  We prove that for rotating shallow water equations on a surface of revolution
with variable Coriolis parameter and vanishing Rossby and Froude numbers, the
classical solution satisfies uniform estimates on a fixed time interval with no
dependence on the small parameters. Upon a transformation using the solution
operator associated with the large operator, the solution converges strongly to
a limit for which the governing equation is given. We also characterize the
kernel of the large operator and define a projection onto that kernel. With
these tools, we are able to show that the time-averages of the solution are
close to longitude-independent zonal flows and height field.

Analysis of PDEs;Dynamical Systems;Atmospheric and Oceanic Physics;Fluid Dynamics