Coarse-grained incompressible magnetohydrodynamics: analyzing the turbulent cascades

Abstract

We formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales ℓ. In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the 'bare' velocity and magnetic fields are 'renormalized' by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. Particular attention is given to the effects of these subscale terms on the balances of the quadratic invariants of ideal incompressible MHD-energy, cross-helicity and magnetic helicity. At large coarse-graining length-scales, the direct dissipation of the invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length ℓ lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stress and subscale EMF, we derive rigorous upper bounds on the effective dissipation they produce in terms of scaling exponents of the velocity and magnetic fields. These bounds provide exact constraints on phenomenological theories of MHD turbulence in order to allow the nonlinear cascade of energy and cross-helicity. On the other hand, we prove a very strong version of the Woltjer-Taylor conjecture on conservation of magnetic helicity. Our bounds show that forward cascade of magnetic helicity to asymptotically small scales is impossible unless 3rd-order moments of either velocity or magnetic field become infinite.

Decoupled Cascades of Kinetic and Magnetic Energy in Magnetohydrodynamic Turbulence

“…This can be understood by considering that in a turbulent flow, the strain S, being a derivative, is dominated by the small-scales, whereas B is dominated by the largescales, near the magnetic spectrum's peak, leading to decorrelation effects. To analyze how magnetic field-line stretching operates at different length-scales, we utilize a coarsegraining approach for diagnosing multi-scale dynamics [7,16]. A coarse-grained field which contains modes at length-scales > is defined by…”

“…where ∇·[· · · ] represents spatial transport terms. Dissipation terms, ν|∇u| 2 and η|∇B| 2 , are mathematically guaranteed to be negligible [16,19] at scales ( ν , η ), with ν and η the viscous and resistive length scales, respectively.…”

Local approach to the study of energy transfers in incompressible magnetohydrodynamic turbulence

“…Finally, let us highlight the fact that space filtering has attracted a lot of interest in the past few years, in order to investigate the local physics of turbulent MHD flows. For instance, separate energy balances for the kinetic and magnetic energies have been derived in order to study the scale locality of the double cascade, the effect of subscale terms, and kinetic scales [62][63][64][65]. This approach has also been used in Hall-MHD numerical data in order to quantify the correlation between regions of nonzero scale-to-scale energy transfers and the presence of coherent structures in these areas [66].…”

3D turbulent reconnection: Theory, tests, and astrophysical implications

“…Other crucial properties are also quite distinct. For example, it is well-known that magnetic helicity undergoes an inverse cascade but no forward cascade in MHD turbulence and is thus very well-conserved (Berger, 1984;Aluie, 2017). This conservation is the cornerstone of the theory of large-scale astrophysical dynamos, as shown by Vishniac and Cho (2001).…”