Download Convection scaling and subduction on Earth and super

Earth and Planetary Science Letters 286 (2009) 492–502
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Convection scaling and subduction on Earth and super-Earths
Diana Valencia ⁎, Richard J. O'Connell
Department of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138, USA
a r t i c l e
i n f o
Article history:
Received 5 July 2008
Received in revised form 20 June 2009
Accepted 7 July 2009
Available online 12 August 2009
Editor: T. Spohn
Keywords:
plate tectonics
super-Earths
parameterized convection
a b s t r a c t
Super-Earths are the smallest class of discovered extra-solar planets. Owing to their relatively small mass, some
might resemble Earth and perhaps be habitable. Because of the connection to habitability through thermal
evolution, we investigate the tectonic regime of massive terrestrial planets. Two independent studies [O'Neill, C.,
Lenardic, A., Oct 2007. Geological consequences of super-sized earths. GRL 34, L19204, Valencia, D., O'Connell, R.J.,
Sasselov, D. D., Nov 2007a. Inevitability of plate tectonics on super-earths. ApJL 670, L45–L48.] have reached
opposing conclusions about the likelihood of plate tectonics. Here, we offer possible reasons for the discordant
findings and address three key aspects of sustaining plate tectonics: deformation on faults, negative buoyancy and
energy dissipation during subduction. We show that in general, the ratio of driving force to plate resistance
increases with planetary mass. This is a consequence of increasing convective stresses, thinning plates and similar
plate structure. We conclude that even though the strength of dry and wet of faults increases for massive
terrestrial planets, the convective stress increases even more allowing deformation to take place. Also, despite
shorter timescales for plate cooling, rocky super-Earths achieve negative buoyancy at subduction zones. Finally, by
investigating the effects of energy dissipation during subduction we find that massive terrestrial planets dissipate
less energy during subduction and hence provide a positive feedback to sustain active-lid tectonics. In conclusion,
rocky super-Earths have more favorable conditions than Earth for the subduction of plates, and hence, for
sustaining plate tectonics.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Despite the formidable observational challenges, fifteen superEarths have been discovered from the ground in the last four years.
Super-Earths are extrasolar planets made primarily of rock and/or ices
and voided of a massive hydrogen or helium envelope. There is no hard
limit on planetary mass that separates the gaseous from the (mostly)
solid planets, but a reasonable value may be in the vicinity of 10M⊕ (Ida
and Lin, 2004). Two important characteristics make super-Earths
interesting objects to study: the ones that are rocky might be similar
to Earth, and depending on their thermal state be habitable; and, the bias
in detection ensures that larger Earth-like planets will be discovered
before a true Earth analog. Furthermore, in the near future several dozen
super-Earths will be discovered with space missions underway—CoRoT
(Borde et al., 2003), Kepler (Borucki et al., 2003)—and others under
construction (Automated Planet Finder, JWST, etc.). To interpret the data
that will be available, it is timely to set the framework for understanding
the properties of these planets. In this study we address the relevant
topic of tectonics in rocky massive planets.
⁎ Corresponding author. Now at the Observatoire de la Cote d'Azur, BP 4229, 06304
Nice Cedex 4, France. Tel.: +33 492 00 30 52; fax: +33 492 00 31 21.
E-mail addresses: [email protected] (D. Valencia), [email protected]
(R.J. O'Connell).
0012-821X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2009.07.015
The thermal evolution of a planet is intrinsically related to the
mode of convection, which can be in a state with plate tectonics or else
with an immobile stagnant lid. There have been attempts to predict
the mode of convection of rocky super-Earths with opposite
conclusions drawn. Valencia et al. (2007) show that terrestrial
super-Earths can exhibit plate tectonics, while O'Neill and Lenardic
(2007) (hereonafter OL07) determine that super-sized Earths will
most likely be in a stagnant lid regime and that only in some cases will
they experience episodic plate tectonics.
Plate tectonics is a complicated process and one that we only evidence
on Earth. Despite the geological data available on Earth, the details of
active lid tectonics are not completely understood. Nevertheless, the
general features of plate tectonics have been laid out over the past
30 years and are well recognized. Determining how the onset of plate
tectonics takes places is a challenge still unresolved. But once subduction
has started, it is much easier to maintain because deformation can happen
along pre-existing faults. Thus, we investigate if plate tectonics can be
sustained in massive versions of Earth. We approach three questions: is
the increase in fault strength larger than the convective stress as to hinder
subduction (as suggested by O'Neill and Lenardic (2007))? Is negative
buoyancy reached at subduction zones in rocky super-Earths to ensure
foundering? Is dissipation during subduction large enough to slow down
plates to the point of halting plate tectonics?
The discovery of super-Earths has recently opened an opportunity for
comparative planetology. It promises to widen the current geophysical
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
theories that have been developed to explain Earth's and the other
terrestrial planets' properties. This places thinking about the solid planets
in our solar system within a wider, more complete planetary context. We
consider this study to be a step in that direction.
2. Different approaches
The subject of plate tectonics on Earth has been approached in three
different ways: with analytical theories, numerical modeling and
experimental work, all complementary to one another. The approach by
OL07 to study tectonics on massive Earth-like planets was to adapt the
numerical model by Moresi and Solomatov (1998). This finite element
code was developed to reproduce plate-like behaviour on Earth. It
considered a temperature-dependent viscous mantle heated from below
overlain by a lithosphere that deforms according to its stress regime. In the
upper cold part, the lithosphere is brittle and may deform as long as the
convective stress exceeds the yield stress. To circumvent the problem of
having plate-like coherent structure while inducing subduction, they
adopted a Non-Newtonian behaviour to a near-surface layer. This
formalism reinforces zones of weakness that then can fail under a Byerlee
criterion and produce trenches. Due to the well known limitations of
modeling systems with large Rayleigh numbers (Ra), Moresi and
Solomatov (1998) considered values in the range Ra=105–108. They
successfully reproduce plate-like behaviour, episodic foundering and
stagnant lid occurring as a function of increasing fault strength. In the
regime of plate tectonics they reach an almost exact agreement with the
theoretical prediction between the Nusselt number (Nu) and Ra from
boundary layer theory.
The conclusions drawn by OL07 rest on how the scaling to bigger
planets from this model was done. They consider super-sized Earths
that have the same density and conclude that larger planets (scaling
ratio of R/RE N 1) would require lower yield stresses to achieve at least
episodic stages of plate tectonism. Thus, they attribute the prevalence
of stagnant lid on rocky super-Earths to the locking of faults caused by
higher pressures from higher gravity values. Furthermore, Fig. 3 in
their paper suggests that smaller planets than Earth can have mobile
plates. In a personal communication with A. Lenardic, this puzzling
conclusion was attributed to lower temperatures on smaller planets.
The thermal age of the small planets they model is thus, older than
their massive counterparts. Lower temperatures increase the viscosity,
so that the deviatoric stress increases to aid subduction. This is the
first clue to the different findings between the two different groups.
On the other hand, the work we presented in Valencia et al. (2007a)
used a parameterized analysis of convection in a system heated from
within to predict the convective properties on massive terrestrial
planets. We investigated Earth-like planets in the sense that they would
have a similar composition: concentration of radioactive elements and
Fe/Si ratio (expressed as similar core-mass fraction). We accounted for
compression so that massive planets would have larger average
densities and gravities that scale accordingly. The conclusion was that
due to thinner plates and larger convective stresses, the conditions for
subduction, would be more favorable on bigger planets.
There are two basic assumptions behind our work, and that of OL07
and Moresi and Solomatov (1998): (1) that the stress needed to cause
deformation is available from convection. This is justified by relating
Earth's plate motions to the tractions on plates (Bird, 1998; Becker et al.,
1999; Rucker and Bird, 2007). And (2) that a Byerlee criterion for failure in
the brittle part of the lithosphere adequately captures the behaviour of
large-scale faults that represent the collective behaviour of randomly
oriented smaller faults. However, a key result that is different in our
findings is that the pressure–temperature structure of the plates remains
almost invariant to mass.
Having that the methods employed by the two groups do not differ in
their core reasoning, we first set out to investigate if the “increase in fault
strength drastically outweighs the change in convective stress” in massive
Earth-like planets as suggested by (O'Neill and Lenardic, 2007).
