Physics of the Earth and Planetary Interiors 149 (2005) 259–278
The combined influences of variable thermal conductivity,
temperature- and pressure-dependent viscosity and
core–mantle coupling on thermal evolution
A.P. van den Berga,∗ , E.S.G. Raineyb,c , D.A. Yuenc
a
Department of Theoretical Geophysics, Institute of Earth Sciences, Utrecht University, 3508 TA Utrecht, Netherlands
b Planetary Sciences Division, Code-150-21, Caltech, Pasadena, CA 91125, USA
c Department of Geology and Geophysics and University of Minnesota Supercomputing Institute,
University of Minnesota, Minneapolis, MN 55455-0219, USA
Received 21 July 2003; received in revised form 27 August 2004; accepted 13 October 2004
Abstract
Most convection studies of thermal history have not considered explicitly the thermal interaction between the mantle flow and
the core. We have investigated the influences of variable thermal conductivity and variable viscosity (temperature- and pressuredependent) on the boundary layer and thermal characteristics of the D"" layer, and the evolution of the thermo-mechanical profiles
of horizontally averaged viscosity and thermal conductivity. Viscosity contrast due to temperature dependence of up to 30,000 has
been considered. Our results show clearly that variable thermal conductivity, though small in magnitude as compared to variations
in the viscosity, does exert a significant delaying influence on mantle cooling, thereby keeping the Urey ratio low, reducing the
growth of the bottom thermal boundary layer, and changing the viscosity profiles over time. A higher temperature at the core–
mantle boundary increases the overall time-dependent behavior of the thermal boundary layers. Enhanced radiative conductivity
results in faster cooling, opposite to the effect of the phonon conductivity component and a superadiabatic temperature gradient
in the deep lower mantle. Finally, the initial value of the core–mantle boundary temperature can be inferred to wield a strong
influence on the subsequent mantle thermal evolution in this model with both variable thermal conductivity and viscosity. We
may conjecture that other rheological and conductivity complexities, such as grain-size dependence of mantle properties, would
also have an impact on the current state of the mantle resulting from the primordial thermal condition.
© 2004 Elsevier B.V. All rights reserved.
1. Introduction
∗
Corresponding author.
E-mail addresses: [email protected] (A.P. van den Berg);
[email protected] (E.S.G. Rainey); [email protected]
(D.A. Yuen).
0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.pepi.2004.10.008
For over three decades now, the thermal evolution of the mantle has been studied with the convection paradigm with the feedback mechanism of
temperature-dependent viscosity being emphasized
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A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
(e.g. Tozer, 1972). The most extensive work using numerical convection models on thermal history has been
conducted with depth-dependent viscosity (Butler and
Peltier, 2002).
While the effects of temperature- and pressuredependent viscosity are well known in the steady state
(e.g. Christensen, 1985), not much work has been carried out in thermal evolution with variable viscosity, save for the work by DeLandro-Clarke and Jarvis
(1997). Moreover, a model for mantle thermal conductivity based on phonon solid-state physics and infrared
spectroscopy was developed by Hofmeister (1999).
Contributions from both phonon and photon conductivity are included in this model. Recently on the basis
of spectroscopic work, Badro et al. (2004) have argued
for the importance of radiative thermal conductivity in
the deep mantle.
van den Berg et al. (2002) and van den Berg and
Yuen (2002) have shown that variable thermal conductivity can delay the secular cooling of the mantle with
a constant viscosity model. Since variations of viscosity in the course of thermal evolution are much greater
than changes in the thermal conductivity, it is therefore
important to evaluate the influence of variable viscosity
on the effects of delayed secular cooling and also the
stabilization of boundary layer activities at the core–
mantle boundary (CMB) from increasing the radiative
contribution to the thermal conductivity (Dubuffet et
al., 2002).
Another important aspect of thermal history is the
influence from thermal coupling of the mantle to the
core. This was first studied within the framework of
one-dimensional parameterized convection models by
Sharpe and Peltier (1978) and Schubert et al. (1979)
and in fully two-dimensional (Steinbach et al., 1993;
Honda and Yuen, 1994) and three-dimensional (Yuen
et al., 1994) convection models. Buffett et al. (1992)
and recently Buffett (2003) pointed out the importance
of core–mantle interactions in thermal-chemical evolution.
The heat flux at the core–mantle boundary (CMB) is
of particular interest for planetary evolution. It controls
the relative partitioning between bottom heating and
internal heating in the lower mantle, and it also has important implications for the geodynamo and the chemical composition the core. The current CMB heat flux is
relatively poorly constrained, but recent estimates indicate that the heat flux may be much higher than early
estimates, perhaps as high as 12 TW (Buffett, 2003).
An important minimum constraint on core heat flux is
the heat necessary to drive the geodynamo and generate a magnetic field, which has probably existed for at
least 3.5 Gyr. Best estimates indicate that prior to the
solidification of the inner core, known sources of heat
in the core are insufficient to drive a magnetic dynamo.
40 K, which is depleted in the mantle, was suggested
as a possible source of radioactivity in the core that
could provide heat necessary for the geodynamo (e.g.,
Hall and Murthy, 1971; Gessman and Wood, 2002).
Although potassium was not thought be a siderophile,
recent experimental evidence shows that 40 K can enter
iron sulphide melts under core conditions (Murthy et
al., 2003). The amount of 40 K in the core can be constrained by the core heat budget, which depends on how
much heat is conducted from the core into the mantle.
For obtaining better estimates of the heat flux at the
CMB, it is necessary to use a model that includes a
realistic mantle thermal conductivity, especially in the
lower mantle, where radiative conductivity effects can
be stronger than the phonon conductivity (Yuen et al.,
2000) and can also be enhanced by iron concentration
in the D"" layer (Manga and Jeanloz, 1996).
In Section 2 we describe the models for 2D mantle convection and the thermal conductivity. In subsequent sections we focus respectively on the the effects
of varying initial CMB temperatures, the enhancement
of radiative thermal conductivity and the temporal development of the thermal structure of the mantle in the
core-coupling model.
