St Andrews Research Repository

St Andrews University Home
View Item 
  •   St Andrews Research Repository
  • University of St Andrews Research
  • University of St Andrews Research
  • University of St Andrews Research
  • View Item
  •   St Andrews Research Repository
  • University of St Andrews Research
  • University of St Andrews Research
  • University of St Andrews Research
  • View Item
  •   St Andrews Research Repository
  • University of St Andrews Research
  • University of St Andrews Research
  • University of St Andrews Research
  • View Item
  • Register / Login
JavaScript is disabled for your browser. Some features of this site may not work without it.

The motion of point vortices on closed surfaces

Thumbnail
View/Open
dritschel2015procroysoca20140890.pdf (1.384Mb)
Date
04/2015
Author
Dritschel, David Gerard
Boatto, S
Funder
EPSRC
Grant ID
EP/H001794/1
Keywords
Vortex dynamics
Point vortices
Closed surfaces
QA Mathematics
NDAS
BDC
R2C
Metadata
Show full item record
Abstract
We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech. 157 , 95–134 (doi:10.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255 , 35–64 (doi:10.1017/S0022112093002381)).
Citation
Dritschel , D G & Boatto , S 2015 , ' The motion of point vortices on closed surfaces ' , Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences , vol. 471 , no. 2176 , 20140890 , pp. 1-25 . https://doi.org/10.1098/rspa.2014.0890
Publication
Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Status
Peer reviewed
DOI
https://doi.org/10.1098/rspa.2014.0890
ISSN
1364-5021
Type
Journal article
Rights
Copyright. 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author andvsource are credited.
Collections
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/6297

Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.

Advanced Search

Browse

All of RepositoryCommunities & CollectionsBy Issue DateNamesTitlesSubjectsClassificationTypeFunderThis CollectionBy Issue DateNamesTitlesSubjectsClassificationTypeFunder

My Account

Login

Open Access

To find out how you can benefit from open access to research, see our library web pages and Open Access blog. For open access help contact: openaccess@st-andrews.ac.uk.

Accessibility

Read our Accessibility statement.

How to submit research papers

The full text of research papers can be submitted to the repository via Pure, the University's research information system. For help see our guide: How to deposit in Pure.

Electronic thesis deposit

Help with deposit.

Repository help

For repository help contact: Digital-Repository@st-andrews.ac.uk.

Give Feedback

Cookie policy

This site may use cookies. Please see Terms and Conditions.

Usage statistics

COUNTER-compliant statistics on downloads from the repository are available from the IRUS-UK Service. Contact us for information.

© University of St Andrews Library

University of St Andrews is a charity registered in Scotland, No SC013532.

  • Facebook
  • Twitter