25 
DISPERSION. AMPLIFICATION DISPERSION. AMPLIFICATION 
Fig. 5.3.1: Transformation at the continental slope (linear Boussinesq equations, ij surface 
elevation, h undisturbed water depth, 10.5: PEDpT) 
In the continued simulation using linear Boussinesq equations (Fig. 5.3.1, 10.5: PEDpT), 
intensification of the leading signal with decreasing depth (shoaling) prevails over weakening 
due to dispersion as the tsunami travels into shallower water. 
In comparison, consider the two-dimensional barotropic model of the North-East Atlantic in 
section 7.2. As its analytical basis is hydrostatic non-linear equations, the topographic 
modification of the propagation velocity is computed according to ^fgh . The model also 
simulates the reduction of wave length ifghT with decreasing depth. As the model equations 
include friction terms, T does not remain constant, and the simple linear estimation 
L = T dee P *fgh f° r a sin 9 le wave is only approximately valid, as in real nature. In the model, 
the height of the individual waves increases at the continental slope. However, local 
increases are mainly due to superposition of single waves. 
The energy balance in the model regarding reflection, non-linearity, and dissipation was not 
explicitly considered. It is known, though, that the numerical approximation of analytical 
equations and of non-linearities, in particular, has an influence on the magnitude of 
dissipation. 
All three approaches - linear hydrostatic equations, linear Boussinesq equations, and non 
linear hydrostatic equations in numerical form - are barotropic modelling concepts. In 
stratified media, tsunami may also cause internal waves, which will also be modified at the 
continental shelf. Their forcing by tsunami has been studied by Hammack (1974). 
5.4 Modification and attenuation on the continental shelf 
5.4.1 Modification 
On the shelf, barotropic equations provide an adequate description for the transformation 
and propagation of tsunami because bottom friction in this area causes strong mixing in the 
water column. In the continuation to the Pedersen simulation (10.5:PEDpT) using linear 
Boussinesq equations, the change in bottom topography is modelled after the depth 
distribution off Portugal, where the shelf is narrow, unlike in the North Sea. The depth profile 
on the entire shelf shows a slope of 500 m per 50 km, i.e. Ah:L b =1:100. On the shelf, 
nearly all partial waves of the simulated tsunami (Fig. 5.2.1) are long waves (h/L<< 0.05), 
and the influence of frequency dispersion is very low. Fig. 5.4.1 shows primarily a further 
reduction of wave length and an increase in the wave height of the leading signal due to the 
decrease in undisturbed depth (shoaling).