Sub-seafloor hydrothermal convection at mid-ocean ridges transfers 25% of the Earth's heat flux and can form massive sulfide ore deposits. Their three-dimensional (3D) structure and transient dynamics are uncertain. Using 3D numerical simulations, we demonstrated that convection cells self-organize into pipelike upflow zones surrounded by narrow zones of focused and relatively warm downflow. This configuration ensures optimal heat transfer and efficient metal leaching for ore-deposit formation. Simulated fluid-residence times are as short as 3 years. The concentric flow geometry results from nonlinearities in fluid properties, and this may influence the behavior of other fluid-flow systems in Earth's crust.