An investigation is made of the properties of inertial boundary currents in a stably stratified, inviscid, non-diffusive ocean. The Boussinesq and β-plane approximations are adopted. The formalism developed by Robinson (1965) is used: the equations are transformed so that density replaces the vertical coordinate as an independent variable, and after a suitable non-dimensionalization of variables, the various fields are expanded as power series in the downstream co-ordinate η. The motion is shown to conserve potential vorticity. The equations and boundary conditions are obtained to order η2. Solutions are obtained in the region of formation of the coastal jet (i.e. the case of no mass flux through the plane η = 0) for several cases in which the potential vorticity function depends on stream function and density in a simple way. For these cases, it is found that in a constant depth ocean, a boundary current can exist only if the geostrophic drift at the boundary-layer edge is westward at all depths. This constraint is relaxed if the depth increases rapidly enough in the downstream (northward) direction. For slopes just in excess of the critical value, a deep onshore counter-current is predicted. Solutions of the first-order problem, using realistic values of the various parameters, have been computed and are found to be in qualitative agreement with observed features of the Florida Current.
In an appendix, it is shown that the constraint of westward geostrophic drift at all levels must hold in a flat-bottomed ocean for arbitrary potential vorticity distributions consistent with stable stratification.