Effect of the vertical component of diffusion on passive scalar transport in an isolated vortex model

Clustering of floating tracers in weakly divergent velocity fields

This work deals with buoyant tracers floating at the ocean surface, where the geostrophic velocity component is two dimensional and rotational (nondivergent) and the ageostrophic component can contain rotational and potential (divergent) contributions that are comparable in size. We consider a random kinematic flow model and study the process of clustering, that is, aggregation of the floating tracer in localized spatial patches. In the long-time limit and in the cases of strongly and weakly divergent flows, the existing analytical theory predicts the process of exponential clustering, which is the emergence of spatial singularities containing all the available tracer. Here we confirm this analytical prediction, in numerical model solutions spanning different combinations of rotational and potential surface velocity components, and report that exponential clustering persists even in weakly divergent flows, however at significantly slower rates. For a wide range of parameters, we analyze not only the exponential clustering but also the other type of tracer aggregation, referred to as fragmentation clustering, as well as the coarse-graining effects on clustering. For the presented analysis we consider ensembles of Lagrangian particles and introduce and apply the statistical topography methodology.

Vortex Interactions Subjected to Deformation Flows: A Review

Deformation flows are the flows incorporating shear, strain and rotational components. These flows are ubiquitous in the geophysical flows, such as the ocean and atmosphere. They appear near almost any salience, such as isolated coherent structures (vortices and jets) and various fixed obstacles (submerged obstacles and continental boundaries). Fluid structures subject to such deformation flows may exhibit drastic changes in motion. In this review paper, we focus on the motion of a small number of coherent vortices embedded in deformation flows. Problems involving isolated one and two vortices are addressed. When considering a single-vortex problem, the main focus is on the evolution of the vortex boundary and its influence on the passive scalar motion. Two vortex problems are addressed with the use of point vortex models, and the resulting stirring patterns of neighbouring scalars are studied by a combination of numerical and analytical methods from the dynamical system theory. Many dynamical effects are reviewed with emphasis on the emergence of chaotic motion of the vortex phase trajectories and the scalars in their immediate vicinity.

Nonlinear dynamics of an elliptic vortex embedded in an oscillatory shear flow

The nonlinear dynamics of an elliptic vortex subjected to a time-periodic linear external shear flow is studied numerically. Making use of the ideas from the theory of nonlinear resonance overlaps, the study focuses on the appearance of chaotic regimes in the ellipse dynamics. When the superimposed flow is stationary, two general types of the steady-state phase portrait are considered: one that features a homoclinic separatrix delineating bounded and unbounded phase trajectories and one without a separatrix (all the phase trajectories are bounded in a periodic domain). When the external flow is time-periodic, the ensuing nonlinear dynamics differs significantly in both cases. For the case with a separatrix and two distinct types of phase trajectories: bounded and unbounded, the effect of the most influential nonlinear resonance with the winding number of 1:1 is analyzed in detail. Namely, the process of occupying the central stability region associated with the steady-state elliptic critical point by the stability region associated with the nonlinear resonance of 1:1 as the perturbation frequency gradually varies is investigated. A stark increase in the persistence of the central regular dynamics region against perturbation when the resonance of 1:1 associated stability region occupies the region associated with the steady-state elliptic critical point is observed. An analogous persistence of the regular motion occurs for higher perturbation frequencies when the corresponding stability islands reach the central stability region associated with the steady-state elliptic point. An analysis for the case with the resonance of 1:2 is presented. For the second case with only bounded phase trajectories and, therefore, no separatrix, the appearance of much bigger stability islands associated with nonlinear resonances compared with the case with a separatrix is reported.

Resonance phenomena in a two-layer two-vortex shear flow

The paper deals with a dynamical system governing the motion of two point vortices embedded in the bottom layer of a two-layer rotating flow experiencing linear deformation and their influence on fluid particle advection. The linear deformation consists of shear and rotational components. If the deformation is stationary, the vortices can move periodically in a bounded region. The vortex periodic motion induces stirring patterns of passive fluid particles in the both layers. We focus our attention on the upper layer where the bottom-layer singular point vortices induce a regular velocity field with no singularities. In the upper layer, we determine a steady-state regime featuring no closed fluid particle trajectories associated with the vortex motion. Thus, in the upper layer, the flow's streamlines look like there is only external linear deformation and no vortices. In this case, fluid particles move along trajectories of almost regular elliptic shapes. However, the system dynamics changes drastically if the underlying vortices cease to be stationary and instead start moving periodically generating a nonstationary perturbation for the fluid particle advection. Then, we demonstrate that this steady-state regime transits to a perturbed state with a rich phase portrait structure featuring both periodic and chaotic fluid particle trajectories. Thus, the perturbed state clearly manifests the impact of the underlying vortex motion. An analysis, based on comparing the eigenfrequencies of the steady-state fluid particle rotation with the ones of the vortex rotation, is carried out, and parameters ensuring effective fluid particle stirring are determined. The process of separatrix reconnection of close stability islands leading to an enhanced chaotic region is reported and analyzed.

Local parametric instability near elliptic points in vortex flows under shear deformation

The dynamics of two point vortices embedded in an oscillatory external flow consisted of shear and rotational components is addressed. The region associated with steady-state elliptic points of the vortex motion is established to experience local parametric instability. The instability forces the point vortices with initial positions corresponding to the steady-state elliptic points to move in spiral-like divergent trajectories. This divergent motion continues until the nonlinear effects suppress their motion near the region associated with the steady-state separatrices. The local parametric instability is then demonstrated not to contribute considerably to enhancing the size of the chaotic motion regions. Instead, the size of the chaotic motion region mostly depends on overlaps of the nonlinear resonances emerging in the perturbed system.