26.2.2.2 Computing density in partial bottom cells

The dependence of density on pressure is linear below a few hundred meters. As was shown above, there are no pressure gradients induced by partial bottom cells due to linear variations in density with depth. With a linear equation of state, bottom flows induced by partial bottom cells are within machine roundoff. Density anomoly using a non-linear equation of state as given by Equation (15.7) is not correct at grid points within partial cells because the polynomial coefficients, reference temperatures and reference salinities are defined only at the depth of grid points within discrete model levels. Effectively, partial bottom cells imply an infinite number of vertical levels. Density anomoly at the grid point within a partial cell can be approximated by linearly interpolating the polynomial coefficients, reference temperatures and reference salinities used in the equation of state (Equation 15.7) to the depth of the partial bottom cell grid point.

Without correcting the density anomoly for the depth of the partial bottom cells, a linear stratification of temperature in a non-linear equation of state generates error flows of about 1 cm/sec along topography (gaussian bump) in a $1^\circ$ resolution model simulation. However, by accounting for the depth of the grid point within the polynomial approximation to the equation of state, the error flows are reduced by an order of magnitude.