Buckling of a columnar vortex

Abstract

The instability of a columnar vortex under axial compression is studied analytically. Prediction of the change with time of an initial perturbation in the form of a small-amplitude helical wave on the vortex core is reduced to solution of a system of two linear, second-order ordinary differential equations. For small rates of compression, a closed-form asymptotic solution of these equations is obtained using a two-variable expansion in time. The results indicate that a vortex under compression may bend (or ‘‘buckle’’) or it may remain columnar with an increase in core radius. The choice of which of these possible outcomes will occur is found to depend on the value of a single dimensionless parameter given by the product of the wave number k of the helical perturbation and the radius σ of the vortex core. When kσ is greater than a critical value, found to be approximately 1.77, the perturbation amplitude will decrease with axial compression and no ‘‘buckling’’ will occur. When kσ is less than this critical value, the perturbation amplitude will increase as the vortex axis is compressed and ‘‘buckling’’ will be observed.