The evolution of stable barotropic vortices in a rotating free-surface fluid

Laboratory experiments reveal that, for increasing time, barotropic cyclones typically show an increasing steepness in their flow profiles. This implies that such vortices become barotropically more unstable. This has been confirmed by observations which are further discussed in a companion paper (Kloosterziel & VAN Heijst 1991). In the present paper the evolutionary process is discussed. It is shown that the observed steepening of the flow profiles is mainly a nonlinear effect due to the advection of relative vorticity by the interior Ekman circulation. The linear Ekman pumping law is found to be a good approximation in the core of the vortices for O(1) Rossby numbers but at larger radii the Ekman pumping is stronger in reality. Free-surface effects are shown to have a broadening effect, which can balance the nonlinear steepening if the Rossby number becomes sufficiently small or the Froude number sufficiently large. In addition it is shown that for the classical spin-down problem with a free surface no expansion in the Froude number needs to be introduced.