a question regarding steady state solutions of the alpha Euler equations

a question regarding steady state solutions of the alpha Euler equations

OK. The $\alpha $ Euler equations are a variant of the Euler equations of
incompressible fluid dynamics and are defined as follows.
The details are available in this paper for example.
http://prl.aps.org/pdf/PRL/v80/i19/p4173_1
(Euler-Poincaré Models of Ideal Fluids with Nonlinear Dispersion by Holm
Marsden and Ratiu)
$(\partial / \partial t + u . \nabla ) v + (\nabla u)^T v + \nabla \pi = 0$
where $v$ is the fluid velocity. $u$ is the filtered velocity given by the
formula $v= (1- \alpha^2 \Delta) u $ using the so called Helmholtz
operator and $\alpha$ is a fixed positive real number.
$\pi$ is the modified pressure given by the formula $\pi = p - 1/2 |u|^2 -
\alpha^2/2 |\nabla u|^2 $. where $p$ is the pressure of the original
fluid.
We assume a 2 dimensional domain to simplify the dynamics.
In vorticity form the equation reads $(\partial /\partial t + u. \nabla )
\omega = 0$ where $\omega = \nabla \times v$ is the vorticity.
We recall that the vorticity form of the Euler equation is given by
$(\partial /\partial t + v. \nabla ) \omega = 0$
i.e the only difference between the Euler equation and the alpha Euler is
that there is a much smoother velocity in the vorticity form of the
equation.
My question is as follows. Assume a steady state of the Euler equations of
the form $v= (v_0,0)$, $\omega_0$ and $p=0$ (a shear flow profile for
example). assume everything is a function purely of y for example (in a
two dimensional domain with coordinates x and y)
Compute $u_0 $ using the formula above. it is clear that this is a steady
state solution of the vorticity form of the equation but somehow doesn't
seem to satisfy the original fluid equations.
This seems to me a paradox unless the pressure is redefined of the form
$\pi = p - 1/2 |u|^2 + \alpha^2/2 |\nabla u|^2 $ i.e a plus instead of a
minus.