A Numerical Technique for Coupling the Momentum and the Continuity Equations for Semi-Implicit 3D Ocean Models

TL;DR

The paper addresses efficient coupling of momentum and depth-integrated continuity in 3D semi-implicit ocean models by introducing an $O(N)$ algorithm that expresses horizontal velocities in terms of neighboring free-surface elevations via simple recursive relations. The method discretizes the hydrostatic primitive equations with an implicit barotropic pressure gradient and vertical viscosity, forming a five-diagonal system for the free-surface that is solved efficiently, while horizontal velocities are recovered through recursive decompositions and sweeps. Validation on standing-wave and wind-driven circulatory flows demonstrates accuracy, mass/momentum conservation, and linear scaling of the coupling cost with the number of horizontal layers, offering substantial speedups over conventional $O(N^{2.33})$ approaches. The work provides a scalable framework for fast, stable 3D coastal-ocean simulations that can handle large time steps without sacrificing fidelity.

Abstract

Semi-implicit methods are powerful and efficient tools for the three-dimensional modeling of coastal and oceanic processes. A semi-implicit finite difference method for 3D hydrostatic primitive equations is presented in this paper. The governing equations are time-discretized with an implicit treatment of barotropic pressure gradient and vertical viscosity. The discretized momentum equations along a water column are coupled with the depth-integrated continuity equation of the column to construct a linear system for free-surface elevations. The novelty of this work lies in formulating an efficient method with the order of complexity O(N) for coupling the momentum and the continuity equations. In this method, the horizontal velocity components are expressed in terms of neighbor free-surface elevations by some simple recursive formulas and then are substituted in the integrated continuity equation.

Figures (5)

• Figure 1: Computational mesh and notations
• Figure 2: The initial condition of the free-surface in the 3D standing wave test case
• Figure 3: Comparisons of simulated free-surface and horizontal velocity $u$ with the analytical solution for the 3D linear standing wave test case: (a) free-surface elevations at $y=0$ and $t=6T$; (b) free-surface elevation at $(x, y)=(0, 0)$; (c) horizontal velocity $u$ at $(x, y)=(250, 0)$.
• Figure 4: Comparison of the simulated horizontal velocity profile with the analytical solution at $x=1250$m, $y=1250$m for the wind-driven circulatory flow case
• Figure 5: Computation time versus the number of horizontal layers