Geoscience Reference
In-Depth Information
where
+ 2). The form of formula (4.42) corresponds to the known rela-
tion for the radius (diameter) of a crater [Schmidt, Holsapple (1982)]
R
S
C
=
R
I
1
3
.
22
δ
= 1
/
(2
α
β
ρ
I
ρ
T
1
/
3
C
T
1
.
24
,
V
I
g
R
I
(4.43)
where
and
C
T
are parameters depending on the properties of the target (wa-
ter, in this case). For water their values are
β
β
≈
0
.
22 (i.e.
α
= 1
/
(2
β
)
−
1
≈
1
.
27),
C
T
≈
1
.
88. By comparison of formulae (4.42) and (4.43) one can note that about
16% of the kinetic energy of the falling body is transformed into tsunami energy
(
0
.
16). Of course, this is an approximate estimate, and it is cor-
rect only if the quantity
ε
= 1
/
(2
·
3
.
22)
≈
is actually not subject to strong variations.
The quantity
q
present in formula (4.41) varies weakly with the size of the falling
celestial body,
R
I
, and of the density ratio
ε
ρ
w
. By comparison of formulae (4.42)
and (4.43) it is not difficult to obtain the following approximate dependence:
ρ
I
/
0
.
39
ρ
w
ρ
0
.
26
1
R
0
.
27
I
q
≈
.
(4.44)
I
In the case, when the density of the celestial body is three times that of the density
of water, (
ρ
I
/
ρ
w
= 3),thevalueof
q
varies between 0.1 (
R
I
= 50 m) and 0.054
(
R
I
= 500 m).
To simplify the calculations it is possible, instead of the cumbersome expres-
sions (4.41) and (4.43) to use approximate formulae that are valid for
V
I
= 20 km/s
and
ρ
I
/
ρ
W
= 3 [Ward, Asphaug (2002)],
R
3
/
4
I
R
C
≈
98
·
,
(4.45)
D
C
≈
0
.
64
·
R
C
.
(4.46)
In Fig. 4.15 the dependences (4.41) and (4.43) are shown by solid lines, the ap-
proximate relationships (4.45) and (4.46) by dotted lines. The cavity diameter is
usually 2.5-3 times greater than its depth. Thus, for example, a cosmic body of ra-
dius 200 m falling into the ocean creates a cavity of diameter about 10 km and depth
of the order of 3.5 km. Note that in the case of celestial bodies of radius
R
I
>
300 m
the calculated cavity depth
D
C
will, as a rule, exceed the ocean depth
H
. In this case,
a crater will not only form in the water, but also in the ocean bottom. To avoid over-
complicating the problem we shall further assume an effective cavity depth
D
ef
C
,
equal to the ocean depth, to be applicable, when
D
C
>
H
. The cavity radius is cal-
culated as previously, in this case.
Figure 4.16 presents a comparison of cavity shapes calculated using the proposed
parametrization (4.41) and (4.43) and obtained by detailed numerical simulation of
the process performed in [Crawford, Mader (1998)]. The complex non-linear model
and the parameterizations proposed are seen to be in quite reasonable agreement.
Noticeable divergence is only observed in the external circular structure, but for
preforming tsunami calculations at a level of estimations it is not too important.