41 
In contrast, numerical models like the BSH models (Dick et al. 2001) are based on so-called 
primitive equations. "Primitive" implies that the equations are solved for the complete 
quantities. In general, however, statistically averaged equations are solved in which 
correlations resulting from the non-linear terms of the equation are expressed by averaged 
quantities. An additional approximation in numerical models is the parameterisation of 
processes with length scales below the model's grid resolution (in the BSH models, this also 
includes sea and swell). 
The starting point for the following discussion is equations 1 (section 5.1), the non-averaged, 
three-dimensional mass and momentum balance equations in co-ordinate-free form (Pichler 
1984). The approximations, which in their simplest form were already given in section 5.1.1, 
are now presented using the so-called p -equation, which is derived as divergence of the 
impulse equation, solved to obtain V 2 /?: 
V 2 p - —— (V-/7v)-V- V pvv + VF -V ■ pV {(/) + (f) G ) -V ■ (2i2x pv) 
dt 
From conservation of mass, it follows: 
V 2 p = V- V/tvv + V- F - V-/?V(^ + ^ G )-V-(2axpv). 
dt 
With p = p(p), a prognostic equation for p is obtained. The compressibility of water has to 
be taken into account only in special cases, though, e.g. in simulations of meteoritic impacts 
(e.g. Mader 2004), volcanic eruptions involving ash admixture, or gas admixture in the 
course of slope failure. 
In most cases, tsunami can be simulated using the Boussinesq approximation, which 
considers water incompressible, V-v = 0, and density differences are only considered in 
connection with gravity impacts. (This is not the approximation used to deduce the 
Boussinesq equations.) Besides, with V-V(^ + ^ G ) = 0 according to the general rules of 
vector calculation, the p -equation in the Boussinesq approximation takes the form: 
V 2 p = -p 0 V ■ Vvv + V • F -Vp ■ V (if) + (f) G ) — V • (2Qxp 0 v). 
Although the tidal potential <f> G is taken into account in the BSH’s models (Muller-Navarra 
2002), the influence of tides in the North Sea is determined primarily by the lateral boundary 
conditions. The vertical component of V^ G is generally neglected. From the p -equation it 
vanishes if density differences are considered strictly in connection with V0. 
In analytical discussions, earth rotation is often neglected and flow is assumed to be 
irrotational, i.e. Vxv = 0, which simplifies the first term on the right side of the equation to 
/? 0 0.5V 2 (v 2 ). Furthermore, the vector v , which in general can be represented as a sum of a 
scalar and a vector potential, in that case can be expressed by a scalar potential alone. With 
respect to the North Sea, the assumption of irrotationality is not reasonable, though, because 
bottom friction, wind forcing and the influence of its variable topography are not negligible. 
Therefore, no potentials are introduced here. 
Also the neglect of earth rotation is not justifiable in simulations of tsunami impacts in the 
North Sea. As tsunami periods are smaller than the inertial period, there is no formation of 
Kelvin waves, as in the case of long external surges, but earth rotation may influence phase 
velocity (Gill 1982, Akylas 1994). Besides, a tsunami does not enter a North Sea at rest but 
interacts with tidal and wind-forced currents, both of which are influenced by earth rotation. 
Although strong interaction is not to be expected because of different periods, tsunami may 
undergo refraction due to such currents. Neglecting the horizontal component of Q., 
however, is rather common use.