Europa and Enceladus have liquid water oceans beneath ice shells. The heat comes from below — tidal dissipation warms the rocky core, and this heat must reach the ice-water interface to maintain the ocean. How it gets there determines whether the ocean is well-mixed or layered, which determines whether any chemistry at the bottom could reach the surface.
Wang, Kang, and Li (arXiv:2603.17185) identify a tug-of-war between two transport mechanisms. Baroclinic eddies — horizontal swirling motions driven by temperature differences between equator and poles — tend to flatten temperature gradients laterally while preserving vertical stratification. Convective plumes — vertical columns driven by bottom heating — punch through stratification and mix the ocean top to bottom. The winner depends on the ratio of heating strengths.
The scaling law is sharp: a stratified layer persists near the ice surface as long as the vertical heating parameter stays below the horizontal parameter raised to the 5/2 power. Below this threshold, eddies redirect bottom heat sideways before it can rise, trapping a cold stable layer at the top. Above the threshold, convective plumes break through.
This matters for habitability. A stratified ocean sequesters nutrients and energy in separate layers — the chemical ingredients from hydrothermal vents at the bottom never reach the ice-water interface where they might feed a biosphere or be sampled by a spacecraft. A well-mixed ocean delivers them throughout the water column.
The question of life on icy moons reduces, in part, to a scaling exponent. Whether bottom heat reaches the top depends not on the heat itself but on whether it exceeds a threshold set by the planet's rotation and size. The ocean knows only one number, and that number decides whether it mixes.