Small scales and anisotropy in low $Rm$ magnetohydrodynamic turbulence

TL;DR

This paper provides rigorous estimates for small-scale sizes and attractor dimensions in low magnetic Reynolds number MHD turbulence, linking mathematical bounds to known physical phenomena and anisotropy conditions.

Contribution

It introduces a mathematically rigorous method to estimate the attractor dimension and small scales in low Rm MHD turbulence, incorporating dissipation modes and anisotropy conditions.

Findings

Derived upper bounds for attractor dimension based on phase space volume growth rates.

Identified mode distributions that minimize dissipation, reflecting known MHD turbulence properties.

Established a necessary condition for flow anisotropy and tridimensionality.

Abstract

In this paper, we derive estimates for size of the small scales and the attractor dimension in low $Rm$ magnetohydrodynamic turbulence by deriving a rigorous upper bound of the dimension of the attractor representing this flow. To this end, we find an upper bound for the maximum growth rate of any $n$-dimensional volume of the phase space by the evolution operator associated to the Navier-Stokes equations. As explained by Constantin et al. (J. Fluid Mech., 1985), The value of $n$ for which this maximum is zero is an upper bound for the attractor dimension. In order to use this property in the more precise case of a 3D periodical domain, we are led to calculate the distribution of $n$ modes which minimises the total (viscous and Joule) dissipation. This set of modes turns out to exhibit most of the well known properties of MHD turbulence, previously obtained by heuristic considerations such as the existence of the Joule cone under strong magnetic field. The sought estimates for the small scales and attractor dimension are then obtained under no physical assumption as functions of the Hartmann and the Reynolds numbers and match the Hartmann number dependency of heuristic results. A necessary condition for the flow to be tridimensional and anisotropic (as opposed to purely two-dimensional) is also built.