An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

TL;DR

This paper introduces an explicit, asymptotic-preserving low Froude scheme for multilayer shallow water models with density stratification, ensuring stability, accuracy, and robustness for large-scale oceanic simulations.

Contribution

It develops a novel explicit scheme that maintains stability and consistency across different regimes, including low Froude numbers, for multilayer stratified shallow water models.

Findings

Stable in multilayer and nonlinear regimes

Consistent with asymptotic limits at small and large scales

Effective for large-scale oceanic circulation simulations

Abstract

We present an explicit scheme for a two-dimensional multilayer shallow water model with density stratification, for general meshes and collocated variables. The proposed strategy is based on a regularized model where the transport velocity in the advective fluxes is shifted proportionally to the pressure potential gradient. Using a similar strategy for the potential forces, we show the stability of the method in the sense of a discrete dissipation of the mechanical energy, in general multilayer and non-linear frames. These results are obtained at first-order in space and time and extended using a simple second-order MUSCL extension. With the objective of minimizing the diffusive losses in realistic contexts, sufficient conditions are exhibited on the regularizing terms to ensure the scheme's linear stability at first and second-order in time and space. The other main result stands in the consistency with respect to the asymptotics reached at small and large time scales in low Froude regimes, which governs large-scale oceanic circulation. Additionally, robustness and well-balanced results for motionless steady states are also ensured. These stability properties tend to provide a very robust and efficient approach, easy to implement and particularly well suited for large-scale simulations. Some numerical experiments are proposed to highlight the scheme efficiency: an experiment of fast gravitational modes, a smooth surface wave propagation, an initial propagating surface water elevation jump considering a non trivial topography, and a last experiment of slow Rossby modes simulating the displacement of a baroclinic vortex subject to the Coriolis force.