24 
The simulation begins with a prescribed initial distribution (Fig. 5.2.1, left) of the surface 
elevation that is typical of a tsunami generated by vertical bottom excursion. It was modelled 
on the basis of the 1969 earthquake off Portugal (earthquake of magnitude 7.9 at 5,000 m 
depth). The initial disturbance propagates in all directions. Thus, in this simulation, a wave of 
only half the initial wave height moves towards the coast. The right part of Fig. 5.2.1 (right) 
shows this kind of separation and the change in surface elevation under the impact of 
dispersion weakening the initial signal. 
In this simulation, the bottom of the deep ocean is relatively flat in the direction away from the 
continental slope. Transatlantic tsunami may be modified or deflected by submarine ridges 
and sea mounts. Mofjeld et al. (2000) defined a parameter, on the basis of linear theory, 
which characterises the relevance of dispersion and reflection by different submarine 
structures to tsunami. 
5.3 Modification on the continental slope 
The first transition to shallower water is at the continental slope. Off Sumatra, the continental 
slope is extremely close to the coastline. With respect to the German North Sea coast, it is a 
distant and dynamically special feature. On the continental slope, ^Jgh decreases strongly, 
while the phase velocity gT short l2k of short waves is independent of depth. With 
gT short /2K> «[gh , the short waves may catch up to the long waves (Mirchina et al. 2001), 
strengthening the leading signal. Under real ocean conditions, however, very short waves 
are dampened out, and the originally medium-length waves turn into long waves with 
decreasing depth due to the decreasing h/L quotient. Therefore, dispersion normally will 
further weaken the leading signal. 
In the long-wave dispersion-free part, dominant waves are overtaken by following waves, 
and superposition then produces a higher signal. In addition, the individual waves are 
shortened at the shelf edge, with their energy concentrated on a smaller area (shoaling). 
Flowever, part of the energy potentially available to shoaling is reflected back into the deep 
ocean at the continental slope. 
Linear hydrostatic theory is appropriate for estimating the behaviour of single long waves. It 
provides simple formulas for changes in wave height, length, and steepness with decreasing 
depth (Masselink 2005). In the absence of friction, the energy flow E<Jgh remains constant 
as the wave enters shallower water. In this hydrostatic case, the energy propagates with 
velocity *Jgh, and it changes according to E/E deep = (h deep /h) 05 . By the same 
approximation, energy is proportional to wave height squared, and hence 
HjHdeep = (h d£ep /h) 025 . In this approximation, a transition from 4,000 m to 1,000 m thus 
would lead to an increase in wave height by the factor V2 . As energy dissipation is entirely 
neglected, the wave period is also retained in the transition to shallower water. Wave length 
L - T deep ifgh is reduced according to LjL deep = (h/h deep ) 0 ' 5 , and wave steepness S = H/L 
increases according to SjS deep = (h deep / h) 015 . With the depths shown in Fig. 5.3.1,4,000 and 
1,000 m, wave length would decrease by half and wave steepness would increase three-fold. 
Flowever, that applies only to gentle bottom slopes. The continental slope represents a rather 
abrupt change in depth and, for an impact perpendicular to the slope, HjH deep is estimated 
better by HlH deep =2h de 5 ep /(h de 5 ep + h 05 ) (Camfield 1990). With the above values, FI thus 
would increase by the factor 1.33 (instead of 1.41). Camfield (1990) also provides suitable 
equations for other bottom profiles and impact angles.