On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles

Abstract

AbstractThis paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles.A point model is used in the vertical, with a no‐slip condition at the sea bed. In the finite difference approach the high‐shear bottom layer is resolved using either a regular grid on a logarithmic or log‐linear transformed co‐ordinate or an irregular grid, varying in such a manner as to retain second‐order accuracy. The accuracy of these various grid schemes is considered in detail.The relative merits of using either the Crank‐Nicolson or Dufort‐Frankel time integration methods are considered; in the case of a fine grid in a high‐viscosity region, some numerical problems are found with the Dufort‐Frankel method.An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high‐shear bottom boundary layer.Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers.The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near‐bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.