider in the following E i , α i j and η i j as well as ˜ E i , ˜ α i j and ˜ η i j only for 1 ≤ i , j ≤ x and y only (and not on z ) we refer to the aforemen-tioned studies (R¨adler et al. 2002a,b, R¨adler & Brandenburg2003). Following the ideas explained there we may conclude thatˆ α i j ( ζ ) = ˆ α ( ζ ) δ i j and ˆ η i j ( ζ ) = ˆ η t ( ζ ) δ i j with functions ˆ α and ˆ η t of . Brandenburg et al.: Scale dependence of alpha e ff ect 3 ζ , and analogously ˜ α i j ( k ) = ˜ α ( k ) δ i j and ˜ η i j ( k ) = ˜ η t ( k ) δ i j withfunctions ˜ α and ˜ η t of k , all for 1 ≤ i , j ≤
2. For obvious reasonsthe same is true for homogeneous isotropic turbulence.
We calculate the ˜ α i j ( k ) and ˜ η i j ( k ), or ˜ α ( k ) and ˜ η ( k ), numericallyby employing the test field method of Schrinner et al. (2005,2007). It has been originally developed for the calculation of thefull α and η tensors [in the sense of (1)] for convection in a spher-ical shell. Brandenburg (2005) employed this method to obtainresults for stratified shear flow turbulence in a local cartesian do-main using the shearing sheet approximation. More recently, Suret al. (2008) calculated in this way the dependences of α and η t for isotropic turbulence on the magnetic Reynolds number, andBrandenburg et al. (2008) have calculated the magnetic di ff usiv-ity tensor for rotating and shear flow turbulence. However, in allthese cases no nonlocality in the connection between E and B has been taken into account.Following the idea of Schrinner et al. we first derive expres-sions for E with several specific B , which we call “test fields”.We denote the latter by B pq and define B = B (cos kz , , , B = B (0 , cos kz , , B = B (sin kz , , , B = B (0 , sin kz , , (15)with any constant B and any fixed value of k . We then replace B and J in (12) by B p c and ∇ × B p c or by B p s and ∇ × B p s .Denoting the corresponding E by E p c or by E p s , respectively,and using (14) we find E p c i ( z ) = B h ˜ α ip ( k ) cos kz − ˜ η † ip ( k ) k sin kz i , E p s i ( z ) = B h ˜ α ip ( k ) sin kz + ˜ η † ip ( k ) k cos kz i , (16)for 1 ≤ i , p ≤
2, where˜ η † ip = ˜ η il ǫ lp = (cid:18) − ˜ η ˜ η − ˜ η ˜ η (cid:19) . (17)From this we conclude˜ α i j ( k ) = B − (cid:20) E j c i ( z ) cos kz + E j s i ( z ) sin kz (cid:21) , ˜ η † i j ( k ) = − ( kB ) − (cid:20) E j c i ( z ) sin kz − E j s i ( z ) cos kz (cid:21) (18)for 1 ≤ i , j ≤ α i j and ˜ η i j if the E pqi with 1 ≤ i , p ≤ q = c and q = s are known. Inpreparing the numerical calculation we start from the inductionequation. Its uncurled form reads ∂ A ∂ t = U × B − η J , (19)where A is the magnetic vector potential, B = ∇ × A , and J = ∇ × B . Here the Weyl gauge of A is used. Taking the average of(19) we obtain ∂ A ∂ t = U × B + u × b − η J . (20) The notation used here di ff ers slightly from that in Brandenburg etal. (2008), where first test fields B p were introduced and only later thetwo versions B p c and B p s are considered. From (19) and (20) we conclude ∂ a ∂ t = U × b + u × B + u × b − u × b − η j , (21)where a = A − A and j = J − J = ∇ × b .For the calculation of the E pq we are interested in the b pq = ∇ × a pq which occur in response to the test fields B pq . Specifying(21) in that sense we obtain ∂ a pq ∂ t = U × b pq + u × B pq + u × b pq − u × b pq − η j pq . (22)Equations of this type are called “test field equations”.So far no approximation such as the second order correla-tion approximation (SOCA), also known as first order smooth-ing approximation, has been made. If we were to make this as-sumption, terms that are nonlinear in the fluctuations would beneglected and (22) would simplify to ∂ a pq ∂ t = U × b pq + u × B pq − η j pq (for SOCA only) . (23)In the following SOCA results will be shown in some cases forcomparison only.In either of the two cases the ˜ α i j and ˜ η i j are to be calculatedfrom E pq = u × b pq . More details of the numerical calculationsof the E pq will be given below in Sect. 2.4.Returning once more to (18) we note that the E pq dependon both k and z introduced with the B pq . As a consequence ofimperfect simulations of the turbulence they may also dependon the time t . The ˜ α i j and ˜ η i j however should depend on k butno longer on z and t . We remove the latter dependences of ourresults by averaging ˜ α i j and ˜ η i j over z and t . For the Roberts flowthere should be no such z or t dependences.The relations (18) allow the determination of all componentsof ˜ α i j and ˜ η i j with 1 ≤ i , j ≤
2. We know already that ˜ α i j = ˜ αδ i j and ˜ η i j = ˜ η t δ i j , that is, ˜ α = ˜ α = ˜ α , ˜ η = ˜ η = ˜ η and˜ α = ˜ α = ˜ η = ˜ η =
0. We may therefore determine ˜ α and˜ η t according to ˜ α = ˜ α and ˜ η t = ˜ η by using the two test fields B q and the relations (18) with i = j = We consider here a special form of a steady flow which, inview of its dynamo action, has already been studied by Roberts(1972). It has no mean part, U = , and u is given by u = − ˆ z × ∇ ψ + k f ψ ˆ z , (24)where ψ = ( u / k ) cos k x cos k y , k f = √ k (25)with some constant k . The flow is fully helical, ∇ × u = k f u .The component form of u as defined by (24) and (25) reads u = u (cid:16) − cos k x sin k y , sin k x cos k y , √ k x cos k y (cid:17) . (26)We note that u = u . In the corresponding expression (27) of Brandenburg (2005) the U term is incorrect. This did not a ff ect his results because U =
0. A. Brandenburg et al.: Scale dependence of alpha e ff ect Next, we consider isotropic, weakly compressible turbulenceand use an isothermal equation of state with constant speed ofsound, c s . Considering first the full velocity field U = U + u wethus accept the momentum equation in the form ∂ U ∂ t = − U · ∇ U − c ∇ ln ρ + f + ρ − ∇ · ρν S , (27)where f is a random forcing function consisting of circularly po-larized plane waves with positive helicity and random direction,and S i j = ( U i , j + U j , i ) − δ i j ∇ · U is the traceless rate of strain ten-sor. The forcing function is chosen such that the magnitude of thewavevectors, | k f | , is in a narrow interval around an average value,which is simply denoted by k f . The corresponding scale, k − , isalso referred to as the outer scale or the energy-carrying scale ofthe turbulence. More details concerning the forcing function aregiven in the appendix of Brandenburg & Subramanian (2005b).With the intention to study the mean electromotive force in thepurely kinematic limit the Lorentz force has been ignored.In addition to the momentum equation we use the continuityequation in the form ∂ ln ρ∂ t = − U · ∇ ln ρ − ∇ · U . (28)In all simulations presented in this paper the strength of theforcing is adjusted such that the flow remains clearly subsonic,that is, the mean Mach number remains below 0.2. Hence forall practical purposes the flow can be considered incompressible(Dobler et al. 2003). In these simulations no mean flow develops,that is U = , so U = u . The relevant equations are solved in a computational domain ofsize L × L × L using periodic boundary conditions. In the caseof the Roberts flow (26) we fix L by L = π/ k . Only four of thetest field equations (22) (those with p = q = c and q = s) aresolved numerically. With turbulence in the kinematic regime thefour equations (27) and (28) for U and ln ρ are solved togetherwith four of the test field equations (22) (again with p = q = cand q = s).Due to the finiteness of the domain in z direction andthe periodic boundary conditions, quantities like E and B have to be considered as functions that are periodic in z .The Fourier integrals used for representing these quantities, Q ( z ) = R ˜ Q ( k ) exp(i kz ) d( k / π ), turn into Fourier series, Q ( z ) = P Q n exp(i k n z ) / L , where k n = π n / L and the summation is over n = , ± , ± , . . . . By this reason only discrete values of k , thatis k = k n , are admissible in (13)–(18). In this framework we maydetermine the ˜ α and ˜ η t only for these k n .As explained above, the test field procedure yields ˜ α and˜ η t not as functions of k alone. They may also show some de-pendence on z and t . After having averaged over z , time aver-ages are then taken over a suitable stretch of the full time serieswhere these averages are approximately steady. We use the timeseries further to calculate error bars as the maximum departurebetween these averages and the averages obtained from one ofthree equally long subsections of the full time series.In all cases the simulations have been carried out using theP encil C ode which is a high-order finite-di ff erence code (sixth order in space and third order in time) for solving the compress-ible hydromagnetic equations together with the test field equa-tions. In the case of the Roberts flow, of course, only the test fieldequations are being solved.
