Large-scale dynamo produced by negative magnetic eddy diffusivities

The existence of incompressible fiow producing negative magnetic eddy diffusivities is demonstrated. This provides for a dynamo mechanism, alternative to a-type effects, requiring neither the presence of mean helicity nor the breaking of parity invariance. In the kinematic dynamo phase, the magnetic field grows exponentially with a growth rate proportional to the square of the wavenumber. The concrete example, analyzed by means ofmultiscale techniques, is a parity-invariant fiow of the Taylor-Green type,


INTRODUCTION
The generation of large-scale magnetic fi elds is one of the key issues in magnetohydrodynamics (e.g., Moffatt, 1978). The nature and the properties of self-inductive mechanisms capable of maintaining a per sistent magnetization are of particular importance for cosmical ob jects, such as stars or galaxies (Parker, 1979;Zeldovich et al., 1983). For the investigation of magnetic fields growth, the Lorentz reaction of the magnetic field B on the velocity v can be neglected (at least initially) and the resulting dynamics is commonly known as the *Corresponding author. e-mail: lanotte@obs-nice.fr 1 kinematic dynamo problem. The velocity field v is indeed prescribed a priori as a function (deterministic or random) of the space-time vari ables x and t. The issue is to determine the characteristics of the pos sible growth of the large-scale components of B(x, t) (large-scale dynamo) in terms ofv properties. The major effect of the incompressible flow v on the mean magnetic field B is to generate a mean electro motive force (e.m.f.) £, infl uencing the evolution of B according to the induction equation ( 1) In a mean-fi eld framework (Moffatt, 1978), the e.m.f. is expanded in a gradient series as (2) where the coefficients aij and f3ijt are functionals of the velocity fi eld v. Higher-order terms involve more than one spatial gradient of B . The gradient expansion has a fully systematic status in scale-separated situations, i.e., when the velocity v has finite typical space-time scales and the scales of interest for B are much larger than all of them. The first term in the expansion (2) is responsible for the well-known a-effect (Steenbeck et al., 1966). If the a tensor does not identically vanish, the term associated to it in (1) (first-order in space variables) will indeed become dominant with respect to the second-order molecular diffusivity term at sufficiently large scales. In three dimensions, this leads to an ex ponential growth of B with a growth rate proportional to the wave number k. Lack of parity-invariance is the central ingredient for a-type instabilities. By parity-invariant flow we mean those fi elds v admit ting a center of symmetry relative to which the simultaneous reversal xi--+ -x and vi--+ -v leaves the flow invariant (deterministically or statis tically, depending on the case considered). When no such center can be found, parity invariance is broken. Since the e.m.f. £ is a vector and B is a pseudo-vector, such breaking is a necessary condition for aij not to vanish identically. By the same argument, the average value of the magnetic helicity A(x, t) • B(x, t) is guaranteed to vanish in parity invariant situations. 1 When the fl.ow does not admit any center of 1 In the relation above, A is the vector potential satisfying 8 = V x A. symmetry, the mean value of (kinetic) helicity can vanish or not. The latter case generally leads to stronger dynamos at large scales, with growth rates proportional to the wavenumber k (and not k? as in the former case). It is therefore commonly accepted that helicity plays an important role for o:-type instabilities (Gilbert et al., 1988). Note also that o: instabilities have a vanishing threshold in the magnetic Rey nolds number, as the molecular diffusivity is subdominant at sufficiently large scales.
For parity-invariant fl ow, a possible large-scale dynamo cannot take place via the a-effect and second-order terms in (2) become relevant. Inserting them into (1), a formally diffusive equation is obtained, where "formally" is meant to stress that the second-order operator on the right-hand side is not guaranteed to be (semi) negative-defi nite. It is certainly negative definite when the molecular diffusivity is large enough to dominate turbulent effects. But when the magnetic Reynolds number is increased, it is conceivable that the turbulent term associ ated to the f3 tensor could destabilize the large scales and quanti tatively overwhelm the stabilizing molecular contribution, thus leading to dynamo, i.e., the growth of B. This circumstance would corre spond to a negative magnetic eddy diffusivity and would provide for a dynamo mechanism not requiring any breaking of parity invariance and/or mean helicity. The purpose of the present paper is to address the question formulated in Section 7.4 of (Moffatt, 1978) whether this dynamo mechanism is physically realizable.
