Konstantin V. Koshel

Baroclinic multipole evolution in shear and strain

In a two-layer quasi-geostrophic model, the evolution of a symmetric baroclinic multipole, composed of a central vortex with strength μκ in the upper layer, and A satellites with strength κ in the lower layer, is studied. This multipole is imbedded in a center-symmetric shear/strain field, either steady or time-periodic. Special attention is given to the case of the tripole (A = 2). The stability of this tripole is assessed and its oscillations in the external field are analyzed. Conditions for resonance of these oscillations are derived and transition to chaos is described.

Global chaotization of fluid particle trajectories in a sheared two-layer two-vortex flow

In a two-layer quasi-geostrophic approximation, we study the irregular dynamics of fluid particles arising due to two interacting point vortices embedded in a deformation flow consisting of shear and rotational components. The two vortices are arranged within the bottom layer, but an emphasis is on the upper-layer fluid particle motion. Vortices moving in one layer induce stirring of passive scalars in the other layer. This is of interest since point vortices induce singular velocity fields in the layer they belong to; however, in the other layer, they induce regular velocity fields that generally result in a change in passive particle stirring. If the vortices are located at stagnation points, there are three different types of the fluid flow. We examine how properties of each flow configuration are modified if the vortices are displaced from the stagnation points and thus circulate in the immediate vicinity of these points. To that end, an analysis of the steady-state configurations is presented with an emphasis on the frequencies of fluid particle oscillations about the elliptic stagnation points. Asymptotic relations for the vortex and fluid particle zero-oscillation frequencies are derived in the vicinity of the corresponding elliptic points. By comparing the frequencies of fluid particles with the ones of the vortices, relations between the parameters that lead to enhanced stirring of fluid particles are established. It is also demonstrated that, if the central critical point is elliptic, then the fluid particle trajectories in its immediate vicinity are mostly stable making it harder for the vortex perturbation to induce stirring. Change in the type of the central point to a hyperbolic one enhances drastically the size of the chaotic dynamics region. Conditions on the type of the central critical point also ensue from the derived asymptotic relations.

Parametric instability of a many point-vortex system in a multi-layer flow under linear deformation

The paper deals with a dynamical system governing the motion of many point vortices located in different layers of a multi-layer flow under external deformation. The deformation consists of generally independent shear and rotational components. First, we examine the dynamics of the system’s vorticity center. We demonstrate that the vorticity center of such a multi-vortex multi-layer system behaves just like the one of two point vortices interacting in a homogeneous deformation flow. Given nonstationary shear and rotational components oscillating with different magnitudes, the vorticity center may experience parametric instability leading to its unbounded growth. However, we then show that one can shift to a moving reference frame with the origin coinciding with the position of the vorticity center. In this new reference frame, the new vorticity center always stays at the origin of coordinates, and the equations governing the vortex trajectories look exactly the same as if the vorticity center had never moved in the original reference frame. Second, we studied the relative motion of two point vortices located in different layers of a two-layer flow under linear deformation. We analyze their regular and chaotic dynamics identifying parameters resulting in effective and extensive destabilization of the vortex trajectories.

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 flows 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.

Entrapping of a vortex pair interacting with a fixed point vortex revisited. II. Finite size vortices and the effect of deformation

We investigate the evolution of a pair of two-dimensional, opposite-signed, finite-size vortices interacting with a fixed point vortex. The present paper builds on the accompanying study by Koshel et al. [“Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices,” Phys. Fluids 30, 096603 (2018)] focusing on the motion of a pair of point vortices impinging on a fixed point vortex. Here, by contrast, the pair of opposite-signed finite-size vortices, or vortex dipole for simplicity, can deform. This deformation has an impact on the dynamics. We show that, as expected, finite size vortices behave like point vortices if they are distant enough from each other. This allows one to recover the rich and diverse set of possible trajectories for the dipole. This includes the regimes of intricate bounded motion when the finite-size vortices remain stable near the fixed vortex for a long time. On the other hand, we show that large finite-size vortices can deform significantly and deviate from the trajectories of equivalent point vortices. When the shear that the vortices induce on each other is large enough, the finite size vortices may break into smaller structures or may even be completely strained out.We investigate the evolution of a pair of two-dimensional, opposite-signed, finite-size vortices interacting with a fixed point vortex. The present paper builds on the accompanying study by Koshel et al. [“Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices,” Phys. Fluids 30, 096603 (2018)] focusing on the motion of a pair of point vortices impinging on a fixed point vortex. Here, by contrast, the pair of opposite-signed finite-size vortices, or vortex dipole for simplicity, can deform. This deformation has an impact on the dynamics. We show that, as expected, finite size vortices behave like point vortices if they are distant enough from each other. This allows one to recover the rich and diverse set of possible trajectories for the dipole. This includes the regimes of intricate bounded motion when the finite-size vortices remain stable near the fixed vortex for a long time. On the other hand, we show that large finite-size vortices can deform significantly and ...

