ISOTOPOLOGICAL RELAXATION, COHERENT STRUCTURES, AND GAUSSIAN TURBELENCE IN 2-DIMENSIONAL (2-D) MAGNETOHYDRODYNAMICS (MHD)

The long-time relaxation of ideal two-dimensional (2-D) magnetohydrody namic (MHD) turbulence subject to the conservation of two infinite fam ilies of constants of motion-the magnetic and the ''cross'' topology i nvariants-is examined. The analysis of the Gibbs ensemble, where all i ntegrals of motion are respected, predicts the initial state to evolve into an equilibrium, stable coherent structure (the most probable sta te) and decaying Gaussian turbulence (fluctuations) with a vanishing, but always positive temperature. The nondissipative turbulence decay i s accompanied by decrease in both the amplitude and the length scale o f the fluctuations, so that the fluctuation energy remains finite. The coherent structure represents a set of singular magnetic islands with plasma flow whose magnetic topology is identical to that of the initi al state, while the energy and the cross topology invariants are share d between the coherent structure and the Gaussian turbulence. These co nservation laws suggest the variational principle of isotopological re laxation that allows one to predict the appearance of the final state from a given initial state. For a generic initial condition having x p oints in the magnetic field, the coherent structure has universal type s of singularities: current sheets terminating at Y points. These stru ctures, which are similar to those resulting from the 2-D relaxation o f magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of D. Biskamp and H. Welter [Phys . Fluids B 1, 1964 (1989)] and are likely to form during the nonlinear stage of the kink tearing mode in tokamaks. The Gibbs ensemble method developed in this work admits extension to other Hamiltonian systems with invariants not higher than quadratic in the highest-order-derivat ive variables. The turbulence in 2-D Euler fluid is of a different nat ure: there the coherent structures are also formed, but the fluctuatio ns about these structures are non-Gaussian.