  Constructing accurate, flexible, and efficient parametrizations is one of the
great challenges in the numerical modelling of geophysical fluids. We consider
here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz
63 model and derive a parametrization using a recently developed statistical
mechanical methodology based on the Ruelle response theory. We derive an
expression for the deterministic and the stochastic component of the
parametrization and we show that the approach allows for dealing seamlessly
with the case of the Lorenz 63 being a fast as well as a slow forcing compared
to the characteristic time scales of the Lorenz 84 model. We test our results
using both standard metrics based on the moments of the variables of interest
as well as Wasserstein distance between the projected measure of the original
system on the Lorenz 84 model variables and the measure of the parametrized
one. By testing our methods on reduced phase spaces obtained by projection, we
find support to the idea that comparisons based on the Wasserstein distance
might be of relevance in many applications despite the curse of dimensionality.
