Numerical studies of the Hasegawa-Mima equation, derived in the context of drift waves but equivalent to the quasi-geostrophic vortex potential equation for Rossby waves, show the stable properties of solitary vortices which are two dimensional, localized, steady and translating solutions of this same equation. A solitary vortex can propagate only in the direction ( x -direction) perpendicular to the density gradient. When this solitary vortex solution is inclined at some angle with respect to the x -axis, its propagation direction oscillates in the x and y plane. In two dimensional collisions, i.e. head-on collision and overtaking, solitary vortices interact two-dimensionally and recover their initial shapes at the end of both types of collisions.

Dynamics of Two-Dimensional Solitary Vortices in a Low-β Plasma with Convective Motion

/pdf/dynamics-of-heton-like-vortices-28rpo56547.pdf

Dynamics of heton-like vortices

Based on experimental evidence, explicit vorticity-distributed solutions to the Euler equations in two dimensions are constructed describing the tripole vortex. The vortex form and the solution outside the region of nonzero vorticity are derived analytically, while the interior is solved numerically. The continuous-vorticity solution reproduces the main features of the tripoles observed in laboratory experiments and numerical simulations-their shape, flow pattern, and the form of the nonlinear vorticity vs streamfunction relation. The approach followed in the construction of a tripole proves to be beneficial in the search for higher-order multipoles, an example being a smooth quadrupole solution.

The tripole vortex: Experimental evidence and explicit solutions.

A new concept, a hetonic quartet, is presented. A hetonic quartet is a two-layer ensemble of four synchronously translating quasigeostrophic discrete vortices aligned perpendicularly to the axis of their translation. On the f-plane, a hetonic quartet is made up of conventional point vortices (confined to either the upper or the lower layer) with specially fitted circulations and distances between each other, the upper- and lower-layer vortices being opposite in sign and located symmetrically about the translation axis. On the β-plane, a two-layer version of the modulated point vortex model [Phys. Fluids 25, 2175 (1982)] is applied. A necessary and sufficient condition for the stability of f-plane hetonic quartets to perturbations that do not violate the above-mentioned symmetry is established using the analytical methods of Hamiltonian dynamics, whereas on the β-plane, a necessary (linear) stability condition is determined. As distinct from hetons, a stable hetonic quartet is not rigid: when slightly move...

Stability and transitions of hetonic quartets and baroclinic modons

This paper studies the joint influence that rotation and the earth’s sphericity have on the dynamics of localized synoptic scale vortices within the quasi-geostrophic barotropic model in the beta-plane approximation. Rossby solitons (two-dimensional vortices exponentially localized in space which propagate without changing their form along the latitude circles) are considered in the first part of the article. The general properties of such solutions are discussed. The simplest examples are presented, and a brief review of the main results is given. The second part is dedicated to the theory of nonstationary monopoles. The physical mechanisms governing the evolution of such vortices are described; different stages of this evolution are determined for intense vortices. Analytical and numerical results are used to confirm the qualitative explanations.

Dynamics of localized vortices on the beta plane

The objective of this work is a numerical study of the stability properties and the evolution of the eastward-travelling baroclinic modons - coherent vortex structures specific to stratified geophysical fluids where differential rotation (the β-effect) is of the essence. In the vortices under study, the initial dependence of the potential vorticity (PV) upon the streamfunction is piecewise-linear, the barotropic component is dipolar, the baroclinic component is circularly symmetric about the vertical axis, and the boundary of the trapped-fluid region (in which the vorticity contours are closed) is a circular cylinder. These modons are shown to be stable for a wide range of parameters. In two- and three-layer fluids, modons of this type are shown to be able to transit to even more durable states, in which the trapped-fluid area is oval in shape and the PV versus streamfunction dependence in this domain is nonlinear. Possible transition mechanisms and linkage between the circular and oval modons are discussed.

Baroclinic modon equilibria on the beta-plane: stability and transitions

Conditions determining the existence of localized steadily translating two-layer vortices (modons) of arbitrary symmetric form on the β-plane are considered. A numerical method for direct construction of modon solutions is suggested and its accuracy is analysed in relation to the parameters of the computational procedure and the geometrical and physical parameters of the modon sought. Using this method, several non-circular baroclinic solutions are constructed marked by nonlinearity of the dependence of the potential vorticity (PV) on the streamfunction in the trapped-fluid area of the modon, i.e. where the streamlines are closed. The linearity of this dependence and the circularity of the trapped-fluid area are shown to be equivalent properties of a modon. Special attention is given to elliptical modons - extended both in the direction of the modon propagation and in the orthogonal direction, the baroclinic PV component being assumed continuous. The differences between the two types of elliptical modons are discussed. The simplest vortical couples and shielded modons are considered. In the context of the continuity of the baroclinic PV field, the stability of modons is discussed based on numerical simulations.

Non-circular baroclinic beta-plane modons: constructing stationary solutions

Explicit stationary solutions to the equations of vorticity conservation on the f- and β-planes are considered, describing barotropic dipoles (modons) with elliptical frontiers. The far field - outside the elliptical separatrix demarcating the regions of closed and open streamlines - is given analytically, whereas the interior is solved numerically using a successive linearization algorithm. Both ellipses extended along the translation axis and those extended in the transverse direction are considered. In the latter case, it is shown that, among the possible solutions, there exist the so-called supersmooth modons marked by continuity of the vorticity derivatives at the separatrix. On the β-plane, the separatrix aspect ratio that allows a supersmooth solution varies depending on the modon translation speed and size, while on the f-plane, there is only one such separatrix aspect ratio. In this context, the limiting transition from the β-plane to the f-plane is discussed. The dependence between the absolute (or relative) vorticity q and the 'co-moving' streamfunction Ψ, which is nonlinear in the interior of non-circular modons, is analysed in detail for both β- and f-planes, the main concern being the relation between the separatrix form, on the one hand, and the shape of the q against Ψ scattergraph on the other

The structure and evolution of elliptical barotropic modons

A theory of viscous evolution and selection of symmetric two-dimensional dipoles is suggested, based on a combination of numerical simulations and an asymptotic analysis, where the slow time scale associated with the vorticity diffusion due to viscosity is incorporated. It is shown that viscosity first brings a dipole to an intermediate asymptotic state, which is independent of the initial conditions, and then slowly takes the dipole away from this state. We demonstrate that, among the variety of possible ideal-fluid dipole solutions, viscosity going to zero selects a unique solution, which is described to high accuracy by the elliptical dipole solution with a separatrix aspect ratio of 1.037.

Viscous selection of an elliptical dipole

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