Geoscience Reference
In-Depth Information
u
0
(0)
1 +
4
C
B
u
0
(0)
3
u
0
(
t
)=
,
(5.25)
t
π
H
where
u
0
(0) is the velocity amplitude at time moment
t
= 0. Taking advantage of
the
relat
ionship between the free-surface d
ispl
acement and the flow velocity
u
≈
ξ
g
/
H
and taking into account that
x
=
t
√
g
H
, we obtain an expression describing
variation of the wave amplitude along the horizontal coordinate,
ξ
0
(0)
1 +
x
/
L
2
,
ξ
0
(
x
)=
(5.26)
H
2
4
C
B
ξ
0
(0)
3
π
where
L
2
=
is the distance, along which the wave amplitude becomes
two times smaller.
1
We shall call this quantity the non-linear dissipation length.
We shall point out a number of special features, distinguishing viscous (linear)
and non-linear damping of long waves from each other. First, the actual character of
damping is different: in the first case it is exponential, while in the second it is hy-
perbolic. Second, the characteristic distance, along which noticeable wave damping
occurs (
L
1
and
L
2
), is related to different parameters of the problem. The quantity
L
1
depends on the wave frequency and on the basin depth, while the quantity
L
2
de-
pends on the wave amplitude and depth. In both cases the distance
L
i
increases with
the depth
H
, but in the case of non-linear damping this dependence is stronger.
Figure 5.8 presents the dependences of dissipation lengths
L
1
and
L
2
upon
the ocean depth. Calculations are performed for characteristic ranges of tsunami
wave frequencies (10
−
4
-10
−
2
Hz) and amplitudes (0.1-10 m) for the coefficient
C
B
= 0
,
0025, viscosity
= 10
−
6
m
2
/s. From the figure it is seen that for condi-
tions of the open ocean,
H
>
10
3
m, viscous and non-linear friction cannot influ-
ence tsunami wave propagation in any noticeable way. For dissipative effects to
ν
Fig. 5.8 Tsunami wave dis-
sipation length versus ocean
depth. Curves 1, 2—viscous
(linear) dissipation, 3, 4—
non-linear dissipation. The
calculation is performed for
C
B
= 0
.
0025,
= 10
−
6
m
2
/s.
Curve 1—10
−
4
Hz, 2—
10
−
2
Hz, 3—0.1 m, 4—10 m.
For comparison, the dotted
line shows a distance equal
to the length of the Earth's
equator
ν
(m)
1
With a precision up to a numerical coefficient, this quantity is in accordance with the result
obtained in the topic [Pelinovsky (1996)].
