The Prandtl-Batchelor theorem for high Reynolds number flow of a homogeneous fluid is generalized to laminar flows of vorticies moving in stratified fluids. Density is homogenized in the vortices as well as the vorticity is. We pay attention to the mass flux by means of the basic equations governing the density distribution with/without diffusions. The theorem is also applicable to a flow with a passive scalar contaminant.