The transient and stationary flow fields produced when initial zonal solid body rotation currents flow over topography is studied in spherical barotropic models for eastward and westward flow directions and with variable and constant Coriolis parameters (f). Fully nonlinear analytical solutions are obtained using the methods of equilibrium statistical mechanics and are compared with linearized solutions and with the qualitative results of previous laboratory and numerical experiments.
The equilibrium solutions show that westward flow with variable f is stable in the sense that only very small amplitude transients are produced and the initial flow is practically unchanged. The stationary stream-function dominates the total flow field and there is excellent agreement with linear steady state solutions and with the qualitative results of laboratory and numerical experiments.
For eastward flow with variable f the equilibrium solutions demonstrate that the flow is unstable; that is, large amplitude transients are produced and the zonal flow direction changes to westward. The transients dominate the total eddy flow field while the smaller amplitude stationary eddy streamfunction is essentially a filtered version of the topography and is very similar to that for initial westward flow. In contrast, the linear steady state solutions are resonant or near resonant and differ dramatically from the nonlinear stationary solutions. The meaning of the linear steady state solutions is discussed and comparisons are made of linear transient and nonlinear equilibrium solutions with laboratory and numerical experiments.
With constant f, the equilibrium solutions show that eastward flow again reverses to westward while westward flow remains that way but decreases its strength. The amplitudes of the transients for the two flow directions are now comparable and are considerably less than for the case of eastward flow with variable f. For both flow directions the transient and stationary eddy energy spectra are strongly peaked at zonal wavenumber |m| = 1 with the stationary spectrum being, dominant. The corresponding linear steady state solutions have a resonance at |m| = 1 when the flow is inviscid and large amplitude there when viscosity is included.