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The Lagrangian-averaged model for magnetohydrodynamics turbulence and the absence of bottleneck

We demonstrate that, for the case of quasi-equipartition between the velocity and the magnetic field, the Lagrangian-averaged magnetohydrodynamics alpha-model (LAMHD) reproduces well both the large-scale and small-scale properties of turbulent flows; in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes alpha-model is somewhat limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. No such regions are found in LAMHD, making this method capable of large reductions in required numerical degrees of freedom; specifically, we find a reduction factor of 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Reynolds number.


I. INTRODUCTION
When large-scale numerical simulations of astrophysical or geophysical magnetohydrodynamics (MHD) are desired, all dynamical scales of the physical system are rarely, if ever, resolved. For this reason, sub-grid-scale (SGS) modeling of MHD dynamics in the context of computations in the geophysical and astrophysical context is required. This modeling can be achieved implicitly, in the simplest example by employing a dissipative numerical scheme, or it can be done explicitly by creating a Large Eddy Simulation (LES-see [1] for a recent review). Explicit methods for MHD are not as pervasive as they are in engineering, or for geophysical and atmospheric flows.
In fact, modeling for MHD is a relatively new field (see [2,3]). One problem with extending the LES methodology for hydrodynamic turbulence to MHD is that most LES are based upon eddy-viscosity concepts [1], which can be related to a known power law of the energy spectrum [4] (although generalizations can be devised, see e.g. [5]), or upon self-similarity. For MHD, the underlying assumption of locality of interactions in Fourier space is not necessarily valid [6,7] (a contradiction of self-similarity) and spectral eddy-viscosity concepts [8] cannot be applied in a straightforward manner as neither kinetic nor magnetic energy is a conserved quantity and the general expression of the energy spectrum is not known at this time [9,10,11,12,13,14,15].
Purely dissipative models [16,17] are inadequate as they ignore the exchange of energy at subfilter scales between the velocity and magnetic fields and such models have been shown to suppress small-scale dynamo action [18] and any inverse cascade from the sub-filter scales [19]. A satisfactory LES for MHD has been proposed for the case starting with some degree of alignment between the velocity and magnetic fields [19,20]. Other restricted-case MHD-LES are applicable to low magnetic Reynolds number [21,22,23]. Extensions of spectral models to MHD based on two-point closure formulations of the dynamical equations proposed recently look promising in the analysis of turbulent flows and of the dynamo mechanism [5]. Finally, though technically not an LES, there are also hyper-resistive models for MHD which require rescaling of the length (wavenumber) scales to a known direct numerical simulation (DNS) [18].
One model which can be written as an LES is the Lagrangian-averaged MHD (LAMHD) equations [24,25,26]. It has been shown to reproduce a number of features of DNS. In two dimensions (2D) for Taylor Reynolds numbers (R λ ) up to ≈ 5000 it has been shown to reproduce selective decay, the inverse cascade of mean-square vector potential, and dynamic alignment between the velocity and magnetic fields [27] as well as the statistics of small-scale cancellation [28] and inter-mittency [29]. In three dimensions (3D) at Reynolds numbers (Re) of ≈ 500, LAMHD reproduced the inverse cascade of magnetic helicity (associated with the development of force-free magnetic field) and the helical dynamo effect [30]. It has also been tested (up to kinetic Re ≈ 3000, magnetic Re ≈ 300) for its ability to predict the critical magnetic Reynolds number for a non-helical dynamo at low magnetic Prandtl number [31]. LAMHD performed well in all these tests. Its equivalent hydrodynamic model, the Lagrangian-averaged Navier-Stokes (LANS) equations, also performed well in tests at R λ 300 (see [32] and references in [33]). However, above Re ≈ 3000 (R λ ≈ 800), it was shown that placing the filter width in the inertial range leads to contamination of the super-filter-scale properties (such as the spectra) for LANS. We refer here to this effect as the super-filter-scale bottleneck, which may be different in nature from the viscous bottleneck observed in some DNS of the Navier-Stokes equations. The contamination may be linked to the formation of spatial regions in the flow with no internal degrees of freedom (so-called "rigid bodies") [33], which also correspond to the development of a secondary inertial range of the LANS equations at sub-filter scales. This super-filter-scale contamination provides an effective constraint on the filter size and, hence, on the available reduction of the total number of the (numerical) degrees of freedom (N dof ) needed to reproduce the large-scale dynamics of the flow at a given Reynolds number; a factor of ≈ 10 can be achieved. This limitation is not apparent in low and moderate Reynolds number (resolution) simulations (e.g., 64 3 LANS compared with 256 3 DNS) as the scale separation is not enough for the above-mentioned phenomenon of contamination of small-scale spectra because of rigid body regions in the flow to appear. The bottleneck (and superfilter-scale contamination) was not studied as such but neither was it observed in 2D LAMHD for high Reynolds number [27,28,29]. 3D LAMHD has only been tested at more moderate Reynolds number [30] (see also [34] for a recent review). The aim of the present work is, thus, to determine if LAMHD in three space dimensions, for higher Reynolds number develops problems similar to that of LANS. Specifically, we test for the existence of spatial regions with no available internal degrees of freedom. We show in the following that LAMHD behaves better in this respect than LANS, and, thus, continues to appear as a promising model for MHD flows.

