Statistical mechanics of Arakawa`s discretizations

The results of statistical analysis of simulation data obtained from long-time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. Statistical mechanical theories are constructed for three discretizations of the quasi-geostrophic model due to Arakawa (1966), each having different conservation properties. It is shown that the three statistical theories accurately explain the differences observed in statistics derived from the discretizations. The effect of the time discretization is also considered, and it is shown that projection is insufficient if the underlying spatial discretization is not conservative.