  In this article we study the $H^1(du)$-gradient flow for the energy $E[X] =
Q[X]/A[X]$ where $Q[X]$ is the Dirichlet energy of $X$, $A[X]$ is the
signedenclosed area of $X$, and $X:\mathbb{S}\rightarrow\mathbb{R}^2$ is a
$H^1(du)$ map. We prove that solutions with initially positive signed enclosed
area exist eternally, and converge as $t\rightarrow\infty$ to a (possibly
multiply-covered) circle. In this way we recover a parametrised isoperimetric
inequality for $H^1(du)$ maps.
