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On the existence of localized rotational disturbances which propagate without change of structure in an inviscid fluid

Published online by Cambridge University Press:  21 April 2006

H. K. Moffatt
H. K. Moffatt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

A wide class of solutions of the steady Euler equations, representing localized rotational disturbances imbedded in a uniform stream U0 is inferred by considering the process of magnetic relaxation to analogous magnetostatic equilibria. These solutions, which may be regarded as generalizations of vortex rings, are characterized by their streamline topology, distinct topologies giving rise to distinct solutions.

Particular attention is paid to the class of axisymmetric solutions described by Stokes stream function ψ(s, z). It is argued that the appropriate topological ‘invariant’ characterizing the flow is the function Vψ representing the volume inside toroidal surfaces ψ = const, in the region of closed streamlines where ψ > 0. This function is described as the ‘signature’ of the flow, and it is shown that in a certain sense, flows with different signatures are topologically distinct. The approach yields a method by which flows of arbitrary signature V(ψ) may in principle be found, and the corresponding vorticity ωφ = sFψ calculated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 289 - 302
Copyright
© 1986 Cambridge University Press

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References

Arnol'D, V.1974 The asymptotic Hopf invariant and its applications. In Proc. Summer School in Differential Equations, Armenian SSR Acad. Sci. (in Russian).
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29–62.Google Scholar
Fraenkel, L. E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135.Google Scholar
Friedman, A. & Turkington, B. 1981 Vortex rings: existence and asymptotic estimates. Trans. Am. Math. Soc. 268, 1–37.Google Scholar
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213–245.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. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359–378.Google Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417–431.Google Scholar
Novikov, E. A. 1984 Generalised dynamics of three dimensional vortex singularities. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), pp. 259–263. North-Holland.
Saffman, P. G. 1961 The final stage of decay of a localized disturbance in a conducting fluid in a uniform magnetic field. Q. J. Mech. Appl. Maths 14, 20.Google Scholar