  In this article we consider the Euler-$\alpha$ system as a regularization of
the incompressible Euler equations in a smooth, two-dimensional, bounded
domain. For the limiting Euler system we consider the usual non-penetration
boundary condition, while, for the Euler-$\alpha$ regularization, we use
velocity vanishing at the boundary. We also assume that the initial velocities
for the Euler-$\alpha$ system approximate, in a suitable sense, as the
regularization parameter $\alpha \to 0$, the initial velocity for the limiting
Euler system. For small values of $\alpha$, this situation leads to a boundary
layer, which is the main concern of this work. Our main result is that, under
appropriate regularity assumptions, and despite the presence of this boundary
layer, the solutions of the Euler-$\alpha$ system converge, as $\alpha \to 0$,
to the corresponding solution of the Euler equations, in $L^2$ in space,
uniformly in time. We also present an example involving parallel flows, in
order to illustrate the indifference to the boundary layer of the $\alpha \to
0$ limit, which underlies our work.