3. Results
Let us first recall some findings of earlier work, which are pre-sented, e.g., by R¨adler (2002a,b). We use here the definitions α = − u , η t0 = u / k f , R m = u /η k f . (29)Adapting the results of analytic calculations in the frameworkof SOCA to the assumptions and notations of this paper (seeAppendix B) we have α/α = η t /η t0 = R m . (30)Moreover, in the general case, also beyond SOCA, it was foundthat α = α R m φ ( √ R m ) (31)with a function φ satisfying φ (0) = α and η t obtained both by generaltest field calculations using (22) and under the restriction toSOCA using (23). These results for α agree completely withboth (30) and (31), and those for η t agree completely with(30). Unfortunately we have no analytical results for η t beyondSOCA.Proceeding now to the ˜ α ( k ) and ˜ η t ( k ) we first note that inSOCA, as shown in Appendix C,˜ α ( k ) = α R m + ( k / k f ) , ˜ η t ( k ) = η t0 R m + ( k / k f ) . (32)The corresponding ˆ α ( ζ ) and ˆ η t ( ζ ), again in SOCA, readˆ α ( ζ ) = α k f R m exp( − k f | ζ | ) , ˆ η t ( ζ ) = η t0 k f R m exp( − k f | ζ | ) . (33)In Fig. 2 results of test field calculations for the functions˜ α ( k ) and ˜ η t ( k ) with R m = / √ ≈ . η t becomes negative for small k . The same has been ob-served with another but similar flow of Roberts type (R¨adler &Brandenburg 2003). For comparison, SOCA results obtained intwo di ff erent ways are also shown: those according to the ana-lytic relations (32) and those calculated numerically by the testfield method with (23). Both agree very well with each other.In order to obtain the results for the kernels ˆ α ( ζ ) and ˆ η t ( ζ )we have calculated integrals as in (14) numerically using the dataplotted in Fig. 2. The results are represented in Fig. 3. Again, an-alytical and numerical SOCA results are shown for comparison.Note that the profiles of ˆ α ( ζ ) and ˆ η t ( ζ ) beyond SOCA are rathernarrow compared with those under SOCA, and that of ˆ η t ( ζ ) evenmore narrow than that of ˆ α ( ζ ). Results for homogeneous isotropic turbulence have been ob-tained by solving the hydrodynamic equations (27) and (28) si-multaneously with the test field equations (22) in a domain of . Brandenburg et al.: Scale dependence of alpha e ff ect 5 Fig. 1.
Dependences of the normalized α and η t on R m for theRoberts flow in the general case, i.e. independent of SOCA (solidlines), and in SOCA (dotted lines). Fig. 2.
Dependences of the normalized ˜ α and ˜ η t on k / k f for theRoberts flow with R m = / √ ≈ . α/ R m and ˜ η t / R m , which areindependent of R m (dotted lines).size L . The forcing wavenumbers k f are fixed by k f / k = α = − u rms , η t0 = u rms / k f , R m = u rms /η k f . (34)Within this framework the dependence of α and η t on R m has been studied by Sur et al. (2008). They considered two Fig. 3.
