Stability of Vortices

Stability of vortex rings

Moore and Saffman (1975) and Tsai and Widnall (1976) discovered that a vortex tube subjected to a strain field in a plane perpendicular to its axis is unstable. This is Moore-Saffman-Tsai-Widnall (MSTW) instability. The strain field is self-induced as a quadrupole field and it feeds resonance between two Kelvin waves with azimuthal wavenumbers separated by 2. For curved vortex tubes, e.g. vortex rings, the curvature comes in as a dipole field. The analysis by Hattori and Fukumoto (2003) found that the resonance between two Kelvin waves m and m+1 fed by the dipole field is more unstable. Theoretical calculations for the effects of density and surface tension on these instabilities can be found in [4] and [5].

Stability of Kida's equilibrium solution

Kida (1981) found a family of vortex filaments preserving their shape while being evolved by local induction approximation in inviscid Euler flows. The linear stability of these knotted and unknotted filaments is examined. This is an ongoing project.

Asymmetric perturbations to Hill's spherical vortex

Hill's spherical vortex (HSV) is a classical solution to inviscid Euler equations. It is showed that HSV is the end member of a one-parameter family of vortex rings (Norbury 1973). I examined the stability of HSV when subjected to inviscid perturbations with azimuthal modes. The linearized perturbation equations was derived and further reduced to a set of coupled ODEs with its eigenvalue being the growth rate. This project was carried out during summer in 2015.