Interactions of vortices with rarefaction solitary waves in a Bose-Einstein condensate and their role in the decay of superfluid turbulence

There are several ways to create the vorticity-free solitary waves -- rarefaction pulses -- in condensates: by the process of strongly nonequilibrium condensate formation in a weakly interacting Bose gas, by creating local depletion of the condensate density by a laser beam, and by moving a small object with supercritical velocities. Perturbations created by such waves colliding with vortices are studied in the context of the Gross-Pitaevskii model. We find that the effect of the interactions consists of two competing mechanisms: the creation of vortex line as rarefaction waves acquire circulation in a vicinity of a vortex core and the loss of the vortex line to sound due to Kelvin waves that are generated on vortex lines by rarefaction pulses. When a vortex ring collides with a rarefaction wave, the ring either stabilises to a smaller ring after emitting sound through Kelvin wave radiation or the entire energy of the vortex ring is lost to sound if the radius of the ring is of the order of the healing length. We show that during the time evolution of a tangle of vortices, the interactions with rarefaction pulses provide an important dissipation mechanism enhancing the decay of superfluid turbulence.Comment: Revised paper accepted by Phys. Rev.

Vortex Splitting in Subcritical Nonlinear Schrodinger Equation

Vortices and axisymmetric vortex rings are considered in the framework of the subcritical nonlinear Schrodinger equations. The higher order nonlinearity present in such systems models many-body interactions in superfluid systems and allows one to study the effects of negative pressure on vortex dynamics. We find the critical pressure for which the straight-line vortex becomes unstable to radial expansion of the core. The energy of the straight-line vortices and energy, impulse and velocity of vortex rings are calculated. The effect of a varying pressure on the vortex core is studied. It is shown that under the action of the periodically varying pressure field a vortex ring may split into many vortex rings and the conditions for which this happens are elucidated. These processes are also relevant to experiments in Bose-Einstein condensates where the strength and the sign of two-body interactions can be changed via Feshbach resonance.Comment: Invited submission to the special issue on Vortex Rings, Journal of Fluid Dynamics Researc

Spontaneous rotating vortex lattices in a pumped decaying condensate

Injection and decay of particles in an inhomogeneous quantum condensate can significantly change its behaviour. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation and analyse the density and currents in the steady state. With homogeneous pumping, rotationally symmetric solutions are unstable. Stability may be restored by a finite pumping spot. However if the pumping spot is larger than the Thomas-Fermi cloud radius, then rotationally symmetric solutions are replaced by solutions with spontaneous arrays of vortices. These vortex arrays arise without any rotation of the trap, spontaneously breaking rotational symmetry.Comment: Updated title and introduction. 4 pages, 3 figure

On the stability of the wind-driven circulation

This work examines the instabilities of steady circulations driven by stationary single-gyre wind forcing in closed rectangular basins with different aspect ratios. The stratified ocean is modeled with quasi-geostrophic 1.5-layer (equivalent-barotropic) and two-layer models. As friction is reduced, a stability threshold is encountered. In the vicinity of this threshold, unstable steady states and their unstable eigenmodes are determined. The structures of the eigenmodes and their associated energy conversion terms allow us to characterize the instabilities. In each case, the loss of stability is associated with an oscillatory instability. Several different instability mechanisms are observed. Which of these is responsible for the onset of instability depends upon the basin aspect ratio and the choice of stratification (1.5- or two-layer). The various mechanisms include instability of the western boundary current, baroclinic instability of the main recirculation gyre, instability of a standing meander located downstream of the main recirculation gyre and a complex instability involving several recirculations and the standing meander. The periods of the eigenmodes range from several months to several years depending upon the kind of instability and type of model. Additional insight into the western boundary current and baroclinic gyre instabilities is provided by an exploration of the stability of (a) the Munk boundary layer flow in 1.5- and two-layer models in an unbounded north-south channel, and (b) an isolated baroclinic vortex on an f-plane

Evolution of rarefaction pulses into vortex rings

The two-dimensional solitary waves of the Gross-Pitaevskii equation in the Kadomtsev-Petviashvili limit are unstable with respect to three-dimensional perturbations. We elucidate the stages in the evolution of such solutions subject to perturbations perpendicular to the direction of motion. Depending on the energy (momentum) and the wavelength of the perturbation different types of three-dimensional solutions emerge. In particular, we present new periodic solutions having very small energy and momentum per period. These solutions also become unstable and this secondary instability leads to vortex ring nucleation.Comment: 5 pages, 5 figure

Instabilities of a steady, barotropic, wind-driven circulation

We explore the stability characteristics of a single, barotropic, wind-driven gyre as a function of the strength of the wind forcing and the size and shape of the basin. We find steady solutions for the barotropic flow in a basin driven by a steady wind stress over a range of values of the Reynolds number and the strength of the wind stress. For those solutions that are close to the stability boundary, we examine the form of the most unstable normal mode. We find that for sufficiently weak forcing, the form of the first instability seen is an instability of the western boundary current. However, for larger values of the forcing, the first instability to set in, as the Reynolds number is reduced, is centered on a standing meander that forms on the continuation of the boundary current after it has left the boundary. Both types of instability are oscillatory. There are several different modes of standing meander instability each associated with Rossby wave-like disturbances in the eastern half of the basin. Each of these modes is most unstable when its frequency is close to a resonance with a basin mode with similar spatial scales

Barotropic, wind-driven circulation in a small basin

We study the asymptotic behavior (large time) of a simple, wind-driven, barotropic ocean model, described by a nonlinear partial differential equation with two spatial dimensions. Considered as a dynamical system, this model has an infinite-dimensional phase space. After discretization, the equivalent numerical model has a phase space of finite but large dimension. We find that for a considerable range of friction, the asymptotic states are low-dimensional attractors. We describe the changes in the structure of these asymptotic attractors as a function of the eddy viscosity of the model. A variety of different types of attractor are seen, with chaotic attractors predominating at higher Reynolds numbers. As the Reynolds number is increased, we observe a slow increase in the dimension of the chaotic attractors. Using an energy analysis, we examine the nature of the instability responsible for the Hopf bifurcation that initiates the transition from asymptotically steady states to time-dependent states

Pade approximations of solitary wave solutions of the Gross-Pitaevskii equation

Pade approximants are used to find approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials. Rational function and generalised rational function approximations of axisymmetric solitary waves of the Gross-Pitaevskii equation are obtained in two and three dimensions. These approximations are used to establish a new mechanism of vortex nucleation as a result of solitary wave interactions.Comment: In press by Journal of Physics: Mathematics and Genera