Barotropic-Baroclinic Splitting for Multilayer Shallow Water Models with Exchanges

Multilayer ocean models (see e.g. Audusse et al., ESAIM: Mathematical Modelling and Numerical Analysis 2011) are popular approximations to the 3D Euler and Navier-Stokes equations. Computational cost obviously increases with the number of layers, which is often chosen to be around 50 in ocean simulations. The barotropic-baroclinic splitting is an important strategy used in numerical ocean models to reduce this computational cost (see e.g. Killworth et al., Journal of Physical Oceanography 1991).

In the present contribution, we focus on the numerical analysis of the barotropic-baroclinic splitting in the context of finite volume schemes. We reformulate the splitting strategy within the nonlinear multilayer framework using terrain-following coordinates, and present it as an exact operator splitting. The barotropic step captures the evolution of free surface and depth averaged velocity with a well-balanced one-layer shallow water model. The baroclinic step incorporates vertical exchanges between layers and adjusts velocities around their mean vertical value.

Our scheme is numerically robust, i.e. no filters or corrections are needed. The numerical solution inherently observes a discrete maximum principle for the tracer and hence guarantees non-negative tracer concentrations. In the language of applied mathematics, we prove a discrete entropy inequality. In the language of geophysics, this guarantees dissipation of kinetic and potential, and therefore of total energy. This is the key stability property for the class of finite volume schemes under consideration. Last, but not least, the gain in terms of computational cost is large, especially in low Froude simulations.

Currently, this work addresses the constant density case; however, ongoing work extends the barotropic-baroclinic splitting to variable density scenarios and models situations such as coastal upwelling. The paper is submitted for publication (Aguillon, Hörnschemeyer, Sainte-Marie, International Journal for Numerical Methods in Fluids, January 2026).