**Appendix B**

The Stony Brook Parallel Ocean Model (sbPOM) follows the principles of the POM. The advantage of the sbPOM is that computational efficiency is improved due to its use of a parallel computing environment. However, because the σ coordinate system is used in the vertical direction in the sbPOM, *z* coordinates must be converted into σ coordinates. This conversion is performed as follows:

$$
\sigma = \frac{\mathbf{z} - \boldsymbol{\eta}}{\mathbf{H} + \boldsymbol{\eta}} \,\tag{A18}
$$

where H(x,y) is the bottom terrain in the horizontal x and y dimensions; η(x,y,t) is the sea level fluctuation in horizontal dimensions x and y at time dimension t, which is integrated from the bottom (z = −H) to the sea surface (z = η); and σ is set from −1 to 0. Using this equation, the basic equations of the sbPOM under the σ-coordinate system can be expressed as

$$\frac{\partial \mathbf{u} \mathbf{D}}{\partial \mathbf{t}} + \frac{\partial \mathbf{u}^2 \mathbf{D}}{\partial \mathbf{x}} + \frac{\partial \mathbf{u} \mathbf{v} \mathbf{D}}{\partial \mathbf{y}} + \frac{\partial \mathbf{u} \mathbf{w} \mathbf{D}}{\partial \sigma} - \mathbf{f} \mathbf{v} \mathbf{D} + \mathbf{g} \mathbf{D} \frac{\partial \eta}{\partial \mathbf{x}} + \frac{\mathbf{g} \mathbf{D}^2}{\rho\_0} \int\_{\sigma}^{0} \left[ \frac{\partial \rho}{\partial \mathbf{x}} - \frac{\sigma}{\mathbf{D}} \frac{\partial \mathbf{D}}{\partial \mathbf{x}} \frac{\partial \rho}{\partial \sigma} \right] d\sigma = \frac{\partial}{\partial \sigma} \left[ \frac{\mathbf{K}\_{\mathbf{M}}}{\mathbf{D}} \frac{\partial \mathbf{u}}{\partial \sigma} \right] + \mathbf{F}\_{\mathbf{x}} \tag{A19}$$

$$\frac{\partial \mathbf{v} \mathbf{D}}{\partial \mathbf{t}} + \frac{\partial \mathbf{u} \mathbf{v} \mathbf{D}}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}^{2} \mathbf{D}}{\partial \mathbf{y}} + \frac{\partial \mathbf{v} \mathbf{w}}{\partial \sigma} + \text{f} \mathbf{u} \mathbf{D} + \mathbf{g} \mathbf{D} \frac{\partial \eta}{\partial \mathbf{y}} + \frac{\mathbf{g} \mathbf{D}^{2}}{\rho\_{0}} \int\_{\sigma}^{0} \left[ \frac{\partial \rho}{\partial \mathbf{y}} - \frac{\sigma}{\mathbf{D}} \frac{\partial \mathbf{D}}{\partial \mathbf{y}} \frac{\partial \rho}{\partial \sigma} \right] \mathbf{d} \sigma = \frac{\partial}{\partial \sigma} \left[ \frac{\mathbf{K}\_{\mathbf{M}}}{\mathbf{D}} \frac{\partial \mathbf{v}}{\partial \sigma} \right] + \mathbf{F}\_{\mathbf{y}} \tag{A20}$$

where

$$\mathbf{D} = \mathbf{H} + \mathbf{\eta} \tag{A21}$$

and where u, v, and ω represent the velocities under the respective σ-coordinates; ρ is the mean fluctuation value; g is the gravitational acceleration; KM is the vertical viscosity coefficient; and Fx and Fy are the horizontal viscosity terms. The use of these equations

should reduce the truncation errors associated with the calculation of the pressure gradient term in an σ-coordinate system over steep topography.
