  We address the question of parameterizing 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 parameterization coupling each
resolved variable only to a few neighboring cells. The closure improves the
results of the low-resolution model and outperforms two purely empirical
stochastic parameterizations. 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 parameterization, if additional tuning of the
coefficients is performed. In addition, the scale-awareness of the
parameterizations is studied.
