Efficient kinetic Lattice Boltzmann simulation of three-dimensional Hall-MHD Turbulence

TL;DR

This paper introduces a new kinetic Lattice Boltzmann code called FLAME for simulating three-dimensional Hall-MHD turbulence, demonstrating its stability, accuracy, and ability to capture the transition from MHD to Hall-MHD regimes.

Contribution

The paper develops and validates a novel LB-based simulation code for Hall-MHD turbulence, efficiently handling the complex Hall electric field term and capturing key plasma spectral features.

Findings

The FLAME code is stable and second-order accurate.

Spectral properties match those from established pseudo-spectral codes.

The code successfully captures the transition from MHD to Hall-MHD regimes.

Abstract

Simulating plasmas in the Hall-MagnetoHydroDynamics (Hall-MHD) regime represents a valuable {approach for the investigation of} complex non-linear dynamics developing in astrophysical {frameworks} and {fusion machines}. Taking into account the Hall electric field is {computationally very challenging as} it involves {the integration of} an additional term, proportional to $\bNabla \times ((\bNabla\times\mathbf{B})\times \mathbf{B})$ in the Faraday's induction {law}. {The latter feeds back on} the magnetic field $\mathbf{B}$ at small scales (between the ion and electron inertial scales), {requiring} very high resolution{s} in both space and time {in order to properly describe its dynamics.} The computational {advantage provided by the} kinetic Lattice Boltzmann (LB) approach is {exploited here to develop a new} code, the \textbf{\textsc{F}}ast \textbf{\textsc{L}}attice-Boltzmann \textbf{\textsc{A}}lgorithm for \textbf{\textsc{M}}hd \textbf{\textsc{E}}xperiments (\textsc{flame}). The \textsc{flame} code integrates the plasma dynamics in lattice units coupling two kinetic schemes, one for the fluid protons (including the Lorentz force), the other to solve the induction equation describing the evolution of the magnetic field. Here, the newly developed algorithm is tested against an analytical wave-solution of the dissipative Hall-MHD equations, pointing out its stability and second-order convergence, over a wide range of the control parameters. Spectral properties of the simulated plasma are finally compared with those obtained from numerical solutions from the well-established pseudo-spectral code \textsc{ghost}. Furthermore, the LB simulations we present, varying the Hall parameter, highlightthe transition from the MHD to the Hall-MHD regime, in excellent agreement with the magnetic field spectra measured in the solar wind.