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Resonant interactions between topographic planetary waves in a continuously stratified fluid

Published online by Cambridge University Press:  12 April 2006

Lawrence A. Mysak
Lawrence A. Mysak
Affiliation:
National Center for Atmospheric Research, Boulder, Colorado 80307
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Abstract

The resonant interactions between topographic planetary waves in a continuously stratified fluid are investigated theoretically. The interacting waves form a resonant triad and travel along a channel with a uniformly sloping bottom. The basic state stratification in the channel is characterized by a constant buoyancy frequency. The existence of solutions to the quadratic resonance conditions is established graphically. Each wave by itself is a bottom-intensified oscillation of the type discovered by Rhines (1970) except for the addition of a small positive frequency correction. This correction must be included to satisfy higher-order terms in the bottom boundary condition. For strong stratification (r2 [Gt ] L2, where r = internal deformation radius and L = channel width), the waves are strongly bottom-trapped and this frequency correction is negligible. For weak stratification (r2 [Lt ] L2) the waves are barotropic and the frequency correction is O(δ), where δ = fractional change in depth across the channel. In many oceanic contexts, δ lies in the range 0·1-0·4 and therefore this correction can produce a significant change in the phase speed. The amplitudes of the waves in the triad obey the classical gyroscopic equations usually encountered in quadratic resonance problems. In particular, the amplitudes evolve on the slow time scale \[ t=O(1/f_0\delta^2), \] which for our scaling assumptions is also O(1/f0Ro), where Ro is the Rossby number.

The results are applied to the Norwegian continental slope region. It is shown that, in this vicinity, there may exist resonant triads consisting of two short, high-frequency waves (periods around 3-4 days) and one long, low-frequency wave (period around 9 days).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 84 , Issue 4 , 27 February 1978 , pp. 769 - 793
Copyright
© 1978 Cambridge University Press

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References

Bretherton, F. P. 1964 Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20, 457–479.Google Scholar
Horn, W. & Schott, F. 1976 Measurements of stratification and currents at the Norwegian continental slope. ‘Meteor’ Forsch.-Ergebn A 18, 23–62.Google Scholar
Huppert, H. E. & Bryan, K. 1976 Topographically generated eddies. Deep-Sea Res. 23, 655–679.Google Scholar
Kenyon, K. 1964 Non-linear energy transfer in a Rossby-wave spectrum. Woods Hole Oceanographic Inst., Summer Study Program in Geophysical Fluid Dynamics. Student Lectures, vol. II, pp. 69–83.
Leblond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.
Loesch, A. Z. 1974 Resonant interactions between unstable and neutral baroclinic waves. J. Atmos. Sci. 31, 1177–1201 (part I), 1202–1217 (part II).Google Scholar
Loesch, A. Z. & Domaracki, A. 1977 Dynamics of N resonantly interacting waves. J. Atmos. Sci. 34, 22–35.Google Scholar
Longuet-Higgins, M. S. & Gill, A. E. 1967 Resonant interactions between planetary waves. Proc. Roy. Soc. A 299, 120–140.Google Scholar
Mcgoldrick, L. F. 1965 Resonant interactions among capillary–gravity waves. J. Fluid Mech. 21, 305–331.Google Scholar
Mcgoldrick, L. F. 1970 On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193–200.Google Scholar
Milne-Thomson, L. M. 1950 Jacobian Elliptic Function Tables. Dover.
Mysak, L. A. 1977 On the stability of the California Undercurrent off Vancouver Island. J. Phys. Ocean. 7, 904–917.Google Scholar
Mysak, L. A. & Schott, F. 1977 Evidence for baroclinic instability of the Norwegian Current. J. Geophys. Res. 82, 2087–2095.Google Scholar
Newell, A. C. 1969 Rossby wave packet interactions. J. Fluid Mech. 35, 255–271.Google Scholar
Newell, A. C. 1972 The post bifurcation stage of baroclinic instability. J. Atmos. Sci. 29, 64–76.Google Scholar
Owens, W. B. & Bretherton, F. P. 1977 A numerical study of mid-ocean mesoscale eddies. Deep-Sea Res. (in press).
Pedlosky, J. 1964 The stability of currents in the atmosphere and ocean: Part I. J. Atmos. Sci. 21, 201–219.Google Scholar
Pedlosky, J. 1975 The amplitude of baroclinic wave triads and mesoscale motion in the ocean. J. Phys. Ocean. 5, 608–614.Google Scholar
Plumb, R. A. 1977 The stability of small amplitude Rossby waves in a channel. J. Fluid Mech. 80, 705–720.Google Scholar
Rhines, P. B. 1970 Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dyn. 1, 273–307.Google Scholar
Richman, J. G. 1976 Kinematics and energetics of the mesoscale mid-ocean circulation: MODE. Ph.D. thesis, Woods Hole Oceanographic Institution/Massachusetts Institute of Technology.
Smith, P. C. 1976 Baroclinic instability in the Denmark Strait overflow. J. Phys. Ocean. 6, 355–371.Google Scholar
Thompson, R. O. R. Y. & Luyten, J. R. 1976 Evidence for bottom-trapped topographic Rossby waves from single moorings. Deep-Sea Res. 23, 629–635.Google Scholar