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balanced and unbalanced Components of Moist Atmospheric Flows with Phase Changes

Chinese Annals of Mathematics,Series B 2019年第6期40卷 1005-1038页

作者： Alfredo N.WETZEL Leslie M.SMITH Samuel N.STECHMANN Jonathan E.MARTIN Department of Mathematics University of Wisconsin-MadisonMadisonWI 53706USA Department of Mathematics and Department of Engineering Physics University of Wisconsin-MadisonMadisonWI 53706USA Department of Mathematics and Department of Atmospheric and Oceanic Sciences University of Wisconsin-MadisonMadisonWI 53706USA Department of Atmospheric and Oceanic Sciences University of Wisconsin-MadisonMadisonWI 53706USA

Atmospheric variables(temperature, velocity, etc.) are often decomposed into balanced and unbalanced components that represent low-frequency and high-frequency waves, respectively. Such decompositions can be defined, for instance, in terms of eigenmodes of a linear operator. Traditionally these decompositions ignore phase changes of water since phase changes create a piecewise-linear operator that differs in different phases(cloudy versus non-cloudy). Here we investigate the following question: How can a balanced–unbalanced decomposition be performed in the presence of phase changes? A method is described here motivated by the case of small Froude and Rossby numbers,in which case the asymptotic limit yields precipitating quasi-geostrophic equations with phase changes. Facilitated by its zero-frequency eigenvalue, the balanced component can be found by potential vorticity(PV) inversion, by solving an elliptic partial differential equation(PDE), which includes Heaviside discontinuities due to phase changes. The method is also compared with two simpler methods: one which neglects phase changes, and one which simply treats the raw pressure data as a streamfunction. Tests are shown for both synthetic, idealized data and data from Weather Research and Forecasting(WRF) model simulations. In comparisons, the phase-change method and no-phase-change method produce substantial differences within cloudy regions, of approximately 5K in potential temperature, due to the presence of clouds and phase changes in the data. A theoretical justification is also derived in the form of a elliptic PDE for the differences in the two streamfunctions.

关键词： Potential vorticity inversion Moist atmospheric dynamics Slow-fast systems balanced-unbalanced decomposition Elliptic partial differential equations

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