Crude closure dynamics through large scale statistical theories

Crude closure algorithms based on equilibrium large scale statistical theories involving only a few constraints are developed here. These algorithms involve the nonlinear evolution of either a single parameter, the energy, for dynamic closore based on equilibrium energy-enstrophy statistical theory, or two parameters, the energy and circulation, for crude dynamic closure based on the equilibrium point vortex statistical theory. The crude closure algorithms are tested systematically through numerical experiments with the Navier-Stokes equations in two dimensions on a rectangular domain with stress-free boundary conditions and strong small scale forcing at moderately large Reynolds numbers. 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 based on the two-parameter theory yields at most 5% velocity errors for all of these examples. The velocity errors can be as large as 10% for the one-parameter closure theory but are often comparable to those obtained with the two-parameter closure. The results of numerical experiments with Ekman drag are also discussed here.