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The degree of knottedness of tangled vortex lines

Published online by Cambridge University Press:  28 March 2006

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

Abstract

Let u(x) be the velocity field in a fluid of infinite extent due to a vorticity distribution w(x) which is zero except in two closed vortex filaments of strengths K1, K2. It is first shown that the integral \[ I=\int{\bf u}.{\boldmath \omega}\,dV \] is equal to αK1K2 where α is an integer representing the degree of linkage of the two filaments; α = 0 if they are unlinked, ± 1 if they are singly linked. The invariance of I for a continuous localized vorticity distribution is then established for barotropic inviscid flow under conservative body forces. The result is interpreted in terms of the conservation of linkages of vortex lines which move with the fluid.

Some examples of steady flows for which I ± 0 are briefly described; in particular, attention is drawn to a family of spherical vortices with swirl (which is closely analogous to a known family of solutions of the equations of magnetostatics); the vortex lines of these flows are both knotted and linked.

Two related magnetohydrodynamic invariants discovered by Woltjer (1958a, b) are discussed in ±5.

Type
Research Article
Copyright
© 1969 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Chandrasekhar, S. 1956 Proc. Nat. Acad. Sci. U.S.A. 42, 15.
Crowell, R. H. & Fox, R. H. 1964 Introduction to Knot Theory. Ginn.
Gauss 1833 Werke. Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1877, 5, 605.
Prendergast, K. 1957 Astrophys. J. 123, 498.
Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics. Longmans.
Shercliff, J. A. 1965 A Textbook of Magnetohydrodynamics. Oxford: Pergamon.
Tait, P. G. 1898 Scientific Papers I. Cambridge University Press.
Thomson, W. (Lord Kelvin) 1868 Trans. Roy. Soc. Edin. 25, 217260.
Woltjer, L. 1958a Proc. Nat. Acad. Sci. U.S.A. 44, 489491.
Woltjer, L. 1958b Proc. Nat. Acad. Sci. U.S.A. 44, 833841.
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