Wave-mean interaction theory
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.
KeywordsGravity Wave Rossby Wave Momentum Flux Critical Line Boussinesq Equation
Unable to display preview. Download preview PDF.
- 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
- 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. 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
- 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. 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