Purpose. 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. The scope is to enlarge the number of known (semi)analytical solutions of nonstationary, nonlinear problems referring to geophysical problems and even to pave the way to their extension to broader geometries and/or velocity fields. Methods and Results. The used method to obtain the solutions is based on the decomposition of the original equations in a part expressing their prescribed spatial structure, so that they can be transformed into ordinary differential equations depending on time only. Based on that analytical procedure, the solutions are then found numerically. 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 and to put them in the context of observed dynamics in the World Ocean, 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. The possibility of solving (semi)analytically complex systems of differential equations representing observed physical phenomena is rare and very valuable. Conclusions. 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.
Solution of Fractional Autonomous Ordinary Differential Equations