Two-layer rotating exchange flows through channels of rectangular cross-section are modelled using semi-geostrophic, zero-potential-vorticity theory. For a given channel cross-section the full range of possible flow states is considered. The interface always has a uniform slope across the channel, but may separate from one or both of the sidewalls to attach to the upper or lower boundary. The flow may be subcritical, critical or supercritical. These different states are identified in a pseudo-Froude-number plane analogous to that developed by Armi (1986) for non-rotating flows. If the ratio of the channel width to the Rossby radius is constant along the length of the channel, then the solution may be traced along the entire channel using a single diagram. Several examples of maximal and submaximal exchanges are considered. This graphical method of solution is contrasted with the functional approach of Dalziel (1988, 1990).<br/>The exchange flux is determined as a function of the channel geometry, the strength of rotation and the difference in Bernoulli potential between the two layers
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