Manuscript number: 2014GL060023
Auxiliary Material for
Effects of various lithospheric yield stresses and different mantle heating modes on the
breakup of the Pangea supercontinent
Masaki Yoshida
(Department of Deep Earth Structure and Dynamics Research, Japan Agency for
Marine–Earth Science and Technology (JAMSTEC), 2-15 Natsushima-cho, Yokosuka,
Kanagawa 237-0061, Japan)
Geophysical Research Letters, 2014
1
1. Model
1.1 Basic equations and parameters
The dimensionless conservation equations for mass, momentum, and energy, which
govern mantle convection under the Boussinesq approximation, and the advection
equation for the composition, are expressed, respectively, as:
  v  0,
(1)
N
N


p      v  v    RaT   Ra ph ( i )  i   Rach ( j )C j   3er  0,
i 1
j 1


(2)
T
 v  T   2T  Q(t ) 2 ,
t
(3)
ph
ch
tr
C j
t
 v  C j  0,
(4)
where v represents the velocity; p the dynamic pressure; η the viscosity; t the time; T the
temperature; Q the time-dependent radioactive heat production rate per unit mass; Γi the
phase function (0 ≤ Γi ≤1); Cj the composition (0 ≤ Cj ≤1); i the index of each phase in
the mantle; j the index of each material compositionally different from the mantle; and er
the unit vector in the radial direction.
The dimensionless parameters are the thermal Rayleigh number Ra, the phase
Rayleigh number Raph(i), the compositional Rayleigh number Rach(j), the internal
heating number Q, and the mantle/shell radius ratio ζ:
Ra 
 ph (i )
ch ( j )
0 0 Tgb3
Hb 2
b
, Ra ph ( i ) 
Ra, Rach ( j ) 
Ra, Q 
,   , (5)
0 0
0 0 T
0 0 T
 0 c p 0 T
r1
where the meanings and values of the other symbols are as listed in Table S1.
The time-dependent mantle heating rate is given by:
H  t   0.9928C0U H U
238
 t ln 2 
 t ln 2 
235
exp  U 238   0.0071C0U H U exp  U 235 




 1/2 
 1/2 
 t ln 2 
40
 t ln 2 
C0Th H Th exp  Th   1.19 104 C0K H K exp  K 40  ,


  1/2 
 1/2 
(6)
2
where C0U , C0Th , C0K are the concentrations of each radioactive isotope (uranium,
238
235
thorium, and potassium), H U , H U , H Th , H K
U
U
Th
K
isotope, and 1/2
, 1/2
,  1/2
,  1/2
238
235
40
40
are the rates of heat release of each
are the half-lives of each isotope [Turcotte and
Schubert, 2002]. The values are listed in Table S2.
References
Turcotte, D. L., and G. Schubert (2002), Geodynamics, 2nd ed., 456 pp., Cambridge Univ.
Press, U.K.
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Table S1. Model parameters used in the present study.
Symbol
g
Definition
Gravitational acceleration
Value
9.81
Unit
m s–2

Reference density of mantle
3300
kg m–3

Reference thermal expansivity of mantle
3 × 10–5
K–1
Ttop, Tbot
Temperatures at the top and bottom
surfaces
Temperature difference across the mantle
273, 2773
K
2500
K
r1 , r0
Radii of Earth and its core
6371, 3504
km
b (= r1−r0)
Thickness of mantle
2867
km
T (=
Tbot−Ttop)
0
Reference thermal diffusivity of mantle
10
m2 s–1
0
Reference viscosity of upper mantle
1021
Pa s
cp
Reference specific heat at constant pressure
1250
of mantle
J kg–1 K–1
Radioactive heat production rate per unit
mass
Clapeyron slope at 410-, 520-, and 660-km
W kg–1
H
-
phase transitions
Density contrast at 410-, 520-, and 660-km
phase transitions
–6
Timedependent
1.6, 4.3, −2.5 MPa K–1
7%, 3% 10%
Viscosity contrast between continents (C1 =
100
1) and oceans (C1 = 0)
Density contrast between continents and
100
surrounding mantle
Free
Yield stress for oceanic lithosphere
y
parameter
Yield stress contrast between continents
100
y
(C1 = 1) and oceans (C1 = 0)
Dimensionless parameters
c
kg m–3
MPa
-
Ra
Thermal Rayleigh number
5.72 × 107
Raph(i)
Phase Rayleigh numbers for 410-km (i =
1), 520-km (i = 2), and 660-km (i = 3)
phase transitions
5.34 × 107,
2.29 × 107, 7.63 × 107
-
4
Rach(j)
Compositional Rayleigh numbers for
continents (j = 1)
Q
Internal heating number

