Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-02T20:50:31.314Z Has data issue: false hasContentIssue false
  • English
  • Français

Spreading of two-dimensional axisymmetric vortices exposed to a rotating strain field

Published online by Cambridge University Press:  10 July 2009

M. R. TURNER  and
A. D. GILBERT
M. R. TURNER*
Affiliation:
Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 4QF, UK
A. D. GILBERT
Affiliation:
Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 4QF, UK
*
†Present address: School of Environment and Technology, University of Brighton, Cockcroft Building, Lewes Road, Brighton BN2 4GJ, UK. Email address for correspondence: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper examines the evolution of an axisymmetric two-dimensional vortex in a steadily rotating strain field and the dynamical interactions that can enhance vortex spreading through resonant behaviour. Starting with a point vortex localized at the origin, the applied strain field generates a cat's eye topology in the co-rotating streamfunction, localized around a radius rext. Now the vortex is allowed to spread viscously: initially rext lies outside the vortex, but as it spreads, vorticity is advected into the cat's eyes, leading to a local flattening of the mean profile of the vortex and so to enhanced mixing and spreading of the vortex. Together with this is a feedback: the response of the vortex to the external strain depends on the modified profile. The feedback is particularly strong when rext coincides with the radius rcat at which the vortex can support cat's eyes of infinitesimal width. There is a particular time at which this occurs, as these radii change with the viscous spread of the vortex: rext moves inwards and rcat outwards. This resonance behaviour leads to increased mixing of vorticity, along with a rapid stretching of vorticity contours and a sharp increase in the amplitude of the non-axisymmetric components. The dynamical feedback and enhanced diffusion are studied for viscously spreading vortices by means of numerical simulations of their time evolution, parameterized only by the Reynolds number R and the dimensionless strength A of the external strain field.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 630 , 10 July 2009 , pp. 155 - 177
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 2001 Disturbing vortices. J. Fluid Mech. 426, 95–133.CrossRefGoogle Scholar
Bassom, A. P. & Gilbert, A. D. 1998 The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109–140.Google Scholar
Briggs, R. J., Daugherty, J. D. & Levy, R. H. 1970 Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fluids 13, 421–432.Google Scholar
Charlton, A. J., O'Neill, A., Lahoz, W. A. & Berrisford, P. 2004 The splitting of the stratospheric vortex in the southern hemisphere, September 2002: dynamical evolution. J. Atmos. Sci. 62, 590–602.CrossRefGoogle Scholar
Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303.Google Scholar
Dritschel, D. G. & Waugh, D. W. 1992 Quantification of the inelastic interaction of unequal vortices in two-dimensional vortex dynamics. Phys. Fluids 4 (8), 1737–1744.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139–161.Google Scholar
Hall, I. M., Bassom, A. P. & Gilbert, A. D. 2003 The effect of fine structure on the stability of planar vortices. Eur. J. Mech. B 22 (2), 179–198.CrossRefGoogle Scholar
Hosokawa, I. & Yamamoto, K. 1989 Fine structure of a directly simulated isotropic turbulence. J. Phys. Soc. Jpn 58, 20–23.CrossRefGoogle Scholar
Jiménez, J., Moffatt, H. K. & Vasco, C. 1996 The structure of the vortices in freely decaying two-dimensional turbulence. J. Fluid Mech. 313, 209–222.Google Scholar
Koh, T.-Y. & Plumb, R. A. 2000 Lobe dynamics applied to barotropic Rossby-wave breaking. Phys. Fluids 12 (6)1518–1528.Google Scholar
Le Dizès, S. 2000 Non-axisymmetric vortices in two-dimensional flows. J. Fluid Mech. 406, 175–198.Google Scholar
Lingevitch, J. F. & Bernoff, A. J. 1995 Distortion and evolution of a localized vortex in an irrotational flow. Phys. Fluids 7 (5), 1015–1026.Google Scholar
Lundgren, T. S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25 (12), 2193–2203.CrossRefGoogle Scholar
Mariotti, A., Legras, B. & Dritschel, D. G. 1994 Vortex stripping and the erosion of coherent structures in two-dimensional flows. Phys. Fluids 6 (12), 3954–3962.CrossRefGoogle Scholar
Moffatt, H. K., Kida, S. & Ohkitani, K. 1994 Stretched vortices: the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241–264.CrossRefGoogle Scholar
Polvani, L. M. & Plumb, R. A. 1992 Rossby wave breaking, microbreaking, filamentation, and secondary vortex formation: the dynamics of a perturbed vortex. J. Atmos. Sci. 49 (6), 462–476.Google Scholar
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31, 55–93.Google Scholar
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.Google Scholar
Schecter, D. A., Dubin, D. H. E., Cass, A. C., Driscoll, C. F., Lansky, I. M., & O'Neil, T. M. 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12 (10), 2397–2412.CrossRefGoogle Scholar
Thuburn, J. & Lagneau, V. 1999 Eulerian mean, contour integral, and finite-amplitude wave activity diagnostics applied to a single-layer model of the winter Stratosphere. J. Atmos. Sci. 56 (5), 689–710.Google Scholar
Turner, M. R., Bassom, A. P. & Gilbert, A. D. 2009 Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices. J. Fluid Mech. Submitted.Google Scholar
Turner, M. R. & Gilbert, A. D. 2007 Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles. J. Fluid Mech. 593, 255–279.Google Scholar
Turner, M. R., Gilbert, A. D. & Bassom, A. P. 2008 Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes. Phys. Fluids 20 (2), 027101-1–027101-10.Google Scholar