II. THE EQUATIONS OF MOTION
We consider the incompressible MHD equations for a fluid with constant density, where v and b denote respectively the velocity and magnetic fields, p the pressure divided by the density, ν the kinematic viscosity, and η the magnetic diffusivity. As is well known, in incompressible MHD, Alfvén waves will travel along a uniform background field, b 0 . From linear perturbation analysis the dispersion relation between wavenumber, k, and frequency, ω, is The wave speed is |b 0 | and, assuming η = ν, the amplification factor is given by exp(−ηk 2 t). The ideal (η = ν = 0) quadratic invariants for MHD are in the L 2 norm. For example, the total energy is given by: The LAMHD equations [25] are given by where u (b) denotes the filtered component of the velocity (magnetic) field and Π the modified pressure. Filtering is accomplished by the application of a normalized convolution filter L : f →f where f is any scalar or vector field. By convention, we define u ≡v. LAMHD in the form given in Eqs. (4) is both computationally efficient and makes clear that Alfvén's theorem is preserved by the model: the smoothed magnetic field is advected by the smoothed velocity. In the remainder of this paper, we take η = ν (unit magnetic Prandtl number) and, thus, it is sufficient to introduce the same filtering for the velocity and magnetic fields in this case. This allows us to write LAMHD in LES form, We choose as our filter the inverse of a Helmholtz operator, Therefore, u = g α ⊗ v where g α is the Green's function for the Helmholtz operator, g α (r) = exp(−r/α)/(4πα 2 r) (i.e., the Yukawa potential), or in Fourier space,û(k) =v(k)/(1 + α 2 k 2 ).
The effective filter width is, thus, approximately α. With this choice, the Reynolds (turbulent) SGS stress tensor is given bȳ τ = α 2 ∇u · ∇u T + ∇u · ∇u − ∇u T · ∇u − ∇b · ∇b T − ∇b · ∇b + ∇b T · ∇b (6) and the divergence of the electromotive-force (emf) SGS stress tensor by In this form, the expression of the SGS tensors make explicit the fact that u = ±b Alfvén waves are preserved even in the subgrid scales. These u = ±b waves travel alongb 0 (the smoothed and unsmoothed fields are identical for uniform b 0 ) and the dispersion relation is where α K and α M are the filter widths for the smoothing of the velocity and magnetic fields, respectively. For α ≡ α K = α M and η = ν (the case we study), the wave speed , the strength of the smoothed background magnetic field reduced by an order α 4 k 4 term. The amplification factor is given by exp (−ηk 2 t(1 + α 2 k 2 /2)) for both u = −b waves traveling in the direction ofb 0 and u =b waves traveling anti-parallel tob 0 . Finally, the ideal quadratic invariants for LAMHD are in the H 1 α (f ) norm. For example, the total energy is given by a mixture of the smooth and rough fields, namely: We solve both sets of equations, Eqs. (1) and (4), for one specific instance of a decaying MHD flow, using a parallel pseudospectral code [35,36] in a three-dimensional (3D) cube with periodic boundary conditions. The initial conditions for the velocity and magnetic fields are constructed from a superposition of three Beltrami (helical) ABC flows to which smaller-scale random fluctuations are added with initial kinetic and magnetic energy E K = E M = 0.5, magnetic helicity where a is the vector potential and the brackets denote volume average), and the initial co-alignment of the fields, v · b |v||b| −1 ≈ 10 −4 (see [14,38] for details). A MHD-DNS with a resolution N 3 = 1536 3 (i.e., 1536 grid points in real space in each direction) and η = ν = 2 · 10 −4 is used as our high Reynolds number test case for the LAMHD model. The DNS computation is stopped when the growth of the total dissipation begins to enter the saturation phase (t = 3.7), at which time the Reynolds number based on the mechanical integral scale is Re ≈ 9200 and the Taylor Reynolds number ≈ 1100. The MHD flow resulting from the initial conditions employed has previously been analyzed for its spectral