493
3. Convective parameters
We elaborate on the parameterized convection analysis to show
the implicit dependence of the convective parameters on the mass of
the planet (M). This is done via the Rayleigh number (Ra). In the case
of a fluid layer of depth D that is internally heated, the Rayleigh
number can be defined in terms of the surface heat flux q:
4
Ra =
ραgD q
κηk
ð1Þ
where g is the gravity and the material properties are: the density ρ,
coefficient of thermal expansion α, thermal diffusivity κ, thermal conductivity k, and viscosity η. These parameters are constant and unambiguously defined in the case of a homogenous fluid layer. On the other hand, in
the case of a planetary mantle, the material properties as well as gravity,
are a function of temperature (T) and pressure (P) and therefore a function
of depth. Given that we are investigating the conditions the plates are
subject to, the relevant parameters in Eq. (1) are those in the vicinity of the
plate. For a discussion on the applicability of boundary layer theory and
our choice of boundary condition (that of heat flux) see Appendix A.
3.1. Shear stress
In mantle convection, the deviatoric horizontal shear stress
underneath the lithosphere depends on the velocity of the plate and
the depth of the convective layer ℓ,
uplate
;
ℓ
Δτxz ~ η
with ℓ ~ D for whole mantle convection. The velocity of a plate is
related to the Rayleigh number (O'Connell and Hager, 1980; Turcotte
and Schubert, 2002a)
2
uplate ea1
κ Ra 1 = 2
λ;
D Rac
where a1 is a coefficient of order unity, λ is the ratio of width to depth
of the convective cell (of order unity in the mantle), and Rac ~ O(1000)
is the critical Ra for convection to occur. Thus, the deviatoric
horizontal stress has no direct dependence on the depth of the mantle
Δτxz ~
κα
kRac
1 = 2
ðηρgqÞ
1=2
:
ð2Þ
Nevertheless, there is still an indirect dependence because an
increase in planetary size due to an increase in mass is accompanied
by an increase in heat content, gravity, average density and possibly
other material properties. Moreover, in a scenario well described as a
heated-from-within system, the stress depends on the heat flux,
which is a temporal quantity that changes as the planet evolves and
cools. In Section 4 we show how we evaluate super-Earths' heat fluxes.
3.2. Boundary layer
In a planet with plate tectonics, the lithosphere is the boundary
layer. For planets with a stagnant lid, the colder and stronger part of
the plate sits above the boundary layer that participates in the
convection (Davaille and Jaupart, 1993; Solomatov, 1995). According
to classical boundary layer theory, the boundary layer thickness is
effectively independent of the depth of the convective layer (depth of
mantle) but depends on the vigour of convection, Ra (O'Connell and
Hager, 1980; Turcotte and Schubert, 2002a),
δ=
1 D Ra −s
a1 2 Rac
ð3Þ
494
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
where s = 1/4 when the boundary condition is the heat flux and s = 1/3
when it is the temperature difference across the mantle. The more
vigorous the convection, the thinner the boundary layer. In terms of the
heat flux the thickness of the boundary layer or plate thickness is
α
κkRac
δ~
−1 = 4 ρgq −1 = 4
:
η
ð4Þ
Eqs. (2) and (4) show the dependence of deviatoric shear stress
and plate thickness on the material properties of the convective fluid
and the heat flux. To obtain the dependence on mass, we obtain the
values of these fluid properties (density, gravity, viscosity) and the
heat flux for super-Earths. We note that α, κ and k might vary among
terrestrial planets, but that the largest uncertainty comes from the
viscosity which can change by orders of magnitude. In addition, while
viscosity depends on pressure, temperature and stress (Karato and
Wu, 1993), the dominant effect is temperature and thus, we consider a
temperature-dependent viscous mantle.
4. Planetary heat flux
A planet's surface heat flow Q, reflects the amount of heat being
transported into the mantle from the core Qcore, generated within
from radioactive heat sources Qrh and its cooling rate at a given point
in time dT(t)/dt:
Q ðtÞ = ρCp
4πR3 dTðtÞ
rh
core
+ Q ðtÞ + Q
ðtÞ:
dt
3
ð5Þ
The contribution from radioactive sources to the heat flow is
rh
0
∑Ri γi
i
Q ðtÞ = ∑Ri γi ðtÞ =
i
where Ri is the amount of heat produced per mass by each of the
major radioactive elements 238U, 235U, 232Th, and 40 K, γi is the amount
of each radioactive element in the planet's mantle at a given time t,
and λi is their decay constant.
238U
Provided that these isotopes have very long half-lifes (i.e. t1/2
=
232Th
4.4 Gy, t1/2
=14 Gy), their concentration in the galaxy is not expected to
vary wildly. Thus, it is reasonable to consider planetsthat
have the same
M
M⊕
. This assump-
tion can subsequently be relaxed to explore its effects on the results.
To calculate the other two terms in Eq. (5) and obtain the total heat
flow of a planet Q at any point in time, we note that a thermal
evolution model is required. Such a model would yield in principle the
exact contribution of radioactive sources to total heat flow (the Urey
ratio—U). Instead, we limit our study to planets mostly dominated by
radioactive heat production—as in the case of Earth—which are
expected to have mass-scaled versions of total heat flow
2
4πR qp = Q p = Q ⊕
M
:
M⊕
Δτxz ~ M
δ~ M
ða + b + c + dÞ = 2
ða + b + c−dÞ = 4
ð7Þ
:
ð8Þ
Even without the implementation of an internal structure model,
qualitative results can be obtained from the scaling of the most simplified
scenario: that of a family of isoviscous rocky super-Earths scaled with
constant density. In this simple case, a=c=0, and R~M1/3, so that the
heat flux q~M1/3 and gravity scales as g=GM1/3. The stress dependence
on mass is then Δτxz ~M1/3 and δ~M− 1/6. This elemental scaling shows
that the ratio of shear stress to plate thickness increases with planetary
mass.
We investigate if this qualitative conclusion holds true in a general
case. In particular, it is not known a-priori the effect of viscosity on shear
stress. Even a modest increases in temperature, could significantly lower
the tractions underneath the plates effectively decreasing the driving
force behind subduction.
We model the viscosity after Davies
(1980) by using a power law fit to
the Arrhenius law, so that η = η0
exp −λi ðt−t0 Þ;
rh
radioactive concentration as Earth, Q rh
p ðtÞ = Q ⊕ ðtÞ
number and plate thickness convergence is reached to yield a
consistent thermal profile. It also includes all major phase transitions
known to occur on Earth including post-perovskite. This model
successfully reproduces Earth and the Terrestrial Planets' structure.
The mass dependence of the properties relevant to this study is
adequately expressed in a power law relation: ρ~Ma, g~Mb, η~Mc, q~Md,
where the exponents a, b, c and d are determined from the results
of the internal structure model. The expressions for deviatoric horizontal shear stress and plate thickness depend on these exponents (from
Eqs. (2) and (4))
ð6Þ
5. Scaling with mass
To get the dependence of plate thickness and shear stress on mass,
we use a detailed internal structure model to obtain the mantle size,
average mantle density, average gravity, and viscosity under the plate
for rocky super-Earths. Since the details of the model have been
explained elsewhere (Valencia et al., 2006, 2007b), we only briefly
describe it here. It solves the differential equations of density, gravity,
mass, pressure and temperature with depth. It uses a Vinet equation of
state (Vinet et al., 1989) and a thermal profile that is conductive
throughout the boundary layers and adiabatic in the convective
interiors of the mantle and core. Through iteration in the Rayleigh
T −n
T0
where n=30 and η0 is the
viscosity value at the reference temperature T0. We are interested in the
viscosity underneath the lithosphere, so that T is the potential
temperature. The results from the internal structure model are that
the planetary radius, average density, gravity, heat flux and viscosity
have a dependence on mass of R ~ M0.262, ρ ~M0.196, g ~M0.503, q ~ M0.476,
and η ~M− 0.64. The viscosity decreases with mass, because the temperature underneath the plate slightly increases from 1566 K for Earth to
1631 K for a 10M⨁-planet. Thus, the dependence of deviatoric stress and
plate thickness is Δτxz ~M0.33 and δ ~ M− 0.42. While the exact values for
the exponents may vary depending on the viscosity formalism adopted,
the robust result is that the ratio of shear stress to plate thickness
increases with mass.
