A Topological Defect Model of Superfluid Vortices

This paper introduces a nonlinear Schrr odinger model for superruid that captures the process of mutual friction between the superruid and normal uid components of helium II. Superruid vortices are identiied as topological defects in the solution of this equation. A matched asymptotic analysis of Neu 7] is adapted to derive an asymptotic dynamics for the vortices in the case they are widely separated compared with their core size. This motion agrees with the classical Hall and Vinen motion 5] in which phenomenological drag terms are added, ad hoc, to the motion of vortices in an inviscid uid. Several simple examples are considered to illustrate the unique character of the motion of superruid vortices. Finally, the motion of vortices in uniformly rotating helium II is considered, and a continuum approximation to their dynamics is obtained in the case of very many vortices. 1. Introduction When liquid helium is cooled to a temperature close to absolute zero it undergoes a phase transition from helium I to helium II. In the 1930's and 1940's it was discovered that helium II possesses some remarkable properties, collectively known as superru-idity. To account for these properties, Landau introduced the two-uid theory which conceives of helium II as being composed of two uids: a normal uid with velocity