# The Matsuno-Gill model on the sphere

Published in *Journal of Fluid Mechanics*, 2023

Official Version to appear.

We extend the Matsuno-Gill model, originally developed on the equatorial β-plane, to the surface of the sphere. While on the β-plane the non-dimensional model contains a single parameter, the damping rate γ, on a sphere the model contains a second parameter, the rotation rate ε1/2 (Lamb number). By considering the different combinations of damping and rotation we are able to characterize the solutions over the (γ, ε1/2) plane. We find that the β-plane approximation is only accurate for fast rotation rates, where gravity waves traverse a fraction of the sphere’s diameter in one rotation period. The particular solutions studied by Matsuno and Gill are only accurate for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere’s diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient, and non-rotating limits approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of γ and ε 1/2. The particular solutions studied by Matsuno and Gill apply to Earth’s oceans while the more general β-plane solutions are only somewhat relevant to Earth’s troposphere. In Earth’s stratosphere, Venus and Titan, only the spherical solutions applies.