An energy consistent discretization of the nonhydrostatic equations in primitive variables

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981) vertical differencing which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term-by-term balance in the kinetic, internal and potential energy budgets, ensuring an energy-consistent discretization with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a HEVI IMEX timestepping algorithm