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Timescales of chemical equilibrium between the convecting solid mantle and over- and underlying magma oceans

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After accretion and formation, terrestrial planets go through at least one magma ocean episode. As the magma ocean crystallises, it creates the first layer of solid rocky mantle. Two different scenarios of magma ocean crystallisation involve that the solid mantle either (1) first appears at the core–mantle boundary and grows upwards or (2) appears at mid-mantle depth and grows in both directions. Regardless of the magma ocean freezing scenario, the composition of the solid mantle and liquid reservoirs continuously change due to fractional crystallisation. This chemical fractionation has important implications for the long-term thermo-chemical evolution of the mantle as well as its present-day dynamics and composition. In this work, we use numerical models to study convection in a solid mantle bounded at one or both boundaries by magma ocean(s) and, in particular, the related consequences for large-scale chemical fractionation. We use a parameterisation of fractional crystallisation of the magma ocean(s) and (re)melting of solid material at the interface between these reservoirs. When these crystallisation and remelting processes are taken into account, convection in the solid mantle occurs readily and is dominated by large wavelengths. Related material transfer across the mantle–magma ocean boundaries promotes chemical equilibrium and prevents extreme enrichment of the last-stage magma ocean (as would otherwise occur due to pure fractional crystallisation). The timescale of equilibration depends on the convective vigour of mantle convection and on the efficiency of material transfer between the solid mantle and magma ocean(s). For Earth, this timescale is comparable to that of magma ocean crystallisation suggested in previous studies , which may explain why the Earth's mantle is rather homogeneous in composition, as supported by geophysical constraints.
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Solid Earth, 12, 421–437, 2021
https://doi.org/10.5194/se-12-421-2021
© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.
Timescales of chemical equilibrium between the convecting solid
mantle and over- and underlying magma oceans
Daniela Paz Bolrão1, Maxim D. Ballmer2,1, Adrien Morison3, Antoine B. Rozel1, Patrick Sanan1, Stéphane Labrosse4,
and Paul J. Tackley1
1Institute of Geophysics, ETH Zurich, 8092 Zurich, Switzerland
2Department of Earth Sciences, University College London, London WC1E 6BT, UK
3University of Exeter, Physics and Astronomy, EX4 4QL Exeter, UK
4Université de Lyon, ENSL, UCBL, Laboratoire LGLTPE, 15 parvis René Descartes, BP7000,
69342 Lyon, CEDEX 07, France
Correspondence: Daniela Paz Bolrão (daniela.bolrao@erdw.ethz.ch)
Received: 10 April 2020 Discussion started: 17 April 2020
Revised: 16 December 2020 Accepted: 24 December 2020 Published: 22 February 2021
Abstract. After accretion and formation, terrestrial planets
go through at least one magma ocean episode. As the magma
ocean crystallises, it creates the first layer of solid rocky man-
tle. Two different scenarios of magma ocean crystallisation
involve that the solid mantle either (1) first appears at the
core–mantle boundary and grows upwards or (2) appears at
mid-mantle depth and grows in both directions. Regardless of
the magma ocean freezing scenario, the composition of the
solid mantle and liquid reservoirs continuously change due
to fractional crystallisation. This chemical fractionation has
important implications for the long-term thermo-chemical
evolution of the mantle as well as its present-day dynam-
ics and composition. In this work, we use numerical mod-
els to study convection in a solid mantle bounded at one or
both boundaries by magma ocean(s) and, in particular, the
related consequences for large-scale chemical fractionation.
We use a parameterisation of fractional crystallisation of the
magma ocean(s) and (re)melting of solid material at the inter-
face between these reservoirs. When these crystallisation and
remelting processes are taken into account, convection in the
solid mantle occurs readily and is dominated by large wave-
lengths. Related material transfer across the mantle–magma
ocean boundaries promotes chemical equilibrium and pre-
vents extreme enrichment of the last-stage magma ocean (as
would otherwise occur due to pure fractional crystallisation).
The timescale of equilibration depends on the convective
vigour of mantle convection and on the efficiency of material
transfer between the solid mantle and magma ocean(s). For
Earth, this timescale is comparable to that of magma ocean
crystallisation suggested in previous studies (Lebrun et al.,
2013), which may explain why the Earth’s mantle is rather
homogeneous in composition, as supported by geophysical
constraints.
1 Introduction
The early Earth experienced at least one episode of extensive
silicate melting, also known as magma ocean (e.g. Abe and
Matsui, 1988; Abe, 1993, 1997; Solomatov and Stevenson,
1993a; Solomatov, 2000; Drake, 2000; Elkins-Tanton, 2012).
A magma ocean was likely formed due to the energy released
during the Moon-forming giant impact (Tonks and Melosh,
1993; ´
Cuk and Stewart, 2012; Canup, 2012), core formation
(Flasar and Birch, 1973), radiogenic heating (Urey, 1956),
electromagnetic induction heating (Sonett et al., 1968), and
tidal heating (Sears, 1992). Due to the presence of an early
atmosphere (Abe and Matsui, 1986; Hamano et al., 2013), it
was sustained for thousands (Solomatov, 2000) to millions of
years (Abe, 1997; Lebrun et al., 2013; Salvador et al., 2017;
Nikolaou et al., 2019).
As the magma ocean cools and its temperature drops be-
low the liquidus, crystals start to appear and consolidate a
first layer of solid cumulates, i.e. the solid part of the man-
tle. Because the shape of the liquidus (and solidus) rela-
tive to the isentropic temperature profile is not well con-
Published by Copernicus Publications on behalf of the European Geosciences Union.
422 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
strained, the depth at which initial crystallisation occurs re-
mains unknown: this depth may be anywhere between the
core–mantle boundary (CMB) (e.g. Abe, 1997; Solomatov,
2015) and mid-mantle depths (Labrosse et al., 2007; Stixrude
et al., 2009; Nomura et al., 2011; Labrosse et al., 2015;
Boukaré et al., 2015; Caracas et al., 2019). Depending on
this depth, several distinct scenarios of magma ocean evolu-
tion are expected to occur.
