We consider the case of homogeneous turbulence in a conducting fluid that is exposed to a uniform external magnetic field at low to moderate magnetic Reynolds numbers (by moderate we mean here values as high as 20). When the magnetic Reynolds number is vanishingly small (Rm 1), it is customary to simplify the governing magnetohydrodynamic (MHD) equations using what is known as the quasi-static (QS) approximation. As the magnetic Reynolds number is increased, a progressive transition between the physics described by the QS approximation and the MHD equations occurs. We show here that this intermediate regime can be described by another approximation which we call the quasi-linear (QL) approximation. For the numerical simulations performed, the predictions of the QL approximation are in good agreement with those of MHD for magnetic Reynolds number up to Rm ? 20.

Magnetohydrodynamics (MHD) applies to many conductive fluid and plasma flows encountered in nature and in industrial applications. In numerous circumstances, the flow is subject to a strong mean magnetic field. This happens in the earth?s liquid core and is ubiquitous in solar physics for topics such as sunspots, solar flares, solar corona, solar wind, etc. Mean magnetic fields play an important role on even larger scales, for instance in the dynamics of the interstellar medium. Among the industrial applications involving applied external magnetic fields are drag reduction in duct flows, design of efficient coolant blankets in tokamak fusion reactors, control of turbulence of immersed jets in the steel casting process and advanced propulsion and flow control schemes for hypersonic vehicles.

Depending on the application, the magnetic Reynolds number, Rm, can vary
tremendously. In astrophysical problems, Rm can be extremely high as a result of
the dimensions of the objects studied. On the contrary, for most industrial flows
involving liquid metal, Rm is very low, usually less than 10?2. When an external
magnetic field is present, it is customary at such low values of Rm to make use of the
so-called quasi-static (QS) approximation. In this approximation, induced magnetic
fluctuations are much smaller than the applied magnetic field and the overall magnetic
effect amounts to adding in the Navier?Stokes equations an extra damping term which
only affects Fourier modes having a component parallel to the magnetic field (more
details below). The derivation of the QS approximation involves taking the limit of
vanishing Rm and its domain of validity is thus an interesting question. Indeed certain
applications, such as advanced schemes for the control of magnetogasdynamic flows
around hypersonic vehicles, involve values of Rm of the order 1 to 10

The limit of vanishing Rm (with mean magnetic field) has been the subject of
several theoretical studies in the past.

To our knowledge, the first numerical study of MHD turbulence in the regime
Rm 1 is due to

Performing MHD simulations in the limit of low Rm is impractical. Aside from the
increased complexity arising from having to carry a separate evolution equation for
the magnetic field, the main problem lies in the time scales involved in the problem.
Indeed, at vanishing magnetic Reynolds number, the magnetic diffusion time scale
tends to zero. The only possibility in that case is to resort to the QS approximation for
which this time scale is not explicitly relevant. Simulations of MHD have thus been
restricted so far to cases where the magnetic and kinetic time scales are of the same
order. This is the case when the magnetic Prandtl number (see below) is close to 1.
Among the numerous previous numerical studies of MHD in this regime, we mention
the work of Oughton, Priest & Matthaeus (1994) which is the most relevant to the
present discussion. They consider the same three-dimensional periodic geometry with
an applied external magnetic field as in

In the present article, we will consider the decay of MHD turbulence under the influence of a strong external magnetic field at moderate magnetic Reynolds numbers. Typical values of Rm that are considered here range from ? 0.1 to ? 20. As Resolution Box size (lx × ly × lz) Rms velocity Viscosity Integral length-scale (3?/4 × ( ??1E(?) d? E(?) d?)) Re = uL/? Dissipation ( ) Dissipation scale (? = (?3/ )1/4)) kmax? Microscale Reynolds number (R? = ?15/(? )u2) Eddy turnover time (? = (3/2)u/ )) a comparison, the initial kinetic Reynolds number common to all our simulations is ReL = 199. This means that the range of Prandtl numbers explored is 5 × 10?4 to 10?1. Our motivation is mainly to exhibit how the transition from the QS approximation to MHD occurs. At the lowest values of Rm studied here, the QS approximation is shown to model the flow faithfully. However, for the higher values of Rm considered, it is clearly inadequate but can be replaced by another approximation which will be referred to as the quasi-linear (QL) approximation. Another objective of the present study is to describe how variations in the magnetic Reynolds number (while maintaining all other parameters constant) affect the dynamics of the flow. This complements past studies where variations in either the strength of the external magnetic field or the kinetic Reynolds number were considered.

