An illustration of the "traditional approximation" of the Coriolis force. Only the locally vertical component of the Earth's rotation vector is included, so any given location on the surface is treated as if it was rotating about a locally vertical axis.

Anticipating that this process should be strongly affected by "non-traditional" effects, I have included the complete Coriolis force in the widely-used multi-layer shallow water equations. These equations serve as a useful idealised model of the ocean, capturing the something of the interaction between effects due to the Earth's rotation and density stratification.

I have used these equations to study the Antarctic Bottom Water, via both both analytical solutions and numerical simulations. The output from one such simulation is illustrated below. My results show that the cross-equatorial flow of the Antarctic Bottom Water is indeed strongly influenced by the complete Coriolis force.

3D rendering of a numerical simulation of Antarctic Bottom Water crossing the equator. The brown surface shows the variations in the height of the ocean bed, whilst the blue surface shows the height of the current. I have used a smoothed bathymetry to remove small-scale features that can not be resolved by the numerical grid.

Related publications:

1. •Cross-equatorial channel flow with zero potential vorticity under the complete Coriolis force, A.L. Stewart and P.J. Dellar, IMA Journal of Applied Mathematics (2012), published online.
   

1. •Multi-layer shallow water equations with complete Coriolis force. Part 2: Linear plane waves, A.L. Stewart and P.J. Dellar, Journal of Fluid Mechanics (2012), 690, 16-50.
   

1. •Cross-equatorial flow through an abyssal channel under the complete Coriolis force: two-dimensional solutions, A.L. Stewart and P.J. Dellar, Ocean Modelling (2011), 40, 87-104.
   

1. •The role of the complete Coriolis force in cross-equatorial flow of abyssal ocean currents, A.L. Stewart and P.J. Dellar, Ocean Modelling (2011), 38, 187-202.
   

1. •Two-layer shallow water equations with complete Coriolis force and topography, A.L. Stewart and P.J. Dellar, Progress in Industrial Mathematics at ECMI 2008, Mathematics in Industry Series, Springer (2010), 1033--1038.
   

1. •Multi-layer shallow water equations with complete Coriolis force. Part 1: Derivation on a non-traditional beta-plane, A.L. Stewart and P.J. Dellar, Journal of Fluid Mechanics (2010), 651, 387-413.
   

In some situations, the complete, unapproximated Coriolis force substantially influences the behaviour of the ocean. These "non-traditional" effects are particularly important close to the equator, where the traditional component of the Coriolis force vanishes, and in regions where density variations are small.

My work is focused on one such region: the deep western equatorial Atlantic ocean. Here the Antarctic Bottom Water, a dense current that originates in Antarctica from melting ice water, crosses the equator through a fracture zone.

3D rendering of the bathymetry where the Antarctic Bottom Water crosses the equator. The arrows represent schematically the path of the current through this region. Data obtained from the NOAA ETOPO1 1 Arc-Minute Global Relief Model.