Stochastic subgrid‐scale parametrization for one‐dimensional shallow‐water dynamics using stochastic mode reduction

Abstract

We address the question of parametrizing the subgrid scales in simulations of geophysical flows by applying stochastic mode reduction to the one‐dimensional stochastically forced shallow‐water equations. The problem is formulated in physical space by defining resolved variables as local spatial averages over finite‐volume cells and unresolved variables as corresponding residuals. Based on the assumption of a time‐scale separation between the slow spatial averages and the fast residuals, the stochastic mode reduction procedure is used to obtain a low‐resolution model for the spatial averages alone with local stochastic subgrid‐scale parametrization coupling each resolved variable only to a few neighbouring cells. The closure improves the results of the low‐resolution model and outperforms two purely empirical stochastic parametrizations. It is shown that the largest benefit is in the representation of the energy spectrum. By adjusting only a single coefficient (the strength of the noise) we observe that there is a potential for improving the performance of the parametrization, if additional tuning of the coefficients is performed. In addition, the scale‐awareness of the parametrizations is studied.