We explore the role of n in this qualitative result (Appendix B). We
conclude that the condition for the shear stress to increase with mass
is satisfied as long as n N 2
a+b
+1
d
a+b
−1
d
, which always holds true. In the
case of constant density scaling a = 0, b =d, so that any n N 0 satisfies the
inequality. For a general case, two conditions are met: 1) the heat flux
roughly scales with gravity q ~g, and 2) the density always increases
with mass (a N 0). Therefore, the inequality holds true for any positive n,
and even slightly negative values, which are perhaps unrealistic.
The very modest increase in potential temperature can be easily seen
from its relation to viscosity and heat flux (Eq. 65 of O'Connell and Hager,
1980)
ðT−Ts Þ4 ðq=KÞ3 κ
Rac ;
≈
16ρgα
η
ð9Þ
where Ts is the surface temperature. By replacing the power law relation
between viscosity and temperature, it is clear that the internal temperature weakly depends on heat flux, gravity and material properties, all
quantities that vary with mass:
!1 = ðn + 4Þ
ðq=KÞ3 κ
Rac
T≈T0 η0
:
16ρgα
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
495
Given that heat flux scales as gravity and both scale as ~M1/2, the
potential temperature (for constant density planets) scales as T ~M1/34
given n = 30. Including the effects of compression on density would
make T more insensitive to mass.
In conclusion, in internally-heated mantles with same radioactive
concentrations, shear stress underneath the plates increases with
planetary mass. Furthermore, the ratio of shear stress to plate
thickness increases for terrestrial massive planets undergoing plate
tectonics. This results provides a possible future benchmark with the
work by OL07. It is unclear from their paper if we agree on this matter.
6. Fault strength
We arrive at the discussion of fault strength and investigate
whether or not it outweighs the increase in convective stresses as
suggested by OL07. Pressure does increases the frictional strength on
faults and hence, can potentially hinder deformation on large planets.
We determine how the stress and the strength on the faults varies
with planetary mass. For deformation to happen, the shear stress on
the fault (τ) has to overcome its strength (τrock). In the same manner
as (Moresi and Solomatov, 1998), we invoke a Byerlee criterion for
deformation on the brittle part of the lithosphere. The Coulomb
strength of rocks is expressed using Byerlee's law
τrock = S + μðσ−λσzz Þ
where S is the internal cohesion parameter, σ is the normal stress on
the fault, σzz = ρgz is the lithostatic pressure, and λσzz expresses the
effect of water on reducing the fault's strength. The normal (σ) and
shear (τ) components of the stress on the fault are
σ = σzz +
τ=
1
Δσxx ð1 + cos2θÞ
2
1
Δσxx sin2θ
2
where θ is the angle of faulting which is related to the friction
coefficient μ by tan 2θ = 1/μ (Turcotte and Schubert, 2002b), Δσxx =L/
δΔτxz is the deviatoric shear stress available from convection, and L
the length of the plate. Moresi and Solomatov (1998) did not consider
the angle of faulting, however, it does not constitute a major difference
in our treatments. In fact, by considering an angle of faulting our
estimates of τrock/τ are more conservative.
We find that despite higher gravity surface values, rocky super-Earths
experience similar pressures under their lithospheres. This is because the
increase in gravity is offset by a decrease in plate thickness, thereby
producing a somewhat constant average pressure on the plate of different
planets. Thus, the lithostatic pressure does not increase for bigger planets,
and consequently does not increase the fault strength. Furthermore,
because the temperature beneath the boundary layer is also somewhat
insensitive to mass as noted before, the P–T plate structure is almost
invariant to planetary mass. This result constitutes a major difference with
OL07.
We calculate the strength of a fault for the family of rocky super-Earths
and show how it compares to the fault shear stress in Fig.1. We take Earth's
strength and stress values as the reference case, and scale them according
to the equations described above. The general feature in Fig. 1 is that the
strength of both dry (long-dashed lines) and wet (short-dashed lines) faults
indeed increases with mass, as well as the shear stress experienced on the
fault (solid lines). The former happens because of a small contribution of
the convective stress on the fault's normal stress that hinders sliding. This
is due to our choice of including the angle of faulting. Had we not, as is the
case used in the model of OL07, the fault strength would have remained
constant (i.e. constant yield stress) independent of planetary mass.
The interesting result is that the increase in the fault's shear stress is
steeper than the modest increase in fault strength, so that terrestrial
Fig. 1. Faults' stress and strength. The shear stress (solid lines) on the fault increases linearly
with mass, while the strength of a dry (long dashed lines) and wet (short dashed lines) fault
increases more slowly. This behaviour is independent of choice of coefficient of friction We
considered two friction coefficient values (top): μ=0.2 and (bottom) μ=0.8. The shaded
region shows the range of values for earthquake stress release (Weins, 2001).
planets can accommodate deformation and experience subduction. The
reason for this is that the deviatoric horizontal normal stress (Δσxx)
increases almost linearly with mass from an increase in deviatoric
horizontal shear stress (Δτxz), and a decrease in plate thickness despite a
modest increase in plate length.
The following expression relates the ratio of deviatoric shear stress
to plate thickness, and the ratio of driving force to plate resistance
Driving Force
=
Plate Resistance
τ
τrock
=
1
2
S + μð1−λÞ
L
δ
1
σ̄ zz +
2
Δτxz sin2θ
L
δ
μΔτxz ð1 + cos2θÞ
:
We find a positive relation between this ratio and planetary mass,
which means that subduction and therefore plate tectonics can be
sustained more easily on terrestrial super-Earths. Fig. 3 shows the
inverse of this ratio as being less than 1 and decreasing with increasing
planetary mass (for the case of s = 0.25). It is clear that the reason
proposed by OL07 to argue that super-Earths cannot exhibit plate
tectonics stands in disagreement with our results. What lies at the core is
that the plate structure they obtain is not invariant with mass. They
claim that the pressure is higher in larger planets, and therefore the plate
thickness they obtain must be larger than that predicted by classical
boundary layer theory. We explore this in Section 8.1.
We complete our investigation of the behaviour of faults in massive
terrestrial planets by considering the uncertainty in the values of the
friction coefficient and the fluid contribution to pore pressure (i.e. 0bλb
0.9—Ranalli, 1995). We compare planets that exhibit the same coefficient
of friction justified in that we consider planets with the same composition.
On a micro-scale, the friction coefficient of different minerals will depend
on their structure (Morrow et al., 2000). However, by looking at planets
that share the same rock composition, we circumvent the problem of
addressing deformation on individual faults and focus on the collective
behaviour that is likely to share the same average value of μ .
Fig. 1 considers an effect of water/volatiles on pore pressure of λ =0.6.
With a relatively high coefficient of friction of μ=0.8 (bottom), the fault
shear stress on Earth required to exceed the strength of a wet fault is
496
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
~100 MPa (available if the deviatoric normal stress is Δσxx =300 MPa). If
we consider the coefficient of friction to be lower μ =0.2 (top), the shear
stress required is ~20 MPa (available if Δσxx =50 MPa). This low value of
μ has been recently proposed as a better value for the effective friction
coefficient on pre-existing faults (Di Toro et al., 2006), which is also more
consistent with the values of stress release during earthquakes on Earth
(shaded area in Fig. 1). We also examine the case where water is less
effective at increasing pore pressure λ=0.2 (and μ=0.8). This low value
of λ requires shear stresses of ~200 MPa for Earth to exceed the strength of
a wet fault (available if Δσxx =550 MPa). The deviatoric shear stress from
convection has been modeled to be 10–100 MPa (Becker et al., 1999), so
such a low value of λ seems unlikely.
More importantly, Fig. 1 shows that the bigger terrestrial superEarths will not require water for sliding to occur. The slope of shear
stress on faults (black line), which describes its dependence on mass,
is steeper than the curve for strength of wet and dry faults, so that at
some point these curves are crossed. The transition between superEarths that require water to the ones that do not, will depend on the
efficiency of water to increase pore pressure on faults (i.e. λ) and the
friction coefficient. Conversely, smaller planets' tectonic regime would
be more affected by the presence or absence of water.
7. Lithospheric density
At convergent margins on Earth, the oldest, coldest and densest
plate is expected to subduct under the younger, hotter and buoyant
plate. Knowing that super-Earths have faster convective velocities
(Valencia et al., 2007a) and hence, younger plates in general, it is
appropriate to establish the conditions in which negative buoyancy
can still be achieved at subduction zones.
