Tc

Aa

b

c

a

ARRA

EVPM

1

eR(lDrp

(

0d

Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Contents lists available at ScienceDirect

Physics of the Earth and Planetary Interiors

journa l homepage: www.e lsev ier .com/ locate /pepi

he dynamical impact of electronic thermal conductivity on deep mantleonvection of exosolar planets

.P. van den Berga,∗, D.A. Yuenb,c, G.L. Beebec, M.D. Christiansenc

Department of Theoretical Geophysics, Institute of Earth Sciences, Utrecht University, 3508 TA Utrecht, NetherlandsDepartment of Geology and Geophysics, University of Minnesota, Minneapolis, MN 55455-0219, USAUniversity of Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0219, USA

r t i c l e i n f o

rticle history:eceived 29 October 2007eceived in revised form 9 November 2009ccepted 11 November 2009

xosolar planetsariable conductivityhase transitionantle convection

a b s t r a c t

We have modelled the time-dependent dynamics of exosolar planets within the framework of a two-dimensional Cartesian model and the extended-Boussinesq approximation. The mass of the super-Earthmodels considered is 8 times the Earth’s mass and the thickness of the mantle is 4700 km, based on aconstant density approximation and a similar core mass fraction as in the Earth. The effects of depth-dependent properties have been considered for the thermal expansion coefficient, the viscosity andthermal conductivity. The viscosity and thermal conductivity are also temperature-dependent. The ther-mal conductivity has contributions from phonons, photons and electrons. The last dependence comesfrom the band-gap nature of the material under high pressure and increases exponentially with tem-perature and kicks in at temperatures above 5000 K. The thermal expansivity decreases by a factor of 20across the mantle because of the high pressures, greater than 1 TPa in the deep mantle. We have varied thetemperatures at the core–mantle boundary between 6000 and 10,000 K. Accordingly the Rayleigh num-ber based on the surface values varies between 3.5 × 107 and 7 × 107 in the different models investigated.Three phase transitions have been considered: the spinel to perovskite, the post-perovskite transitionand the post-perovskite decomposition in the deep lower mantle. We have considered an Arrhenius typeof temperature dependence in the viscosity and have extended the viscosity contrast due to temperatureto over one million. The parameter values put us well over into the stagnant lid regime. Our numericalresults show that because of the multiple phase transitions and strongly depth-dependent properties,

particularly the thermal expansitivity, initially most of the planetary interior is strongly super-adiabaticin spite of a high surface Rayleigh number, because of the presence of partially layered and penetrativeconvective flows throughout the mantle, very much unlike convection in the Earth’s mantle. But withthe passage of time, after several billion years, the temperature profiles become adiabatic. The notableinfluence of electronic thermal conductivity is to heat up the bottom boundary layer quasi-periodically,giving rise to strong coherent upwellingss, which can punch their way to the upper mantle and break up

ttern

the layered convective pa

. Introduction

Up to now, well over 350 exosolar planets have been discov-red and their habitability has been emphasized (Haghighipour andaymond, 2007). According to the Extrasolar Planets Encyclopaediahttp://www.exoplanet.eu) and this list keeps on growing relent-

essly. Most of these planetary bodies have been unveiled byoppler analysis from the scattered light of the central star which

eveals the wobbling of the star due to the gravitational pull oflanets in close orbits (Perryman, 2000; Marcy et al., 2005; ?; Bean

∗ Corresponding author. Tel.: +31 30 253 5072.E-mail addresses: berg@geo.uu.nl (A.P. van den Berg), daveyuen@gmail.com

D.A. Yuen).

031-9201/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.pepi.2009.11.001

.© 2009 Elsevier B.V. All rights reserved.

et al., 2008). Most of these planets found up to now are muchlarger than the Earth, typically, the size of Jupiter. But recently sev-eral ‘super-Earth’ planets estimated at 5–10 Earth masses, dubbedsuper-Earths, have been uncovered, near the star Gliese 876, whichhave been estimated about 7 Earth masses. A nearby star Gliese581, hosts several planets. One of these labeled planet-c has a massestimated to be around five Earth masses (http://www.solstation.com/stars/gl581.htm). Recently both size and mass were obtainedof a super-Earth planet of five Earth masses, in the CoRot-7 starsystem, resulting in the first density estimated of a terrestrial exo-

planet, similar to Earth’s mean density (Queloz et al., 2009).

These astronomical discoveries have elicited tremendous inter-est from the geophysical and also astrophysical communities (e.g.Kokubo and Ida, 2008; Bean et al., 2008) because planets of about5 Earth masses are not much different from Earth in size, less than

th and

aateTsamraapc(

rrtsLdvcpt

otsptE

cB6cpumfoppdoe

atpiYfpmctabadp

bm

A.P. van den Berg et al. / Physics of the Ear

factor two, whereas they have much higher internal pressurend interior temperature. Material behavior of solid state man-le silicates under extreme P, T condition differs from the behaviorxperienced under Earth like deep mantle conditions P < 136 GPa,< 4000 K. In particular the perovskite phase MgSiO3 may dis-

ociate in the deep mantle of super-Earth planets (Umemoto etl., 2006). The endothermic phase transition corresponding to thisineral dissociation appears to have a strongly negative Clapey-

on slope which could induce convective layering (e.g. Honda etl., 1993) leading to reduced cooling rate for the deep mantlend core. A somewhat similar configuration with an endothermichase transition from spinel to perovskite could occur close to theore–mantle boundary of Mars at pressure values of about 24 GPaBreuer et al., 1997).

Due to the increased surface gravity (g∼30 m s−2) for a rep-esentative model planet of eigth times Earth’s mass and outeradius (R∼1.6R⊕), and corresponding pressure gradient, the effec-ive lithopheric strength of these super-Earth planets is probablyignificantly higher than for Earth-like conditions. O’Neill andenardic (2007) have investigated this poignant effect on mantleynamics where they found a clear trend towards stagnant lid con-ection for increasing planetary radius. This impacts the mantleonvective regime and thermal history of the planet resulting inroducing a higher internal temperatures and slower cooling ofhe interior.

Our simple density model is constrained by the given massf the planet and we consider compressed mantle density upo 8267 kg/m3. Valencia et al. (2007a) investigated the internaltructure of super-Earth planets using values for the internal tem-erature derived from parameterized convection models, leadingo mantle temperatures that are not much higher than for thearth’s mantle.

Here we take a different tack and use the unknown temperatureontrast across the mantle �T as a control parameter (e.g. van denerg and Yuen, 1998) and we investigate a range of �T values from000 to 10,000 K. A tendency towards higher mantle temperaturean be expected when the planet is in the stagnant lid regime asredicted by O’Neill and Lenardic (2007). At the same time the vol-metric density of radiogenic heating scales with the compressedantle density resulting in amplification of the internal heating

or a given mass concentration of radioactive isotopes. In this wayur model includes the (P, T) range where Umemoto et al. (2006)redict a new phase transition where the MgSiO3 post-perovskitehase dissociates into the oxides. This new phase transition is pre-icted to be endothermic with a strongly negative Clapeyron slopef −18 MPa/K and a strong impact on mantle circulation can bexpected from this phase transition.

Very high pressure, as exists in a deep super-Earth mantle,lso strongly reduces thermal expansivity in the planetary man-le. This increases the phase buoyancy parameter of deep mantlehase boundaries and re-enforces the degree of convective layer-

ng induced by an endothermic phase boundary (Christensen anduen, 1985). Estimates based on a phonon thermodynamic model

or the MgO–SiO2 system, including the MgSiO3 post-perovskitehase, (Jacobs and de Jong, 2007) predict a decrease in ˛ across theantle with a factor of twenty (Jacobs, 2007, personal communi-

ation). The role of pressure dependent ˛ has been investigated forhe Earth’s mantle (Hansen et al., 1993; Matyska and Yuen, 2007)nd precise values of the decrease of ˛ across the mantle is stilleing investigated (Katsura et al., 2009). The stronger ˛ contrastcross a super-Earth mantle, considered here results in a larger ten-

ency towards layering of deep mantle convection especially in theresence of the dissociation of post-perovskite.

Since the seminal experimental work on thermal conductivityy Hofmeister (1999) the role played by thermal conductivity inantle convection has received a rejuvenating interest (Dubuffet

Planetary Interiors 178 (2010) 136–154 137

et al., 1999; van den Berg et al., 2001; Dubuffet et al., 2002). Butthe thermal conductivity considered in those works has been con-cerned with modest temperature conditions. Thermal conductivityat high temperature conditions behave differently (Umemoto etal., 2006). Above about 5000 K it is strongly impacted by an elec-tronic contribution, which behaves exponentially in temperatureand becomes dominant in a composite conductivity model for thesehigh temperatures (Umemoto et al., 2006).

