Using an iterative algorithm, a family of stationary two-dimensional vortical dipoles is constructed, including translational (symmetric and asymmetric about the translation axis) and orbital (i.e. moving in circles) dipoles. The patches of uniform vorticity comprising a dipole possess symmetry about the axis passing through their centroids and are, generally, unequal in area and absolute value of vorticity. The solutions are discriminated by three parameters, the ratio of the areas of individual vortices, the ratio of their vorticities and the separation between the centroids of the patches. The dipole stability and evolution of unstable states are studied numerically with a contour dynamics method, where the perturbations allowed are, generally, asymmetric. The diagrams of convergence of the iterative algorithm (without any symmetry constraints) are built in three cross-sections of the parameter space: at opposite vorticity of the individual vortices, at equal areas of the vortices and at zero net circulation of the vortex pairs (when inequality of areas of the individual vortices is offset by inequality of the absolute values of vorticity). The convergence bound is shown to be close to the stability bound in the parameter space, and the larger is the separation, the stronger are the perturbations needed to move the dipole out of equilibrium. Typical scenarios of the evolution of unstable symmetric translational dipoles and weakly stable dipoles of other kinds are described, including the transition of a dipole into an oscillating tripole -the scenario that has not been discussed so far.

Instabilities and long-term evolution of two-dimensional circular flows around a rigid circular cylinder (island) are studied analytically and numerically. For that we consider a base flow consisting of two concentric neighbouring rings of uniform but different vorticity, with the inner ring touching the cylinder. We first study the inviscid linear stability of such flows to perturbations of the free edges of the rings. For a given ratio of the vorticity in the rings, the governing parameters of the problem are the radii of the inner and outer rings scaled on the cylinder radius. In this two-dimensional parameter space, we determine analytically the regions of linear stability/instability of each azimuthal mode m = 1, 2, . . . . In the physically most meaningful case of zero net circulation, for each mode m > 1, two regions are identified: a regular instability region where mode m is unstable along with some other modes, and a unique instability region where only mode m is unstable. After the conditions of linear instability are established, inviscid contour-dynamics and high-Reynolds-number finite-element simulations are conducted. In the regular instability regions, simulations of both kinds typically result in the formation of vortical dipoles or multipoles. In the unique instability regions, where the inner vorticity ring is much thinner than the outer ring, the inviscid contour-dynamics simulations do not reveal dipole emission. In the viscous simulation, because viscosity has time to widen the inner ring, the instability develops in the same manner as in the regular instability regions.

A finite-core heton is a baroclinic $f$-plane modon of a special type: it is composed of two patches of uniform quasi-geostrophic potential vorticity (PV) residing in different layers of a two-layer rotating fluid. This paper focuses on numerical construction of steadily translating, doubly symmetric, finite-core hetons and testing their stability. Such a heton, which possesses symmetry about the translation axis and the transverse axis, is a stationary solution to the equations of PV conservation in each of the layers when considered in a comoving frame of reference. When constructing the heton solutions and examining their bifurcations, we identify a heton by a pair of independent non-dimensional parameters, the half-length (in the translation direction) of a PV patch and the distance of the front point of the upper patch from the translation axis. The advantage of this method over other tried approaches is that it allows one to obtain solutions of new, previously unknown types. The results of testing the heton stability are presented on the plane made by a mean radius of a PV patch and the (horizontal) separation between the centroids of the patches. Two kinds of stability are tested separately, the stability to arbitrary perturbations that do not preserve the symmetry of the initial state and the stability to so-called symmetric perturbations that do not violate the initial symmetry. The hetons comparable in size with the Rossby radius, and smaller, are always stable in both senses. However, when some critical size is exceeded, the heton stability becomes dependent on the separation, and the larger the heton, the higher the separation required for stability. The separation guaranteeing the stability to symmetric perturbations is smaller than that required for the stability to arbitrary perturbations. Interrelations between instabilities and bifurcations are briefly discussed.