This article is organized as follows. In § 2, we review the derivation of the quasistatic approximation and stress the assumptions that might pose a problem as the magnetic Reynolds number is increased. Section 3 is devoted to the description of the numerical experiments performed using the quasi-static approximation and MHD. In § 4, we describe the quasi-linear approximation and test it numerically against full MHD. A concluding summary is given in § 5. 2. MHD equations in the presence of a mean magnetic field

Two dimensionless parameters are usually introduced to characterize the effects of a uniform magnetic field applied to unstrained homogeneous turbulence in an electrically conductive fluid. They are the magnetic Reynolds number Rm and the interaction number N (also known as the Stuart number):

vL Rm ? ? =

L2 ?

L v ,

N ? ? B2L ?v = ? ?m In the above expressions, v = uiui /3 is the r.m.s. of the fluctuating velocity ui; L is the integral length scale of the flow (defined in table 1); ? = 1/(? µ) is the magnetic diffusivity where ? is the electric conductivity of the fluid, and µ is the fluid magnetic permeability; ? is the fluid density and B is the strength of the applied external magnetic field. The magnetic Reynolds number represents the ratio of the characteristic time scale for diffusion of the magnetic field L2/? to the time scale of the turbulence ? = L/v. Related to Rm, we can also define a magnetic Prandtl number representing the ratio of Rm to the hydrodynamic Reynolds number ReL, Pm ? ?? = RRemL ,

ReL =
vL
?
The interaction number N represents the ratio of the large-eddy turnover time ? to the
Joule time ?m = ?/(? B2), i.e. the characteristic time scale for dissipation of turbulent
kinetic energy by the action of the Lorentz force

In this section, we review the derivation of the quasi-static approximation

If the external magnetic field Bext is explicitly separated from the fluctuations bi, i the incompressible MHD equations can be written as 1 (µ?) ?t ui = ??i(p/?) ? uj ?j ui +

Bext + bj ?j Biext + bi + ? ui, (2.3) j ?t Biext + bi = ?uj ?j Biext + bi + Bext + bj ?j ui + ? j

Bext + bi , i (2.4) where p is the sum of the kinematic and magnetic pressures and ? is the kinematic viscosity. Since we consider initially isotropic freely decaying homogeneous turbulence, there is no mean velocity field. Therefore, for the sake of simplicity, we avoid introducing explicitly a decomposition of the velocity field into mean and fluctuating parts. Also, the external magnetic field is taken to be homogeneous and stationary so that (2.3) and (2.4) reduce to

1 1 ?t ui = ??i(p/?) ? uj ?j ui + (µ?) bj ?j bi + (µ?) Bjext ?j bi + ? ui, ?t bi = ?uj ?j bi + bj ?j ui + Bjext ?j ui + ? bi.

As pointed out in

?b ? bi = L2 (2.5) (2.6) where b =

bibi/3, and noting that In place of (2.6) we thus have, in the limit Rm 1, Rm = vL ? = uj ?j bi ? bi = bj ?j ui .

? bi

The so-called quasi-static (QS) approximation is obtained by further assuming that ?t bi ? 0 in (2.9). To understand how this comes about, we consider the time scales of the two terms on the right-hand side of (2.9). Since Bext is independent of time, the time scale of Bext ?j ui is T = L/v, while the time scale of the diffusion term can j be identified with the damping time T? = L2/?. The ratio of these two time scales is then

T? = Rm, (2.10)

