On Vortex Tube Stretching and Instabilities in an Inviscid Fluid

We study instabilities that are present in two models that retain some of the dynamics of vortex tube stretching in the motion of a fluid in 3 dimensions. Both models are governed by a 2-dimensional PDE and are hence more tractable than the full 3-dimensional Euler equations. The first model is the so called surface quasi-geostrophic equation. The second model is a class of 3-dimensional flows that are invariant with respect to one spatial coordinate. Both models are constructed in the context of a rapidly rotating fluid. Instabilities due to an effect analogous to vortex tube stretching are detected: these instabilities are in the linearised equations in the first model and in the nonlinear equations in the second model. Such instabilities are absent, or weaker, in strictly 2-dimensional fluid motion.