Geoscience Reference
In-Depth Information
Fig. 5.12 Formulation of the problem of a tsunami run-up on the coast
In determining boundary conditions for practical tsunami calculations the so-
called 'vertical wall' approximation has become widespread. A boundary condition
of this type provides for total reflection of the wave at a fixed isobath. Note that
the vertical wall approximation is not a purely academic abstraction, it imitates quite
a type of coast, encountered quite often—a rocky precipice, falling off to the water.
In the notation, given in Fig. 5.12, the vertical wall corresponds to
= 90
◦
or to
β
L
= 0.
From elementary theory of linear waves it is known that, if a channel of fixed
depth ends in a vertical wall, then the height of the run-up onto the wall is deter-
mined as twice the incident wave amplitude,
R
L
= 2
ξ
0
.
Actually, when approaching the coast, the tsunami amplitude may be commen-
surable with the depth. Therefore, to determine the run-up height one must, gen-
erally speaking, apply non-linear theory. Omitting the details of resolving the non-
linear problem of long-wave theory, expounded in the topic [Pelinovsky (1996)],
we present the resulting analytical formula that relates the run-up onto a vertical
wall,
R
N
, and the wave amplitude far from the coast,
ξ
0
,
R
N
= 4
H
1 +
ξ
0
1
/
2
,
1 +
ξ
0
H
H
−
(5.46)
where
H
is the basin depth. It is readily verified that the relation
R
L
= 2
ξ
0
is a partial
case of formula (5.46) given the condition
1. Comparison of quantities
R
L
and
R
N
shows that taking into account non-linearity enhances the run-up amplitude
insignificantly. As the non-linearity (of quantity
ξ
0
/
H
ξ
0
/
H
) increases, the ratio
R
N
R
L
grows monotonously, but this growth is not without limit,
R
N
R
L
= 2
.
lim
ξ
H
→
∞
This means, the run-up amplitude, calculated with account of non-linearity, can-
not be superior to twice the amplitude corresponding to linear theory.
We further consider the one-dimensional problem of a long wave moving along
a slope (0
<
/
2). We write the non-linear equations for shallow water, taking
into account that the basin depth is a linear function of the horizontal coordinate,
H
=
H
0
−
α
β
<
π
x
,
