A transport approach to the convolution method for numerical modelling of linearized 3D circulation in shallow seas

Abstract

AbstractA new method for solving the linearized equations of motion is presented in this paper, which is the implementation of an outstanding idea suggested by Welander: a transport approach to the convolution method. The present work focuses on the case of constant eddy viscosity and constant density but can be easily extended to the case of arbitrary but time‐invariant eddy viscosity or density structure. As two of the three equations of motion are solved analytically and the main numerical ‘do‐loop’ only updates the sea level and the transport, the method features succinctness and fast convergence.The method is tested in Heaps' basin and the results are compared with Heaps' results for the transient state and with analytical solutions for the steady state. The comparison yields satisfactory agreement. The computational advantage of the method compared with Heaps' spectral method and Jelesnianski's bottom stress method is analysed and illustrated with examples.Attention is also paid to the recent efforts made in the spectral method to accelerate the convergence of the velocity profile. This study suggests that an efficient way to accelerate the convergence is to extract both the windinduced surface Ekman spiral and the pressure‐induced bottom Ekman spiral as a prespecified part of the profile.The present work also provides a direct way to find the eigenfunctions for arbitrary eddy viscosity profile. In addition, mode‐trucated errors are analysed and tabulated as functions of mode number and the ratio of the Ekman depth to the water depth, which allows a determination of a proper mode number given an error tolerance.