Hamiltonian contour dynamics and applications*

It is known that contour dynamics (CD) can be viewed as a Hamiltonian system  defined on a phase space of parameterization invariant functionals of closed curves [1-3].  This structure can be shown to follow from a general theory that reduces to both the noncanonical Hamiltonian structure of the two-dimensional Euler equations and CD.  Generalizations to 2D and 3D quasigeostrophic  systems also fit this general theory.  This  opens the way to use tools from Hamiltonian dynamics to interpret CD results.  We will explore generalizations of the Kirchhoff vortex dynamics as integrable and non-integrable Hamiltonian systems,  and Hamiltonian bifurcations.   In addition, perturbation  theory in terms of amplitude expansions for single and multi-contours will be discussed in several contexts  including two contour barotropic instability  and interactions between baroclinic and barotropic waves on a barotropic vortex.

[1] Bell, G.I., The nonlinear evolution of a perturbed axisymmetric eddy; in GFD Summer Program, 1990, pp. 232–249.

[2] Morrison, P.J. and Flierl, G.R., Hamiltonian Contour Dynamics. Bull. Amer. Phys. Soc., 2001, 46, ED4.

[3] Morrison, P.J., Hamiltonian and Action Principle Formulations of Plasma Physics. Phys. Plasmas, 2005, 12,

058102.