Stochastic homogenization for an energy conserving multi-scale toy model of the atmosphere

• Introduces a toy model for the large-scale dynamics of the atmosphere.
• Constructs a stochastic parametrization of unresolved fast processes which is energy conserving.
• Perform stochastic homogenization to distill the effective slow dynamics.

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.