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Block-Iterative Method of Solving the Nonhydrostatic Pressure in Terrain-Following Coordinates: Two-Level Pressure and Truncation Error Analysis

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  • 1 Mesoscale and Microscale Meteorology Division, National Center for Atmospheric Research,* Boulder, Colorado
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Abstract

A technique for the treatment of the pressure in anelastic, nonhydrostatic terrain-following coordinates is described. It involves the use of two levels of pressure in such a manner so as to ensure that the anelastic mass-continuity equation is satisfied to round-off level. This procedure significantly improves model stability and accuracy. In the presence of modestly steep topography, the computational burden of the diagnostic elliptic pressure solver is equivalent to that of a direct solver. The two-level pressure approach is viewed as inappropriate for iterative schemes. A pressure truncation error analysis is described for calculating the second-order truncation error fields Γ associated with kinetic energy conservation for arbitrary formulations of the pressure gradient terms. The full transformed equation set is used, such that the combined effect of all of the equations contributing to the error is considered. Truncation error equations are derived for two specific formulations containing terms of Ox2, Δy2, Δz2). These equations are used to validate a more general field analysis technique applicable for any numerical formulation. The kinetic energy errors that result specifically from the application of the two-level pressure technique are compared with the second-order Γ errors and are shown to be 5–10 times as small. Simulations show the stabilizing effect of the two-level pressure technique where comparisons between the two-level approach using a single block iteration and the same approach using a fully converged solution show negligible differences. The particular cases chosen were numerically unstable using a single block iteration without the two-level approach. The error analysis showed modest errors in the kinetic energy budget resulting from the numerical formulation of the pressure gradient terms with little difference between the formulations tested. The cases presented all had well-resolved topography.

Corresponding author address: Terry L. Clark, NCAR, P.O. Box 3000, Boulder, CO 80307-3000. clark@ucar.edu

Abstract

A technique for the treatment of the pressure in anelastic, nonhydrostatic terrain-following coordinates is described. It involves the use of two levels of pressure in such a manner so as to ensure that the anelastic mass-continuity equation is satisfied to round-off level. This procedure significantly improves model stability and accuracy. In the presence of modestly steep topography, the computational burden of the diagnostic elliptic pressure solver is equivalent to that of a direct solver. The two-level pressure approach is viewed as inappropriate for iterative schemes. A pressure truncation error analysis is described for calculating the second-order truncation error fields Γ associated with kinetic energy conservation for arbitrary formulations of the pressure gradient terms. The full transformed equation set is used, such that the combined effect of all of the equations contributing to the error is considered. Truncation error equations are derived for two specific formulations containing terms of Ox2, Δy2, Δz2). These equations are used to validate a more general field analysis technique applicable for any numerical formulation. The kinetic energy errors that result specifically from the application of the two-level pressure technique are compared with the second-order Γ errors and are shown to be 5–10 times as small. Simulations show the stabilizing effect of the two-level pressure technique where comparisons between the two-level approach using a single block iteration and the same approach using a fully converged solution show negligible differences. The particular cases chosen were numerically unstable using a single block iteration without the two-level approach. The error analysis showed modest errors in the kinetic energy budget resulting from the numerical formulation of the pressure gradient terms with little difference between the formulations tested. The cases presented all had well-resolved topography.

Corresponding author address: Terry L. Clark, NCAR, P.O. Box 3000, Boulder, CO 80307-3000. clark@ucar.edu

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