On the completeness and the expansion theorem for eigenfunctions of the Sturm-Liouville-Rossby type

Author:
Akira Masuda

Journal:
Quart. Appl. Math. **47** (1989), 435-445

MSC:
Primary 34B25; Secondary 76C20, 86A05

DOI:
https://doi.org/10.1090/qam/1012268

MathSciNet review:
MR1012268

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Rossby waves in a closed ocean form an unfamiliar kind of eigenvalue problem. An investigation is made of the one-dimensional generalized version, which is called here the Sturm--Liouville--Rossby (SLR) problem. The eigenvalue and the eigenfunction of the SLR equation are calculated from those of the associated Sturm--Liouville (SL) equation and vice versa. The expansion theorem and the completeness theorem for the SLR eigenfunctions are proved to be valid in parallel to the SL problem. Two representations based on the SL eigenfunctions and on the SLR eigenfunctions are provided for Green's function, which gives an example of a reproducing kernel.

**[1]**D. L. T. Anderson and A. E. Gill,*Spin-up of a stratified ocean, with application to upwelling*, Deep-Sea Research**22**, 583 (1975)**[2]**George Bachman and Lawrence Narici,*Functional analysis*, Academic Press, New York-London, 1966. MR**0217549****[3]**R. Courant and D. Hilbert,*Methods of mathematical physics. Vol. I*, Interscience Publishers, Inc., New York, N.Y., 1953. MR**0065391****[4]**E. Firing and R. C. Beardsley,*The behavior of a barotropic eddy on a beta-plane*, J. Phys. Oceanogr.**6**, 67 (1976)**[5]**Joel N. Franklin,*Axisymmetric inertial oscillations of a rotating fluid*, J. Math. Anal. Appl.**39**(1972), 742–760. MR**0307580**, https://doi.org/10.1016/0022-247X(72)90195-3**[6]**H. P. Greenspan,*On the transient motion of a contained rotating fluid*, J. Fluid Mech.**20**(1964), 673–696. MR**0178666**, https://doi.org/10.1017/S002211206400146X**[7]**H. P. Greenspan,*On the general theory of contained rotating fluid motions*, J. Fluid Mech.**22**(1965), 449–462. MR**0180046**, https://doi.org/10.1017/S0022112065000897**[8]**P. H. LeBlond and L. A. Mysak,*Waves in the Ocean*, Elsevier, 1978**[9]**M. J. Lighthill,*Dynamic response of the Indian ocean to the onset of the south west monsoon*, Phil. Trans. Roy. Soc. London**265**, 45 (1969)**[10]**M. S. Longuet-Higgins,*Planetary waves on a rotating sphere*, Proc. Roy. Soc. London**A 279**, 446 (1964)**[11]**A. Masuda,*A proof and applications of the expansion theorem for the Rossby normal modes in a closed rectangular basin*, J. Oceanogr. Soc. Japan**43**, 237 (1987)**[12]**A Masuda,*A supplementary note on the completeness of the Rossby normal modes in a rectangular basin*, J. Oceanogr. Soc. Japan**44**, 40 (1988)**[13]**A. J. Miller,*Nondivergent planetary oscillations in midlatitude ocean basins with continental shelves*, J. Phys. Oceanogr.**16**, 1914 (1986)**[14]**David G. Schaeffer,*On the existence of discrete frequencies of oscillation in a rotating fluid*, Studies in Appl. Math.**54**(1975), no. 3, 269–274. MR**0452103**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
34B25,
76C20,
86A05

Retrieve articles in all journals with MSC: 34B25, 76C20, 86A05

Additional Information

DOI:
https://doi.org/10.1090/qam/1012268

Article copyright:
© Copyright 1989
American Mathematical Society