  We describe the long-term dynamics of sustained stratified shear flows in the
laboratory. The Stratified Inclined Duct (SID) experiment sets up a two-layer
exchange flow in an inclined duct connecting two reservoirs containing salt
solutions of different densities. This flow is primarily characterised by two
non-dimensional parameters: the tilt angle of the duct with respect to the
horizontal, $\theta$ (a few degrees at most), and the Reynolds number $Re$, an
input parameter based on the density difference driving the flow. The flow can
be sustained with constant forcing over arbitrarily long times and exhibits a
wealth of dynamical behaviours representative of geophysically-relevant
sustained stratified shear flows. Varying $\theta$ and $Re$ leads to four
qualitatively different regimes: laminar flow; mostly laminar flow with
finite-amplitude, travelling Holmboe waves; spatio-temporally intermittent
turbulence with substantial interfacial mixing; and sustained, vigorous
interfacial turbulence (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp.
242-253). We seek to explain the scaling of the transitions between flow
regimes in the two-dimensional plane of input parameters $(\theta, Re)$. We
improve upon previous studies of this problem by providing a firm physical
basis and non-dimensional scaling laws that are mutually consistent and in good
agreement with the empirical transition curves we inferred from 360 experiments
spanning $\theta \in [-1^\circ, 6^\circ]$ and $Re \in [300, 5000]$. To do so,
we employ state-of-the-art simultaneous volumetric measurements of the density
field and the three-component velocity field, and analyse these experimental
data using time- and volume-averaged potential and kinetic energy budgets. We
show that regime transitions are caused by ...
