A particle diffusing around a point of stable mechanical equilibrium in a static but non-conservative force field enters into a steady state characterized by circulation in the probability flux. Circulation in such a Brownian vortex is not simply a deterministic response to the solenoidal component of the force, but rather reflects an interplay between force-driven probability currents and diffusion. As an example of this previously unrecognized class of stochastic heat engines, we consider a colloidal sphere stably trapped in a conventional optical tweezer. Rather than coming into thermodynamic equilibrium with the surrounding heat bath, the particle's Brownian fluctuations are biased into a toroidal roll. We demonstrate both theoretically and experimentally that the circulation in this practical realization of the Brownian vortex can undergo flux reversal.