Therefore the deep mantle dynamics of planetary evolutionunder high temperature and very large pressure conditions, suchas may be valid for super-Earth planets involves complex materialbehavior and may be rather different from an Earth-like case, whereshallow tectonics may play a more prominent role. This makesthe investigation of deep mantle dynamics in exosolar planets avery interesting subject for modelling studies within a comparativeplanetological framework.

2. Model description

2.1. Density model

As a first approximation, we will not consider density strati-fication due to self-compression, though we certainly recognizetheir importance in modifying flow structures in the deep mantle(e.g. Jarvis and McKenzie, 1980, Zhang and Yuen, 1996). Instead weapply a simple density model of a two-layer planet with separateuniform densities of the mantle and core �m and �c respectively. Auniform mantle density is consistent with the use of an extendedBoussinesq approximation (e.g. Christensen and Yuen, 1985) in ourconvection equations as explained in Section 2.2. In Appendix Aexpressions are given for the mantle and core radius in terms theplanetary mass, known from astronomic data, and the unknownmass fraction of the core Xc . These expressions have been used totabulate the planetary outer radius R(�c, �m) for a relevant range of�c, �m values. This was carried out for models constrained by theplanetary mass M = 8M⊕ as in the recently discovered super-Earthexosolar planet orbiting the star Gliese 876.

To constrain the model further, we make some necessaryassumptions concerning the overall composition and state of coremantle differentiation. We assume the overall composition to besimilar as for the Earth. We further assume that early core/mantledifferentiation resulted in the same core mass fraction as for EarthXc = 0.315. A contour diagram of planet outer radius R(�c, �m) isshown in Fig. 1, for an Earth-like value of Xc = 0.315.

Several special model cases are shown in Fig. 1, labeled m1, . . .,m5, indicated by separate symbols. Dimensional parameter valuesfor these models are listed in Table 1.

Model m1 represents a uniform density case, �c = �m = �⊕,without core mantle differentiation, where the uniform density isequal to Earth’s mean density, with an outer radius of twice theEarth radius.

The other models m2, . . ., m5 are located on the dashed lineof constant �c/�m = 2.35. This ratio corresponds to the value for atwo-layer model with homogeneous mantle and core, constrainedby the Earth’s total mass and moment of inertia factor. Model m2 isdefined by setting the constraint R′ = 2 and �c/�m = 2.35. Substi-tution of the known parameters in the expression (15) for R′

c givesR′

c = 1.094. Substitution of the value for R′c in the expression for

�′c (16) gives �′

c = 1.9246. Finally substituting in the expression forthe mantle density (17) gives �′

m = 0.81905. Models m3, . . ., m5are obtained by applying an increasing uniform compression fac-

tor f > 1 to the core and mantle densities, keeping a constant ratio�c/�m = 2.35. f is defined to be the ratio of the core or mantle den-sity of the different model cases listed in Table 1, mj, j = 1, . . . , 5,with respect to the corresponding core or mantle density of modelm3.

138 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Table 1Dimensional parameters of models m1–m5, corresponding to the discrete symbols in Fig. 1. The table includes the surface- and core–mantle boundary values of the gravityacceleration g(R) and g(Rc) and the values of the static pressure at the CMB and in the centre of the planet P(Rc) and P(0).

Model # f R (per 1000 km) Rc (per 1000 km) �m (kg/m3) �c (kg/m3) g(R) (m/s2) g(Rc) (m/s2) P(Rc) (GPa) P(0) (GPa)

m1 – 12.742 8.6698 5511.4 5511.4 19.63 13.36 370.168 689.2739m2 0.819 12.742 6.9699 4514.1m3 1.0 11.922 6.5212 5511.4m4 1.5 10.415 5.6968 8267.1m5 2.0 9.4623 5.1759 11023.

Fig. 1. Planetary radius as a function of non-dimensional core and mantle density(scale value �⊕ = 5511 kg m−3) for models constrained by M = 8M⊕ and Xc = Xc⊕ =0c2

tTaits

Rbj

Faf

.315. The dashed line represents the subset of models with increasing uniformompression factor f (see column 2 in Table 1) with a fixed density ratio �c/�m =.35. The labeled symbols m1, . . ., m5 refer to the model cases listed in Table 1.

The scaling of the surface gravity with the planet radius R forhe models located on the dashed line in Fig. 1 is shown in Fig. 2(a).he surface gravity increases non-linearly with the planetary radius

s shown the left hand frame. Model case m4 with surface grav-ty 29.38 m/s2 and a CMB pressure of 1.105 TPa has been used inhe mantle convection experiments. At this CMB pressure the dis-ociation of post-perovskite can be expected to have taken place

ig. 2. (a) Left: Surface gravity as a function of planet size (outer radius R) for the modelsfixed density ratio �c/�m = 2.35. Discrete symbols correspond to the models m1, . . ., m

or model m4.

10607. 19.36 20.67 493.226 1257.11712951. 22.42 23.60 643.673 1640.58019426. 29.38 30.93 1105.25 2816.93425901. 35.60 37.47 1621.88 4133.753

for sufficiently high temperature (Umemoto et al., 2006). Fig. 2(b)shows the radial distribution of the gravity acceleration g(r) in themantle and top of the core, for model case m4. The peak to peakvariation of g is less than 10% in the mantle and we have used auniform value of g equal to the surface value in our convectioncalculations. A relatively uniform mantle gravity value reflects theconnection between our model and the Earth’s mantle resultingfrom the given model assumptions, in particular for an Earth-likecore mass fraction. Planets with a much smaller core mass frac-tion show a significantly greater variation of g(r), with zero valuedcentral gravity for a planet without a core. In general, low gravityvalues will impact the style of convection and may lead to increasedlayering in a similar way as a strong decrease of thermal expansiv-ity with pressure from considerations of experimental equation ofstate Katsura et al., 2009.

2.2. Convection model

The governing equations of the numerical model expressconservation of mass momentum and energy in the extendedBoussinesq formulation. We have also used a 2-D Cartesian modelfor a planetary body, which is another gross simplification. Symbolsused are explained in Table 2.

∂juj = 0 (1)

⎛ ∑ ⎞

−∂i�P + ∂j�ij = Ra⎝˛T −

jRa

�j⎠ ıi3 (2)

�ij = �(T, P)(

∂jui + ∂iuj

)(3)

constrained by M = 8M⊕ , Xc = Xc⊕ = 0.315 and varying uniform compression with5 shown in Fig. 1 and Table 1. (b) Right: Radial distribution of gravity acceleration

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 139

Table 2Physical parameters

Symbol Definition Value Unit

h Depth of the mantle model 4.7 × 106 mz Depth coordinate aligned with gravity – –P Thermodynamic pressure – –�P = P − �0gz Dynamic pressure – –T Temperature – –Tsurf Surface temperature 573 K�T Temperature scale 6000 ≤ �T ≤ 10,000 Kui Velocity field component – –eij = ∂jui + ∂iuj Strain rate tensor – –

e = [1/2eijeij]1/2 Second invariant of strain rate – –

w Vertical velocity aligned with gravity – –�exp(T, z) = �0 exp(cz − bT) Exponential viscosity model b = ln(��T ), c = ln(��P )

where ��T = 300, ��P = 100 are viscosity contrasts dueto T and P

– Pa s

�0 Viscosity scale value 2.5 × 1023 Pa s

�Arr (P, T) = B exp

(E∗ + PV ∗

RT

)Arrhenius viscosity model – Pa s

B Viscosity prefactor 4.98 × 1017 Pa sE∗ Activation energy 3 × 105 J/molV ∗ Activation volume 0.5 × 10−6 m3/molR Gass constant 8.3144 J K−1 mol−1

�ij = �eij Viscous stress tensor – –� = �e2 Viscous dissipation function –˛(z) = �˛

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

�˛ = ˛(1) Expansivity contrast across the layer 0.1–0.05 –c = �˛1/3 − 1 – –˛0 Thermal expansivity scale value 4 × 10−5 K−1

� Density – –�0 Density scale value 8.267 kg m−3

cp Specific heat 1250 J K−1 kg−1

k Thermal conductivity – –k0 Conductivity scale value 4.7 W m−1 K−1

= k�cp

Thermal diffusivity – –g Gravitational acceleration 30 m s−2

Ra = �0˛0g�Th3

0�0Thermal Rayleigh number – –

Rbj = ı�jgh3

0�0Rayleigh number j th phase transition – –

ı�j Density increment jth phase transition – kg m−3

�j = 12

(1 + sin

((z − z0j)