We calculate the mean density of a plate (ρ̄¯lit) composed of basaltic
crust (ρbas =2880 kg/m3) and lithospheric mantle (ρman =3330 kg/m3)
at the maximum age of the plate (when its thickness is ~double the mean
calculated thickness),
ρ̄lit =
1 hc
∫ ρ ½1 + αðzÞðTm −TðzÞÞdz
hc 0 bas
+
2δ
1
∫ ρ ½1 + αðzÞðTm −TðzÞÞdz
ð2δ−hc Þ hc man
where hc is crustal thickness, the coefficient of thermal expansion is
̂
α(z) =â +bT(z)
+ ĉ/T(z)2 with the values for â, b̂ and ĉ determined by
(Poudjom-Djomani et al., 2001), and Tm is the temperature beneath the
plate. We find that the mean lithospheric density will be denser for every
Earth-like super-Earth provided the crustal thickness does not exceed
16% of the total plate thickness during subduction (Fig. 2).
Oceanic crustal thickness depends on the extent of melting, which
depends on the depth at which the solidus intersects the thermal
adiabatic profile of the planet. The expression for crustal thickness hc is
hc =
Fmax
ðz0 −zf Þ;
2
where Fmax =r(P0 −Pf) is the maximum fraction of melting, r is the melt
production per unit of decompression; and z0 and zf are the initial and
final depths of decompression melting corresponding to pressures P0 and
̃
Pf respectively. For the solidus, we used the expression Tsol =ãP2 +bP+c
̃
with the coefficients experimentally determined by Hirschmann (2000).
The temperature beneath the plate and hence, at ridges varies little
among rocky super-Earths (for the same surface temperature). Owing
to larger gravity values, the intersecting pressure is achieved at
shallower depths, so that we anticipate the extent of melting to be
reduced with increasing planetary mass. This suggests that the crust
will remain under 16% of the plate's thickness. We perform a simple
calculation to obtain the crustal thickness of each super-Earth by
obtaining the depth at which the magma solidus is intersected by an
Fig. 2. Negative buoyancy in super-Earths. Top: We show the relative density of the plate
with respect to the underlying mantle for different melt fraction production from 1 to 2.5%
melt per kbar. Even at 2.2% melt per kbar the plate for the largest super-Earths is negatively
buoyant. Bottom: We show the fraction of crust to the total lithospheric thickness at the
time of subduction. The crustal thickness decreases with increasing planetary mass, while
the lithosphere decreases even more, so that the fraction of crustal thickness increases for
more massive planets.
ascending parcel and the amount of maximum fractional melting.
Fig. 2 shows the results. The conclusion is that as long as the change in
melt fraction r with pressure is below 2.4% melt per kbar, buoyancy
will be negative at subduction zones for terrestrial super-Earths. The
value of 2.4% melt per kbar is larger than the limits considered
appropriate for Earth (1–2%), although unknown for other planets.
Recent studies (Hynes, 2005; Afonso et al., 2007) show that on Earth,
the lithospheric mean density is not larger than the underlying mantle's at
subduction zones. However, Afonso et al. (2007) note that the plate's
density is larger than the adiabatic mantle (or ridge column density) and
Hynes (2005) shows that flooding can easily cause negative buoyancy.
Furthermore, Becker et al. (1999) showed that compression could thicken
the cold lithosphere to the point of inducing subduction. These ideas
suggest that perhaps negative buoyancy before subduction might not play
as important a role as previously thought. In any case, we find that superEarths are able to achieve negative buoyancy at convergent margins.
8. Energy dissipation during subduction
In this section we explore the effects of energy dissipation during
subduction as a scenario that might halt plate tectonics. Conrad and
Hager (1999a) recognized that a major source of dissipation can be the
subduction of thick strong plates. If the energy required in subducting
the plate is large, the plate will slow down, thicken even more making
subduction more difficult, until eventually it can no longer be sustained.
We investigate the effects of this process by modeling subduction in two
ways: bending of the plate—after Conrad and Hager (1999a), and
continuous shearing of the plate into the mantle.
8.1. Plate bending
Conrad and Hager (1999b, a) modeled the lithosphere as bending
into the mantle at subduction zones (with some prescribed radius of
curvature) and calculated the energy dissipated in this process.
Through an energy balance they derived a relationship between Nu
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
and Ra that would capture the effects of plate bending. They argued
that the exponent s in the Nu–Ra relation would be effectively
decreased in the presence of strong dissipative subduction zones,
when most of the energy is dissipated by lithospheric bending instead
of in the shearing mantle. Conrad and Hager (1999b) described Earth's
mantle as heated from below and proposed s to be 0.15.
By considering a system with a known surface heat flux and
correspondingly using an exponent of s = 1/4 (see Appendix A), we
have shown that the plates of more massive terrestrial planets are
thinner and thus, the effect of dissipation during subduction is small.
This means that if the planet avoids having a thick lithosphere to begin
with, it will be easier for the lithosphere to remain thin. The question
that follows is to determine how thick the plates can be without
causing too much energy dissipation during lithospheric bending, as
to avoid continuously thickening into a stagnant lid.
Another way to answer this question is to calculate the minimum
value for s that would still allow for the strength of the fault to be less
than the shear applied—τrock/τ b 1—so that deformation can still take
place. To assess this, we considered different values of s and obtain the
values for plate thickness, horizontal deviatoric shear and normal
stresses on the fault and with these, calculate the fault's shear stress τ
and strength τrock as shown above. The results are shown in Fig. 3 for two
different coefficients of friction μ = 0.2 (top) and μ = 0.8 (bottom). The
nominal case of s = 1/4 described in Section 6 shows how the fault's
strength is more easily overcome by the stress on the fault as the mass of
the planet increases. There are three regimes for deformation depending
on the value of s: i) if s is not too small (s ≳ 0.19) the ratio of strength to
stress decreases with mass, suggesting that subduction would take place
more easily in more massive planets, ii) if s is very small (s ≲ 0.13) this
ratio increases dramatically to suggest that only Earth would have
subduction, and iii) for intermediate values of s (0.13 ≲s b 0.19), the value
of μ plays an important role.
These results imply that as long as the relation of heat transport to
the vigour of convection is not negligible (s N 0.16), planets with larger
masses could also experience subduction and hence, plate tectonics.
497
Fig. 4. Cartoon of convection and subduction. Subduction is modeled as a continuous
shearing of blocks that sink into the mantle at convergent margins. The velocity profile
of the core of the shearing mantle is taken to vary linearly (following Turcotte and
Schubert, 2002a). The potential energy is equated to balance the energy dissipated at
shearing the core of the mantle and at subduction zones.
8.2. Subduction shear
Another way to model energy consequences of subduction is to
consider continuous vertical shearing of the plate into the mantle.
Fig. 4 shows a cartoon explaining this approach. At subduction zones,
an infinitely small block is sheared from the plate into the mantle. This
process is continuous so that the work rate of this system per length
along the subduction zone (ℓs) is
ẇ = f τy δv0
where τy is the yield stress at which the block slides past the plate into
the mantle, v0 is the vertical velocity of the flow and fδ is the fraction
of the plate that experiences this shear. The rest of the plate can
deform as a viscous layer. In this formalism we assume that the stress
at which the block yields is independent of depth. This assumption is
partly supported by the fact that the stress release in shallow and deep
earthquakes is estimated to be of the same order. It has been
estimated to be of 3–10 MPa for shallow earthquakes and 10–44 MPa
for deep earthquakes (N300 km) (Weins, 2001).
In terms of total energy per unit time consumed during subduction, Ẇ =fτ y δ(u0ℓs)D/L, which amounts to ~ 1010–1011 W from
τy = 107–108 Pa, δ = 100 km, f = 1/2, D = 3000 km, λ = D/L ~ 2, and
with a plate production rate at mid-ocean ridges u0ℓs = 3 km2/s.
We follow the same energy analysis by Turcotte and Schubert
(2002a), to balance the potential energy (Φpe) with the dissipation in
the core of the shearing mantle (Φman
vd ) and in subduction zones
(Φsub
vd ). The expression for each of the quantities per strike length is
1= 2
Φpe = ρgαðT−Ts ÞDu0
man
2
Φvd = 4ηu0
sub
Φvd = f τy δ
D ′
λ
L
κL
π
1 = 2
L
u
D 0
where T is the temperature of the core of the mantle, L =u0δ2/(κ π) is the
length of the plate, u0 is the horizontal velocity, and λ′ =λ2 +λ− 2.
