  An equation of Monge-Amp\`ere type has, for the first time, been solved
numerically on the surface of the sphere in order to generate optimally
transported (OT) meshes, equidistributed with respect to a monitor function.
Optimal transport generates meshes that keep the same connectivity as the
original mesh, making them suitable for r-adaptive simulations, in which the
equations of motion can be solved in a moving frame of reference in order to
avoid mapping the solution between old and new meshes and to avoid load
balancing problems on parallel computers.
  The semi-implicit solution of the Monge-Amp\`ere type equation involves a new
linearisation of the Hessian term, and exponential maps are used to map from
old to new meshes on the sphere. The determinant of the Hessian is evaluated as
the change in volume between old and new mesh cells, rather than using
numerical approximations to the gradients.
  OT meshes are generated to compare with centroidal Voronoi tesselations on
the sphere and are found to have advantages and disadvantages; OT
equidistribution is more accurate, the number of iterations to convergence is
independent of the mesh size, face skewness is reduced and the connectivity
does not change. However anisotropy is higher and the OT meshes are
non-orthogonal.
  It is shown that optimal transport on the sphere leads to meshes that do not
tangle. However, tangling can be introduced by numerical errors in calculating
the gradient of the mesh potential. Methods for alleviating this problem are
explored.
  Finally, OT meshes are generated using observed precipitation as a monitor
function, in order to demonstrate the potential power of the technique.
