Nonlinear tidal waves in channels: A perturbation method adapted to the importance of quadratic bottom friction

Bottom friction plays an important role in propagation and damping of long waves in shallow water. A perturbation method, well adapted to the importance of this bottom stress, is presented and applied for channels with constant mean depth and mean cross-section area. The main idea results from a development of quadratic bottom friction in a Fourier series up to the third order of approximation. A quasi-linearization of the damping effect of bottom stress is deduced from this expansion, which allows one to introduce the second-order damping effects of friction into the first-order system defining the fundamental component of the spectrum, and the main part of the third-order damping terms into the computation of the second-order harmonic components. From this expansion, the generating role of the harmonic played by friction is also identified and taken into account in the second-order approximation. Analytical and analytico-numerical solutions are presented and compared to numerical solutions of the full nonlinear equations obtained by the Lax-Wendroff finite-difference integration technique. These comparisons show that the analytical solutions limited to the second-order approximation fit very well with the L.W. solutions.