## Vortex Problems, Rotating Spiral Structures, and the Hannay-Berry Phase

Citations: | 1 - 1 self |

### BibTeX

@MISC{Newton_vortexproblems,,

author = {Paul Newton and Banavara Shashikanth},

title = {Vortex Problems, Rotating Spiral Structures, and the Hannay-Berry Phase},

year = {}

}

### OpenURL

### Abstract

This paper describes the occurence of phase anholonomies in the context of point vortex problems for two-dimensional incompressible flows. After giving a brief description of anholonomic effects in other contexts, we focus attention on the restricted three-vortex problem and a simpler modified problem where the `Hannay-Berry' phase can be computed using multi-scale asymptotic methods. Our main emphasis in this paper is to show how the Hannay-Berry phase arises as the leading term in an asymptotic expansion as the result of a non-uniform limit process. We show how it arises when computing the long time growth rate of passive scalar interfaces as they wrap around vortex cores in the presence of a slowly varying background field due to other vortices, and discuss the results in the context of `spiral-vortex' models for 2D turbulence. 1 Introduction Anholonomic effects arise in mechanical systems both on the classical as well as quantum levels. Simply put, anholonomy is the failure of a s...

### Citations

156 |
Quantal phase factors accompanying adiabatic changes
- Berry
- 1984
(Show Context)
Citation Context ...s not situated on the Equator or one of the Poles [5, 6]. Phase anholonomies have become the focus of intensive research since the discovery of such an effect in quantum mechanics by M. Berry in 1984 =-=[2, 3, 4]-=-. Classical analogues were then discovered by J. Hannay [10], and since that time, several review papers have been written and many new insights have been gained [8, 12, 18]. Our purpose in this paper... |

39 |
Strained spiral vortex model for turbulent fine structure
- Lundgren
- 1982
(Show Context)
Citation Context ...elta` 0 = ~ ` 0 (T ) = 2�� R 0 cos(OE 0 ): (30) 4 Slowly rotating spirals In this section we make contact with some work on spiral vortex models for 2D turbulence. Starting with the paper by Lundg=-=ren [11]-=- and subsequently developed by Moffat [14] and Gilbert [9], the following point of view was advocated. 2D incompressible turbulence, in its simplest form, can be viewed as a collection of `vortex spir... |

36 |
An analytical study of transport, mixing and chaos in an unsteady vortical flow
- Rom-Kedar, Leonard, et al.
- 1990
(Show Context)
Citation Context ...mented on by Everson and Sreenivasan [7]. If one is interested in the `diffusion' of vorticity in the spiral regions, or in the mixing of passive scalars in these regions as discussed, for example in =-=[1, 13, 16]-=-, the most relevant quantity to compute is the long time growth rate of the vorticity interface. We focus on that question in this section and show how this is intimately related to the H-B phase for ... |

14 | Classical adiabatic angles and quantal adiabatic phase
- Berry
- 1985
(Show Context)
Citation Context ...s not situated on the Equator or one of the Poles [5, 6]. Phase anholonomies have become the focus of intensive research since the discovery of such an effect in quantum mechanics by M. Berry in 1984 =-=[2, 3, 4]-=-. Classical analogues were then discovered by J. Hannay [10], and since that time, several review papers have been written and many new insights have been gained [8, 12, 18]. Our purpose in this paper... |

14 |
Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian
- Hannay
- 1985
(Show Context)
Citation Context ...nholonomies have become the focus of intensive research since the discovery of such an effect in quantum mechanics by M. Berry in 1984 [2, 3, 4]. Classical analogues were then discovered by J. Hannay =-=[10]-=-, and since that time, several review papers have been written and many new insights have been gained [8, 12, 18]. Our purpose in this paper is to describe how such phase anholonomies arise in vortex ... |

11 |
Anticipations of the geometric phase
- Berry
(Show Context)
Citation Context ...has to re-orient itself before coming in contact with the floor [12], and when a Foucault pendulum oscillates through a 24 hour period as long as it is not situated on the Equator or one of the Poles =-=[5, 6]-=-. Phase anholonomies have become the focus of intensive research since the discovery of such an effect in quantum mechanics by M. Berry in 1984 [2, 3, 4]. Classical analogues were then discovered by J... |

