Mode splitting

The external system consisting of the surface elevation equation (57) and the transport equations (61) and (62) underlies a strict time step constraint if the discretisation is carried out explicitly:

$$\Delta t < \left[\frac12 \left(\frac{1}{\Delta x}+\frac{1}{\Delta y}\right) \sqrt{2gD}\right]^{-1}.$$ (41)

In contrast to that, the time step of the internal system is only depending on the Courant number for advection,

$$\Delta t < \min\left\{\frac{\Delta x}{u_{\max}},\frac{\Delta y}{v_{\max}} \right\},$$ (42)

which in the case of sub-critical flow is a much weaker constraint. In order not to punish the whole model with a small time step resulting from the external system, two different approaches of mode splitting have been developed in the past.

The first approach, in which the external mode is calculated implicitly, has been proposed by Madala and Piacsek (1977). This method is numerically stable (if advection is absent) for unconditionally long time steps. The temporal approximation is of second order if semi-implicit treatment is chosen. In such models, the external and internal mode are generally calculated with the same time steps (see e.g. Backhaus (1985)). The introduction of interactions terms like (63) - (70) is thus not necessary in such models.

Another approach is to use different time steps for the internal (macro times steps $\Delta t$) and the external mode (micro time steps $\Delta t_m$). One of the first free surface models which has adopted this method is the Princeton Ocean Model (POM), see Blumberg and Mellor (1987). This method has the disadvantage that interaction terms are needed for the external mode and that the consistency between internal and external mode is difficult to obtain. The advantage of this method is that the free surface elevation is temporally well resolved which is a major requirement for models including flooding and drying. That is the reason why this method is adopted here.

The micro time step $\Delta t_m$ has to be an integer fraction $M$ of the macro time step $\Delta t$. $\Delta t_m$ is limited by the speed of the surface waves (41), $\Delta t$ is limited by the current speed (42). The time stepping principle is shown in figure 5. The vertically integrated transports are averaged over each macro time step:

$$\bar U_{i,j}^{n+1/2} = \frac{1}{M}\sum_{l=n+0.5/M}^{n+(M-0.5)/M} U^l_{i,j}$$ (43)

and

$$\bar V_{i,j}^{n+1/2} = \frac{1}{M}\sum_{l=n+0.5/M}^{n+(M-0.5)/M} V^l_{i,j}$$ (44)

such that

$$\begin{array}{l} \displaystyle \frac{\zeta_{i,j}^{n+1}-\zeta_... ... V_{i,j}^{n+1/2}-\bar V_{i,j-1}^{n+1/2}}{\Delta y}. \end{array}$$ (45)

Figure 5: Sketch explaining the organisation of the time stepping.