Flows between ocean basins are often controlled by narrow channels and shal
low sills. A multi-layer hydraulic control theory is developed for exchange
flow through such constrictions. The theory is based on the inviscid shall
ow-water equations and extends the functional approach introduced by Gill(1
977) and developed by Dalziel (1991). The flows considered are those in rec
tangular-cross-section channels connecting two large reservoirs, with a sin
gle constriction (sill and/or narrows). The exchange flow depends on the st
ratification in the two reservoirs, represented as a finite number of immis
cible layers of(different) uniform density. For most cases the flow is 'con
trolled' at the constriction and often at other points along the channel (v
irtual controls) too. As with one- and two-layer hydraulics, controls are l
ocations at which the flow passes from one solution branch to another, and
at which (at least) one internal wave mode is stationary. The theory is app
lied to three-layer flows, which have two internal wave modes, predicting i
nterface heights and layer fluxes from the given reservoir conditions. The
theoretical results for three-layer flows are compared to a comprehensive s
et of laboratory experiments and found to give good agreement. The laborato
ry experiments also show other features of the flow, such as the formation
of waves on the interfaces. The implications of the results for oceanograph
ic flows and ocean modelling are discussed.