Singular Casimir elements: their mathematical justification and physical implications
Seminar Room 1, Newton Institute
The problem of singular Poisson operator, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is studied in the context of a Casimir deficit, where Casimir elements constitute the center of the Lie-Poisson algebra underlying the Hamiltonian formulation. The nonlinearity of the evolution equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the "dimension" of the center) changes. Singular Casimir elements stemming from this singularity are unearthed using a generalized functional derivative (which may be regarded as hyperfunctions generated by an "infinite-dimensional partial differential operator" with a singularity). A singular perturbation (introduced by finite dissipation) destroys the leaves foliated by Casimir elements, removing the topological constraint and allowing the state vector to move towards lower-energy state in unconstrained phase space. The dynamics of an ideal plasma is constrained by the helicity as well as helical-flux Casimir elements pertinent to resonant singularities. A finite-resistivity singular perturbation gives rise to negative-energy "tearing modes" by destroying the helical-flux Casimir leaves.