## Abstract

Using a scaling that allows us to separate the effects of the gravity wave speed from those of boundary location, we reduce the equations for linear waves in a zonal channel on the equatorial beta-plane to a single-parameter eigenvalue problem of the Schrödinger type with parabolic potential. The single parameter can be written δ = (Δφ)^{2}/α^{1/2}, where α = gH(2ΩR)^{-2}, Δφ is half the channel width, g is the acceleraticn due to gravity, H is the typical height of the troposphere or ocean, Ω is the Earth's rotational frequency, and R is the Earth's radius. The Schrödinger-type equation has exact analytical solutions in the limits δ → 0 and δ → ∞, and one can use these to write an approximate expression for the solution that is accurate everywhere to within 4%. In addition to the simple expression for the eigenvalues, the concise and unified theory also yields explicit expressions for the associated eigenfunctions, which are pure sinusoidal in the δ → 0 limit and Gaussian in the δ → ∞ limit. Using the same scaling, we derive an eigenvalue formulation for linear waves in an equatorial channel on the sphere with a simple explicit formula for the dispersion relation accurate to O{(Δφ)^{2}}. From this, we find that the phase velocity of the anti-Kelvin mode on the sphere differs by as much as 10% from -α^{1/2}. Integrating the linearized shallow-water equations on the sphere, we find that for for larger α and Δφ, the phase speeds of all of the negative modes differ substantially from their phase speeds on the beta-plane. Furthermore, the dispersion relations of all of the waves in the equatorial channel on the sphere approach those on the unbounded sphere in a smooth asymptotic fashion, which is not true for the equatorial channel on the beta-plane.

Original language | English |
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Pages (from-to) | 571-577 |

Number of pages | 7 |

Journal | Quarterly Journal of the Royal Meteorological Society |

Volume | 133 |

Issue number | 624 PART A |

DOIs | |

State | Published - Apr 2007 |

## Keywords

- Anti-Kelvin mode
- Equatorial waves
- Equivalent barotropic
- Schrödinger equation