On the inertial range bounds of K-41 like Magnetohydrodynamics turbulence

TL;DR

This paper rigorously establishes bounds for the inertial range in K-41 like MHD turbulence, proving the spectral slope is proportional to -5/3 within a specific wave number range, and analyzes the spectral energy decay of solutions.

Contribution

It provides the first rigorous mathematical bounds for the inertial range in K-41 like MHD turbulence and analyzes spectral energy decay of Leray solutions.

Findings

Inertial range bounds for K-41 like MHD turbulence are explicitly formulated.

Spectral energy of the MHD system is bounded and decreases over time.

Leray weak solutions are shown to be bounded in Fourier space.

Abstract

The spectral slope of Magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; $ -3/2 $ is the spectral slope in Kraichnan-Iroshnikov-Dobrowolny (KID) theory, $ -5/3 $ in Marsch-Matthaeus-Zhou's and Goldreich-Sridhar theories also called Kolmogorov-like (K-41 like) MHD theory, combination of the $-5/3$ and $-3/2$ scales in Biskamp and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig [Physica D 241(2012) 426-438], we establish inertial range bounds for K-41 like phenomenon in MHD turbulent flow through a mathematical rigour; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to $-5/3$ is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.