ız

))Phase parameter j th phase transition – –

z0j =(

P(j)ref

+ �j

(T − T (j)

ref

))/(�0g) Depth j th phase transition – –

�j Clapeyron slope j th phase transition – –ız Half width of the phase transitions 100 kmDi = ˛0gh

cpSurface dissipation number 4.53 –

H0h2

TpttacppapfaE

RH =cp0�T

Internal heating number

H0 Density internal heating

DT

Dt= ∂j

((T, P)∂jT

)+ ˛Diw(T + T0) +

∑j

�jRbj

RaDi

D�j

Dt(T + T0)

+ Di

Ra� + RHH(t) (4)

he continuity Eq. (1) describes mass conservation of the incom-ressible fluid model. Conservation of momentum is expressed inhe Stokes Eq. (2) based on the infinite Prandtl number assump-ion. The rheological constitutive Eq. (3) defines the temperaturend pressure dependent linear viscous rheology. Two types of vis-osity models are applied in different model cases, an exponentialarameterization in terms of layer viscosity contrasts and the tem-erature, commonly called the Frank–Kamenetski approximation

nd an Arrhenius type parameterization in terms of activationarameters of the form exp(E∗/RT), where E∗ is the activationree energy, R is the gas constant and T is the absolute temper-ture. Details of the parameterizations used are given in Table 2.nergy transport is governed by (4), where the righthand side

10 ≤ RH ≤ 17

2.55 × 10−12 W kg−1

terms are for thermal diffusion, adiabatic heating, latent heat ofphase transitions, viscous dissipation and radiogenic internal heat-ing respectively.

Multiple phase transitions are implemented using an extendedBoussinesq formulation through phase parameter functions�j(P, T), which parameterize the Clapeyron curves in the phase dia-gram. This is expressed in the equation of state (Christensen andYuen, 1985),

� = �0

⎛⎝1 − ˛(T − Tsurf) +

∑j

�jı�j

�0

⎞⎠ (5)

We apply a composite temperature and pressure dependent

conductivity model which contains contributions from phonon(klat), photon (krad), (Hofmeister, 1999) and electron kel transport(Umemoto et al., 2006).

k(T, P) = klat(T, P) + krad(T) + kel(T) (6)

140 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Table 3Parameters describing the conductivity model

Symbol Definition Value Unit

k Thermal conductivity – –k0 Conductivity scale

value4.7 W m−1 K−1

a Conductivitypower-law index

0.3 –

� Grueneisen parameter 1.2 –K0 Bulk modulus 261 GPaK ′

0 Pressure derivative ofbulk modulus

5 –

ktrunc Truncation value of krad 15 W m−1 K−1

Ti, i = 1, . . . , 4 Corner temperatureskrad taper

573,4000,4500,7000 K

Eel Activation energyelectron transport(3 ev)

3 × 1.6 × 10−19 J

Tw(

k

Seddts4okp

k

TsTpkt(

e

k

wto4

aFioacb

Table 4Parameters of the phase transitions.

Pref (GPa) Tref (K) � (MPa/K) ı�/�0 (%)

kB Boltzmann constant 1.38 × 10−23 J/Kkep Electron conductivity

prefactor487.54 W m−1 K−1

he contributions of klat and krad are modified versions of an earlierork (van den Berg et al., 2005), which was based on the Hofmeister

1999) model.

∗lat = k0

(Tsurf

T

)a

× exp[−

(4� + 1

3

)˛(P)(T − Tsurf)

]

×(

1 + K′0P

K0

)(7)

ymbols used in the parameterization of the conductivity arexplained in Table 3. The parameterization of the pressure depen-ence of the phonon conductivity in (7) would make it theominant contribution for Earth-like deep mantle conditions. Inhe present model applied to super-Earth conditions we assumeaturation of the phonon contribution above a temperature of about000 K where we apply a smooth truncation to an upper limit valuef ktrunc = 15Wm−1 K−1. To this end the original lattice component∗lat has been modified by introduction of a smooth truncation at areset upper limit ktrunc,

lat =((

k∗lat

)−2 + (ktrunc)−2)−1/2

(8)

he original polynomial representation of krad of Hofmeister (1999)hows serious anomalies for temperatures above about 5000 K.herefore we replaced this polynomial parameterization by a sim-le parameterization, where krad(T) is set to a uniform value ofrad = 2 W m−1 K−1 in an operational temperature window andapered off to zero on both sides of this window using harmonicsine function) tapers with corner temperatures specified in Table 3.

The conductivity component from electron transport is param-terized as,

el = kep exp(

− Eel

kBT

)(9)

here kB is Boltzmann’s constant and the activation energy is seto Eel = 3 eV. The prefactor value for kep, defined in Table 3, isbtained from an estimated reference conductivity at 10,000 K of0 W m−1 K−1 (Umemoto et al., 2006)).

The different components of the composite conductivity modelre shown for relevant temperature and pressure values in Fig. 3.rames (a) and (b) show the distributions of the electron conductiv-

ty and radiative conductivity respectively which are independentf pressure. The electron conductivity becomes operational abovetemperature of 5000 K. In the parameterization of krad the upper

ut-off temperature has been set at 5000 K. Frame (c) shows iso-aric temperature distributions at four different pressure values,

P1 23.6 2000 −1.8 7.5P2 125 2750 7.2 1.5P3 1000 7000 −18 2

of the lattice conductivity component. The lattice conductivity hasbeen truncated at high pressure at a value of 15 W m−1 K−1. The cor-responding total effective conductivity values are shown in frame(d).

The coupled Eqs. (1), (2), (4), are solved on a 2-D rectangulardomain of aspect ratio 2.5. Finite element methods are used (vanden Berg et al., 1993) based on the SEPRAN finite element package(Segal and Praagman, 2000) to solve the equations. The bound-ary conditions used are: free slip, impermeable conditions on allboundaries. The temperature at the horizontal boundaries are con-stant with a fixed temperature contrast �T across the layer drivingthermal convection. Zero heat-flux conditions apply to the verti-cal boundaries. By prescribing a constant temperature at the CMBin our model we have not included thermal coupling between themantle and the core heat reservoir.

The initial condition for the temperature for all model runs wasobtained from a non-dimensional temperature snapshot of a timedependent convection run with representative parameters. Profilesof the initial temperature are shown in the results section in Fig. 12.

The finite element mesh consists of 45,451 nodal points, 151points vertical and 301 points in the horizontal direction. The nodalpoints span 22,500 quadratic triangular elements for the Stokesequation. Each Stokes element is subdivided in four linear trianglesfor the energy equation.

The time-dependent energy equation drives the time integra-tion which is based on a predictor corrector method employingan implicit Euler predictor and a Crank–Nicolson corrector step ina second order correct integration scheme. An adaptive time stephas been used equal to half the Courant time step.

3. Results from numerical experiments

Six models with contrasting parameters will be investigatedwhich show characteristic features for the extremely high temper-ature and pressure regime, which prevails in the deep mantle ofa putative super-Earth model with a mass eight times the Earth’smass. Three phase transitions involving the silicates are included inthese models the parameters of which are listed in Table 4. Theseare an endothermic phase transition near 24 GPa correspondingto the transition from the � spinel polymorph of olivine to per-ovskite and an exothermic transition near 125 GPa from perovskiteto post-perovskite Tsuchiya et al., 2004. The third transition frompost-perovskite to a new extremely high pressure phase 3PO, whichconsists of periclase and silicon oxide, is strongly endothermic� = −18 MPa/K and occurs at a reference pressure and tempera-ture of 1000 GPa and 7000 K, i.e. near the core–mantle boundary(CMB) of the super-Earth models, situated at 1.17 TPa, (Umemotoet al., 2006).

In the presentation of the modelling experiments, we havedevoted our attention on the role of the 3PO phase transition in thedynamics of mantle convection. We investigate in particular theinterplay of several thermal conductivity and thermal expansivitymodels with the 3PO phase transition controlling the convective

layering.

Our emphasis is to study the influence of the variations of phys-ical properties on the style of exosolar planetary convection underextreme conditions.

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 141

Fig. 3. Components of the composite conductivity model. Frames (a) and (b) show the distributions of the electron conductivity and radiative conductivity respectivelyw rent v( lues P

w

mibvtwBri

TC

hich are independent of pressure. Frame (c) shows the lattice conductivity, at diffed) shows the corresponding effective conductivity. Color coding of the pressure va

The characteristic parameters of the models being investigated,hich are labeled A, B, C, D, E and F, are given in Table 5.

