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M
0
ρ
asth
ρ
w
M
1
ʔρ
ρ
0
¼
ʔρ
T
þ ʔρ
F
¼
ʱ
S
¼
ð
12
9
Þ
T
þ ʲ
F
ð
Þ
:
12
:
15
Sediment physical properties
The porosity varies with depth z:
where
ρ
0
is the mantle density at the surface,
ʱ
¼
3.3
10
5
K
1
is the
coefficient for change in density due to melt depletion (see
below). Melt depletion
F
is tracked by a parameter
X
, which
is a hypothetical completely compatible trace element. The
value of
X
increases from 1 as melting progresses. The trace
element can be related to melt depletion,
F
, assuming that
the melt and solid remain in equilibrium with each other, by
a simple mass balance,
F
is the thermal expansion coefficient and
ʲ
ðÞ
¼
˕
0
exp
z
z
c
˕
z
ð
12
10
Þ
:
where
˕
0
is the porosity at the surface and
z
c
the compaction
length. The thickness of a cell varies accordingly:
1)/
X
. We can then base
the initial density variation with depth due to melt depletion
on density changes predicted from melting experiments of
natural and model rocks. We assume that change in density
is a linear function of the removal of melt (Equation
12.15
).
This allows for
¼
(
X
ð
1
˕
0
Þ
ʔ
z
¼
ʔ
z
0
ð
12
:
11
Þ
ð
1
˕
ðÞ
z
Þ
where
z
0
are the compacted and decompacted
thicknesses of a cell. The thermal conductivity follows a
geometric mean in the model:
ʔ
z
and
ʔ
ʲ
to be calculated from a reference state of
melt depletion (Scott
1992
),
ʻ
bulk
¼
ʻ
˕ ðÞ
1
˕
ðÞ
matrix
z
ʻ
ð
12
12
Þ
:
ʔρ
ref
X
ref
ρ
0
X
ref
water
ʲ
¼
ð
12
16
Þ
:
1
0.6Wm
1
K
1
where
ʻ
bulk
is the bulk conductivity,
ʻ
water
¼
is the conductivity of water, and
ʻ
matrix
is the conductivity of
where X
ref
¼
23 %. From
the melting of model fertile lherzolite compositions and
estimates from natural rocks, complete clynopy-roxene
removal occurs at melt depletion of 23 %. This is our
reference state of melt depletion, and using
1.3, which is equivalent to F
¼
the rock matrix.
The density (and the specific volumetric heat) follow
harmonic means:
34 kg
m
3
as estimated for Proterozoic lithosphere (Poudjom
Djomani et al.
2001
),
ʔˁ
ref
¼
ρ
bulk
¼
ρ
water
˕
ðÞþρ
matrix
1
z
ð
˕
ðÞ
z
Þ
ð
12
13
Þ
:
0.04.
We consider a simple temperature dependent diffusion
creep that is a reasonable approximation for the deformation
of the Earth based on geological observations (Watts and
Zhong
2000
). Viscosity is given by,
ʲ
¼
where
ʻ
bulk
is the bulk density,
ʻ
water
is the density of water,
and
ʻ
matrix
is the density of the rock matrix.
Appendix 2: Model of Lithospheric Instabilities
E
RT
ʷ
¼
ʷ
0
ʷ
ref
exp
ð
12
17
Þ
:
To understand how a change in lithosphere thickness at the
margins of the Congo Basin will affect subsidence we form
an idealised model of viscous flow within a 2-D Cartesian
domain. We assume that deformation of the lithosphere and
asthenosphere can be effectively captured as a Stokes flow
with the Boussinesq approximation. The numerical model is
as outlined in Armitage et al. (
2008
,
2013
). The key equation
is the momentum balance,
120 kJmol
1
(Watts and Zhong
2000
). The scal-
ing viscosity,
where
E
¼
ʷ
0
to dimensionalise dimensionless viscosity,
is set from the thermal Rayleigh number,
Td
3
¼
ʱ
g
ρ
m
ʔ
¼ 4
:
877
10
6
Ra
ð
12
:
18
Þ
ʺʷ
0
∂
˄
ij
∂
p
x
j
þ
∂
where d
¼
700 km is the depth to the base of the model;
x
i
¼
ʔρ
g
ʻ
i
ð
12
:
14
Þ
10
6
m
2
s
1
∂
ʺ
¼
is the thermal diffusivity and
ʔ
T
¼
1,315 K is the temperature difference between the surface
and depth d. Using this Rayleigh number, the scaling mantle
viscosity is
where
˄
ij
is the deviatoric stress,
p
is pressure,
g
is gravity
and
ʻ
i
is a unit vector in the vertical direction. The change in
density,
ʷ
0
¼
10
20
Pa s. Finally
ʷ
ref
¼
1.129, which is
ʔρ
is a function of temperature and melt depletion,
calculated from a reference state (T
ref
¼
1,588 K) such that
F
,
viscosity
ʷ
¼
10
20
Pa s at 200 km depth. The relatively
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