  We study generalised quasilinear (GQL) approximations applied to turbulent
plane Couette flow. The GQL framework is explored in conjunction with a
Galerkin reduced-order model (ROM) recently developed by Cavalieri & Nogueira
(Phys. Rev. Fluids 7, L102601, 2022), which considers controllability modes of
the linearised Navier-Stokes system as basis functions, representing coherent
structures in the flow. The velocity field is decomposed into two groups: one
composed by high-controllability modes and the other by low-controllability
modes. The former group is solved with the full nonlinear equations, whereas
the equations for the latter are linearised. We also consider a modified GQL
framework, wherein the linearised equations for the low-controllability modes
are driven by nonlinear interactions of modes in the first group, which are
characterised by large-scale coherent structures. It is shown that GQL-ROMs
successfully recover the statistics of the full model with relatively high
controllability thresholds and sparser nonlinear operators. Driven GQL-ROMs
were found to converge more rapidly, providing accurate description of the
statistics with a larger number of linearised modes with respect to standard
GQL approximations. The results presented here show that further model
reductions are attainable with GQL-ROMs, which can be valuable to extend these
models to larger Reynolds numbers.
