We study a possibly integrable model of abelian gauge fields on a
two-dimensional surface M, with volume form mu. It has the same phase space as
ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism
group of M, SDiff(M,mu). Gauge field Poisson brackets differ from the
Heisenberg algebra, but are reminiscent of Yang-Mills theory on a null surface.
Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant
observables. Some symplectic leaves of the Poisson manifold are identified. The
Hamiltonian is a magnetic energy, similar to that of electrodynamics, and
depends on a metric whose volume element is not a multiple of mu. The magnetic
field evolves by a quadratically non-linear `Euler' equation, which may also be
regarded as describing geodesic flow on SDiff(M,mu). Static solutions are
obtained. For uniform mu, an infinite sequence of local conserved charges
beginning with the hamiltonian are found. The charges are shown to be in
involution, suggesting integrability. Besides being a theory of a novel kind of
ideal flow, this is a toy-model for Yang-Mills theory and matrix field
theories, whose gauge-invariant phase space is conjectured to be a coadjoint
orbit of the diffeomorphism group of a non-commutative space.