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 dis- cretization chosen. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are con- structed for the discrete dynamical systems arising from three discretizations due to Arakawa (1966) which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.
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Academic Press
Journal of Computational Physics
Symplectic Integration of Atmospheric Dynamics: Long-term Statistical Accuracy for Ensemble Climate Simulations
Computational Dynamics

Dubinkina, S, & Frank, J.E. (2007). Statistical mechanics of Arakawa`s discretizations. Journal of Computational Physics, 227(2), 1286–1305. doi:10.1016/