Magnetohydrodynamic waves in non-uniform flows II: stress-energy tensors, conservation laws and Lie symmetries

Abstract

The interaction of magnetohydrodynamic (MHD) waves in a non-uniform, time-dependent background plasma flow is investigated using Lagrangian field theory methods. The analysis uses Lagrangian maps, in which the exact position of the fluid element ${\bf x}^*$ is expressed as a vector sum of the position vector ${\bf x}$ of the background plasma fluid element plus a Lagrangian displacement $\xib({\bf x},t)$ due to the waves. An exact theory for the wave and background stress energy tensors is developed based on the exact Lagrangian and the exact Lagrangian map. Noether's theorems are used in conjunction with the exact action and Lagrangian maps to determine the general form of conservation laws for the total system of waves and background plasma, corresponding to divergence symmetries of the action. The energy and momentum conservation laws of the system are derived from Noether's first theorem corresponding to the time and space translation symmetries of the action, respectively. As examples of the use of Noether's first theorem, we derive the conservation laws associated with the 10-parameter Galilean group admitted by the MHD equations. This includes the space and time translation symmetries, the space rotations, and the Galilean boosts. A class of solutions of the Lie determining equations for the infinite-dimensional MHD fluid relabeling symmetries are used to discuss the corresponding conservation laws. Ertel's theorem for the conservation of potential vorticity for the system of waves and background gas in ideal gas dynamics is derived from an infinite-dimensional fluid relabeling symmetry of the action.