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Stable and unstable shear modes of rotating parallel flows in shallow water

Published online by Cambridge University Press:  21 April 2006

Y.-Y. Hayashi
Geophysical Institute, University of Tokyo, Tokyo 113, Japan
W. R. Young
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02143, USA


This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial β-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is ½ h(u2 + v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uhu). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability.

A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously.

For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.

Research Article
© 1987 Cambridge University Press

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Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609646.Google Scholar
Benjamin, T. B. 1963 The threefold classification of unstable disturbances in flexible surfaces bounding inviscid fluids. J. Fluid Mech. 16, 436450.Google Scholar
Blumen, W. 1970 Shear layer instability of an inviscid compressible fluid. J. Fluid Mech. 40, 769781.Google Scholar
Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear layer instability of a compressible fluid. Part 2. J. Fluid Mech. 71, 305316.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 114.Google Scholar
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343377.Google Scholar
Held, I. M. 1985 Pseudomomentum and orthogonality of modes in shear flows. J. Atmos. Sci. 42, 22802288.Google Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Met. Soc. 111, 877946.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.Google Scholar
Landahl, M. T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13, 609632.Google Scholar
Marinone, S. G. & Ripa, P. 1984 Energetics and instability of a depth independent Equatorial jet. Geophys. Astrophys. Fluid Dyn. 30, 105130.Google Scholar
Paldor, N. 1983 Linear stability and stable modes of geostrophic fronts. Geophys. Astrophys. Fluid Dyn. 24, 299326.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer. 624 pp.
Rayleigh, Lord 1896 The Theory of Sound, vol. II. Dover.
Ripa, P. 1983 General stability conditions for zonal flows in a one layer model on the beta-plane or the sphere. J. Fluid Mech. 126, 463487.Google Scholar
Salmon, R. 1982 The shape of the main thermocline. J. Phys. Oceanogr. 12, 14581479.Google Scholar
Sardeshmukh, P. D. & Hoskins, B. J. 1985 Vorticity balances in the tropics during the 1982–83 El Nino–Southern Oscillation event. Q. J. R. Met. Soc. 111, 261278.Google Scholar
Satomura, T. 1981a An investigation of shear instability in a shallow water. J. Met. Soc. Japan 59, 148167.Google Scholar
Satomura, T. 1981b Supplementary note on shear instability in a shallow water. J. Met. Soc. Japan 59, 168171.Google Scholar
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. A 215, 126.Google Scholar