Monopolar and dipolar vortex solitons in two space dimensions

Abstract

We compute solitary waves which solve Au _y2 U-(U +yu2)/C = 0 where c is the phase speed in the x direction and y = 0 is the equator. This equation is a heuristic model for Rossby waves on the "equatorial beta-plane" in geophysical fluid dynamics. For positive c only, there are one-signed solutions ("monopole vortices") which are centered in the middle latitudes. When c c [-oG-1/3] or c > 0, there are dipoles which have matching vortices of opposite sign in each hemisphere. A third class of solutions is composed of equator-spanning monopoles that are unsymmetric in y. In addition to these families of strict solitary waves, there are also quadrapole vortices which are "weakly non-local solitons" in the sense that they almost meet the usual criterion for solitary waves except for a weak-very weak-radiation to infinity. Orthogonal rational Chebyshev functions and Newton's iteration are used to compute numerical solutions, but four analytic approximations are also derived. Although the equation is only a crude model of geophysical waves, its rich diversity offers a good education in solitary waves in two space dimensions.

Hamiltonian–Dirac simulated annealing: Application to the calculation of vortex states

“…Issues pertaining to the existence of solutions in quasigeostrophic dynamics for monopolar and dipolar states [32][33][34][35][36][37]28] would be a natural next set of applications. One could also pursue reduced magnetohydrodynamics, since it possesses a dynamics and Hamiltonian structure [38] similar to the vortex examples presented here, but with one additional scalar field that describes the magnetic field.…”

“…Fig. 3 shows the vorticity and contours of the streamfunction for the H-dynamics of (37). Observe that the system sheds filaments, evolves toward an elliptical vortex state close to the Kirchhoff vortex in a time t 20, and then rotates uniformly for a long period of time.…”

“…Eqs. (37) and (38) are ideal and conserve energy in principle, although this will not be precisely the case when numerically solved. In (39) and (40) the parameters σ and α can be used to weight the contributions of the ideal and dissipative dynamics, respectively.…”

“…With the above assumptions on F , the dynamics generated by (37) clearly conserves all Casimir invariants for any H, and the (37) and (38) to possess asymptotically stable equilibria in any frame of reference. However, these systems are infinite-dimensional systems in which filamentation is known to occur (see e.g.…”