In the final section we will state our conclusions and
offer our perspectives for the role played by variable
viscosity acting in concert with variable thermal conductivity in particular in view of the effects of enhanced
radiative conductivity as indicated in recent mineral
physics results (Badro et al., 2004), and core–mantle
coupling in shaping the thermal history.
2. Description of the convection, conduction
and viscosity models
We use a 2D mantle convection model including
thermal coupling to the core. Fig. 1 shows a diagram
of the cartesian computational domain, illustrating the
thermal coupling between mantle and core included
in our model. An aspect-ratio of 2.5 for the compu-
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
261
Fig. 1. Domain diagram showing the earth’s thermally coupled mantle and core in a spherical configuration (top) and in a cartesian 2D box of
aspect ratio 2.5, used in the numerical mantle convection model. The core is represented by an isothermal heat reservoir, thermally coupled to
the convecting mantle. This core reservoir is cooled by the heat flux into the mantle driven by the temperature contrast δT across the bottom
boundary layer of the mantle. In this model the temperature contrast across the convecting mantle TCMB (t) − Tsurface decreases with the cooling
of the core.
tational domain has been considered throughout. This
same aspect-ratio was employed in our previous works
(van den Berg and Yuen, 1998; van den Berg et al.,
2002).
The mantle convection model is based on the
extended Boussinesq approximation for an infinite
Prandtl number, incompressible fluid (Steinbach et al.,
1989). In this model conservation of mass, momentum
and energy and the constitutive rheological relation are
expressed in the following non-dimensional equations
∂j u j = 0
(1)
− ∂i #P + ∂j τij = αRaTδi3
(2)
τij = η(T, P)(∂j ui + ∂i uj )
(3)
DT
= ∂j (κ(T, P)∂j T ) + αDiw(T + T0 )
Dt
Di
+
( + RH(t)
Ra
(4)
Symbols used in (1)–(4) are defined in Table 1. In
Eq. (4) D/Dt denotes the substantive derivative. H(t)
is an exponential decaying function and R is a nondimensional measure of radiogenic strength. For the
internal heating of the model we use a uniform distribution with exponential time dependence H(t) characterized by a half-life time of 2.5 Gyr, and an initial
value of the internal heating number R equal to 20,
corresponding to an internal heating, which is about a
factor of two stronger than the present-day chondritic
value.
For the non-dimensionalization scheme we used the
depth of the convecting layer h as the spatial scale and
h2 /κ0 , a thermal diffusion time of the layer, as the time
scale. The temperature scale #T corresponds to the
initial temperature contrast across the layer, of an intermediate case of several models with different initial CMB temperature TCMB (0). We used temperature
contrasts of 3000, 3500 and 4000 K, corresponding to
initial core–mantle boundary temperatures TCMB (0) of
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Table 1
Physical parameters
Symbol
Definition
Value
Unit
h
z
P
#P
T
Tsurface
#T
ui
eij = ∂j ui + ∂i uj
"1/2
!
1
e=
2eij eij
w
η(T, z) = η0 exp(cz − bT )
η0
τij = ηeij
( = ηe2
#α
α(z) =
[c(1 − z) + 1]3
#α = α(1)
c = #α1/3 − 1
α0
ρ
ρ0
cp
k
k0
a
γ
K0
K0"
b0
b1
b2
b3
k
κ=
ρcp
g
# $
−t
H(t) = H0 exp
τ
τ
H0
H 0 h2
R=
cp κ0 #T
ρ0 α0 g#Th3
Ra =
κ0 η0
α0 gh
Di =
cp
qC (t)
X
Height of the mantle model
Depth coordinate aligned with gravity
Static pressure
Dynamic pressure
Temperature
Surface temperature
Temperature scale
Velocity field component
Strain rate tensor
3 × 106
–
–
–
–
273
3500
–
–
m
–
–
–
–
K
K
–
–
Second invariant of strain rate
–
–
Vertical velocity aligned with gravity
Temperature and pressure/depth dependent viscosity
Viscosity scale value
Viscous stress tensor
Viscous dissipation function
–
–
–
–
–
–
Pa s
–
–
Depth dependent thermal expansivity
–
–
–
–
K−1
–
kg m−3
J K−1 kg−1
–
W m−1 K−1
–
–
GPa
–
Thermal expansivity scale value
Density
Density scale value
Specific heat
Thermal conductivity
Conductivity scale value
Conductivity powe-law index
Grueneisen parameter
Bulk modulus
Pressure derivative of bulk modulus
Coefficient photon conductivity
2 × 10−5
–
4000
1250
–
4.7
0.3
1.2
261
5
1.7530 × 10−2
−1.0365 × 10−4
2.2451 × 10−7
−3.4071 × 10−11
Thermal diffusivity
–
–
Gravitational acceleration
9.8
m s−2
Time-dependent internal heating
–
W kg−1
Dimensional decay time of radioactive heating
Dimensional value of internal heating
3.6
–
Gyr
W kg−1
Non-dimensional internal heating number
20
–
Rayleigh number
–
–
Dissipation number
0.47
–
Average heatflow density at the CMB
Ratio of core to mantle heat capacity
–
0.44
–
–
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
3273, 3773 and 4273 K. The initial temperature field
of the mantle in the cases with different TCMB (0) were
computed by applying an appropriate uniform scaling
factor to the initial mantle temperature of the intermediate case (TCMB (0) = 3773 K). The latter is obtained
from a statistically steady-state equilibrium model run
with enhanced (R = 40) and constant internal heating
with a zero heat flux bottom boundary.