T indicating that at low magnetic Reynolds number, diffusion time is much smaller than large-eddy turnover time. This justifies the assumption ?t bi ? 0 since the magnetic fluctuations then adapt instantaneously to the slowly varying velocity field and reach their asymptotic values for which ?t bi ? 0 (see § 4 for more details). In the QS approximation, we thus have Using a Fourier representation for ui and bi, this equation is readily solved and yields ? bi = ?Bjext ?j ui. bm(k, t) = i Bjext kj um(k, t), ?k2 (2.8) (2.9) (2.11) (2.12) where we have defined um(k, t) = um(x, t) exp(?ik · x), bm(k, t) =

bm(x, t) exp(?ik · x). (2.13) x x Since bi is now expressed completely in terms of ui, the evolution equation for the velocity field can be explicitly closed. In Fourier representation we obtain, (B ext · k)2 ?k2 ?t um(k, t) = ?ikmp (k, t) ? [uj ?j ui]m(k, t) ? ? um(k, t) ? ?k2um(k, t), (2.14) where p = p/? (consistently with the small magnetic fluctuations assumption, the second-order term bj ?j bi does not appear in (2.14)).

To summarize, two simplifications are required in order to reach (2.14). The first consists in neglecting the nonlinear terms uj ?j bi and bj ?j ui in (2.6). The second is obtained by discarding the time derivative of bi in (2.9). These two simplifications are consequences of the assumption Rm 1 and we should thus expect them to break down when the magnetic Reynolds number is increased. In the next sections, we test the QS approximation by comparing its predictions to those obtained using the MHD equations (2.5) and (2.6). Run number

To test the domain of validity of the QS approximation, we have used two different pseudospectral codes. The first one simulates the MHD equations (2.5) and (2.6), while the second one simulates (2.14). All the runs presented here have a resolution of 2563 Fourier modes in a (2?)3 computational domain.

The initial condition for the velocity field is common to both codes. It consists
of a developed turbulence field that is adequately resolved in the computational
domain adopted. Some of its characteristics are given in table 1. For the MHD
case, an initial condition for bi has to be chosen at t = t0. Here, we have made the
choice bi (t0) = 0. In other words, our simulations describe the response of an initially
non-magnetized turbulent conductive fluid to the application of a strong magnetic
field. The corresponding completely linearized problem has been described in detail
in

In order to distinguish between our numerical runs, we will vary the values of the interaction parameter and the magnetic Reynolds number (at t = t0). When these two quantities are set, the only free parameters in the evolution equations (2.5), (2.6) and (2.14) are completely determined, i.e.

BAext =

N v2 Rm , ? = vL Rm (3.1) where Bext is the external magnetic field strength in Alfve´ n units Bext = Bext /?µ?

A A and the values of v and L are given in table 1. The values of Rm and N for all our runs are given in table 2 along with the corresponding values of ? and BAext . Because of the finite computer resources available, we note the following two restrictions about the present work. First, because of the limitation in the achievable kinetic Reynolds number, the simulations (except in the quasi-static limit) do not reach values of the N = 1

Prandtl number as low as we could wish and present in traditional liquid?metal applications. Secondly, at the later stage of the kinetic energy decay, flow structures tend to be elongated strongly in the direction of the imposed magnetic field. In that case, the appropriateness of the periodic boundary conditions is questionable. This issue is unavoidable, but thankfully arises only in one direction.

In this section we present some results obtained by performing the simulations detailed in § 3.1.

In figure 1, we plot the time evolution of the normalized kinetic energy, EK =

1 EK (0) dx 12 ui(x)ui(x).

In this and subsequent figures, time has been non-dimensionalized using the Joule
timescale (see (2.10)). Keeping N constant, it is clear from the figure that as the
magnetic Reynolds number is decreased, the decays converge to the quasi-static
limit (dotted curve). At Rm = 0.1, MHD and the QS approximation are barely
distinguishable for the cases run; at Rm = 1, differences are clearly observed. As
expected, the discrepancy between MHD and the QS approximation is quite severe
at intermediate values of the Rm. We also note here the presence of oscillations in the
kinetic energy at long times for the case N = 10. Their origin is well known

The next diagnostic we examine is the evolution of the energy contained in the magnetic fluctuations. This quantity is defined through, 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6

For each set of runs at fixed N two graphs are presented. The left-hand side graphs represent ?zoomed? versions of the right-hand side graphs and have been included in order to show with more detail the initial magnetic growths for the different runs. At t = t0, the evolution of the magnetic field is determined by, ?t bi |t=0 = Bjext ?j ui |t=0 , (3.4) with Bext given by (3.1). As the magnetic Reynolds number is decreased, the initial slope of (3.3) should thus increase and this is exactly what is observed in the lefthand side graphs of figure 2. After some time, the magnetic energies all reach their maximum value and then start to decrease. The rate of decay increases at lower magnetic Reynolds numbers since in the limit chosen, ? = vL/Rm. Related to the oscillations in the kinetic energy we observe for N = 10 some oscillations in the magnetic energy at long times.

