Numerical Schemes for Stochastic Backscatter in the Inverse Cascade of Quasigeostrophic Turbulence

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of two-dimensional and quasigeostrophic turbulence, where the net transfer of energy is from small to large scales. A numerical scheme for stochastic backscatter in the two-dimensional and quasigeostrophic inverse kinetic energy cascades is developed and analyzed. Its essential properties include a local formulation amenable to implementation in finite difference codes and non-periodic domains, smooth behavior at the coarse grid scale, and realistic temporal correlations. The scheme is related to a simplified version of stochastic superparameterization [Grooms and Majda, J. Comp. Phys. 271, (2014)]; the simplified version allows detailed numerical analysis, focusing on the spatial and temporal correlation structure of the modeled backscatter. The method is demonstrated in an idealized setting of quasigeostrophic turbulence using a low-order finite difference code, where it produces a good approximation to the results of a spectral code with more than 5 times higher nominal resolution. When the essential properties of the new scheme are integrated into the full stochastic superparameterization, which includes downscale potential energy transfer in addition to kinetic energy backscatter, the results are in excellent agreement with the high resolution reference simulation. The method is of particular importance to eddy-permitting ocean modeling applications in the inverse cascade regime, but it should also serve to improve stochastic backscatter algorithms in regimes of turbulence with a net downscale cascade of energy.