### Abstract

Recent numerical simulations reveal remarkably different behavior in rotating stably stratified fluids at low Froude numbers for finite Rossby numbers as compared with the behavior at both low Froude and Rossby numbers. Here the reduced low Froude number limiting dynamics in both of these situations is developed with complete mathematical rigor by applying the theory for fast wave averaging for geophysical flows developed recently by the authors. The reduced dynamical equations include all resonant triad interactions for the slow (vortical) modes, the effect of the slow (vortical) modes on the fast (inertial gravity) modes, and also the general resonant triad interactions among the fast (internal gravity) waves. The nature of the reduced dynamics in these two situations is compared and contrasted here. For example, the reduced slow dynamics for the vortical modes in the low Froude number limit at finite Rossby numbers includes vertically sheared horizontal motion while the reduced slow dynamics in the low Froude number and low Rossby number limit yields the familiar quasigeostrophic equations where such vertically sheared motion is completely absent-in fact, such vertically sheared motions participate only in the fast dynamics in this quasigeostrophic limit. The use of Ertel's theorem on conservation of potential vorticity is utilized, for example, in studying the limiting behavior of the rotating Boussinesq equations with general slanted rotation and unbalanced initial data. Other interesting physical effects such as those of varying Prandtl number on the limiting dynamics are also developed and compared here.

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

Pages (from-to) | 1-50 |

Number of pages | 50 |

Journal | Geophysical and Astrophysical Fluid Dynamics |

Volume | 87 |

Issue number | 1-2 |

State | Published - 1998 |

### Fingerprint

### Keywords

- Gravity waves
- Resonances
- Strongly stratified flow
- Vortical modes

### ASJC Scopus subject areas

- Geochemistry and Petrology
- Geophysics
- Space and Planetary Science
- Computational Mechanics
- Mechanics of Materials
- Astronomy and Astrophysics

### Cite this

*Geophysical and Astrophysical Fluid Dynamics*,

*87*(1-2), 1-50.

**Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers.** / Embid, Pedro F.; Majda, Andrew J.

Research output: Contribution to journal › Article

*Geophysical and Astrophysical Fluid Dynamics*, vol. 87, no. 1-2, pp. 1-50.

}

TY - JOUR

T1 - Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers

AU - Embid, Pedro F.

AU - Majda, Andrew J.

PY - 1998

Y1 - 1998

N2 - Recent numerical simulations reveal remarkably different behavior in rotating stably stratified fluids at low Froude numbers for finite Rossby numbers as compared with the behavior at both low Froude and Rossby numbers. Here the reduced low Froude number limiting dynamics in both of these situations is developed with complete mathematical rigor by applying the theory for fast wave averaging for geophysical flows developed recently by the authors. The reduced dynamical equations include all resonant triad interactions for the slow (vortical) modes, the effect of the slow (vortical) modes on the fast (inertial gravity) modes, and also the general resonant triad interactions among the fast (internal gravity) waves. The nature of the reduced dynamics in these two situations is compared and contrasted here. For example, the reduced slow dynamics for the vortical modes in the low Froude number limit at finite Rossby numbers includes vertically sheared horizontal motion while the reduced slow dynamics in the low Froude number and low Rossby number limit yields the familiar quasigeostrophic equations where such vertically sheared motion is completely absent-in fact, such vertically sheared motions participate only in the fast dynamics in this quasigeostrophic limit. The use of Ertel's theorem on conservation of potential vorticity is utilized, for example, in studying the limiting behavior of the rotating Boussinesq equations with general slanted rotation and unbalanced initial data. Other interesting physical effects such as those of varying Prandtl number on the limiting dynamics are also developed and compared here.

AB - Recent numerical simulations reveal remarkably different behavior in rotating stably stratified fluids at low Froude numbers for finite Rossby numbers as compared with the behavior at both low Froude and Rossby numbers. Here the reduced low Froude number limiting dynamics in both of these situations is developed with complete mathematical rigor by applying the theory for fast wave averaging for geophysical flows developed recently by the authors. The reduced dynamical equations include all resonant triad interactions for the slow (vortical) modes, the effect of the slow (vortical) modes on the fast (inertial gravity) modes, and also the general resonant triad interactions among the fast (internal gravity) waves. The nature of the reduced dynamics in these two situations is compared and contrasted here. For example, the reduced slow dynamics for the vortical modes in the low Froude number limit at finite Rossby numbers includes vertically sheared horizontal motion while the reduced slow dynamics in the low Froude number and low Rossby number limit yields the familiar quasigeostrophic equations where such vertically sheared motion is completely absent-in fact, such vertically sheared motions participate only in the fast dynamics in this quasigeostrophic limit. The use of Ertel's theorem on conservation of potential vorticity is utilized, for example, in studying the limiting behavior of the rotating Boussinesq equations with general slanted rotation and unbalanced initial data. Other interesting physical effects such as those of varying Prandtl number on the limiting dynamics are also developed and compared here.

KW - Gravity waves

KW - Resonances

KW - Strongly stratified flow

KW - Vortical modes

UR - http://www.scopus.com/inward/record.url?scp=0040010955&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040010955&partnerID=8YFLogxK

M3 - Article

VL - 87

SP - 1

EP - 50

JO - Geophysical and Astrophysical Fluid Dynamics

JF - Geophysical and Astrophysical Fluid Dynamics

SN - 0309-1929

IS - 1-2

ER -