Fast Escape in Incompressible Vector Fields

Swimmers caught in a rip current flowing away from the shore are advised to swim orthogonally to the current to escape it. We describe a mathematical principle in a similar spirit. More precisely, we consider flows $\gamma$ in the plane induced by incompressible vector fields $\textbf{v}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ satisfying $c_1 < \|v\| < c_2.$ The length $\ell$ a flow curve $\dot \gamma(t) = \textbf{v}(\gamma(t))$ until $\gamma$ leaves a disk of radius 1 centered at the initial position can be as long as $\ell \sim c_2/c_1$. The same is true for the orthogonal flow $\textbf{v}^{\perp} = (-\textbf{v}_2, \textbf{v}_1)$. We show that a combination does strictly better: there always exists a curve flowing first along $\textbf{v}^{\perp}$ and then along $\textbf{v}$ which escapes the unit disk before reaching the length $\sqrt{4\pi c_2 / c_1}$. Moreover, if the escape length of $\textbf{v}$ is uniformly $\sim c_2/c_1$, then the escape length of $\textbf{v}^{\perp}$ is uniformly $\sim 1$ (allowing for a fast escape from the current). We also prove an elementary quantitative Poincar\'{e}-Bendixson theorem that seems to be new.