  We describe first integrals of geostrophic equations, which are similar to
the enstrophy invariants of the Euler equation for an ideal incompressible
fluid. We explain the geometry behind this similarity, give several equivalent
definitions of the Poisson structure on the space of smooth densities on a
symplectic manifold, and show how it can be obtained via the Hamiltonian
reduction from a symplectic structure on the diffeomorphism group.
