  It is widely believed that statistical closure theories for dynamical systems
provide statistics equivalent to those of the governing dynamical equations
from which the former are derived. Here, we demonstrate counterexamples in the
context of the widely used mean-field quasilinear (QL) approximation applied to
2D fluid dynamical systems. We compare statistics of QL numerical simulations
with those obtained by direct statistical simulation via a cumulant expansion
closed at second order (CE2). We observe that, though CE2 is an exact
statistical closure for QL dynamics, its predictions disagree with the
statistics of the QL solution for identical parameter values. These
disagreements are attributed to instabilities, which we term rank
instabilities, of the second cumulant dynamics within CE2 that are unavailable
in the QL equations.
