  We consider a family of approximations to the Euler equations obtained by
adding $(-\Delta)^{-\alpha/2}$ to the non-locality in the Biot-Savart kernel
together with a mollification (with parameter $\varepsilon$). We consider the
evolution of a thin vortex tube. We show that the velocity on the filament
(core of the tube) in the limit as $\varepsilon\to 0$ is given
$\frac{C(\alpha,t)}{\alpha} \kappa B + \mathcal O(1)$ where $\kappa$ and $B$
are the curvature and binormal of the curve, and $C$, $C^{-1}$ are uniformly
bounded.
