van den Berg et al., 2005, Earth Planetary Science Letters.

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 Berg a,∗ , E.S.G. Rainey b,c , D.A. Yuen c 

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: berg@geo.uu.nl (A.P. van den Berg); 

emma@gps.caltech.edu (E.S.G. Rainey); davey@msi.umn.edu 

(D.A. Yuen). 

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 

0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved. 

doi:10.1016/j.pepi.2004.10.008

260 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 T CMB (t) − T surface decreases with the cooling 

of the core. 

tational domain has been considered throughout. This Symbols used in (1)–(4) are defined in Table 1. In 

same aspect-ratio was employed in our previous works Eq. (4) D/Dt denotes the substantive derivative. H(t) 

(van den Berg and Yuen, 1998; van den Berg et al., is an exponential decaying function and R is a nondimensional 

measure of radiogenic strength. For the 

2002). 

The mantle convection model is based on the internal heating of the model we use a uniform distribution 

with exponential time dependence H(t) char- 

extended Boussinesq approximation for an infinite 

Prandtl number, incompressible fluid (Steinbach et al., acterized by a half-life time of 2.5 Gyr, and an initial 

1989). In this model conservation of mass, momentum value of the internal heating number R equal to 20, 

and energy and the constitutive rheological relation are corresponding to an internal heating, which is about a 

expressed in the following non-dimensional equations factor of two stronger than the present-day chondritic 

DT 

termediate case of several models with different initial 

CMB temperature T CMB (0). We used temperature 

= ∂ j (κ(T, P)∂ j T ) + αDiw(T + T 0 ) 

Dt 

+ Di 

contrasts of 3000, 3500 and 4000 K, corresponding to 

Ra + RH(t) (4) initial core–mantle boundary temperatures T CMB (0) of 

∂ j u j = 0 (1) 

value. 

For the non-dimensionalization scheme we used the 

− ∂ i P + ∂ j τ ij = αRaTδ i3 (2) depth of the convecting layer h as the spatial scale and 

τ ij = η(T, P)(∂ j u i + ∂ i u j ) (3) 

h 2 /κ 0 , 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 in-


Table 1 

Physical parameters 

Symbol Definition Value Unit 

h Height of the mantle model 3 × 10 6 m 

z Depth coordinate aligned with gravity – – 

P Static pressure – – 

P Dynamic pressure – – 

T Temperature – – 

T surface Surface temperature 273 K 

T Temperature scale 3500 K 

u i Velocity field component – – 

e ij = ∂ j u i + ∂ i u j Strain rate tensor – – 

[ 

e = 

1 

2e ij e ij 

] 1/2 

Second invariant of strain rate – – 

w Vertical velocity aligned with gravity – – 

η(T, z) = η 0 exp(cz − bT ) Temperature and pressure/depth dependent viscosity – – 

η 0 Viscosity scale value – Pa s 

τ ij = ηe ij Viscous stress tensor – – 

= ηe 2 Viscous dissipation function – 

α(z) = 

α 

[c(1 − z) + 1] 3 Depth dependent thermal expansivity – 

α = α(1) – – 

c = α 1/3 − 1 – 

α 0 Thermal expansivity scale value 2 × 10 −5 K −1 

ρ Density – – 

ρ 0 Density scale value 4000 kg m −3 

c p Specific heat 1250 J K −1 kg −1 

k Thermal conductivity – – 

k 0 Conductivity scale value 4.7 W m −1 K −1 

a Conductivity powe-law index 0.3 – 

γ Grueneisen parameter 1.2 – 

K 0 Bulk modulus 261 GPa 

K 0 ′ Pressure derivative of bulk modulus 5 – 

b 0 Coefficient photon conductivity 1.7530 × 10 −2 

b 1 

−1.0365 × 10 −4 

b 2 

2.2451 × 10 −7 

b 3 

−3.4071 × 10 −11 

κ = 

k 

ρc p 

Thermal diffusivity – – 

g Gravitational acceleration 9.8 m s −2 

( ) −t 

H(t) = H 0 exp 

τ 

Time-dependent internal heating – W kg −1 

τ Dimensional decay time of radioactive heating 3.6 Gyr 

H 0 Dimensional value of internal heating – W kg −1 

R = H 0h 2 

c p κ 0 T 

Non-dimensional internal heating number 20 – 

Ra = ρ 0α 0 gTh 3 

κ 0 η 0 

Rayleigh number – – 

Di = α 0gh 

c p 

Dissipation number 0.47 – 

q C (t) Average heatflow density at the CMB – – 

X Ratio of core to mantle heat capacity 0.44 –


3273, 3773 and 4273 K. The initial temperature field 

of the mantle in the cases with different T CMB (0) were 

computed by applying an appropriate uniform scaling 

factor to the initial mantle temperature of the intermediate 

case (T CMB (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)/ρc p we use 

the temperature- and pressure-dependent conductivity 

Hofmeister model (Hofmeister, 1999) 

