Rearrangement theory is about reorganizing a given function (or map) in some specific order (monotonicity, cycle monotonicity etc...). This is somewhat similar to the convection phenomenon in fluid mechanics, where fluid parcels are continuously reorganized in a stabler way (heavy fluid at bottom and light fluid at top). Convection theory is one of the most important piece of geo-sciences, related to weather forecasting, oceanography, volcanism, earthquake etc... In our talk, we make these analogies more precise by analyzing the Navier-Stokes equations with buoyancy and Coriolis forces. We will see how these approximations are related to the concept, well known in optimal transport theory, of rearrangement of maps as gradient of convex functions.