9 |
turbulent vortex motion in two-dimensional flows, in
- Aref, Integrable
- 1983
(Show Context)
Citation Context ...mented on by Everson and Sreenivasan [7]. If one is interested in the `diffusion' of vorticity in the spiral regions, or in the mixing of passive scalars in these regions as discussed, for example in =-=[1, 13, 16]-=-, the most relevant quantity to compute is the long time growth rate of the vorticity interface. We focus on that question in this section and show how this is intimately related to the H-B phase for ... |

9 | Vortex pairing: The mechanism of turbulent mixing layer growth at moderate Reynolds number - Winant, Browand - 1974 |

8 |
Quantum phase corrections from adiabatic iteration
- Berry
- 1987
(Show Context)
Citation Context ...s not situated on the Equator or one of the Poles [5, 6]. Phase anholonomies have become the focus of intensive research since the discovery of such an effect in quantum mechanics by M. Berry in 1984 =-=[2, 3, 4]-=-. Classical analogues were then discovered by J. Hannay [10], and since that time, several review papers have been written and many new insights have been gained [8, 12, 18]. Our purpose in this paper... |

8 |
Spiral structures and spectra in two-dimensional turbulence
- Gilbert
- 1988
(Show Context)
Citation Context ...tating spirals In this section we make contact with some work on spiral vortex models for 2D turbulence. Starting with the paper by Lundgren [11] and subsequently developed by Moffat [14] and Gilbert =-=[9]-=-, the following point of view was advocated. 2D incompressible turbulence, in its simplest form, can be viewed as a collection of `vortex spirals' which arise from the merging of a strong vortex core ... |

5 |
1991] Symmetry, stability, geometric phases, and mechanical integrators, Nonlinear Science Today
- Marsden, O’Reilly, et al.
(Show Context)
Citation Context ...inally, let your arm drop back to your side. Your palm should now be facing your body, a 90 degree rotation from its initial configuration, despite the fact that your arm has completed one full cycle =-=[12]-=-. A similar anholonomy occurs when a cat is dropped from an upsidedown position and has to re-orient itself before coming in contact with the floor [12], and when a Foucault pendulum oscillates throug... |

3 |
Accumulation rates of spiral-like structures in fluid flows
- Everson, Sreenivasan
- 1992
(Show Context)
Citation Context ...cting vortex spirals with a characteristic vorticity distribution, as discussed in Gilbert [9]. The presence of such structures in careful experiments has been commented on by Everson and Sreenivasan =-=[7]-=-. If one is interested in the `diffusion' of vorticity in the spiral regions, or in the mixing of passive scalars in these regions as discussed, for example in [1, 13, 16], the most relevant quantity ... |

1 |
Unsteady Models for the Nonlinear Evolution of the Mixing Layer
- Meiburg, Newton, et al.
- 1995
(Show Context)
Citation Context ...mented on by Everson and Sreenivasan [7]. If one is interested in the `diffusion' of vorticity in the spiral regions, or in the mixing of passive scalars in these regions as discussed, for example in =-=[1, 13, 16]-=-, the most relevant quantity to compute is the long time growth rate of the vorticity interface. We focus on that question in this section and show how this is intimately related to the H-B phase for ... |

1 |
Simple Topological Aspects of Vorticity Dynamics
- Moffat
- 1984
(Show Context)
Citation Context ... (30) 4 Slowly rotating spirals In this section we make contact with some work on spiral vortex models for 2D turbulence. Starting with the paper by Lundgren [11] and subsequently developed by Moffat =-=[14]-=- and Gilbert [9], the following point of view was advocated. 2D incompressible turbulence, in its simplest form, can be viewed as a collection of `vortex spirals' which arise from the merging of a str... |

1 |
Hannay-Berry Phase and the Restricted Three Vortex Problem
- Newton
- 1994
(Show Context)
Citation Context ...oblems, Rotating Spiral Structures, and the Hannay-Berry Phase 256 2 The Berry phase as a non-uniform limit process The Hannay-Berry phase for the restricted 3 vortex problem was treated in detail in =-=[15]-=-. The basic configuration, shown in Fig. 1, has two co-rotating point vortices of equal strength a distance D apart, with a tracer particle orbiting around one of the point vortices with coordinates (... |

1 |
Berry's Phase
- Zwanziger, Koenig, et al.
- 1990
(Show Context)
Citation Context ...echanics by M. Berry in 1984 [2, 3, 4]. Classical analogues were then discovered by J. Hannay [10], and since that time, several review papers have been written and many new insights have been gained =-=[8, 12, 18]-=-. Our purpose in this paper is to describe how such phase anholonomies arise in vortex dynamics problems, and in particular how they play an important role in determining the long time growth rates of... |