Crude closure for flow with topography through large-scale statistical theory

Crude closure algorithms based on equilibrium statistical theories are developed here for prototypical geophysical flows involving barotropic flow over topography. These algorithms are developed utilizing the simplest energy-enstrophy statistical theory for flow with topography. Only a single parameter, the energy, is tracked by the algorithm and the entire flow structure is predicted through the equilibrium statistical state. In particular, no explicit parametrization of a sub-grid scale energy spectrum is utilized in the algorithm. The predictions of the crude closure algorithm are compared with direct pseudo-spectral numerical simulations of the barotropic flow equations with random small-scale forcing and dissipation for a variety of random topographies in basin, channel and periodic geometries. In most situations studied here, the energy is tracked within small errors by the crude closure, while the velocity errors rarely exceed 10% provided that the enstrophy/energy ratio is not large or growing significantly in time. Examples are also introduced where the crude closure algorithm based on the energy-enstrophy theory fails; in these circumstances, a crude closure algorithm based on more sophisticated equilibrium statistical theories is introduced as a possible remedy.