# Wave-mean interaction theory

## 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.

## Keywords

Gravity Wave Rossby Wave Momentum Flux Critical Line Boussinesq Equation## Preview

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## Bibliography

- D. G. Andrews, J. R. Holton, and C. B. Leovy.
*Middle Atmosphere Dynamics*. Academic Press, 1987.Google Scholar - M. P. Baldwin, L. J. Gray, T. J. Dunkerton, K. Hamilton, P. H. Haynes, W. J. Randel, J. R. Holton, M. J. Alexander, I. Hirota, T. Horinouchi, D. B. A. Jones, J. S. Kinnersley, C. Marquardt, K. Sato, and M. Takahashi. The quasi-biennial oscillation.
*Revs. Geophys.*, 39:179–229, 2001.CrossRefGoogle Scholar - F. P. Bretherton. On the mean motion induced by internal gravity waves.
*J. Fluid Mech.*, 36:785–803, 1969.MATHCrossRefGoogle Scholar - O. Bühler. On the vorticity transport due to dissipating or breaking waves in shallow-water flow.
*J. Fluid Mech.*, 407:235–263, 2000.MATHCrossRefMathSciNetGoogle Scholar - O. Bühler and T. E. Jacobson. Wave-driven currents and vortex dynamics on barred beaches.
*J. Fluid Mech.*, 449:313–339, 2001.MATHCrossRefMathSciNetGoogle Scholar - O. Bühler and M. E. Mclntyre. Remote recoil: a new wave-mean interaction effect.
*J. Fluid Mech.*, 492:207–230, 2003.MATHCrossRefMathSciNetGoogle Scholar - O. Bühler and M. E. McIntyre. Wave capture and wave-vortex duality. Submitted to
*J. Fluid Mech*. Preprint available at http://www.cims.nyu.edu/~obuhler., 2004.Google Scholar - P. G. Drazin and W. H. Reid.
*Hydrodynamic Stability*. Cambridge University Press, 1981.Google Scholar - D. C. Fritts and M. J. Alexander. Gravity-wave dynamics and effects in the middle atmosphere.
*Revs. Geophys.*, 41(1), doi:10.1029/2001RG000106., 2003.Google Scholar - P. H. Haynes. Nonlinear instability of a rossby-wave critical layer.
*J. Fluid Mech.*, 161: 493–511, 1985.MATHCrossRefMathSciNetGoogle Scholar - P. H. Haynes. On the instability of sheared disturbances.
*J. Fluid Mech.*, 175:463–478, 1987.MATHCrossRefGoogle Scholar - P. H. Haynes. Critical layers. In J. R. Holton, J. A. Pyle, and J. A. Curry, editors,
*Encyclopedia of Atmospheric Sciences*. London, Academic/Elsevier, 2003.Google Scholar - I Held. The general circulation of the atmosphere.
*GFD lecture notes, ed. Rick Salmon*.*http://gfd.whoi.edu/proceedings/2000/PDF/lectures2000.pdf*2000.Google Scholar - C. O. Hines. Earlier days of gravity waves revisited.
*Pure and Applied Geophysics*, 130: 151–170, 1989.CrossRefGoogle Scholar - P. D. Killworth and M. E. McIntyre. Do rossby-wave critical layers absorb, reflect or over-reflect? j. fluid mech.
*449–492*., 1985.Google Scholar - Y.-J. Kim, S. D. Eckermann, and H.-Y. Chun. An overview of the past, present and future of gravity-wave drag parametrization for numerical climate and weather prediction models.
*Atmos.-Oc.*, 41:65–98, 2003.Google Scholar - M. J. Lighthill.
*Waves in Fluids*. Cambridge University Press, Cambridge, 1978.MATHGoogle Scholar - M. E. McIntyre. On global-scale atmospheric circulations. In G. K. Batchelor, H. K. Moffatt, and M. G. Worster, editors,
*Perspectives in Fluid Dynamics: A Collective Introduction to Current Research*, pages 557–624. Cambridge, University Press, 631 pp., 2000.Google Scholar - M. E. McIntyre and W. A. Norton. Dissipative wave-mean interactions and the transport of vorticity or potential vorticity.
*J. Fluid Mech.*, 212, 403–435. Corrigendum 220, 693., 1990.MATHCrossRefMathSciNetGoogle Scholar - J. Pedlosky.
*Geophysical Fluid Dynamics*(*2nd edition*). Springer-Verlag, New York, 1987.MATHGoogle Scholar - R. A. Plumb. The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation.
*J. Atmos. Sci.*, 34:1847–1858, 1977.CrossRefGoogle Scholar - C. Staquet and J. Sommeria. Internal gravity waves: From instabilities to turbulence.
*Ann. Rev. Fluid Mech.*, 34:559–594, 2002.CrossRefMathSciNetGoogle Scholar