properties and for the spatial structures it develops [14,37,38]. In this paper, we perform a simulation with similar initial conditions and parameters but now using LAMHD at a resolution of 512 3 grid points; we also perform for comparison purposes a Navier-Stokes LANS run with the same initial velocity field but with b ≡ 0, on a grid of 512 3 points. In both cases, the filter width is α = 2π/18 (k α = 18) and is, thus, large enough to preclude any artifact of numerical resolution altering the results. Based on previous analyses [33,39], we estimate k max /k α η ≈ 2.4 (where k max is the maximum wavenumber resolved in the simulation, and k α η is the LAMHD dissipation scale) using computations conducted for η = ν = 6 · 10 −4 with a Reynolds number of Re ≈ 2200. However, the main point of using such a large filter is to test if LAMHD fails in the same way as LANS. We finally perform a LES simulation in a 256 3 grid using the LAMHD equations with the same viscosity and diffusivity as the 1536 3 DNS used for the comparison. In this way, we extend the Re ≈ 9200 computation in time by a factor of 3.

A. Spectral contamination in LANS for an ABC flow and its absence in the MHD case
One of the main findings of our preceding work with LANS on the Navier-Stokes equations is that a k +1 scaling develops in the (kinetic) energy spectrum at sub-filter scales; this leads to a contamination of super-filter scales because of detailed energy conservation (per triadic interactions). This LANS k +1 spectrum (together with super-filter-scale spectral contamination) has only recently been recognized, in the case of one specific forcing function at large Reynolds number [33], but such a spectral contamination has not yet been generally demonstrated (although theoretical arguments for the k +1 spectrum have been given in [33]). Thus, we first confirm its presence in a LANS simulation with the same viscosity and the (nearly) same initial conditions for the velocity field as for the MHD-DNS (and LAMHD runs) examined in this paper, and based on large-scale ABC flows with superimposed random noise at small scale. Due to the presence of random noise and considering the differences in resolution and the presence of a filter in the LAMHD runs, the initial conditions were not exactly reproduced, although the same procedure was used to generate them. In the present Navier-Stokes case, we find again what can be called an enhanced (super-filter-scale) bottleneck: the positive power-law spectral contamination of the kinetic energy spectrum E K (k) in the LANS run is observed for times after the peak of dissipation (see dotted line, Fig. 1a). The fitted spectrum is k +0.5 (note that k +1 requires the entire LANS spectrum to be resolved, and therefore has only been observed for much larger values of k max /k α η ). For the given parameters and initial conditions, we find the super-filter-scale bottleneck for LANS. However, when integrating the MHD equations with the Lagrangian model (dashed line, Fig. 1a) with these same parameters, no such contamination is present. Note that the spectra for the DNS-MHD are shown at the time of peak dissipation, while the spectra for the Lagrangianaveraged models are for a slightly later time in order to allow for the possible formation of rigid bodies, which are known to be at the source of the spectral contamination close to the filter wavenumber in the Navier-Stokes case. For this reason, and due to the slight differences in initial conditions, we have chosen to plot spectra normalized to that of the DNS at k = 14 to emphasize the scaling. For most of the inertial range (also in an approximate sense below the filter width α) the scaling of E K (k) is reproduced by the LAMHD simulation. The sub-filter scaling for LAMHD is not as steep as MHD, but is not a positive scaling law. The agreement for E M (k) is remarkable. More importantly, neither positive-power-law spectra nor contamination of the super-filter-scale spectra are evidenced at all.