Normalized integral kernels ˆ α and ˆ η t versus k f ζ for theRoberts flow with R m = / √ ≈ . α/ R m and ˆ η t / R m , which areindependent of R m (dotted lines). The full width half maximumvalues of k f ζ for ˆ α and ˆ η t are about 0 . .
2, respectively.cases, one with ν/η = . u rms /ν k f = . α/α and η/η approach unity for R m ≫ α ( k ) and ˜ η t ( k ) with ν/η =
1. Both˜ α and ˜ η t decrease monotonously with increasing | k | . The two val-ues of ˜ α for a given k / k f but di ff erent k f / k and R m are alwaysvery close together. The functions ˜ α ( k ) and ˜ η t ( k ) are well repre-sented by Lorentzian fits of the form˜ α ( k ) = α + ( k / k f ) , ˜ η t ( k ) = η t0 + ( k / k f ) . (35)In Fig. 5 the kernels ˆ α ( ζ ) and ˆ η t ( ζ ), again with ν/η = α ( ζ ) = α k f exp( − k f | ζ | ) , ˆ η t ( ζ ) = η t0 k f exp( − k f | ζ | ) . (36)Evidently, the profile of ˆ η t is half as wide as that of ˆ α . This cor-responds qualitatively to our observation with the Roberts flowbeyond SOCA, see the crosses in Fig. 3. There is however nocounterpart to the negative values of ˆ η t that occur in the exampleof the Roberts flow.The results presented in Figs 4 and 5 show no noticeabledependences on R m . Although we have not performed any sys-tematic survey in R m , we interpret this as an extension of theabove–mentioned results of Sur et al. (2008) for α and η t to theintegral kernels ˆ α and ˆ η t . Of course, this should to be checkedalso with larger values of R m . Particularly interesting would be aconfirmation of di ff erent widths of the profiles of ˆ α and ˆ η t .
4. Discussion
Our results are important for calculating mean–field dynamomodels. The mean–field induction equation governing B , here A. Brandenburg et al.: Scale dependence of alpha e ff ect Fig. 4.
Dependences of the normalized ˜ α and ˜ η t on the nor-malized wavenumber k / k f for isotropic turbulence forced atwavenumbers k f / k = R m =
10 (squares) and k f / k = R m = . ν/η =
1. The solid lines givethe Lorentzian fits (35).
Fig. 5.
Normalized integral kernels ˆ α and ˆ η t versus k f ζ forisotropic turbulence forced at wavenumbers k f / k = R m =
10 (squares) and k f / k =
10 with R m = . ν/η =
1. The solid lines are defined by (36).defined as average over x and y , with E according to (8), allowssolutions of the form ℜ h B exp(i kz + λ t ) i , B z =
0, with thegrowth rate λ = − (cid:2) η + ˜ η t ( k ) (cid:3) k ± ˜ α ( k ) k . (37) A dynamo occurs if λ is non–negative. Since ˜ α ≤ E and B the ˜ η t ( k ) and ˜ α ( k ) turn into ˜ η t (0) and ˜ α (0), re-spectively.When using the definitions (29) for the Roberts flow, or (34)for isotropic turbulence, we may write (37) in the form λ = η t0 k ( − " γ R m + ˜ η t ( k / k f ) η t0 kk f + ˜ α ( k / k f ) α ) kk f , (38)where γ = γ = η t and ˜ α depend only via k / k f on k we have chosenthe arguments k / k f .Consider first the Roberts flow, that is (38) with γ = λ is non–negative in some interval 0 ≤ k / k f ≤ κ and ittakes there a maximum. Dynamos with k / k f > κ are impossible.Of course, κ depends on R m . With the analytic SOCA results(32) we find that κ = R for small R m and that it grows mono-tonically with R m and approaches unity in the limit of large R m .For small R m (to which the applicability of SOCA is restricted)a dynamo can work only with small k / k f , that is, with scalesof the mean magnetic field that are much smaller than the sizeof a flow cell. Furthermore, κ never exceeds the correspondingvalues for vanishing nonlocal e ff ect, which is R / (1 + R ).In that sense the nonlocal e ff ect favors smaller k , that is, largerscales of the mean magnetic field. With the numerical results be-yond SOCA represented in Fig. 2, with R m = / √
2, we have κ ≈ . ... .
