Journal Article

An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables

Ullrich, Paul A., Hall, David M., Eldrid, Christopher, Taylor, Mark A., Guba, Oksana, Steyer, Andrew

2020

Overview

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, https://doi.org/10.1175/MWR-D-14-00069.1) 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, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) 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 up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit‐explicit time stepping algorithm. Plain Language Summary Energy consistent discretizations have proven useful in guiding the development of numerical methods for simulating fluid dynamics. They ensure that the discrete method does not have any spurious sources of energy, which can lead to unstable and unrealistic simulations. Here we provide an energy consistent discretization of the equations used by global models of the Earth's atmosphere. The discretization is written in terms of standard variables in spherical coordinates and supports a wide variety of terrain following vertical coordinates. It can be used with any horizontal discretization that has a discrete version of the integration‐by‐parts identity. Key Points We give a discrete Hamiltonian formulation of the nonhydrostatic equations in primitive variables The formulation supports mass or height terrain following coordinates The Lorenz staggered vertical discretization obeys a derivative product rule

• Publisher: John Wiley & Sons, Inc,American Geophysical Union (AGU)
• Subject: Approximation / Atmosphere / Boundary conditions / dynamical core / Energy / Energy conservation / hamiltonian / mimetic / nonhydrostatic / Potential energy / Variables / Vertical advection