A discrete Nambu bracket for 2D extended Magnetohydrodynamics

TL;DR

This paper introduces a new Hamiltonian formulation for 2D extended Magnetohydrodynamics using a discrete Nambu bracket, enabling structure-preserving numerical simulations with high conservation accuracy.

Contribution

It presents a novel trilinear bracket formulation for 2D XMHD and develops a finite difference discretization that preserves key conservation laws.

Findings

The scheme accurately conserves invariants in simulations.

The discretization employs the Arakawa Jacobian for stability.

Simulations demonstrate high fidelity in planar dynamics.

Abstract

In this note we propose a trilinear bracket formulation for the Hamiltonian extended Magnetohydrodynamics (XMHD) model with homogeneous mass density. The corresponding two-dimensional representation is derived by performing spatial reduction on the three-dimensional bracket, upon introducing a symmetric representation for the field variables. Subsequently, the trilinear bracket of the resulting two-dimensional, four-field model is discretized using a finite difference scheme, which results in semi-discrete dynamics that involve the Arakawa Jacobian. Simulations of planar dynamics show that this scheme respects the desired conservation properties to high precision.