Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-21T18:04:05.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Spherical vortices in rotating fluids

Published online by Cambridge University Press:  08 May 2018

M. M. Scase Open the ORCID record for M. M. Scase [Opens in a new window]  and
H. L. Terry
M. M. Scase*
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
H. L. Terry
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
*
†Email address for correspondence: matthew.scase@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

A popular model for a generic fat-cored vortex ring or eddy is Hill’s spherical vortex (Phil. Trans. R. Soc. A, vol. 185, 1894, pp. 213–245). This well-known solution of the Euler equations may be considered a special case of the doubly infinite family of swirling spherical vortices identified by Moffatt (J. Fluid Mech., vol. 35 (1), 1969, pp. 117–129). Here we find exact solutions for such spherical vortices propagating steadily along the axis of a rotating ideal fluid. The boundary of the spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and hence carry fluid along with the spherical vortex. As the rotation rate is further increased, further concentric ‘sibling’ vortex rings are formed.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , R4
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Eisenga, A. H. M.1997 Dynamics of a vortex ring in a rotating fluid. PhD thesis, Eindhoven University of Technology.Google Scholar
Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29–62.Google Scholar
Fraenkel, L. E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135.CrossRefGoogle Scholar
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. A 185, 213–245.Google Scholar
Lighthill, M. J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27 (4), 725–752.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35 (1), 117–129.CrossRefGoogle Scholar
Moffatt, H. K. & Moore, D. W. 1978 The response of Hill’s spherical vortex to a small axisymmetric disturbance. J. Fluid Mech. 87 (4), 749–760.CrossRefGoogle Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57 (3), 417–431.CrossRefGoogle Scholar
Pozrikidis, C. 1986 The nonlinear instability of Hill’s vortex. J. Fluid Mech. 168, 337–367.Google Scholar
Protas, B. & Elcrat, A. 2016 Linear stability of Hill’s vortex to axisymmetric perturbations. J. Fluid Mech. 799, 579–602.Google Scholar
Stewartson, K. 1958 On the motion of a sphere along the axis of a rotating fluid. Q. J. Mech. Appl. Maths 11 (1), 39–51.Google Scholar
Taylor, G. I. 1922 The motion of a sphere in a rotating liquid. Proc. R. Soc. Lond. A 102 (715), 180–189.Google Scholar
Verzicco, R., Orlandi, P., Eisenga, A. H. M., van Heijst, G. J. F. & Carnevale, G. F. 1996 Dynamics of a vortex ring in a rotating fluid. J. Fluid Mech. 317, 215–239.Google Scholar