Advanced Ocean Modelling: Using Open-Source Software

168 5 3D Level ModellingFig. 5.30 Arakawa C-grid for one-dimensional shallow-water applicationsbarotropic), that frictional effects are negligibly small, and that sea-level anomaliesη are small compared with undisturbed water depth h o (leading to simplification ofthe volume conservation equation Eq. (5.41)).On the basis of the one-dimensional version of the Arakawa C-grid (Fig. 5.30),the explicit numerical finite-difference scheme of the above equations can be formulatedin three subsequent steps given by:u ∗ k = un k − Δt g( ηk+1 n − ) ηn k Δx (5.42)η n+1k= ηk n − Δt h (o u∗k − u ∗ )k−1 Δx (5.43)u n+1k= u ∗ k (5.44)where n is the time level, Δt is the numerical time step, and Δx is the grid spacing.We assume that the computational domain covers the grid cells from k = 1tok = nx and that the cells k = 0 and k = nx + 1 are reserved for the implementationof boundary conditions. Care has to be taken here, given that the above equations arenot symmetric with respect to the boundary conditions. The prediction for η doesnot use data of u nx+1 and the prediction of u does not use values of η 0 . If we wantto prescribe boundary conditions for η but not for u, this implies that u also needsto be predicted in the grid cell k = 0, which is assumed in the following.5.13.5 Zero-Gradient ConditionsZero-gradient conditions, also called von Neumann conditions, are sometimesemployed for dynamic pressure at open boundaries for elimination for geostrophicflow components parallel to a boundary. For the barotropic surface gravity wavemode, being embedded in the dynamics, this condition implies vanishing flow normalto the boundary (u nx = 0 and u 0 = 0). Hence, these conditions imply full wavereflection at lateral boundaries resulting is a standing wave that can significantlybias the predictions in the interior of the model domain.