The mixed Rossby–gravity wave on the spherical Earth

Nathan Paldor*, Itzhak Fouxon, Ofer Shamir, Chaim I. Garfinkel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


This work revisits the theory of the mixed Rossby–gravity (MRG) wave on a sphere. Three analytic methods are employed in this study: (a) derivation of a simple ad hoc solution corresponding to the MRG wave that reproduces the solutions of Longuet-Higgins and Matsuno in the limits of zero and infinite Lamb's parameter, respectively, while remaining accurate for moderate values of Lamb's parameter, (b) demonstration that westward-propagating waves with phase speed equalling the negative of the gravity-wave speed exist, unlike the equatorial β-plane, where the zonal velocity associated with such waves is infinite, and (c) approximation of the governing second-order system by Schrödinger eigenvalue equations, which show that the MRG wave corresponds to the branch of the ground-state solutions that connects Rossby waves with zonally symmetric waves. The analytic conclusions are confirmed by comparing them with numerical solutions of the associated second-order equation for zonally propagating waves of the shallow-water equations. We find that the asymptotic solutions obtained by Longuet-Higgins in the limit of infinite Lamb's parameter are not suitable for describing the MRG wave even when Lamb's parameter equals 104. On the other hand, the dispersion relation obtained by Matsuno for the MRG wave on the equatorial β-plane is accurate for values of Lamb's parameter as small as 16, even though the equatorial β-plane formally provides an asymptotic limit of the equations on the sphere only in the limit of infinite Lamb's parameter.

Original languageAmerican English
Pages (from-to)1820-1830
Number of pages11
JournalQuarterly Journal of the Royal Meteorological Society
Issue number715
StatePublished - Jul 2018

Bibliographical note

Publisher Copyright:
© 2018 The Authors. Quarterly Journal of the Royal Meteorological Society published by John Wiley & Sons Ltd on behalf of the Royal Meteorological Society.


  • Yanai wave
  • mixed Rossby–gravity
  • shallow-water waves
  • waves on a sphere


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