The computation of invariant solutions and the visualisation of the associated state space
have played a key role in the understanding of transition and the self-sustaining process
in wall-bounded shear flows. In this study, an extension of this approach is sought for a
turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multiscale near-wall turbulence, which resolves two adjacent spanwise integral length scales
of motion, is considered using a shear stress-driven flow model (Doohan et al., J. Fluid
Mech., vol. 913, 2021, A8). The edge state, twenty-six travelling waves and two periodic
orbits are computed, which represent either the large- or small-scale self-sustaining
processes. Given that the spanwise length scales are not widely separated here, it could be
envisaged that turbulent trajectories visit these solutions in the state space. Considering
the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined,
but the turbulent state is not found to approach any of these solutions. A detailed
analysis reveals that this is due to the lack of scale interaction processes captured by
the invariant solutions, including the mean-fluctuation interaction, the energy cascade in
the streamwise wavenumber space and the cascade-driven energy production discovered
recently. There is a single solution that resembles turbulence much more than the others,
which captures two-scale energetics and a scale interaction process involving energy
feeding from small to large spanwise scales through the subharmonic sinuous streak
instability mode. Based on these observations, it is conjectured that the state space view
of turbulent trajectories wandering between solutions would need suitable refinement to
model multi-scale turbulence, when each solution does not represent multi-scale processes
of turbulence. In particular, invariant solutions that are inherently multi-scale would be
required.