  A stochastic mode reduction strategy is applied to multiscale models with a
deterministic energy-conserving fast sub-system. Specifically, we consider
situations where the slow variables are driven stochastically and interact with
the fast sub-system in an energy-conserving fashion. Since the stochastic terms
only affect the slow variables, the fast-subsystem evolves deterministically on
a sphere of constant energy. However, in the full model the radius of the
sphere slowly changes due to the coupling between the slow and fast dynamics.
Therefore, the energy of the fast sub-system becomes an additional hidden slow
variable that must be accounted for in order to apply the stochastic mode
reduction technique to systems of this type.
