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.
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Cooperative Institute for Mesoscale
Meteorological Studies
University of Oklahoma
NOAA National Severe Storms Laboratory
National Weather Center
Norman, OK
United States