Scattering of acoustic waves by a superfluid vortex

Abstract

Abstract. The scattering of plane acoustic waves by a vortex in a two-dimensional superfluid is examined for small Mach number M . The solution is developed from a systematic expansion of the governing equations in three separate regions: an inner vortical region which scales as the healing length, an interaction region governed by irrotational hydrodynamics, and an outer wave region. The solutions in the different regions are matched together. The leading-order scattered field occurs at O(M 2 δ) in the wave region, where δ is the small non-dimensional amplitude of the incoming acoustic wave. The far-field behaviour of the wave-region solution shows that a different form of the expansion is required in the forward scatter direction: this corresponds to the expression previously derived for the acoustic Magnus force.

Wave–vortex interactions in the nonlinear Schrödinger equation

“…Here, * denotes complex conjugation. If ϕ = 0, then the mass is still conserved, but the momentum and energy may change as described in (14) below.…”

“…A special conceptual place is therefore reserved for the very simple but powerful superfluid models based on the (defocusing) nonlinear Schrödinger equation (NLS), also known as the Gross-Pitaevskii equation in this field. 13,14 Whatever the context, the NLS equation has a well-known fluiddynamical interpretation via the Madelung transformation, which allows comparing and contrasting classical and quantum fluid dynamics on an even footing. 15 Strictly speaking, the NLS equation can only model the Bose-Einstein condensate, which is closely related to superfluid behaviour 16 at zero temperature, but as a model for this limiting case it is of striking simplicity.…”

Sound scattering by a vortex: case of exponentially decaying velocity

“…Coste, Lund & Umeki (1999) considered the scattering of surface gravity waves by vertical vorticity in shallow water governed by linearised shallow water equations. Smith (2002) studied acoustic wave scattering by a superfluid vortex using the linearised Gross–Pitaevskii equation, whereas Wei et al (2017) investigated electromagnetic waves in a liquid using Maxwell's equations. In the framework of the Born approximation, which takes place for weak vortices, the inhomogeneous wave equation governing the scattered field is equivalent for equivalent incident waves and vortices of different nature if rewritten in corresponding quantities.…”