  We consider the Euler equation of quasi-geostrophic fluids which is widely
used in weather forecast. Our goal is to study explicit volume-preserving
numerical methods for very long simulations on an energy and enstrophy
preserving discretization. To this purpose, we compute the average fields and
estimate statistical parameters. It is observed that the statistical parameters
depend on the integrals of the discretization and that the computed parameters
are affected by the error in those, even if the volume of the phase space is
correctly preserved. We conclude that monitoring the error in the integrals by
the explicit volume preserving method can be used as an indication of how good
the estimated parameters are.
