Vortex Stretching by a Simple Hyperbolic Saddle

Abstract

We study solutions to the 3D Euler vorticity equation of the form $\omega = \tilde \omega (x,t)\left( {\frac{{\partial t}}{{\partial {x_2}}},\frac{{\partial t}}{{\partial {x_1}}},0} \right)$ in a neighborhood U. When the curvesf (x ₁ x ₂ t) = const are circles then these solutions are the well known axisymmetric 3D flow without swirl, and for this case there is no vortex stretching. If we assumef(x _1,x_2, t) = const to be a set of curves that contain a simple hyperbolic saddle then vortex stretching may take place. We show that the angle of the saddle can not close faster than a double exponential in time and there is no breakdown. Similar results are obtain in two dimensional models.