1.1 Crystallisation of a magma ocean from the bottom
If crystallisation of the magma ocean starts at the CMB, the
first layer of solid cumulates forms at the bottom of this
magma ocean (Fig. 1a). As the temperature of the ocean de-
creases, the crystallisation front steadily progresses upwards,
creating more and more solid cumulates. When the crystalli-
sation front reaches the surface of the planet, the solid mantle
of the Earth is fully formed.
Assuming that the temperature of solid cumulates stays
close to that of the solidus, these solid cumulates are ther-
mally unstable as the solidus is steeper than the isentrope.
Also assuming that some degree of fractional crystallisa-
tion occurs (Solomatov and Stevenson, 1993b; Brown et al.,
2014; Elkins-Tanton et al., 2003), the magma ocean becomes
progressively enriched in iron silicates (FeO), as iron be-
haves like a mildly incompatible element (Murakami and
Bass, 2011; Nomura et al., 2011; Andrault et al., 2012;
Tateno et al., 2014). Accordingly, the solid cumulates (ini-
tially enriched in MgO) that form in chemical equilibrium
with the overlying magma ocean incorporate progressively
more FeO with time and, as a result, become denser with
time (Fig. 1a). Therefore, on top of being thermally un-
stable, the solid cumulates are also gravitationally unsta-
ble due to their composition. This leads to a large-scale
overturn after magma ocean crystallisation (Elkins-Tanton
et al., 2003, 2005) or multiple small-scale overturns during
crystallisation (Maurice et al., 2017; Ballmer et al., 2017b;
Boukaré et al., 2018; Morison et al., 2019; Miyazaki and Ko-
renaga, 2019b). Such overturn(s) may lead to remelting of
FeO-enriched material at depth, as the isentrope of such ma-
terial is steeper than its melting curve through most of the
mantle.
This dense remelted material may form a basal magma
ocean (BMO) (Labrosse et al., 2015), join an existing one
(Labrosse et al., 2007) (see Sect. 1.2), or react with the un-
derlying solid mantle (Ballmer et al., 2017b). Hence, the
solid mantle may evolve from being bounded above by only
one magma ocean, the top magma ocean (TMO), to being
bounded by two magma oceans, the TMO and BMO, de-
pending on the fate of overturned cumulates. Ultimately, the
TMO is expected to completely crystallise, potentially leav-
ing a long-lived BMO after the final overturn of the most
FeO-enriched cumulates. Because the overturning events are
potentially swift and of large-scale nature, the resulting solid
mantle and magma oceans are not necessarily in chemical
equilibrium.
1.2 Crystallisation of a magma ocean from the middle
If crystallisation of the magma ocean instead starts some-
where at mid-mantle depths and the crystals formed are
near-neutrally buoyant (Labrosse et al., 2007; Boukaré et al.,
2015), the first layer of solid mantle forms and separates the
magma ocean into a TMO and BMO (Fig. 1b). The two crys-
tallisation fronts then move in opposite directions: the TMO
front progresses upwards until it reaches the surface of the
planet, and the BMO front progresses downwards until it
reaches the CMB. In this process, both the TMO and BMO,
as well as the related cumulates, become progressively en-
riched in FeO (Fig. 1b). In contrast to the TMO cumulates
(see Sect. 1.1), BMO cumulates are likely formed over much
longer timescales (Labrosse et al., 2007) and are expected
to form a stable density profile. By the time the BMO is
fully crystallised, a dense stable solid layer may persist at
the base of the mantle. This dense layer may explain seismic
observations that point to the existence of thermo-chemical
piles near the CMB (Masters et al., 2000; Ni and Helm-
berger, 2001; Garnero and McNamara, 2008; Deschamps
et al., 2012; Labrosse et al., 2015; Ballmer et al., 2016).
1.3 Motivation
Along these lines, the chemical evolution of the solid mantle
depends on the history of early planetary melting and crys-
tallisation. This is a history with either one or two magma
oceans and with convection in the solid mantle driven by un-
stable thermal and/or chemical stratification, probably while
magma ocean(s) at the top and/or bottom are still present.
While any such convection would imply remelting of solid
cumulates, the related consequences for mantle evolution are
poorly understood. Only a few numerical modelling studies
have explicitly incorporated coupled remelting and crystalli-
sation at the magma ocean–mantle boundary or boundaries
(Labrosse et al., 2018; Morison et al., 2019; Agrusta et al.,
2019), and none of these studies have explored the conse-
quences for chemical evolution.
In this paper, we use a numerical model to investigate
the thermo-chemical evolution of the solid mantle in contact
with a TMO and/or a BMO. We consider that convection in
the solid mantle starts before the end of magma ocean crys-
tallisation; therefore, dynamic topographies that may form
at one or both solid mantle–magma ocean boundaries can
melt or crystallise. We do not explicitly account for the pro-
gression of the crystallisation front(s). However, we test sev-
eral evolution scenarios and different magma oceans thick-
nesses. We determine the timescales of chemical equilibrium
between the magma ocean(s) and the solid mantle, and we
compare them with those of progression of the crystallisa-
tion front (e.g. Lebrun et al., 2013). For simplicity, in the
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 423
Figure 1. Sketches of magma ocean (MO) crystallisation scenarios. As cooling of the MO proceeds, adiabats (cyan dashed lines) cross the
liquidus (yellow curve), and the solid mantle (S) appears, either (a) near the core–mantle boundary (CMB) or (b) somewhere at mid-mantle
depths. In panel (b) the MO is divided into a top magma ocean (TMO) and a basal magma ocean (BMO) as soon as the solid appears. In both
scenarios, liquid and solid cumulates become enriched in FeO with time, which may lead to an overturn of solid material (not depicted, see
text for details). Mush is not considered.
following we use the term solid–liquid phase changes inter-
changeably with fractional crystallisation and melting pro-
cesses at the interface between the solid mantle and TMO
and/or BMO.
2 Numerical model
2.1 Problem definition
We use the finite volume or finite difference method with
the StagYY (Tackley, 2008) convection code to model the
thermo-chemical evolution of the solid mantle during magma
ocean crystallisation. We test three different evolution sce-
narios, as the solid mantle may be bounded above by a
TMO and/or below by a BMO (Fig. 2). We assume steady
crystallisation front(s) and test different magma ocean thick-
nesses: when only one ocean is present, it can be 100, 500, or
1000 km thick; when both oceans are present, the thickness
of each ocean is 100 and/or 500 km.