International audienceThe so-called carousel tripoles are constructed and characterized in the framework of two-layer quasi-geostrophic contour dynamics, and their stability is examined. Such a tripole is a steadily rotating doubly symmetric ensemble of three collinear vortices, or more specifically, uniform-potential-vorticity patches, with the central, core vortex, located in the upper layer, and the two remaining, satellite vortices, in the lower layer, or vice versa. The carousel tripole solutions are obtained with the use of a numerical iterative procedure. A tripole with zero total potential vorticity can be generally identified by a point in the plane spanned by two parameters, namely, the typical size of the patches relative to the Rossby deformation radius, and some shape parameter. We consider two kinds of the parameter plane by taking as the second parameter either the distance d between the centroids of the core and one of the satellites (termed also separation) or, alternatively, the minimal distance h between the core centroid and the satellite contour, measured along the symmetry axis that passes through the centroids of the core and satellites. Accordingly, to capture the stationary tripoles, we use two alternative numerical procedures, which are based on fixing the first or the second pair of parameters. This is done because the areas of convergence of the two procedures differ somewhat from each other. The areas of convergence are plotted in the parameter planes, and in each of the planes, two branches of solutions are found bifurcating from some segments of the lines bounding the convergence areas. Stability is tested in numerical simulations with the numerical noise taken as a perturbation factor. Stability/instability of a tripole is determined by examining the oscillations in the perimeter of one of the vortex satellites. For each tripole size, both stable and unstable solutions exist. The stability bounds coincide with the bifurcation lines, so that one branch of the solutions is stable while the other is not. As a whole, tripoles with considerable separation behave stably

By applying a theoretical approach, we propose a hypothetical scenario that might explain some features of the movement of a long-lived mesoscale anticyclone observed during 1990 in the Bay of Biscay [R. D. Pingree and B. Le Cann, “Three anticyclonic slope water oceanic eddies (SWODDIES) in the southern Bay of Biscay in 1990,” Deep-Sea Res., Part A 39, 1147 (1992)]. In the remote-sensing infrared images, at the initial stage of observations, the anticyclone was accompanied by two cyclonic eddies, so the entire structure appeared as a tripole. However, at later stages, only the anticyclone was seen in the images, traveling generally west. Unusual for an individual eddy were the high speed of its motion (relative to the expected planetary beta-drift) and the presence of almost cycloidal meanders in its trajectory. Although surface satellites seem to have quickly disappeared, we hypothesize that subsurface satellites continued to exist, and the coherence of the three vortices persisted for a long time. A significant perturbation of the central symmetry in the mutual arrangement of three eddies constituting a tripole can make reasonably fast cycloidal drift possible. This hypothesis is tested with two-layer contour-dynamics f-plane simulations and with finite-difference beta-plane simulations. In the latter case, the interplay of the planetary beta-effect and that due to the sloping bottom is considered.

We investigate numerically the transitions and oscillatory regimes in two-layer quasigeostrophic hetons and tripoles composed of patches of uniform potential vorticity (PV). The contour-surgery algorithms are employed, in which either some symmetries are preserved, or asymmetric evolution of the vortex structures is allowed, induced by generally asymmetric numerical noise. The fluid layers are assumed equally thick. First, the evolution of hetons is considered. A heton, a steadily translating pair of vortices residing in different layers, is antisymmetric in the sense that the two PV patches are opposite in sign and symmetric in shape about the axis of translation. A feebly stable heton, when exposed to weak antisymmetric perturbations, responds by developing an oscillation, which culminates in a transition to a new, substantially robust oscillating heton. The results obtained reinforce our earlier findings regarding the modon-to-modon transition (Kizner et al., J. Fluid Mech., vol. 468, 2002, pp. 239–270; Kizner, Phys. Fluids, vol. 18 (5), 2006, 056601; Kizner, UTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (ed. Borisov et al.), IUTAM Bookseries, vol. 6, 2008, pp. 125–133. Springer) and clarify the transition mechanism. Asymmetric perturbations might cause a heton-to-tripole transition. Next we consider the transitions and oscillations in carousel tripoles exposed to weak, generally asymmetric perturbations. A carousel tripole is a steadily rotating centrally symmetric ensemble of three PV patches, with the central vortex being located in one layer and the two remaining, satellite vortices in the other layer. Depending on the tripoles’ size, hence also on the shape of the satellite vortices, three different types of transition are revealed, the transition to a ringed (shielded) monopole being one of them. Whereas the transition of a ringed monopole into a tripole is a known phenomenon, the reverse transition in baroclinic flows is detected for the first time.

A two-dimensional problem of the motion of a single vortex near an infinite straight wall with singular gaps is solved both analytically, using a point-vortex approach, and numerically based on the method of contour surgery for a vortex patch. The background irrotational flow was generated by a balanced point source-sink system located at the gaps. Three different regimes of vortex evolution were detected and studied in detail: (i) Complete or partial transit, i.e., continuation of the motion along the wall; (ii) complete destruction, i.e., the “penetration” through the sink gap; and (iii) capture in a certain area against the wall between the gaps. These regimes are controlled by three parameters: the ratio of the vortex size and the distance between the gaps, the remoteness of the vortex trajectory from the wall, and the ratio of the intensities of the background flow and the vortex. A bifurcational character of the transition between the regimes was observed. Steady-state solutions were found numerically, including the orbital O state, where the vortex's centroid moves along a constant orbit, while the shape of the vortex changes periodically. Capturing the vortex was usually carried out in a form close to this state.

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