Stochastic Superparameterization in a One-dimensional Model for Wave Turbulence

Superparameterization is a multiscale numerical method wherein solutions of prognostic equations for small scale processes on local domains embedded within the computational grid of a large scale model are computed and used to force the large scales. It was developed initially in the atmospheric sciences, but stands on its own as a nascent numerical method for the simulation of multiscale phenomena. Here we develop a stochastic version of superparameterization in a difficult one dimensional test problem involving self-similarly collapsing solitons, dispersive waves, and a turbulent inverse cascade of energy from small to large scales. We derive the nonlinear model equations by imposing a formal scale separation between resolved large scales and unresolved small scales; this allows the use of subdomains embedded within the large scale grid to describe the local small scale processes. To decrease the computational cost, we make a systematic quasi-linear stochastic approximation of the nonlinear small scale equations and use the statistical mean of the nonlinear small scale forcing (the covariance) in the large scale equations. The stochastic approximation allows the embedded domains to be formally infinite (unrealistically large scales are suppressed on the embedded domains). Further simplifications allow us to precompute the small scale forcing terms in the large scale equations as functions of the large scale variables only, which results in significant computational savings. The results are positive. The method increases the energy in overdamped simulations, decreases the energy in underdamped simulations, and improves the spatial distribution and frequency of collapsing solitons.