Entrapping of a vortex pair interacting with a fixed point vortex revisited. I. Point vortices

The problem of a pair of point vortices impinging on a fixed point vortex of arbitrary strengths [E. Ryzhov and K. Koshel, “Dynamics of a vortex pair interacting with a fixed point vortex,” Europhys. Lett. 102, 44004 (2013)] is revisited and investigated comprehensively. Although the motion of a pair of point vortices is established to be regular, the model presents a plethora of possible bounded and unbounded solutions with complicated vortex trajectories. The initial classification [E. Ryzhov and K. Koshel, “Dynamics of a vortex pair interacting with a fixed point vortex,” Europhys. Lett. 102, 44004 (2013)] revealed that a pair could be compelled to perform bounded or unbounded motion without giving a full classification of either of those dynamical regimes. The present work capitalizes upon the previous results and introduces a finer classification with a multitude of possible regimes of motion. The regimes of bounded motion for the vortex pair entrapped near the fixed vortex or of unbounded motion, when the vortex pair moves away from the fixed vortex, can be categorized by varying the two governing parameters: (i) the ratio of the distances between the pair’s vortices and the fixed vortex and (ii) the ratio of the strengths of the vortices of the pair and the strength of the fixed vortex. In particular, a bounded motion regime where one of the pair’s vortices does not rotate about the fixed vortex is revealed. In this case, only one of the pair’s vortices rotates about the fixed vortex, while the other oscillates at a certain distance. Extending the results obtained with the point-vortex model to an equivalent model of finite size vortices is the focus of Paper II [J. N. Reinaud et al., “Entrapping of a vortex pair interacting with a fixed point vortex revisited. II. Finite size vortices and the effect of deformation,” Phys. Fluids 30, 096604 (2018)].The problem of a pair of point vortices impinging on a fixed point vortex of arbitrary strengths [E. Ryzhov and K. Koshel, “Dynamics of a vortex pair interacting with a fixed point vortex,” Europhys. Lett. 102, 44004 (2013)] is revisited and investigated comprehensively. Although the motion of a pair of point vortices is established to be regular, the model presents a plethora of possible bounded and unbounded solutions with complicated vortex trajectories. The initial classification [E. Ryzhov and K. Koshel, “Dynamics of a vortex pair interacting with a fixed point vortex,” Europhys. Lett. 102, 44004 (2013)] revealed that a pair could be compelled to perform bounded or unbounded motion without giving a full classification of either of those dynamical regimes. The present work capitalizes upon the previous results and introduces a finer classification with a multitude of possible regimes of motion. The regimes of bounded motion for the vortex pair entrapped near the fixed vortex or of unbounded motion, wh...

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.

Toroidal vortices over isolated topography in geophysical flows

This work deals with a model of a topographically trapped vortex appearing over isolated topography in a geophysical flow. The main feature of the study is that we pay special attention to the vertical structure of a topographically trapped vortex. The model considered allows one to study the vertical motion which is known not to be negligible in many cases. Given topography in the form of an isolated cylinder, and radial symmetry and stationarity of a uniform flow, in the linear approximation, we formulate a boundary value problem that determines all the components of the velocity field through a six-order differential operator, and nonincreasing boundary conditions at the center of the topography, and at infinity. The eigenvalues of the boundary value problem correspond to bifurcation points, in which the flow becomes unstable, hence non-negligible vertical velocities occur. We formulate a condition for the boundary value problem to have a discrete spectrum of these bifurcation points, and hence to be solvable. Conducting a series of test calculations, we show that the resulting vortex lies in the vicinity of topography, and can attain the distance up to half of the topography characteristic radius.

Interaction of an along-shore propagating vortex with a vortex enclosed in a circular bay

A simple dynamical model of vortex interactions taking place near a curved boundary mimicking a circular bay is formulated and examined. An initial configuration consisting of a point vortex in the bay and of an incident point vortex moving toward the bay along the straight part of the boundary is considered. Both vortices are of equal strengths. Typical stationary regimes of the bay-bound vortex when the incident vortex is far from the bay are obtained. When the incident vortex comes near the bay, its interaction with the bay-bound one may result in irregular motion of both vortices. Typical outcomes of the interaction are established to be (i) the incident vortex passes over the bay without forcing the bay-bound vortex to leave the bay; (ii) the incident vortex becomes entrapped within the bay, whereas the bay-bound vortex leaves it; (iii) both vortices leave the bay shortly after the interaction as separate vortices or as a bound leap-frogging pair; (iv) both vortices exhibit convoluted dynamics being ...

Advection of passive scalars induced by a bay-trapped nonstationary vortex

A simple model of fluid particle advection induced by the interaction of a point vortex and incident plane flow occurring near a curved boundary is analyzed. The use of the curved boundary in this case is aimed at mimicking the geometry of an isolated bay of a circular shape. An introduction of such a boundary to the model results in the appearance of retention zones, where the vortex can be permanently trapped being either stationary or periodically oscillating. When stationary, it induces a steady velocity field that in turn ensures regular advection of nearby fluid particles. When the vortex oscillates periodically, the induced velocity field turns unsteady leading to the manifestation of chaotic advection of fluid particles. We show that the size of the fluid region engaged into chaotic advection increases almost monotonically with the increased magnitude of the vortex oscillations provided the magnitude remains relatively small. The monotonicity is accounted for the fact that the frequency of the vortex oscillations incommensurable with the proper frequency of fluid particle rotations in the steady state. Another point of interest is that it is demonstrated that bounded regions, in which the vortex may be trapped, can appear even at a significant distance from the bay. Making use of a Lagrangian indicator, examples of fluid particle advection induced by the periodic motion of the vortex inside the bay are adduced.