38 
Owing to the linearity of the equations, also the partial solutions satisfy the shallow water 
equations. Likewise, solutions for different initial conditions are additive. Consider an initial 
condition consisting of two spatially separate signals. 
Let t] ext , u ext and rj. mt , u- mt each be solutions for one of the spatially separate initial 
conditions. Like the total solution, they consist of partial solutions running in separate 
directions. At any arbitrary point, the total solution can also be composed of these solutions 
to form 
1 = it, + It, + it + It and u = u + ext + u~ ext + «+ + . 
Let the external initial condition in space be to the left of the internal condition. Then the area 
between the two initial signals is first reached only by the external partial solution running in 
positive direction and by the internal partial solution running in negative direction because the 
other partial solutions are moving away from this area. In an infinitely extended area, it also 
remains like that. In the area between the two initial signals, the composed solution in this 
case reduces to 
V b = it, + lt and u b = u + ext + ut , which implies rf^ = 0 and /7+ =0. 
For the partial solutions 1] + ,/7“ of the total solution /7, 
it = it,> Il = It and u + b = u + ext , u~ b = m“ applies in this area. 
Expressed by the external and the internal total solution, the following is true for t] b in the 
area between the assumed initial conditions in a one-dimensional, infinitely extended area: 
lb = 0-5(7** + u ex ,4Ws) + 0.5(/7 int -u M Jhfg) or equivalently 
0 = Q.5{rj ext - u ext h/Jgh) and 0 = 0.5(i] int + u int h/Jgh). The solution for u b is: 
u b = °- 5 (l ex , + u ext ) + 0.5(/7 int yfgjh - u iat ). 
In case the external initial condition in space is to the right of the internal initial condition 
l~b = It,. it = it and u~ b = u~ ext , u + b = is valid respectively. 
At open boundaries, especially in case of a variable topography, such simple approaches 
quickly cease to be valid. Nevertheless, the boundary conditions formulated by Flather 
(Flather et al. 1975) on that basis have been used with some success in water level 
predictions. 
To allow a smooth transition from known external values to a computed internal solution left 
of the external solution, often the condition 77^ - it, is enforced. Then, to satisfy continuity, 
either the computed rj. mt is used to derive u iat from 7^ =/7^ (e.g. Jones et al. 2003) or rj. mt 
is derived using computed u iat (Kowalik 2003). 
The purpose of the BSFI’s North-East Atlantic model is to simulate atmospherically forced 
water level changes entering the North Sea as external surges. It is assumed at its open 
boundary (Atlantic, Norwegian Sea) that no signal enters from the Atlantic Ocean and 
Norwegian Sea, i.e. l ext ,u ext = 0. The incoming signal used for the simulations in section 7.2 
is a prescribed u ext , and t] ext is described by ij ext ~ = 0 at the western and southern 
boundaries and by rj ext + = 0 at the northern boundary. As the signal is assumed to enter an 
ocean at rest, rj int , u iat = 0 is used in the initial stage of the simulation.