Mean field theory and coherent structures for vortex dynamics on the plane

We present a new derivation of the Onsager-Joyce-Montgomery (OJM) equilibrium statistical theory for point vortices on the plane, using the Bogoliubov-Feynman inequality for the free energy, Gibbs entropy function and Landau's approximation. This formulation links the heuristic OJM theory to the modern variational mean field theories. Landau's approximation is the physical counterpart of a large deviation result, which states that the maximum entropy state does not only have maximal probability measure but overwhelmingly large measure relative to other macrostates.Comment: PACS: 47.15.Ki, 67.40.Vs, 68.35.Rh 16 page

Quasi-periodic dynamics of desingularized vortex models

Sufficient conditions for the existence of quasi-periodic solutions of two different desingularized vortex models for 2-dimensional Euler flows are derived. One of these models is the vortex blob model for the evolution of a periodic vortex sheet and the other is a second order elliptic moment model (DEMM) for the evolution of widely separated vortex regions. The method involves the identification of the well-known point vortex Hamiltonian term in both models. A transformation to new canonical variables (the JL-coordinates) and the definition of special open sets in phase space (the cone sets) puts the Hamiltonians considered into nearly integrable form. KAM-theory is used to prove the desired results for arbitrary degrees of freedom and almost arbitrary circulations in these models. A rigorous validification of the DEMM assumption is obtained. In view of the lack of a rigorous theory for vortex sheet roll-up past the critical time, the dynamical system approach presented here provides an alternative method for studying the macroscopic structures formed in the post-critical period.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27871/1/0000285.pd

Singular manifolds and quasi-periodic solutions of Hamiltonians for vortex lattices

A general method for establishing the existence of quasi-periodic solutions of Hamiltonian systems for vortex lattices is illustrated in a simple example involving two degrees of freedom. The geometry of intersecting singular manifolds of the Hamiltonians introduces suitable canonical transformations which put the Hamiltonian into the form of singular weakly coupled oscillators. As by-products of this procedure, additional integrals of motion are found for the leading term in the transformed Hamiltonian. These extra integrals are approximate invariants for the full Hamiltonians.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27359/1/0000384.pd