Partition functions and equilibrium measures in two-dimensional and quasi-three-dimensional turbulence

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Abstract
An attempt is made to construct numerically equilibrium measures for the Euler equations by first examining measures for discretized approximate systems and then searching on the computer for the limit of vanishing discretization. First the partition function is evaluated for two-dimensional discretized incompressible fields with a hydrodynamical energy function and an infinite number of invariants; the behavior of the partition functions is examined as the discretization is refined. The results are contrasted with those of recent mean-field theories, which are seen to be reasonable approximations only at moderate temperatures. The two-dimensional vortex system has no phase transitions and no states invariant under refinement of the discretization, except at zero temperature. Finite-temperature equilibrium measures may appear if a simple representation of vortex stretching is added to the system, in agreement with recent work on three-dimensional turbulence, where these equilibrium measures are used as key building blocks.