Monotone Finite-Difference Approximations to the Advection-Condensation Problem

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  • 1 National Center for Atmospheric Research, Boulder, Colorado
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Abstract

We discuss herein numerical difficulties with finite-difference approximations to the thermodynamic conservation laws near sharp, cloud-environment interfaces. The Conservation laws for entropy and water substance variables are coupled through the phase change processes. This coupling of the thermodynamic equations may lead to spurious numerical oscillations that, in general, are not prevented by direct application of traditional monotone methods developed for the uncoupled equations. In order to suppress false oscillations in the solutions, we consider special techniques which derive from the flux-corrected-transport (FCT) methodology. In these, we incorporate physical information about condensation-evaporation processes directly into the limiters constraining the antidiffusive fluxes of the FCT methods. We elaborate upon two different advection-condensation schemes relevant to the two formulations of the advection-condensation problem, commonly used in cloud modeling. For the fractional-time-steps formulation, we develop an FCT advevtion scheme which preserves the monotone character of transported thermodynamic variables and also ensures monotonicity of the relative humidity field diagnosed after the advection step. This results in monotone solutions after the entire advection-condensation cycle. For the conservative variables formulation, we derive a simple and efficient FCT scheme that ensures nonoscillatory thermodynamic fields diagnosed from the advected variables. Theoretical considerations are illustrated with idealized one-dimensional kinematic tests and with examples of two-dimensional simulations of a small cumulus cloud. Dynamical calculations show that morphology of the cloud-environment interface strongly depends on the numerical scheme applied.

Abstract

We discuss herein numerical difficulties with finite-difference approximations to the thermodynamic conservation laws near sharp, cloud-environment interfaces. The Conservation laws for entropy and water substance variables are coupled through the phase change processes. This coupling of the thermodynamic equations may lead to spurious numerical oscillations that, in general, are not prevented by direct application of traditional monotone methods developed for the uncoupled equations. In order to suppress false oscillations in the solutions, we consider special techniques which derive from the flux-corrected-transport (FCT) methodology. In these, we incorporate physical information about condensation-evaporation processes directly into the limiters constraining the antidiffusive fluxes of the FCT methods. We elaborate upon two different advection-condensation schemes relevant to the two formulations of the advection-condensation problem, commonly used in cloud modeling. For the fractional-time-steps formulation, we develop an FCT advevtion scheme which preserves the monotone character of transported thermodynamic variables and also ensures monotonicity of the relative humidity field diagnosed after the advection step. This results in monotone solutions after the entire advection-condensation cycle. For the conservative variables formulation, we derive a simple and efficient FCT scheme that ensures nonoscillatory thermodynamic fields diagnosed from the advected variables. Theoretical considerations are illustrated with idealized one-dimensional kinematic tests and with examples of two-dimensional simulations of a small cumulus cloud. Dynamical calculations show that morphology of the cloud-environment interface strongly depends on the numerical scheme applied.

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