Regular and chaotic flows in paraboloidal basins and eddies

Shallow water equations (including the effect of earthʹs rotation) governing the flow in a paraboloidal basin or an eddy bounded by a free surface are studied both analytically and numerically. The system of governing PDE is first exactly reduced to a twelfth order system of nonlinear ODE. Several subsystems of the latter are derived under varying conditions and studied. Some of these subsystems admit exact analysis. Previous exact solutions of the PDE system are recovered as special cases. A fairly general sub-system of eighth order, which follows from the twelfth order system from the fact that, in the deformation modes, the internal motions of the fluid are unaffected by the motion of the center of mass, is studied numerically both when there is a supporting basin and when the outer boundary is free. It is found that when there is an elliptical basin, this system has both periodic and chaotic solutions. Time series, power spectra and Lyapunov exponents for different sets of parameters are computed and depicted. The present study may have application in warm core rings in the Gulf stream as well as tidal motions in basins.