## Abstract

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 10^{4}. 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 language | American English |
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Pages (from-to) | 1820-1830 |

Number of pages | 11 |

Journal | Quarterly Journal of the Royal Meteorological Society |

Volume | 144 |

Issue number | 715 |

DOIs | |

State | Published - 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.

## Keywords

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