Vortex Dynamics: The Legacy of Helmholtz and Kelvin

Abstract

The year 2007 will mark the centenary of the death of William Thomson (Lord Kelvin), one of the great nineteenth-century pioneers of vortex dynamics. Kelvin was inspired by Hermann von Helmholtz’s [7] famous paper “Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen”, translated by P. G. Tait and published in English [17] under the title “On Integrals of the Hydrodynamical Equations, which Express Vortex-motion”. Kelvin conceived his “Vortex theory of Atoms” (1867–1875) on the basis that, since vortex lines are frozen in the flow of an ideal fluid, their topology should be invariant. We now know that this invariance is encapsulated in the conservation of helicity in suitably defined Lagrangian fluid subdomains. Kelvin’s efforts were thwarted by the realisation that all but the very simplest three-dimensional vortex structures are dynamically unstable, and his vortex theory of atoms perished in consequence before the dawn of the twentieth century. The course of scientific history might have been very different if Kelvin had formulated his theory in terms of magnetic flux tubes in a perfectly conducting fluid, instead of vortex tubes in an ideal fluid; for in this case, stable knotted structures, of just the kind that Kelvin envisaged, do exist, and their spectrum of characteristic frequencies can be readily defined. This introductory lecture will review some aspects of these seminal contributions of Helmholtz and Kelvin, in the light of current knowledge.