### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 1183-1234 |

Number of pages | 52 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 50 |

Issue number | 12 |

State | Published - Dec 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*50*(12), 1183-1234.

**Dissipation, topography, and statistical theories for large-scale coherent structure.** / Majda, Andrew J.; Holen, Margaret.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 50, no. 12, pp. 1183-1234.

}

TY - JOUR

T1 - Dissipation, topography, and statistical theories for large-scale coherent structure

AU - Majda, Andrew J.

AU - Holen, Margaret

PY - 1997/12

Y1 - 1997/12

N2 - 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.

AB - 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.

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M3 - Article

AN - SCOPUS:0031507314

VL - 50

SP - 1183

EP - 1234

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 12

ER -