The answer to the equivalent question for other physical situations is known. For the transport of passive scalars, the effect of an incom pressible velocity v is always stabilizing, i.e., the eddy diffusivity is larger than the molecular diffusivity (Mc Laughlin et al., 1985). For time-independent potential flow, the eddy diffusivity is smaller than the molecular diffusivity, but it cannot become negative (Vergassola and Avellaneda, 1997). Another relevant situation is momentum trans port in the Navier-Stokes equations. The effect analogous to the a effect is the so-called AKA effect (Frisch et al., 1987), which also disappears in the presence of parity-invariance. The existence of negative eddy viscosities is now well established, in two and in three dimensions, both for isotropic and anisotropic flow (Meshalkin and Sinai, 1961;Nepomnyashchy, 1976;Sivashinsky, 1985;Gama et al., 1994;Wirth et al., 1995). The existence of negative eddy viscosities points in favor of negative magnetic eddy diff usivities, but this indication should be taken with caution. Indeed, for simple parallel flows (depending on a single coordinate), the eddy viscosity is for example known to be negative, but the origin of the instability (at least in this case) comes entirely from the non-local pressure term (e.g., Meshalkin and Sinai, 1961;Dubrulle and Frisch, 1991 ), which is absent in the kinematic dynamo problem. This important difference is confi rmed in the Appendix, where the same problem of parallel flow is solved for magnetic fi elds and no dynamo is indeed present.
Arguments in favor of negative magnetic eddy diffusivities were previously given in Roberts (1972 ) and Kraichnan (1976). The atten tion in Roberts (1972) was mainly concentrated on a-type dynamo, but some evidence was also given that magnetic fields could grow when orders higher than the first in (2) are relevant. A three-scale argu ment was considered in Kraichnan (1976) to show that fluctuations of the a-coefficient at intermediate scales could give a negative contri bution to the eddy diffusivity on large scales, but no defi nite con clusion could be drawn. Here, we shall analyze the problem using multiscale techniques, presented in Section 2. The advantage is that the nature and the order of the large-scale dynamics can be systematically identifi ed and the calculation of the eddy-diffusivity tensor is reduced to the solution of auxiliary equations on the elementary periodicity cell. In Section 3, numerical simulations of the auxiliary equations are used to investigate a steady variant of the Taylor-Green vortex, which is found to produce a negative magnetic eddy diffusivity above a critical Reynolds number.

MULTISCALE FORMALISM
The aim of this section is to present the multiscale formalism that will be later exploited for the calculation of magnetic eddy diffusivities.
The incompressible velocity fi eld v(x, t) in the kinematic dynamo equation (3) is supposed to have a finite typical length scale £ 0 , e.g., to be £ 0periodic, and we shall be interested in the dynamics of the magnetic field B at scales L » l0. The ratio l0/ L = c «: I provides a small parameter that can be exploited for a perturbative solution. The per turbation being singular (e.g., Bender and Orszag, 1978), it is neces sary to treat the problem by singular perturbation methods (see Nayfeh, 1973;Van Dyke, 1975). The most convenient method for the case at stake is the multiscale technique, also known as homogeniza tion (Bensoussan et al., 1978). In addition to the original variables (x, t) ("fast" variables) characterizing the basic flow v, a new set of space-time variables X, T ("slow" variables) is introduced. The ration ale in their choice is that the large-scale dynamics should be 0(1) in the new variables, e.g., X = ex. The prescription that resolves the singularity is to treat the two sets of variables, fast and slow, as inde pendent throughout the perturbative expansion. The enlargement of degrees of freedom is compensated by the presence of non-trivial solvability conditions in the resulting equations, giving the large-scale evolution equations.
Let us implement the previous procedure for the kinematic dynamo equation (3). Since we are interested in eddy diffusivities we shall impose the velocity field v to be parity-invariant. This ensures the ab sence of the o: effect and makes the dynamics at large scales expected to be second-order in the space variables. The appropriate rescalings for the slow variables 2 are then ( 4) Treating fast and slow variables as independent implies the following chain rule for the derivatives (5) Here, we denote by the symbol {) the derivatives with respect to fast variables and by (V, or) those with respect to slow variables. The solution is then sought in the form of a perturbative series in c: where the B (n) functions depend on both fast and slow variables. Inserting the expansion (6) and the rule for derivatives (5) into the original equation (3), we derive the following hierarchy of equations: The operator A is simply the induction operator acting on fast variables: Note that these conditions mix different orders of B (n), implying, for example, that ox (v x u<l)) ;.b v · au<I) _ B0)·8v.
The operator A having derivatives at the left of all terms, the average (A f) vanishes for any f having the same periodicities as v.