We assume that the solid mantle is an infinite Prandtl num-
ber fluid. We assume mechanical stability between the solid
mantle and magma oceans, i.e. ρTMO < ρS< ρBMO, where
ρTMO,ρS, and ρBMO are the densities of the TMO, solid
mantle, and BMO respectively. We set constant values for
gravitational acceleration (g), viscosity (η), thermal diffu-
sivity (κ), heat capacity (Cp), the thermal expansion coef-
ficient (α), and the compositional expansion coefficient (β).
The values of these parameters can be found in Table 1.
We make equations dimensionless to reduce the number of
parameters that describe the physical problem. Dimensions
of distance, time, and temperature can be recovered using
the thickness of the solid mantle(hS), the thermal diffusive
timescale h2
S
κ, and the temperature difference between bot-
tom and top solid mantle boundaries (1T =TT+) re-
spectively. The dimensionless temperature (T) is defined as
follows:
T=T0T+
1T .(1)
We assume incompressibility in the Boussinesq approx-
imation (e.g. Chandrasekhar, 1961). Therefore, mass, en-
ergy, composition, and momentum conservation equations
are written as follows:
·u=0 (2)
T
t +u·T=2T(3)
XS
FeO
t +u·XS
FeO =0 (4)
p+2u+RaThTi B(XS
FeO hXS
FeOi)ˆ
r=0,(5)
https://doi.org/10.5194/se-12-421-2021 Solid Earth, 12, 421–437, 2021
424 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
Figure 2. Sketches of three possible evolution scenarios: solid mantle (S) bounded by a (a) top magma ocean (TMO), (b) basal magma ocean
(BMO), and (c) TMO and BMO. Solid mantle is taken as a spherical shell with density ρS, thickness hS, inner radius R, and outer radius
R+. TMO and BMO are taken with respective densities of ρTMO and ρBMO and respective thicknesses of hTMO and hBMO. Superscripts
+ and at the boundaries refer to the boundary between the TMO and the solid mantle and the boundary between the solid mantle and
the BMO respectively.
Table 1. Parameters used in the simulations.
Parameter (dimension) Symbol Value
Radius of the core–mantle boundary (km) RCMB 3480
Total radius of the planet (km) RP6370
Thickness of the solid mantle today (km) hM2890
Gravitational acceleration (m s2)g9.81
Viscosity (Pa s) η
Thermal diffusivity (m2s1)κ5×107
Heat capacity (J kg1K1)Cp1200
Thermal conductivity (W m1K1)k3.0
Thermal expansion coefficient (1 /K) α105
Reference density (kg m1)ρ5000
Buoyancy number (–) B1.0
Solid–liquid partition coefficient of FeO (–) K0.3
Phase change number (–) 8101105
Thickness of top magma ocean (km) hTMO 0 if only BMO present, 100, 500, 1000
Thickness of basal magma ocean (km) hBMO 0 if only TMO present, 100, 500, 1000
Supercriticality (–) SC 102105
FeO concentration of the bulk (–) XBulk
FeO 0.12
where uis the velocity field, hTiis the lateral average of the
temperature field T,tis the time, XS
FeO is the FeO molar con-
tent in the solid mantle, hXS
FeOiis the lateral average of XS
FeO,
pis the dynamic pressure, Ra is the Rayleigh number, and B
the buoyancy number. The last two respective parameters are
defined as follows:
Ra =ρgα1T h3
S
ηκ ,(6)
B=β
α1T .(7)
In this study, we consider that magma oceans and solid
mantle are only made of (Fe, Mg)O (see Sect. 2.3 for more
details). We set temperature to 1.0 and 0.0 at the bottom
and top solid domain boundaries respectively. Regarding the
buoyancy number, Earth-like models point to a value of
B3 for the present-day mantle (i.e. based on the density
difference between Mg- and Fe-rich silicate endmembers as
well as CMB temperature estimates). However, there are sig-
nificant uncertainties associated with the value of α(which
is temperature and pressure dependent, e.g. Tosi et al., 2013)
and that of 1T in the solid part of the primitive mantle. For
example, 1T increases with the thickness of the solid layer.
Within these uncertainties, at least a range of 1 B3 is
acceptable. In this paper, we choose B=1.0 in order to limit
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 425
the impact of the initial condition on the onset of convection
(see Sect. 4).
The solid domain is represented using the spherical an-
nulus geometry (Hernlund and Tackley, 2008), composed of
a grid of 512 ×128 cells, in which Eqs. (2)–(5) are solved.
Composition is advected by tracers. We assume that each
magma ocean is well-mixed and that its dynamics are fast
compared with that of the solid mantle. In our set-up, magma
oceans are treated as simple 0-D compositional reservoirs at
solid mantle boundaries. In the following we use the super-
scripts + and to refer to top and bottom solid mantle
boundaries respectively. In equations, the sign ± reads as
+ if a TMO is considered and as if a BMO is consid-
ered. The subscript MO refers to magma ocean. Thus, when
we introduce a quantity, e.g. ξ, related to a magma ocean, we
introduce it as ξ±
MO, with ξ+
MO =ξTMO relating to the TMO
and ξ
MO =ξBMO relating to the BMO.
2.2 Dynamic topography and the phase change
boundary condition
As convection in the solid mantle likely starts before the
end of magma ocean crystallisation (Maurice et al., 2017;
Ballmer et al., 2017b; Boukaré et al., 2018; Morison et al.,
2019; Miyazaki and Korenaga, 2019b), dynamic topogra-
phies are supported at one or both solid mantle boundaries.
The timescale for producing dynamic topography is noted τη.
This topography can be eroded by solid–liquid phase changes
on a timescale related to the transfer of energy and FeO
through the magma ocean, from material that is crystallising
to material that is melting. We denoted this timescale using
τφ.
