Exact free surfaces in constant vorticity flows

Solitary waves on constant vorticity flows with an interior stagnation point

“…The simplest case of flows with critical levels is that of constant negative vorticity, and there are several advantages of studying this case. Flows with constant vorticity are more easily tractable mathematically (see Ehrnström 2008;Wahlén 2009;Constantin, Strauss & Vȃrvȃrucȃ 2016;Hur & Wheeler 2020). Moreover, these flows are of substantial practical importance being pertinent to a wide range of hydrodynamic phenomena (see Constantin et al 2016, p. 196); an important example are currents producing shear near the sea bed.…”

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

“…2013) also obviated the need for Jacobi elliptic functions in describing Pocklington's cotravelling hollow-vortex pair (Pocklington 1895) and facilitated the calculation of its linear stability properties. The novel prime function approach to Kinnersley's solutions in Crowdy (1999 b ) might similarly uncover new solutions for water waves with vorticity on fluid sheets thereby generalizing the results of Hur & Wheeler (2020).…”

“…In view of these analytical connections (Crowdy 2000) between Crapper's capillary waves and hollow vortices with surface tension (Crowdy 1999 a ; Wegmann & Crowdy 2000), and since the same mappings (3.1) used in the latter problem also solve the H-state problem, it is natural to ask if the new H-state results in this ‘radial geometry’ might produce analogous exact solutions to some problem in Crapper's periodic water wave geometry. It turns out that such solutions have very recently been discovered by Hur & Wheeler (2020), whose work was motivated by a string of other recent contributions (Hur & Dyachenko 2019 a , b ; Hur & Vanden-Broeck 2020) where it was noticed that Crapper's capillary wave profiles were emerging in numerical simulations of rotational water waves. A similar thing happened here: noticing that the critical shape from Nelson et al.…”

“…On making the identifications with related to via (3.1) it can be shown that (4.1) is retrieved (to within unimportant additive constants) as . That is, the limit of the H-states reproduces the water wave solutions of Hur & Wheeler (2020).…”

“…Actually, the surprising reappearance of Crapper's profiles in a problem different from the original problem of capillary water waves was noticed earlier by Crowdy & Roenby (2014) who found exact solutions, given by Crapper's profiles (4.1), for steady water waves with vorticity: in their problem, the vorticity in each period window is concentrated in a submerged cotravelling point vortex. In view of the recent results of Hur & Wheeler (2020) the potential theoretic concept of ‘balayage’ (Shapiro 1992) comes to mind: one imagines that the uniform vorticity in each period window in the Hur & Wheeler (2020) solutions is ‘swept’ into a single point vortex to give the solutions of Crowdy & Roenby (2014) and without changing the wave profile.…”