For the thermal diffusivity κ = k(T, P)/ρcp we use
the temperature- and pressure-dependent conductivity
Hofmeister model (Hofmeister, 1999)
#
$
298 a
k(T, P) = k0
T
! #
$
"
1
× exp − 4γ +
α(P)(T − 298)
3
#
$
3
%
K" P
× 1+ 0
+
fbi T i
(5)
K0
i=0
In (5) the first term gives the phonon contribution to the
effective conductivity and the second term is the contribution from photon transport. The amplification factor
f (e.g. van den Berg et al., 2002) of the photon term,
valued f = 1 in the Hofmeister model (Hofmeister,
1999), is used here as a control parameter to vary the
relative contribution of both mechanisms in the effective thermal conductivity. We consider in particular
models with f values of 0, 1, 2 and 5 to investigate
the impact of the radiative thermal conductivity on the
model behavior. We have not considered the grain-size
dependence of thermal conductivity, which can vary
non-monotonically with depth (Hofmeister, 2004).
The phonon term decreases with increasing temperature, ∂klat /∂T < 0 and increases with increasing pressure, ∂klat /∂P > 0. The photon term on the other hand
increases with temperature ∂krad /∂T > 0 and is insensitive to pressure ∂krad /∂P = 0.
To interpret the numerical modelling results we will
also use the thermal resistivity, defined as the inverse of
the thermal conductivity r = 1/k, in analogy with the
theory of electricity. One-dimensional depth profiles of
horizontally averaged resistivity can then be integrated
from a boundary point zb to obtain a resistance profile
R(z) through thermal boundary layers of the mantle,
the lithosphere and the CMB region
(
& z'
1
R(z) =
(6)
dz"
")
k(z
zb
263
This thermal resistance has been in use in geothermics as a means of obtaining reliable heatflow estimates from bore holes with strongly fluctuating conductivity profiles (Beardsmore and Cull, 2001; Bullard,
1939).
For the rheological model we have chosen an exponential temperature and depth (pressure) dependent
viscosity for Newtonian rheology
η(T, z) = η0 exp(cz − bT )
(7)
where c, b are defined in Table 1 in terms of the viscosity contrasts across the convecting layer due to depth
(pressure) (#ηP ) and temperature (#ηT ), respectively.
The value of #ηP is fixed at 100. For most cases, the
value of #ηT is 3000, but for comparison we also show
some contrasting cases.
Eqs. (1), (2) and (4) are solved by using finite element methods for the spatial discretization, and applying a penalty function method for the continuity equation and Stokes momentum equations (1) and (2). The
energy equation (4) which drives the time dependent
system is integrated in time using a predictor corrector method (van den Berg et al., 1993). The finite element mesh consists of 150 × 140 nodal points in the
horizontal and vertical direction, respectively. Mesh refinement was applied near the horizontal boundaries,
where the vertical nodal point spacing was reduced to
6 km from a value of 30 km in the interior domain.
Mesh refinements near the thermal boundary layers
is essential in calculations using variable conductivity due to the occurrence of strong temperature gradients and similar sharp variations in the effective thermal conductivity in the boundary layer (see Yuen et
al., 2000), which need to be resolved numerically. Especially the computation of the surface heat-flux requires a very high resolution of the finite element mesh
(van den Berg et al., 2001).
We use free-slip impermeable boundaries and a
fixed top surface temperature of 273 K. On the vertical boundaries a zero heat flux symmetry condition was applied. The model runs were started
from a statistically steady state obtained for a zero
heat flux bottom boundary and constant internal
heating.
Thermal coupling between mantle and core is represented by an isothermal heat reservoir of the core,
shown in Fig. 1, where the temperature TC is controlled
by the average heat-flow from the core–mantle bound-
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A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
ary qC described by the ordinary differential equation
(Steinbach et al., 1993)
dTC
A
qC (t)
=−
dt
ρC cPC VC
(8)
where A is the area of the core–mantle boundary surface
and CC = ρC cPC VC is the total heat capacity of the
core. The core heat capacity is expressed as a fraction
X of the mantle heat capacity CM , resulting in an O.D.E.
for TC (t):
dTC
1
=−
qC (t)
dt
Xρcp h
(9)
where ρ and cp are the mantle values of density and
heat capacity. Parameter values are given in Table 1.
At each time step TC is updated by integrating (9),
using the average heatflow value qC , computed from
the finite element solution for the mantle temperature
field. This is done by forward extrapolation in time
of the heat flux in an Euler type scheme. The updated
uniform core temperature is then taken as a time dependent boundary condition for the finite element computation of the mantle temperature in the next time step, so
TCMB (t) = TC (t) in (9). We note that analytical asymptotic methods (Solomatov and Zharkov, 1990) for treating thermal history, though capable of handling variable viscosity, may be hard-pressed to apply in the case
of variable thermal conductivity.
3. Results
A comparison of a series of temperature snapshots
for typical secular cooling runs for two different values
of the TCMB (0) = 3273 and 4273 K is shown in Fig. 2.
The snapshots represent a time span of about the age
of the Earth, illustrating the effects of different initial
TCMB (0), showing faster cooling for the hotter initial
temperature case.
In previous work we have investigated the impact
of variable conductivity on the secular cooling of the
convecting mantle, restricted to isoviscous models (van
den Berg and Yuen, 2002) and to models with temperature dependence of the viscosity limited to around
1000 (van den Berg et al., 2004). Here we consider
models including pressure and temperature dependent
viscosity given by (7) with a higher value of the temperature dependence. In contrast to previous work we
also focus on the opposite role of the different heat
transport mechanisms (phonons versus photons) in the
composite conductivity model. Finally we introduce
here thermal coupling between the convecting mantle
and a thermal reservoir representing the core. Photon
conductivity is likely to be dominant in the lower mantle and hence important for thermal core/mantle coupling. Phonon conductivity is suppressed by the 1/T
temperature dependence.
The effect of varying #ηT on the overall cooling
history of the coupled mantle and core system is presented in Figs. 3 and 4. The cooling curves in Fig. 3,
show the volume averaged temperature of the mantle,
comparing also the variable conductivity cases with the
corresponding constant conductivity cases. The corresponding constant conductivity cases have the same
surface conductivity value as the variable conductivity
cases. For the conductivity model with f = 1 this also
corresponds to an approximately similar value of the
volume average conductivity. This was also shown in
previous work, Fig. 7 of van den Berg and Yuen (2002).
