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The stability of planetary waves on a sphere

Published online by Cambridge University Press:  29 March 2006

P. G. Baines
P. G. Baines
Affiliation:
CSIKO, Division of Atmospheric Physics, Aspendale, Victoria 3195, Australia
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Abstract

The stability of individual inviscid barotropic planetary waves and zonal flow on a sphere to small disturbances is examined by means of numerical solution of the algebraic eigenvalue problem arising from the spectral form of the governing equations. It is shown that waves with total wavenumber n (the lower index of the Legendre function Pmn which describes the waves’ meridional structure) less than 3 are stable for all amplitudes, whereas those with n ≥ 3 are unstable if their amplitudes are sufficiently large. For travelling waves (m ≠ 0) with n = 3 and 4 and with disturbances comprised of 30 modes, the amplitudes required for instability are approximated by those obtained from triad interactions, and are smaller than those given by Hoskins (1973). For the zonal-flow modes (m = 0) the critical amplitudes are smaller than those predicted by triad interactions, and are close to those obtained from Rayleigh's classical criterion.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 73 , Issue 2 , 27 January 1976 , pp. 193 - 213
Copyright
© 1976 Cambridge University Press

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References

Bourke, W. 1972 An efficient one-level, primitive equation spectral model Mon. Weather Rev. 100, 683.Google Scholar
Craig, R. A. 1945 A solution of the non-linear vorticity equation for atmospheric motion J. Met. 2, 175.Google Scholar
Eberlein, P. J. 1962 Jacobi-like method for the automatic computation of eigenvalues and eigenvectors of an arbitrary matrix J. Soc. Ind. Appl. Math. 10, 74.Google Scholar
Fjortoft, R. 1953 On changes in the spectral distribution of kinetic energy in two-dimensional non-divergent flow Tellus, 5, 225.Google Scholar
Gill, A. E. 1974 The stability of planetary waves Geophys. Fluid Dyn. 6, 29.Google Scholar
Grad, J. & Brebner, M. A. 1968 Algorithm 343, eigenvalues and eigenvectors of a real general matrix Comm. A.C.M. 11, 820.Google Scholar
Hasselmann, K. 1967a Non-linear interactions treated by the methods of theoretical physics (with application to the generation of waves by wind). Proc. Roy. Soc. A 299, 77.Google Scholar
Hasselmann, K. 1967b A criterion for non-linear wave stability. J. Fluid Mech. 30, 737.Google Scholar
Hoskins, B. J. 1973 Stability of the Rossby-Haurwitz wave Quart. J. Roy. Met. Soc. 99, 723.Google Scholar
Hoskins, B. J. & Hollingsworth, A. 1973 On the simplest example of the barotropic instability of Rossby wave motion J. Atmos. Sci. 30, 150.Google Scholar
Jahnke, E. & Emde, F. 1945 Tables of Functions with Formulae and Curves. Dover.
Karunin, A. B. 1970 On Rossby waves in barotropic atmosphere in the presence of zonal flow. Izv. Atmos. Ocean. Phys. 6, 1091 (English trans.).Google Scholar
Lilly, D. K. 1972 Numerical simulation studies of two-dimensional turbulence. II. Stability and predictability studies Geophys. Fluid Dyn. 4, 1.Google Scholar
Lilly, D. K. 1973 A note on barotropic instability and predictability J. Atmos. Sci. 30, 145.Google Scholar
Longuet-Higgins, M. S. & Gsc>ILL, A. E. 1967 Resonont interactions between planetary waves. Proc. Roy. Soc. A 299, 120.Google Scholar
Lorenz, E. 1972 Barotropic instability of Rossby wave motion J. Atmos. Sci. 29, 258.Google Scholar
Mcewan, A. D. 1971 Degeneration of resonantly-excited standing internal gravity waves J. Fluid Mech. 50, 431.Google Scholar
Mcewan, A. D. & Robinson, R. 1975 Parametric instability of internal gravity waves J. Fluid Mech. 67, 667.Google Scholar
Merilees, P. E. & Warn, H. 1975 On energy and enstrophy exchanges in two-dimensional non-divergent flow J. Fluid Mech. 69, 625.Google Scholar
Neamtan, S. M. 1946 The motion of harmonic waves in the atmosphere J. Met. 3, 53.Google Scholar
Phillips, N. A. 1959 Numerical integration of the primitive equations on the hemisphere Mon. Weather Rev. 87, 333.Google Scholar
Platzman, G. W. 1962 The analytical dynamics of the spectral vorticity equation J. Atmos. Sci. 19, 313.Google Scholar
Puri, K. & Bourke, W. 1974 Implications of horizontal resolution in spectral model integrations Mon. Weather Rev. 102, 333.Google Scholar
Rhines, P. B. 1970 Wave propagation in a periodic medium with application to the ocean Rev. Geophys. Space Phys. 8, 303.Google Scholar
Silberman, I. 1954 Planetary waves in the atmosphere J. Met. 11, 27.Google Scholar
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon Press.