Linear waves in a symmetric equatorial channel

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 (2Ω R ) −2 , Δϕ is half the channel width, g is the acceleration 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. Copyright © 2007 Royal Meteorological Society