  We derive a formula for the entropy of two dimensional incompressible
inviscid flow, by determining the volume of the space of vorticity
distributions with fixed values for the moments Q_k= \int_w(x)^k d^2 x. This
space is approximated by a sequence of spaces of finite volume, by using a
regularization of the system that is geometrically natural and connected with
the theory of random matrices. In taking the limit we get a simple formula for
the entropy of a vortex field. We predict vorticity distributions of maximum
entropy with given mean vorticity and enstrophy; also we predict the
cylindrically symmetric vortex field with maximum entropy. This could be an
approximate description of a hurricane.