The temperature at the CMB is kept at a constant value in theseodels. The unknown temperature contrast across the mantle �T

s treated as a control parameter and we have investigated modelehavior for the cases specified in Table 5 at three contrasting �Talues, 6000, 8000, 10,000 K respectively. Higher temperature con-rasts than the estimated 4000 K for the Earth’s mantle are in line

ith a higher volume/surface ratio of the larger planet investigated.esides bottom heating, we have included internal heating fromadioactive decay at a constant value of 2.5 × 10−12 W/kg, result-ng values of the internal heating number RH , defined in Table 2,

able 5haracteristic parameters of the convection models investigated.

Model Expansivitycontrast, �˛

Conductivitymodel, k(T, P)

Clapeyronslope 3PO, �(MPa/K)

Viscosity type

A 0.1 k0 −18 ExponentialB 0.05 k0 −18 ExponentialC 0.05 klat + krad + kel −18 ExponentialD 0.05 klat + krad + kel 0 ExponentialE 0.05 k(z) −18 ExponentialF 0.05 klat + krad + kel −18 Arrhenius

alues of pressure, using a truncation value at high pressure of 15 W m−1 K−1. Framei, 0, 390, 780, 1170 GPa is indicated in the plot legend.

vary between 10 and 17. We regard this range to be a speculativeconjecture.

In Fig. 4 representative snapshots of the temperature field andcorresponding streamfunction field are shown for model casesA, B, C defined in Table 5, with the same temperature contrast�T = 10,000 K.

Instantaneous positions of the three phase boundaries areindicated by the white lines. The depths of the first two phasetransitions are much smaller than in the Earth’s mantle due to themuch greater pressure gradient resulting from the high density(� = 8.29 kg/m3) and gravity acceleration g = 29.4 m s−2 derivedin Section 2.1. Phase boundary topography is also directly affectedby the large pressure gradient of the exosolar planet. This is thereason that the peak to peak amplitude of the topography on the3PO phase boundary is limited to a value of 170 km in spite of thefact that the relevant Clapeyron slope has a very high magnitude of18 MPa K−1.

The different degree in the style of layered convection nearthe bottom 3PO phase boundary can be clearly discerned in these

frames, in particular in the distribution and spacing of the con-tour lines in the streamfunction. Model A with a decrease factorof thermal expansivity across the mantle �˛ = 0.1 shows limitedinteraction between the endothermic 3PO phase boundary, withhot plumes crossing the phase boundary in several locations. A

142 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

F ) for dt e phas

wod

twcmwaseB3pYi

msb(rdttt

ltsataop

ig. 4. Snapshots of non-dimensional temperature (left) and streamfunction (righthe layer �T = 10,000 K. White lines indicate the instantaneous position of the thre

hole mantle flow pattern is indicated by the complete circuitsf the streamfunction with highest velocities related to three coldescending flows emerging from the top boundary.

The middle row of snapshots for model B with, �˛ = 0.05, illus-rates a penetrative convective flow regime in the streamfunctionith flow concentrated in the top half of the mantle and the bottom

irculation is being driven by the convection cells in the shallowantle. As in the A model the flow is driven by the cold down-ellings. The temperature snapshots reveal an increased layering

t the 3PO boundary at the bottom, as indicated by the mushroomhaped plumes, which show only limited penetration of the deep-st phase boundary. The increase in layering between models A andis related to the smaller value of the thermal expansivity near thePO phase transition resulting in a higher effective phase buoyancyarameter that scales inversely with respect to ˛ (Christensen anduen, 1985). Similar layered penetrative convection was observed

n Breuer et al. (1997) in a model for the Martian mantle.In contrast to the constant thermal conductivity models A, B

odel C has a variable thermal conductivity k(T, P) model with atrong contribution from electron thermal conductivity kel near theottom of the mantle, where temperatures are in excess of 5000 Ksee Figs. 13–15). The high conductivity at the base of the mantleesults in an increased heat flow into the bottom 3PO layer, pro-ucing episodic occurrence of massive mantle plumes that breakhrough the phase boundary, as illustrated by the hot plume nearhe middle of the temperature frame and corresponding high ver-ical velocity shown by the close spacings in the streamfunction.

In order to investigate the contrasting dynamics of the bottomayer of the different models we have applied passive tracer par-icles to monitor the relevant physical fields. To this end we havetaged a set of 2000 randomly distributed tracer particles which are

dvected by the convective flow. The instantaneous coordinates,emperature and mineral phase in the tracer positions were storedt every integration time step. To reinforce further the impactf different parameterizations on the deep mantle dynamics, inarticular on the convective layering near the bottom 3PO phase

ifferent models A, B, C defined in Table 5, for the same temperature contrast acrosse boundaries included in the model.

region, we present statistics and space-time trajectories of tracerparticles in Fig. 5, for the B and C models. Tracer residence timesfor the bottom 3PO layer were calculated for the duration of themodel runs of 3.2 Gyr. Histograms illustrating different statisticsof the tracer residence times are shown in frames (a) and (b). Resi-dence times attain higher values in model case B in agreement witha more layered type of convection. This agrees also with the highervalues of the (clipped) first bin for case B, representing tracers thathave been confined to the layers overlying the 3PO phase region.

Trajectories for four representative tracers from representativebins of �∼1 Gyr (C model) and �∼3 Gyr (B model) are shown inframes (c) and (d). For the same tracers of (c) and (d), correspond-ing time series of the height above the CMB are displayed in frames(e) and (f). From (c) and (f) we see that the C-model tracers with�∼1 Gyr have been deposited in the 3PO layer in the early his-tory where they remain for about 1 Gyr. After this time they wereadvected into the overlying post-perovskite layer where they mixthroughout the domain without returning to the 3PO region exceptfor a short excursion of a single tracer at 2.5 Gyr. The removalof the four tracers from the 3PO layer coincides with a flushingevent related to the hot plume penetrating the 3PO phase bound-ary shown in Fig. 4(C). In contrast to this frame (d) and (f) showtrajectories from the B model results for tracers taken from the res-idence time bin at 3 Gyr. These tracers are moved into the bottom3PO layer in the early history where they remain for the remainderof the model run time, in agreement with the more isolated char-acter of the bottom layer corresponding to the greater convectivelayering observed for this case.

From the tracer records the coordinates and time values ofthe 3PO phase boundary crossings were determined for upwardand downward crossings separately. An estimate of the volume

flux through the phase boundary is given by the number of phaseboundary crossings in a given time window normalized by the totalnumber of tracers in the domain (van Summeren et al., 2009). Fig. 6ashows an x, t plot of the tracer phase boundary crossings for modelC. This figure illustrates that the 3PO transition is a leaky bound-

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 143

Fig. 5. Comparison of different convective layering characteristics between models C (lefthand column of frames) and B (righthand) defined in Table 5. Frames (a) and (b)show histograms of residence time values in the bottom 3PO phase layer for 2000 randomly distributed particle tracers that are advected by the convective flow. Histogramsh idencb en fror ersus

atabFbt∼3ii

ictflctp

i

ave been normalized normalized identically to obtain a unit integral value over reseen clipped at unit value. Frames (c) and (d) show trajectories for four tracers takespectively. Frames (e) and (f) show the non-dimensional height above the CMB v

ry for mantle convective flow for this model case. Mass transporthrough the boundary is characterized by localized hot plumesnd broader regions of return flow by cold downwellings. Episodicehavior of the plume dynamics suggested by the Vrms curve ofig. 16 later in this section is reflected in the clusters of hot upwardoundary crossing events that emerge at several x-locations. Thewo event clusters at t∼1.1 Gyr and dimensionless x-coordinate1.2 and 1.8 are associated with two plumes breaking through thePO boundary shown in Fig. 4. This dynamic episode can also be

dentified in a corresponding peak value of the Vrms curve shownn Fig. 16.

Temporal variation of mass transport through the 3PO boundarys quantified in frame (b). This figure shows the number of upwardrossing events accumulated in 100 Myr time bins normalized byhe total number of particles as an estimate of the fractional volumeux involved. The maximum flux values coincide with the episodic

lusters of hotplumes breaking through the boundary. Up to 5% ofhe mantle volume crosses the 3PO boundary per 100 Myr for thisarticular case.

Fig. 6c shows the temperature distribution of upward flow-ng tracers crossing the depth of 90% of the domain depth, about

e time. The maximum values of the first bins 2.45 (model C) and 4.87 (model B) hasm representative residence time bins, � = 1 Gyr (model C) and � = 3 Gyr (model B)model time for the same tracers of (c) and (d).

halfway the 3PO layer. There is a clear correlation with the (red)upward 3PO boundary crossings in frame (a). The temperatureof the emerging hot upwellings at x = 1.2 and x = 1.8 increasestowards the time of plume breakthrough t∼1.1 Gyr. This is followedby a drop of the observed temperature and subsequent gradualincrease in agreement with an episodic convective style.