Equating the terms and using the definition of Ra in terms of the
heat flux (Eq. (1)) yields
Fig. 3. Strength–stress ratio for different convection regimes. The values of s in the relation
Nu ~ (Ra/Rac)s are shown for different coefficients of friction μ = 0.2 (top) and μ = 0.8
(bottom). s=0.25 corresponds to the nominal case with no dissipation during subduction.
Smaller values of s indicate increasingly larger energy dissipation contributions from
lithosphere bending during subduction.
Nu =
#
"
3 −1 = 4
1
Ra 1 = 4
f τy δ
1
+
:
2 4π2 λ′
4πλ′ κη L
ð10Þ
It is clear that when τy or f is zero, we recover the classical boundary
layer result. Eq. (10) shows the form in which dissipation from subduction
498
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
appears in the Nu–Ra relation (e.g. Conrad and Hager (1999b)). The
characterization of the effect as simply changing the exponent (Korenaga,
2006) obscures this.
f τy δ3
If we define ϕ = 4πλ′
κη L , the result can be used to obtain the
thickness of the plate δ,
1=4
δ = δ0 ½1 + ϕðδÞ
;
where δ0 is the plate thickness of a system with no dissipation during
subduction. This is a quartic equation that expresses a correction to
plate thickness from dissipation of energy during subduction. It is
clear that the effect of subduction of a strong plate is to increase its
size (ϕ N 0). This happens because the plate slows down (Nu decreases
with increasing δ), has more time to cool and thicken. This equation
also captures the fact that there is a positive feedback, the thickening
effect of subduction is more pronounced when the plates are thick (ϕ
increases with δ). The range of the correction term is 0.2 b ϕ b 2 on
Earth, calculated from the values given above and the range in τy,
κ = 10− 6 m2/s, and η = 1021 Pa s. This yields a corrected plate thickness
on Earth of 1.005δ0 b δ b 1.05δ0—at most 5%.
It is obvious from this result that rocky super-Earths with their thinner
plates have a very small correction term and hence, their subduction
zones are only weakly dissipative.
9. Discussion
Plate tectonics is an evolutionary phenomenon and the conditions
for it will change as the planet evolves and cools. At present, Earth has
enough heat flux to drive vigorous convection needed for an active
plate state. A planet's heat flux varies with time and while on short
timescales it might increase (Sleep, 2000; Van Keken et al., 2001),
over long timescales it decreases as the planet cools. This means, that
at some stage, when the heat flux falls below a threshold, plate
tectonics will cease to operate on a planet. The timing for this event
depends on the thermal evolution. Although, extra sources of heat,
such as tidal heating, can aid in achieving and prolonging the lifetime
of active-lid tectonics.
We find that planets of the same mass would have the same potential
to sustain plate tectonics as dictated by their ratio of shear stress to plate
thickness. Venus is not the counter example because of two important
differences with Earth: the lack of water (Mian and Tozer, 1990; Nimmo
and McKenzie, 1998) and a higher surface temperature (Lenardic et al.,
2008).
Our results are in agreement with the findings of (O'Neill et al.,
2007) where they state that Venus and tidally-heated Io are the only
other objects in the solar system that might have had plate tectonics in
the past by being in an episodic regime.
R = R⨁(M/M⨁)0.262–0.274. Because of possible unknown phase transitions and uncertainties in the equation of state, the exponents are an
upper limit. Thus, a convenient approximation for the radius scaling is
R ~ M1/4, which translates to gravity scaling as g ~M1/2. We recommend
these scalings for comparing Earth to massive Earth analogs.
As for the more fundamental difference, the main question is how
to best apply a (quasi) steady state boundary condition to an evolving
planet. Heat comes from radioactive sources in the interior; this
suggest scaling for constant heat sources, which turns out to be the
same as that for imposed heat flux. In an internally heated mantle, the
interior temperature adjusts to conform with the imposed fixed heat
flux (which is the boundary condition). In a system with a fixed
temperature difference driving convection, the heat flux adjusts.
Given the uncertainties in all such models, the most accurate choice is
unclear. In a mixed-system, the results might depend on which case is
dominant (basal to heating ratio).
In a system that is heated-from-below—that is, with a fixed
temperature difference (ΔT) across its convective layer—and constant
surface temperature, the deviatoric stresses are expected to increase
with the size of the planet, due to an increase in gravity and density
basal
Δτxz
1=3 1=3
= ηðTÞ
κ
ραgΔT
ac
R
2 = 3
:
If, however, the viscosity decreases, convection stresses would decrease as well.
In the case of internally heated planets, we obtain a very slight increase
in temperature drop within the boundary layer for massive planets (see
Eq. (9)). This qualitatively agrees with the treatment of OL07, in that larger
planets are hotter. However, the subsequent effect of lower viscosities is
further accompanied by an increase in heat flux for massive rocky superEarths, so that the net effect is that the deviatoric stress increases—
opposite to what OL07 find. The relevant equation for this case is
internal
Δτxz
1=2
= ηðTÞ
κραgq 1 = 2
ac
:
kR
Therefore, the differences in the results may be coming from the
different boundary conditions. Our model considers a steady state
scenario with heat coming from radioactive sources. It predicts that for
the same concentration, bigger planets will have larger convective
stresses and thinner lithospheres—a favorable scenario for plate tectonics.
In addition, smaller planets are expected to cool more rapidly
because of their area to volume ratio. Being that the ability to sustain
plate tectonics decreases as the planet cools, small planets will cease
to have mobile lids faster than massive rocky ones.
9.2. Treatment for viscosity
9.1. Differences between the two models
Through a personal communication with A. Lenardic, we learned
that their choice of scaling yields smaller planets to be colder than the
bigger counterparts. Lower interior temperatures translate into higher
viscosities and larger driving stresses, which can drive subduction
more easily. Thus, they conclude that small planets can achieve plate
tectonics while rocky super-Earths cannot.
There are two important differences in our treatment and that of
OL07. We consider (1) scalings that take into account the compression
effects on radius, gravity and average density; and more importantly,
(2) heat flux as the boundary condition to describe the system.
With respect to the first difference, a constant density scaling is not
adequate in comparing Earth to massive Earth-like planets. Independent models (Valencia et al., 2006, 2007c,b; Sotin et al., 2007; Seager
et al., 2007) clearly show that the relation between radius and mass
deviates from a constant density scaling. The different models, which
vary in sophistication and treatment agree that the radius scales as
A natural question that arises in the treatment of terrestrial
massive planets, is the effect of pressure in convection and specifically
on viscosity. The pressure at the core–mantle boundary of massive
Earth-like planets increases almost linearly with mass (Valencia et al.,
2006), and thus, has a profound effect on viscosity at such depths.
By its temperature and pressure dependent nature, viscosity is
effectively a function of depth, lateral variations are comparatively
minor except in plates, which have a different rheology anyway. The
limits on convection come from the transfer of heat through the
boundary layer. This is the (or at least one) basis for the derivation of
the parameterized convection models used. These seem to work
reasonably quantitatively for the Earth with a viscosity of ~1020 Pa s
for the upper mantle, even though the lithosphere is not characterized
well by such a rheology, and that the lower mantle viscosity is higher
(~1022–1023 Pa s). Therefore, we argue that if the temperature and
pressure regime in the uppermost mantle of a terrestrial super-Earth
is similar to the Earth, it should behave similarly. Consequently, the
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
pressure effects in the lowermost mantle of rocky super-Earths can be
considered a secondary effect.
In any case, we think this to be a first step towards understanding
convection and tectonics in massive planets and consider that
pressure effects on viscosity, depth-dependent thermal expansivity,
conductivity and diffusivity to be future research.
10. Summary and conclusions
It is important to establish the tectonic regime of a planet when
determining its thermal state, which in turn is fundamental to the
question of habitability. Here, we examine from classical boundary
layer theory, how the convective parameters and fault properties scale
for terrestrial super-Earths. We address three key issues of plate
tectonics: deformation on faults, negative buoyancy, and energy
dissipation during subduction. Even though the mantle is a complicated system (with chemical heterogeneities, chaotic dynamics,
complex rheology, etc.), simple models like ours are useful in
illuminating relations between important parameters. Moreover, to
assess the effect of mass on the characteristics of planets, it is vital to
appropriately scale the relevant properties with mass. For this we use
a detailed internal structure model that renders simple and robust
relations: heat flux scales with gravity and radius scales as R ~ M1/4.
