Stochastic parameterizations are broadly used in climate modeling to represent
subgrid scale processes. While different parameterizations are being developed
considering different aspects of the physical phenomena, less attention is
given to the technical and numerical aspects. In particular, the use of
Empirical Orthogonal Functions (EOFs) is well established whenever a spatial
structure is required, without considering its possible drawbacks. By applying
an energy consistent parameterization to the 2-layer Quasi-Geostrophic (QG)
model, we investigate the model sensitivity to the \emph{a priori} assumptions
made on the parameterization. In particular, we consider here two methods to
prescribe the spatial covariance of the noise. First, by using climatological
variability patterns provided by EOFs, and second, by using time-varying
dynamics-adapted Koopman modes, approximated by Dynamic Mode Decomposition
(DMD). The performance of the two methods are analyzed through numerical
simulations of the stochastic system on a coarse spatial resolution, and the
outcomes compared to a high-resolution simulation of the original
deterministic system. The comparison reveals that the DMD based noise
covariance scheme outperforms the EOF based. The use of EOFs leads to a
significant increase of the ensemble spread, and to a meridional misplacement
of the bi-modal eddy kinetic energy (EKE) distribution. On the other hand,
using DMDs, the ensemble spread is confined and the meridional propagation of
the zonal jet stream is accurately captured. Our results highlight the
importance of the systematic design of stochastic parameterizations with
dynamically adapted spatial correlations, rather than relying on statistical
spatial patterns.