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The nonlinear dynamics of baroclinic wave ensembles

Published online by Cambridge University Press:  20 April 2006

Joseph Pedlosky
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543

Abstract

A theory is developed to describe the weakly nonlinear dynamics which applies in the simultaneous presence of several, long, baroclinic waves. The geometry is flat (i.e. β = 0) and dissipation is modelled by Ekman friction in the context of the quasi-geostrophic two-layer model. Three main problems are discussed.

  1. For free, unstable waves it is shown that the wave which is realized in finite amplitude is not the linearly most unstable wave. Rather a longer wave, capable of achieving the single largest steady amplitude, is favoured in the competition for the potential energy of the basic state. This result is shown necessary if the end state is steady and numerous numerical calculations indicate the pre-eminence of the same wave if the final state is vacillatory. The notion of conjugate waves, capable of identical final amplitude, is also discussed.

  2. If the free waves are subject to time-varying supercriticality so that intervals of stability ensue, the response is asymmetric over the period of the forcing. Sufficiently rapid ‘seasonal’ forcing leads to long-term aperiodic response.

  3. If each wave in the spectrum is directly forced a wave hysteresis phenomenon occurs. Sudden jumps in the wave amplitude at critical values of the forcing are intrinsic to the wave response. Again, sufficiently rapid wave forcing produces an aperiodic response.

The forced wave problem exhibits multiple equilibria. Each solution branch corresponds to a different dominant wave. The determination of the realized branch depends on the relative stability criteria developed for the free waves.

Type
Research Article
Copyright
© 1981 Cambridge University Press

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References

Buzyna, G., Pfeffer, R. L. & Kung, R. 1978 Cyclic variations of the imposed temperature contrast in a thermally driven rotating annulus of fluid. J. Atmos. Sci. 35, 859881.Google Scholar
Drazin, P. G. 1970 Non-linear baroclinic instability of a continuous zonal flow. Quart. J. Roy. Met. Soc. 96, 667676.Google Scholar
Drazin, P. G. 1972 Non-linear baroclinic instability of a continuous zonal flow of a viscous fluid. J. Fluid Mech. 55, 577588.Google Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1, 3352.Google Scholar
Pedlosky, J. 1970 Finite amplitude baroclinic waves. J. Atmos. Sci. 27, 1530.Google Scholar
Pedlosky, J. 1971 Finite amplitude baroclinic waves with small dissipation. J. Atmos. Sci. 28, 587597.Google Scholar
Pedlosky, J. 1972 Limit cycles and unstable baroclinic waves. J. Atmos. Sci. 29, 5363.Google Scholar
Pedlosky, J. & Frenzen, C. 1979 Chaotic and periodic behaviour of baroclinic waves. J. Atmos. Sci. 37, 11771196.Google Scholar
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