Coherent structures and self-consistent transport in a mean field Hamiltonian model

A study of coherent structures and self-consistent transport is presented in the context of a Hamiltonian mean field, single wave model. The model describes the weakly nonlinear dynamics of marginally stable plasmas and fluids, and it is related to models of systems with long-range interactions in statistical mechanics. In plasma physics the model applies to the interaction of electron "holes" and electron "clumps," which are depletions and excesses of phase-space electron density with respect to a fixed background. In fluid dynamics the system describes the interaction of vortices with positive and negative circulation in a two-dimensional background shear flow. Numerical simulations in the finite-N and in the N → ∞ kinetic limit (where N is the number of particles) show the existence of coherent, rotating dipole states. We approximate the dipole as two "macroparticles" (one hole and one clump) and consider the N = 2 limit of the model. We show that this limit has a family of symmetric, rotating integrable solutions described by a one-degree-of-freedom nontwist Hamiltonian. A perturbative solution of the nontwist Hamiltonian provides an accurate description of the mean field and rotation period of the dipole. The coherence of the dipole is explained in terms of a parametric resonance between the rotation frequency of the macroparticles and the oscillation frequency of the self-consistent mean field. This resonance creates islands of integrability that shield the dipole from regions of chaotic transport. For a class of initial conditions, the mean field exhibits an elliptic-hyperbolic bifurcation that leads to the filamentation, chaotic mixing and eventual destruction of the dipole.