Unfreezing Casimir invariants: singular perturbations giving rise to forbidden instabilities

TL;DR

This paper interprets Casimir invariants in fluid and plasma mechanics as adiabatic invariants and shows how their unfreezing through singular perturbations can lead to instabilities like tearing modes.

Contribution

It introduces a physical interpretation of Casimir invariants as adiabatic invariants and demonstrates how singular perturbations can unfrozen these invariants, leading to new dynamical phenomena.

Findings

Casimir invariants can be viewed as adiabatic invariants.

Unfreezing Casimirs via singular perturbations increases degrees of freedom.

Application to magnetohydrodynamics reveals a mechanism for tearing-mode instability.

Abstract

The infinite-dimensional mechanics of fluids and plasmas can be formulated as "noncanonical" Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that foliate the phase space, imposing topological constraints on the dynamics. Here we proffer a physical interpretation of Casimir elements as \emph{adiabatic invariants} ---upon coarse graining microscopic angle variables, we obtain a macroscopic hierarchy on which the separated action variables become adiabatic invariants. On reflection, a Casimir element may be \emph{unfrozen} by recovering a corresponding angle variable; such an increase in the number of degrees of freedom is, then, formulated as a \emph{singular perturbation}. As an example, we propose a canonization of the resonant-singularity of the Poisson bracket operator of the linearized magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir element) constraining the dynamics is unfrozen, giving rise to a tearing-mode instability.