In order to describe the transition from QS behaviour to MHD in the simulations as the magnetic Reynolds is increased, a second measure of the magnetic energy can be introduced:

EM1 =

1 EM1(t0)

2 dk 12 Bjext kj ?2k4 |ui(k, t)|2.

(3.5) This is the (normalized) energy of the magnetic fluctuations when they are computed from the QS expression (2.12). When the dynamics are governed by the QS approximation, (3.5) should coincide with EM /EM1(t0). This is illustrated in figure 3. At magnetic Reynolds number Rm = 0.1, EM /EM1(t0) and EM1 are very close soon after the external magnetic field is switched on. This indicates that, in a very short time, the magnetic fluctuations forget their initial state and ?align? with the predictions of the QS approximation (2.12). This is entirely in the spirit of the assumption that at low magnetic Reynolds number the time derivative in (2.9) can be neglected (or rather that it is significant only during a very short transient time). At higher values 1.0 3 t N = 1, Rm = 0.1

N = 1, Rm = 5 0.2 0.4 0.6 0.8 1.0 3 t 1 2 4 5 6 of the magnetic Reynolds number, such as Rm = 5, this transient time becomes longer and the ?true? energy content of the fluctuations reaches values comparable to those predicted by the QS expression only after several Joule times.

As we expect, this discussion indicates that neglecting the time derivative of bi in the induction equation is problematic when the magnetic Reynolds number reaches moderate values.

A characteristic feature of MHD flows subject to a strong external magnetic field
is the appearance of a strong anisotropy in the flow. In the QS approximation this
is easily seen by observing that in (2.14) only Fourier modes with wave vectors
having a non-zero projection onto Bext are affected by the extra Joule damping. In
i
order to quantify the anisotropy we follow the approach of

When the flow is completely isotropic, we have tan2 ?u = 2 implying ?u 54.7?. If the flow becomes independent of the z-direction then tan2 ?u ? ? or equivalently ?u ? 90?. Figure 4 shows the evolution with time of ?u for the different runs. At where Q(k) = ui(k, t0) 2 depends only on the norm of k (note that for calculus purposes we have replaced summations by integrals). Switching to spherical coordinates (kz = k cos ? , kx = k sin ? cos ?, ky = k sin ? sin ?), we observe that the only difference between Ax, Ay and Az comes from the angular integrations: Ax = I ax, Ay = I ay, Az = I az where,

N = 1, the anisotropy is only important for the QS, Rm = 0.1 and Rm = 1 runs. For N = 10, all the runs become highly anisotropic.

The initial anisotropy in the magnetic field can also be computed exactly. At time t0 + t ( t 1), bi(k) is computed using (3.4): bi(k, t0 + t) = iBzext kzui(k, t0) t. This means that at t0 + t, (3.7) can be rewritten as: with

, Ax = Ay = Az = dkxdkydkz kx2 kz2Q(k), dkxdkydkz ky2 kz2Q(k), dkxdkydkz kz4Q(k), 100 80 to its initial value since the velocity field remains largely isotropic (as it is at the beginning of the simulation). Instead, ?b evolves to a value compatible with an isotropic magnetic field. In the case N = 10, some strong oscillations in the magnetic anisotropy are observed. For the times considered, no clear trends in the mean are present. Note that some oscillations are also present in the velocity anisotropy, but are much less pronounced.

The preceding section indicates that, for our numerical simulations at magnetic Reynolds numbers of the order 10?1 ? 1, the QS approximation and MHD produce nearly identical results. For higher values of Rm, the QS approximation is not valid and has to be replaced to predict the flow accurately. Since magnetic fluctuations remain small in all the runs performed, it is natural still to consider a linearized induction equation. However, results reported in § 3.2.2 support the idea that neglecting the time derivative of bi in the induction equation is not appropriate in the context of moderate Rm.