B. The lack of rigid bodies in LAMHD in the large−α limit for unforced flows
Evidence for the development of rigid bodies in LANS (which led to its limited use as a LES) has only been shown for l ≪ α [33]. Since investigation of the large−α limit is not as computationally demanding as the small−l limit, it is interesting to look at this limit as a rough indication of what occurs for small α and smaller l. This approach has been employed both for the LANS Navier-Stokes case in two dimensions [40] and in three dimensions [33]. In such a case, the purpose is to examine the properties of the model itself, as opposed to trying to reproduce large-scale properties, the large-scale behavior being reduced to a very small span of wavenumbers. With this practice, the properties of the sub-filter-scales can be studied, to better understand the origin (or lack) of super-filter-scale contamination. We now use this limit to further explore the differences between LAMHD and LANS. We employ simulations for the two models with the same initial conditions as before, with η = ν = 5 · 10 −5 (Re ≈ 26, 000 at peak of dissipation for LAMHD), and a resolution of 256 3 grid points. Note that these dissipative coefficients are four times smaller than what was considered in the previous section since, for a fixed resolution, the achievable Reynolds number goes as α 2/3 . This follows for LANS from the predicted (and verified) degrees [33,39]. The scaling of LAMHD may differ, but the same value of the viscosity is employed for the two models, regardless.
For LANS, we observe the expected k +1 zero flux inertial range (see Fig. 2) which is followed by a viscous (sub-filter-scale) bottleneck feature, k +1.5±.2 , before the dissipative range proper. We conducted a second simulation with ν = 10 −4 and found a k 1.4±.3 spectrum. This is analogous to results for DNS of the Navier-Stokes equations where only the viscous bottleneck is observed at moderate Reynolds number and is preceded by an inertial range only for higher Reynolds. These viscous bottlenecks may be different in nature from the (super-filter-scale) bottlenecks discussed before, which are not associated to the onset of the dissipative range but to the development of a secondary inertial range in LANS below the filtering length, and result in contamination of the large (resolved) scales when the LANS equations are used as an LES. Having confirmed that our analysis from the forced LANS case extends to the decaying LANS simulation, we now apply it to LAMHD. The large−α LAMHD spectra are given in Fig. 3. Notably, there is no positive-powerlaw spectrum.
FIG. 2: Spectrum of kinetic energy for a 256 3 grid with k α = 3 (ν = 5·10 −5 ) LANS, b ≡ 0 (Navier-Stokes case). The fitted grey line, k +1.1±.4 , agrees with the rigid-body hypothesis for the inertial range [33]. This slope is followed by a steeper slope attributed to a bottleneck, with k +1.5±.2 . order structure functions for isotropic, homogeneous turbulence. Exact results for these structure functions have been found for incompressible MHD [41] and for LAMHD [29]. The latter are, in terms of both the smooth fieldsz ± ≡ u ±b and the rough fields z ± ≡ v ± b (where the z-fields are called the Elsässer variables): where . denotes volume averaging, For sub-filter scales (l ≪ α),z ± ∼ l 2 α −2 z ± and the scaling law becomes dimensionallyzzz ∼ εl.
This implies a sub-filter scale spectrum corresponding to the invariants E α ± ≡ ||z ± || 2 α /2 for the ideal non-dissipative case. We then have E α as for LANS [39]. Recall that in the flux relation, Eq. (10), ε α ± stands for the energy transfer and dissipation rate of E α ± . Hence, the prediction, Eq. (11), for the spectra, The spectra shown in Fig. 3 for large−α LAMHD do not exclude, due to the large uncertainties of the fitted power laws, the predicted k −1 spectra. A spectral prediction for LAMHD can also be arrived at by dimensional analysis of the spectrum which follows the scaling ideas originally due to Kraichnan [42] and which is developed for LANS in Ref. [43]. Here, the energy dissipation rate, ε α ± = dE α ± /dt, is related to the spectral energy density by where t k is the turnover time for an eddy of size ∼ k −1 . This turnover time is related to a "velocity," or, for αk ≫ 1, In the Iroshnikov-Kraichnan [11,12] (hereafter, IK) phenomenology, Alfvén waves (corresponding to either z ∓ = 0) can only interact nonlinearly when they collide along field lines (along which they travel in opposite directions). The characteristic time for an Alfvén wave is t A ∼ (kB 0 ) −1 . If this is less than t k , the effective transfer time t T is increased, t T ∼ t 2 k /t A . Substitution of this new transfer time into Eq. (12) yields, instead of Eq. (13) or, for αk ≫ 1, The spectra shown in Fig. 3 for large−α LAMHD also agree with the IK predicted spectra, Eq.
