Abstract
A popular model for a generic fat-cored vortex ring or eddy is Hill’s spherical vortex (Phil. Trans. R. Soc. A, vol. 185, 1894, pp. 213–245). This well-known solution of the Euler equations may be considered a special case of the doubly infinite family of swirling spherical vortices identified by Moffatt (J. Fluid Mech., vol. 35 (1), 1969, pp. 117–129). Here we find exact solutions for such spherical vortices propagating steadily along the axis of a rotating ideal fluid. The boundary of the spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and hence carry fluid along with the spherical vortex. As the rotation rate is further increased, further concentric ‘sibling’ vortex rings are formed.
Highlights
In 1894 Hill published his famous solution for the steady flow of a spherical vortex in an ideal fluid (Hill 1894)
We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and carry fluid along with the spherical vortex
Hill’s spherical vortex may be viewed as a special non-swirling member of a doubly-infinite family of swirling spherical vortices identified by Moffatt (1969) that may be matched onto an oncoming irrotational stream
Summary
In 1894 Hill published his famous solution for the steady flow of a spherical vortex in an ideal fluid (Hill 1894). Hill’s solution was later shown to be the end member of a family of steadily propagating vortex rings (Norbury 1973) of varying core thickness that includes ‘thin-cored’ rings (see Fraenkel 1970, 1972) where the rotational fluid is confined within a narrow region that does not extend to the axis. The inner solution, given by either Hill (1894) or Moffatt (1969), is shown in grey This inner solution for the spherical vortex is matched onto a modified Taylor (1922) solution in the outer region by enforcing continuity of velocity and pressure across the boundary.
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