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Constructing accurate, flexible, and efficient parametrizations is one of thegreat challenges in the numerical modelling of geophysical fluids. We considerhere the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz63 model and derive a parametrization using a recently developed statisticalmechanical methodology based on the Ruelle response theory. We derive anexpression for the deterministic and the stochastic component of theparametrization and we show that the approach allows for dealing seamlesslywith the case of the Lorenz 63 being a fast as well as a slow forcing comparedto the characteristic time scales of the Lorenz 84 model. We test our resultsusing both standard metrics based on the moments of the variables of interestas well as Wasserstein distance between the projected measure of the originalsystem on the Lorenz 84 model variables and the measure of the parametrizedone. By testing our methods on reduced phase spaces obtained by projection, wefind support to the idea that comparisons based on the Wasserstein distancemight be of relevance in many applications despite the curse of dimensionality.

A Statistical Mechanical Approach for the Parametrization of the Coupling in a Fast-Slow System