Abstract
Abstract. 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 lies 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 (2Δx) 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.
Highlights
The accurate representation of vertical wave motion is essential for models of the atmosphere
The 10th-order vertical discretization with fourth-order flowdependent viscosity produces a flow that more closely approximates the reference solutions at a resolution that would otherwise be considered too poor for the dynamical features considered
The authors have not found a dynamical reason for correlation involving high-order vertical discretization coupled with high-order dissipation schemes and the reference solution with uniform damping
Summary
The accurate representation of vertical wave motion is essential for models of the atmosphere. As observed by Thuburn and Woollings (2005), Thuburn (2006), and Toy and Randall (2007), the choice of vertical coordinate (whether height based, mass based or entropy based) implies an optimal vertical staggering of prognostic variables for maintaining correct behavior for wave motions relevant to the atmosphere. Unstaggered discretizations (that is, discretizations where all prognostic variables are stored on model levels) possess stationary computational modes, which represent gross errors in the dispersion properties of the solution (Melvin et al, 2012; Ullrich, 2014b). Unlike the horizontal, these wave modes can be dramatically enhanced by an implicit treatment of the vertical at high Courant number
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