The main feature of the results shown in Figs. 3 and
4 is a similar cooling delay of the variable conductivity models with respect to the corresponding constant
conductivity models. This amounts to an accumulated
delay time of almost two billion years at a model time of
4.5 Gyr. These results indicate that the delay we found
earlier in secular cooling in isoviscous models with
variable conductivity is a robust phenomenon, also in
models including variable viscosity. The results also
show that the trend in the cooling rate, for increasing #ηT , is non-monotonic. It appears that cooling is
slightly slower for #ηT = 3 × 103 (Fig. 3b) than for
#ηT = 3 × 102 (Fig. 3a). Increasing the temperature
to #ηT = 3 × 104 the cooling rate increases also (Fig.
3c). This non-monotonic trend is the same for constant
conductivity models represented by the dashed curves.
The small difference in the overall cooling behavior between the three contrasting viscosity cases is surprising
inview of the significant difference of the interior viscosity shown for these cases shown in Fig. 4. It appears
from these results that one cannot simply apply the temperature dependence of the viscosity to predict an increased cooling rate for increasing #ηT . The pressure
dependence of the viscosity complicates the details of
the resulting temperature distribution which feeds back
into the viscosity, resulting in this non-monotonic behavior.
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
265
Fig. 2. Snapshots of the temperature field for two different initial CMB temperatures TCMB = 3273 and 4273 K. The variable conductivity model
used is the same in both cases with f = 1. Different temperature scales have been used between the initially hotter and cooler model cases. The
difference in the thermal evolution of the convecting mantle is illustrated by a series of snapshots spanning the age of the earth. The hot model
in the righthand column shows a significantly faster cooling than the initially cooler model. The hot model also shows smaller scale convective
features then the cooler model.
Depth profiles of temperature and viscosity are
shown in Fig. 4 for three cases with contrasting temperature dependence of the viscosity #ηT = 300, 3000
and 30,000. The conductivity model used is the same in
all three cases, corresponding to the Hofmeister (1999)
model with f = 1. The initial value TCMB (0) is 3773 K
in all cases and the snapshot corresponds to an integration time of 4.428 Gyr. Internal temperatures for the
three cases shown are roughly similar, with a larger
temperature difference of several hundred degrees in a
layer of 500 km above the CMB. The variation in the
corresponding horizontally averaged viscosity is between one and two orders of magnitude between the different model cases inline with the differences in #ηT .
In order to investigate the mechanism behind the
cooling delay of the variable conductivity models we
have applied a 1D depth dependent conductivity model.
The corresponding conductivity, ka (z), is computed
from the horizontally averaged conductivity, taken
from a variable (Hofmeister, 1999) conductivity model
with f = 1 substituted in (5). The 1D profile is defined as the time-averaged value, for an averaging time
window of 5 Gyr, of horizontally averaged conductivity snapshots. The result of this space and time averaging of the conductivity is shown in Fig. 5a. The
small variation of the effective conductivity profiles
over time, due to the secular cooling, is illustrated by
the width of the bundle of black curves. The time averaged profile ka (z) is represented by the red curve.
We compared the thermal history of the variable conductivity model with the model based on the 1D profile
ka (z). The viscosity model is kept the same in this com-
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A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
Fig. 3. Thermal evolution curves showing volume averaged temperature of the convecting mantle against integration time %T (t)&, for three
models characterized by different values of the temperature dependence of the viscosity, #ηT = 300, 3000 and 30,000. The variable
conductivity model used is the same in all cases with f = 1. Results
for variable k, f = 1 are compared with corresponding constant conductivity models. A strong delaying effect on secular cooling of variable conductivity is clearly shown by these results. Furthermore this
delay is robust for increasing values of the temperature dependence
of the viscosity.
parison, #ηT = 3 × 103 , #ηP = 102 . Time series of
global quantities for both cases are shown in Fig. 5b
and c. Fig. 5b shows the evolution of the volume averaged mantle temperature and the temperature of the
core heat reservoir, indicating almost identical thermal
evolution. Fig. 5c shows a corresponding times series of
the CMB and surface heat flux. The equivalence of the
heat flux level for both models shown in this frame is in
agreement with the coincidence of the thermal history
curves in Fig. 5b. Note that the heat flux level has the
right order of magnitude for Earth. Experiments with
purely pressure dependent conductivity, using the same
surface value, have shown a negligible cooling delay
compared to the full pressure and temperature dependent conductivity models (van den Berg et al., 2004).
These results show that the delay in secular cooling
of the variable conductivity models can be ascribed
to the average 1D structure of the conductivity profile
and in particular to the low conductivity zone (LCZ) at
shallow depth. The effect of this LCZ is to increase the
thermal resistance of the lithosphere, which suppresses
conductive heat transport through the lithosphere and
results in delayed secular cooling.
3.1. Effect of initial CMB temperature TCMB (0) on
mantle evolution
In Fig. 6 we show the evolution over time of the
volume averaged temperature %T (t)&, for three differ-
Fig. 4. 1D depth profiles of horizontally averaged temperature (left) and viscosity (right), for the same models as in Fig. 3. The results shown
correspond to an integration time of 4.428 Gyr and an initial CMB temperature TCMB (0) = 3773 K for all models. The main differences in the
temperature are in the bottom boundary layer, which is also reflected in the shape of the viscosity hill in the deep lower mantle. The difference in
the viscosity profiles are up to about two orders of magnitude, in line with the small temperature differences and the viscosity parameter values.
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
267
Fig. 5. Lefthand frame: (a) (black) Snapshots, evenly spaced in time
between 0 and 4.5 Gyr, of horizontally averaged conductivity profiles, (red) time averaged conductivity profile computed from the
black profiles. Righthand frames: Time series comparing the thermal evolution of a variable conductivity model (f = 1 denoted by
black curves) and a time averaged 1D depth dependent conductivity
model ka (z), shown in frame (a), including a shallow low conductivity zone, denoted by red curves. (b) Core temperature (top curves)
and average mantle temperature (bottom curve). (c) Heat flux through
CMB (lower curves) and Earth’s surface (top curves).
