Starting from the collisionless Vlasov equation, we derive two simple coupled two-dimensional partial differential equations describing the radial-longitudinal beam vortex motion associated with space charge effects in isochronous cyclotrons. These equations show that the vortex motion can be intuitively understood as the nonlinear advection of the beam by the E × B velocity field, where E is the electric field due to the space charge and B is the applied magnetic field. The partial differential equations are also formally identical to the two-dimensional Euler equations for a fluid of uniform density. From this analogy, we explain why elongated beams develop spiral halos and a stable round core while round beams are always stable. Solving the coupled equations numerically, we find good agreement between our model and Particle-In-Cell simulations.