  A semi-Lagrangian Characteristic Mapping method for the solution of the
tracer transport equations on the sphere is presented. The method solves for
the solution operator of the equations by approximating the inverse of the
diffeomorphism generated by a given velocity field. The evolution of any tracer
and mass density can then be computed via pullback with this map. We present a
spatial discretization of the manifold-valued map using a projection-based
approach with spherical spline interpolation. The numerical scheme yields $C^1$
continuity for the map and global second-order accuracy for the solution of the
tracer transport equations. Error estimates are provided and supported by
convergence tests involving solid body rotation, moving vortices,
deformational, and compressible flows. Additionally, we illustrate some
features of computing the solution operator using a numerical mixing test and
the transport of a fractal set in a complex flow environment.
