Lattice Boltzmann Large Eddy Simulation Model of MHD

TL;DR

This paper develops a lattice Boltzmann large eddy simulation model for 2D magnetohydrodynamic turbulence, enabling accurate subgrid modeling while preserving divergence-free magnetic fields.

Contribution

It extends lattice Boltzmann LES to MHD turbulence, deriving closed equations that incorporate subgrid effects without divergence cleaning.

Findings

Successfully models MHD turbulence with LES using lattice Boltzmann method.

Maintains divergence-free magnetic field without additional cleaning.

Provides a framework for subgrid modeling in MHD simulations.

Abstract

The work of Ansumali \textit{et al.}\cite{Ansumali} is extended to Two Dimensional Magnetohydrodynamic (MHD) turbulence in which energy is cascaded to small spatial scales and thus requires subgrid modeling. Applying large eddy simulation (LES) modeling of the macroscopic fluid equations results in the need to apply ad-hoc closure schemes. LES is applied to a suitable mesoscopic lattice Boltzmann representation from which one can recover the MHD equations in the long wavelength, long time scale Chapman-Enskog limit (i.e., the Knudsen limit). Thus on first performing filter width expansions on the lattice Boltzmann equations followed by the standard small Knudsen expansion on the filtered lattice Boltzmann system results in a closed set of MHD turbulence equations provided we enforce the physical constraint that the subgrid effects first enter the dynamics at the transport time scales. In particular, a multi-time relaxation collision operator is considered for the density distribution function and a single relaxation collision operator for the vector magnetic distribution function. The LES does not destroy the property that $\nabla \cdot \mathbf B = 0$ automatically without the need for divergence cleaning.