A conservative numerical scheme for the multilayer shallow‐water equations on unstructured meshes

Abstract The multilayer shallow‐water equations (mSWEs) constitute a valid and realistic model for many large‐scale oceanic and atmospheric dynamical processes by enabling vertical variations in density and momentum across the layers. A Hamiltonian formulation is given for the mSWEs written in vorticity and divergence variables. An energy‐conserving scheme is derived from the Hamiltonian formulation, utilizing the direct connection between energy conservation and the skew symmetry of the Poisson bracket. The scheme operates on unstructured meshes, implements physically relevant boundary conditions, and possesses optimal dispersive wave relations. A new mechanism is proposed to prevent layer interface outcropping. With appropriately set parameters, the mechanism is effective in inhibiting outcropping, with only limited impact on the overall dynamics of the flow and no noticeable change in the stiffness of the model or the courant‐friedrichs‐lewy mandated time‐step sizes.