(16). In fact, the spectra more closely correspond to this prediction; this is consistent with the fact that, for this flow, an IK spectrum E(k) ∼ k −3/2 is observed at large scale (followed by a weak turbulence anisotropic spectrum E(k ⊥ ) ∼ k −2 ⊥ at small scale) [14]. Again, simulations at higher resolution are needed for a definite answer and the result may not be universal as shown for example in the context of reduced MHD dynamics due to the presence of a strong uniform magnetic field B 0 [44] or for MHD with a strong B 0 [13].
Another indication of the zero-flux regions in LANS is found by examining the spatial variation of the cubed increments associated with the scaling laws δu (l)δu i (l)δv i (l) for LANS and Why does LAMHD not exhibit the same spectral contamination as LANS? One possible cause is the hyperdiffusivity term seen in the LES form for LAMHD, Eq. (7), whereas there is no hyperviscosity-like term in LANS. To test if this hyperdiffusion is responsible for the lack of spectral contamination in LAMHD, we removed the hyperdiffusion by settingτ b = 0 in Eqs. (5) or, equivalently, by substituting η∇ 2b for η∇ 2 b in Eqs. (4). We then start the run from the same initial conditions but now with these new equations employing α = 2π/33 and ν = η = 2 · 10 −4 at a resolution of 384 3 (with hyperdiffusion, a smaller resolution of 256 3 is possible, see Section III D). Note that such a modified LAMHD model is not expected to, nor found to, perform well as a SGS model; this numerical experiment is performed here only in order to assess the effect of the hyper-diffusive term introduced by the α modeling. We find that hyperdiffusion is not responsible for the lack of a k +1 spectral contamination in LAMHD (see Fig. 5). Other possible causes for LAMHD not exhibiting the super-filter-scale bottleneck as does LANS are the actual physical differences between the two fluids that are modeled, Navier-Stokes and MHD. First, unlike incompressible Navier-Stokes, MHD supports oscillatory solutions (Alfvén waves) which are linked to enhanced spectral nonlocality of energy transfer [6,45] leading to dynamic interactions between widely separated scales. For Navier-Stokes, the depletion of energy transfer due to local interactions at some cutoff in wavenumber is believed to bring about the bottleneck effect [46,47,48,49]. However, related to the spectrally nonlocal energy transfer via Alfvén waves, MHD does not seem to exhibit a bottleneck in its spectra between the inertial and dissipative ranges [14]. As LAMHD supports Alfvén waves at all scales (and alters their dissipation and wave speed appreciably only for sub-filter scales), the same physics could be behind the lack of a super-filter-scale bottleneck in LAMHD.
Another difference between the fluid and MHD cases is the geometry of the dissipative structures: one finds vortex filaments for Navier-Stokes at high value of the vorticity, and current and vorticity sheets for MHD; sheets which are found to roll-up at high Reynolds number [38]. It has been claimed that the development of helical filaments in the fluid case can lead to the depletion of nonlinearity and the quenching of local interactions [50,51] and, hence, to the viscous bottleneck. A similar energy transfer depletion may occur in LANS. In [33] evidence is presented that Taylor's frozen-in turbulence hypothesis applied to Lagrangian averages leads to the formation of "rigid bodies" in the flow wherein there are no internal degrees of freedom and no transfer of energy to smaller scales (i.e. regions with ε ∼ δu 3 /l = 0 as well as ω × v = 0). These regions are likely related to the shorter, thicker vortex filaments formed and the suppression of vortex stretching dynamics as α is increased [52]. As MHD has spectrally non-local transfer (e.g., velocity at large scales does stretching of magnetic field lines at small scales) this leads to the break up of these rigid bodies in the LAMHD case and the breakup of the viscous bottleneck in the MHD case. The magnetic field interaction with the large scale velocity can re-enable transfer of energy to smaller scales of the velocity field. Indeed, defining the kinetic spectral transfer due to the Lorentz force as for LAMHD , and as for MHD, we see in MHD, this conservation is broken by the Lorentz force, where C is any material curve. As a result, while in ideal Navier-Stokes a material curve C defines the boundary of a vorticity tube with fixed strength, in MHD these structures are deformed and their vorticity content changed by the Lorentz force. A similar result follows for LAMHD and Breaking the conservation of circulation in this way can prevent the formation of a bottleneck.
