Three-dimensional momentum equations

For geophysical coastal sea and ocean dynamics, usually the three-dimensional hydrostatic equations of motion with the Boussinesq approximation and the eddy viscosity assumption are used (Bryan (1969), Cox (1984), Blumberg and Mellor (1987), Haidvogel and Beckmann (1999), Kantha and Clayson (2000b)). In the flux form, the dynamic equations of motion for the horizontal velocity components can be written in Cartesian coordinates as:

$$\begin{array}{l} \displaystyle \partial_t u +\partial_z(uw) -... ...eta}\partial_x b\,dz' \bigg) = - g\partial_x \zeta, \end{array}$$ (1)

$$\begin{array}{l} \displaystyle \partial_t v +\partial_z(vw) -... ...zeta}\partial_y b\,dz' \bigg)= - g\partial_y \zeta. \end{array}$$ (2)

The vertical velocity is calculated by means of the incompressibility condition:

$$\partial_x u +\partial_y v +\partial_z w = 0.$$ (3)

Here, $u$, $v$ and $w$ are the ensemble averaged velocity components with respect to the $x$, $y$ and $z$ direction, respectively. The vertical coordinate $z$ ranges from the bottom $-H(x,y)$ to the surface $\zeta(t,x,y)$ with $t$ denoting time. $\nu_t$ is the vertical eddy viscosity, $\nu$ the kinematic viscosity, $f$ the Coriolis parameter, and $g$ is the gravitational acceleration. The horizontal mixing is parameterised by terms containing the horizontal eddy viscosity $A_h^M$, see Blumberg and Mellor (1987). The buoyancy $b$ is defined as

$$b=-g\frac{\rho-\rho_0}{\rho_0}$$ (4)

with the density $\rho$ and a reference density $\rho_0$. The last term on the left hand sides of equations (1) and (2) are the internal (due to density gradients) and the terms on the right hand sides are the external (due to surface slopes) pressure gradients. In the latter, the deviation of surface density from reference density is neglected (see Burchard and Petersen (1997)). The derivation of equations (1) - (3) has been shown in numerous publications, see e.g. Pedlosky (1987), Haidvogel and Beckmann (1999), Burchard (2002b).

In hydrostatic 3D models, the vertical velocity is calculated by means of equation (3) velocity equation. Due to this, mass conservation and free surface elevation can easily be obtained.

Drying and flooding of mud-flats is already incorporated in the physical equations by multiplying some terms with the non-dimensional number $\alpha$ which equals unity in regions where a critical water depth $D_{crit}$ is exceeded and approaches zero when the water depth $D$ tends to a minimum value $D_{min}$:

$$\alpha=\min\left\{1,\frac{D-D_{min}}{D_{crit}-D_{min}}\right\}.$$ (5)

Thus, $\alpha=1$ for $D\geq D_{crit}$, such that the usual momentum equation results except for very shallow water, where simplified physics are considered with a balance between tendency, friction and external pressure gradient. In a typical wadden sea application, $D_{crit}$ is of the order of 0.1 m and $D_{min}$ of the order of 0.02 m (see Burchard (1998), Burchard et al. (2004)).