The relative values of the two timescales, τηand τφ, con-
trol the dynamical behaviour of the boundary. If τητφ,
dynamic topography can build before being erased by the
phase change. In this case, dynamic topography is only lim-
ited by the balance between viscous stress in the solid and
the buoyancy associated with the topography. In the limit of
small topographies, this leads to the classical non-penetrating
free-slip boundary condition in which the radial velocity of
the solid effectively goes to zero at the boundary (Ricard
et al., 2014). On the other hand, if τητφ, the topography is
erased faster by phase changes than it can be built by viscous
stress in the solid. Consequently, this removes the stress im-
posed by the topography and the associated limit to the radial
velocity. These processes are incorporated into our boundary
condition, described by the phase change number,
8=τφ
τη
,(8)
considering that when 8 , dynamic topography is built
(or relaxes) by viscous forces, and when 80, it is eroded
by melting or fractional crystallisation processes (Deguen,
2013; Deguen et al., 2013). The related phase change bound-
ary condition in dimensionless form, at one or both solid
mantle boundaries is
2ur
r p±8±ur=0,(9)
where uris the vertical velocity of the flow in the solid
mantle. On the one hand, this boundary condition can act
like a non–penetrating free–slip boundary condition when
8 , as vertical velocities of the solid flow tend to zero
at the boundaries. Under this boundary condition, transfer of
material across a solid mantle–magma ocean boundary can-
not occur. On the other hand, this boundary condition can act
as being “open” to phase changes when 80, as these ver-
tical velocities will be non-zero at the solid mantle–magma
ocean boundary, and a significant flux of solid and liquid ma-
terial can cross it to melt and crystallise. Hence, transfer of
material across the phase change boundary is efficient. In the
extreme case of 8=0, this boundary condition corresponds
to free in- and outflow.
The specific value of 8is difficult to constrain (because
τφis non-trivial to determine) and is also expected to vary
with time (i.e. because τηdepends on the thickness of the
solid mantle) (Deguen, 2013; Deguen et al., 2013). How-
ever, for a purely thermal case, Morison et al. (2019) and
Morison (2019) estimate 8+105and 8103for the
Earth respectively. Therefore, significant transfer of mate-
rial across the solid mantle–magma ocean boundaries is ex-
pected. However, also consider that real multi-phase rocks
typically melt over large pressure ranges, except for truly
eutectic bulk compositions. The depleted residue of mantle
melting may restrict the efficiency of material transfer across
the solid mantle–magma ocean boundaries, depending on the
efficiency of melt–solid segregation near the boundaries. In
addition to the expected temporal evolution of 8±, this po-
tential restriction motivates our exploration of a broad range
of 8±. In this study we use seven values of 8±that range
from 101to 105. We use 8=101as the lowest possible
value of 8±because the resolution of the thermal boundary
layer is computationally demanding once 8±decreases be-
low 101.
Deguen et al. (2013) and Labrosse et al. (2018) found that
the critical Rayleigh number, Rac, for the solid mantle is
strongly sensitive to 8and the set-up considered, i.e. hav-
ing a TMO and/or a BMO, as well as to the thickness of the
solid layer. For instance, if the solid mantle is bounded by a
TMO of 100 km and 8 ,Racis of the order of 103, but
for small 8, it is of the order of 102.Raccan even decrease
to arbitrarily small values of the order of 8if a TMO and
BMO are both considered. Therefore, we also systematically
vary the Rayleigh number, Ra, which controls the convec-
tive vigour of the mantle. We choose Ra as multiples of Rac,
according to the supercriticality factor, SC:
Ra =Rac×SC.(10)
We use four values of SC ranging from 102to 105.
https://doi.org/10.5194/se-12-421-2021 Solid Earth, 12, 421–437, 2021
426 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
2.3 Compositional treatment
In this study, we consider a simplified compositional model
with only two components, FeO and MgO, which are thought
to be the Fe-rich and Mg-rich endmembers of mantle sili-
cates. We denote the FeO and MgO molar content in the solid
and magma ocean parts by XS
FeO and XMO
FeO and using XS
MgO
and XMO
MgO respectively. We consider mass balance between
FeO and MgO in the solid mantle and magma oceans; there-
fore, XS
FeO +XS
MgO =1, and XMO
FeO +XMO
MgO =1.
Our model simulates melting and crystallisation of mate-
rial depending on the influx and outflux of material at the
solid mantle boundary. Melting of solid material is simulated
when dynamic topography develops outside the solid do-
main, i.e. when there is an outflux of material from the solid
domain. It is assumed that no fractionation operates when the
solid melts, i.e. all (Fe,Mg)O present in this topography goes
into the magma ocean. Therefore, tracers that leave the solid
domain pass their information (about mass and composition)
to the magma ocean and are deleted.
We simulate crystallisation of the magma ocean when neg-
ative dynamic topography develops in the solid domain, i.e.
when there is an influx of mass in the solid domain. When
this happens, the influx of material pushes tracers, and cells
near the boundary are left with no tracers. To ensure mass
conservation, new ntracers are introduced in those cells, and
this process simulates solid mantle being created. To deter-
mine n, i.e. the amount of tracer that needs to be introduced
in the solid part, we calculate the influx of mass correspond-
ing to this dynamic topography and divide it by the tracer
ideal mass. We then equally distribute the influx of mass by
ntracers. The composition of the solid created is related to
that of the liquid by fractional crystallisation; therefore, only
a fraction of FeO goes into the solid. This fraction is given
by the distribution coefficient, K:
K=XS
FeOXMO
MgO
XMO
FeOXS
MgO
.(11)
We assume that K=0.3 (e.g. Corgne and Wood, 2005; Lieb-
ske et al., 2005). The difference between the influx and out-
flux of material through the boundary is of the order of 1015,
meaning that conservation of mass in the solid mantle is en-
sured.