For example, for the fluid case in the Clark−α model (which differs from LANS only in the conservation of Γ), it was also found that no super-filter-scale bottleneck was present [53].

D. LES Application
Having now shown that LAMHD does not suffer the same drawbacks with regards to energy spectra as LANS, we may turn our attention to a practical application. The purpose of a SGS model or LES is to make predictions about large Reynolds number flows at a reduced computational Time evolution of total enstrophy, j 2 + ω 2 ( j 2 + ω ·ω for LAMHD and j ·j + ω ·ω for the nonhyperdiffusive case). Note that LAMHD gives a better agreement to the total dissipation rate up to the maximum time that the high resolution DNS is performed. Also note that the DNS equivalent to the LAMHD run presented here is not feasible on presentday computers at a reasonable cost.
expense. From the scaling arguments in Refs. [33,39], using simulations conducted at Re ≈ 2200, and assuming a k −1 scaling, we can estimate α = 1/33 for a 256 3 LAMHD-LES "prediction" of our 1536 3 MHD-DNS. Time evolution of the energies and the total enstrophy are shown in Fig. 7 for much later times than reasonably attainable with the MHD DNS with present-day computers.
Also shown are results for solving the MHD equations, Eqs. (1) with ν = 2 · 10 −4 and a resolution of 256 3 : a so-called "unresolved DNS" and the non-hyperdiffusive modified-LAMHD from the previous section. Before the peak of dissipation, t ≈ 4, the unresolved DNS gives a poorer prediction of the total dissipation and total energy which is then followed by a significantly larger and somewhat later peak of dissipation, at t ≈ 5 than the resolved DNS and the LAMHD LES.
The non-hyperdiffusive LAMHD is not expected to perform well as a SGS model and it is seen to be clearly under-dissipative. The ratio of magnetic to kinetic dissipation is ≈ 1.5 for the DNS, Together with Fig. 7 (b) these ratios show that LAMHD achieves accurate total dissipation by an excess of magnetic dissipation and a reduction of kinetic dissipation (both at the small scales).
This feature has already been depicted in Fig. 15 of Ref. [27]. Compensated energy spectra for the peak of dissipation (t ∈ [2.7, 3.7]) are shown in Fig. 8. For the under-resolved DNS, we observe the appearance of a tail at large wavenumbers with a k 2 spectrum as predicted using statistical mechanics arguments for truncated systems in the ideal (ν = 0, η = 0) case [54].
The under-resolved spectra are not significantly different from the resolved DNS, but note that a reliable and convincing determination of spectral indices, beyond visual inspection, does require high resolutions. Comparing now the resolved DNS and the LAMHD run, the quality of the spectra are similar for scales larger than α. Recall that differences at the largest scales, stem from the differences in initial conditions as stated in Section III A, and from time evolution of the flow. Finally, noting that the computer saving here is 6 3 in memory and 6 4 in running time, we conclude that the LAMHD continues to behave satisfactorily, as already shown both in two space dimensions [27,28,29] and in 3D [30], in particular in the context of the dynamo problem of generation of magnetic fields by velocity gradients; thus, LAMHD may prove to be a useful tool in many astrophysical contexts where magnetic fields are dynamically important, such as in the solar and terrestrial environments, or in the interstellar and intergalactic media. We also computed a 512 3 LAMHD-LES (α = 1/85) which retains more of the small scales than the 256 3 LAMHD-LES while still yielding significant computational savings over the 1536 3 DNS.
We compare this with the result for α = 1/18 (chosen not as a LES but to stress the model) in Fig.   9. The structure of sheets observed in MHD dissipative structures is preserved in the LAMHD simulations, although current and vortex sheets become thicker in LAMHD as a result of the filter as α is increased. This is necessary to achieve reduced resolution computations. Note that these sheets are different in nature from the fat 'rigid bodies' observed in LANS, as the turbulent energy transfer to small scales is not quenched and there is no super-filter-scale bottleneck.