One of the main results is that convective stresses increase, plate
thickness decreases, while the pressure–temperature structure of the
plate is almost the same for massive planets with the same radioactive
heat concentration. These qualitative results differ from the findings of
OL07.
Consequently, fault strength modestly increases with planetary
mass, while the shear stress that can cause sliding increases even
more, such that it exceeds the strength of wet and even dry faults of
massive planets. Deformation is accommodated more easily in bigger
Earth-like planets. Conversely, smaller planets require weakening
agents, like water, much more critically if deformation on faults is to
be sustained and subduction to occur.
Furthermore, by determining the crustal thickness in massive planets,
we determine that they reach negative buoyancy at subduction zones.
Crustal thickness decreases because, due to increasing gravity values, the
solidus of terrestrial super-Earths is crossed at shallower depths.
Nevertheless and without surprise, not all scenarios are conducive to
plate tectonics. In particular, we examine the case when the planet
develops thick plates when the first lithosphere is formed. If the plates
are thick enough, the strength of the plate would increase due to a
substantial increase in the lithostatic confining pressure and this would
prevent any sliding from occurring. On the other hand, if the plates are
initially formed thin—perhaps after the solidification of the magma
ocean—the scenario for plate tectonics is favorable.
Through an energy balance, we find that rocky super-Earths that
have achieved plate tectonics, have thinner plates. Hence, they dissipate
less energy during subduction, and can maintain plate tectonics easier
than on Earth.
Appendix A. Convection models
Simple models of the average properties of convecting systems have
been derived on the basis of conservation of momentum and energy;
these lead to parameterizations of the heat flow and temperature from
the parameters that describe the system, primarily the Rayleigh number.
These are well known and presented in textbooks (e.g. Turcotte and
Schubert, 2002a). Although they are simple, and are sometimes referred
to as “box models”, they can accurately capture the interdependence of
parameters and behaviour of convecting systems. In such models the
properties of variables such as temperature should be regarded as
appropriate averages over a spatially variable temperature field, possibly
averaged over time as well; as such the models accurately reflect
conservation of momentum and energy, on which they are ultimately
499
based. Such models have long been used to study the thermal evolution
of the Earth (e.g. McKenzie and Weiss, 1975; Davies, 1980; Conrad and
Hager, 1999b; Korenaga, 2008) and planets.
The governing equations
The Navier–Stokes equation follows from conservation of momentum. In dimensionless form it is
3
κρ dυi
1 dυi
ℓ
=
= υi;jj +
ð−p;i + fi Þ:
η dt
Pr dt
κη
ð11Þ
Here the length scale is ℓ, the time scale is τ=ℓ2/κ, with κ=k/ρcp the
thermal diffusivity, k thermal conductivity, ρ density, cp the heat capacity
and η the viscosity. The velocity υi, xi and t are dimensionless, index
notation is used: υi;j ≡
∂υi
,
∂xj
and fi and p are the dimensional body force and
pressure. The Prandtl number Pr=η/ρκ is the ratio of momentum
diffusivity to thermal diffusivity, and is ~1024 for the Earth.
The body force is fi = ρgδi3 where gravity is in the x3 direction. We
can subtract the mean hydrostatic pressure that satisfies p,i = ρ̄¯g, with
ρ̄¯ the horizontally averaged density, Eq. (11) becomes
1 dυi
Δρgℓ3
= υi;jj −p;i +
δi3
Pr dt
κη
Here p is now the dimensionless nonhydrostatic pressure, scaled
using the viscosity η, and Δρ is the density variation from the
hydrostatic state. For thermal density variations Ti, Δρ = ραTi, where
α is the coefficient of thermal expansion. With reference temperature
T0, which we take as the scaling temperature, we then get
1 dυi
ρgαT0 ℓ3 Ti
= υi;jj −p;i +
δ
Pr dt
κη
T0 i3
The combination
3
ρgαT0 ℓ
= Ra
κη
is the Rayleigh number; this gives the ratio of buoyancy forces to viscous
forces, and is the main parameter governing thermal convection.
For the Earth, with Pr ~ 1024 the equation is then
0 = υi;jj −p;i + Ra δT
ð12Þ
T
where δT = i is the dimensionless temperature variations. Note that
T0
the Rayleigh number is the only parameter in the equation.
The dimensionless energy equation that governs heat transfer is
2
∂T
αgℓ Ta ∂p
gℓ ′ ′
ℓ
ð
τ e˙ +
A
+ υi T;i −
+ υi p;i Þ = T;ii +
cp T0 ∂t
cp T0 ij ij
kT0
∂t
ð13Þ
where we scale temperature, length and time as before with T0, ℓ and
ℓ2/κ, and scale pressure p and deviatoric stress τij′ with ρgℓ.
Using the Gruneisen parameter γ = (αKs)/(ρcp) this becomes
∂T
ρgℓγ Ta ∂p
ρgℓγ ′ ′
ℓ2
ð
τij e˙ij +
A
+ υi T;i −
+ υi p;i Þ = T;ii +
K
K
T
αT
kT
∂t
s
0 ∂t
s
0
0
ð14Þ
The Gruneisen ratio is usually of order unity for Earth materials, so
the terms with γ are small if the hydrostatic pressure ρgℓ is small
compared to the adiabatic bulk modulus Ks. In this case the volumetric
compression of the material will be small and essentially incompressible (i.e. υi,i = 0). Correspondingly, the effects of adiabatic temperature changes and viscous heating will be small. These terms balance
each other when integrated over the entire volume of the region
500
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
(Backus, 1975), i.e. they cannot contribute to the energy of the whole
system, although they can appear locally.
Hence, for the incompressible case
2
q
q
D
=
=
qcond
kT0 = D
2δ0
Nu =
2
∂T
ℓ
∂T
ℓ
A or
A:
+ υi T;i = T;ii +
= ðT;i −υi TÞ;i +
kT0
kT0
∂t
∂t
isothermal on average. The heat transport can be written q =kT0/δ0
where δ0 is the thickness of an equivalent conductive boundary layer.
ð15Þ
The term T,i represents conductive heat flow, and the term υiT
represents advective heat flow. The latter must vanish when the
normal velocity approaches zero at a boundary, near which heat
transport will be dominated by conduction. Similarly, in the interior of
a convecting fluid, the temperature gradients will be small and heat
transport dominated by convection.
This must be a function of the Rayleigh number and the aspect
ratio λ, and if these are separable so that Nu = f(Ra)a0(λ) then
Nu =
q
qcond
δ0 =
The only parameter in the dimensionless equations (12) and (15) is
the Rayleigh number and the heat sources, which also appears in the
Rayleigh number when they are present. Consequently the solution to
the equations will depend on the Rayleigh number, together with
those that describe the geometry of the boundary–value problem and
the boundary conditions. For the case of a layer with thickness D we
take the length scale ℓ = D. The horizontal scale of convection may be
undetermined a priori, or constrained by lateral boundary conditions;
it may be characterized as λD where λ is the aspect ratio. Consequently, any solution of the problem must be, in dimensionless form,
a function of the Rayleigh number and parameters such as λ describing
the geometry of the system.
We can distinguish three cases with different thermal cases and
boundary conditions:
(1) Fixed temperature T = T0 at the bottom and T = 0 at the top.
In this case the scaling temperature is the temperature
difference across the layer T0, and the Rayleigh number is
Ra = ρgαT0 D3/κ η.
(2) Fixed heat flux q0 at the top with a fixed temperature at the top, and
no heat flux at the top. In this case the scaling temperature can be
taken to be T0 =q0D/k, which is the temperature difference across
the layer for the conductive state. The Rayleigh number is
Ra=ρgαq0 D4/kκη.
(3) Uniform heat sources A distributed in the layer with fixed
temperature at the top. The Rayleigh number can be taken as
Ra = ρgαAD5/kκ η with T0 = AD2/k, which is twice the conductive temperature across the layer in the conductive state.
Recognizing that the heat flux out of the layer is q0 =AD, this
becomes the same as case 2 above for the heat flux boundary
condition.
