Internal hydraulic theory is often used to describe idealized bi-directional exchange
flow through a constricted channel. This approach is formally applicable to layered
flows in which velocity and density are represented by discontinuous functions that
are constant within discrete layers. The theory relies on the determination of flow
conditions at points of hydraulic control, where long interfacial waves have zero phase
speed. In this paper, we consider hydraulic control in continuously stratified exchange
flows. Such flows occur, for example, in channels connecting stratified reservoirs and
between homogeneous basins when interfacial mixing is significant. Our focus here is
on the propagation characteristics of the gravest vertical-mode internal waves within
a laterally contracting channel.
Two approaches are used to determine the behaviour of waves propagating through
a steady, continuously sheared and stratified exchange flow. In the first, waves are
mechanically excited at discrete locations within a numerically simulated bi-directional
exchange flow and allowed to evolve under linear dynamics. These waves are then
tracked in space and time to determine propagation speeds. A second approach, based
on the stability theory of parallel shear flows and examination of solutions to a sixth-order
eigenvalue problem, is used to interpret the direct excitation experiments. Two
types of gravest mode eigensolutions are identified: vorticity modes, with eigenfunction
maxima centred above and below the region of maximum density gradient, and density
modes with maxima centred on the strongly stratified layer. Density modes have phase
speeds that change sign within the channel and are analogous to the interfacial waves
in hydraulic theory. Vorticity modes have finite propagation speed throughout the
channel but undergo a transition in form: upwind of the transition point the vorticity
mode is trapped in one layer. It is argued that modes trapped in one layer are not
capable of communicating interfacial information, and therefore that the transition
points are analogous to control points. The location of transition points are identified
and used to generalize the notion of hydraulic control in continuously stratified flows.