Backscatter in Large-Scale Flows

Downgradient mixing of potential-voriticity and its variants are commonly employed to model the effects of unresolved geostrophic turbulence on resolved scales. This is motivated by the (inviscid and unforced) particle-wise conservation of potential-vorticity and the mean forward or down-scale cascade of potential enstrophy in geostrophic turubulence. By examining the statistical distribution of the transfer of potential enstrophy from mean or filtered motions to eddy or sub-filter motions, we find that the mean forward cascade results from the forward-scatter being only slightly greater than the backscatter. Downgradient mixing ideas, do not recognize such equitable mean-eddy or large scale-small scale interactions and consequently model only the mean effect of forward cascade; the importance of capturing the effects of backscatter---the forcing of resolved scales by unresolved scales---are only beginning to be recognized. While recent attempts to model the effects of backscatter on resolved scales have taken a stochastic approach, our analysis suggests that these effects are amenable to being modeled deterministically.