  The dynamics of nonstationary, nonlinear, axisymmetric, warm-core geophysical
surface frontal vortices affected by Rayleigh friction is investigated
semi-analytically using the nonlinear, nonstationary reduced-gravity
shallow-water equations. In this frame, it is found that vortices characterized
by linear distributions of their radial velocity and arbitrary structures of
their section and azimuthal velocity can be described exactly by a set of
nonstationary, nonlinear coupled ordinary differential equations. The
first-order problem (i.e., that describing vortices characterized by a linear
azimuthal velocity field and a quadratic section) consists of a system of 4
differential equations, and each further order introduces in the system three
additional ordinary differential equations and two algebraic equations. In
order to illustrate the behavior of the nonstationary decaying vortices, the
system's solution for the first-order and for the second-order problem is then
obtained numerically using a Runge-Kutta method. The solutions demonstrate that
inertial oscillations and an exponential attenuation dominate the vortex
dynamics: Expansions and shallowings, contractions and deepenings alternate
during an exact inertial period while the vortex decays. The dependence of the
vortex dissipation rate on its initial radius is found to be non-monotonic: It
is higher for small and large radii. Our analysis adds realism to previous
theoretical investigations on mesoscale vortices, represents an ideal tool for
testing the accuracy of numerical models in simulating nonlinear, nonstationary
frictional frontal phenomena in a rotating ocean, and paves the way to further
extensions of (semi-) analytical solutions of hydrodynamical geophysical
problems to more arbitrary forms and more complex density stratifications.