Several physical effects interact in creating the dynamics ofthe 3PO layer. In particular the conductivity model and stronglyendothermic phase transition, reinforced by a low value of thethermal expansivity, ∼5% of the surface value, play a role in con-trolling the episodic behavior. To separate these different effectswe have investigated two contrasting models were the parame-ters were chosen to exclude a single effect. These are the constantconductivity model B and a variable conductivity model D with azero Clapeyron slope of the 3PO transition. Fig. 7 shows results forboth these models B (left) and D (right). Mass transport through

the 3PO boundary is significantly smaller for the constant conduc-tivity model B (Fig. 7 left hand) than for the variable conductivitymodel C (Fig. 6). This is quantified by the total number of phaseboundary crossing events for the B model which is less then 50%of the C model value. This is in agreement with the snapshot of the

144 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Fig. 6. (a) Phase boundary crossing events at the 3PO transition displayed by the horizontal and time coordinates of passive tracers monitoring the local mineral phasefor model C. Red (1864) and blue (1877) symbols represent upward and downward crossings respectively. (b) Upward mass flux through the 3PO boundary estimated by(100 Myr) time binned values of the number of upward phase boundary crossings normalized by the total number of tracers in the domain. Red dots indicate results basedon the full data set of 2000 tracer particles. The black crosses showing test results computed from half the data set indicate that a sufficient number tracers were used. (c)Color coded temperature values of upward monitor crossings of the depthlevel 0.9 times the layer depth, inside the 3PO layer, for model C. The number of upward crossinge betwc ws is ip nterpw

taws

dCcib

sftcl

FTt

vents is 1387. A rainbow palette has been used indicating temperature variationooler off-central parts of the upwellings. Episodic broadening of the upwelling floositive feedback related to the temperature dependence of the conductivity. (For ieb version of the article.)

emperature and streamfunction field shown in Fig. 4B, suggestinglayered convective flow regime for the bottom 3PO region witheak leakage through the boundary near x-coordinate 0.7 at the

napshot time 1.2 Gyr.The right hand frames of Fig. 7 show the result of removing the

ynamical effect of the endothermic phase boundary by setting thelapeyron slope to zero, while maintaining the variable thermalonductivity. The number of phase boundary crossings has stronglyncreased with respect to model C and the dynamics are controlledy stable cold downwellings instead of episodic hot plumes.

Comparison of the results of models B, C and D shows that thetrongly endothermic phase transition is the essential ingredient

or the layering of the flow in the 3PO region and that the effect ofhe highly variable conductivity is to produce a weakening of theonvective layering and introduction of episodic behavior of theayering.

ig. 7. Similar results as in Fig. 6 the uniform conductivity model B (left). The number ohe two rightand frames show similar data for a modelrun with the same parameters as transition has been set to zero (right hand results). The number of up- and downward ph

een 7000 and 10,000 K. The hotter cores of plumes can be distinguished from then agreement with the episodic dynamics of the 3PO layer which is reinforced by aretation of the references to color in this figure legend, the reader is referred to the

The latter effect can be understood by comparing the contrast-ing wavelength structure of the lateral variations of temperaturein the 3PO layer between models B and C as shown in Fig. 4B,C.The temperature field of the variable conductivity results is clearlydominated by longer wavelengths (Matyska et al., 1994) thanthe corresponding constant conductivity snapshot. An increasein the dominant wavelength of lateral variations of tempera-ture decreases the stability of phase induced convective layering(Tackley, 1995).

The mechanism described above results from a more or lessconstant (in time) high effective conductivity value in the bottomlayer including the phase boundary, comparable to a downward

shift of the local Rayleigh number with well known impact on thephase induced convective layering (Christensen and Yuen, 1985).To test this explanation we did a model run, labeled E in Table 5,with a purely depth dependent conductivity model defined by a

f up- and downward phase boundary crossing events is 772 and 774 respectively.he variable conductivity model C except that the Clapeyron slope of the 3PO phaseaseboundary crossing events is 2506 and 2514 respectively.

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 145

F ependd ed fos he rea

dtatrctFae

stbcpcF

6twcs

Fid

ig. 8. (a) (3PO) Phase boundary crossing events for model E with purely depth downward events is 1819 and 1821 respectively. (b) Conductivity depth profile ushown in black. (For interpretation of the references to color in this figure legend, t

epth profile obtained from time averaging snapshots of horizon-ally averaged conductivity profiles of model C (van den Berg etl., 2005). The results of this model run, shown in Fig. 8, confirmhe equivalent dynamics of the C and E models, and the dominantole of the horizontal average conductivity profile. The dynami-al similarity between the C and E models is further illustrated inhe temperature snapshot of Fig. 9. This figure, almost identical toig. 4C, shows two hot plumes breaking through the 3PO bound-ry at the model time value 1.24 Gyr corresponding to the two redvent clusters near x-coordinate values of 1.1 and 2.0.

Fig. 10 shows the representative snapshots of temperature andtreamfunction for different model runs with a temperature con-rast �T = 8000 K. The contrast between the B and C models hasecome smaller compared to Fig. 4 due to the smaller value of theonductivity near the bottom. This is a result of the reduced tem-erature and the strong temperature dependence of the electroniconductivity component which is dominant in the results shown inig. 4.

Model results for a smaller temperature difference of �T =000 K are shown in Fig. 11. The same distinction holds between

he A, B and C model cases as in the previous two figures with ahole mantle convection regime for the A case and penetrative

onvection for the models B and C. Overall the flow velocities areignificantly smaller than in the previous cases for higher values of

ig. 9. Temperature snapshot of the depth dependent conductivity model E, show-ng very similar dynamics of large plume breakthrough as in Fig. 4 for the P, Tependent variable conductivity model C.

ent conductivity showing similar results as in Fig. 6a. The number of upward andr model E shown in red, computed by timeaveraging k(z, t) profiles from model Cder is referred to the web version of the article.)

�T indicated by the streamfunction plots. The episodic hot plumeswhich occur for the C model at higher temperature contrasts areabsent at the reduced �T = 6000 K. This is a result of the muchsmaller contribution of the electron conductivity for this case asillustrated in Fig. 18.

Because of their larger sizes, the exosolar planet convectionmodels are characterized by strongly depth-dependent propertiesand show clear signs of layered penetrative convection. We havetherefore investigated the level of adiabaticity in the temperaturedistribution for two end-member values of the mantle temperaturecontrast �T of 10,000 and 5000 K respectively, for the model case Cof Table 5. We have compared the instantaneous temperature pro-files for the two models and compared them to a reference adiabatchosen by visually matching with the time dependent geotherm.We also computed the deviation between the local temperaturegradient and the adiabatic gradient

ˇ = d〈T〉/dz − ˛(z)Di(〈T〉 + T0) (10)

We display the results in Figure 12. Three frames are shown foreach value of �T , (1) for the horizontally averaged temperatureprofile, (2) the temperature difference with the reference adiabatand (3) the deviation from the adiabatic gradient ˇ. The referenceadiabat which was chosen by trial and error visual matching withthe geotherms is added as the red-dashed line in the left-handtemperature frame. The potential temperature for the cold case is1500 and 2750 K for the hot case. The results show that the differ-ence between the geotherm and an ‘equilibrium adiabat’ decayswith time for both the end-member cases, thus suggesting thatwith sufficient mixing of the early state an adiabatic profile willbe obtained but only after a time interval of two or more billionyears, which is longer for lower temperature contrast across themantle.

It looks like in the initial spin-up of the model cold material isdumped and hot material rises. Hot material is mixed rapidly inthe upper half of the model as a result of the penetrative layer-ing illustrated in the temperature and streamfunction snapshots ofthe previous figures. This is illustrated by the rapid convergence

of the geotherms to the reference adiabat. At the same time thebottom half is convecting much slower at least partly due to thehigh bottom conductivity, such that the bottom half is heating upslowly towards the adiabat. This trend is supported by the T − Ta

frames, showing convergence to the adiabat above a relatively sub-

146 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Fig. 10. Temperature and streamfunction snapshots for models similar to Fig. 4 but with a temperature contrast �T = 8000 K across the mantle. The contrast between the Band C models has become significantly smaller due to the much smaller value of the conductivity near the bottom. This is a result of the reduced temperature and the strongtemperature dependence of the electronic conductivity component which is dominant in the results shown in Fig. 4.

Fig. 11. Temperature and streamfunction snapshots for models similar to Fig. 4 but with a reduced temperature contrast �T = 6000 K across the mantle. The contrastbetween the B and C models has become significantly smaller due to the much smaller value of the conductivity near the bottom. This is a result of the reduced temperatureand the strong temperature dependence of the electronic conductivity component which is dominant in the results shown in Fig. 4.

