Linear analysis on the non-hydrostatic pressure field of depth-integrated models

In the development of depth-integrated models, certain assumptions are made on the vertical profile of the horizontal velocity, and the pressure field is eliminated through vertical integration of the vertical momentum equation. In this study, analytical forms of the approximated non-hydrostatic pressure field are derived from five most representative depth-integrated models, which include three Boussinesq-type models, a weighted residual type model and a non-hydrostatic model. A Stokes wave-type Fourier analysis is then conducted to examine their capability in reproducing the vertical profile of the non-hydrostatic pressure field, which is compared with linear wave theory. We found the Boussinesq-type models are able to reproduce the non-hydrostatic pressure field from shallower water up to the kd values similar to that of concerning linear wave phase velocity. Specifically, the two-layer Boussinesq model generally outperforms the higher-order (μ⁴) Boussinesq model, especially for higher kd values. The weighted residual type model can reasonably capture the non-hydrostatic pressure near to the free surface even at high kd values, but undulations appear in the lower part of the water column where non-hydrostatic pressure is close to zero. The non-hydrostatic model shows rapid convergence of linear wave phase velocity by using few vertical layers, however, its capability in reproducing the non-hydrostatic pressure field is limited, especially at the highest layer interface, where nonphysical phenomena of reversed pressure could appear. This suggests that it is necessary to employ more layers when detailed information on the pressure field is needed, e.g. wave–structure interaction problems.