In this paper, we attempt to estimate the characteristic
timescale to establish chemical equilibrium between the solid
mantle and the magma ocean(s). Assuming a full equilib-
rium between the solid mantle and magma oceans (super-
script Eq”), the FeO content in the bulk, XBulk
FeO , can be ex-
pressed as function of the volumes (VS,VTMO, and VBMO)
and the FeO content (XS,Eq
FeO ,XTMO,Eq
FeO , and XBMO,Eq
FeO ) in the
solid mantle and magma oceans:
XBulk
FeO =XTMO,Eq
FeO VTMO +XBMO,Eq
FeO VBMO +XS,Eq
FeO VS
VTMO +VBMO +VS
.(12)
From Eqs. (11) and (12), one can find the FeO content in
the solid mantle when it is in chemical equilibrium with the
magma ocean(s):
XS, Eq
FeO =b+b24ac
2a,(13)
where
a=VS(1K),
b=VTMO +VBMO +VSKXBulk
FeO (VTMO +VBMO +VS)
(1K),
c= XBulk
FeO K(VTMO +VBMO +VS).
However, because chemical equilibrium would take too
long to reach in a reasonable runtime, we look for the
timescale to reach chemical half-equilibrium. Starting with
an FeO content in the solid mantle equal to XS, Ini
FeO , the half-
equilibrium is reached when the solid mantle reaches the
content XS, Eq/2
FeO , which is defined as
XS, Eq/2
FeO =XS, Ini
FeO +XS,Eq
FeO
2.(14)
We denote by tS, Eq/2 the time at which the solid mantle
reaches chemical half-equilibrium.
Previous studies suggest that the Fe content of the present-
day bulk silicate Earth is 0.113 (Taylor and McLennan, 1985)
or 0.107 (McDonough and Sun, 1995). We suppose that some
of the Fe could migrate to the core with time (e.g. Nguyen
et al., 2018); therefore, in this study, we use XBulk
FeO =0.120.
We start the simulations with a homogeneous FeO content in
the solid mantle and magma ocean(s), XS, Ini
FeO =XTMO, Ini
FeO =
XBMO, Ini
FeO =0.120. Although this initial composition is not
consistent with the fractional crystallisation assumed in this
problem, it effectively serves our goal of measuring the
timescale to reach chemical equilibrium between solid and
liquid reservoirs.
3 Results
3.1 Chemical evolution of the mantle bounded on top
by a TMO
In this subsection, we investigate how the chemical evolution
of the solid mantle is affected by the efficiency of mass trans-
fer across the phase change boundary, as controlled by 8. As
mentioned in the previous section, low values of 8corre-
spond to efficient material transfer across the phase change
boundaries, and high values of 8correspond to inefficient
material transfer, similar to classical convection. We first
analyse the case of a solid mantle bounded above by a TMO,
as the most straightforward scenario for early planetary evo-
lution. At the end of this subsection, we briefly compare this
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 427
scenario with those where the solid mantle is in contact with
just a BMO and with both magma oceans.
Because the parameter space explored in this paper is vast,
as an example, we illustrate the chemical evolution of a solid
mantle bounded by a TMO of 500 km, under three different
values of phase change number, 8+=101,102, and 103,
at the same supercriticality value of SC =105. As the criti-
cal Rayleigh number, Rac, decreases as 8+decreases, these
three cases have different values of Rayleigh number, Ra.
Hence, for 8+=101,102, and 103,Ra =100 ×105,635 ×
105, and 687 ×105respectively. According to Eqs. (12) and
(13), the expected FeO content in each reservoir in chemi-
cal equilibrium would be approximately XS,Eq
FeO =0.082 and
=XTMO,Eq
FeO =0.229. As we initialise each reservoir with an
FeO content of XS, Ini
FeO =XTMO, Ini
FeO =0.120, the system does
not start in chemical equilibrium. We determine the time
needed to reach chemical half-equilibrium.
Figure 3 shows the chemical evolution in dimensionless
time units of these three cases. Our models predict that re-
gardless of the value of 8+, the FeO content in the solid
mantle decreases towards XS,Eq
FeO , and the FeO content in the
TMO increases towards XTMO,Eq
FeO , thereby bringing the solid
mantle and the TMO close to chemical equilibrium (but not
chemical homogeneity as seen later). However, the lower
the value of 8+, the faster half-equilibrium is reached, as
it effectively increases the exchange of material between
reservoirs. We calculate the time needed to reach chemical
half-equilibrium, and half-equilibrium is reached 10 times
faster for 8=101than for 8=102and 200 times faster
for 8=101than for 8=103.
In Fig. 4, we present snapshots of the FeO content in the
solid mantle for these three cases. Our models show that dy-
namics in the solid mantle is very different between cases.
With 8+=101(Fig. 4a), mantle flow is dominated by
degree-1 convection, which persists stably for the whole sim-
ulation time. With this pattern of convection, there is an up-
welling of primordial material (in yellow) that melts on one
hemisphere, while material from the TMO crystallises at the
boundary and forms a downwelling on the other hemisphere
(in blue). This downwelling is FeO depleted, which intro-
duces a strong heterogeneity in the solid mantle. Degree-1
convection involves very little deformation, which explains
the existence of a considerable amount of primordial mate-
rial in the solid mantle, even around the time at which chem-
ical half-equilibrium occurs (snapshot inside the red box).
As 8+increases (Fig. 4b, c), higher-degree modes of con-
vection with several convection cells appear. Although the
composition of the TMO and the average composition of the
solid mantle tend to mutual chemical equilibrium in all three
cases, chemical homogeneity across the solid mantle is not
necessarily reached.
Our models show that for other evolution scenarios, i.e.
solid mantle in contact with just a BMO and with a TMO
and BMO, the system also evolves to a state close to chemi-
cal equilibrium but not chemical homogeneity. In Fig. 5, we
present snapshots of the FeO content in the solid mantle for
different evolution scenarios at about the time of chemical
half-equilibrium. When the solid mantle is in contact with
just a BMO, material from the magma ocean crystallises at
the boundary and forms upwellings (in blue). This material
is FeO depleted and, similarly to the TMO case, introduces
a strong heterogeneity in the solid mantle around the half-
equilibrium time. When the solid mantle is in contact with
both oceans, convection occurs with degree-1, i.e. material
of the TMO and of the BMO crystallises at the correspond-
ing boundary and forms a downwelling (in blue) and an up-
welling (in blue and green) respectively. Note that in this
scenario, as the volume of the BMO is smaller than that of
the TMO, the BMO composition changes rapidly. Therefore,
the composition of the upwelling changes rapidly as well
(colours from blue to green). The degree-1 pattern of con-
vection persists stably for the whole simulation time.
