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
=−∇·
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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