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 147

Fig. 12. Vertical profiles illustrating the time dependent degree of adiabaticity of the mantle for two endmember model cases, of type C defined in Table 5, with differenttemperature contrast across the mantle, �T = 10,000 K top row (a)–(c) and �T = 5000 K bottom row (d)–(f). The lefthand column ((a) and (d)) shows snapshots of thehorizontally averaged temperature at different time instances indicated in the legend. The red dashed line indicates an adiabatic temperature profile with potential (surface)temperature of 2750 K (a) and 1500 K (d) visually selected to overlap with the evolving geotherms. The middle column (b) and (e) shows snapshots of the difference betweent howsa figure

so

coFin(tTphpatv

At(�lw

he mantle temperature profile and the reference adiabat. The righthand column sdiabatic value defined in (10). (For interpretation of the references to color in this

tantial thermal boundary layer at the bottom with a thicknessf 700 km.

Fig. 13 shows horizontally averaged profiles of temperature, vis-osity and the thermal conductivity for models A, B and C. The valuef the temperature contrast is �T = 10,000 K in all cases shown.rames (a)–(c) are for the variable conductivity model C. Frame (c)llustrates the strong increase of k(T, P) under high T, P conditionsear the bottom of the mantle. Frames (d)–(e) illustrate the effect of3PO) phase induced layered convection, for the constant conduc-ivity case B with layer contrast of thermal expansivity �˛ = 0.05.he layering results in a steep increase of temperature across thehase boundary (indicated by the black Clapeyron curve in the left-and frames) and associated decrease of the slope of the viscosityrofile. Frames (f)–(g) illustrate the model case A with constant knd �˛ = 0.1. There is no layering in this case with higher ˛ andhe depth profiles show little variation with respect to the initialalue profiles.

In Fig. 14 similar profiles as in Fig. 13 are shown for models, B, C for a smaller temperature contrast �T = 8000 K. The effec-

ive conductivity value for the variable k case C, shown in framec), is substantially smaller than in the corresponding model with

T = 10,000 K of Fig. 13. The difference in the degree of convectiveayering between the three cases A, B, C is reflected in the profiles,

hich is similar to Fig. 13.

corresponding profiles of the deviation of the local temperature gradient from thelegend, the reader is referred to the web version of the article.)

Fig. 15 shows corresponding profiles as in Figs. 13 and 14 formodel runs with �T = 6000 K. The effective thermal conductiv-ity shown in frame (c) is now substantially smaller than in theprevious figures and the degree of convective layering is smallerthan for the corresponding cases shown in Figs. 13 and 14. Theresults here argue for the importance of electronic thermal conduc-tivity in causing layered convection in the deep mantles of exosolarplanets.

Next we show the time-series of several global quantities forvariable conductivity models of type C for different values of thetemperature contrast across the mantle in Fig. 16. The basal heat-flow in Fig. 16a varies strongly between the different conductivitymodels, in line with the extremely strong temperature dependenceof the electron conductivity. Especially for the �T = 10,000 K casethere is a clear correlation between the local maxima in the bot-tom heat flux and corresponding maxima in the rms velocity shownin the bottom frame. This is indicative of the episodic hot plumesbreaking through the 3PO phase boundary as shown in the topframe of Fig. 4.

The volumetrically averaged temperature is shown in Fig. 16b.The rate of temperature increase is highest for the model with�T = 10,000 K (∼200k/Gyr) and (∼30k/Gyr) for the �T = 6000 case.This is a consequence of the strong asymmetry in the conductiv-ity depth profile with low surface k values and high CMB values

148 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Fig. 13. Horizontal average depth profiles of temperature (left column) viscosity (middle) and thermal conductivity (right), for three models A (bottom row), B (middle row)and C (top row) shown in Fig. 4. The temperature contrast across the mantle is fixed at �T = 10,000 K for all cases. The Clapeyron curve of the 3PO phase is indicated by thes

ottotfi�tc

caoruf

traight black line in frames (a), (d), (f).

f k resulting from the strong temperature dependence of elec-ron thermal conductivity kel (see Figs. 13 and 15). Mantle heatransport is then controlled by the high thermal resistance (low k)f the top thermal boundary layer van den Berg et al., 2005 andhe high conductivity near the core mantle boundary - allowingor rapid mantle heating from the core. This is clearly illustratedn the bottom heat-flow with highest values for the model with

T = 10,000 K. This effect is amplified by the negative tempera-ure dependence ∂k/∂T < 0 of the dominant (lattice) conductivityomponent klat at shallow depth.

Rms velocities are shown in Fig. 16c. The Vrms curves show ahange in dynamics after a initial time interval characterized by rel-

tively low velocities below 1 cm/yr the length of which dependsn the conductivity model. For the model with �T = 10,000 K thiselatively quiet period ends at about 1 Gyr with a velocity spinp directly related to the plume break through shown in the toprame of Fig. 4. For the models with lower temperature contrast

A and B the initial quiet period lasts longer and the episodes ofvelocity spin-up show a smaller Vrms amplitude. The correla-tion between the different quantities suggests that the episodesof increased convective vigor must be related to an increase inthe temperature of the bottom 3PO layer with positive feedbackfrom the temperature dependent electron conductivity. The inter-action with the 3PO phase boundary with a strongly negativeClapeyron slope resisting whole mantle flow leads to the flushingevents (e.g. Honda et al., 1993; Tackley et al., 1993; Steinbach etal., 1993) when large plumes break through the resisting phaseboundary. At lower values of �T the effective bottom thermalconductivity is smaller and the time period necessary for heat-

ing up the bottom layer and reaching critical temperature takeslonger.

We give the corresponding results to Fig. 16 for the constantconductivity model B in Fig. 17. These models are character-ized by a penetrative convective regime with feeble convective

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 149

a temp

vdfla

tTccdttkb

oe3

Fig. 14. Similar profiles as in Fig. 13 but now for

igor in the bottom half of the model and a low uniform con-uctivity, in agreement with low values of the surface heatux, the rate of internal temperature and the rms velocitymplitudes.

Fig. 18 shows the overall breakdown of the composite conduc-ivity for the different �T values used in the convection results.he corresponding temperature profiles are shown in Fig. 18a. Theontrolling effect of temperature and pressure on the differentomponents of k is clearly illustrated. The electron thermal con-uctivity kel becomes operational at T > 5000 K (see Fig. 3). Sincehe radiative component krad tapers off for T > 5000 K, there is ahresholding effect and represents an effective switch from krad toel at this temperature. kel is the dominant mechanism near the

ottom boundary layer for the model with �T = 10,000 K.

The results presented sofar show clearly that an episodic typef deep mantle dynamics is produced by the interplay of thermalxpansivity, thermal conductivity and the Clapeyron slope of thePO phase transition.

erature contrast across themantle �T = 8000 K.

O’Neill and Lenardic (2007) have argued that super-Earth typeterrestrial planets have a larger propensity for stagnant lid dynam-ics of the upper mantle as a result of the high pressure gradientin the shallow mantle leading to surpressing brittle deforma-tion mechanisms. This may have an impact also on episodic deepmantle dynamics. In the following we therefore investigate theimpact of a stagnant lid on the model results. Stagnant lid con-vection occurs for sufficiently high viscosity contrasts around105 to 106 (e.g. Moresi and Solomatov, 1995). To investigate theeffects of stagnant lid convection in our models we have used anArrhenius parameterization of the viscosity, defined in Table 2,in a model, labeled F in Table 5, with identical thermophysi-cal parameters as in model C and a temperature contrast across

the mantle �T = 10,000 K. Snapshots of depth profiles of rele-vant quantities for model F are given in Fig. 19. The stagnant lidcharacter of model F is illustrated in the profiles of the viscos-ity and the amplitude of the velocity field. The viscosity profilesshow more than six orders of viscosity variation and a stag-

150 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Fig. 15. Similar profiles as in Fig. 13 but now for a temperature contrast across the mantle �T = 6000 K.

ntalrsdl

ceirbFa

ant lid of about 300 km thickness. This is in agreement withhe velocity profiles showing an effectively rigid outer surfacend rapid increase of the flow velocity in the mantle below theid. Temperature and conductivity profiles are similar to the cor-esponding profiles of model C shown in Fig. 13, except for aystematic increase of mantle temperature over time due to theecrease in mantle cooling capacity in the presence of the stagnant

id.Time series of global quantities for model F are shown in Fig. 20,

orresponding to similar data for model C shown in Fig. 16. Thepisodic nature of the mantle dynamics for model F is most clearly

llustrated by the Vrms curve in Fig. 20c which is much like the cor-esponding curve for the C model and �T = 10,000 K. Differencesetween results of the two models are; shorter time scale for model, related to a combination of higher temperature and conductivitynd lower viscosity in the deep mantle, compared to model C. Fur-

ther, the mantle temperature increases more, in the stagnant lidcase F, as a consequence of the thermal resistance of the stagnantlid.