3.2 Timescales of chemical half-equilibrium between
the solid mantle and magma ocean(s)
Figure 6 shows the timescales of chemical half-equilibrium
for the scenarios explored in the previous subsection. For
a wide range of SC and 8±, these timescales are shown
for a solid mantle bounded by a TMO of 500 km thickness
(Fig. 6a), by a BMO of 500 km thickness (Fig. 6b), and by
a TMO and a BMO of 500 and 100 km thickness respec-
tively (Fig. 6c). For all evolution scenarios, models predict
that timescales of chemical half-equilibrium decrease for de-
creasing 8±. In other words, chemical half-equilibration is
more efficient for efficient material transfer across the solid
mantle–magma ocean boundaries. Our results also show that
the timescales of chemical half-equilibration are similar (i.e.
of the same order of magnitude) for a given SC and 8±rang-
ing between 101and 101, independent of the evolution sce-
nario. This shows that below 8±=101a regime with effi-
cient material transfer across the solid mantle-magma ocean
boundaries is established. The transition to the regime of in-
efficient material transfer (i.e. in which mantle flow is limited
by viscous building of dynamic topography) occurs some-
where between 8±=101and 102. In this regime, timescales
of half-equilibration systematically increase with 8±. Our
models predict that this transition between regimes occurs
over a similar interval of 8±for other thicknesses of TMO
and/or BMO.
To obtain an empirical scaling law, we fit the predicted
timescales, tS, Eq/2
pred , for all simulations that reached chemi-
cal half-equilibrium. The fitting equation provides tS, Eq/2
pred in
dimensionless form as a function of Ra,8±and VS:
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428 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
Figure 3. (a) Evolution of the FeO content, XFeO, in the solid mantle (S, green) and top magma ocean of 500 km (TMO, pink), with
dimensionless time units. We test three different values of phase change number: 8+=101,102, and 103(different lines). Green stars
correspond to the point where the FeO content in chemical half-equilibrium in the solid mantle, XS, Eq/2
FeO , is reached. In these simulations,
supercriticality is SC =105.(b) A zoomed in view of panel (a) at the beginning of evolution.
Figure 4. Snapshots of the FeO content in the solid mantle, XS
FeO, as a function of dimensionless time (factor of 106inside each annulus)
for the cases presented in Fig. 3: (a) 8+=101,(b) 8+=102, and (c) 8+=103. In these simulations, the top magma ocean thickness is
500 km and there is no basal magma ocean. Supercriticality is SC =105in all three cases shown. Snapshots within a red box indicate that
the model time is close to the chemical half-equilibrium time, tS, Eq/2. For 8+=101, 102, and 103,tS, Eq/2 =9.6×106, 114.9×106,
and 1895.5×106respectively. Magenta contours correspond to the streamlines of the flow.
tS, Eq/2
pred =maxa0Ra a18±
10 a2VS
VMa3
,
a4Raa58±
10 a6VS
VMa7,(15)
where VMis the volume of the present-day Earth’s mantle.
Coefficients of this equation can be found in Table 2. In the
Appendix A of this paper, we explain the regression method
and show a good agreement between the timescales to reach
chemical half-equilibrium, obtained with our model predic-
tions and our empirical scaling law (Fig. A1).
Equation (15) presents two branches, each correspond-
ing to a different regime: the left branch corresponds to the
regime of efficient material transfer across the solid mantle–
magma ocean boundaries, and the right branch corresponds
to the regime of inefficient material transfer. Our models
predict that in the regime of efficient material transfer (i.e.
for low values of 8), timescales to reach chemical half-
equilibrium are only loosely affected by the volume of the
solid mantle (or in other words, by the volume of the magma
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 429
Figure 5. Snapshots of the FeO content in the solid mantle, XS
FeO,
close to the time at which the system reaches chemical half-
equilibrium (time is dimensionless with a factor of 106inside
each annulus), for (a) 8=101,(b) 8=102, and (c) 8=103.
The solid mantle is in contact with (left) a top magma ocean (TMO)
of 500 km, (middle) a basal magma ocean (BMO) of 500 km, and
(right) a TMO of 500 km and a BMO of 100 km. Supercriticality
is SC =105in all cases shown. Magenta contours correspond to
the streamlines of the flow. Cases with 8=103for a solid mantle
in contact with only BMO and with TMO and BMO did not reach
chemical half-equilibrium. Note that the difference between the as-
pect ratio of each evolution scenario is too small to be noticed in
these annuli.
ocean(s)). The volume of the solid only systematically affects
the timescales once the regime shifts to the one of inefficient
material transfer. This conclusion is independent of the evo-
lution scenario. One possible explanation for the weak influ-
ence of the solid mantle’s volume is that convection occurs
with a low degree at low values of 8, so the geometry of the
problem is less important.
We find that timescales to reach chemical half-equilibrium
are about a factor of 3 larger for a solid mantle in contact
with just a TMO (Fig. 6a) than for a solid mantle with just
a BMO (Fig. 6b). This can be explained by the fact that the
geometry of the problem is different in both cases. Although
the TMO and the BMO have the same thickness, the volume
of the TMO is larger than that of the BMO by roughly a factor
of 3, which explains the increased duration to reach the half-
equilibrium FeO content in the magma ocean.
When it comes to a solid mantle bounded by both oceans
(Fig. 6c), models predict that timescales are roughly 2 orders
of magnitude smaller than those of a solid mantle in contact
with just a TMO (Fig. 6a), for a given Rayleigh number. This
result is explained by two effects. The critical Rayleigh num-
ber is much lower when two magma oceans are present than
when only one is present. In principle, when both magma
oceans are present, the critical Rayleigh number can even
be arbitrarily low as 8±decreases towards zero (Labrosse
et al., 2018). Moreover, Agrusta et al. (2019) showed that
the heat flow and root mean square (RMS) velocity in the
solid mantle vary linearly with Ra when both magma oceans
are present, whereas heat flow and RMS velocity in the solid
mantle vary as Ra1/3and Ra2/3respectively when only one
magma ocean present. This further increases the difference
between the two scenarios at a given value of the Rayleigh
number. Therefore, one should expect that the timescales
to reach chemical half-equilibrium may be arbitrarily low,
depending on the efficiency of material transfer across the
BMO–mantle and TMO–mantle boundaries.
