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We present an entropy stable nodal discontinuous Galerkin spectral elementmethod (DGSEM) for the two-layer shallow water equations on two dimensionalcurvilinear meshes. We mimic the continuous entropy analysis on thesemi-discrete level with the DGSEM constructed on Legendre-Gauss-Lobatto (LGL)nodes. The use of LGL nodes endows the collocated nodal DGSEM with thesummation-by-parts property that is key in the discrete analysis. Theapproximation exploits an equivalent flux differencing formulation for thevolume contributions, which generate an entropy conservative split-form of thegoverning equations. A specific combination of an entropy conservativenumerical surface flux and discretization of the nonconservative terms is thenapplied to obtain a high-order path-conservative scheme that is entropyconservative and has the well-balanced property for discontinuous bathymetry.Dissipation is added at the interfaces to create an entropy stableapproximation that satisfies the second law of thermodynamics in the discretecase. We conclude with verification of the theoretical findings throughnumerical tests and demonstrate results about convergence, entropy stabilityand well-balancedness of the scheme.

An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes