Hamiltonian contour dynamics: theory and applications

It is known that contour dynamics (CD) can be viewed as a Hamiltonian system (a field theory) defined on a phase space of parameterization invariant functionals of closed curves [1-4]. 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, to point vortices, and to CD. The structure includes 2D quasigeostrophic systems as well as layer models. Given the Hamiltonian structure, well-developed tools become available. One such tool is simulated annealing [4,5], a numerical technique for calculating equilibrium states that relies on the Hamiltonian structure, and an associated technique called Dirac constraint theory. Various single and multi-contour states will be obtained by this technique, including V-states and other (at least) nearly integrable configurations.

[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.

[4] Flierl, G.R., Morrison, P.J., and Swaminathan, R.V. Jovian Vortices and Jets. Fluids: Topical Collection "Geophysical Fluid Dynamics" 4, 104 2019.

[5] G. R. Flierl and P. J. Morrison, Hamiltonian-Dirac Simulated Annealing: Application to the Calculation of Vortex States,” Physica D 240, 212 (2011).