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Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals

Published online by Cambridge University Press:  20 April 2006

H. K. Moffatt
H. K. Moffatt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge
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Abstract

The well-known analogy between the Euler equations for steady flow of an inviscid incompressible fluid and the equations of magnetostatic equilibrium in a perfectly conducting fluid is exploited in a discussion of the existence and structure of solutions to both problems that have arbitrarily prescribed topology. A method of magnetic relaxation which conserves the magnetic-field topology is used to demonstrate the existence of magnetostatic equilibria in a domain [Dscr ] that are topologically accessible from a given field B0(x) and hence the existence of analogous steady Euler flows. The magnetostatic equilibria generally contain tangential discontinuities (i.e. current sheets) distributed in some way in the domain, even although the initial field B0(x) may be infinitely differentiable, and particular attention is paid to the manner in which these current sheets can arise. The corresponding Euler flow contains vortex sheets which must be located on streamsurfaces in regions where such surfaces exist. The magnetostatic equilibria are in general stable, and the analogous Euler flows are (probably) in general unstable.

The structure of these unstable Euler flows (regarded as fixed points in the function space in which solutions of the unsteady Euler equations evolve) may have some bearing on the problem of the spatial structure of turbulent flow. It is shown that the Euler flow contains blobs of maximal helicity (positive or negative) which may be interpreted as ‘coherent structures’, separated by regular surfaces on which vortex sheets, the site of strong viscous dissipation, may be located.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 159 , October 1985 , pp. 359 - 378
Copyright
© 1985 Cambridge University Press

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References

Arnol'D, V. 1965 Sur la topologie des écoulements stationnaires des fluids parfaits. C. R. Acad. Sci. Paris 261, 17–20.Google Scholar
Arnol'D, V. 1966 Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier Grenoble 16, 319–361.Google Scholar
Arnol'D, V. 1974 The asymptotic Hopf invariant and its applications (in Russian). In Proc. Summer School in Differential Equations, Erevan. Armenian SSR Acad. Sci.
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1985 Chaotic streamlines and Lagrangian turbulence; the ABC flows. Preprint.
Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963 Finite-resistivity instabilities of a sheet pinch. Phys. Fluids 6, 459–484.Google Scholar
Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris 262, 312–314.Google Scholar
Levich, E. & Tsinober, A. 1983 On the role of helical structures in three-dimensional turbulent flow. Phys. Lett. 93 A, 293–297.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.Google Scholar
Moffatt, H. K. 1977 Six lectures on general fluid dynamics and two on hydromagnetic dynamo theory. In Fluid Dynamics (Les Houches lectures 1973) (ed. R. Balian & J.-L. Peube), pp. 151–233. Gordon & Breach.
Moffatt, H. K. 1984 Simple topological aspects of turbulent vorticity dynamics. In Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), pp. 223–230. Elsevier.
Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics, §3.2. Longmans.
Sweet, P. A. 1969 Mechanisms of solar flares. Ann. Rev. Astron. Astrophys. 7, 149–176.Google Scholar
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139–1141.Google Scholar
Tsinober, A. & Levich, E. 1983 On the helical nature of three-dimensional coherent structures in turbulent flows. Phys. Lett. 99 A, 321–324.Google Scholar
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natn. Acad. Sci. 44, 489–491.Google Scholar