Vortex Interactions Subjected to Deformation Flows: A Review

Koshel, K. V.; Ryzhov, Eugene A.; Carton, Xavier

doi:10.3390/fluids4010014

Cited by 17 publications

(20 citation statements)

References 154 publications

(274 reference statements)

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“…Further, oceanic motions typically have a non-zero background flow component. Previous studies of the interaction of background flows with two-point-vortex systems in the two-layer case [6] and the continuously stratified case [25] suggest the possibility of relative motion between the constituents of the heton, inducing as a result chaotic passive particle trajectories. Finally, on very large spatial scales, the Coriolis frequency is no longer constant, and the so-called β-effect needs to be taken into account.…”

Section: Discussionmentioning

confidence: 99%

“…Finally, on very large spatial scales, the Coriolis frequency is no longer constant, and the so-called β-effect needs to be taken into account. Point-vortices on the β-plane display richer dynamics than those on the f -plane, including the ability to selfadvect [6] and radiate Rossby waves [14]. A systematic study of these and other interesting perturbations-a fluid with more point vortices than a single heton, a generalization of the point-vortex heton to hetons with distributed vortices-is left to future work.…”

Section: Discussionmentioning

confidence: 99%

“…Mesoscale coherent vortices are ubiquitous in the ocean and play important roles in the dynamics and transport of the ocean and climate [1][2][3][4][5][6]. Idealized modeling of coherent vortices is a powerful tool in understanding their behavior and impact.…”

Section: Introductionmentioning

confidence: 99%

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Volume Transport by a 3D Quasigeostrophic Heton

Sivakumar

Weiss

2022

Fluids

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Oceanic flows self-organize into coherent vortices, which strongly influence their transport and mixing properties. Counter-rotating vortex pairs can travel long distances and carry trapped fluid as they move. These structures are often modeled as hetons, viz. counter-rotating quasigeostrophic point vortex pairs with equal circulations. Here, we investigate the structure of the transport induced by a single three-dimensional heton. The transport is determined by the Hamiltonian structure of the velocity field induced by the heton’s component vortices. The dynamics display a sequence of bifurcations as one moves through the heton-induced velocity field in height. These bifurcations create and destroy unstable fixed points whose associated invariant manifolds bound the trapped volume. Heton configurations fall into three categories. Vertically aligned hetons, which are parallel to the vertical axis and have zero horizontal separation, do not move and do not transport fluid. Horizontally aligned hetons, which lie on the horizontal plane and have zero vertical separation, have a single parameter, the horizontal vortex half-separation Y, and simple scaling shows the dimensional trapped volume scales as Y3. Tilted hetons are described by two parameters, Y and the vertical vortex half-separation Z, rendering the scaling analysis more complex. A scaling theory is developed for the trapped volume of tilted hetons, showing that it scales as Z4/Y for large Z. Numerical calculations illustrate the structure of the trapped volume and verify the scaling theory.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Volume Transport by a 3D Quasigeostrophic Heton

Sivakumar

Weiss

2022

Fluids

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“…Mesoscale coherent vortices are ubiquitious in the ocean and play important roles in the dynamics and transport of the ocean and climate [1][2][3][4][5][6]. Idealized modeling of coherent vortices is a powerful tool in understanding their behavior and impact.…”

Section: Introductionmentioning

confidence: 99%

Volume Transport by a 3D Quasigeostrophic Heton

Sivakumar¹,

Weiss²

2022

Preprint

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Oceanic flows self-organize into coherent vortices which strongly influence their transport and mixing properties. Counter-rotating vortex pairs can travel long distances and carry trapped fluid as they move. These structures are often modeled as hetons, viz. counter-rotating quasigeostrophic point vortex pairs with equal circulations. Here, we investigate the structure of the transport induced by a single three-dimensional heton. The transport is determined by the Hamiltonian structure of the velocity field induced by the heton's component vortices. The dynamics displays a sequence of bifurcations as one moves through the heton-induced velocity field in height. These bifurcations create and destroy unstable fixed points whose associated invariant manifolds bound the trapped volume. Heton configurations fall into three categories. Vertically aligned hetons do not move and do not transport fluid. Horizontally aligned hetons have a single parameter, the horizontal vortex half-separation Y, and simple scaling shows the dimensional trapped volume scales as Y 3 . Tilted hetons are described by two parameters, Y and the vertical vortex half-separation Z, rendering the scaling analysis more complex. A scaling theory is developed for the trapped volume of tilted hetons showing that it scales as Z 4 /Y for large Z. Numerical calculations illustrate the structure of the trapped volume and verify the scaling theory.

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“…The model of point vortices outside the circular domain considered by us is a limiting case of more complex models, for example, the movement of discrete vortices in an annular, in a deformation flow [25], the model of moving cylinder interacting with the configuration of N point vortices [26]. Therefore, our research is a necessary step in the analysis of similar problems in all these models.…”

Section: Introductionmentioning

confidence: 99%

On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon

Kurakin

Ostrovskaya

2020

Mathematics

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The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation Γ around the circle is arbitrary. The problem has three parameters N, q, Γ , where q = R 2 / R 0 2 . This old problem of vortex dynamics is posed by Havelock (1931) and is a generalization of the Kelvin problem (1878) on the stability of a regular vortex polygon (Thomson N-gon) on the plane. In the case of Γ = 0 , the problem has already been solved: in the linear setting by Havelock, and in the nonlinear setting in the series of our papers. The contribution of this work to the solution of the problem consists in the analysis of the case of non-zero circulation Γ ≠ 0 . The linearization matrix and the quadratic part of the Hamiltonian are studied for all possible parameter values. Conditions for orbital stability and instability in the nonlinear setting are found. The parameter areas are specified where linear stability occurs and nonlinear analysis is required. The nonlinear stability theory of equilibria of Hamiltonian systems in resonant cases is applied. Two resonances that lead to instability in the nonlinear setting are found and investigated, although stability occurs in the linear approximation. All the results obtained are consistent with those known for Γ = 0 . This research is a necessary step in solving similar problems for the case of a moving circular cylinder, a model of vortices inside an annulus, and others.

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Vortex Interactions Subjected to Deformation Flows: A Review

Cited by 17 publications

References 154 publications

Volume Transport by a 3D Quasigeostrophic Heton

Volume Transport by a 3D Quasigeostrophic Heton

Volume Transport by a 3D Quasigeostrophic Heton

On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon

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