IV. DISCUSSION
In this paper, we have tested the LAMHD model against high Reynolds number direct numerical simulations (up to Reynolds numbers of ≈ 9200) and in particular we have focused our attention on the dynamics of small scales near the α cut-off. We find that the small-scale spectrum presents no particular defect; specifically, we find that, unlike in the hydrodynamical case, the Lagrangian-averaged modeling for MHD exhibits, even at large Reynolds numbers, neither a positive-power-law spectrum nor any contamination of the super-filter-scale spectral properties.
This difference between LANS and LAMHD is not due to the inclusion of a hyper-diffusive term in LAMHD that stems from the derivation of the model; rather, it stems from fundamental differences between hydrodynamics and MHD. (19)). The forcing term is associated with the Alfvén waves, and represents the removal of circulation (and of kinetic energy) that is transfered to the magnetic field. Note that in Fourier space, the term scales as kE M (k) and is dominant compared to the dissipation in the inertial range. This term precludes the formation of rigid bodies, giving as a result a larger net flux towards smaller scales and a resulting larger dissipation in MHD/LAMHD. This is illustrated in Fig. 4. This sink of circulation may also be the cause of the lack of a viscous-scale bottleneck in MHD. In LANS it was shown [33,53] that conservation of the circulation (except for viscosity) leads to the formation of rigid bodies that fill a substantial volume of the fluid, and that in turn substantially decrease the energy flux to small scales, reduce dissipation, and create the super-filter scale bottleneck. In LAMHD, the destruction of sub-filter-scale rigid bodies by large scale magnetic field and shear results as the presence of a magnetic field permits the development of long-range interactions in spectral space [6,7,45]. This can also explain why α−models for other non-local equations, or for problems that do not preserve the circulation provide good SGS models. As an example, the use of LANS in primitive equations ocean modeling gives satisfactory results, e.g. in its reproducing the Antarctic circumpolar current baroclinic instability that can be seen only at substantially higher resolutions when using direct numerical simulations [55].
Energy is dissipated in MHD flows through two different processes. Viscosity is responsible for the dissipation of mechanical energy, while Ohmic losses are responsible for dissipation of magnetic energy. Mechanical and magnetic energy are not conserved separately, but rather coupled as illustrated by the existence of Alfvén waves, which correspond to oscillations of the magnetofluid with the velocity field parallel or anti-parallel to the magnetic field, and associated to the interchange of magnetic and kinetic energy. In MHD, it is believed that most of the total energy in the flow is finally dissipated (mediated by this interchange) through Ohmic losses, in a process that involves reconnection of magnetic field lines. This is supported by several simulations of MHD turbulence [56,57] and is consistent with phenomenology. While in hydrodynamics small scales are permeated by a myriad of vortex filaments, in MHD the dominant dissipative structures are current sheets, where strong gradients of the magnetic field and their associated strong currents lead to rapid Ohmic dissipation. Sub-grid models attempt to replace the physical processes of smallscale dissipation by processes that mimic the non-linear transfer of energy to smaller scales (where energy is in reality dissipated, but now in scales that are not resolved by the model). In traditional LES, this is done with enhanced turbulent viscosities. Note that the eddy viscosity is not obtained from the linear dissipative term (the term that describes the actual physical process responsible for the dissipation) but from the non-linear terms in the equations (the terms that describe the coupling between fields at different scales). The final goal is not to capture the dissipation processes, but to be able to preserve (with computational gains) the large scale dynamics.
Lagrangian averaged models take a different (although related, see e.g., [29]) approach. Besides adding (in some cases, as in the case of MHD) an enhanced viscosity, the non-linear terms are modified at small scales. This modification changes the time-scale of the energy cascade, and as a result changes the scaling law of the energy spectrum E(k) at sub-filter scales. This change leads to changes in the dissipation, as the dissipation is in the original equations proportional to k 2 E(k). The end result (an enhanced dissipation that is intended to mimic the transfer of energy to smaller scales in the unresolved scales) should be the same as in a traditional LES: gains in computing costs preserving as much information of the large scale flow as possible. As in the case of LES, the actual dissipation process is not as important as the fact that large-scale dynamics should be reproduced with minimal contamination by the sub-grid model. We believe the results presented here (and in earlier work [27,28,29,30,31]) show this is the case, and allow the use of the LAMHD equations as a subgrid model of MHD turbulence. However, considering the differences observed between LANS and LAMHD, we discuss the dissipation processes in LAMHD. Two mechanisms for dissipation can be identified in LAMHD: dissipation of mechanical energy through the viscosity, and dissipation of magnetic energy through (enhanced) Ohmic losses.
