The response of Hill's spherical vortex to a small axisymmetric disturbance

H. K. Moffatt, D. W. Moore
1978 Journal of Fluid Mechanics  
The response of a Hill's spherical vortex to an irrotational axisymmetric small perturbation is examined on the assumption that viscous diffusion of vorticity is negligible. The problem of determining the response is first reduced to a system of differential equations for the evolution of the Legendre coefficients of the disturbance stream function. This system is then solved approximately and it is shown that, if the initial disturbance is such as to make the vortex prolate spheroidal, the
more » ... spheroidal, the vortex detrains a fraction $c of its original volume, where e is the fractional extension of the axis of symmetry in the imposed distortion. The detrained fluid forms a thin spike growing from the rear stagnation point. If the vortex is initially oblate, irrotational fluid is entrained a t the rear stagnation point to the interior of the vortex. t During the course of this work, it was drawn to our attention that Bliss (1973) has studied the problem by methods similar to those described here, and has obtained equations equivalent to (2.19) below. He did not attempt an analytical solution of these equations, but drew similar conclusions to ours, from a numerical study, concerning the failure of a truncation technique.
doi:10.1017/s002211207800186x fatcat:yepcgubxmbbebk34bevm6izc5m