Mantle/shell radius ratio
Timedependent
0.45
E
Dimensionless activation energy
11.51
2.31× 107
-
Table S2. Rates of heat release H, half-lives 1/2, and concentrations C0 of the
radioactive isotopes in chondritic meteorites [Turcotte and Schubert, 2002].
Isotope
H [W kg1]
 [yr]
C0 [kg kg1]
238
U
9.46×105
4.47×10
-
U
5.69×104
7.04×10
235
U
232
40
Th
K
K
9.81×10
2.64×105
2.92×105
1.40×10
1.25×10
8×10
29×10
-
3.48×109
-
56×10
5
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Figure captions for supplemental figures
2014GL060023-fs01.jpg
Figure S1. Time sequence of the temperature anomaly (i.e., the deviation from the
horizontally averaged temperature) of the upper mantle at a depth of 493 km for models
with mixed heating mode of the mantle (Case Q1). The lithospheric yield stress y are: (a)
80 MPa, (b) 100 MPa, (c) 120 MPa, and (d) 140 MPa.
2014GL060023-fs02.jpg
Figure S2. Time sequence of the temperature anomaly (i.e., the deviation from the
horizontally averaged temperature) of the upper mantle at a depth of 493 km for models
with a purely basal heating mode of the mantle (Case Q2). The lithospheric yield stress
y are: (a) 80 MPa, (b) 100 MPa, (c) 120 MPa, and (d) 140 MPa.
2014GL060023-fs03.jpg
Figure S3. Time sequence of the drifting continents for the model with mixed heating
mode of the mantle (Case Q1). The lithospheric yield stress y is 200 MPa. The first, third,
and fourth panels of each figure show the configuration of drifting continents at 200, 100,
and 0 Ma. The blue region indicates the region of oceanic lithosphere, and the color-coded
region indicates the region of the continents. The white line contour shows the
temperature anomaly (i.e., the deviation from the horizontally averaged temperature) of
the upper mantle at a depth of 493 km. The contour intervals are 50 °C. The solid and
dashed lines represent positive and negative temperature anomalies, respectively. The
second panel of each sequence shows the distribution of viscosity (see color bars at the
bottom of each figure) and velocity at 200 Ma.
2014GL060023-fs04.jpg
Figure S4. Time sequence of the drifting continents for the model with mixed heating
mode of the mantle (Case Q2). The lithospheric yield stress y is 200 MPa. The first, third,
and fourth panels of each figure show the configuration of drifting continents at 200, 100,
and 0 Ma. The blue region indicates the region of oceanic lithosphere, and the color-coded
region indicates the region of the continents. The white line contour shows the
temperature anomaly (i.e., the deviation from the horizontally averaged temperature) of
the upper mantle at a depth of 493 km. The contour intervals are 50 °C. The solid and
dashed lines represent positive and negative temperature anomalies, respectively. The
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second panel of each sequence shows the distribution of viscosity (see color bars at the
bottom of each figure) and velocity at 200 Ma.
2014GL060023-fs05.jpg
Figure S5. Time sequence of the temperature anomaly (i.e., the deviation from the
horizontally averaged temperature) of the upper mantle at a depth of 493 km the model
with mixed heating mode of the mantle (Case Q1). The lithospheric yield stress y is 200
MPa.
2014GL060023-fs06.jpg
Figure S6. Time sequence of the temperature anomaly (i.e., the deviation from the
horizontally averaged temperature) of the upper mantle at a depth of 493 km for the model
with a purely basal heating mode of the mantle (Case Q2). The lithospheric yield stress
y is 200 MPa.
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