An important practical issue for modeling fluid flows which exhibit organized large scale structures, such as geophysical flows in a wide variety of circumstances (McWilliams, JFM 1984, Holloway, Ann. Rev. FM, 1986), involves the development of crude closure models. Such crude closure dynamics should quantitatively capture evolving nonlinear large scale behavior with reasonable accuracy via the evolution of a small number of parameters.
Here we propose and develop crude dynamic closure based on equilibrium large scale statistical theories involving only a few constraints. The crude closure involves the nonlinear evolution of a single parameter, the energy, for dynamic closure based on equilibrium energy-enstrophy statistical theory (Salmon, Holloway, and Henderschott, JFM 1976), with possibly additional parameters for crude closure based on more sophisticated statistical theories. The crude closure algorithms are tested systematically through numerical experiments with barotropic flow over varying topography driven by small scale random forcing at moderately large Reynolds numbers. The following two-dimensional domains are considered: the basin, the channel, and the torus. A series of successively more stringent tests is devised with conditions ranging from freely decaying flows to spin-up from rest by random forcing with like signed vortices, and finally to random forcing by vortices with alternating or opposite signs. Comparison of standard spectral simulations with crude dynamic closure yields remarkably robust and accurate prediction of the evolution of bulk features of the flow, such as the energy, while relative errors in velocity usually do not exceed 10%.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics