Investigating Inherent Numerical Stabilization for the Moist, Compressible, Non‐Hydrostatic Euler Equations on Collocated Grids

Abstract

This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite‐Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth‐order‐accurate reconstructions with and without Weighted Essentially Non‐Oscillatory limiting. Acoustic‐only upwinding for pressure and mass flux terms and advective‐only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st‐order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k−5/3 to smaller spatial scales by reducing dissipation either with higher‐order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k−5/3 exhibit physically realizable solutions, even though the spectra appear to be physical.

A new WENO-based momentum advection scheme for simulations of ocean mesoscale turbulence

“…For this we follow Ringler (2011), who proposed an upwind reconstruction of vorticity in the context of two-dimensional turbulence. Upwind reconstructions are diffusive: an upwind reconstruction of quantity a with respect to a velocity u leads to a truncation error that is proportional to |u|∂ n a, with n an odd exponent equal to the upwinding order (Norman et al, 2023). The error of upwind vorticity reconstruction is proportional to an odd derivative of vorticity -hence an even derivative of the velocity field -and acts as a diffusion of momentum in the momentum equations:…”

“…Diffusive numerical schemes has seen application in various computational fluid dynamics fields, especially in combination with the conservative (or "flux-form") formulation of the advection operator (Karaca et al, 2012;Maulik & San, 2018;Zeng et al, 2021), including in atmospheric models (Smolarkiewicz & Margolin, 1998;Souza et al, 2023;Norman et al, 2023) and regional ocean models (Shchepetkin & McWilliams, 1998a;Holland et al, 1998;Mohammadi-Aragh et al, 2015). However, finite-volume general circulation models (GCMs) often favor the rotational formulation of the advection operator due to its ease of implementation with non-regular grids, such as the cubed sphere grid (Ronchi et al, 1996), the latitude-longitude capped grid (Fenty & Wang, 2020), or the tripolar grid (Madec & Imbard, 1996).…”

A New WENO‐Based Momentum Advection Scheme for Simulations of Ocean Mesoscale Turbulence

“…For this we follow Ringler (2011), who proposed an upwind reconstruction of vorticity in the context of two-dimensional turbulence. Upwind reconstructions are diffusive: an upwind reconstruction of quantity a with respect to a velocity u leads to a truncation error that is proportional to |u|∂ n a, with n an odd exponent equal to the upwinding order (Norman et al, 2023). The error of upwind vorticity reconstruction is proportional to an odd derivative of vorticity-hence an even derivative of the velocity field-and acts as a diffusion of momentum in the momentum equations:…”

“…We seek to improve the numerical error associated with vorticity reconstruction both by reducing its magnitude and also by computing the reconstruction so that the error is diffusive, and therefore “smooth,” rather than dispersive and “noisy.” For this we follow Ringler (2011), who proposed an upwind reconstruction of vorticity in the context of two‐dimensional turbulence. Upwind reconstructions are diffusive: an upwind reconstruction of quantity a with respect to a velocity u leads to a truncation error that is proportional to | u | ∂ n a , with n an odd exponent equal to the upwinding order (Norman et al., 2023). The error of upwind vorticity reconstruction is proportional to an odd derivative of vorticity—hence an even derivative of the velocity field—and acts as a diffusion of momentum in the momentum equations: $$${\mathcal{N}}_{\zeta }\sim {{\Delta }}^{n}\vert v\vert {\partial }_{y}^{n}\zeta \ \ \text{in}\,(1),\ {\mathcal{N}}_{\zeta }\sim {{\Delta }}^{n}\vert u\vert {\partial }_{x}^{n}\zeta \ \ \text{in}\,(2)\,.$$$ …”

“…Diffusive numerical schemes have seen application in various computational fluid dynamics fields, especially in combination with the conservative (or "flux-form") formulation of the advection operator (Karaca et al, 2012;Maulik & San, 2018;Zeng et al, 2021), including in atmospheric models (Norman et al, 2023;Smolarkiewicz & Margolin, 1998;Souza et al, 2023) and regional ocean models (Holland et al, 1998;Mohammadi-Aragh et al, 2015;Shchepetkin & McWilliams, 1998a). However, finite-volume general circulation models (GCMs) often favor the rotational formulation of the advection operator due to its ease of implementation with non-regular grids, such as the cubed sphere grid (Ronchi et al, 1996), the latitudelongitude capped grid (Fenty & Wang, 2020), or the tripolar grid (Madec & Imbard, 1996).…”