Fig. 6. Thermal evolution curves showing volume averaged temperature of the convecting mantle against integration time %T (t)&. The
three curve groups are labeled with the corresponding values of the
initial CMB temperature, TCMB (0) = 3273 K, 3773 K and 4273 K.
The different curves in each group represent models with different conductivity models. The effect of increasing TCMB (0) is an increased slope representing a higher cooling rate. Furthermore the
constant conductivity model cases (dashed lines) show the fastest
secular cooling. The largest cooling delay is obtained for the models with f = 0, corresponding to absence of radiative conductivity.
The effect of increasing the radiative component (increasing f ) is to
speed up the cooling rate.
ent values of the initial CMB temperature TCMB (0). For
each value of TCMB (0), represented by the three curve
groups in Fig. 6, we also show a comparison of results
for different conductivity models, including a constant
conductivity case represented by the dashed curves and
four variable conductivity cases for different values of
the multiplication factor f for the radiative component
of thermal conductivity. In general, the constant conductivity models show faster mantle cooling than most
of the variable conductivity cases, only surpassed by the
variable k models with strongly enhanced conductivity
(f = 10). Among the variable conductivity models the
model with f = 10 shows the fastest cooling which is
related to the different structure of the low conductivity zone (shown below) for shallow depth. The models
with f = 0 and 1 show a similar cooling history with
an overall cooling delay of about 2 Gyr with respect to
the corresponding constant conductivity case.
The effect of the initial CMB temperature, TCMB (0),
results in an increase of the overall cooling rate as is
apparent from the increasing slope of the %T (t)& curves.
At the same time the cooling delay between the variable
k and constant k remains fairly constant. These results
with variable viscosity corrobarate our earlier results
for simpler models with constant viscosity and without
thermal coupling between mantle and core (van den
Berg et al., 2002; van den Berg and Yuen, 2002).
Cooling rates of the convecting mantle are shown
in Fig. 7 for the same three values of the initial CMB
temperature TCMB (0). The cooling rates, d%T &/dt, were
computed by a central difference approximation of
the time series of the volume averaged temperature
%T (t)&. For each value of TCMB (0) a variable conductivity model with f = 1 is compared with a corresponding
constant conductivity case. The trend between the different panels is an increase of the cooling rate in line
with a similar trend in the slopes of %T (t)& shown in Fig.
6. Increasing values of the initial CMB temperature are
also reflected in the degree of time dependence. The
higher TCMB (0) cases are characterized by rapid fluctuations of the cooling rate, in contrast to the smooth
curves of the volume averaged temperature. This is re-
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A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
Fig. 7. Time evolution of the rate of secular cooling, d%T &/dt, for different initial CMB temperatures, TCMB (0) = 3273 K (a), 3773 K (b) and
4273 K (c). A comparison is made between constant conductivity (black curves) and variable conductivity (f = 1). The lower absolute values
of the variable conductivity models is consistent with the delay in secular cooling of the corresponding curves in Fig. 6. Increasing the initial
CMB temperature results in a higher cooling rate in line with Fig. 6 and a much stronger time dependence of the the cooling rate.
lated to the increasing fluctuations in the surface heat
flow, which contribute to the secular cooling, with increasing Rayleigh number (van den Berg et al., 1993;
van den Berg and Yuen, 2002). The delay in secular
cooling of the variable conductivity cases, apparent in
Fig. 6, is reflected in Fig. 7 by the consistently lower
cooling rates for the variable conductivity curves labeled f = 1.
In Fig. 8 we show time series of the temperature of
the core heat reservoir for the same models as used for
Fig. 6, i.e. three different initial core temperatures and
five different conductivity models. These core cooling
results show a similar trend as in Fig. 6 for the mantle.
However, there is a difference in the sensitivity to the
contribution of the radiative conductivity expressed in
the amplification factor f. The core temperature seems
more sensitive for increased f values. This is due to the
fact that models with f = 5 (red curves) show already
faster core cooling than the constant conductivity runs,
whereas Fig. 6 shows that the switch to faster mantle cooling for increased f occurs later, between f = 5
and 10. An explanation of this different behavior is that
the accelerating effect of f on mantle cooling, resulting in an increasing effect on the temperature contrast
between mantle and core, is compounded with the enhanced cooling by radiative heat transport across the
CMB for increased values of f. This sensitivity is also
increasing for higher initial core temperatures, in agreement with the temperature dependence of the radiative
conductivity.
Time evolution of the CMB heat flux is shown in
Fig. 9, for several model cases with the same initial core
temperatures as in Figs. 6 and 8. The results show that
the variable conductivity with f = 0 produces the lowest core heat flux. For the constant conductivity case,
represented by the dashed line, the core heat flux is
relatively high and for the variable conductivity cases
with enhanced conductivity, for 10 the highest core heat
flux is obtained. The trend in these core heat flux results
is consistent with the corresponding core temperature
curves shown in Fig. 8, in agreement with the fact that
core temperature is obtained in our model by integrating the cmb heat flux according to (9).
Evolution of the heat flux from the core has been
studied mainly in parameterized models characterized
by smooth time variations (Buffett, 2003). Our model
results show a remarkably high fluctuation level of the
core heat flux and one could speculate that such fluctu-
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
Fig. 8. Thermal histories of the core for the same models as shown
in Fig. 5. Three curve groups are labeled with the initial core temperature. Each group includes results for different conductivity models.
The trends in these curves illustrate the key role of radiative conductivity in controlling core cooling. The slowest core cooling is
obtained for the f = 0 model without any radiative conductivity,
krad = 0, and the cooling rate increases with increasing relative contribution of krad controlled by the amplification factor f. Furthermore
this effect is stronger in a hotter earth in line with the temperature
dependence ∂krad /∂T > 0.
ations could impact the geodynamo process and leave
their marks in the paleomagnetic intensity record.
