Isotopological relaxation, coherent structures, and Gaussian turbulence in two-dimensional (2-D) magnetohydrodynamics (MHD)

The long-time relaxation of ideal two-dimensional (2-D) magnetohydrodynamic (MHD) turbulence subject to the conservation of two infinite families of constants of motion - the magnetic and the "cross" topology invariants - is examined. The analysis of the Gibbs ensemble, where all integrals of motion are respected, predicts the initial state to evolve into an equilibrium, stable coherent structure (the most probable state) and decaying Gaussian turbulence (fluctuations) with a vanishing, but always positive temperature. The nondissipative turbulence decay is accompanied by decrease in both the amplitude and the length scale of 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 initial state, while the energy and the cross topology invariants are shared between the coherent structure and the Gaussian turbulence. These conservation laws suggest the variational principle of isotopological relaxation that allows one to predict the appearance of the final state from a given initial state. For a generic initial condition having x points in the magnetic field, the coherent structure has universal types of singularities: current sheets terminating at Y points. These structures, which are similar to those resulting from the 2-D relaxation of 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-derivative variables. The turbulence in 2-D Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.