Rivers empty into oceans and lakes as turbulent sediment-laden jets, which can be characterized by a Gaussian horizontal velocity profile that spreads and decays downstream because of shearing and lateral mixing at the jet margins. Recent experiments demonstrate that this velocity field controls river-mouth sedimentation patterns. In nature, diffuse jets are associated with mouth bar deposition forming bifurcating distributary networks, while focused jets are associated with levee deposition and the growth of elongate channels that do not bifurcate. River outflows from elongate channels are similar in structure to cold filaments observed in ocean currents, where high potential vorticity helps to preserve coherent structure over large distances. Motivated by these observations, we propose a hydrodynamic theory that seeks to predict the conditions under which elongate channels form. Our approach models jet velocity patterns using the flow vorticity. Both shearing and lateral spreading are directly related to the vertical component of vorticity. We introduce a new kind of potential vorticity that incorporates sediment concentration and thus allows study of jet sedimentation patterns. The potential vorticity equation reduces the number of fluid momentum equations to one without losing generality. This results in a compact analytical solution capable of describing the streamwise evolution of the potential vorticity of a sediment-laden jet from initial conditions at the river mouth. Our theory predicts that high potential vorticity is a necessary condition for focused levee deposition and the creation of elongate channels. Comparison to numerical, laboratory, and field studies indicates that potential vorticity is a primary control on channel morphology. Our results may be useful for designing river delta restoration schemes such as the proposed Mississippi Delta diversion.