Abstract
Weakly nonlinear quasi-geostrophic planetary waves on a beta-plane and topographic waves over a linearly inclined bottom are examined by use of shallow water equations for a small beta parameter. Long solitary wave solutions missed by the use of the traditional quasi-geostrophic approximation are found in a channel ocean with neither a sheared current nor a curved (non-linearly inclined) bottom topography. The solutions are missed in the traditional approach because the irrotational motion associated with the geostrophic divergence is neglected by the quasi-geostrophic approximation. Another example which calls attention to the limitation of the traditional quasi-geostrophic approximation is the nonlinear evolution of divergent planetary eddies whose scale is much larger than the Rossby's radius of deformation. Some aspects of a new evolution equation are briefly discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Benney, D. J. (1966): Long nonlinear waves in fluid flows. J. Math. Phys. (Cambridge, MA),45, 52–63.
Boyd, J. P. (1977): Solitary waves on the equatorial betaplane.In, Review Papers of Equatorial Oceanography FINE Workshop Proceedings, ed. byD. Moore. Nova Univ., Dania, Florida, 13 pp.
Boyd, J. P. (1980): Equatorial solitary waves. Part I: Rossby solitons. J. Phys. Oceanogr.,10, 1699–1717.
Charney, J. G. (1973): Planetary fluid dynamics.In, Dynamic Meteorology, ed. byP. Morel, D. Reidel, Dordrecht, pp. 97–351.
Clarke, R. A. (1971): Solitary and cnoidal planetary waves. Geophys. Fluid Dyn.,2, 343–354.
Flierl, G. R. (1979): Baroclinic solitary waves with radial symmetry. Dyn. Atmos. Oceans,3, 15–38.
Flierl, G. R. (1980): Introduction to coherent features. Woods Hole Oceanogr. Inst. Tech. Rep. WHOI-80-53, 1–33.
Grimshaw, R. (1977): Nonlinear aspects of long shelf waves. Geophys. Astrophys. Fluid Dyn.,8, 3–16.
Hukuda, H. (1979): Solitary Rossby waves in a two-layer system. Tellus,31, 161–169.
Larsen, L. H. (1965): Comments on “solitary waves in the Westerlies”. J. Atmos. Sci.,22, 222–224.
Long, R. R. (1964): Solitary waves in the Westerlies. J. Atmos. Sci.,21, 197–200.
Malanotte Rizzoli, P. andM. C. Hendershott (1980): Solitary Rossby waves over variable relief and their stability. Part I: the analytical theory. Dyn. Atmos. Oceans,4, 247–260.
Matsuura, T. andT. Yamagata (1982): On the evolution of nonlinear planetary eddies larger than the radius of deformation. J. Phys. Oceanogr.,12, 440–456.
Maxworthy, T. andL. G. Redekopp (1976): A solitary wave theory of the Great Red Spot and other observed features in the Jovian atmosphere. Icarus,29, 261–271.
Meyers, G. (1979): On the annual Rossby wave in the tropical North Pacific Ocean. J. Phys. Oceanogr.,9, 663–674.
Miles, J. W. (1979): On solitary Rossby waves. J. Atmos. Sci.,36, 1236–1238.
Odulo, A. B. andYe. N. Pelinovskiy (1978): Nonlinear topographic Rossby waves. Oceanology,18, 9–11.
Redekopp, L. G. (1977): On the theory of solitary Rossby waves. J. Fluid Mech.,82, 725–745.
Stumpf, H. G. andR. V. Legeckis (1977): Satellite observations of mesoscale eddy dynamics in the eastern tropical Pacific Ocean. J. Phys. Oceanogr.,7, 648–658.
Smith, R. (1972): Nonlinear Kelvin and continental-shelf waves. J. Fluid Mech.,52, 376–391.
White, W. B. (1977): Annual forcing of baroclinic long waves in the tropical North Pacific Ocean. J. Phys. Oceanogr.,7, 50–61.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yamagata, T. On nonlinear planetary waves: A class of solutions missed by the traditional quasi-geostrophic approximation. Journal of the Oceanographical Society of Japan 38, 236–244 (1982). https://doi.org/10.1007/BF02111106
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02111106