From the equations, the total variation of energy goes as [27]: dE/dt = −ν ω ·ω −η j 2 and as a result the mechanical energy dissipation scales as k 2 E V (k) while the magnetic energy dissipation scales as (1 + α 2 k 2 )k 2 E M (k). The extra k 2 factor in the latter gives more dissipation than in the LANS case. This excess of magnetic dissipation in LAMHD mimics, as previously mentioned, the dominant contribution to dissipation by Ohmic losses in MHD. This hyperdiffusion is required in the sub-filter scales to accurately model the total energy dissipated at the unresolved scales. This was demonstrated by our experiments with a modified LAMHD, where we (non-consistently) removed the hyperdiffusive term and found the resulting model to fail as a LES.
Yet another way to understand the differences between LANS (for incompressible isotropic and homogeneous flows) and LAMHD is to consider the derivation of these models [25] using the generalized Lagrangian-mean (GLM) formalism [58]. This form of Lagrangian averaging describes wave, mean-flow interactions. For the case of weak turbulence, where the nonlinear transfer is dominated by waves, GLM requires in principle no closure. As a result, GLM gives an exact closed theory for the evolution of the wave activity. On the other hand, when there are no waves (as in incompressible Navier-Stokes) or when eddies dominate the transfer, a closure is required. One possible closure assumes that fast fluctuations are just advected by the mean flow (basically, Taylor's frozen-in hypothesis for the small scale turbulent fluctuations) and leads to the several "α-models" that include LANS and LAMHD. In this context, it is not surprising for subgrid models based on GLM to perform better in the presence of Alfvén waves (for LAMHD) or Rossby and gravity waves (for the Lagrangian-averaged primitive equations [55]). The more relevant the waves are to the dynamics, and to the non-linear coupling of modes in the system, the less relevant is the hypothesis behind the closure. Furthermore, the α-model equations can then be expected to be a better approximation to the problem at hand, that is, to be closer to an exact closure of the original system of equations.
In the fluid case, the application of the " Taylor It was noted in [30] when assessing the properties of LAMHD in the dynamo context that the overall temporal evolution was satisfactory, e.g. with a correct growth rate, although the growth of the magnetic seed field started slightly earlier in the LAMHD run than in the DNS. One can speculate as to whether this delay is linked to the super-bottleneck effect of LANS (which prevails when the magnetic field is negligible compared to the velocity, the two modeling approaches, LAMHD and LANS, being dynamically consistent). This point is left for future work; one could determine as well at what ratio of magnetic to kinetic energy the overshooting of spectra in LANS disappears for LAMHD.
Also deserving of a separate study is to investigate the behavior of LAMHD when anisotropies that appear at small scales [14] are present; this would be essential when a uniform magnetic field is imposed to the overall flow. The evaluation of the behavior of the model when computing spectra in the perpendicular and parallel directions (with respect to a quasi-uniform magnetic field, computed by locally averaging the field in a sphere of radius comparable to the integral scale) remains to be done but is somewhat time consuming. An analysis of the structures that develop in the highly turbulent LAMHD flow studied in the preceding section is also left for future work; of particular interest is the occurrence of Kelvin-Helmholtz like roll-up of current sheets as observed at high resolution [14]; however, the choice of the parameter α in the present paper was made on the basis of questioning the existence or lack thereof of a rigid-body high-wavenumber k +1 spectrum and, thus, was not optimized for the study of the inertial range properties of the flow for which a much smaller value of the length α could be used.
Finally, how far resolution can be reduced when using LAMHD as a LES for various statistics of interest will also require further detailed study. The present study shows that, to reproduce the super-filter-scale energy spectrum in three dimensions, gains by a factor of 1300 in computing time can be achieved. The need to reproduce higher order statistics can decrease these gains.
As an example, in two-dimensional MHD, it was shown that gains when using LAMHD as a subgrid model depend for high order moments on the order that one wants to see to be accurately reproduced [29].