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.

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