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

## Abstract

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.

- Authors:

- Pacific Oceanological Institute, FEB RAS, 43, Baltiyskaya Street, Vladivostok 690041 (Russian Federation)
- (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22596478

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 26; Journal Issue: 8; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; CHAOS THEORY; NONLINEAR PROBLEMS; PARAMETRIC INSTABILITIES; STEADY-STATE CONDITIONS; TRAJECTORIES; VORTEX FLOW; VORTICES

### Citation Formats

```
Koshel, Konstantin V., E-mail: kvkoshel@poi.dvo.ru, Institute of Applied Mathematics, FEB RAS, 7, Radio Street, Vladivostok 690022, Far Eastern Federal University, 8, Sukhanova Street, Vladivostok 690950, and Ryzhov, Eugene A., E-mail: ryzhovea@gmail.com.
```*Local parametric instability near elliptic points in vortex flows under shear deformation*. United States: N. p., 2016.
Web. doi:10.1063/1.4961123.

```
Koshel, Konstantin V., E-mail: kvkoshel@poi.dvo.ru, Institute of Applied Mathematics, FEB RAS, 7, Radio Street, Vladivostok 690022, Far Eastern Federal University, 8, Sukhanova Street, Vladivostok 690950, & Ryzhov, Eugene A., E-mail: ryzhovea@gmail.com.
```*Local parametric instability near elliptic points in vortex flows under shear deformation*. United States. doi:10.1063/1.4961123.

```
Koshel, Konstantin V., E-mail: kvkoshel@poi.dvo.ru, Institute of Applied Mathematics, FEB RAS, 7, Radio Street, Vladivostok 690022, Far Eastern Federal University, 8, Sukhanova Street, Vladivostok 690950, and Ryzhov, Eugene A., E-mail: ryzhovea@gmail.com. Mon .
"Local parametric instability near elliptic points in vortex flows under shear deformation". United States.
doi:10.1063/1.4961123.
```

```
@article{osti_22596478,
```

title = {Local parametric instability near elliptic points in vortex flows under shear deformation},

author = {Koshel, Konstantin V., E-mail: kvkoshel@poi.dvo.ru and Institute of Applied Mathematics, FEB RAS, 7, Radio Street, Vladivostok 690022 and Far Eastern Federal University, 8, Sukhanova Street, Vladivostok 690950 and Ryzhov, Eugene A., E-mail: ryzhovea@gmail.com},

abstractNote = {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.},

doi = {10.1063/1.4961123},

journal = {Chaos (Woodbury, N. Y.)},

number = 8,

volume = 26,

place = {United States},

year = {Mon Aug 15 00:00:00 EDT 2016},

month = {Mon Aug 15 00:00:00 EDT 2016}

}