Energy of eigen-modes in magnetohydrodynamic flows of ideal fluids

TL;DR

This paper derives an analytical expression for the energy of eigen-modes in ideal magnetohydrodynamic flows, revealing that unstable modes have zero energy, while stable modes can have positive or negative energy, with implications for understanding instabilities.

Contribution

It provides a novel analytical formula for eigen-mode energies in ideal MHD flows, highlighting the role of non-symmetric modes and negative energy waves in instabilities.

Findings

Unstable modes have zero energy.

Stable oscillatory modes can have positive or negative energy.

Negative energy waves are linked to non-symmetric eigen-modes.

Abstract

Analytical expression for energy of eigen-modes in magnetohydrodynamic flows of ideal fluids is obtained. It is shown that the energy of unstable modes is zero, while the energy of stable oscillatory modes (waves) can assume both positive and negative values. Negative energy waves always correspond to non-symmetric eigen-modes -- modes that have a component of wave-vector along the equilibrium velocity. These results suggest that all non-symmetric instabilities in ideal MHD systems with flows are associated with coupling of positive and negative energy waves. As an example the energy of eigen-modes is calculated for incompressible conducting fluid rotating in axial magnetic field.