95, again a value smaller than unity. In this case,too, a dynamo does not work with scales of the mean magneticfield smaller than that of a flow cell. There is no crucial impactof the negative values of ˜ η t for k / k f < . γ =
3. Again λ is non–zero in the interval 0 ≤ k / k f ≤ κ and ittakes there a maximum. Some more details are shown in Fig. 6.With the Lorentzian fits (35) of the results depicted in Fig. 4 wefind κ ≈ .
60 for R m =
10 (and 0.45 for R m = . ff ects it turns out that κ ≈ . R m =
10 (and 0.59 for R m = . ff ect favors smaller k , or largerscales of the mean magnetic field.These findings may become an important issue especially fornonlinear dynamos or for dynamos with boundaries. Examplesof the last kind were studied, e.g., by Brandenburg & Sokolo ff (2002) and Brandenburg & K¨apyl¨a (2007). In these cases how-ever the underlying turbulence is no longer homogeneous andtherefore the kernels ˆ α and ˆ η t are no longer invariant under trans-lations, that is, depend not only on ζ but also on z . The finitewidths of the ˆ α and ˆ η t kernels may be particularly important ifthere is also shear, because then there can be a travelling dy-namo wave that may also show strong gradients in the nonlinearregime (Stix 1972, Brandenburg et al. 2001).For another illustration of the significance of a finite widthof the kernels ˆ α and ˆ η t we consider a one-dimensional nonlin-ear mean–field model with periodic boundary conditions. Wemodify here the model of Brandenburg et al. (2001, Sect. 6),with a dynamo number of 10 (corresponding to 5 times super-critical) and R m =
25, by introducing the integral kernels (36).Figure 7 shows the components of the mean magnetic field fortwo di ff erent values of k f / k and for the conventional case wherethe kernels are delta–functions. Note that k corresponds to thelargest scale of the magnetic field compatible with the bound- . Brandenburg et al.: Scale dependence of alpha e ff ect 7 Fig. 6.
Normalized growth rate λ ( k ) for isotropic turbulence, cal-culated according to relation (38) with γ = η t /η t0 and˜ α/α as given in (35), for R m → ∞ (upper solid line), as wellas R m =
10 and 3.5 (next lower solid lines). For comparison λ is also shown for the case in which ˜ η t /η t0 coincides with ˜ α/α as given in (35) (dotted lines) and for that of vanishing nonlocale ff ect, in which ˜ η t /η t0 = ˜ α/α = R m . Fig. 7.
Mean magnetic field components B x and B y , normalizedby the equipartition field strength B eq , in the one-dimensionalnonlinear dynamo model characterized in the text, for di ff erentvalues of k f / k and for vanishing nonlocal e ff ects.ary condition. It turns out that the magnetic field profiles are notdrastically altered by the nonlocal e ff ect. Small values of k f / k ,however, correspond to smoother profiles.Let us start again from E in the form (8), specify there, inview of isotropic turbulence, ˆ α and ˆ η t according to (36), and rep-resent B ( z − ζ ) and J ( z − ζ ) by Taylor series with respect to ζ . Astraightforward evaluation of the integrals provides us then with E ( z ) = X n ≥ (cid:18) α k n f ∂ n B ( z ) ∂ z n − η t0 (2 k f ) n ∂ n J ( z ) ∂ z n (cid:19) . (39) This corresponds to relations of the type (1) or (2), simply gener-alized by taking into account higher than first–order derivativesof B .The terms with derivatives of J in (39) can be interpreted inthe sense of hyperdi ff usion. While all of them have the samesigns in real space, the signs of the corresponding terms inFourier space alternate, which implies that every second termacts in an anti-di ff usive manner. Thus, a truncation of the expan-sion should only be done such that the last remaining term hasan even n , as otherwise anti-di ff usion would dominate on smalllength scales and cause B to grow beyond any bound.There are several investigations in various fields in whichhyperdi ff usion has been considered. In the purely hydrody-namic context, R¨udiger (1982) derived a hyperviscosity term andshowed that this improves the representation of the mean veloc-ity profile in turbulent channel flows. In the context of passivescalar di ff usion, Miesch et al. (2000) determined the hyperdif-fusion coe ffi cients for turbulent convection and found that theyscale with n like in Eq. (39). We are however not aware of earlierstudies di ff erentiating between di ff usive and anti-di ff usive terms.We have investigated the nonlocal cases presented in Fig. 7using truncations of the expansion (39). However, two problemsemerged. Firstly, terms with higher derivatives produce Gibbsphenomena, i.e. wiggles in B , so the results in Fig. 7 are not wellreproduced. Secondly, high–order hyperdi ff usion terms tend togive severe constraints on the maximum admissible time step,making this approach computationally less attractive. It appearstherefore that a direct evaluation of the convolution terms is moste ff ective.
