Vertically integrated mode

In order to provide the splitting of the model into an internal and an external mode, the continuity equation and the momentum equations are vertically integrated. The vertical integral of the continuity equation together with the kinematic boundary conditions (6) and (7) gives the sea surface elevation equation:

$$\partial_t \zeta = - \partial_x U- \partial_y V.$$ (57)

with

$$U=\int_{-H}^{\zeta} u\,dz,\qquad V=\int_{-H}^{\zeta} v\,dz.$$ (58)

Integrating the momentum equations (1) and (2) vertically results in:

$$\begin{array}{l} \displaystyle \partial_tU+ \tau_b^x+\alpha\b... ...a}\partial_x b\,dz'\,dz \bigg) = -gD\partial_x\zeta \end{array}$$ (59)

and

$$\begin{array}{l} \displaystyle \partial_tV+ \tau_b^y+\alpha\b... ...}\partial_y b\,dz'\,dz \bigg) = -gD\partial_y\zeta. \end{array}$$ (60)

Here, $\tau_b^x$ and $\tau_b^y$ are bottom stresses. Their calculation is discussed in section 8.13.9. As a first preparation for the mode splitting, these integrals of the momentum equations can be formally rewritten as

$$\begin{array}{l} \displaystyle \partial_tU+\frac{R}{D^2}U\sqr... ...-fV +S^x_{A}-S^x_D+S^x_B\bigg) = -gD\partial_x\zeta \end{array}$$ (61)

and

$$\begin{array}{l} \displaystyle \partial_tV+\frac{R}{D^2}V\sqr... ...U +S^y_{A}-S^y_D+S^y_{B}\bigg) = -gD\partial_y\zeta \end{array}$$ (62)

with the so-called slow terms for bottom friction

$$S^x_{F}=\tau^x_b-\frac{R}{D^2}U\sqrt{U^2+V^2},$$ (63)

$$S^y_{F}=\tau^y_b-\frac{R}{D^2}V\sqrt{U^2+V^2},$$ (64)

horizontal advection

$$S^x_{A}=\int_{-H}^{\zeta}\left(\partial_x u^2+\partial_y(uv)\right)\,dz- \partial_x\left(\frac{U^2}{D}\right)-\partial_y\left(\frac{UV}{D}\right),$$ (65)

$$S^y_{A}=\int_{-H}^{\zeta}\left(\partial_x (uv)+\partial_yv^2\right)\,dz- \partial_x\left(\frac{UV}{D}\right)-\partial_y\left(frac{V^2}{D}\right),$$ (66)

horizontal diffusion

$$\begin{array}{l} \displaystyle S^x_D= \int_{-H}^{\zeta}\big( ... ... +\partial_x\left(\frac{V}{D}\right)\right)\right), \end{array}$$ (67)

$$\begin{array}{l} \displaystyle S^y_D= \int_{-H}^{\zeta}\big( ... ... +\partial_x\left(\frac{V}{D}\right)\right)\right), \end{array}$$ (68)

and internal pressure gradients

$$S^x_B=-\int_{-H}^{\zeta}\int_z^{\zeta}\partial_x b\,dz'\,dz$$ (69)

and

$$S^y_B=-\int_{-H}^{\zeta}\int_z^{\zeta}\partial_y b\,dz'\,dz.$$ (70)

The drag coefficient $R$ for the external mode is calculated as (this logarithmic dependence of the bottom drag from the water depth and the bottom roughness parameter $z_b^0$ is discussed in detail by Burchard and Bolding (2002)):

$$R = \left(\frac{\kappa} {\ln\left(\frac{\frac{D}{2}+z^b_0}{z^b_0}\right)}\right)^2.$$ (71)

It should be noted that for numerical reasons, an additional explicit damping has been implemented into GETM. This method is based on diffusion of horizontal transports and is described in section 7.4.14 on page .