document: The need to represent scale interactions in weather and climate prediction models has, for many decades, motivated research into the use of adaptive meshes [3,34,38]. R-adaptivity – mesh redistribution – involves deforming a mesh in order to vary local resolution and was first considered for atmospheric modelling more than twenty years ago by Dietachmayer and Droegemeier [14]. It is an attractive form of adaptivity since it does not involve altering the mesh connectivity, does not create load balancing problems because points are never created or destroyed, does not require mapping of solutions between meshes [26], does not lead to sudden changes in resolution and can be retro-fitted into existing models. Variational methods exist which attempt to control resolution in different directions for r-adaptive meshes (e.g. [23,25]). Alternatively, the solution of the Monge–Ampère equation to generate an optimally transported (OT) mesh based on a scalar valued monitor function is a useful form of r-adaptive mesh generation because it generates a mesh equidistributed with respect to a monitor function and does not lead to mesh tangling [7]. We will see that the optimal transport problem on the sphere leads to a slightly different equation of Monge–Ampère type, which has not before been solved numerically on the surface of a sphere, which would be necessary for weather and climate prediction using r-adaptivity.