5. Conclusions
The test field procedure turned out to be a robust method fordetermining turbulent transport coe ffi cients (see Brandenburg2005, Sur et al. 2008 and Brandenburg et al. 2008). The presentpaper shows that this also applies to the Fourier transforms of theintegral kernels which occur in the nonlocal connection betweenmean electromotive force and mean magnetic field, in otherwords to the more general scale–dependent version of thosetransport coe ffi cients. For isotropic turbulence the kernels ˆ α andˆ η t have a dominant large-scale part and decline monotonouslywith increasing wavenumbers. This is consistent with earlierfindings (cf. Brandenburg & Sokolo ff ff usivity is about half as wide as that for thealpha e ff ect. This result is somewhat unexpected and would beworthwhile to confirm before applying it to more refined meanfield models. On the other hand, the e ff ects of nonlocality be-come really important only when the scale of the magnetic fieldvariations is comparable or smaller than the outer scale of theturbulence.One of the areas where future research of nonlocal turbulenttransport coe ffi cients is warranted is thermal convection. Herethe vertical length scale of the turbulent plumes is often com-parable to the vertical extent of the domain. Earlier studies byMiesch et al. (2000) on turbulent thermal convection confirmedthat the transport of passive scalars is nonlocal, but it is also moreadvective than di ff usive. It may therefore be important to alsoallow for nonlocality in time. This would make the expansionof passive scalar perturbations more wave-like, as was show byBrandenburg et al. (2004) using forced turbulence simulations. A. Brandenburg et al.: Scale dependence of alpha e ff ect Acknowledgements.
We acknowledge the allocation of computing resourcesprovided by the Centers for Scientific Computing in Denmark (DCSC), Finland(CSC), and Sweden (PDC). We thank Matthias Rheinhardt for stimulating dis-cussions. A part of the work reported here was done during stays of K.-H. R. andM. S. at NORDITA. They are grateful for NORDITA’s hospitality.
