  Continuing from Part 1 (Hern\'andez \emph{et al.}, \emph{arXiv:2108.12395},
2021), a generalized quasilinear (GQL) approximation is studied in turbulent
channel flow using a flow decomposition defined with spanwise Fourier modes:
the flow is decomposed into a set of low-wavenumber spanwise Fourier modes and
the rest high-wavenumber modes. This decomposition leads to the nonlinear
low-wavenumber group that supports the self-sustaining process within the given
integral length scales, whereas the linearised high-wavenumber group is not
able to do so, unlike the GQL models in Part 1 which place a minimal
mathematical description for the self-sustaining process across all integral
scales. Despite the important physical difference, it is shown that the GQL
models in this study share some similarities with those in Part 1: i.e. the
reduced multi-scale behaviour and anisotropic turbulent fluctuations.
Furthermore, despite not being able to support the self-sustaining process in
the high-wavenumber group, the GQL models in the present study are found to
reproduce some key statistical features in the high-wavenumber group solely
through the `scattering' mechanism proposed by previous studies. Finally, using
the nature of the GQL approximation, a set of numerical experiments suppressing
certain triadic nonlinear interactions are further carried out. This unveils
some key roles played by the certain types of triadic interactions including
energy cascade and inverse energy transfer in the near-wall region. In
particular, the inhibition of inverse energy transfer in the spanwise direction
leads to suppression of the near-wall positive turbulent transport at large
scales.
