Skip to main content Accessibility help
×
Home

Islands in three-dimensional steady flows

  • C. C. Hegna (a1) and A. Bhattacharjee (a1)

Abstract

We consider the problem of steady Euler flows in a torus. We show that in the absence of a direction of symmetry the solution for the vorticity contains δ-function singularities at the rational surfaces of the torus. We study the effect of a small but finite viscosity on these singularities. The solutions near a rational surface contain cat's eyes or islands, well known in the classical theory of critical layers. When the islands are small, their widths can be computed by a boundary-layer analysis. We show that the islands at neighbouring rational surfaces generally overlap. Thus, steady toroidal flows exhibit a tendency towards Beltramization.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Islands in three-dimensional steady flows
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Islands in three-dimensional steady flows
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Islands in three-dimensional steady flows
      Available formats
      ×

Copyright

References

Hide All
Arnol'D, V 1974 The asymptotic Hopf invariant and its applications In Proc. Summer School in Differential Equations, Erevan. Armenian SSR Acad. Sci.
Bauer, F., Betancourt, F. & Garabedian, P. 1978 A Computational Method in Plasma Physics. Springer.
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181204.
Bhattacharjee, A. 1984 Variational method for toroidal equilibria with imperfect flux surfaces. Plasma Phys. Contr. Fusion 26, 977990.
Bhattacharjee, A., Wiley, J. C. & Dewar, R. L. 1984 Variational method for three-dimensional toroidal equilibria. Comput. Phys. Commun. 31, 213225.
Boozer, A. H. 1981 Plasma equilibrium with rational magnetic surfaces. Phys. Fluids 24, 19992003.
Boozer, A. H. 1983 Evaluation of the structure of ergodic fields. Phys. Fluids 26, 12881290.
Cary, J. R. & Kotschenreuther, M. 1985 Pressure induced islands in three-dimensional toroidal plasmas. Phys. Fluids 28, 13921401.
Cary, J. R. & Littlejohn, R. G. 1983 Noncanonical Hamiltonian mechanics and its applications to magnetic field line flow. Ann. Phys. 151, 134.
Childress, S. & Soward, A. M. 1989 Scalar transport and alpha-effect for a family of cat's eye flows. J. Fluid Mech. 205, 99133.
Chirikov, B. V. 1979 A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263401.
Dewar, R. L. 1985 Optimal oscillation-center transformations. Physica 17D, 3753.
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids 10, 137154.
Greene, J. M. & Johnson, J. L. 1961 Determination of hyromagnetic equilibria. Phys. Fluids 4, 875890.
Haberman, R. 1972 Critical layers in parallel flows Stud. Appl. Maths 51, 139161.
Hegna, C. C. & Bhattacharjee, A. 1989 Magnetic island formation in three-dimensional plasma equilibria.. Phys. Fluids B 1, 392397.
Hegna, C. C. & Bhattacharjee, A. 1990 Island formation in magnetostatics and Euler flows. In Proc. IUTAM Symp. on Fluid Mechanics, Cambridge, 13–18 August, 1989 (ed. H. K. Moffatt & A. Tsinober), pp. 206215. Cambridge University Press.
Hirshman, S. P. & Whitson, J. C. 1983 Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Phys. Fluids 26, 35533568.
Kerr, R. M. & Hussain, F. 1989 Simulation of vortex reconnection. Physica 37D, 474484.
Kraichnan, R. H. & Panda, R. 1988 Depression of nonlinearity in decaying isotropic turbulence. Phys. Fluids 31, 23952398.
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1, 265274.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergammon.
Lao, L. L., Greene, J. M., Wang, T. S., Helton, F. J. & Zawadzki, E. M. 1985 Three-dimensional toroidal equilibria and stability by a variational spectral method. Phys. Fluids 28, 869877.
Moffatt, H. K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359378.
Moffatt, H. K. 1986 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations. J. Fluid Mech. 166, 359378.
Moffatt, H. K. 1988 On the existence, structure and stability of MHD equilibrium states In Proc. Cargese Workshop on Turbulence and Nonlinear Dynamics in MHD flows.
Pumir, A. & Siggia, E. D. 1990 Collapsing solutions in the 3-D euler equations. In Proc. IUTAM Symp. on Fluid Mechanics, Cambridge, 13–18 August, 1989 (ed. H. K. Moffatt & A. Tsinober), pp. 469477. Cambridge University Press.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Islands in three-dimensional steady flows

  • C. C. Hegna (a1) and A. Bhattacharjee (a1)

Metrics

Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed