Linear internal waves and the control of stratified exchange flows

Internal hydraulic theory is often used to describe idealized bi-directiona l exchange flow through a constricted channel. This approach is formally ap plicable to layered flows in which velocity and density are represented by discontinuous functions that are constant within discrete layers. The theor y relies on the determination of flow conditions at points of hydraulic con trol, 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 foc us 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 thr ough a steady, continuously sheared and stratified exchange flow. In the fi rst, waves are mechanically excited at discrete locations within a numerica lly simulated bi-directional exchange flow and allowed to evolve under line ar dynamics. These waves are then tracked in space and time to determine pr opagation speeds. A second approach, based on the stability theory of paral lel shear flows and examination of solutions to a sixth-order eigenvalue pr oblem, is used to interpret the direct excitation experiments. Two types of gravest mode eigensolutions are identified: vorticity modes, with eigenfun ction maxima centred above and below the region of maximum density gradient , and density modes with maxima centred on the strongly stratified layer. D ensity modes have phase speeds that change sign within the channel and are analogous to the interfacial waves in hydraulic theory. Vorticity modes hav e finite propagation speed throughout the channel but undergo a transition in form: upwind of the transition point the vorticity mode is trapped in on e layer. It is argued that modes trapped in one layer are not capable of co mmunicating interfacial information, and therefore that the transition poin ts are analogous to control points. The location of transition points are i dentified and used to generalize the notion of hydraulic control in continu ously stratified flows.