A discrete Nambu bracket for 2D extended Magnetohydrodynamics

M Kraus,G N Throumoulopoulos,D A Kaltsas

doi:10.1088/1742-6596/1391/1/012037

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.

Highlights

The Hamiltonian formulation of ideal fluid models [1] (e.g. Magnetohydrodynamics) in Eulerian viewpoint involves noncanonical variables and Poisson operators that are degenerate and inhomogeneous in phase space P
In view of (19), (12) and (17)–(18) we find that the semi-discrete dynamical equations are provided by (13)–(16) with the field variables on the lhs being replaced by their values at (i, j) node and the Jacobians on the rhs by the Arakawa Jacobian Jij as it is given in [9]
Employing a discretization algorithm, which has been introduced in the context of fluid dynamics, we derived a conservative, semi-discrete set of equations that involve the well known Arakawa Jacobian

Summary

Introduction

The Hamiltonian formulation of ideal fluid models [1] (e.g. Magnetohydrodynamics) in Eulerian viewpoint involves noncanonical variables and Poisson operators that are degenerate and inhomogeneous in phase space P. Unlike the Hamiltonian functional, being conserved as a consequence of the necessary property of antisymmetry of the associated Poisson bracket (dH/dt = {H, H} = 0), the Casimirs are conserved due to its degeneracy, i.e.

Results

Conclusion