Evaluation of horizontal gradients in sigma-coordinate shallow water models

Evaluating horizontal gradients in three-dimensional shallow water models that use bottom-following sigma coordinates can lead to large errors near steep bathymetry. A technique that has been proposed to minimize this problem involves computing horizontal gradients in cartesian coordinates, while treating all other terms in a sigma coordinate framework. We study this technique through both truncation error analysis and numerical experimentation, and compare it to the direct application of sigma coordinates. While the Cartesian coordinate method has better convergence properties and generally smaller truncation errors when horizontal gradients are zero, the sigma coordinate method can be more accurate in other physically relevant situations. Also, the Cartesian coordinate method introduces significant numerical diffusion of variable sign near the bottom (where physical diffusion is particularly small), thus potentially leading to instabilities. Overall, we consider the sigma coordinates to be the best approach.