Perturbation theory for Rankine vortices

Gorshkov, K. A.; Ostrovsky, Lev A.; Soustova, I. A.

doi:10.1017/s0022112099007211

Cited by 9 publications

(5 citation statements)

References 13 publications

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“…This is supported by the field observations of Smith & Largier (1995). Further, it is known that dipolar vortex structures are generally more robust than their monopolar counterparts (Gorshkov, Ostrovsky & Soustova 2000) and are therefore more likely to be of importance in the dynamics of the surf zone. The ability of dipoles to self-propagate (and so transport fluid) also provides a natural mechanism for the dipole to approach and interact with the topography.…”

Section: Introductionmentioning

confidence: 99%

Surf-zone vortices over stepped topography

Johnson

McDonald

2004

J. Fluid Mech.

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The problem of vortical motions in the surf zone is simplified by taking the bottom topography to be piecewise flat while allowing finite-height jumps in depth between flat regions. The motion of an arbitrary number of singular vortices is cast into Hamiltonian form and the rule for relating Hamiltonians in conformally equivalent domains derived. Examples are given of a singular vortex pair colliding head-on with a step, of a vortex propagating along a curved coast to cross a step, and of a vortex being swept past a circular island straddling a step. Surf-zone vortices are then modelled as finite-area vortex patches and their motion followed by contour dynamics. It is shown that the paths of singular vortices can yield highly accurate explicit predictions of the paths of the centroids of vortex patches. Possible applications to surf-zone rip currents are noted.

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Section: Introductionmentioning

confidence: 99%

Surf-zone vortices over stepped topography

Johnson

McDonald

2004

J. Fluid Mech.

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“…Asymptotic approaches have been attempted [33], but they are limited to small times. The approach r dr dr r dr dr rh − of [10] may show promise, but has so far only been developed in two dimensions and for small perturbations to boundaries. A final conclusion is that when contour dynamics can be extended to include additional physics, the outcome is a coupled vortex patch-vortex sheet system, and numerical simulations of the evolution of vortex sheets are known to be complicated by the existence of singularities.…”

Section: Discussionmentioning

confidence: 99%

Generalized Contour Dynamics: A Review

et al. 2018

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Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular buoyancy effects and magnetic fields that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow.

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“…A. Malomed for the helpful notes concerning the book plan. In addition to the published sources, in parts of Chapters 6 and 7, some materials from the doctoral thesis of K. A. Gorshkov [10] were used with his kind consent. Many thanks to S. Westgaard, who helped to improve the style of the manuscript, and R. Babu, M. Judge and T. D'Cruz for their crucial technical help.…”

Section: Parameters Including the Functions A(t) And ( ) T ωmentioning

confidence: 99%

Front Matter

2014

Asymptotic Perturbation Theory of Waves

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vii PrefaceAs far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. Albert EinsteinUsing approximations to find solutions of differential equations has a long history, especially in the areas of applied science and engineering, where relatively simple ways to obtain quantitative results are crucial. Since the 18th century, perturbation methods have been actively used in celestial mechanics for describing planetary motion as a three-body interaction governed by gravitation. As was shown by Poincaré [19], the three-body problem is not completely integrable, and its solution was often based on finding a small perturbation to the solution of the integrable two-body problem, when the mass of at least one of the three bodies is small. One famous example is the prediction of an existing and the position of a new planet by Adams and Leverrier based on calculation of perturbations of Uranus, which resulted in the discovery of the planet Neptune by Galle in 1846. A more recent overview of this topic can be found, for example, in [6,7]. In the 20th century the notion of a "theory of oscillations" as a unifying concept, meaning the application of similar equations and methods of their solution to quite different physical problems, came into being. In particular, the phase plane method and Poincaré mapping proved to be very efficient. For early work in this area, see the classic book by Andronov et al. [3]. viii Preface Regular perturbation schemes allow one to find a deviation from the basic, unperturbed solution, when the deviation remains sufficiently small indefinitely or at least at a time interval at which the solution should be determined. However, in many important cases, a small initial perturbation can strongly change the solution even if the basic equations are only slightly disturbed. In these cases the smallness of the expansion parameter is reflected in the slowness, but not necessarily smallness, of the deviation from the solution existing when that parameter is zero. The slowness means that the time of significant deviation from the unperturbed solution is much longer than the characteristic time scale of the process, such as the period of oscillations or the duration of a pulse. An adequate tool for solving such problems is the asymptotic perturbation theory, which constructs a series in a small parameter in which the main term of the expansion differs from the unperturbed solution in that it contains slowly varying parameters; their variation can be found from "compatibility," or "orthogonality," conditions which secure the finiteness of the higher-order perturbations. In the framework of an asymptotic method, the series may not even converge, but a sum of a finite number of terms in the series approaches the exact solution when the expansion parameter tends to zero, which is sufficient in most applications. Still more involved is singular perturbation theory, in which the asymptotic series does not converge to the unperturbed so...

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Perturbation theory for Rankine vortices

Cited by 9 publications

References 13 publications

Surf-zone vortices over stepped topography

Surf-zone vortices over stepped topography

Generalized Contour Dynamics: A Review

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