A Numerical Study of Methods for Moist Atmospheric Flows: Compressible Equations

Max Duarte Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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Ann S. Almgren Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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Kaushik Balakrishnan Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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John B. Bell Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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David M. Romps Department of Earth and Planetary Science, University of California, Berkeley, and Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California

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Abstract

Two common numerical techniques for integrating reversible moist processes in atmospheric flows are investigated in the context of solving the fully compressible Euler equations. The first is a one-step, coupled technique based on using appropriate invariant variables such that terms resulting from phase change are eliminated in the governing equations. In the second approach, which is a two-step scheme, separate transport equations for liquid water and water vapor are used, and no conversion between water vapor and liquid water is allowed in the first step, while in the second step a saturation adjustment procedure is performed that correctly allocates the water into its two phases based on the Clausius–Clapeyron formula. The numerical techniques described are first validated by comparing to a well-established benchmark problem. Particular attention is then paid to the effect of changing the time scale at which the moist variables are adjusted to the saturation requirements in two different variations of the two-step scheme. This study is motivated by the fact that when acoustic modes are integrated separately in time (neglecting phase change related phenomena), or when soundproof equations are integrated, the time scale for imposing saturation adjustment is typically much larger than the numerical one related to the acoustics.

Corresponding author address: Max Duarte, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., MS 50A-1148, Berkeley, CA 94720. E-mail: mdgonzalez@lbl.gov

Abstract

Two common numerical techniques for integrating reversible moist processes in atmospheric flows are investigated in the context of solving the fully compressible Euler equations. The first is a one-step, coupled technique based on using appropriate invariant variables such that terms resulting from phase change are eliminated in the governing equations. In the second approach, which is a two-step scheme, separate transport equations for liquid water and water vapor are used, and no conversion between water vapor and liquid water is allowed in the first step, while in the second step a saturation adjustment procedure is performed that correctly allocates the water into its two phases based on the Clausius–Clapeyron formula. The numerical techniques described are first validated by comparing to a well-established benchmark problem. Particular attention is then paid to the effect of changing the time scale at which the moist variables are adjusted to the saturation requirements in two different variations of the two-step scheme. This study is motivated by the fact that when acoustic modes are integrated separately in time (neglecting phase change related phenomena), or when soundproof equations are integrated, the time scale for imposing saturation adjustment is typically much larger than the numerical one related to the acoustics.

Corresponding author address: Max Duarte, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., MS 50A-1148, Berkeley, CA 94720. E-mail: mdgonzalez@lbl.gov
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