We thus consider here an intermediate approximation which is defined by the following simplified MHD equations: 1 ?t ui = ??i(p/?) ? uj ?j ui + (µ?) Bjext ?j bi + ? ui, ?t bi = Bjext ?j ui + ? bi. (4.1) (4.2) This approximation will be referred to as the quasi-linear (QL) approximation since only the nonlinear terms involving the magnetic field are discarded whereas the nonlinear convective term in the velocity equation is retained. Of course, if ?t bi is neglected in (4.2) we immediately recover the quasi-static approximation.

Equation (4.2) is nothing other than a diffusion equation for the magnetic field with a source term given by Bjext ?j ui. In Fourier space, the solution of this equation is easily obtained and reads, t0 t bi(k, t) = bi(k, t0) exp(??k2t) + i d? kj Bjext ui(k, ? ) exp(??k2(t ? ? )).

(4.3) From the first term on the right-hand side of (4.3), we see that the initial condition for bi is damped more rapidly with increasing magnetic diffusivity (if v and L are constant this is equivalent to a decrease in the magnetic Reynolds number). Note that with the initial condition we have chosen for the magnetic field, i.e. bi(t0) = 0, this first term vanishes. At high magnetic diffusivity, the integral in (4.3) converges to (2.12) and the QS approximation holds. More explicitly, we have the situation depicted in figure 6. The interval between the dashed lines represents the support in which exp(??k2(t ? ? )) is significant and thus where there is some contribution to the integral of (4.3). As ? increases, this interval becomes smaller, and ui(k, t) may be assumed constant in that short period of time. The integration is then immediate and we obtain, bi(k, t) = bi(k, 0) exp(??k2t) + ik?jBk2jext [1 ? exp(??k2(t ? t0))]ui(k, t), (4.4) ui 0

ui(k,?) t which converges to (2.12) in the limit ? ? ?. Thus, the time history of ui(k, t) plays a role only when ? is relatively small, in which case the exponential has a wider support.

In order to compare the QL approximation with MHD, we have performed the same numerical simulations as described in § 3, but this time using (4.1) and (4.2) instead of the QS approximation.

In figures 7 and 8, we present the time history of the kinetic energy (as defined by (3.2)) obtained from both MHD and the QL approximation. For reference, we have also included the predictions obtained using the QS approximation. For N = 1, the QL approximation and MHD agree nearly perfectly for all values of the magnetic Reynolds number. For N = 10, the agreement is still very good.

Figures 9 and 10 represent the time evolution of the energy of the magnetic fluctuations (defined by (3.3)) for the different runs. In all cases, there is a systematic overestimate of the peak energies by the QL approximation. This overestimate is more pronounced for the case at N = 1 than for those at N = 10. 10 20

In figures 11 and 12, the anisotropy angle ?u computed from the QL approximation and MHD is displayed. For reference, we have also included the anisotropy evolutions predicted using the QS approximation, which, as expected, are inadequate, especially for Rm = 10 and 20. In the runs with N = 1, the anisotropy predicted by the QL approximation is always more pronounced than for MHD. For the runs at N = 10, the same remark holds for the beginning of the decay. After a certain time, the trend inverses and the anisotropy is more pronounced in the case of MHD. This appears to be due to a rapid saturation of anisotropy in the QL runs.

The comparison of the anisotropy angles ?b are presented in figures 13 and 14. Here, the trend is given by an underestimate of ?b by the QL approximation. The discrepancy is somewhat more important for the runs where N = 1.

The initial trends observed for both ?u and ?b are to be expected. Indeed, it is clear that the additional nonlinear terms present in the MHD equations tend to restore isotropy. This effect will be more pronounced at the beginning of the decay when the flow is more turbulent. In the case of ?u, it is therefore natural to observe an initial overestimate of ?u by the QL approximation. Similarly, we know from MHD results discussed earlier that ?b starts from an initial value of 39.2? and evolves progressively towards values close to the isotropic value of 54.7?. This trend should be slower in the QL case because of the absence of the nonlinear terms and this is exactly what is observed in figures 13 and 14.

