Geoscience Reference
In-Depth Information
Fig. 3.7 Maximum
amplitudes of surface
displacements for incom-
pressible (1) and compress-
ible (2) liquids at the centre
of the active zone versus
the propagation velocity of
the bottom displacement,
v
3.1.5 Peculiarities of Wave Excitation in a Basin of Variable Depth
Analytical resolution of the problem of movements of a compressible liquid in
a basin with an irregular bottom encounters significant complications, while in
the general case it is not even possible. Therefore, in studying peculiarities of the ex-
citation of elastic gravitational waves in a basin of
variable depth
it is expedient to
apply numerical simulation [Nosov, Kolesov (2003)]. It must be noted that numer-
ical methods have also to be applied in dealing with analytical solutions (for cal-
culating integrals). Given all the obvious advantages of analytical solutions, direct
numerical simulation often turns out to be much more efficient.
We shall consider the plane problem (3.4)-(3.6).
Numerical resolution implies using a region of finite dimensions for calculations.
Thus, besides the boundary conditions on the bottom and on the surface, condi-
tions must be formulated for the left and right boundaries of the calculation region.
As such conditions, the conditions for free second-order transition (of elastic waves)
were chosen [Marchuk et al. (1983)]:
2
F
2
F
∂
+
c
2
2
2
F
c
∂
t
−
∂
∂
= 0
,
x
=
x
min
,
x
max
.
(3.34)
∂
x
∂
t
2
∂
z
2
Equation (3.4) and the boundary conditions (3.5), (3.7) and (3.34) were re-
duced to a dimensionless form in accordance with formulae (
x
∗
,
z
∗
)=(
x
,
z
)
H
−
1
max
,
t
∗
=
tH
−
1
max
c
, where
H
max
is the maximum depth of the basin.
The distribution of depths chosen for calculations imitated transition from
the shelf zone through the continental slope towards the abyssal plain (Fig. 3.8a).
The parameter
L
= 80 km was not varied. The depths
H
1
and
H
2
were varied be-
tween 0.5 and 8.5 km. The maximum steepness of the slope amounted to 0.1. The
tsunami source was located on the slope and represented a displacement involving
residual deformation. The form of the space-time law of motion of the bottom
deformation,
(
x
,
t
)=
X
(
x
)
T
(
t
), is shown in Fig. 3.8b. Movement of the bottom
occurred in a direction normal to the surface (of the bottom). The displacement
duration varied between 1 and 100 s.
η