( ) 298 a 

k(T, P) = k 0 

T 

× exp 

× 

[ ( 

− 4γ + 1 ) 

] 

α(P)(T − 298) 

3 

( 

1 + K′ 0 P ) 

+ 

K 0 

3∑ 

fb i T i (5) 

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, 

∂k lat /∂T < 0 and increases with increasing pressure, 

∂k lat /∂P > 0. The photon term on the other hand 

increases with temperature ∂k rad /∂T > 0 and is insensitive 

to pressure ∂k rad /∂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 z b to obtain a resistance profile 

R(z) through thermal boundary layers of the mantle, 

the lithosphere and the CMB region 

R(z) = 

∫ z 

z b 

〈 1 

k(z ′ ) 

〉 

dz ′ (6) 

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) = η 0 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) (η P ) and temperature (η T ), respectively. 

The value of η P is fixed at 100. For most cases, the 

value of η T 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 T C is controlled 

by the average heat-flow from the core–mantle bound-


ary q C described by the ordinary differential equation 

(Steinbach et al., 1993) 

dT C 

dt 

A 

= − q C (t) (8) 

ρ C c PC V C 

where A is the area of the core–mantle boundary surface 

and C C = ρ C c PC V C is the total heat capacity of the 

core. The core heat capacity is expressed as a fraction 

X of the mantle heat capacity C M , resulting in an O.D.E. 

for T C (t): 

dT C 

dt 

= − 1 

Xρc p h q C(t) (9) 

where ρ and c p are the mantle values of density and 

heat capacity. Parameter values are given in Table 1. 

At each time step T C is updated by integrating (9), 

using the average heatflow value q C , 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 

T CMB (t) = T C (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 T CMB (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 

T CMB (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 η T 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 

η T , is non-monotonic. It appears that cooling is 

slightly slower for η T = 3 × 10 3 (Fig. 3b) than for 

η T = 3 × 10 2 (Fig. 3a). Increasing the temperature 

to η T = 3 × 10 4 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 η T . 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.


Fig. 2. Snapshots of the temperature field for two different initial CMB temperatures T CMB = 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 η T = 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 T CMB (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 η T . 

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, k a (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 k a (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 

k a (z). The viscosity model is kept the same in this com-


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, η T = 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, η T = 3 × 10 3 , η P = 10 2 . 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 T CMB (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 T CMB (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.


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 k a (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). 

ent values of the initial CMB temperature T CMB (0). For 

each value of T CMB (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, T CMB (0), 

results in an increase of the overall cooling rate as is 

apparent from the increasing slope of the 〈T (t)〉 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, T CMB (0) = 3273 K, 3773 K and 4273 K. 

The different curves in each group represent models with different 

conductivity models. The effect of increasing T CMB (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. 

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 T CMB (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 T CMB (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 T CMB (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-


Fig. 7. Time evolution of the rate of secular cooling, d〈T 〉/dt, for different initial CMB temperatures, T CMB (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-


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, 

k rad = 0, and the cooling rate increases with increasing relative contribution 

of k rad controlled by the amplification factor f. Furthermore 

this effect is stronger in a hotter earth in line with the temperature 

dependence ∂k rad /∂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 

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 k rad 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 k rad 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 

T CMB (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 T CMB 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


Fig. 10. Temperature (left) and streamfunction (right) snapshots for time t = 4.428 Gyr for models with initial CMB temperature T CMB (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 T CMB (0) = 3273, 3773 and 4273 K.


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 T CMB (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) = T CMB (t) − 

T A (z CMB , t), between the actual CMB temperature and 

the extrapolated CMB temperature T A (z CMB , t) of a 

mantle adiabat obtained by a least squares estimate, 

for the depth range between 1000 and 1800 km depth,


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 T CMB (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.


Fig. 14. Evolution in time of the temperature contrast across the thermal 

boundary layer at the CMB. Different initial CMB temperature 

are shown T CMB (0) = 3273, 3773, 4273. Each panel shows results 

for different conductivity models, constant k, f = 0, 1 and 5. 

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, k rad = 0, f = 0 (green), amplified k rad , 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.


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 

k rad 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 properties 

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


intensity record, through a possible influence on the 

geodynamo. 

4. A higher value of the initial CMB temperature, 

T CMB (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). 

5. A higher initial core–mantle boundary temperature 

leads to faster secular cooling of the mantle and 

faster core cooling. 

6. 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 T CMB (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. 

7. There is a greater predominance of smaller convective 

upwellings for variable k, especially in cases 

with a high initial T CMB (0). In this model, large 

plumes are less prevalent, and the lower mantle may 

have less capacity to create superplumes. 

8. 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 conductivity. 

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-


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 T CMB (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 10 8 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 thermal 

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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