**Appendix A**

The OGCMs solve the three-dimensional primitive equations, including Navier–Stokes equation, static equilibrium equation, continuity equation, temperature and salinity conservation equation, and state equation. The equations governing the dynamics of coastal circulation contain fast-moving, external gravity waves, and slow-moving, internal gravity waves. It is desirable in terms of computer economy to separate the vertically integrated equations (barotropic mode) from the vertical structure equations (baroclinic mode). This technique, known as mode splitting, permits the calculation of the free surface elevation with little sacrifice in computational time by solving the velocity transport separately from the three-dimensional calculation of the velocity and the thermodynamic properties. The split-explicit free surface formulation used in the NEMO model follows the one proposed by [21]. The general idea is to solve the free-surface equation and the associated barotropic velocity equations with a smaller time step than Δ*t*, which is the time step used in the baroclinic mode for the three-dimensional prognostic variables (Figure A1).

The barotropic mode solves the following depth-integrated equations:

$$\frac{\partial \overline{\mathcal{U}}}{\partial t} = -\mathfrak{fc} \times \overline{\mathcal{U}} - \mathfrak{g} \nabla \eta - \frac{\mathfrak{c}}{H + \eta} \overline{\mathcal{U}} + \overline{\mathcal{G}} \tag{A1}$$

$$\frac{\partial \overline{\eta}}{\partial t} = -\nabla \cdot \left[ (H + \eta) \overline{\Pi} \right] + P - E \tag{A2}$$

where *U* is the depth-integrated barotropic velocity, *η* is the sea surface height, *f* is the Coriolis parameter, and *g* is the acceleration of gravity. **G** is a forcing term held constant, containing coupling term between modes, surface atmospheric forcing, and slowly varying barotropic terms not explicitly computed to improve efficiency. The third term on the right-hand side of Equation (A1) represents the bottom stress. *H* represents the depth of the ocean. *P* and *E* represent the precipitation and evaporation, respectively.

Time filtering is eventually applied to barotropic quantities to avoid aliasing of fast barotropic motions into three-dimensional equations. When the filtered sea surface height option is used, the momentum equation's force is solved implicitly. Thus, an elliptic equation, which is solved using the PCG solver in the NEMO model, is formulated.

**Figure A1.** The split time stepping is used in the model. The m represents the time of the barotropic step, and M is the total number of steps among each baroclinic step.
