Cross-equatorial channel flow with zero potential vorticity under the complete Coriolis force

We investigate the cross-equatorial flow of deep ocean currents through a channel along the sea floor using the 1½-layer or equivalent-barotropic shallow water equations. We restrict our attention to flows with zero potential vorticity, motivated by measurements of the deep Atlantic ocean, and focus on the role of the so-called non-traditional components of the Coriolis force due to the locally horizontal component of the Earth's rotation vector. These components are typically neglected in theoretical studies of ocean dynamics. We first obtain steady asymptotic solutions in a straight-walled channel by assuming that the channel half-length is much larger than the intrinsic lengthscale, the equatorial Rossby deformation radius. The leading-order solution describes a current that switches from the western to the eastern side of the channel as it crosses from the southern to the northern hemisphere. Including the non-traditional component of the Coriolis force substantially increases the cross-equatorial transport as long as the flow is everywhere northwards, but substantially decreases the transport if part of the flow retroflects and returns to the southern hemisphere. We compare our steady asymptotic solutions with time-dependent numerical solutions of the non-traditional shallow water equations. We impose the a priori assumption of zero potential vorticity by writing the fluid's absolute angular momentum as the gradient of a scalar potential. The time-average of our numerical solutions converges to our steady asymptotic solutions in the limit of large channel length, with an error of only around 1% even when our asymptotic parameter ε is equal to 1. However, our solutions diverge when the layer depth is sufficiently small that portions of the flow become super-critical. The resulting formation of steady shocks prevents the time-dependent solution from approaching the asymptotic solution.