3.2. Influence from enhanced thermal radiative
conductivity
As discussed above, the parameter f represents a
measure of the radiative contribution to the thermal
conductivity, with f = 1 having the same value as the
model presented by Hofmeister (1999). Fig. 10 compares the two-dimensional temperature fields for values of constant thermal conductivity, and variable conductivity with f = 1 and 0 (purely lattice conductivity) and f = 5 (enhanced radiative conductivity). It is
clear that with variable k, the entire convective region
is hotter, in comparison to constant k (Dubuffet et al.,
1999, 2002). As was observed in the previous work using constant viscosity (van den Berg et al., 2002), we
find that the Hofmeister variable conductivity results
in a reduced overall convective vigor. The constant
k case has stronger downwellings and earlier, betterdeveloped plumes. For different values of f, there is
269
Fig. 9. Heat flow through the core–mantle boundary against time,
for the same model cases as in Fig. 7. The core heatflow increases
with the relative contribution of the radiative conductivity krad in
agreement with the core cooling histories shown in Fig. 7. Core heat
flux is highly time dependent in these models with peak to peak
values of about 100%. This is a result of the strong time dependence
of cold downwellings cooling the hot core in these models which are
largely cooled from above.
still present a noticeable difference in the temperature
fields between the constant k and variable k models, although less so in the enhanced krad case f = 5, which
has the fastest cooling rate of the variable k models
considered here (Fig. 6).
The viscosity profiles for the entire mantle are
shown in Fig. 11 for constant conductivity, f = 1 and
5 and three different values of initial CMB temperature
TCMB (0). It is interesting to note that a sharper low viscosity valley is produced by a lower temperature at the
CMB.
For constant viscosity Dubuffet et al. (2002) have
noted that there is a bifurcation in the behavior of the
convective solution, as f is increased beyond a certain value, which depends on the TCMB and on the
amount of internal heating. In Fig. 12 we show the
one-dimensional profiles of the horizontally averaged
temperature %T &, the viscosity %η& and the thermal conductivity %k& for f = 0, 1 and 5.
The temperature profiles show that the effect of introducing variable conductivity is to make the thermal
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Fig. 10. Temperature (left) and streamfunction (right) snapshots for time t = 4.428 Gyr for models with initial CMB temperature TCMB (0) =
3773 K, and different thermal conductivity.
Fig. 11. Global 1D depth profiles of horizontally averaged viscosity, for integration time t = 5.161 Gyr, for different conductivity models,
constant k (a), variable conductivity with f = 1 (b) and f = 5 (c). In each frame different curves are shown for models with different initial
CMB temperature TCMB (0) = 3273, 3773 and 4273 K.
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
271
Fig. 12. Global 1D depth profiles of horizontally averaged temperature (left), conductivity (middle) and viscosity (right). Results are for the
same initial CMB temperature of TCMB (0) = 3773 K and for an integration time t = 2.952 Gyr.
boundary layer thinner at the top and thicker at the bottom, compared to the constant conductivity case. This
can be interpreted in terms of the different systematics
of the thermal resistance profiles defined in (6), through
the top and bottom boundary layers, as discussed in
more detail below. Increased thermal resistance of the
top boundary layer has in increasing effect on the temperature contrast across the lithosphere. Similarly the
decreased resistance near CMB results in a decreased
temperature contrast across CMB.
We see that the viscosity profiles are all very similar
but that the conductivity and temperature profiles reveal sharp changes with the amount of enhanced radiative conductivity from subadiabatic to superadiabatic
gradient. Thus one cannot casually employ a constant
value thermal gradient in the lower mantle for determining the viscosity profile (Yamazaki and Karato,
2001). There is a dramatic variation in the shape of
%k& for values of f exceeding 3. This ‘transition’ is also
reflected in the character of the temperature gradient in
the bottom part of the mantle. The temperature gradient
for f = 5 shows a superadiabatic character, in contrast
to the models with lower f values, which show a subadiabatic geotherm in the bottom parts of the mantle.
We obtained similar results in models with a zero heat
flux bottom boundary condition (van den Berg et al.,
2002). The super adiabatic geotherm is consistent with
the mineral physics result of da Silva et al. (2000) on
the basis of the bulk modulus variation with depth. This
superadiabatic character of the geotherm may indicate
an enhanced radiative heat transfer in the deep mantle.
This high temperature gradient in the lower mantle, due
to enhanced radiative heat transfer, is also reminiscent
of temperature distributions resulting from an abyssal
source of radiogenic heating invoked in the deep mantle
model by Kellogg et al. (1999).
The evolution of the thermal-mechanical structure
near the CMB is shown in Fig. 13, where we plot
the %T &, %η& and %k& profiles for constant conductivity,
f = 0, 1 and 5. The effect of variable thermal conductivity is to retard the growth of the thermal boundary
layer. With larger values of f the growth rate of the
boundary layer approaches that associated with a constant thermal conductivity.
We see that the more efficient heat transfer in the
case of f = 5 gives rise to a cooler lower mantle temperature and hence a shallower trough in the viscosity
at the CMB.
The time evolution of the temperature contrast
δT across the thermal boundary layer at the CMB
is shown in Fig. 14. These temperature contrasts
where calculated as the difference, δT (t) = TCMB (t) −
TA (zCMB , t), between the actual CMB temperature and
the extrapolated CMB temperature TA (zCMB , t) of a
mantle adiabat obtained by a least squares estimate,
for the depth range between 1000 and 1800 km depth,
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A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
Fig. 13. Profiles of horizontally averaged temperature (left column), conductivity (middle column) and viscosity (righthand column) for the
bottom 300 km of the mantle. Models shown are for an initial CMB temperature TCMB (0) = 3773 K. Different curves in each panel correspond
to different integration times, t = 0, 2.36 and 5.02 Gyr, and the direction of increasing time is indicated by the arrows. The columns shown
correspond to different conductivity models, constant k (top), f = 1, 0 and 5.
