Stochastic models of shear-flow turbulence with enstrophy transfer to subgrid scales

A stochastic model for shear-flow turbulence is constructed under the const raint that the parameterized nonlinear eddy-eddy interactions conserve ener gy but dissipate potential enstrophy. This parameterization is appropriate for truncated models of quasigeostrophic turbulence that cascade potential enstrophy to subgrid scales. The parameterization is not closed but constit utes a rigorous starting point for more thorough parameterizations. A major simplification arises from the fact that independently forced spatial stru ctures produce covariances that can be superposed linearly. The constrained stochastic model cannot sustain turbulence when dissipation is strong or w hen the mean shear is weak because the prescribed forcing structures extrac t potential enstrophy from the mean flow at a rate too slow to sustain a tr ansfer to subgrid scales. The constraint therefore defines a transition she ar separating states in which turbulence is possible from those in which it is impossible. The transition shear, which depends on forcing structure, a chieves an absolute minimum value when the forcing structures are optimal, in the sense of maximizing enstrophy production minus dissipation by large- scale eddies. The results are illustrated with a quasigeostrophic model with eddy dissipa tion parameterized by spatially uniform potential vorticity damping. The tr ansition shear associated with spatially localized random forcing and with reasonable eddy dissipation is close to the correct turbulence transition p oint determined by numerical simulation of the fully nonlinear system. In c ontrast, the transition sheer corresponding to the optimal forcing function s is unrealistically small, suggesting that at weak shears these structures are weakly excited by nonlinear interactions. Nevertheless, the true forci ng structures must project on the optimal forcing structures to sustain a t urbulent cascade. Because of this property and their small number, the lead ing optimal forcing functions may be an attractive basis set for reducing t he dimensionality of the parameterization problem.