Hamiltonian description and stability of vortex flows in interchange mode turbulence

It is generally recognized that nonlinear interaction of short scale fluctuations in a magnetized plasma can generate large scale nonlinear structures. In the present paper we extend the theory of large-scale structure generation on the flute mode turbulence. For this turbulence the contribution of density fluctuations and finite ion Larmor radius effects are significant and must be taken into account. In the course of the analysis, we first derive from the two-fluid macroscopic equations a pair of coupled nonlinear equations for the perturbed density and potential that describe the nonlinear dynamics of flute modes. The Hamiltonian structure of these model equations has been identified and used to find a complete set of invariants, including so called Casimirs. Explicit nonlinear stationary solutions to the model equations describing the large scale vortex flow which is localized in the direction of plasma inhomogeneity and periodic in direction of plasma symmetry have been found. These solutions are "breathers" and Kelvin-Stuart "cat's eyes" and well known in 2-D incompressible fluid dynamics. Under some restrictions on free parameters they correspond, physically, to so-called "vortex streets". We consider the stability of these stationary solutions. To this end the Lyapunov factional was constructed from the complete set of invariants. By varying this functional we found that the "vortex street" solutions are linearly stable to long wavelength perturbations.