  We consider a three-dimensional slow-fast system arising in fluid dynamics
with quadratic nonlinearity and additive noise. The associated deterministic
system of this stochastic differential equation (SDE) exhibits a periodic orbit
and a slow manifold. We show that in presence of noise, the deterministic slow
manifold can be viewed as an approximate parameterization of the fast variable
of the SDE in terms of the slow variables, for certain parameter regimes. We
exploit this fact to obtain a two dimensional reduced model from the original
stochastic system, which results into a Hopf normal form with additive noise.
Both, the original as well as the reduced system admit ergodic invariant
measures describing their respective long-time behaviour. It is then shown that
for a suitable Wasserstein metric on a subset of the space of probability
measures on the phase space, the discrepancy between the marginals along the
radial component of each invariant measure is controlled by a parameterization
defect measuring the quality of the parameterization.
  An important technical tool to arrive at this result is Girsanov's theorem
that allows us to derive such error estimates in presence of an oscillatory
instability. This approach is finally extended to parameter regimes for which
the variable to parameterize is no longer evolving on a faster timescale than
that of the resolved variables. There also, error estimates involving the
Wasserstein metric are derived but this time for reduced systems obtained from
stochastic parameterizing manifolds involving path-dependent coefficients to
cope with such challenging regimes.