The anisotropy angles ?u and ?b provide a scalar measure of anisotropy. For the
purpose of modelling MHD flows, especially flows with mean deformation, it is also
important to have a directional (tensorial) one-point description of anisotropy. A
one-point statistical measure of anisotropy for the hydrodynamic field is possible
in terms of the structure anisotropy tensors, introduced by Kassinos, Reynolds and
Rogers (2001) and

ui = ijk?k,j , ?i,kk = ??i , ?k,k = 0. (4.5) In the above relations, the superscript is used to specify the fluctuating parts of each variable used. Furthermore, ?i is the fluctuation of the vorticity of the flow: ? = ? × u . Note also that ? is a local quantity that contains non-local turbulence information.

Here, we make use of simpler definitions that are valid for homogeneous turbulence. The Reynolds stress tensor can be expressed as

Rij =

Eij (k) d3k,
(4.6)
where Eij (k) ? u? i(k)u? j?(k) is the velocity spectrum tensor, k is the wavenumber vector,
hats denote Fourier coefficients and ? denotes a complex conjugate. (In homogeneous
fields, discrete Fourier expansions can be used to represent individual realizations
in a box of length L; then the discrete cospectrum of two fields f and g is given
by X? ij (k) = (L/2?)3f?i(k)g?j?(k), where the bar represents an ensemble average over
the box. The cospectrum of two fields Xij (k) is the limit of the discrete cospectrum
X? ij as L ? ?. Here we use Xij (k) ? f?i(k)g?j?(k) as a shorthand notation, but the
exact definition should be kept in mind.) Note that if u1 = 0 everywhere, then
R11 = 0. We have emphasized

The structure dimensionality tensor is

Dij = kikj k2 Enn(k) d3k.

Fij = 1 k2 Wij (k) d3k.

Like Rij , Dij is dominated by the large-scale energy-containing turbulence. We see that Dij is determined by the energy distribution along rays in k-space. If the turbulence is independent of x1 then D11 = 0, since there is no energy associated with modes having a non-zero k1 component of the wavenumber vector.

The circulicity tensor is where Wij (k) is the (fluctuating) vorticity spectrum tensor. Hence, Fij is determined by the vorticity of the large-scale energy-containing turbulence. If the large-scale vorticity is aligned with the x1-axis, then Fij = 0 except for F11. Fij provides information on the large-scale circulation of the turbulence.

For homogeneous turbulence, the traces of all three tensors are equal to twice the kinetic energy:

Rii = Dii = Fii = q2 = 2Ek.

Rij + Dij + Fij = q2?ij .

Moreover, for homogeneous turbulence these tensors are not linearly independent; they satisfy a constitutive relationship: It is convenient to normalize each of these tensors by their trace: rij = Rij /Rkk, dij = Dij /Dkk, fij = Fij /Fkk.

Of course, rii = dii = fii = 1 and any particular component of the normalized tensors can vary only between 0 and 1. For isotropic turbulence, rij = dij = fij = 13 ?ij . (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) 1 2 3 4 5 6 1 2 3 4 5

6 33

Anisotropy invariant maps, such as the one introduced by

The evolution of the componentality and dimensionality tensors is shown in figures 15?18. The general trends are consistent with the picture of anisotropy obtained from ?u and ?b. For example, anisotropy is more pronounced at N = 10 than at N = 1, while at a given N , anisotropy becomes progressively suppressed as Rm is increased and nonlinear effects become important. The QS approximation (which implicitly assumes Rm ? 0) is of course not able to capture this trend, and the quality of its predictions is unsatisfactory especially for Rm = 10 and Rm = 20. It is therefore important that the QL approximation captures enough of the nonlinear effects to be able to adequately represent this trend for the range of Rm values that we have tested.

The levels of componentality anisotropy (see figures 15 and 16) predicted by the
QL approximation are in excellent agreement with those produced by the MHD
runs. Perhaps the fact that nonlinear terms are retained in the momentum equation
contributes to this excellent agreement. Also, it was shown in

As shown in figures 17 and 18, the overall agreement between the dimensionality anisotropy levels predicted by the QL approximation and those obtained via MHD is satisfactory but not perfect. The QL approximation consistently overpredicts the dimensionality anisotropy at N = 1, whereas this trend is reversed at N = 10, where, at least at larger times, MHD seems to predict higher anisotropy levels for the dimensionality. However, the QL approximation exhibits the correct trends as Rm is varied and provides a systematic improvement over the QS approximation for all the runs performed.

