37 
and the quantities computed in NEA are then transferred to N10 as boundary values. Both a 
positive and a negative initial signal were used, each with a one-hour period. In a second 
report (Smallman 2006) NEA is replaced by an extended form of the British operational storm 
surge model CS23 (Flather 2000, 2D, horizontal resolution 12 km) and the simulations are 
started from known initial conditions. Boundary conditions are then transferred to a model 
with variable grid spacing (TELEMAC-2D, finite elements, horizontal resolution 12 km down 
to 1 km). 
In its operational applications, the BSH’s model system also uses nested models. For the 
simulations in section 7.2, three positive signals with a half-hour period were prescribed for 
the North-East Atlantic model (North-East Atlantic model, 2D, horizontal resolution 10 km). 
Flowever, the signal modification toward the North Sea was not considered fully reliable (cf. 
section 7.1), and analytical signals were used as boundary conditions for the North Sea 
model (North Sea model, 2 km, 2D, horizontal resolution 2 km) as well. 
It is remarkable that in all simulations using an analytical boundary the signal comes from a 
single direction and enters perpendicular to the open boundary. This is attributable to the fact 
that the mathematical and numerical treatment of open (non-physical) boundaries in 
numerical models is problematic (e.g. Durran 1998, Blayo et al. 2005). The numerical 
problem involves the requirement to have an unfalsified incoming signal and an unhindered 
outward transportation of a signal from the interior, both in arbitrary directions with respect to 
the boundary. Furthermore, on short time scales, the transported outward signal should be 
saved and allowed to be transported back if required. The first two aspects are explained on 
the basis of a one-dimensional example (cf. Kowalik 2003 and Flather and Davis 1975). 
For flat bottom and initial conditions u - 0 and tj 0 = 2rjâf{x) for .*<0, rj 0 for 
x>0, the linear, hydrostatic, one-dimensional shallow water equations du/dt = -gdrj/dx 
and dîj/dt = —hdu/dx provide the solutions (e.g. Gill 1982): 
7 = 7" +7 + =7o/(* + V^) + 7o/(-*-V£^)’ 
u=u~ +u + = -№/1 + 4^1 ) + / (-* “ ) ■ 
Fig. 6.1.7 shows the temporal evolution of the simple initial condition, 2^ 0 = 2ij+, f(x) = 1 
for \x\ < L and f(x) = 0 otherwise. 
vsv///////////vr/K7/zrzt, i ■■ravyçsxys. 
Fig. 6.1.7: Solution of one-dimensional shallow water equations for an initially constant 
elevation confined to a finite region (Gill 1982, Fig. 5.9 b). 
The solution thus is an additive superposition of two signals propagating in opposite 
directions. Vice versa, the two partial solutions can be represented by the total solution: 
tj + -0.5(Tj + uh/^fgh) and u + =0.5(rj^fgh/h + u), 
1) =0.5(ij-uhJJgh) and u =0.5(ij^fgh/h-u).