Statistical equilibrium measures and coherent states in two-dimensional turbulence
✍ Scribed by Bruce Turkington
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 373 KB
- Volume
- 52
- Category
- Article
- ISSN
- 0010-3640
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✦ Synopsis
The equilibrium statistics of the Euler equations in two dimensions are studied, and a new continuum model of coherent, or organized, states is proposed. This model is defined by a maximum entropy principle similar to that governing the Miller-Robert model except that the family of global vorticity invariants is relaxed to a family of inequalities on all convex enstrophy integrals. This relaxation is justified by constructing the continuum model from a sequence of lattice models defined by Gibbs measures whose invariants are derived from the exact vorticity dynamics, not a spectral truncation or spatial discretization of it. The key idea is that the enstrophy integrals can be partially lost to vorticity fluctuations on a range of scales smaller than the lattice microscale, while energy is retained in the larger scales. A consequence of this relaxation is that many of the convex enstrophy constraints can be inactive in equilibrium, leading to a simplification of the mean-field equation for the coherent state. Specific examples of these simplified theories are established for vortex patch dynamics. In particular, a universal relation between mean vorticity and stream function is obtained in the dilute limit of the vortex patch theory, which is different from the sinh relation predicted by the Montgomery-Joyce theory of point vortices.