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Statistical mechanics of Arakawa’s discretizations. (English) Zbl 1130.82005
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 used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exist. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to A. Arakawa [J. Comput. Phys. 1, 119–143 (1966; Zbl 0147.44202)], which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the difference observed in statistics derived from the discretizations.
The authors of the present paper first briefly recall the quasi-geostrophic potential vorticity equation and its conservation properties. Then, they recall Arakawa’s three discretizations, their conservation properties, and they prove that all of these define divergence-free vector fields. Generalizations to Arakawa’s discretizations with the Nambu bracket approach are considered. Equilibrium statistical mechanical theories are developed for the three discretizations for an ideal fluid in vorticity-stream function form. After that the results are extended to the cases in which only one of the quantities (energy or enstrophy) is conserved. Time integration aspects (symmetric implicit midpoint method) are discussed in detail. Finally, numerical experiments confirming the statistical predictions are presented. The results of the paper make a strong argument for the use of conservative discretizations in weather and climate simulations.

MSC:
82-08 Computational methods (statistical mechanics) (MSC2010)
82-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics
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