### Abstract:

The long-term evolution of Gaussian eddies is studied in an equivalent barotropic model using both linear and nonlinear quasi-geostrophic theory in an attempt to understand westward propagating satellite altimetry tracked mesoscale eddies. By examining both individual eddies and a large basin seeded with eddies, it is shown that long term eddy coherence and the zonal wavenumber-frequency power spectral density are best matched by the nonlinear model. Individual characteristics of the eddies including amplitude decay, length decay, zonal and meridional propagation speed of a previously unrecognized quasi-stable state are examined to provide baseline properties for comparison with extended models.
An analytical technique is then used for evaluating scales of motion of typical mesoscale eddies in order evaluate the success of existing models and find other more appropriate theories. Starting from the spherical shallow water equations and assuming geostrophic dominance, a potential vorticity conservation law is derived in terms of all four non-dimensional parameters inherent in the equations while retaining the spherical geometry. By retaining freedom in the parameters, the scales can be determined at which various theories remain valid. It is argued that the FP equation equation and a new extension to the FP equation are required to describe the mid-latitude mesoscale eddies.
Analytical solutions to the FP equation are sought using the classical and exterior differential systems methods of group foliation. Both methods of group foliation are used to find the cnoidal solution of the Korteweg-de Vries equation, a one-dimensional form of the FP equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest, and a reduced quasi-linear hyperbolic system is derived for the two-dimensional FP equation.
The forces driving the slow westward propagation of mesoscale eddies also underly a particle constrained to the surface of the earth, but are quantitatively misunderstood. Starting with a free particle and successively adding constraints, it is shown that the particle's motion is inertial, despite literature to the contrary, and that an accelerometer trapped in inertial motion would not measure an acceleration.