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Statistical equilibrium measures and coherent states in two-dimensional turbulence. (English) Zbl 0990.76029

The author proposes a new model of statistical equilibrium for the description of coherent states in a freely evolving two-dimensional incompressible inviscid fluid. The theory is based on a real invariance for the ideal vorticity dynamics. Other well-known models, such as the Joyce-Montgomery theory or Miller-Robert theory, use an approximation procedure based on the lattice discretization (a spatial truncation, or a spectral truncation). Consequently, the conservation of dynamical invariants is lost, and it has to be restored in an artificial way.
The model presented in the paper avoids this problem, since the vorticity dynamics of the discrete model is obtained by “windowing” the exact continuum dynamics on a lattice. The windowing is performed by the convolution operator defined by a Fejér kernel. A consequence of the windowing operation is that many vorticity dynamics may corresponds to a single initial condition on the lattice. In this way the model takes into account the fluctuations of vorticity at the scales smaller than the characteristic lattice scale. The preceding models implicitly assumed the dynamics to be constant on the scales smaller than the lattice scale. The main novelty of the theory is the relaxation of enstrophy constraints on the maximum entrophy state to convex inequalities. The enstrophies are partially lost in favour of degrees of freedom of the vorticity below the lattice scale. This allows the ideal dynamics to dissipate any convex enstrophy in the continuum model. The paper starts with a review of principal models in the literature, with the aim to make a comparison with the model presented. The author then gives a construction of a discrete model on the lattice, where the “windowing” operations are used. A Gibbs measure is given for the dynamics on the lattice. As lattice scales becomes smaller and smaller, the discrete model converges to the continuum model. The rigorous derivation of the continuum limit can be found in [C. Boucher, R. S. Ellis and B. Turkington, J. Stat. Phys. 98, No. 5-6, 1235-1278 (2000; Zbl 0966.76039)].
Finally, some particular cases are explicitly solved, in order to make a comparison with Joyce-Montgomery and the Miller-Robert models, and different relation are obtained between the mean vorticity and the stream function.

MSC:

76F55 Statistical turbulence modeling
76M35 Stochastic analysis applied to problems in fluid mechanics

Citations:

Zbl 0966.76039
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