### Abstract

Here a mathematically rigorous framework is developed for deriving new reduced simplified dynamical equations for geophysical flows with arbitrary potential vorticity interacting with fast gravity waves. The examples include the rotating Boussinesq and rotating shallow water equations in the quasi-geostrophic limit with vanishing Rossby number. For the spatial periodic case the theory implies that the quasi-geostrophic equations are valid limiting equations in the weak topology for arbitrary initial data. Furthermore, simplified reduced equations are developed for the fashion in which the vortical waves influence the gravity waves through averaging over specific gravity wave/vortical resonances.

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

Pages (from-to) | 619-658 |

Number of pages | 40 |

Journal | Communications in Partial Differential Equations |

Volume | 21 |

Issue number | 3-4 |

State | Published - 1996 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*Communications in Partial Differential Equations*,

*21*(3-4), 619-658.

**Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity.** / Embid, Pedro F.; Majda, Andrew J.

Research output: Contribution to journal › Article

*Communications in Partial Differential Equations*, vol. 21, no. 3-4, pp. 619-658.

}

TY - JOUR

T1 - Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity

AU - Embid, Pedro F.

AU - Majda, Andrew J.

PY - 1996

Y1 - 1996

N2 - Here a mathematically rigorous framework is developed for deriving new reduced simplified dynamical equations for geophysical flows with arbitrary potential vorticity interacting with fast gravity waves. The examples include the rotating Boussinesq and rotating shallow water equations in the quasi-geostrophic limit with vanishing Rossby number. For the spatial periodic case the theory implies that the quasi-geostrophic equations are valid limiting equations in the weak topology for arbitrary initial data. Furthermore, simplified reduced equations are developed for the fashion in which the vortical waves influence the gravity waves through averaging over specific gravity wave/vortical resonances.

AB - Here a mathematically rigorous framework is developed for deriving new reduced simplified dynamical equations for geophysical flows with arbitrary potential vorticity interacting with fast gravity waves. The examples include the rotating Boussinesq and rotating shallow water equations in the quasi-geostrophic limit with vanishing Rossby number. For the spatial periodic case the theory implies that the quasi-geostrophic equations are valid limiting equations in the weak topology for arbitrary initial data. Furthermore, simplified reduced equations are developed for the fashion in which the vortical waves influence the gravity waves through averaging over specific gravity wave/vortical resonances.

UR - http://www.scopus.com/inward/record.url?scp=0001755104&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001755104&partnerID=8YFLogxK

M3 - Article

VL - 21

SP - 619

EP - 658

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 3-4

ER -