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:
Integrating the momentum equations (1) and (2) vertically results in:
Here, and 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
with the so-called slow terms for bottom friction
and internal pressure gradients
The drag coefficient for the external mode is calculated as (this logarithmic dependence of the bottom drag from the water depth and the bottom roughness parameter is discussed in detail by Burchard and Bolding (2002)):
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 .