A snapshot of the temperature field from the results of theF model case is shown in Fig. 21 for a model time of 1.24 Gyr,close to the first of a double peak in Vrms, corresponding to aplume breaking through the 3PO phase boundary, visible in thesnapshot. Compared to the corresponding mobile lid results inFig. 4c (model C) and Fig. 9 (model E), cold downwellings areless significant in Fig. 21 as a result of the different stagnant lidconditions. This further illustrates the fact that the characteris-

tic episodic deep mantle dynamics is mainly controlled by theparticular thermophysical parameters prevailing in the deep man-tle of exosolar planets. Therefore plate dynamics on the surfacedo not exert any influence on deep mantle processes of exosolarplanets.

A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154 151

Fig. 16. Timeseries of (a) bottom heat flux, (b) volume average temperature and (c)root-mean-squared velocity for the variable conductivity models C. Three values ofthe mantle temperature contrast �T are considered, 6000 K (blue), 8000 K (black)and 10,000 K (red). (For interpretation of the references to color in this figure legend,the reader is referred to the web version of the article.)

Fig. 17. Similar timeseries as in Fig. 16 for the constant conductivity models B withthermal expansivity contrast �˛ = 0.05. These uniform k models show no globalwarming and much lower bottom heat flux than shown in Fig. 16. The low Vrms val-ues are in agreement with penetrative convection without large scale mass transportthrough the 3PO interface.

Fig. 18. Temperature depth profiles and corresponding decomposition of the variable conductivity for three values of the temperature contrast �T across the mantle. (a)Temperature. (b–d) Corresponding conductivity (k-component) profiles.

152 A.P. van den Berg et al. / Physics of the Earth and Planetary Interiors 178 (2010) 136–154

Fig. 19. Depth profiles of horizontally averaged temperature (a), conductivity (b), viscosity (c), and magnitude of velocity (d) for three snapshots labeled by the model timevalue in frame (a). The Clapeyron curve of the 3PO transition is drawn in the lefthand frame to indicate the approximate location of the phase boundary. The viscosity variesby six orders of magnitude due to temperature in the shallower part of the model. This results in stagnant lid convection as illustrated by the vertical distribution of thevelocity amplitude indicating an approximately 300 km thick immobile lid.

Fig. 20. Time series of global quantities for the stagnant lid model F: (a) CMB heat-flow, (b) volume average temperature and (c) rms velocity amplitude. The dynamicsof model F is similar as in model C, as characterized by the episodic behavior of hotplumes breaking through the 3PO phase boundary, producing sharp peaks in Vrms.Model F shows sharper peaks than model C which is related to the greater viscositycontrast due to temperature and lower viscosity in the bottom region near the 3POtransition.

Fig. 21. Temperature snapshot for the stagnant lid model F. Different stages of inter-

action between hot plumes and the 3PO phase boundary are illustrated, varying fromdeflection (right), breakthrough (second from left) and regular flow through (secondfrom right).

4. Discussions and concluding remarks

Super-Earth extrasolar planets are fundamentally different fromthe Earth in the pressure and temperature range of the differentmantles and this particular aspect brings in more phase transitionsand different heat transport mechanisms, unknown in the Earth’smantle and causes noticeable deviation from a thermal evolutionpredicted by parameterized convection Sharpe and Peltier, 1978.

Our calculations address a number of important dynamicalproblems in thermal convection of super-Earth planets. However,we note that our model does have some shortcomings such as lackof density stratification, curvature from a spherical-shell model andalso the assumption of a large core. However, all of these factors willconspire to induce greater degrees of layering in mantle convection.

An interesting outcome from our study is the development of asuper-adiabatic layer of around 700 km, which coincides with thelocation of the 3PO phase transition. This relatively thick super-adiabatic layer will influence the amount of heat delivered fromthe core and the thermal evolution of the planet.

The major result from our work comes from the interplaybetween the strongly endothermic 3PO phase transition and theelectronic thermal conductivity, which allows for the periodicbuild-up of thermal energy to be discharged across the bottomboundary layer in the form of coherent upwellings. This represents

a new type of episodic dynamics in which focused plumes are peri-odically generated, by the steep rise in the exponential temperaturedependence of the thermal conductivity.

We have shown that the high value of the thermal conductiv-ity in the bottom thermal boundary layer is the critical ingredient

th and

clmcitoidddm(

iwtavrawpatlcliVoc

A

Miamfst(

A

atctmWtt

ara

wcIl

A.P. van den Berg et al. / Physics of the Ear

reating this new dynamics. This was done by reproducing simi-ar episodic dynamics with a purely depth dependent conductivity

odel defined from a time averaged conductivity profile of a modelase with a fully temperature and pressure dependent conductiv-ty. This high degree of similarity shows that positive feedback fromhe temperature dependence of conductivity into the dynamicsf plume formation is not a first order effect. This latter findings somewhat similar to our earlier demonstration that significantelay in planetary secular cooling in models with temperatureependent phonon conductivity is mainly due to the average con-uctivity profile characterized by a sub-lithospheric conductivityinimum in the shallow mantle creating a low conductivity zone

LCZ) (van den Berg et al., 2005).Furthermore we have shown that the episodic plume dynam-

cs occurs both in a mobile lid mantle convection regime obtainedith an exponential parameterization of the viscosity, with modest

emperature dependence, and a stagnant lid regime, obtained withn Arrhenius type viscosity with more than six orders of viscosityariation due to temperature. The latter case is considered moreepresentative for exosolar planets of the super-Earth type (O’Neillnd Lenardic, 2007). A stagnant lid regime of planetary convectionith variable viscosity may be linked to plate tectonics through alastic bifurcation (Trompert and Hansen, 1998) and may play a rolelso in exosolar planets (Valencia et al., 2007b). The robustness ofhe episodic mantle dynamics is an important result since it under-ines the fact that parameterized convection models which do notapture episodicity are inadequate to model thermal evolution ofarge super-Earth type exosolar planets. Future work would need toncorporate both realistic equation of states (Umemoto et al., 2006;alencia et al., 2006, 2009) and sphericity. But the major findingsf electronic thermal conductivity should still be valid under theseircumstances.

cknowledgements

We thank discussions with Phil Allen, Michel Jacobs, Radekatyska, Joost van Summeren, Diane Valencia, Renata Wentcov-

tch and K. Umemoto. Constructive reviews of Craig O’Neil andn anonymous reviewer were very helpful for improving theanuscript. This research has been supported by the ITR grant

rom National Science Foundation given to the VLAB at the Univer-ity of Minnesota. Computational resources were provided throughhe Netherlands Research Center for Integrated Solid Earth ScienceISES 3.2.5).

ppendix A. Density parameterization

We consider a simple two layer model of a planet consisting ofhomogeneous core and mantle with contrasting uniform densi-

ies �c and �m respectively. Assuming a uniform density mantle isonsistent with the use of an extended Boussinesq formulation ofhe mantle convection equations applied in our modelling experi-

ents. The two layer model is set up for a planet of given mass M.e further assume the mass fraction of an assumed metal core Xc

o be given. In the modelling experiments we set Xc = 0.315, equalo the Earth value.

The model can then be formulated in terms of the volume aver-ge densities of the core �c and mantle �m. This way the outer planetadius R and the core radius Rc can be expressed in terms of M, Xc

nd �c, �m.

As a first approximation of cases with a self-compressing planet

e consider several models which are related by a simple uniformompression factor, taken to be identical for the mantle and core.n the 2-D �c, �m parameter space these models are located on aine with fixed �c/�m. This parameterization can be used to study

Planetary Interiors 178 (2010) 136–154 153

in a simple way the effect of compression on the internal gravityand pressure distribution.

In the following expressions are derived for the outer radius ofthe planet and its core in terms M, Xc, �c, �m. For the total planetarymass we have,

M = Mc + Mm = XcM + (1 − Xc)M (11)

The core radius is derived from the core mass fraction as,

Rc = R(

1 + �c

�m

(1Xc

− 1))−1/3

(12)

An expression for the outer radius is derived from the total massas,

R3 = 3M

4

(Xc

1�c

+ (1 − Xc)1

�m

)(13)

With the non-dimensionalization scheme, M = M′M0, R = R′R0,�c = �′

c�0, M0 = �0V0 = �043 R3

0, we get,

R′3 = M′(

Xc1�′

c+ (1 − Xc)

1�′

m

)(14)

For the special case of a uniform sphere with M′ = 1 this expres-sion produces a unit non-dimensional outer radius R′ = 1 with�c = �m = �0 and �′

c = �′m = 1.

