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An asymptotic solution of the tidal equations

Published online by Cambridge University Press:  28 March 2006

Stanley J. Jacobs
Stanley J. Jacobs
Affiliation:
Department of Meteorology and Oceanography, The University of Michigan, Ann Arbor, Michigan
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Abstract

The Cauchy problem for the β-plane form of the tidal equations is solved for both oscillatory and delta function initial data. The radius of deformation is assumed to be much less than the radius of the earth, and in accord with this assumption a ray approximation is employed.

It is shown that, owing to the rapid rate of propagation of inertio-gravity waves, the motion in its initial development tends towards geostrophic balance. However, the solution given by the ray approximation is singular on certain surfaces in space and time, the envelopes of the rays. A local boundary-layer theory is employed to correct this deficiency. The existence of these caustics implies that the process of geostrophic adjustment is more complicated than hitherto imagined.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 30 , Issue 3 , 29 November 1967 , pp. 417 - 438
Copyright
© 1967 Cambridge University Press

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References

Buchal, R. N. & Keller, J. B. 1960 Boundary layer problems in diffraction theory. Comm. Pure Appl. Math. 13, 85.Google Scholar
Bretherton, F. P. 1964 Low frequency oscillations trapped near the equator. Tellus, 16, 81.Google Scholar
Carrier, G. F. 1966 Gravity waves on water of variable depth. J. Fluid Mech. 24, 641.Google Scholar
Chester, C., Friedman, B. & Ursell, F. 1957 An extension of the method of steepest descents. Proc. Camb. Phil. Soc. 58, 599.Google Scholar
Courant, R. & Hilbert, D. 1962 Methods of Mathematical Physics, II. Interscience.
Keller, J. B. 1958 A geometrical theory of diffraction. Calculus of Variations and its Applications. Amer. Math. Soc. Ed. by L. M. Graves. New York: McGraw-Hill.
Lamb, H. 1932 Hydrodynamics. New York: Dover.
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Oxford and New York: Pergamon Press.
Lax, P. D. 1957 Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24, 627.Google Scholar
Lewis, R. M. 1964 Asymptotic methods for the solution of dispersive hyperbolic equations. Asymptotic Solutions of Differential Equations and their Applications. Ed. by C. H. Wilcox. New York: John Wiley.
Longuet-Higgins, M. S. 1965a The response of a stratified ocean to stationary or moving wind systems. Deep Sea Res. 12, 973.Google Scholar
Longjuet-Higgins, M. S. 1965b Planetary waves on a rotating sphere. II. Proc. Roy. Soc. A 284, 40.Google Scholar
Mahony, J. J. 1962 An expansion method for singular perturbation problems. J. Aust. Math. Soc. 2, 440.Google Scholar
Obukhov, A. 1949 K voprosu o geostroficheskom vetra. Isv. Akad. Nauk SSSR, Ser. Geograf. Geofiz. 13, no. 4.Google Scholar