Geoscience Reference
In-Depth Information
speaking, necessary to consider two modes—inertial and viscous (quasistation-
ary) [Simpson (1987)]. After the landslide body has suddenly become free (from
its initial state) the flow of the fluid forming the landslide undergoes transition
from the inertial-mode state to the viscous-mode state, when the vertical profile
of the flow has already been established. In the given model we assume the tran-
sition time to be negligible, and the flow to be constantly in the quasistationary-
mode, adapting relatively slowly, in the process of movement, to the shape of
the bottom relief.
4. In this model we neglect mixture effects on the landslide-water boundary. This
means that no exchange of mass takes place between the flow of sedimentary
material and the water.
Owing to the condition of adhesion the tangential velocity component at the ocean
bottom must turn to zero. At the upper boundary of the landslide absence is assumed
of tangential tensions, i.e. the normal component of the velocity gradient turns to
zero. Under such conditions the horizontal velocity of the stationary flow of a fluid
exhibits a parabolic vertical profile,
2
)
,
U(
x
,
y
,
z
,
t
)=U
m
(
x
,
y
,
t
)(2
ξ
−
ξ
(4.1)
=(
z
+
h
s
)
/
D
is the dimensionless vertical coordinate.
The continuity and momentum balance equations for a viscous flow in a land-
slide, obtained from the equations of hydrodynamics by integration along the verti-
cal coordinate with account of formula (4.1), have the following form:
ξ
where
∂
D
∂
+
2
3
(
∇
·
D
U)=0;
(4.2)
t
ρ
2
(
ρ
1
∇η
−
2
3
∂
U
2
15
U
D
∂
D
∂
8
15
(U
g
2
U
D
2
ν
t
−
+
·
∇
)U =
−
ρ
2
−
ρ
1
)
∇
(
D
−
h
s
)+
.
∂
t
(4.3)
Here the condition must be fulfilled that the landslide flow across the boundary of
the coastal line
G
always be zero, and that during its movement the landslide does
not cross the external (free) boundary
.
The upper layer of the fluid (water) is described by non-linear equations of mo-
tion in the approximation of shallow water:
Γ
∂
(
h
+
η
)
+[
∇
·
(
h
+
η
)u]=0;
(4.4)
∂
t
∂
u
+(u
·
∇
)u =
−
g
∇η
.
(4.5)
∂
t
Actually, the generation of surface waves by a moving landslide body is only due to
the continuity equation (4.4). The waves further propagate under the condition that
boundary conditions and the non-linear equation (4.5) be satisfied.
On the open external boundary of the region,
Γ
, the one-dimensional emission
(g
/
h
)
1
/
2
, where
u
n
is the velocity
condition for outgoing waves is applied:
u
n
=
η
