Magnus and Iordanskii Forces in Superfluids

TL;DR

This paper presents a simple thermodynamic argument to derive the superfluid transverse force on a vortex, clarifying the contributions from superfluid and vortex velocities and addressing longstanding theoretical debates.

Contribution

It offers a robust, macroscopic thermodynamic derivation of the superfluid transverse force, reconciling different theoretical results and challenging the Iordanskii force hypothesis.

Findings

Derived the superfluid velocity component of the transverse force as $- ho_s {oldsymbol{ abla}} imes {v}_s$

Showed that the vortex velocity component of the transverse force is $ ho_s {oldsymbol{ abla}} imes {v}_V$

Concluded that there is no transverse force proportional to the normal fluid velocity, conflicting with Iordanskii's theory.

Abstract

The total transverse force acting on a quantized vortex in a superfluid is a problem that has eluded a complete understanding for more than three decades. In this letter I propose a remarkably simple argument, somewhat reminiscent of Laughlin's beautiful argument for the quantization of conductance in the quantum Hall effect, to define the superfluid velocity part of the transverse force. This term is found to be $- \rho_s {\kappa}_s \times {v}_s$. Although this result does not seem to be overly controversial, this thermodynamic argument based only on macroscopic properties of the superfluid does offer a robust derivation. A recent publication by Thouless, Ao and Niu has demonstrated that the vortex velocity part of the transverse force in a homogeneous neutral superfluid is given by the usual form $\rho_s {\kappa}_s \times {v}_V$. A combination of these two independent results and the required Galilean invariance yields that there cannot be any transverse force proportional to the normal fluid velocity, in apparent conflict with Iordanskii's theory of the transverse force due to phonon scattering by the vortex.