Vol. 16, No. 1, 2021

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Spectral steady-state solutions to the 2D compressible Euler equations for cross-mountain flows

Jorge E. Guerra and Paul A. Ullrich

Vol. 16 (2021), No. 1, 99–117
Abstract

We present an algorithm for obtaining reference solutions to the nonhydrostatic, compressible, dry Euler equations in unapproximated form by systematic linearization and solution using the Newton–Raphson method within a bounded, rectangular atmospheric domain. The state fields are expanded in Hermite functions (horizontal) and Chebyshev polynomials (vertical), resulting in a truncated hybrid spectral colocated discretization analogous to quasianalytical Fourier solutions for the linear Boussinesq system available in the literature. Lastly, our method incorporates general background profiles of wind and stratification (including piecewise linear functions), expanding the range of numerical test conditions available for validation. Our model is solved efficiently by direct matrix inversion using modest computing resources. We show an improvement in error estimation using our spectral solution compared to a known approximated analytical reference and introduce solutions under more general conditions converged to steady state within machine precision. Lastly, we demonstrate grid convergence of long-term, independent model integrations to our solution reference.

Keywords
spectral methods, orographic gravity waves, numerical model validation
Mathematical Subject Classification
Primary: 35J56, 76U60, 86-08, 86-10, 86A10
Milestones
Received: 16 December 2020
Accepted: 28 February 2021
Published: 22 June 2021
Authors
Jorge E. Guerra
Cooperative Institute for Mesoscale Meteorological Studies
University of Oklahoma
NOAA National Severe Storms Laboratory
National Weather Center
Norman, OK
United States
Paul A. Ullrich
Department of Land, Air, and Water Resources
University of California, Davis
Davis, CA
United States