An analytical model describing the β-induced drift of isolated nonlinear eddies such as the cold- and warm-core rings observed in the Atlantic Ocean is proposed. The ocean is approximated by two layers and attention is focused on frictionless upper ocean eddies whose surface area is finite. These isolated eddies are nonlinear in the sense that (a) the corresponding Rossby number is relatively large and (b) the interface vertical displacements ("amplitudes") are comparable to the upper layer undisturbed depth. Solutions for steadily translating eddies which carry their entire mass as they move are sought. Examination of the problem in a moving coordinate system enables one to construct such solutions analytically by using the equations of motion in an integrated form and a power series expansion.Significant differences between the behavior of cyclonic and anticyclonic eddies are found. Although both cyclonic and anticyclonic eddies drift to the west due to β, their speeds and dynamical behavior are very different. For some range of parameters the β-induced drift of an anticyclonic eddy differs by as much as 400% from the drift of a cyclonic eddy with similar characteristics. Furthermore, the β-induced translation of cyclonic eddies increases with size and decreases with "amplitude" whereas the speed of anticyclonic eddies decreases with size and increases with increasing amplitude. In addition, the translation of anticyclonic eddies is larger than the long wave speed (based on the undisturbed depth) whereas the translation of cyclonic eddies is smaller than the long wave speed. Since such a dynamical behavior is not revealed by quasi-geostrophic theory (which does not distinguish between cyclonic and anticyclonic eddies) it is suggested that nonlinearity plays an important role in the dynamics of some isolated rings.Application of the theory to the Gulf Stream rings suggests that the self-propelled movement due to β is ≈2 cm sec−1 for cold-core rings and ≈1 cm sec−1 for warm-core rings. Each ring may carry as much as 8–10,000 km3 of upper ocean water as it moves.
Nof, Doron. 1983. "On the migration of isolated eddies with application to Gulf Stream rings." Journal of Marine Research 41, (3). https://elischolar.library.yale.edu/journal_of_marine_research/1688