A necessary solvability condition for the auxiliary equations of the hierarchy is thus where(•) denotes the average over the periodicities of the basic flow v.
In more general terms, by the Fredholm alternative theorem, the right hand side g should be orthogonal to constant functions, that constitute the null-space of the operator A t , adjoint of A.
The first solvability condition at O(c 0 ) is automatically satisfied since (8v) = 0. Using the linearity of the equation, its solution can be generally written in the form with the field Sij having zero mean and satisfying the equation At the next order O(c1), the solvability condition or equivalently using (15) (v ;(x ) S1 1( x )) -( vJ(x ) S u( x )) = 0, would in general be non trivial, corresponding to the a effect. However, if the velocity is parity invariant v(-x) = -v(x), from (16) it follows that Sij(-x) = Sij(x) and the average in (18) therefore vanishes. Using again the linearity of the equation, we can write its solution in the form where the fi eld rij1(x) is the zero-mean solution of Finally, from the solvability condition at O(t:2) we obtain the evolution equation for the large-scale magnetic field (B (O))(X, T): Substituting in (2 1) the expression (19), we obtain (22) with the eddy diffusivity expressed in terms of averages of the r field: The large-scale dynamics is formally diffusive, as expected. Note that, according to the definition (23), the eddy diffusivity T/ E is the sum of the molecular contribution T/ and those due to the presence of the small-scale velocity field v. Since the transported quantity is a vector, the eddy diffusivity is a fourth-order tensor, whose entries are averages of functions depending only on fast variables. For a generic velocity field, the tensor rfutm will not be isotropic, i.e., the eddy diffusivity will depend on the direction. To single out this dependence, it is useful to decompose a generic field Bon the Fourier basis and note that we can look for eigenfunctions of (22) in the form of plane waves: Plugging (24) into (22), for each unit vector k we obtain the following eigenvalue problem: The non-trivial part of this (d x d) matrix lives in fact on the (d-1) dimensional sub-space perpendicular to the vector k, the latter having zero eigenvalue. In three dimensions, once the entries of the T/ E tensor are known, the calculation of the eddy diffusivity in a given direction reduces therefore to the simple diagonalization of a 2 x 2 matrix.

TAYLOR-GREEN FLOWS
In this section we shall discuss the behavior of magnetic eddy diffusivities for the Taylor-Green flow, and a slight variant of it that will be shown to produce a negative magnetic eddy diffusivity. 8 The Taylor-Green flow v T 0 : v;-0 = sinxcosycosz, v;0 = -cosxsinycosz, v;0 = 0, (27) was thoroughly investigated in the study of vorticity enhancement by vortex-line stretching and the consequent production of small-scale eddies (Brachet et al., 1983). The planes x, y or z = mr (with n integer) are of mirror symmetry and they constitute the faces of the so-called impermeable box since no fluid crosses these boundaries. The overall geometry of the flow resembles that of a now classical experimental con fi guration consisting of a shear layer between two counter-rotating disks. The many symmetries of the Taylor-Green flow (see Brachet et al., 1983) ensure that when v T0 is used as initial condition in a Na vier-Stokes simulation, the resulting fl ow admits the following Fourier expansion (when bifurcations breaking these symmetries are neglected): where the vector u vanishes unless the integers (m, n,p) are all even or all odd. The full MHD nonlinear problem with a Taylor-Green forcing on the velocity was recently numerically investigated in for example Nore et al. (1997). For very high viscosities only the basic mode of the forcing is active, while more and more modes other than the basic one are excited in the expansion (28) when the Reynolds number is increased (the non-basic modes will be referred to, in the sequel, as recirculation modes). The important observation made in Nore et al. (1997) is that the magnetic fi eld grows for sufficiently large Reynolds numbers when the forcing is not on the smallest wavenumber of the box. Furthermore, the fastest growing mode of the magnetic field has a component on the smallest wavenumber, suggesting a sort of scale separation. These were our motivations in considering the following generalization (with the two most energetic recirculation modes) of the Taylor-Green flow where A and B are free parameters. The other constants, multiples of 1/ 13 , could also have in principle been left free. For simplicity, we decided to set them to the value they take when the solution of the Navier-Stokes equations with a Taylor-Green forcing is expanded in powers of the Reynolds number. Note that both the original Taylor Green flow (27) and its variant (29) are parity-invariant (with respect to the origin). The vanishing of the mean helicity (h(x)) = (v·w) and the absence of a effect are therefore guaranteed. An important quali tative difference between the two fl ows is that, for (27), h(x) itself van ishes everywhere and not just its average value. This is not the case for the fl ow (29) where helicity fluctuations, although globally vanish ing, are locally quite strong. The magnetic eddy diffusivities for the flows (27) and (29) are calculated using the formalism of Section 2 and numerically solving the resulting auxiliary equations (1 6) and (20). The numerical code is a standard pseudo-spectral one, with the classical 2/3 truncation procedure for dealiasing (Gottlieb and Orszag, 1977). Since magnetic Reynolds numbers (defined here as Rm= 1/ry ) turn out to be of order ten, only moderate resolutions 32 3 and 64 3 are needed. We checked by the energy spectra that truncation or finite size errors are negligible for the range of magnetic ?iffusivities consid e red; it was verified that the results for the two resolutions coincide within relative errors of few per cent.
