Modulated point-vortex pairs on a rotating sphere: Dynamics and chaotic advection

Drótos, Gábor; Tél, Tamás; Kovács, Gergely

doi:10.1103/physreve.87.063017

Cited by 4 publications

(11 citation statements)

References 45 publications

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“…The vortex pair is defined by having opposite vorticities G 0 1r 5 2G 0 2r [ G 0 at the reference latitude u r . The chord distance D 0 of the elements of the pair turns out to be constant during the motion (Drótos et al 2013). The dimensionless equations for the two elements of the pair are found to be…”

Section: The Vortex Pair Modelmentioning

confidence: 90%

“…As a starting point, we briefly summarize the results of Drótos et al (2013). We consider two point vortices whose location is specified by angles l, u in geographical coordinates on the surface of a sphere (l being the longitude, u being the latitude).…”

Section: The Vortex Pair Modelmentioning

confidence: 99%

“…This line, owing to the functional form of the velocity field of any point vortex, is always perpendicular to the velocities of the elements of the pair and, hence, to the center-ofmass velocity u of the pair. As pointed out in Drótos et al (2013), we can restrict our investigations without the loss of generality to cases with an initially strictly zonal center-ofmass velocity. We thus always choose the initial meridional component y 0 of the center-of-mass velocity to be 0.…”

Section: The Vortex Pair Modelmentioning

confidence: 99%

“…Recently, we generalized the principle of modulation for vortex motions on the entire rotating sphere by making the vortex circulation nonlinearly dependent on the latitudinal angle u j of vortex j (Drótos et al 2013). Under the assumption that a point vortex represents a small patch of vorticity of an area a 2 p, the circulation of vortex j with coordinate u j is written as…”

Section: Introductionmentioning

confidence: 99%

“…Two basically different types of trajectories are found: a small-amplitude meandering motion extending to both sides of the equator and traveling to the east, called wobbling, and a self-intersecting large-amplitude motion traveling to the west, called tumbling. In our earlier paper (Drótos et al 2013) we pointed out that the vortex pair trajectories are also of wobbling or tumbling type-just the role of the equator is taken over by the reference latitude u r , defined in (1). This can take on any value, allowing us to focus our attention to the vicinity of an arbitrarily chosen latitude.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the Validity of the β-Plane Approximation in the Dynamics and the Chaotic Advection of a Point Vortex Pair Model on a Rotating Sphere

Drótos

Tél

2015

Journal of the Atmospheric Sciences

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The dynamics of modulated point vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). In this setting the authors point out a qualitative difference between the full spherical dynamics and the one obtained in a β-plane approximation. In particular, dipole trajectories starting at the same location evolve to completely different directions under these two treatments, despite the fact that the deviations from the initial latitude remain small. This is a strong indication for the mathematical inconsistency of the traditional β-plane approximation. At the same time, a consistently linearized set of equations of motion leads to trajectories agreeing with those obtained under the full spherical treatment. The β-plane advection patterns due to chaotic advection in the velocity field of finite-sized vortex pairs are also found to considerably deviate from those of the full spherical treatment, and quantities characterizing transport properties (e.g., the escape rate from a given region) strongly differ.

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Section: The Vortex Pair Modelmentioning

confidence: 90%

Section: The Vortex Pair Modelmentioning

confidence: 99%

Section: The Vortex Pair Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Validity of the β-Plane Approximation in the Dynamics and the Chaotic Advection of a Point Vortex Pair Model on a Rotating Sphere

Drótos

Tél

2015

Journal of the Atmospheric Sciences

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

Ryzhov

Koshel

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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

Ryzhov

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Modulated point-vortex pairs on a rotating sphere: Dynamics and chaotic advection

Cited by 4 publications

References 45 publications

On the Validity of the β-Plane Approximation in the Dynamics and the Chaotic Advection of a Point Vortex Pair Model on a Rotating Sphere

On the Validity of the β-Plane Approximation in the Dynamics and the Chaotic Advection of a Point Vortex Pair Model on a Rotating Sphere

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

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

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