Geoscience Reference
In-Depth Information
Note that the theory of long waves does not always describe processes at
the tsunami source correctly. As it was shown in Chap. 3, three-dimensional
formulation of the problem for describing processes at the source may happen
to be quite essential.
In a Cartesian reference system the equations of non-linear theory of long waves,
with account of the bottom friction and the Coriolis force, has the following form:
C
B
U
√
U
2
+
V
2
D
∂
+
U
∂
+
V
∂
g
∂ξ
∂
U
∂
U
∂
U
∂
=
−
x
−
+
fV
,
(5.29)
t
x
y
C
B
V
√
U
2
+
V
2
D
∂
V
∂
+
U
∂
V
x
+
V
∂
V
g
∂ξ
∂
=
−
y
−
−
fU
,
(5.30)
t
∂
∂
y
x
DU
+
y
DV
= 0
,
∂ξ
∂
∂
∂
∂
∂
+
(5.31)
t
ξ
ξ
1
D
1
D
U
=
u
d
z
,
V
=
v
d
z
,
−
H
−
H
where
U
,
V
are the flow velocity components, averaged over the depth, along
the axes 0
x
and 0
y
, respectively,
is the free-surface displacement from the equilib-
rium position,
D
(
x
,
y
,
t
)=
H
(
x
,
y
)+
ξ
ξ
(
x
,
y
,
t
) is the thickness of the water column,
g is the acceleration of gravity,
f
= 2
ω
sin
ϕ
is the Coriolis parameter,
ω
is the an-
gular velocity of the Earth's rotation,
is the latitude and
C
B
is a dimensionless
empirical coefficient, which is usually set to 0.0025. There also exist more precise
models, which take into account the dependence of quantity
C
B
on the thickness of
the water column. Thus, for example, the following dependence is applied in [Titov
et al. (2003)]:
ϕ
g
n
2
D
1
/
3
,
C
B
=
(5.32)
where
n
is the Manning coefficient, the value of which depends on the roughness of
the bottom surface. A typical value of the Manning coefficient for a coast free from
dense vegetation, amounts to
n
= 0
.
025 s/m
1
/
3
.
Note that formula (5.32) yields the value
C
B
= 0
.
0025 for a water column
of
D
≈
15 m. The dependence of
C
B
(
D
) is weak (
C
B
(1 m)
≈
0
.
006,
C
B
(100 m)
≈
0
.
0013), therefore, the results of calculations of wave dynamics carried out as-
suming the coefficient
C
B
to be constant and with account of the dependence (5.32),
should not differ strongly from one another. We recall that the bottom friction does
practically not influence tsunami propagation at large depths.
In certain models another form is used for writing the non-linear equations of
the theory of long waves ('in total fluxes'),
C
B
M
√
M
2
+
N
2
D
2
M
2
D
+
∂
∂
MN
D
=
∂
M
∂
+
∂
∂
g
D
∂ξ
∂
−
x
−
+
fN
,
(5.33)
t
x
y
C
B
N
√
M
2
+
N
2
D
2
MN
D
+
N
2
D
=
∂
∂
∂
∂
∂
g
D
∂ξ
∂
N
∂
+
−
y
−
−
fM
,
(5.34)
t
x
y
