  The stratified inclined duct (SID) sustains an exchange flow in a long,
gently sloping duct as a model for continuously-forced density-stratified flows
such as those found in estuaries. Experiments have shown that the emergence of
interfacial waves and their transition to turbulence as the tilt angle is
increased appears linked to a threshold in the exchange flow rate given by
inviscid two-layer hydraulics. We uncover these hydraulic mechanisms with (i)
recent direct numerical simulations (DNS) providing full flow data in the key
flow regimes (Zhu & Atoufi et al., arXiv:2301.09773, 2023), (ii) averaging
these DNS into two layers, (iii) an inviscid two-layer shallow water and
instability theory to diagnose interfacial wave behaviour and provide physical
insight. The laminar flow is subcritical and stable throughout the duct and
hydraulically controlled at the ends of the duct. As the tilt is increased, the
flow becomes everywhere supercritical and unstable to long waves. An internal
undular jump featuring stationary waves first appears near the centre of the
duct, then leads to larger-amplitude travelling waves, and to stronger jumps,
wave breaking and intermittent turbulence at the largest tilt angle. Long waves
described by the (nonlinear) shallow water equation are locally interpreted as
linear waves on a two-layer parallel base flow described by the
Taylor-Goldstein equation. This link helps us interpret long-wave instability
and contrast it to short-wave (e.g. Kelvin-Helmholtz) instability. Our results
suggest a transition to turbulence in SID through long-wave instability relying
on vertical confinement by the top and bottom walls.
