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

Abstract

Abstract 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...