Simple convection models
Temperature boundary conditions
In a uniform fluid layer of depth D the only parameter in the
dimensionless momentum and energy equations is the Rayleigh
number. For the case of fixed temperature boundary condition, the
effect of convection will be to increase the heat transport through the
layer above the value for the conductive state. The dimensionless
measure of this is the Nusselt number
Nu =
q
q
=
qcond
kT0 = D
ð16Þ
where q is the heat flux. If there is no convection Nu = 1, and convection
increases its magnitude. For vigorous convection, when the Rayleigh
number is large, a thermal boundary layer forms at the upper and lower
boundaries where heat transport is by conduction. In the interior heat
transport is dominated by convection, and the temperature is nearly
=
1 = 3
q
D
Ra
=
= a0 ðλÞ
KT0 = D
2δ0
Rac
ð18Þ
or
The parameters
ð17Þ
Rac ηκ
ραgT0
1 = 3
1
:
2a0 ðλÞ
ð19Þ
This states that the equivalent boundary layer thickness δ0 is
independent of the layer thickness D.
A direct consideration of the boundary layer by Howard (1966)
came to the same result. He argued that the fluid next to the cold
boundary would cool
off by conduction and the temperature would be
pffiffiffiffiffi
Tðz; tÞ = Ti erf ðz = 2 κt Þ, where erf(x) is the error function and the
temperature in the interior
pffiffiffiffiffiffiffiffi of the layer Ti = T0/2. The heat flow at the
surface is qðtÞ = kTi = πκt , which is pthe
ffiffiffiffiffiffiffiffiheat flow for a conductive
boundary layer of thickness δðtÞ = πκt . The cooled layer would
thicken until it became dynamically unstable and it would sink into
the interior to be replaced by hot fluid. The time is estimated as that
for the boundary layer to become unstable to convection, or when the
layer thickness δ(t) = δc such that the Rayleigh number for the
boundary layer Ra = ραgT iδ3c /κη = Rac where Rac is the critical
Rayleigh number for the onset of convection of a layer.
Assuming that the time for the detachment and replacement of the
layer is short compared to the cooling time, the average heat flow is
q̄¯=2kTi/δc = 2kTi/(D(Rac/Ra)1/3). We have taken Ti = T0/2 since a
boundary layer also exists at the bottom of the convecting layer. The
dimensionless heat flow, the Nusselt number, is then
Nu =
kT = δ
D
q̄
= i 0 =
=
2δ0
kT0 = D
kT0 = D
Ra
Rac
1 = 3
≈ a0
Ra
Rac
1 = 3
ð20Þ
with the equivalent conductive boundary layer thickness
δ0 =
1 D Rac 1 = 3
:
a0 2 Ra
This is the same relation as before, and again the boundary layer
thickness is independent of the layer depth. This is appropriate, since
δc is determined by a local stability criterion which should not be
influenced by a distant boundary. Of course this arises from the
assumption that the boundary layer exists, and by definition is thin
compared to the layer depth. The analysis also ignores the manner of
detachment of the boundary layer, which will be related to the pattern
and aspect ratio λ of convection.
Heat flux boundary conditions
If the heat flux q0 through the layer is specified as a boundary
condition, the Rayleigh number is Ra=ρgαq0D4/(kκ η). Since the heat
flux is specified, the effect of convection will be to lower the temperature
drop across the layer below that for conduction alone, and determine the
interior temperature Ti of the layer. The scaling relation used for the
temperature boundary condition case, or the boundary layer stability
condition (O'Connell and Hager, 1980) yield
Nu =
1 = 4
q0 D = k
D
Ra
=
= a1
2Ti
2δ0
Rac
ð21Þ
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
where we have used the Nusselt number to represent the ratio of the
internal temperature Ti =q0δ0/k to that for the conductive state, and as
before a1 should be of order 1. Again, the boundary layer thickness δ0 is
independent of the layer thickness D, since its stability is determined by
a local stability criterion. The Nusselt number reflects this.
If volumetric heat sources A drive convection, the Rayleigh number
can be taken as Ra =ρgαAD5/(kκ η) with T0 =AD2/k, which is twice the
temperature difference across the layer in the conductive state. However,
recognizing that the heat production per unit area is q0 =AD, this
becomes the same as the case for the heat flux boundary condition. This is
consistent with the boundary layer stability condition, since the boundary
layer will respond only to the heat flux through it, and the boundary layer
thickness δ0 is independent of the depth of the layer.
It is interesting that the expressions for the Nusselt number have
different exponents depending on the boundary conditions, even though
they are derived from the same physical considerations. The scaling
reflects the independence of the boundary layer thickness on the layer
depth; the different exponents reflect that D appears to different powers
in the Rayleigh number. Although this result may appear anomalous, it is
confirmed by the more rigorous derivation of the Nu–Ra from energy
considerations. And even if this relation is only approximate, so long as
the exponents are near these values the boundary layer thickness will
depend only weakly on the layer depth.
For models of mantle convection, plate tectonics defines the
boundary layer. The uniform surface motion of the plates, as well as
the observed heat flow and bathymetry of the plates reflects a
conductively cooling boundary layer that grows thicker with time. The
time the layer is on the surface is related to the horizontal extent of the
plate λD and the plate velocity v0. Using a conductive cooling model of
the plate gives scaling for the velocities v0 corresponding to the scaled
heat flow:
For constant temperature boundary conditions
2
u0 ≈a0 λ
2
κ Ra 3
D Rac
and for constant flux boundary conditions
2
u0 ≈a1 λ
Boundary layer models
The dimensionless equation for conservation of momentum is
ð22Þ
where Re = (ρu0ℓ)/η is the Reynolds' number, u0 is a characteristic
velocity, and the body force fi = Ra(Ti/T0)δi3.
If the Reynolds' number is small, multiplying by vi and integrating
over the volume yields
∫ τ′ij ėij′ dV + ∫ υi fi dV = 0
V
V
υi fi = ðαg = cp Þqadv :
Models based on Eq. (23) allow the consideration of other sources
of dissipation such as a strong lithosphere, which must be deformed
during subduction (Conrad and Hager, 1999b).
Even though Eq. (23) is based on the integral of buoyancy and flux
over the whole region, the scaling results are the same as those based
only on the stability of the boundary layer, which leads to results
(Eqs. (20) and (21)) that are independent of the overall depth of the layer.
This reflects that the limiting process is the conduction of heat across the
thermal boundary layer at the surface. In fact, for either flux or temperature boundary conditions, the boundary layer is characterized by
Ti4
κηRac
≈
ραg
ðq= kÞ3
ð24Þ
with δ0 = Ti/(q/k).
If these results are applied to a system with an increase in viscosity
at a depth well below the boundary layer thickness, the result will not
depend directly on the deeper viscosity, but rather on the heat flux
delivered to the top of the layer, or on the temperature maintained at
great depth, depending on the type of boundary condition specified.
Appendix B
We examine the effect of viscosity in convective stress and adopt a
power law fit η=η0(T/T0)−n to the Arrhenius law. Having that the
q
potential temperature is T = Ts + , (with Ts the surface temperature),
k
viscosity depends on the surface temperature, heat flux and plate thickness, which in turn depends on the viscosity. We obtain a simple expression for viscosity by removing its explicit dependence on plate thickness:
qδ
kT0
−n k
k
2
+ n Ts −… :
1−nTs
qδ
qδ
Replacing the value for δ from Eq. (4) yields,
These relations are kinematic, however, since they depend on the
motion of the surface layer as it cools, and that the layer sinks when its
thickness reaches a critical value.
dυ
∂υ
Re ρ i = Re ρð i + υi υ;i Þ = τij;j + fi
dt
∂t
the buoyancy flux vΔρg can be simply related to the advected heat υTi ρcp,
which constitutes most of the heat flux away from boundary layers:
η
=
η0
1
κ Ra 2
:
D Rac
501
ð23Þ
when there is no work done on the boundaries of the region. Thus the
viscous work done on the system is balanced by the work done by sinking
or rising density heterogeneities. This relation is exact, and applies to the
volume averages of dissipation and vertical momentum (i.e. buoyancy)
flux for any distribution of density heterogeneities in equilibrium.
