  We discuss the applicability of quasilinear-type approximations for a
turbulent system with a large range of spatial and temporal scales. We consider
a paradigm fluid system of rotating convection with a vertical and horizontal
temperature gradients. In particular, the interaction of rotating with the
horizontal temperature gradient drives a ``thermal wind'' shear flow whose
strength is controlled by a horizontal temperature gradient. Varying the
parameters systematically alters the ordering of the shearing timescale, the
convective timescale, and the correlation timescale. We demonstrate that
quasilinear-type approximations work well when the shearing timescale or the
correlation timescale is sufficiently short. In all cases, the Generalised
Quasilinear approximation (GQL) systematically outperforms the Quasilinear
approximation (QL). We discuss the consequences for statistical theories of
turbulence interacting with mean gradients. We conclude with comments about the
general applicability of these ideas across a wide variety of non-linear
physical systems.
