We study a Hamiltonian toy model for a Lagrangian fluid parcel in the semi-geostrophic limit which exhibits slow and fast dynamics. We first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system. In a second step we use stochastic singular perturbation theory to derive an effective reduced stochastic differential equation for the slow dynamics. We verify the results in numerical simulations.

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Physica - D, Nonlinear Phenomena
Thermostat closures for inviscid fluids
Computational Dynamics

Frank, J.E, & Gottwald, G.A. (2013). Stochastic homogenization for an energy conserving multi-scale toy model of the atmosphere. Physica - D, Nonlinear Phenomena, 254, 45–56.