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

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 149 (2005) 259–278 263
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-