Chinese Annals of Mathematics, Series B

, Volume 40, Issue 6, pp 1005–1038 | Cite as

Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes

  • Alfredo N. WetzelEmail author
  • Leslie M. SmithEmail author
  • Samuel N. StechmannEmail author
  • Jonathan E. MartinEmail author


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.


Potential vorticity inversion Moist atmospheric dynamics Slow-fast systems Balanced-unbalanced decomposition Elliptic partial differential equations 

2000 MR Subject Classification

35R05 86A10 


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© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Department of Mathematics and Department of Engineering PhysicsUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.Department of Mathematics and Department of Atmospheric and Oceanic SciencesUniversity of Wisconsin-MadisonMadisonUSA
  4. 4.Department of Atmospheric and Oceanic SciencesUniversity of Wisconsin-MadisonMadisonUSA

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