Jacques Verron

Studies of the properties of vortex motions in a stably stratified and fast rotating fluid that can be described by the equation for the evolution of a potential vortex in the quasi-geostrophic approximation are reviewed. Special attention is paid to the vortices with zero total intensity (the so-called hetons). The problems considered include self-motion of discrete hetons, the stability of a solitary distributed heton, and the interaction between two finite-core hetons. New solutions to the problems of three or more discrete vortices with a heton structure are proposed. The existence of chaotic regimes is revealed. The range of applications of the heton theory and the prospects for its future application, particularly in respect, to the analysis of the dynamic stage in the development of deep ocean convection, are discussed.

The problem of studying the motion of three vortex lines with arbitrary intensities in an unbounded two-dimensional finite-thickness layer of a homogeneous fluid is known [25], [9], [28], [1] to belong to the class of integrable problems. However, a complete classification of possible motions was constructed only recently [10], [28], [41]. In [40], [39], [20] a generalization is given for two-layer rotating fluid in the particular case determined by the conditions of (i) zero total circulation of vortices, and (ii) the equality of the intensities of two vortices. Here, the first of these restrictions is lifted.

The problem of three vortex lines in a homogeneous layer of an ideal incompressible fluid is generalized to the case of a two-layer liquid with constant density values in each layer. For zero-complex-momentum systems the theory of the roundabout two-layer tripole is built. When the momentum is different from zero, based on the phase portraits in trilinear coordinates, a classification of possible relative motions of a system composed of three discrete (or point) vortices is provided. One vortex is situated in the upper layer, and the other two in the lower layer; their total intensity is zero. More specifically, a model of a two-layer tripole is constructed, and existence conditions for stationary solutions are found. These solutions represent a uniform translational motion of the following vortex structures: 1) a stable collinear configuration triton, a discrete analog of the vortex structure modon+rider, 2) an unstable triangular configuration. Features of the absolute motion of the system of three discrete vortices were studied numerically.
We compared the dynamics of a system of three point vortices with the dynamics of three finite-core vortices (vortex patches). In studying the evolution of the vortex patch system, a two-layer version of the Contour Dynamics Method (CDM) was used. The applicability of discrete-vortex theory to the description of the finite-size vortex behavior is dicussed. Examples of formation of vortical configurations are given. Such configurations appear either after merging of vortices of the same layer or as a result of instability of the two-layer vortex structure.

The problem of four vortex lines with zero total circulation and zero impulse on a unlimited fluid plane, as it is known [1,3,4,16], is reduced to a problem of three point vortices and is integrated in quadratures. In the given work these results are transferred on a case of four vortices in a two-layer rotating liquid. The analysis of phase trajectories of relative motion of vortices is made, and the singularities of absolute motion on an example of a head-on, off-center collision of two two-layer vortex pairs are studied. In particular, the new class of quasistationary solutions for the given type of motions is obtained. The problems of interaction of the distributed (or finite-core) two-layer vortices are discussed.