ESRL/PSD Seminar Series

Rossby waves on an aqua-planet: New solutions of Laplace's Tidal Equations over a sphere

Nathan Paldor
Fredy and Nadine Hermann Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel


Presently, the only eigenvalue problem associated with zonally propagating wave solutions of Laplace's Tidal Equations (LTE) was derived over 40 years ago by Longuet-Higgins [1]. The eigenvalue of this equation is the Lamb's parameter (the inverse-square of the nondimensional speed of gravity waves) whereas the waves' frequencies and wavenumbers appear in the coefficients of the linear operator. The complex form of the linear operator of the eigenvalue equation makes it impossible to find its eigenvalues (and possibly eigenfunctions) in general. Even if such explicit solutions could be, somehow, found it is unclear how one can derive from them the dispersion relations of the waves, given that the eigenvalue is Lamb's parameter that has no direct relation to the wave's frequency and wavenumber. LTE were recently solved in a channel (both in mid-latitudes and on the equator) by formulating the eigenvalue equation of its wave solutions in spherical coordinates as an approximate Schrodinger equation. The energy levels of this equation determine the phase speeds of Interia-Gravity (Poincare) waves and Planetary (Rossby) waves via the roots of a simple cubic [2, 3]. Although the channel set-up is consistent mathematically and appealing physically, it is too far removed from the real ocean, where no zonal walls actually exist. The new, Schrodinger equation, approach to propagating wave solutions of LTE was recently extended to a baroclinic ocean on a sphere [4] and it yielded explicit expressions for all three waves in the limit of small gravity wave speed (i.e. such as those in a baroclinic ocean or atmosphere). An more recent extension of the Schrodinger equation formulation of LTE is relevant to Planetary (Rossby) waves in a barotropic ocean over the entire sphere, but this extension does not apply to Inertia-Gravity (Poincare) waves. The accuracy of the new analytical solutions demonstrated by comparing then to exact, numerically computed, dispersion relation of the full system. REFERENCES 1. Longuet-Huggins M.S., The eigenfunctions of Laplace's Tidal equations over a sphere. Philos. Trans. Roy. Soc. London. 1968. V. A262. P. 511-607. 2. De-Leon Y., Erlick C., Paldor N., The eigenvalue equations of equatorial waves on a sphere. 2010. Tellus. V. 62A. P. 62-70. 3. De-Leon Y., Paldor N., Linear waves in midlatitudes on the rotating spherical earth. J. Phys. Oceanogr. 2009. V. 39. P. 3204-3215. 4. De-Leon Y., Paldor N., Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet. 2011. Tellus. V. 63A. P. 348-353

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