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
Fig. 14. Evolution in time of the temperature contrast across the thermal boundary layer at the CMB. Different initial CMB temperature
are shown TCMB (0) = 3273, 3773, 4273. Each panel shows results
for different conductivity models, constant k, f = 0, 1 and 5.
273
based on the horizontally averaged 1D temperature profile. The constant k models show the highest temperature contrast, as are also illustrated in the temperature
profiles of Fig. 12. The temperature contrast for the
variable k cases increases with f between f = 0 and 5.
A more rapid development of a higher temperature contrast δT may explain a larger tendency towards early
plume formation from the CMB in the constant conductivity cases. Considering the effects of the structure
of the bottom thermal boundary layer on the cooling of
the core, we see that the high δT value for the constant
k case is apparently compensated by a lower conductivity value, as shown in Fig. 12(middle), resulting in
an intermediate core heat flux, which can be observed
clearly in Fig. 9.
More insight can be obtained in the trends in the
model results for enhanced radiative conductivity by
comparing profiles of the 1D thermal resistance defined
Fig. 15. Vertical profiles of horizontally averaged conductivity %k& (left), resistivity %1/k& (middle) and corresponding thermal resistance R(z)
(right). Blue and green curves correspond to snapshots, evenly spaced in time from 0 to 4.55 Gyr, for two models with contrasting contribution
of the radiative conductivity, krad = 0, f = 0 (green), amplified krad , f = 10 (blue). The top row of frames shows a zoom in on the top 500 km
of the model. The bottom row shows the bottom boundary layer.
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A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
in (6). This quantity represents an integrated measure of
the variation with depth of the parameters controlling
conductive heat transport, conductivity and resistivity.
Since the depth variation of the resistance is monotonic, its dependence on the parameterization of the
conductivity model, like the f parameter value, is more
straight forward than the conductivity or resistivity profiles. Like the electric resistance of a layer corresponds
to the electric voltage required to drive a unit current
through the layer, the thermal resistance corresponds
to the necessary temperature contrast to drive a unit
heat flow through the layer. The simple behavior of the
thermal resistance in the thermal boundary layers may
provide a basis for the development of parameterized
convection models including effects of variable conductivity on thermal history.
Fig. 15 shows 1D depth profiles of conductivity, resistivity and corresponding thermal resistance through
the top and bottom thermal boundary layers for different mantle convection models. Two cases with contrasting contribution of the radiative conductivity are
shown. The green curves correspond to a purely phonon
conduction case (f = 0) and the blue curves are for
strongly enhanced radiative conductivity (f = 10).
The black curves indicate the constant conductivity
reference case. The trend in the models for enhanced
radiative conductivity is clearly reflected in the monotonic resistance profiles. In the top thermal boundary
layer, top frames, the contribution from the radiative
conductivity controls the resistance profile (c) where
the constant conductivity case is intermediate between
the contrasting variable conductivity models. This reflects the trend in the cooling curves for the mantle
shown in Fig. 5, which explains the strong impact of
krad in speeding up secular cooling, through its influence on the low conductivity zone and the resulting
thermal resistance of the lithosphere. In the bottom
boundary layer, bottom frames, a similar relation exists
except that the constant conductivity case has the highest resistance values, corresponding to the minimum
value of the conductivity shown in frame (d).
4. Discussion and conclusions
We have developed a core–mantle coupling convection model within the framework of a cartesian 2D geometry. This model has many realistic transport proper-
ties built in, such as variable thermal conductivity and
variable viscosity. It does not have surface plates, phase
transitions and chemical heterogeneities. But, nonetheless, this study will shed some light on the nature
of the thermo-mechanical structure in the deep lower
mantle.
4.1. Summary of important findings
Variable thermal conductivity affects both conductive and convective cooling mechanisms in the mantle.
Introducing pressure- and temperature-dependent thermal conductivity along with temperature- and pressuredependent viscosity into the mantle convection model
results in several important changes in the cooling behavior and mantle flow patterns:
1. The secular cooling rate of the mantle is lower, using the Hofmeister conductivity model (Hofmeister,
1999) (f = 1) than with constant thermal conductivity. Heat flux at the surface is reduced. Increased
f (greater radiative contribution to thermal conductivity) tends to increase the cooling rate relative to
the Hofmeister model. This thermal conductivity
mechanism, acting in concert with partial melting
(Korenaga, 2003) can help to retain a lot of the primordial heat of the Earth. Therefore, variable thermal conductivity can keep the Urey ratio low, which
is consistent with highly depleted heat-producing
elements in the mantle (Jochum et al., 1983), favored by geochemists.
2. We have shown that the cooling delay of the variable conductivity models is closely linked to the
formation of a low conductivity zone (LCZ) at
shallow depth. This LCZ results from the negative
temperature derivative of the dominant lattice conductivity. These results imply that purely pressuredependent conductivity models, characterized by a
monotonic increase of the conductivity with depth
are not suitable for long-term thermal history calculation (Anderson, 1987; Steinbach, 1991; Solheim
and Peltier, 1994; Tackley, 1996; van Keken, 2001;
Butler and Peltier, 2002).
3. Our model results show a high fluctuation level of
the heat flow from the core into the mantle, with
higher fluctuations for models with enhanced radiative conductivity. We speculate that such fluctuations could leave an imprint in the paleomagnetic
A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278
4.
5.
6.
7.
8.
intensity record, through a possible influence on the
geodynamo.
A higher value of the initial CMB temperature,
TCMB (0), leads to a more stable boundary layer due
to the increase in radiative conductivity with temperature. It is important to note that core–mantle
boundary temperature is also important for boundary layer stability (Sevre et al., 2002).
A higher initial core–mantle boundary temperature
leads to faster secular cooling of the mantle and
faster core cooling.
