  The long-time relaxation of ideal two dimensional magnetohydrodynamic
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
non-dissipative 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 iso-topological
relaxation which allows us 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 2D relaxation of magnetic field frozen into an ideally
conducting viscous fluid, are observed in the numerical experiment of Biskamp
and Welter. 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 two dimensional Euler
fluid is of a different nature: there the coherent structures are also formed,
but the fluctuations about these structures are non-Gaussian.
