  Upwinded mass fluxes are described and analysed for advection operators
discretised using mixed mimetic spectral elements. This involves a
Petrov-Galerkin formulation by which the mass flux test functions are evaluated
at downstream locations along velocity characteristics. As for the original
mixed mimetic spectral element advection operator, the upwinded mass flux
advection operator is conservative, however unlike the original advection
operator, which is purely hyperbolic, the upwinded advection operator adds
dissipation which is biased towards high wave numbers. The upwinded advection
operator also removes the spectral gaps present in the dispersion relation for
the original advection operator. As for the original advection operator, a
material form advection operator may be constructed by similarly downwinding
the trial functions of the tracer gradients. Both methods allow for the
recovery of exact energy conservation for an incompressible flow field via
skew-symmetric formulations. However these skew-symmetric formulations are once
again purely hyperbolic operators which do not suppress oscillations. The
scheme is implemented within a shallow water code on the sphere in order to
diagnose and interpolate the potential vorticity. In the absence of other
dissipation terms, it is shown to yield more coherent results for a standard
test case of barotropic instability.