The local maximum of viscosity (viscosity hill) in
the lower mantle and the region of low viscosity
below the hill also change with variable k. The viscosity hill is smaller for variable k compared with
constant k. Increased TCMB (0) (or decreased viscosity) also leads to a decrease in the size of the
viscosity hill in the lower mantle and at the same
time maintains a shallower viscosity gradient in the
lower mantle.
There is a greater predominance of smaller convective upwellings for variable k, especially in cases
with a high initial TCMB (0). In this model, large
plumes are less prevalent, and the lower mantle may
have less capacity to create superplumes.
The geotherm in the deep mantle becomes superadiabatic (da Silva et al., 2000) with enhanced values of radiative thermal conductivity, which controls the magnitude of the thermal gradient in the
lower mantle (Yamazaki and Karato, 2001). Recent
suggestion by Badro et al. (2004) has added support to the importance of radiative heat transfer in
the deep mantle.
These results clearly demonstrate that temperatureand pressure-dependent thermal conductivity, though
small in magnitude, can not be neglected in mantle dynamics and planetary thermal evolution, through the
combined nonlinear feedback interactions among thermal conductivity, temperature and viscosity.
4.2. Applications to the core
We have found that with core coupling the heat flux
at the CMB depends on both conductivity in the lower
boundary layer and the evolving temperature difference across the lower boundary layer, both of which
are significantly impacted by variable thermal conduc-
275
tivity. The heat flux at the core–mantle boundary shows
large fluctuations and is reduced for variable thermal
conductivity with a normal amount of radiative conductivity (f = 1). These results would suggest that the
heat flux out of the core may actually be less than that
predicted by models with constant conductivity. If less
heat is conducted from the core into the mantle, this
would delay draining the core’s internal energy supply. This would decrease the minimum necessary concentration of radioactive elements such as 40 K in the
core. However, the CMB heat flux with variable conductivity is strongly dependent on initial CMB temperature and on the relative strength of radiative conductivity in the lower mantle, which is represented in this
model by the parameter f (e.g. Dubuffet et al., 2002).
These parameters are not known with enough certainty
to provide tight enough constraints on the CMB heat
flux.
4.3. The role in 1D parameterized convection
calculations
Because of the nonlinearities in mantle rheology,
mantle thermal conductivity and mantle convection,
thermal evolution with convection is definitely influenced by its initial temperature condition for a period
usually called the thermal adjustment time (Solomatov,
2001). After the thermal adjustment time, thermal
histories calculated from different initial conditions
should converge. Early thermal evolution models based
on parameterized convection, found a thermal adjustment time of less than 1 Gyr (e.g. Schubert et al., 1979);
however, when more complex rheology is taken into
account, the thermal adjustment time for parameterized models becomes much longer (see discussion by
Solomatov, 2001). Our 2D convection results for different values of initial CMB temperature indicate that the
initial thermal state affects mantle thermal evolution
for a period of time greater than the age of the Earth.
After around 5 Gyr, there is still a significant difference
in interior temperature, core temperature, and thermal
profiles for cases with different initial core CMB temperature. This implies that for a fully convective system with strongly temperature- and pressure-dependent
viscosity and variable thermal conductivity, the initial
thermal state is an important parameter to be considered, in modeling thermal evolution and this cannot be
neglected, as in early hypotheses of fast thermal adjust-
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ment with only temperature-dependent viscosity (e.g.,
Tozer, 1972).
4.4. Limitations of the model and perspectives
Since small-scale upwellings appear to be important convective cooling mechanisms for the model with
variable conductivity, especially in cases with a high
initial TCMB (0), it is important to use a grid with high
enough resolution to accurately represent small-scale
features. A major limitation in the applicability of the
results discussed to the real Earth is a result of the
two-dimensional rectangular geometry of the model
domain. For a spherical domain, the ratio of interior
volume to outer surface area is larger than in the rectangular case. This means that in a spherical domain
there will be a reduction in interior temperature (Zhang
and Yuen, 1996). Core–mantle dynamics are also influenced by the particular geometry. In a rectangular domain, the size of the core- mantle boundary interface is
equal to the size of the mantle-surface interface. These
are not equal in a domain with curved geometry: for
cylindrical or spherical geometry, the size ratio of the
core–mantle boundary to the mantle-surface boundary
is less than one. This will impact the coupled cooling of the mantle and core system (van Keken, 2001).
Jarvis (1993) investigated the effects of using a curved
domain in mantle models and found that the relative
thickness of upper and lower boundary layers and the
temperature drop across the boundary layers depended
on the degree of curvature. For the curvature of the
Earth’s mantle boundaries, the interior temperature is
also lower by several hundred degrees (Jarvis, 1993).
Thus, to apply the results of numerical convection models to the Earth with a reasonable degree of accuracy,
it is preferable to use a 3D spherical convection model
(e.g. Tackley et al., 1993; Monnereau and Yuen, 2002)
but with variable properties built-in. But this will be
a computational challenge because of the high spatial
resolution involved, with at least 108 grid points. In
future work the role of the new post-perovskite phase
change near the bottom of the mantle (Murakami et
al., 2004) needs to be explored since large changes in
the physical properties, including thermal conductivity
(Badro et al., 2004), may be expected (Tsuchiya et al.,
2004).
Finally, our results point to the potential role played
by complex transport properties, such as variable ther-
mal conductivity and grain-size dependence of material
properties (Solomatov, 1996, 2001; Hofmeister, 2004),
on making the thermal evolution of the mantle more dependent on the initial condition, because of the slower
secular cooling, which induce other modes of planetary
evolution (Sleep, 2000).
Acknowledgments
We acknowledge thorough and constructive reviews
by Paul Tackley and Thorsten Becker which greatly
helped to improve the manuscript. We thank discussions with Tomo K.B. Yanagawa, Anne M. Hofmeister, Fabien W. Dubuffet, Marc Monnereau, Renata M.
Wentzcovitch, Slava Solomatov and Erik O.D. Sevre.
This work has been supported by the geophysics program of the National Science Foundation and the Dutch
NWO.
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