A High-Order Staggered Finite-Element Vertical Discretization for Non-Hydrostatic Atmospheric Models

Atmospheric modeling systems require economical methods to solve the non-hydrostatic Euler equations. Two major differences between hydrostatic models and a full non-hydrostatic description lie in the vertical velocity tendency and numerical stiffness associated with sound waves. In this work, we introduce a new arbitrary-order vertical discretization entitled the staggered nodal finite element method (SNFEM). Our method uses a generalized discrete derivative that consistently combines the discontinuous Galerkin and spectral element methods on a staggered grid. Our combined method leverages the accurate wave propagation and conservation properties of spectral elements with staggered methods that eliminate stationary (2dx) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work, we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the large-eddy regime. Our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties as the low-order alternative.