Wave–vortex interactions and effective mean forces: three basic problems

Abstract

Three examples of wave–vortex interaction are studied, in analytically tractable weak refraction regimes with attention to the mean recoil forces, local and remote, that are associated with refractive changes in wave pseudomomentum fluxes. Wave-induced mean forces of this kind can be persistent, with cumulative effects, even in the absence of wave dissipation. In each example, a single wavetrain propagates past a single vortex. In the first two examples, in a two-dimensional, non-rotating acoustic or shallow-water setting, the focus is on whether or not the wavetrain overlaps the vortex core. In the overlapping case, the recoil has a local contribution given by the Craik–Leibovich force on the vortex core, the vector product of Stokes drift and mean vorticity. (For a quantum vortex this contribution is called the Iordanskii force arising from the Aharonov–Bohm effect on a phonon current.) However, in all except one special limiting case there are additional “remote” contributions, mediated by Stokes-drift-induced return flows that can intersect the vortex core well away from locations where the waves are refracted. The third example is a non-overlapping, remote-recoil-only example in a rapidly rotating frame, in which the waves are deep-water gravity waves and the mean flow obeys shallow-water quasigeostrophic dynamics. Contrary to what might at first be thought, the Ursell “anti-Stokes flow” induced by the rotation – an Eulerian-mean flow tending to cancel the Stokes drift – fails to suppress remote recoil. There are nontrivial open questions about extending these results to regimes of stronger refraction, especially regarding the scope of the “pseudomomentum rule” for the wave-induced recoil forces.

Keywords:

• Waves
• momentum
• pseudomomentum
• Iordanskii force
• Aharonov–Bohm effect

Acknowledgments

Pavel Berloff provided the first stimulus to embark on this study. I thank him and Natalia Berloff, Oliver Bühler, David Dritschel, Victor Kopiev, Stefan Llewellyn Smith, Hayder Salman, Mike Stone, Jim Thomas, Jacques Vanneste, Bill Young, and the referees for the main paper and for the present paper – all six of them – for their interest and for many useful and challenging comments, which led to extensive revisions of both papers.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The curl of this two-dimensional force field $\mathit{F}$ is just that required to move the vortex core leftward through the fluid at velocity ${\mathit{u}}_{\mathrm{t}\mathrm{r}}$, while the divergence of $\mathit{F}$ sets up the dipolar pressure field required to produce the corresponding changes outside the core, where the velocity field is irrotational. Thus defined, $\mathit{F}$ has the dimensions of acceleration, length/(time)${}^2$, i.e. force per unit mass, since it is a forcing term on the right-hand side of the standard momentum equation having $\mathrm{\partial }\mathit{u}/\mathrm{\partial }t$ on the left, whose curl is the standard vorticity equation. So for instance the resultant force on a two-dimensional vortex core of depth H is $\rho H\iint \mathit{F}\mathrm{d}x\mathrm{d}y$ where ρ is fluid density. The factor $\rho H$ will be ignored in what follows. Strictly speaking, therefore, “resultant force” and “impulse” in the main text should be read as ${\rho }^{-1}$ times resultant force and impulse per unit core depth.