Geoscience Reference
In-Depth Information
Substituting formulae (3.39) into equations (3.37) and (3.38) and averaging over
the period of 'fast' oscillations, we obtain a set of equations for describing the aver-
age movement of the liquid,
−
v
,
∇
v
+
ρ
∇
p
ρ
∂
v
−
∇
p
+(
v
,
∇
)
v
=
+ g
,
(3.40)
ρ
2
∂
t
div
ρ
v
.
∂
ρ
∂
+ div (
ρ
v
)=
−
(3.41)
t
In performing the averaging we applied rules, similar to the Reynolds rules, ap-
plied in turbulence theory.
In the case of an incompressible liquid (range 'II') the mean is calculated from
the period of the ocean bottom oscillations, and the average motion can, obviously,
be described as the flow of an incompressible liquid. If the liquid is compressible
(range 'III'), then as the period for averaging one should take the quantity 4
H
max
c
,
where
H
max
is the maximum depth of the basin. It is known that acoustic modes with
periods superior to 4
H
max
c
do not exist, consequently, in this case, also, the mean
movement can be described as the flow of an incompressible liquid. Taking into
account that
ρ
= const and neglecting the term quadratic in the average velocity,
(
v
,
∇
)
v
, one arrives at the following system:
−
v
,
∇
v
+
ρ
∇
p
∂
v
−
∇
ρ
p
=
+ g
,
(3.42)
2
∂
t
ρ
div
ρ
v
.
1
ρ
div (
v
)=
−
(3.43)
The expressions obtained differ from the usual linearized Euler equations for an
incompressible liquid by the presence of the following new terms:
≡−
v
∇
v
+
∇
p
2
−
v
,
∇
v
+
ρ
∇
p
Φ
=
2
,
(3.44)
2
2
c
2
ρ
ρ
div
ρ
v
≡−
div
p
v
,
1
ρ
1
−
s
=
(3.45)
c
2
ρ
which can be interpreted as a force field
and a distributed source of mass,
s
.
The origin of the new terms is due to the non-linearity of the initial equations. The
combined action of the force field and the distributed source of mass under certain
conditions is capable of causing long gravitational waves. We shall speak of this
action as a 'non-linear tsunami source'.
For calculating the waves caused by the action of the force field and the dis-
tributed source of mass, we shall apply the linear theory of long waves. The expedi-
ence of choosing this theory is, first of all, explained by the fact that we are interested
in large-scale motions correlated in space (i.e. long waves), and, moreover, this way
seems the most simple one.
Φ
