Simulation of Ocean Circulation of Dongsha Water Using Non-Hydrostatic Shallow-Water Model

Abstract

A two-dimensional non-hydrostatic shallow-water model for weakly dispersive waves is developed using the least-squares finite-element method. The model is based on the depth-averaged, nonlinear and non-hydrostatic shallow-water equations. The non-hydrostatic shallow-water equations are solved with the semi-implicit (predictor-corrector) method and least-squares finite-element method. In the predictor step, hydrostatic pressure at the previous step is used as an initial guess and an intermediate velocity field is calculated. In the corrector step, a Poisson equation for the non-hydrostatic pressure is solved and the final velocity and free-surface elevation is corrected for the new time step. The non-hydrostatic shallow-water model is verified and applied to both wave and flow driven fluid flows, including solitary wave propagation in a channel, progressive sinusoidal waves propagation over a submerged bar, von Karmann vortex street, and ocean circulations of Dongsha Atolls. It is found hydrostatic shallow-water model is efficient and accurate for shallow water flows. Non-hydrostatic shallow-water model requires 1.5 to 3.0 more cpu time than hydrostatic shallow-water model for the same simulation. Model simulations reveal that non-hydrostatic pressure gradients could affect the velocity field and free-surface significantly in case where nonlinearity and dispersion are important during the course of wave propagation.

Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method

“…A number of non-hydrostatic coastal ocean models have been developed [44][45][46][47][48][49][50][51][52][53][54]. A depth-averaged (one-layer) non-hydrostatic shallow-water model was developed and applied to model the propagation of a solitary wave and the interactions of the propagating solitary wave with the submerged structure in a previous study [67]. Computed results showed the promising potential of the model.…”

A Weighted-Least-Squares Meshless Model for Non-Hydrostatic Shallow Water Waves

“…The shape of the free surface profile should always be symmetric, no matter how long the solitary wave propagates. Nevertheless, simulating the case with a hydrostatic SWE model does not produce the correct result [1][2][3]. As concluded in [4], such kind of error is due to the exclusion of the dispersion term in the hydrostatic shallow water equations.…”

“…in which U, V and W are the average values of u, v and w by the integration in the z direction while n m is the Manning's coefficient. For finding the values of Q at discretized nodes, or in the elements, solving a time independent Poisson equation is usually employed in most of the already published non-hydrostatic models [2,3,11,15,17,19]. This will form a huge global matrix system when the computational domain is discretized with a great number of nodes.…”

“…The root mean square errors are 0.0155 m and 0.0345 m, respectively. The exact solution and the numerical result of [3] are also plotted for comparison. The comparison shown in Figure 1 indicates that accuracy is improved using smaller nodal resolution and time increment.…”

“…It was shown in [13] that in dealing with the dispersion effects of water waves, the non-hydrostatic shallow water equations outperform the Boussinesq equations [14]. In the recent couple decades, many non-hydrostatic SWE models were developed for weakly nonlinear water wave problems by using the grid/meshbased methods [1][2][3]11,13,[15][16][17][18][19] such as the finite difference method (FDM) and the finite element method (FEM).…”