Geoscience Reference
In-Depth Information
F
∂
=(
η
,
n)
,
z
=
−
H
(
x
,
y
)
,
(5.44)
∂
n
where
τ
F
=
F
d
t
.
0
The initial elevation is determined from the solution of problem (5.42)-(5.44) by
the following formula:
(
x
,
y
)=
F
z
(
x
,
y
,
0)
.
The vector of bottom deformations
ξ
, entering into formulae (5.41) and (5.44),
is calculated making use of the analytical solution of the stationary problem of elas-
ticity theory [Okada (1985)]. The calculation technique is described in detail in
Sect. 2.1.3.
As a rule, the boundary conditions used for simulating tsunami propagation
within the theory of long waves pertain to one of the following three types [Marchuk
et al. (1983)]:
η
1. interaction with the coast
2. non-reflecting
3. perturbation, arriving from external area.
In the most simple case, the interaction of waves with the coast is described as
total reflection from the coast. To this end one considers that on a certain fixed
isobath (usually, 10-20 m) the flow velocity component normal to the coastline (or
the chosen isobath) turns to zero,
V
n
= 0
.
A direct consequence of this condition is the equality to zero of the component
normal to the shoreline (or the chosen isobath) of the derivative of the free surface
displacement,
∂ξ
∂
n
= 0
.
The condition of total reflection is usually applied in those cases, when the main
goal is to investigate wave propagation in the open ocean. In analysing tsunami
dynamics in the shelf zone a more detailed description is necessary of the interaction
of waves with the coast. Here, it has sense to consider partial reflection of waves and
to make use of the formula proposed by A. V. Nekrasov [Nekrasov (1973)],
ξ
√
g
H
H
+
V
n
=
1
r
1 +
r
−
,
ξ
where the parameter
r
, characterizing the degree of reflection, varies within limits
from0upto1.