When a temperature distribution corresponding to a sinking thermal
boundary layer is used (e.g. McKenzie and Weiss, 1975; Turcotte and
Schubert, 2002a) Eq. (23) gives the same results that were obtained above
for either the temperature or heat flux boundary conditions. We note that
η=
η0
n = 4 n n 3n = 4
a1 T0 k
n n=4
α
!4 = ðn
2 κ
+ 4Þ ρg
q3
n = ðn
+ 4Þ
1−nTs
4 = ðn + 4Þ
k
k 2
2
+ n Ts
−…
:
qδ
qδ
The first term on the right hand side is practically a constant. To
leading order, the viscosity has a dependence of
n = ðn +
ρg
η∼ 3
q
4Þ
:
ð25Þ
We examine whether or not there is a case in which viscosity effects
might change the implicit dependence of shear stress with mass, as to
decrease in bigger planets. We find that as long as the viscosity decreases
with temperature, n N 0, the dependence of stress on mass is always
positive, Δτxz ~ M ψ, ψ N 0, where
ψ=
ð2n + 4Þða + bÞ + cð4−2nÞ
:
2ðn + 4Þ
Thus, the condition for shear stress to increase with mass is
nN2
a+b
+ 1Þ
d
;
a+b
−1
d
ð
which is always met.
502
D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502
References
Afonso, J.C., Ranalli, G., Fernandez, M., 2007. Density structure and buoyancy of the
oceanic lithosphere revisited. JRL 34, L10302.
Backus, G.E., 1975. Gross thermodynamics of heat engines in deep interior of Earth. Proc.
Nat. Acad. Sci. 72, 1555–1558.
Becker, T.W., Faccena, C., O'Connell, R.J., Giardini, D., 1999. The development of slabs in
the upper mantle: insights from numerical and laboratory experiments. JGR 104,
15207–15226.
Bird, P., 1998. Testing hypotheses on plate-driving mechanisms with global lithosphere
models including topography, thermal structure, and faults. JGR 103, 10115–10129.
Borde, P., Rouan, D., Leger, A., 2003. Exoplanet detection capability of the COROT space
mission. Astron. Astrophys. 405, 1137.
Borucki, W.J., Koch, D., Basri, G., Brown, T., Caldwell, D., Devore, E., Dunham, E., Gautier, T.,
Geary, J., Gilliland, R., Gould, A., Howell, S., Jenkins, J., 2003. Kepler mission: a mission to
find earth-size planets in the habitable zone. In: Fridlund, M., Henning, T. (Eds.),
Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search
for Extrasolar Terrestrial Planets, ESA SP-539, pp. 69–81. ESA Publication Division,
Noordwijk.
Conrad, c.P., Hager, B.H., 1999a. Effects of plate bending and fault strength at subduction
zones on plate dynamics. JGR 104, 17551–17571 aug.
Conrad, c.P., Hager, B.H., 1999b. The thermal evolution of an earth with strong subduction
zones. GRL 26, 3041–3044 oct.
Davaille, A., Jaupart, C., 1993. Transient high-Rayleigh-number thermal convection with
large viscosity variations. J. Fluid Mech. 253, 141–166.
Davies, G., 1980. Thermal histories of convective earth models and constraints on
radiogenic heat production in the earth. J. Geophys. Res. 85, 2517–2530.
Di Toro, G., Hirose, T., Nielsen, S., Pennacchioni, G., Shimamoto, T., 2006. Natural and
experimental evidence of melt lubrication of faults during earthquakes. Science 311,
647–649.
Hirschmann, M.M., 2000. Mantle solidus: experimental constraints in the effects of
peridotite composition. G-cubed 1.
Howard, L., 1966. Convection at high Rayleigh numbers. In: Görtler, H. (Ed.), Proc. 11th
Int. Cong. Appl. Mech. Munich, 1964. Springer, pp. 1109–1115.
Hynes, A., 2005. Buoyancy of the oceanic lithosphere and subduction initiation. Int. Geol.
Rev. 47, 938–951.
Ida, S., Lin, D., 2004. Toward a deterministic model of planetary formation. I. A desert in
the mass and semimajor axis distributions of extrasolar planets. ApJ 604, 388–413.
Karato, S., Wu, P., 1993. Rheology of the upper mantle: a synthesis. Science 260, 771–778.
Korenaga, J., 2006. Geophysical Monograph 164. AGU, Ch. Archean geodynamics and the
thermal evolution of Earths, pp. 7–32.
Korenaga, J., 2008. Urey ratio and the structure and evolution of earth's mantle. Rev. Geophys.
4, RG2007. doi:10.1029/2007RG000, 241, 2008.
Lenardic, A., Jellinek, A.M., Moresi, L.N., 2008. A climate induced transition in the
tectonic style of a terrestrial planet. EPSL 271, 34–42.
McKenzie, D., Weiss, N., 1975. Speculations on the thermal and tectonic history of the
earth. Geophys. J. R. Astron. Soc. 42, 131–174.
Mian, Z.U., Tozer, D., 1990. No water, no plate tectonics: convective heat transfer and the
planetary surfaces of venus and earth. Terra Nova 2, 455–459.
Moresi, L., Solomatov, V., 1998. Mantle convection with a brittle lithospehre: thoughts
on the global tectonics styles of the earth and venus. Geophys. J. Int. 133, 669–682.
Morrow, C., Moore, D., Lockner, D.A., 2000. The effect of mineral bond strength and
adsorbed water on fault gouge frictional strength. GRL 27, 815–818.
Nimmo, F., McKenzie, D., 1998. Volcanism and tectonics on venus. Annu. Rev. Earth
Planet. Sci. 26 (1), 23–51.
O'Connell, R.J., Hager, B., 1980. On the thermal state of the earth. In: Dziewonski, A.M.,
Boschi, E. (Eds.), Physics of the Earth's Interior. North Holland, pp. 270–317.
O'Neill, C., Lenardic, A., 2007. Geological consequences of super-sized earths. GRL 34,
L19204 Oct.
O'Neill, C., Jellinek, A.M., Lenardic, A., 2007. Conditions for the onset of plate tectonics on
terrestrial planets and moons. EPSL 261, 20–32.
Poudjom-Djomani, Y., O'Reilly, H., Griffin, S.Y., 2001. The density structure of
subcontinental lithosphere through time. Earth Planet. Sci. Lett. 184, 605–621.
Ranalli, G., 1995. Rheology of the Earth. Chapman and Hall, Ch. Faults, pp. 90–116.
Rucker, W.K., Bird, P., 2007. Basal shear-traction, not slab-pull, may be the dominant
plate-driving force. AGU Fall Meet Supl. (abstract) 88 (52).
Seager, S., Kuchner, M., Hier-Majumder, C., Militzer, B., 2007. Mass-radius relationships
for solid exoplanets. ApJ 669, 1279–1297 Nov.
Sleep, N.H., 2000. Evolution of the mode of convection within terrestrial planets. JGR
105, 17563–17568.
Solomatov, V.S., 1995. Scaling of temperature- and stress-dependent viscosity convection.
Phys. Fluids 7, 266–274.
Sotin, C., Grasset, O., Mocquet, A., 2007. Mass-Radius curve for extrasolar Earth-like planets
and ocean planets.
Turcotte, D., Schubert, G., 2002a. Geodynamics. Ch. Fluid Mechanics. Cambridge University
Press, pp. 226–291.
Turcotte, D., Schubert, G., 2002b. Geodynamics. Ch. Faulting. Cambridge University Press,
pp. 339–373.
Valencia, D., O'Connell, R.J., Sasselov, D.D., 2006. Internal structure of massive terrestrial
planets. Icarus 181, 545–554.
Valencia, D., Sasselov, D.D., O'Connell, R.J., 2007c. Detailed models of super-earths: how
well can we infer bulk properties? ApJ 665, 1413 Aug.
Valencia, D., Sasselov, D.D., O'Connell, R.J., 2007a. Radius and structure models of the
first super-earth planet. ApJ 656, 545–551 Feb.
Valencia, D., O'Connell, R.J., Sasselov, D.D., 2007b. Inevitability of plate tectonics on
super-earths. ApJL 670, L45–L48 Nov.
Van Keken, P., Ballentine, C., Porcelli, D., 2001. A dynamical investigation of the heat and
helium imbalance. Earth Planet. Sci. Lett. 188, 421–434.
Vinet, P., Rose, J., Ferrante, J., Smith, J., 1989. Universal features of the equation of state of
solids. J. Phys. Cond. Matter 1, 1941–1963.
Weins, D.A., 2001. Seismological constraints on the mechanism of deep earthquakes:
temperature dependence of deep earthquake source properties. PEPI 127, 145–163.