For the core radius we get with the same non-dimensionalization scheme,

R′c = R′

(1 + �′

c

�′m

(1Xc

− 1))1/3

(15)

Explicit expressions for the core and mantle densities can bederived from the above for R′ and R′

c .

�′c = XcM′

R′3c

(16)

�′m = M′(1 − Xc)

R′3 − R′3c

(17)

References

Bean, J.L., Mc Arthur, B.E., Benedict, G.F., Armstrong, A., 2008. Detection of a thirdplanet in the HD 74156 system using Hobby-Eberly telescope. Astrophys. J. 672,178–184.

Breuer, D., Yuen, D.A., Spohn, T., 1997. Phase transitions in the martian mantle:implications for partially layered convection. Earth Planet. Sci. Lett. 148, 457–469.

Christensen, U.R., Yuen, D.A., 1985. Layered convection induced by phase transitions.J. Geophys. Res. 90, 10291–10300.

Dubuffet, F., Yuen, D.A., Rabinowcz, M., 1999. Effects of a realistic mantle ther-mal conductivity on the patterns of 3-D convection. Earth Planet. Sci. Lett. 171,401–409.

Dubuffet, F., Yuen, D.A., Rainey, E., 2002. Controlling thermal chaos in the mantleby positive feedback from radiative thermal conductivity. Nonlinear Process.Geophys. 9 (3–4), 311–323.

Haghighipour, N., Raymond, S.N., 2007. Habitable planet formation in binary plane-tary systems. Astrophys. J. 666, 436–447.

Hansen, U., Yuen, D.A., Kroening, S.E., Larsen, T.B., 1993. Dynamical consequences ofdepth-dependent thermal expansivity and viscosity on mantle circulations andthermal structure. Phys. Earth Planet. Inter. 77, 205–223.

Hofmeister, A.M., 1999. Mantle values of thermal conductivity and the geothermfrom phonon lifetimes. Science 283, 1699–1706.

Honda, S., Yuen, D.A., Balachandar, S., Reuteler, D., 1993. Three-dimensional insta-bilities of mantle convection with multiple phase transitions. Science 259,1308–1311.

Jacobs, M.H.G., de Jong, B.H.W.S., 2007. Placing constraints on phase equilibriaand thermophysical properties in the system MgO − SiO2 by a thermodynam-ically consistent vibrational method. Geochimica et Cosmochimica Acta 71,3655–3660.

Jarvis, G.T., McKenzie, D.P., 1980. Convection in a compressible fluid with infinite

Prandtl number. J. Fluid Mech. 96, 525–583.

Kokubo, E., Ida, S., 2008. Formation of terrestrial planets from protoplanets. II: Statis-tics of planetary spin. Astrophys. J. 671 (2), 2082–2090.

Marcy, G., Butler, R.P., Fischer, D., Vogt, S., Wright, J.T., Tinney, C.G., Jones, H.R.A.,2005. Observed properties of exoplanets: masses, orbits, and metallicities. Prog.Theoret. Phys. (Suppl. 158), 1–19.

1 th and

M

M

M

K

O

PQ

S

S

S

T

T

T

54 A.P. van den Berg et al. / Physics of the Ear

atyska, C., Moser, J., Yuen, D.A., 1994. The potential in uence of radiative heattransfer on the formation of megaplumes in the lower mantle. Earth PlanetarySci. Lett. 125, 255–266.

atyska, C., Yuen, D.A., 2007. Lower mantle properties and convection models ofmultiscale plunes. In: Foulger, G.R., Jurdy, D.M. (Eds.), Plates, Plumes, and Plan-etary Processes, Special Paper 430. Geological Society of America, pp. 137–163.

oresi, L.-N., Solomatov, V.S., 1995. Numerical investigation of 2D convection withextremely large viscosity variations. Phys. Fluids 7, 2154–2162.

atsura, T., Yokoshi, S., Kawabe, K., Shatskiy, A., Manthilake, M.A.G.M., Zhai, S.,Fukui, H., Hegoda, H.A.C.I., Yoshino, T., Yamazaki, D., Matsuzaki, T., Yoneda, A.,Ito, E., Sugita, M., Tomioka, N., Hagiya, K., Nozawa, A., Funakoshi, K.-I., 2009.P–V–T relations of MgSiO3 perovskite determined by in situ X-ray diffractionusing a large-volume high-pressure apparatus. Geophys. Res. Lett. 36, L01305,doi:10.1029/2008GL035658.

’Neill, C., Lenardic, A., 2007. Geological consequences of super-sized Earths. Geo-phys. Res. Lett. 34, L19204, doi:10.1029/2007GL030598.

erryman, M.A.C., 2000. Extra-solar planets. Rep. Prog. Phys. 63, 1209–1272.ueloz, D., et al., 2009. The CoRoT-7 planetary system: two orbiting super-Earths.

Astronomy Astrophys., 506 (1), 303–319, doi:10.1051/0004-6361/200913096.egal, A., Praagman N.P., 2000. The SEPRAN package. Technical report.

http://ta.twi.tudelft.nl/sepran/sepran.html.harpe, H.N., Peltier, W.R., 1978. Parameterized mantle convection and the Earth’s

thermal history. Geophys. Res. Lett. 5, 737–740.teinbach, V., Yuen, D.A., Zhao, W., 1993. Instabilities from phase transitions and the

timescales of mantle evolution. Geophys. Res. Lett. 20, 1119–1122.ackley, P.J., 1995. On the penetration of an endothermic phase-transition by

upwellings and downwellings. J. Geophys. Res. 15 (B8), 15477–15488.ackley, P.J., Stevenson, D.J., Glatzmaier, G.A., Schubert, G., 1993. Effects of an

endothermic phase transition at 670 km depth in a spherical model of convectionin the Earth’s mantle. Nature 361, 699–704.

rompert, R., Hansen, U., 1998. Mantle convection simulations with rheologies thatgenerate plate-like behavior. Nature 395, 686–689.

Planetary Interiors 178 (2010) 136–154

Tsuchiya, T., Tsuchiya, J., Umemoto, K., Wentzcovitch, R.M., 2004. Phase transitionin MgSiO3 perovskite in the Earths lower mantle. Earth Planet. Sci. Lett. 224,241–248.

Umemoto, K., Wentzcovitch, R.M., Allen, P.B., 2006. Dissociation of MgSiO3 in thecores of gas giants and terrestrial exoplanets. Science 311, 983–986.

Valencia, D., O’Connell, R.J., Sasselov, D., 2006. Internal structure of massive terres-trial planets. Icarus 181, 545–554.

Valencia, D., Sasselov, D., O’Connell, R.J., 2007a. Radius and structure models of thefirst super-Earth planet. Astrophys. J. 656, 545–551.

Valencia, D., O’Connell, R.J., Sasselov, D.D., 2007b. Inevitability of plate tectonics onsuper-Earths. Astrophys. J. 670, 45–48.

Valencia, D., O’Connell, R.J., Sasselov, D.D., 2009. The role of high-pressure exper-iments on determining super-Earth properties. Astrophys. Space Sci. 322,135–139 doi:10.1007/s10509-009-0034-6.

van den Berg, A.P., van Keken, P.E., Yuen, D.A., 1993. The effects of a composite non-Newtonian and Newtonian rheology in mantle convection. Geophys. J. Int. 115,62–78.

van den Berg, A.P., Yuen, D.A., 1998. Modelling planetary dynamics by using the tem-perature at the core–mantle boundary as a control variable: effects of rheologicallayering on mantle heat transport. Phys. Earth Planet. Inter. 108, 219–234.

van den Berg, A.P., Yuen, D.A., Steinbach, V., 2001. The effects of variable thermalconductivity on mantle heat-transfer. Geophys. Res. Lett. 28, 875–878.

van den Berg, A.P., Rainey, E.S.G., Yuen, D.A., 2005. The combined influences ofvariable thermal conductivity, temperature- and pressure-dependent viscosityand core–mantle coupling on thermal evolution Phys. Earth Planet. Inter. 149,259–278.

van Summeren, J.R.G., van den Berg, A.P., van der Hilst, R.D., 2009. Upwellings from

a deep mantle reservoir filtered at the 660 km phase transition in thermo-chemical convection models and implications for intra-plate volcanism. Phys.Earth Planet. Inter. 172, 210–224.

Zhang, S., Yuen, D.A., 1996. Intense local toroidal motion generated by vari- ableviscosity compressible convection in 3-D spherical shell. Geophys. Res. Lett. 23,3135–3138.