The Completeness of Eigenfunctions of the Tidal Equation on an Equatorial Beta Plane

Zhaohua Wu Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland

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Dennis W. Moore Ocean Climate Research Division, Pacific Marine Environmental Laboratory, Seattle, Washington

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Abstract

The approximate tidal theory on an equatorial beta plane has been widely applied to tropical atmospheric dynamics. There are many successful examples of such applications. However, the mathematical and physical origin of the recently discovered continuous spectrum associated with meridional eigenfunctions of negative equivalent depth is yet to be given, and the completeness of the meridional eigenfunctions in the approximate tidal theory remains to be proved.

In this note, a proof of the completeness of the meridional eigenfunction is presented. The differential equation is first transformed into an equivalent integral equation that relates the solution of the differential equation to the corresponding Green's function. It is then shown that the Green's function corresponding to the meridional eigenvalue–eigenfunction problem is linear, self-adjoint, completely continuous, and square integrable over the meridional infinite domain under the principle of analytic continuation. Therefore, the eigenfunctions form a complete Hilbert space. All the eigenvalues and eigenfunctions are then identified using the method of spectral representation of a second-order differential operator. Related physical properties of the eigenfunctions are also discussed.

Corresponding author address: Dr. Zhaohua Wu, Center for Ocean–Land–Atmosphere Studies, 4041 Powder Mill Rd., Suite 302, Calverton, MD 20705. Email: zhwu@cola.iges.org

Abstract

The approximate tidal theory on an equatorial beta plane has been widely applied to tropical atmospheric dynamics. There are many successful examples of such applications. However, the mathematical and physical origin of the recently discovered continuous spectrum associated with meridional eigenfunctions of negative equivalent depth is yet to be given, and the completeness of the meridional eigenfunctions in the approximate tidal theory remains to be proved.

In this note, a proof of the completeness of the meridional eigenfunction is presented. The differential equation is first transformed into an equivalent integral equation that relates the solution of the differential equation to the corresponding Green's function. It is then shown that the Green's function corresponding to the meridional eigenvalue–eigenfunction problem is linear, self-adjoint, completely continuous, and square integrable over the meridional infinite domain under the principle of analytic continuation. Therefore, the eigenfunctions form a complete Hilbert space. All the eigenvalues and eigenfunctions are then identified using the method of spectral representation of a second-order differential operator. Related physical properties of the eigenfunctions are also discussed.

Corresponding author address: Dr. Zhaohua Wu, Center for Ocean–Land–Atmosphere Studies, 4041 Powder Mill Rd., Suite 302, Calverton, MD 20705. Email: zhwu@cola.iges.org

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