40 
Fig. 6.1.8: Propagation of modified North Sea input signal between Cuxhaven and 
Geesthacht (distance in 100 km, 1 positive signal, period 1800 s, wave height 3m 
from the north, MAFITIN model, Plüß 2005, personal communication) 
6.1.8 Run-up and inundation 
Run-up onto dry land was not taken into account by operational tsunami warning models 
prior to the tsunami of December 2004. Since then, both the MOST model and the Japanese 
Tsunami-N2 model (Imamura et al. 2006) have successfully integrated the simulation of land 
inundation and reproduced historical tsunami (Geist et al. 2006). A recent comparison of the 
performance of different models in simulating run-up during the tsunami of December 2004 is 
given by Horrilio et al. (2006). 
The BSH models easily simulate flood and ebb tides in the North German tidal flats, including 
their exposure at low water, but are not capable of simulating the inundation of higher land 
and sea dikes. 
In case a real tsunami should flood the North German marshes, it would have to be taken 
into account that grassland would not dissipate much of the energy of the arriving bore, and 
that a tsunami would propagate faster in narrow channels, where water is deeper, than on 
land. 
However, it appears hardly reasonable to model inundation caused by a hypothetical tsunami 
before achieving a satisfactory simulation of the near-shore processes. 
6.2 Outlook for dispersion modelling in BSH models 
An important reason for the limited suitability of the BSH's models for the simulation of 
tsunami is the hydrostatic assumption implied, and hence the complete neglect of frequency 
dispersion. Non-hydrostatic model computations still are very time-consuming. Boussinesq 
models are non-hydrostatic to the first order and are often used in wave simulations as a 
compromise between high computational demands and the incorporation of modifications 
due to dispersion. (For a comparison with regard to the tsunami of December 2004 see 
Horrilio et al. 2006.) The following section deals with a possible way of including dispersion 
effects in the BSH’s models in order to achieve the level of accuracy that Boussinesq models 
have. For that purpose, the important approximations of analytical wave theory have been 
formulated for primitive equations. 
In the analytical wave theories referred to in section 5.1, terms in the equations of motion and 
their boundary conditions are not simply disregarded. Rather, the dependent variables are 
expanded according to a characteristic parameter, and higher-order terms in the equations 
are expressed by lower-order quantities. Finally, terms are neglected from a certain order 
upwards. In this way, a closed system of equations for lower-order quantities is created (e.g. 
Peregine 1972, Liu et al. 2002). Such systems include the different types of Boussinesq 
equations (e.g. Boussinesq 1871, Voit 1987, Madsen et al. 1991, Madsen et al. 1992), 
mostly two-dimensional approximations to describe long waves taking into account 
dispersion (/t/l 2 ^ 0) and non-linear effects (0.5H/h ^ 0, but mostly h 1 /L 2 0.5H/h ~ 0).