Geology Reference
In-Depth Information
a
local compressibility number
, λ again representing the adiabatic bulk modulus
or incompressibility. The decompression factor, as a function of radius, can be
found by solution of the di
erential equation (6.184). For an initial value of the
decompression factor,
f
i
, at an initial radius
r
i
, its value at radius
r
is
ff
f
i
exp
r
dr
f
i
exp
r
dr
ρ
0
(
r
)
g
0
(
r
)
λ(
r
)
C
loc
r
f
(
r
)
=
−
=
−
.
(6.186)
r
i
r
i
If the initial radius is chosen as
b
, the radius of the core-mantle boundary, and the
value of the decompression factor there is taken as unity, then the decompression
factor as a function of radius becomes
exp
b
r
dr
.
ρ
0
(
r
)
g
0
(
r
)
λ(
r
)
f
(
r
)
=
(6.187)
The programme DECOMP calculates the decompression factor for the outer
core at each of the Earth model integration points for a specific Earth model. The
Earth model file name is entered at the request of the programme, and the output,
for use in subsequent calculations, is stored in the file decomp.dat. The integral in
the exponent of expression (6.187) for the decompression factor is evaluated using
Simpson's rule. For two successive model integration points at radii
r
1
and
r
3
,the
integrand g(
r
) has the values g(
r
1
)
=
g
3
. Through calls to the sub-
routines SPMAT and INTPL, described in Section 1.6, the value of the integrand at
the midpoint (
r
3
=
g
1
and g(
r
3
)
h
is found as g
2
by cubic spline interpolation.
By Simpson's rule, the contribution to the integral is
h
3
(g
1
+
−
r
1
)/2
=
r
2
=
r
1
+
4g
2
+
g
3
).
(6.188)
As can be shown by successive substitution of the functions 1,
r
,
r
2
and
r
3
,
Simpson's rule is third-order accurate; in other words, exact for the cubic spline
used in the approximation of the integrand.
C
PROGRAMME DECOMP.FOR
C
C DECOMP.FOR calculates the scalar decompression factor f for
C a particular Earth model. It first reads in the Earth model,
C converts it to SI units, and uses the subroutines SPMAT and INTPL
C to interpolate outer core values, using spline interpolation.
C The required quadrature for the calculation of the decompression
C factor as a function of radius is performed using Simpson's rule.
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION R(100),RHO(100),GZERO(100),RI(300),RHOI(300),F(100),
1 GZEROI(300),ENAME(10),NM(4),NI(4),NK(4),B(98,198),C(100,100)
DOUBLE PRECISION MU(100),LAMBDA(100),MUI(300),LAMBDAI(300),
1 LAMBDAM
CHARACTER*20 EMODEL
C Read in Earth model file name.
WRITE(6,10)
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