Upscaling of the porous shallow water equations through the use of periodic homogenization

Abstract

Due to the large and costly nature of ocean models, they are often limited in computational resolution - meaning accurate solvers must take into account boundary geometry at the subgrid-scale without explicitly modelling it. One method of characterizing subgrid-scale features is Brinkman penalization, where the solid/fluid interface is modelled as a porous medium, which yields many stability, accuracy, and efficiency benefits. In this work, we aim to extend the Brinkman method by testing a penalization that accounts for a position dependent tensorial permeability which will allow the model to experience friction in a directionally dependent fashion - implicitly preserving roughness that may be lost in a porosity-only approach. We simulate flow through solid/fluid and semi-permeable permeability-defined substructure configurations using the porous shallow water equations at both the subgrid-scale as well as the coarsened scale. Our findings indicate that coarsened simulations well approximate averaged fine-scale simulations in both velocity distribution and total kinetic energy. We coarsen subgrid-scale permeability using the periodic homogenization approach, which we find to be rigorous, fast, and accurate.