Appendix A: Justification of equations (5) and (12)
In view of (5) we start with equation (3) for E , put E = ,assume that K ◦ B is a purely spatial convolution. Applyingthen the Fourier transform as defined by Q ( x ) = R ˜ Q ( k ) exp(i k · x ) d ( k / π ) we obtain˜ E i ( k ) = ˜ K i j ( k ) ˜ B j ( k ) . (A.1)Since E and B have to be real we conclude that ˜ K ∗ i j ( k ) = ˜ K i j ( − k ).Further the assumption of isotropic turbulence requires that ˜ K i j is an isotropic tensor. We write therefore˜ K i j = ˜ α ( k ) δ i j + ˜ α ′ ( k ) k i k j + i˜ η t ( k ) ǫ i jk k k (A.2)with ˜ α , ˜ α ′ and ˜ η t being real functions of k = | k | . Consideringfurther that k · ˜ B = k × ˜ B = ˜ J we find˜ E ( k ) = ˜ α ( k ) ˜ B ( k ) − ˜ η t ( k ) ˜ J ( k ) . (A.3)Transforming this in the physical space we obtain immediately(5).In view of (12) we start again from equation (3) and put E = but we have to consider K ◦ B now as a convolution with respectto z only. Applying a Fourier transformation defined by Q ( z ) = R ˜ Q ( k ) exp(i kz ) d( k / π ) we obtain a relation analogous to (A.1),˜ E i ( k ) = ˜ K i j ( k ) ˜ B j ( k ) . (A.4)and may now conclude that ˜ K ∗ i j ( k ) = ˜ K i j ( − k ). We arrive so at˜ K i j = ˜ α i j ( k ) + i k ˜ η ′ i j ( k ) (A.5)with real tensors ˜ α i j and ˜ η ′ i j , which are even in k . Combining(A.4) and (A.5) and considering that the i k ˜ B i can be expressedby the ˜ J i (i k ˜ B = ˜ J , i k ˜ B = − ˜ J , i k ˜ B =
0) we may confirmfirst (13) and so also (12).
Appendix B: Mean–field results for the Roberts flow
A mean–field theory of the Roberts dynamo, developed in viewof the Karlsruhe dynamo experiment, has been presented, e.g., inpapers by R¨adler et al. (2002a,b), in the following referred to asR02a and R02b. There a fluid flow like that given by (26) is con-sidered but without coupling of its magnitudes in the xy –planeand in the z –direction. The mean fields are defined by averagingover finite areas in the xy –plane so that they may still depend on x and y in addition to z . As shown in the mentioned papers E ,when contributions with higher than first–order derivatives of B are ignored, has then the form E = − α ⊥ h B − (ˆ z · B )ˆ z i − β ⊥ ∇ × B − ( β k − β ⊥ ) h ˆ z · ( ∇ × B ) i ˆ z − β ˆ z × h ∇ (ˆ zB ) + (ˆ z · ∇ ) B i (B.1)with constant coe ffi cients α ⊥ , β ⊥ , β k and β [see (R02a 9) or(R02b 9)]. Reducing this to the case considered above, in which B depends no longer on x and y , we find E = α h B − (ˆ z · B )ˆ z i − η t ∇ × B , (B.2) where ∇ × B = ˆ z × ∂ B /∂ z , and α = − α ⊥ , η t = β ⊥ + β . (B.3)Results for α ⊥ , β ⊥ , β k and β obtained in the second–ordercorrelation approximation are given in (R02a 19) and (R02a 49)as well as in (R02b 19) and (R02b 38). When fitting them with u ⊥ = (2 /π ) u , u k = √ /π ) u , π/ a = k , √ π/ a = k f , R m ⊥ = √ R m and R m k = (8 /π ) R m to the assumptions and notations usedabove we find just (30). Likewise (R02a 20) and also (R02b 20)lead to (31). Appendix C: ˜ α and ˜ η t under SOCA for Roberts flow Let us start with the relation (B.2) and subject it to a Fouriertransformation so that˜ E = u × ˜ b = ˜ α h ˜ B − (ˆ z · ˜ B )ˆ z i − i k ˜ η t ˆ z × ˜ B . (C.1)From the induction equation we have η ( ∇ − k ) ˜ b = − ( ∇ + i k ˆ z ) × ( u × ˜ B ) , k ˜ b z = . (C.2)The solution ˜ b reads˜ b = − k + k (cid:26) ˆ z × ∇ ( ˜ B · ∇ ψ ) − k f ( ˜ B · ∇ ψ )ˆ z + i k h ˆ z × ∇ ψ (ˆ z · ˜ B ) + k f ψ ( ˜ B − (ˆ z · ˜ B )ˆ z ) i(cid:27) , (C.3)where u is according to (24) expressed by ψ . Calculate now ˜ E x or˜ E y and note that ψ = ( u / k ) and ( ∂ψ/∂ x ) = − ψ∂ ψ/∂ x = u . When comparing the result with (C.1) we find immediately(32). Using then relations of the type (14) we find also (33). References
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