A circulation model in which variation of the horizontal grid spacing obtained by employing boundary-fitted coordinates is described. This is a method in which elliptical transformation equations are solved to map an arbitrary grid spacing in physical space onto a squared mesh in a transformed space. The three-dimensional conservation of momentum and mass equations are then solved in this transformation grid. In addition, an algebraic transformation is used in the vertical to map the free surface and bottom onto coordinate surfaces. The resulting equations are solved by a time-centred semi-implicit finite difference method.
The model was first tested on several idealized geometries such as tidal flow in a wedge-shaped basin and geostrophic flow along a straight coastline. The results were compared with available analytical solutions and showed good agreement. For a more complicated topography such as a basin with a headland, the simultation results and computing time for different grid sizes have been compared with results from the Leendertse model, in which square grids are used.
The model has been implemented for the Norwegian continental shelf. Boundary-fitted coordinates with point attraction were employed to decrease the grid spacing at Haltenbanken and along the shelf edge. The two-dimensional version of the model was used to simulate the semi-diurnal M2 constituent, showing good agreement with observed amplitudes and phases, and with results from other numerical models.
In numerical circulation models it is often required to have a large model region to include effects (e.g. meteorological forcing) outside the study area which will influence the flow within the area. In addition, a fine grid resolution is often required in the study area to represent the flow caused by variation in the bottom topography and the coastal boundary, and to resolve the current pattern near areas of special interest (e.g. construction areas). If the grid spacing is made small enough to resolve the smallest spatial scales, and if this grid spacing has to be employed for the whole domain, the computational costs will often become excessive. To overcome this problem, several methods have been used to vary the gird resolution. Examples are : finite element techniques (triangular mesh being most popular, Pinder and Gray, 1977), finite difference methods using conformal curvilinear grids (Reid et al., 1977), orthogonal curvilinear grids (Willemse et al., 1986), stretched rectangular grids (Waldrop and Farmer, 1974), irregular (triangular) grids (Thacker, 1977), and boundary-fitted coordinates (Johnson, 1982).
This chapter presents results from a three-dimensional circulation model in which the variation of the horizontal grid spacing is obtained by employing boundary-fitted coordinates. In the method, a set of coupled quasi-linear elliptic transformation equations are solved to map an arbitrary horizontal multiconnected region from physical space to a rectangular mesh structure in the transformed horizontal plane (Spaulding, 1984). The three-dimensional conservation of mass and momentum equations, with approximations suitable for continental shelf and coastal sea areas, that form the basis of the hydrodynamic model. Are then solved in this transformed space. In additional, an algebraic transformation is used in the vertical direction to map the free surface and bottom onto coordinate surfaces.