  Using a two-dimensional fluid description, we investigate the nonlinear
radial-longitudinal dynamics of intense beams in storage rings and cyclotrons.
With a multiscale analysis separating the time scale associated with the
betatron motion and the slower time scale associated with space-charge effects,
we show that the longitudinal-radial vortex motion can be understood in the
frame moving with the charged beam as the nonlinear advection of the beam by
the $\mathbf{E}\times\mathbf{B}$ velocity field, where $\mathbf{E}$ is the
electric field due to the space charge and $\mathbf{B}$ is the external
magnetic field. This interpretation provides simple explanations for the
stability of round beams and for the development of spiral halos in elongated
beams. By numerically solving the nonlinear advection equation for the beam
density, we find that it is also in quantitative agreement with results
obtained in PIC simulations.