Overall, the structure anisotropy tensors confirm the traditional picture of elongation of the energy-containing structures in the direction x3 of external mean magnetic field. This elongation is evidenced by the fact that d33 is suppressed relative to d11 and d22, and it is more pronounced in the case N = 10. The turbulence tends to become two-dimensional; note, however, that it remains three-component (none of the rij components is appreciably suppressed).

The quasi-static (QS) approximation offers a valuable engineering approximation for the prediction of MHD flows at small magnetic Reynolds numbers Rm 1. However, important technological applications, such as advanced propulsion and flow control schemes for hypersonic vehicles, involve MHD and MGD flows at moderate magnetic Reynolds numbers 1 Rm 20. In order to devise successful schemes for the prediction of these technological flows we need to understand better the intermediate regime that bridges the domain where the QS approximation is valid and the high-Rm regime, where full nonlinear MHD is the only resort.

By studying the case of decaying homogeneous MHD turbulence, we have established that the quasi-static (QS) approximation is valid for Rm 1, but progressively deteriorates as Rm is increased beyond 1. The magnetic Stuart number does not seem to have a strong effect on the accuracy of the QS approximation. That is, at a given Rm, the accuracy of the QS approximation is roughly the same for N = 1 as it is for N = 10.

We have studied another approximation, the QL approximation, for use at higher Rm. As with the QS approximation, this approximation assumes small magnetic fluctuations, but it resolves the time dependence of these fluctuations explicitly. The QL approximation performs like the QS approximation for Rm 1, but has the advantage that it retains good agreement with MHD for 1 Rm 20. It should be noted that Rm = 20 is the highest value of the magnetic Reynolds number that we have tested during this effort. Therefore, our numerical simulations indicate that the QL approximation should be adopted in place of the QS approximation for flows with a moderate value of the magnetic Reynolds number (1 Rm 20). At higher values of the magnetic Reynolds number, we cannot avoid using the full MHD equations (at least for the type of flow considered here). Numerical tests performed at Rm = 50 and Rm = 100 (not reported in the article) show that the QL approximation is inadequate in that regime. Thus, the QL approximation cannot be applied directly to study the dynamo problem.

In terms of computational costs, the QS approximation is clearly the cheapest of the three methods used during our study. It has fewer nonlinear terms to evaluate, and the time step required to advance the flow is governed by the time scale of the velocity field which, for most industrial cases involving liquid metal, is significantly longer than the time scale of the underlying magnetic field. There is no doubt that the QS approximation should be the approximation of choice for the prediction of flows with Rm 1.

The computational cost of solving directly the QL approximation transport equations does not depart enormously from that of solving the MHD equations, but nevertheless allows a reasonable gain since fewer nonlinear terms need be evaluated. The appeal of the QL approximation lies more in the prospect of simpler turbulence models for conductive flows at moderate magnetic Reynolds number. Indeed, the structure of (4.1) and (4.2) is simpler than that of the MHD equations. The gain in simplicity is even more substantial in terms of the Reynolds-averaged equations. We thus have a strong hope that devising turbulence models in the framework of the QL approximation should be an easier task than trying to tackle the MHD equations. In fact, we are currently engaged in the development of structure-based closures of the QL approximation for homogeneous turbulence in a conductive fluid subject to mean deformation and a uniform external magnetic field. This effort builds on earlier work that dealt with the modelling of decaying homogeneous MHD turbulence.

The authors are grateful to the Center for Turbulence Research for hosting and providing financial support for part of this work during the 2002 Summer Program. S. K. wishes to acknowledge partial support of this work by AFOSR. B. K. and D. C. are researchers of the Fonds National pour la Recherche Scientifique (Belgium). This work has also been supported in part by the Communaute´ Fran c¸aise de Belgique (ARC 02/07-283) and by the contract of association EURATOM ? Belgian state. The content of the publication is the sole responsibility of the authors and it does not necessarily represent the views of the Commission or its services.