The eigenvalue equations of equatorial waves on a sphere

Yair De-Leon; Carynelisa Erlick; Nathan Paldor

doi:10.1111/j.1600-0870.2009.00420.x

Authors:

Yair De-Leon,

Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, IL

Carynelisa Erlick,

Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, IL

Nathan Paldor

Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, IL

Abstract

Zonally propagating wave solutions of the linearized shallow water equations (LSWE) in a zonal channel on the rotating spherical earth are constructed from numerical solutions of eigenvalue equations that yield the meridional variation of the waves’ amplitudes and the phase speeds of these waves. An approximate Schr¨odinger equation, whose potential depends on one parameter only, is derived, and this equation yields analytic expressions for the dispersion relations and for the meridional structure of the waves’ amplitudes in two asymptotic cases. These analytic solutions validate the accuracy of the numerical solutions of the exact eigenvalue equation. Our results show the existence of Kelvin, Poincar´e and Rossby waves that are harmonic for large radius of deformation. For small radius of deformation, the latter two waves vary as Hermite functions. In addition, our results show that the mixed mode of the planar theory (a meridional wavenumber zero mode that behaves as a Rossby wave for large zonal wavenumbers and as a Poincar´e wave for small ones) does not exist on a sphere; instead, the first Rossby mode and the first westward propagating Poincar´e mode are separated by the anti-Kelvin mode for all values of the zonal wavenumber.

How to Cite: De-Leon, Y., Erlick, C. and Paldor, N., 2010. The eigenvalue equations of equatorial waves on a sphere. Tellus A: Dynamic Meteorology and Oceanography, 62(1), pp.62–70. DOI: http://doi.org/10.1111/j.1600-0870.2009.00420.x

Accepted on 19 Oct 2009 Submitted on 13 Jul 2009

References

Abramowitz , M. and Stegun , I.A . 1972 . Handbook of Mathematical Functions . Dover Publications, Inc. , NY, USA , 1043 .

Cane , M.A. and Sarachilc , ES . 1979 . Forced baroclinic ocean motions BI: the linear equatorial basin case . J. Mar Res . 37 , 355 – 398 .

De-Leon Y. and Paldor , N . 2009 . Linear waves in mid-latitudes on the rotating spherical Earth . J. Phys. Oceanogr 39 ( 12 ), 3204 – 3215 .

Erlick , C. , Paldor , N. and Ziv , B . 2007 . Linear waves in a symmetric equatorial channel . Q. J. R. Meteorol. Soc . 133 ( 624 ), 571 – 577 .

Gent , P.R. and McWilliams , J.C . 1983 . The equatorial waves of balanced models . J. Phys. Oceanogr 13 , 1179 – 1192 .

Longuet-Higgins , M.S . 1968 . The eigenfunctions of Laplace’s tidal equations over a sphere . Phil. Trans. R. Soc. Lond. A . 262 , 511 – 607 .

Matsuno , T . 1966 . Quasi-geostrophic motions in the equatorial area . J. MeteoroL Soc. Japan . 44 , 25 – 43 .

Trefethen , L.N . 2000 . Spectral Methods in MATLAB . SIAM, Philadel-phia, PA , 165 pp .

[CIT0001] Abramowitz , M. and Stegun , I.A . 1972 . Handbook of Mathematical Functions . Dover Publications, Inc. , NY, USA , 1043 .

[CIT0002] Cane , M.A. and Sarachilc , ES . 1979 . Forced baroclinic ocean motions BI: the linear equatorial basin case . J. Mar Res . 37 , 355 – 398 .

[CIT0003] De-Leon Y. and Paldor , N . 2009 . Linear waves in mid-latitudes on the rotating spherical Earth . J. Phys. Oceanogr 39 ( 12 ), 3204 – 3215 .

[CIT0004] Erlick , C. , Paldor , N. and Ziv , B . 2007 . Linear waves in a symmetric equatorial channel . Q. J. R. Meteorol. Soc . 133 ( 624 ), 571 – 577 .

[CIT0005] Gent , P.R. and McWilliams , J.C . 1983 . The equatorial waves of balanced models . J. Phys. Oceanogr 13 , 1179 – 1192 .

[CIT0006] Longuet-Higgins , M.S . 1968 . The eigenfunctions of Laplace’s tidal equations over a sphere . Phil. Trans. R. Soc. Lond. A . 262 , 511 – 607 .

[CIT0007] Matsuno , T . 1966 . Quasi-geostrophic motions in the equatorial area . J. MeteoroL Soc. Japan . 44 , 25 – 43 .

[CIT0008] Trefethen , L.N . 2000 . Spectral Methods in MATLAB . SIAM, Philadel-phia, PA , 165 pp .

Reading: The eigenvalue equations of equatorial waves on a sphere

Original Research Papers

The eigenvalue equations of equatorial waves on a sphere

Authors:

Yair De-Leon,

Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, IL

X close

Carynelisa Erlick,

Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, IL

X close

Nathan Paldor

Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, IL

X close

Abstract

References