Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-29T14:44:12.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Islands in three-dimensional steady flows

Published online by Cambridge University Press:  26 April 2006

C. C. Hegna  and
A. Bhattacharjee
C. C. Hegna
Affiliation:
Department of Applied Physics, Columbia University, New York, NY 10027, USA
A. Bhattacharjee
Affiliation:
Department of Applied Physics, Columbia University, New York, NY 10027, USA
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 527 - 542
Copyright
© 1991 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

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, 181–204.Google Scholar
Bhattacharjee, A. 1984 Variational method for toroidal equilibria with imperfect flux surfaces. Plasma Phys. Contr. Fusion 26, 977–990.Google Scholar
Bhattacharjee, A., Wiley, J. C. & Dewar, R. L. 1984 Variational method for three-dimensional toroidal equilibria. Comput. Phys. Commun. 31, 213–225.Google Scholar
Boozer, A. H. 1981 Plasma equilibrium with rational magnetic surfaces. Phys. Fluids 24, 1999–2003.Google Scholar
Boozer, A. H. 1983 Evaluation of the structure of ergodic fields. Phys. Fluids 26, 1288–1290.Google Scholar
Cary, J. R. & Kotschenreuther, M. 1985 Pressure induced islands in three-dimensional toroidal plasmas. Phys. Fluids 28, 1392–1401.Google Scholar
Cary, J. R. & Littlejohn, R. G. 1983 Noncanonical Hamiltonian mechanics and its applications to magnetic field line flow. Ann. Phys. 151, 1–34.Google Scholar
Childress, S. & Soward, A. M. 1989 Scalar transport and alpha-effect for a family of cat's eye flows. J. Fluid Mech. 205, 99–133.Google Scholar
Chirikov, B. V. 1979 A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–401.Google Scholar
Dewar, R. L. 1985 Optimal oscillation-center transformations. Physica 17D, 37–53.Google Scholar
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids 10, 137–154.Google Scholar
Greene, J. M. & Johnson, J. L. 1961 Determination of hyromagnetic equilibria. Phys. Fluids 4, 875–890.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows Stud. Appl. Maths 51, 139–161.Google Scholar
Hegna, C. C. & Bhattacharjee, A. 1989 Magnetic island formation in three-dimensional plasma equilibria.. Phys. Fluids B 1, 392–397.Google Scholar
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. 206–215. Cambridge University Press.
Hirshman, S. P. & Whitson, J. C. 1983 Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Phys. Fluids 26, 3553–3568.Google Scholar
Kerr, R. M. & Hussain, F. 1989 Simulation of vortex reconnection. Physica 37D, 474–484.Google Scholar
Kraichnan, R. H. & Panda, R. 1988 Depression of nonlinearity in decaying isotropic turbulence. Phys. Fluids 31, 2395–2398.Google Scholar
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1, 265–274.Google Scholar
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, 869–877.Google Scholar
Moffatt, H. K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359–378.Google Scholar
Moffatt, H. K. 1986 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations. J. Fluid Mech. 166, 359–378.Google Scholar
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. 469–477. Cambridge University Press.