  A deterministic multiscale toy model is studied in which a chaotic fast
subsystem triggers rare transitions between slow regimes, akin to weather or
climate regimes. Using homogenization techniques, a reduced stochastic
parametrization model is derived for the slow dynamics. The reliability of this
reduced climate model in reproducing the statistics of the slow dynamics of the
full deterministic model for finite values of the time scale separation is
numerically established. The statistics however is sensitive to uncertainties
in the parameters of the stochastic model. It is investigated whether the
stochastic climate model can be beneficial as a forecast model in an ensemble
data assimilation setting, in particular in the realistic setting when
observations are only available for the slow variables. The main result is that
reduced stochastic models can indeed improve the analysis skill, when used as
forecast models instead of the perfect full deterministic model. The stochastic
climate model is far superior at detecting transitions between regimes. The
observation intervals for which skill improvement can be obtained are related
to the characteristic time scales involved. The reason why stochastic climate
models are capable of producing superior skill in an ensemble setting is due to
the finite ensemble size; ensembles obtained from the perfect deterministic
forecast model lacks sufficient spread even for moderate ensemble sizes.
Stochastic climate models provide a natural way to provide sufficient ensemble
spread to detect transitions between regimes. This is corroborated with
numerical simulations. The conclusion is that stochastic parametrizations are
attractive for data assimilation despite their sensitivity to uncertainties in
the parameters.
