The elliptical model of two-dimensional vortex dynamics. II: Disturbance equations

In Part I [Phys. Fluids A 3, 845 (1991)] approximate equations were developed that describe the basic evolution of vortices in a general strain field. These equations take the form of a set of coupled, nonlinear ordinary differential equations describing the time evolution of the centroids, aspect ratios, and orientations of a nested set of elliptical contours representing each vortex. Here, in Part II, the model is extended to include disturbances to the elliptical shape of each contour, disturbances that are excited naturally by the interaction with other vortices. This interaction is worked out explicitly for the first time. The final equations obtained decouple into sets of equations for each mode symmetry, allowing for a very simple description of the disturbance evolution. Numerical tests show remarkable agreement between the elliptical model and the full equations of motion in four problems: (1) the equilibrium contour shapes of a multicontour family of vortices, (2) the linear stability of this family, (3) the equilibrium, nonelliptical shapes of two corotating vortex patches, and (4) the interaction between two symmetrical vortex patches, including merging.