Effect of the Coefficient on the Performance of A Two-Layer Boussinesq-Type Model

The coefficients embodied in a Boussinesq-type model are very important since they are determined to optimize the linear and nonlinear properties. In most conventional Boussinesq-type models, these coefficients are assigned the specific values. As for the multi-layer Boussinesq-type models with the inclusion of the vertical velocity, however, the effect of the different values of these coefficients on linear and nonlinear performances has never been investigated yet. The present study focuses on a two-layer Boussinesq-type model with the highest spatial derivatives being 2 and theoretically and numerically examines the effect of the coefficient a on model performance. Theoretical analysis show that different values for α (0.13⩽α⩽0.25) do not have great effects on the high accuracy of the linear shoaling, linear phase celerity and even third-order nonlinearity for water depth range of 0<kh⩽10 (k is wave number and h is water depth). The corresponding errors using different α values are restricted within 0.1%, 0.1% and 1% for the linear shoaling amplitude, dispersion and nonlinear harmonics, respectively. Numerical tests including regular wave shoaling over mildly varying slope from deep to shallow water, regular wave propagation over submerged bar, bichromatic wave group and focusing wave propagation over deep water are conducted. The comparison between numerical results using different values of α, experimental data and analytical solutions confirm the theoretical analysis. The flexibility and consistency of the two-layer Boussinesq-type model is therefore demonstrated theoretically and numerically.