Statistical equilibrium solutions of the shallow water equations

Physical Review Letters

Volumn 86, Issue 9, 2001, Pages 1761-1764

  Weichman, Peter B ^a   Petrich, Dean M ^b  

^a Blackhawk Golden   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

HAMILTONIANS; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PHASE EQUILIBRIA; STATISTICAL METHODS; SURFACES; THERMODYNAMIC PROPERTIES; TURBULENCE;

FREE SURFACE; SHALLOW WATER EQUATIONS;

FLOW OF FLUIDS;

EID: 14344276146     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevLett.86.1761     Document Type: Article

References (14)

1. Real fluids are always viscous, but models of turbulence generally concern themselves with the "inertial range" where loss of energy due to viscous damping is small compared to that due to the cascade process, and an energy conserving model is appropriate.
2. See, e.g., A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, 1971), Vol. 1.
3. See, e.g., V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmoqorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, New York, 1992).
4. For several examples of such systems, see, e.g., D. D. Holm, J. E. Marsden, T. Ratiu, and A. Weistein, Phys. Rep. 123, 1 (1985).
5. Study of these states is partly motivated by the constraints they place on simulations of turbulent flow, e.g., the degree to which they properly preserve the conservation laws.
6. J. Miller, P. B. Weichman, and M. C. Cross, Phys. Rev. A 45, 2328 (1992). Detailed references to the previous literature are contained in this paper.
7. The statistical approach relies on the assumption of ergodicity of the dynamics. This assumption has been explored numerically with mixed results: depending upon the initial condition, the dynamics may get stuck in metastable equilibria. See, e.g., Pei-Long Chen, Ph.D. thesis, Caltech, 1996; D. Z. Jin and D. H. E. Dubin, Phys. Rev. Lett. 80, 4434 (1998).
8. The statistical approach relies on the assumption of ergodicity of the dynamics. This assumption has been explored numerically with mixed results: depending upon the initial condition, the dynamics may get stuck in metastable equilibria. See, e.g., Pei-Long Chen, Ph.D. thesis, Caltech, 1996; D. Z. Jin and D. H. E. Dubin, Phys. Rev. Lett. 80, 4434 (1998).
9. th = ∇ · (h∇φ).
10. 0 increases.
11. See, e.g., A. Balk, Phys. Lett. A 187, 302 (1994).
12. Divergences in these gradients may in fact occur in finite time since the shallow water equations are believed to produce shock wave solutions. Appropriate continuation of the equations nevertheless allows the conservation of ω/h to be maintained even in the presence of shocks [D. D. Holm (private communication)].
13. Dissipation processes, present in any experimental system, also act on the vortical flows, but the conservation laws guarantee that microscopic fluctuations in Ω, unlike Q, remain finite and hence will be dissipated much less strongly. This is expected to lead to an intermediate range of time scales [6], after dissipation of the wave motions, but prior to significant dissipation of vortical flows, where the equilibrium theory provides a valid description.
14. P. H. Chavanis and J. Sommeria, physics/0004056.