  We demonstrate that a separation of the velocity field in large and small
scales according to a streamwise Fourier decomposition identifies subspaces
with stable Lyapunov exponents and allows the dynamics to exhibit properties of
an inertial manifold, such as the synchronization of the small scales in
simulations sharing the same large scales or equivalently the decay of all
small scale flow states to the state uniquely determined from the large scale
flow. This behaviour occurs for deviations with streamwise wavelength smaller
than 130 wall units which was shown in earlier studies to correspond to the
streamwise spectral peak of the cross-flow velocity components of the top
Lyapunov vector of the turbulent flow.
