Skip to main content
    • Aa
    • Aa

Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices


The spreading and diffusion of two-dimensional vortices subject to weak external random strain fields is examined. The response to such a field of given angular frequency depends on the profile of the vortex and can be calculated numerically. An effective diffusivity can be determined as a function of radius and may be used to evolve the profile over a long time scale, using a diffusion equation that is both nonlinear and non-local. This equation, containing an additional smoothing parameter, is simulated starting with a Gaussian vortex. Fine scale steps in the vorticity profile develop at the periphery of the vortex and these form a vorticity staircase. The effective diffusivity is high in the steps where the vorticity gradient is low: between the steps are barriers characterized by low effective diffusivity and high vorticity gradient. The steps then merge before the vorticity is finally swept out and this leaves a vortex with a compact core and a sharp edge. There is also an increase in the effective diffusion within an encircling surf zone.

In order to understand the properties of the evolution of the Gaussian vortex, an asymptotic model first proposed by Balmforth, Llewellyn Smith & Young (J. Fluid Mech., vol. 426, 2001, p. 95) is employed. The model is based on a vorticity distribution that consists of a compact vortex core surrounded by a skirt of relatively weak vorticity. Again simulations show the formation of fine scale vorticity steps within the skirt, followed by merger. The diffusion equation we develop has a tendency to generate vorticity steps on arbitrarily fine scales; these are limited in our numerical simulations by smoothing the effective diffusivity over small spatial scales.

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

G. Boffetta , D. Del Castillo Negrete , C. López , G. Pucacco & A. Vulpiani 2003 Diffusive transport and self-consistent dynamics in coupled maps. Phys. Rev. E 67 (026224), 111.

R. J. Briggs , J. D. Daugherty & R. H. Levy 1970 Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fluids 13, 421432.

D. Del Castillo Negrete 2000 aSelf-consistent dynamics in the single wave model. Physica A 280, 1021.

D. Del Castillo Negrete 2000 bSelf-consistent chaotic transport in fluids and plasmas. Chaos 10, 7588.

D. G. Dritschel & M. E. McIntyre 2008 Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci. 65, 855874.

P. Garaud 2001 Latitudinal shear instability in the solar tachocline. Mon. Not. R. Astron. Soc. 324, 6876.

I. M. Hall , A. P. Bassom & A. D. Gilbert 2003 The effect of fine structure on the stability of planar vortices. Eur. J. Mech. B/Fluids 22, 179198.

P. H. Haynes , D. A. Poet & E. F. Shuckburgh 2007 Transport and mixing in kinematically and dynamically consistent flows. J. Atmos. Phys. 64, 36403651.

M. Heimpel & J. Aurnou 2007 Turbulent convection in rapidly rotating spherical shells: a model for equatorial and high latitude jets on Jupiter and Saturn. Icarus 187, 540557.

A. Mariotti , B. Legras & D. G. Dritschel 1994 Vortex stripping and the erosion of coherent structures in two-dimensional flows. Phys. Fluids 6, 39543962.

M. E. McIntyre & T. N. Palmer 1984 The ‘surf zone’ in the stratosphere. J. Atmos. Terr. Phys. 46, 825849.

W. A. Norton 1994 Breaking Rossby waves in a model stratosphere diagnosed by a vortex-following coordinate system and a technique for advecting material contours. J. Atmos. Sci. 51, 654673.

D. A. Schecter , D. H. E. Dubin , A. C. Cass , C. F. Driscoll , I. M. Lansky & T. M. O'Neil 2000 Inviscid damping of asymmetries on a 2-d vortex. Phys. Fluids 12, 23972412.

R. K. Scott & L. M. Polvani 2007 Forced-dissipative shallow-water turbulence on the sphere and the atmospheric circulation of the giant planets. J. Atmos. Sci. 64, 31583176.

D. W. Waugh & R. A. Plumb 1994 Contour advection with surgery: a technique for investigating fine scale structure in tracer transport. J. Atmos. Sci. 51, 530540.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 131 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th September 2017. This data will be updated every 24 hours.