To solve the auxiliary equations, the most convenient way is to apply repeatedly the evolution operatorA with a sufficiently small timestep until the field settles down to a stationary state. Note that this requires all small-scale modes to be damped (which is certainly guaranteed if the Reynolds numbers is small enough). This very same assumption is indeed also implicitly assumed in the multiscale pro cedure itself: for the dynamo instability to be at large scales, and for multiscale techniques to be applicable, it is necessary for small scales to be stable. The validity of such an assumption can be checked a posteriori.
Let us now tum to the discussion of the results. In Figure 1, to better highlight the effects of the small-scale velocity field on large scale transport, we have plotted the minimum, over the unit wave numbers k, of the difference between the eddy diffusivity 7f (k), defined in (23), and the molecular diffusivity T/· For both the Taylor-Green flow (27) and the flow (29) with only the first recirculating mode excited (A = 1, B = 0), such difference always remains positive, and a fortiori so does the whole eddy diffusivity TJE. In particular, in the former case, the eddy diffusivity has a tendency to saturate to a fi nite value, while in the latter case, the tendency is to decrease. However, when the magnetic Reynolds number is increased above Rm � 11, the kinematic dynamo equation becomes unstable at small scales, comparable to those of the velocity field, indicating that the primary dynamo mechanism is not acting at large scales. The evidence of negative magnetic eddy diffusivities is finally obtained for the flow (29) with A = B = l. As shown in Figure 2, for Rm � K;,, rit � 7.9, there is at least one mode k having a negative eddy diffusivity 7f (k). As follows from (24), the corresponding eigencomponent of the magnetic field is exponentially amplifi ed, with a growth-rate proportional to k 2 . The modes lying in the xy plane are the most unstable for any Rm > �,:jt   In conclusion, the possibility of a dynamo action by negative magnetic eddy diffusivities has been demonstrated. Helicity fluctua tions, although globally vanishing, appear to play an important role in producing the negative eddy diffusivity, confirming the intuition of Kraichnan (1976). The negative eddy diffusivity mechanism does not require any breaking of parity invariance and offers therefore an alternative to a-type effects.

APPENDIX A. SIMPLE ANALYTICALLY SOLVABLE CASES
We shall briefl y discuss here some simple cases where the calculation of the magnetic eddy diffusivity can be carried out analytically. For none of them does the eddy diffusivity turn out to be negative.
Let us start from the case when the magnetic Reynolds number R m is small enough to permit a perturbative expansion of the mag netic eddy diffusivity. The operator A in (10) then reduces to the heat operator 1{ = 8 1 -T/& whose inversion on periodic functions having zero mean and a spectrum decaying sufficiently fast is well defined and easy to perform, e.g., in Fourier space. The solutions at the dominant order of the auxiliary equations (8) and (9)  The expression of the eddy diffusivity tensor immediately follows from (23). For time-independent velocity fields, the turbulent contribu tion turns out to be positive definite. Indeed, the combination I= -r/J1 m k/<.1b;b m [which is the one involved in (26) where the simplifying feature is that they depend on a single coordinate (here arbitrarily chosen to be x) . It follows that, in solving (8) and (9), the term iJ x (v x •) in A identically vanishes and the solution of the equations reduces again to inverting the heat operator. The previous perturbative formulae can therefore be simply carried over and the eddy diffusivity tensor is guaranteed to be positive defi nite by the general result (A.2).
Let us finally recall that in the case when the velocity field is random and has a short-correlation time, an exact equation for the mean magnetic field exists (Katsantzev, 1968). The stretching term B·iJv does not give any contribution and the resulting magnetic eddy diffusivity tensor is simply % i m = 'f/ O; mOj / + O; m fo° (vj( x, t) v1 (x, O) ) dt, which is positive-definite (and position-independent for a homoge neous flow).