  We develop a quasilinear theory of the 2D Euler equation and derive an
integro-differential equation for the evolution of the coarse-grained
vorticity. This equation respects all the invariance properties of the Euler
equation and conserves angular momentum in a circular domain and linear impulse
in a channel. We show under which hypothesis we can derive a H-theorem for the
Fermi-Dirac entropy and make the connection with statistical theories of 2D
turbulence.
