A reduced gravity model which includes variable density is investigated analytically and numerically. A procedure is established to generate families of solutions with the same flow structure of any nondivergent solution of the classical reduced gravity model. Inhomogeneous rodons are then constructed as explicit examples of exact solutions on the f plane. A formal stability [Holm et al., 1985] theorem, valid for the case of circular inhomogeneous rodons, shows that for the vortex to be unstable the density must increase outward somewhere. A particle-in-cell numerical method is used to simulate the behavior of inhomogeneous rodons and circular vortices with realistic decay of velocity toward the edge of the buoyancy front. The results provide information on the long-term evolution of unstable vortices. The unstable vortices show that the limited resolution available is never enough to resolve the convolution of isopycnals due to adiabatic stirring. This result is seen in the numerical model and interpreted physically as a homogenization process of the vortex interior.