Localized dipoles: From 2D to rotating shallow water

The progress made in the theory of localized dipoles over the course of the past century is overviewed. The dependence between the dipole shape, on the one hand, and the vorticity-streamfunction relation in the frame of reference co-moving with the dipole, on the other hand, is discussed. We show that, in 2D non-divergent and quasi-geostrophic dipoles, circularity of the trapped-fluid region and linearity of the vorticity-streamfunction relation in this region are equivalent. The existence of elliptical dipoles of high smoothness is demonstrated. A generalization of the dipole theory to the rotating shallow water model is offered. This includes the construction of localized f-plane dipole solutions (modons) and demonstration of their soliton nature, and derivation of a necessary condition for an eastward-traveling β-plane modon to exist. General properties of such ageostrophic modons are discussed, and the fundamental dissimilarity of fast and/or large dipoles in the rotating shallow water model from quasi-geostrophic dipoles is demonstrated and explained.