Finite-Difference Analysis of a Quasi-3D Wave-Driven Flow Model: Stability, Grid Structure and Parameter Sensitivity

Wave-driven free-surface flows pose numerical challenges due to tensorial radiation stress forcing, anisotropic diffusion, and strong sensitivity to closure parameters. This paper investigates the numerical behavior of a quasi-3D wave-driven flow model using a coupled depth-integrated (2D) solver with a diagnostic three-dimensional (3D) reconstruction employed for consistency verification to evaluate the validity of dimensional reduction. The scheme is implemented on a staggered Arakawa C-grid with a terrain-following vertical coordinate and explicit pseudo-time-stepping, which enables the direct assessment of stability limits. A reference experiment and systematic sensitivity tests are performed for three idealized bathymetries of increasing complexity. Bottom friction primarily controls the free-surface response, with critical thresholds (e.g., f≳0.03) identified via the free-surface displacement Z as markers for the onset of numerical stiffness. Horizontal eddy viscosity Nh has a weak influence on depth-integrated transport over most of the tested range, whereas vertical eddy viscosity Nv governs both transport magnitude and stability through the vertical diffusion constraint, acting as the primary bottleneck for computational efficiency. A stability map in the (Nv,Δt,Nz) space is provided to delineate stable, marginal, and unstable regimes identifying an optimal vertical resolution of Nz≈10 for coastal applications. Grid resolution experiments quantify convergence trends and show that sensitivity increases with bathymetric complexity, revealing that bathymetric aliasing in multi-bar systems can lead to errors of up to 20% if gradients are under-resolved. Finally, a consistent set of diagnostic metrics is proposed for comparing 2D solutions with their vertically resolved counterparts, establishing a validity envelope where 2D models remain reliable versus regimes where explicit vertical shear resolution is mandatory. The results provide a practical roadmap for parameter selection, ensuring numerical robustness in complex, mechanically forced free-surface CFD applications.