  We examine the effectiveness of the Generalised Quasilinear (GQL)
Approximation introduced by Marston et al (2016). This approximation splits the
variables into large and small scales in directions where there is a
translational symmetry and removes nonlinear interactions involving only small
scales. We utilise as a paradigm problem three-dimensional, turbulent, rotating
Couette flow. We compare the results obtained from Direct Numerical Solution of
the equations with those from Quasilinear (QL) and GQL calculations. In this
three-dimensional setting, there is a choice of cut-off wavenumber for the GQL
approximation both in the streamwise and in the spanwise directions. We
demonstrate that the GQL approximation significantly improves the accuracy of
mean flows, spectra and two-point correlation functions over models that are
quasilinear in any of the translationally invariant directions, even if only a
few streamwise and spanwise modes are included. We argue that this provides
significant support for a programme of Direct Statistical Simulation utilising
the GQL approximation.
