### Abstract

This is an informal account of the fluid-dynamical theory describing nonlinear interactions between small-amplitude waves and mean flows. This kind of theory receives little attention in mainstream fluid dynamics, but it has been developed greatly in atmosphere and ocean fluid dynamics. This is because of the pressing need in numerical atmosphere-ocean models to approximate the effects of unresolved small-scale waves acting on the resolved large-scale flow, which can have very important dynamical implications. Several atmosphere ocean example are discussed in these notes (in particular, see §5), but generic wave-mean interaction theory should be useful in other areas of fluid dynamics as well. We will look at a number of examples relating to the basic problem of classical wave-mean interaction theory: finding the nonlinear O(a ^{2}) mean-flow response to O(a) waves with small amplitude a ≪ 1 in simple geometry. Small wave amplitude a ≪ 1 means that the use of linear theory for O(a) waves propagating on an O(1) background flow is allowed. Simple geometry means that the flow is periodic in one spatial coordinate and that the O(1) background flow does not depend on this coordinate. This allows the use of averaging over the periodic coordinate, which greatly simplifies the problem.

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

Title of host publication | CISM International Centre for Mechanical Sciences, Courses and Lectures |

Publisher | Springer International Publishing |

Pages | 95-133 |

Number of pages | 39 |

DOIs | |

State | Published - Jan 1 2005 |

### Publication series

Name | CISM International Centre for Mechanical Sciences, Courses and Lectures |
---|---|

Volume | 483 |

ISSN (Print) | 0254-1971 |

ISSN (Electronic) | 2309-3706 |

### Fingerprint

### Keywords

- Boussinesq equation
- Critical line
- Gravity wave
- Momentum flux
- Rossby wave

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Computer Science Applications
- Modeling and Simulation

### Cite this

*CISM International Centre for Mechanical Sciences, Courses and Lectures*(pp. 95-133). (CISM International Centre for Mechanical Sciences, Courses and Lectures; Vol. 483). Springer International Publishing. https://doi.org/10.1007/3-211-38025-6_4

**Wave-mean interaction theory.** / Buhler, Oliver.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*CISM International Centre for Mechanical Sciences, Courses and Lectures.*CISM International Centre for Mechanical Sciences, Courses and Lectures, vol. 483, Springer International Publishing, pp. 95-133. https://doi.org/10.1007/3-211-38025-6_4

}

TY - CHAP

T1 - Wave-mean interaction theory

AU - Buhler, Oliver

PY - 2005/1/1

Y1 - 2005/1/1

N2 - This is an informal account of the fluid-dynamical theory describing nonlinear interactions between small-amplitude waves and mean flows. This kind of theory receives little attention in mainstream fluid dynamics, but it has been developed greatly in atmosphere and ocean fluid dynamics. This is because of the pressing need in numerical atmosphere-ocean models to approximate the effects of unresolved small-scale waves acting on the resolved large-scale flow, which can have very important dynamical implications. Several atmosphere ocean example are discussed in these notes (in particular, see §5), but generic wave-mean interaction theory should be useful in other areas of fluid dynamics as well. We will look at a number of examples relating to the basic problem of classical wave-mean interaction theory: finding the nonlinear O(a 2) mean-flow response to O(a) waves with small amplitude a ≪ 1 in simple geometry. Small wave amplitude a ≪ 1 means that the use of linear theory for O(a) waves propagating on an O(1) background flow is allowed. Simple geometry means that the flow is periodic in one spatial coordinate and that the O(1) background flow does not depend on this coordinate. This allows the use of averaging over the periodic coordinate, which greatly simplifies the problem.

AB - This is an informal account of the fluid-dynamical theory describing nonlinear interactions between small-amplitude waves and mean flows. This kind of theory receives little attention in mainstream fluid dynamics, but it has been developed greatly in atmosphere and ocean fluid dynamics. This is because of the pressing need in numerical atmosphere-ocean models to approximate the effects of unresolved small-scale waves acting on the resolved large-scale flow, which can have very important dynamical implications. Several atmosphere ocean example are discussed in these notes (in particular, see §5), but generic wave-mean interaction theory should be useful in other areas of fluid dynamics as well. We will look at a number of examples relating to the basic problem of classical wave-mean interaction theory: finding the nonlinear O(a 2) mean-flow response to O(a) waves with small amplitude a ≪ 1 in simple geometry. Small wave amplitude a ≪ 1 means that the use of linear theory for O(a) waves propagating on an O(1) background flow is allowed. Simple geometry means that the flow is periodic in one spatial coordinate and that the O(1) background flow does not depend on this coordinate. This allows the use of averaging over the periodic coordinate, which greatly simplifies the problem.

KW - Boussinesq equation

KW - Critical line

KW - Gravity wave

KW - Momentum flux

KW - Rossby wave

UR - http://www.scopus.com/inward/record.url?scp=70349595200&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349595200&partnerID=8YFLogxK

U2 - 10.1007/3-211-38025-6_4

DO - 10.1007/3-211-38025-6_4

M3 - Chapter

AN - SCOPUS:70349595200

T3 - CISM International Centre for Mechanical Sciences, Courses and Lectures

SP - 95

EP - 133

BT - CISM International Centre for Mechanical Sciences, Courses and Lectures

PB - Springer International Publishing

ER -