We conduct long simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreement with theoretical models, as well as the continuum statistical mechanical theory for ideal flow developed by Eillis, Haven & Turkington (Nonlinearity, 2002). In particular the results verify that the apparently trivial conservation of potential vorticity along particle paths using the HPM method significantly influences the mean state. As a side note, the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean.
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Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Dubinkina, S., & Frank, J. (2009). Statistical relevance of vorticity conservation with the Hamiltonian particle-mesh method. Modelling, Analysis and Simulation [MAS]. CWI.