The RNA world scenario posits replication by RNA polymerases. On early Earth, a geophysical setting is required to separate hybridized strands after their replication and to localize them against diffusion. We present a pointed heat source that drives exponential, RNA-catalyzed amplification of short RNA with high efficiency in a confined chamber. While shorter strands were periodically melted by laminar convection, the temperature gradient caused aggregated polymerase molecules to accumulate, protecting them from degradation in hot regions of the chamber. These findings demonstrate a size-selective pathway for autonomous RNA-based replication in natural nonequilibrium conditions.