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Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes

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Abstract

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 eigen-modes 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 5 K 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Alfredo N. Wetzel

  2. Department of Mathematics and Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Leslie M. Smith

  3. Department of Mathematics and Department of Atmospheric and Oceanic Sciences, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Samuel N. Stechmann

  4. Department of Atmospheric and Oceanic Sciences, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Jonathan E. Martin

Authors
  1. Alfredo N. Wetzel
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  2. Leslie M. Smith
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  3. Samuel N. Stechmann
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  4. Jonathan E. Martin
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Correspondence to Alfredo N. Wetzel, Leslie M. Smith, Samuel N. Stechmann or Jonathan E. Martin.

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Dedicated to Professor Andrew J. Majda on the occasion of his 70th birthday

This work was supported by the National Science Foundation through grant AGS1443325 and DMS-1907667 and the University of Wisconsin-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation.

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Wetzel, A.N., Smith, L.M., Stechmann, S.N. et al. Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes. Chin. Ann. Math. Ser. B 40, 1005–1038 (2019). https://doi.org/10.1007/s11401-019-0170-4

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  • DOI: https://doi.org/10.1007/s11401-019-0170-4

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