Stochastic Models of Shear-Flow Turbulence with Enstrophy Transfer to Subgrid Scales

A stochastic model for shear-flow turbulence is constructed under the constraint that the parameterized nonlinear eddy–eddy interactions conserve energy 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 constitutes a rigorous starting point for more thorough parameterizations. A major simplification arises from the fact that independently forced spatial structures produce covariances that can be superposed linearly. The constrained stochastic model cannot sustain turbulence when dissipation is strong or when the mean shear is weak because the prescribed forcing structures extract potential enstrophy from the mean flow at a rate too slow to sustain a transfer to subgrid scales. The constraint therefore defines a transition shear separating states in which turbulence is possible from those in which it is impossible. The transition shear, which depends on forcing structure, achieves 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 dissipation parameterized by spatially uniform potential vorticity damping. The transition shear associated with spatially localized random forcing and with reasonable eddy dissipation is close to the correct turbulence transition point determined by numerical simulation of the fully nonlinear system. In contrast, the transition shear corresponding to the optimal forcing functions is unrealistically small, suggesting that at weak shears these structures are weakly excited by nonlinear interactions. Nevertheless, the true forcing structures must project on the optimal forcing structures to sustain a turbulent cascade. Because of this property and their small number, the leading optimal forcing functions may be an attractive basis set for reducing the dimensionality of the parameterization problem.