Determining wavenumbers for Hall and electron magnetohydrodynamics turbulence

TL;DR

This paper extends the concept of determining wavenumbers from fluid dynamics to magnetized plasma turbulence, establishing bounds related to dissipation scales in Hall- and electron-MHD models.

Contribution

It proves the existence of time-dependent determining wavenumbers for weak solutions of Hall- and electron-MHD, improving previous results and linking them to phenomenological dissipation scales.

Findings

Existence of time-dependent determining wavenumbers for weak solutions.

Upper bounds on wavenumbers align with Kolmogorov-like dissipation scales.

Uniform bounds established for strong electron-MHD solutions.

Abstract

In turbulent flows, the Kolmogorov wavenumber characterizes the smallest scales at which viscous effects dominate. A mathematical analogue of this notion first introduced by Foias and Prodi [8] -- a determining wavenumber -- quantifies the minimal set of modes that uniquely determine the long-time behavior of solutions. Extending this framework from the Navier-Stokes equations to magnetized plasma models, we focus on the Hall-MHD and Electron-MHD turbulence in sub-ion and dissipation ranges. We prove existence of time-dependent determining wavenumbers for weak solutions of the Hall- and electron-MHD, improving upon previous results that were not optimal and lacked any comparison with phenomenological dissipation scales. Under explicit scale-localized intermittency assumptions, we show that their time averages are bounded above by Kolmogorov-like dissipation wavenumbers predicted by phenomenological studies of plasma turbulence. For strong electron-MHD solutions, we also establish a uniform bound on the magnetic determining wavenumber from Besov regularity.