  We present a semi-Lagrangian characteristic mapping method for the
incompressible Euler equations on a rotating sphere. The numerical method uses
a spatio-temporal discretization of the inverse flow map generated by the
Eulerian velocity as a composition of sub-interval flows formed by $C^1$
spherical spline interpolants. This approximation technique has the capacity of
resolving sub-grid scales generated over time without increasing the spatial
resolution of the computational grid. The numerical method is analyzed and
validated using standard test cases yielding third-order accuracy in the
supremum norm. Numerical experiments illustrating the unique resolution
properties of the method are performed and demonstrate the ability to reproduce
the forward energy cascade at sub-grid scales by upsampling the numerical
solution.
