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The non-hydrostatic (NH) compressible Euler equations for dry atmospherewere solved in a simplified two-dimensional (2-D) slice framework employing aspectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontalSEM, which decomposes the physical domain into smaller pieces with a smallcommunication stencil, a high level of scalability can be achieved. By usingvertical FDM, an easy method for coupling the dynamics and existing physicspackages can be provided. The SEM uses high-order nodal basis functionsassociated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL)quadrature points. The FDM employs a third-order upwind-biased scheme forthe vertical flux terms and a centered finite difference scheme for thevertical derivative and integral terms. For temporal integration, atime-split, third-order Runge–Kutta (RK3) integration technique was applied.The Euler equations that were used here are in flux form based on thehydrostatic pressure vertical coordinate. The equations are the same asthose used in the Weather Research and Forecasting (WRF) model, but a hybridsigma–pressure vertical coordinate was implemented in this model.  We validated the model by conducting the widely used standard tests: linearhydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia–gravity wave, and rising thermal bubble.The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is  accurate and robust providedsufficient diffusion is applied. The results with various horizontalresolutions also showed convergence of second-order accuracy due to theaccuracy of the time integration scheme and that of the vertical direction,although high-order basis functions were used in the horizontal. By usingthe 2-D slice model, we effectively showed that the combined spatialdiscretization method of the spectral element and finite difference methodsin the horizontal and vertical directions, respectively, offers a viablemethod for development of an NH dynamical core

Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects