On the approximation of horizontal gradients in sigma co‐ordinates for bathymetry with steep bottom slopes

Abstract Nowadays the simulation of free surface flow and transport in rivers, estuaries and seas is often based upon three‐dimensional modelling systems. Most of these three‐dimensional modelling systems use sigma co‐ordinates in the vertical. By the use of the sigma transformation the water column can be divided into the same number of layers independently of the water depth. Especially for steep bottom slopes combined with vertical stratification of the density, sigma‐transformed grids impose numerical problems for the accurate approximation of horizontal gradients. This paper deals with algorithms for the approximation in sigma co‐ordinates of the horizontal diffusive fluxes of temperature and salinity and for the approximation of the horizontal pressure gradients. The approximation of the horizontal diffusive fluxes is based upon a finite volume method. The approximation of the pressure gradients is directly related to the approximation of the diffusive fluxes. Artificial vertical diffusion and artificial flow due to truncation errors are minimized. The method described in this paper is not hampered by the so‐called ‘hydrostatic consistency condition’. This will be illustrated by numerical experiments.