Geology Reference
In-Depth Information
For an oscillation with time dependence exp
i
ω
t
, from equation (6.173), the vec-
tor displacement field
u
is found to follow the equation of motion
2
u
−
ω
+
2
i
ω
Ω
×
u
=−∇
χ
−
β
g
0
∇·
u
,
(6.175)
with the generalised potential given as
p
1
χ
=
ρ
0
−
V
1
.
(6.176)
From the equation of state (6.168), the density perturbation is given by
1
α
ρ
1
=−
u
·∇
ρ
0
+
2
u
·∇
p
0
.
(6.177)
Flow pressure fluctuations are given by equation (6.169) as
2
p
1
=−
u
·∇
p
0
−
ρ
0
α
∇·
u
.
(6.178)
Again neglecting the e
ff
ects of flow pressure fluctuations, we have
1
α
2
u
·∇
p
0
=−
ρ
0
∇·
u
,
(6.179)
which, on substitution in the equation of state, yields
ρ
1
=−
u
·∇
ρ
0
−
ρ
0
∇·
u
=−∇·
(ρ
0
u
).
(6.180)
With substitution from the hydrostatic condition (6.157), expression (6.179)
provides
the subseismic form of the equation of continuity
,
1
α
∇·
u
=−
2
g
0
·
u
.
(6.181)
The displacement field obeying (6.181) can be shown to be reduced to an equi-
valent solenoidal vector field by multiplication by a scalar
decompression factor f
(Friedlander, 1985). Thus,
∇·
(
f
u
)
=
u
·∇
f
+
f
∇·
u
=
0,
(6.182)
provided that scalar
f
obeys
∇
f
f
=
g
0
α
2
=−
∇
U
0
α
2
,
(6.183)
where
U
0
is the geopotential. Scalar
f
then obeys the ordinary di
ff
erential equation
in the radius,
df
f
=−
C
loc
dr
r
,
(6.184)
with
dU
0
dr
r
α
ρ
0
g
0
r
λ
,
C
loc
=
2
=
(6.185)
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