The corotating hollow vortex pair: steady merger and break-up via a topological singularity

Abstract

Abstract

‘H-states’: exact solutions for a rotating hollow vortex

“…A hollow vortex is usually defined to be a finite-area constant pressure region having a non-zero circulation around it (Michell 1890; Baker, Saffman & Sheffield 1976; Saffman 1992). Although Pocklington (1895) solved the cotravelling (opposite-signed) hollow-vortex pair problem in the 19th century, the basic problem of the corotating (like-signed) hollow-vortex pair has only recently been treated by Nelson, Krishnamurthy & Crowdy (2020). Indeed, there has been a resurgence of interest in the hollow-vortex model.…”

“…Nelson et al. (2020) instead devised a numerical method tailored to account for the doubly connected nature of the fluid exterior to the two vortices. As the angular velocity increases, each vortex is found to extend a thin finger towards the centre of rotation until the vortices almost touch; this sequence is shown from left to right in figure 1.…”

“…Nelson et al. (2020) adapted their numerical scheme to compute the shape of a single 2-fold rotationally symmetric rotating hollow vortex. In that case, a thin waist forms in the vortex shape which eventually collapses; this sequence is shown from right to left in figure 1.…”

“…Nelson et al. (2020) argue this to be evidence of a topological singularity since the same state is approached from two topologically distinct directions but with no blow-up of any physical quantities.

Figure 1.Shapes of two steadily rotating hollow vortices, in the corotating frame, as computed numerically by Nelson et al.

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“…Shapes of two steadily rotating hollow vortices, in the corotating frame, as computed numerically by Nelson et al. (2020). The present paper shows that the single rotating hollow vortices in the three right-most images can in fact be described in analytical form: they are the case of a class of -fold-symmetric ‘H-states’.…”