Dissipation, topography, and statistical theories for large-scale coherent structure

Andrew J. Majda, Margaret Holen

Research output: Contribution to journalArticlepeer-review


Various facets of the equilibrium statistical theories for large-scale coherent structures for two-dimensional flows with and without topography are studied here. The classical few-constraint statistical theories involving energy-enstrophy principles or point vortices are shown to be statistically sharp in the more recent statistical theories with an infinite number of constraints; in other words, at the macrostates of the few-constraint theories, the many-constraint theory provides no additional statistical information. These results are established through a general link between these statistical theories, generalized "selective decay" principles, and statistical sharpness. Through an asymptotic procedure, the many-constraint statistical theories for flows with topography and small-potential vorticity are shown to yield the simpler energy-enstrophy macroscopic states at leading order with systematic higher-order corrections involving a renormalized topography that includes higher moments of the microscopic potential vorticity distribution. For nonequilibrium flows with and without topography, the utility of crude approximate dynamics based on "adiabatic approximation" through the macrostates of few-constraint statistical theory is developed here. It is established that for nonequilibrium decaying flows with viscous dissipation, the crude dynamics based on macrostates involving statistical point vortices yields an excellent approximation; the role of "selective decay" principles is also clarified and compared quantitatively in this context through both mathematical theory and numerical experiments. Surprisingly, these approximate dynamics yield a much poorer approximation with moderate Ekman drag as the dissipative mechanism, and a simple analytical explanation is provided here. Finally, all of these issues are pursued more briefly for damped and driven flows with topography.

Original languageEnglish (US)
Pages (from-to)1183-1234
Number of pages52
JournalCommunications on Pure and Applied Mathematics
Issue number12
StatePublished - Dec 1997

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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