  We present an entropy stable nodal discontinuous Galerkin spectral element
method (DGSEM) for the two-layer shallow water equations on two dimensional
curvilinear meshes. We mimic the continuous entropy analysis on the
semi-discrete level with the DGSEM constructed on Legendre-Gauss-Lobatto (LGL)
nodes. The use of LGL nodes endows the collocated nodal DGSEM with the
summation-by-parts property that is key in the discrete analysis. The
approximation exploits an equivalent flux differencing formulation for the
volume contributions, which generate an entropy conservative split-form of the
governing equations. A specific combination of an entropy conservative
numerical surface flux and discretization of the nonconservative terms is then
applied to obtain a high-order path-conservative scheme that is entropy
conservative and has the well-balanced property for discontinuous bathymetry.
Dissipation is added at the interfaces to create an entropy stable
approximation that satisfies the second law of thermodynamics in the discrete
case. We conclude with verification of the theoretical findings through
numerical tests and demonstrate results about convergence, entropy stability
and well-balancedness of the scheme.