3.3 Chemical half-equilibrium and crystallisation
timescales
In this subsection, we compare the timescales to reach chem-
ical half-equilibrium between a solid mantle and a TMO of
a given thickness with timescales of crystallisation of such
a TMO, as calculated for the Earth case. The timescales of
TMO crystallisation (i.e. before reaching the mush stage) are
given by Lebrun et al. (2013), hereinafter denoted by tC
L13.
We take the solid mantle bounded on top by a TMO of 100,
500, and 1000 km thickness and use Eq. (15) to determine
the timescales of half-equilibration as a function of phase
change number, 8+. In an attempt to apply our fitting equa-
tion to Earth, we assume that the global Rayleigh number of
the early Earth mantle just after solidification of the TMO is
between RaM=108and RaM=109.RaMis calculated on
the basis of the total thickness of the solid mantle, hM. The
Rayleigh number, Ra, used in Eq. (15) is then rescaled to the
actual thickness of the solid mantle (i.e. before solidification
of the TMO) as follows:
Ra =RaMhS
hM3
.(16)
This rescaling neglects the change in various physical param-
eters (from Eq. 6), but it is sufficient for our discussion.
The comparison between timescales is presented in Fig. 7.
The timescale to crystallise the TMO is loosely dependent
on its thickness, and this time is around 1 Myr. Our mod-
els predict that there are significant chemical exchanges be-
tween the TMO and the solid mantle for 8+<10 and <100
for RaM=108and RaM=109respectively. Therefore, for
8+smaller than these values, the TMO is expected to have
reached (at least) chemical half-equilibrium with much of the
mantle before reaching the mush stage. Thus, a very strong
enrichment of the final-stage TMO as predicted by fractional
crystallisation models (Elkins-Tanton et al., 2003) is not ex-
pected to occur for small to moderate 8+.
Increasing the thickness of the TMO and, hence, decreas-
ing the thickness of the solid mantle, decreases the Rayleigh
number and the ratio VS/VM, which make the dimensionless
time increase (see fitting). Moreover, decreasing the thick-
https://doi.org/10.5194/se-12-421-2021 Solid Earth, 12, 421–437, 2021
430 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
Figure 6. Timescales to reach chemical half-equilibrium (in different colours), tS, Eq/2, between the solid mantle and (a) a top magma ocean
(TMO) of 500 km, (b) a basal magma ocean (BMO) of 500 km, and (c) a TMO of 500 km and BMO of 100 km, for different values of
supercriticality, SC =102105(Ra is indicated by dashed lines), and phase change number, 8=101105. White circles are cases that
did not reach chemical half-equilibrium within a reasonable runtime.
Table 2. Results of the regressions of the timescales of chemical half-equilibration using the following form: tS, Eq/2
pred =
maxha0Raa18±
10 a2VS
VMa3,a4Raa58±
10 a6VS
VMa7i. The regression method is detailed in Appendix A.
Regime Coefficient Solid mantle in contact with
TMO BMO TMO and BMO
Solid–liquid phase changes (80) a0464 850 103 146 100 473
a11.042 1.008 1.000
a20.313 0.176 0.948
a30.994 1.326 0.646
Viscous building (8 )a412 075 20 743 48 481
a50.884 0.972 0.999
a61.214 1.208 1.195
a72.584 7.278 2.583
Error (%) 28.3 21.2 22.4
ness of the solid mantle decreases the scale for time (h2
S),
which partially compensates for the increase mentioned pre-
viously when recovering the dimensional time. As a result,
the thickness of solid in the phase change regime only loosely
affects the dimensional half-equilibration time (Fig. 7). The
effect of the thickness of solid is a bit stronger in the high-8
regime (as mentioned before).
4 Discussion
Our models address the compositional evolution of the
solid mantle bounded by magma oceans above and/or be-
low, and they constrain the time needed to chemically
(half-)equilibrate these reservoirs. While the concept of a
TMO that potentially interacts with the underlying solid
mantle is now well accepted, the idea of a long-lived BMO
remains controversial. Whether or not a BMO can be sta-
bilised depends on the slope of the adiabat versus that of the
melting curve (Labrosse et al., 2007) and/or on the fate of
FeO-rich TMO cumulates that sink to the CMB (Labrosse
et al., 2015; Ballmer et al., 2017b). Regardless of these is-
sues, our study can be applied to various scenarios, includ-
ing those with a solid mantle bounded just by a TMO, and
bounded by a TMO and BMO.
Classical fractional crystallisation models predict strongly
inverse chemical stratification of the initial solid mantle and,
consequently, a global-scale overturn by the end of a TMO
crystallisation (e.g. Elkins-Tanton et al., 2003, 2005). The
propensity of such a massive density-driven overturn de-
pends on whether or not chemical equilibration between the
solid mantle and magma ocean(s) can occur before magma
ocean solidification. Any style of magma ocean crystallisa-
tion leaves a strongly superadiabatic thermal profile, which
should drive convection in the cumulate layers before full so-
lidification (Solomatov and Stevenson, 1993a; Ballmer et al.,
2017b; Maurice et al., 2017; Boukaré et al., 2018), and re-
lated vertical flow should promote (partial) melting near the
TMO–mantle (and BMO–mantle) boundary or boundaries.
In this study, we focus on a phase change boundary condition
that allows material to flow through the boundary or bound-
aries and continuously change the composition of solid and
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 431
Figure 7. Timescales to reach chemical half-equilibrium between the solid mantle and a TMO of 100, 500, and 1000 km (orange, blue, and
purple solid lines respectively; this study, tS, Eq/2
pred ) versus different values of phase change number, 8+, and timescales of crystallisation
of that TMO until it is completely mushy (thicknesses are shown using the same colours and dashed lines, from Lebrun et al., 2013, tC
L13)
versus different values of phase change number, 8+. Panel (a) corresponds to a Rayleigh number of the solid part of RaM=108, and
panel (b) corresponds to RaM=109. The rose shading corresponds to a time higher than the age of the Earth. We note that tC
L13 values are
consistent with the timescales constrained by Nikolaou et al. (2019) and Salvador et al. (2017).
liquid reservoirs. While flow through the boundary or bound-
aries remains penalised by the phase change number, 8, this
approach implies final degrees of melting of at least 40 %,
i.e. beyond the rheological transition, at which crystals be-
come suspended (Abe, 1997; Costa et al., 2009). Future work
using a more realistic melting model is needed to test whether
these high degrees of melting can indeed be reached or to
quantify the effects of partial melting on equilibration be-
tween the mantle and TMO and/or BMO.
Our models show that the composition of the solid man-
tle and magma oceans strongly depends on the phase change
number, 8. In this study, we take 8as being constant through
time, but because this number depends on the dynamics and
thicknesses of the magma oceans, 8may change continu-
ously in a more realistic model with moving boundaries. Al-
though we show that chemical equilibration can occur before
full crystallisation of the magma oceans, variations in 8and
a moving-boundary scheme should be considered in further
studies.
Considering that 8±values are low when TMO and BMO
(or just TMO) crystallisation starts (Morison et al., 2019;
Morison, 2019), mantle convection would first assume a
degree-1 pattern (Fig. 5), possibly with implications for the
origin of crustal dichotomy on the Moon (e.g. Ishihara et al.,
2009) and Mars (e.g. Roberts and Zhong, 2006; Citron et al.,
2018). However, it remains to be shown that such a degree-
1 pattern of convection would be able to survive through all
stages of magma ocean crystallisation.
The crystallisation fronts move at different speeds, as the
TMO can crystallise in a few million years (Lebrun et al.,
2013; Salvador et al., 2017; Nikolaou et al., 2019), whereas
the BMO may persist for much longer (e.g. Labrosse et al.,
2007, 2015). Therefore, 8±would change accordingly. The
efficiency of equilibration during the late-stage magma ocean
depends on the timescale of freezing of this final stage, as
well as on the efficiency of mass transfer (8+) for a thin and
partially mushy TMO.
Once the TMO is fully crystallised, 8+tends to infinity,
while 8assumes a finite value as long as the BMO is still
present. Dynamics in the solid mantle would change accord-
ingly: convection in the solid mantle may be either domi-
nated by degree-1 (low 8) or by higher degrees of convec-
tion (high 8) (Fig. 5). Although our models do not account
for core cooling explicitly, the heat transfer across the man-
tle is expected to be much more efficient for lower values
of 8than for high values. This implies that the BMO is
likely to crystallise much faster than suggested by Labrosse
et al. (2007, 2015) for low 8, at least as long as no dense
FeO-enriched materials accumulate at the bottom of the solid
mantle to prevent efficient mass transfer across the BMO–
mantle boundary. The BMO may even be thermally coupled
to the relatively fast-cooling TMO for low 8+and low 8.
On the other hand, it is conceivable that thermally coupled
TMO and BMO crystallise more slowly than expected for
a thermally isolated TMO (Agrusta et al., 2019). The pres-
ence of a BMO makes heat transfer across the mantle and
out of the thermally coupled BMO and core more efficient
than for cases without a BMO and with a boundary layer
at the CMB instead. Such a situation implies that there is a
larger heat reservoir available to buffer the temperature of
the TMO for a given heat flux through the atmosphere and
to space. Note that the mass and heat capacity of the core
are similar to that of a 1000 km thick magma ocean; hence,
the timescale for full crystallisation could be roughly twice
the value usually computed (cf. Lebrun et al., 2013). How-
ever, the timescales of TMO, BMO and core cooling would
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432 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
be largely unaffected if the BMO were thermo-chemically
stratified (Laneuville et al., 2018). Whether or not material
transfer across the whole mantle, as predicted here for cases
with low 8±, can efficiently cool the core has important im-
plications for the long-term thermal evolution of terrestrial
planets as well as the propensity of an (early) dynamo.
Even though timescales of BMO crystallisation are not
well constrained, chemical exchange between the two
magma oceans (through the solid mantle) is still likely to
occur. Note that the same process (i.e. mantle convection)
that removes heat from the BMO and core is responsible
for this chemical exchange. As an example for the Earth,
if we take a TMO and a BMO with 100 km thickness each,
8±10 and a Rayleigh number of 108, we would expect
a half-equilibrium between solid mantle, TMO, and BMO
in less than 460 kyr, i.e. before TMO crystallisation (and
even more so, before BMO crystallisation). This chemical
exchange, however, does not necessarily imply homogene-
ity between the TMO and BMO, because the relevant phase
diagrams that control fractional crystallisation at the TMO–
solid mantle (low pressures) and BMO–solid mantle (high
pressures) boundaries are very distinct (e.g. Thomas et al.,
2012; Boukaré et al., 2015). For example, while the FeO dis-
tribution coefficient, K, defined in Eq. (11), is taken as a con-
stant in this study, its value is likely to be pressure dependent
(Nomura et al., 2011; Miyazaki and Korenaga, 2019a), po-
tentially causing partitioning of FeO into the BMO. Regard-
less, any such exchange between the TMO, solid mantle, and
BMO could be a way to sequester trace elements (including
heat-producing elements) into the BMO, particularly if the
TMO freezes faster than the BMO.
Once both oceans crystallise and 8± , convection in
the solid mantle likely changes to higher degrees of con-
vection (as already seen with 8=102and 103in Fig. 4b
and c), similar to present-day Earth mantle dynamics. Our
models predict a largely homogeneous solid mantle, with
some regions preserving significant primordial heterogene-
ity for long timescales. The preservation of heterogeneity
is likely to be enhanced once composition-dependent rhe-
ology (i.e. a difference in intrinsic strength of mantle ma-
terials) is considered (Manga, 1996; Ballmer et al., 2017a;
Gülcher et al., 2020). Indeed, primordial cumulates formed
in the lower mantle may be strongly enriched in MgSiO3
bridgmanite (Boukaré et al., 2015) and, hence, intrinsically
strong (Yamazaki and Karato, 2001). In the present day, large
low-shear-velocity provinces (LLSVPs) are perhaps the most
prominent and seismically evident large-scale mantle hetero-
geneities. That they are only rather mildly Fe-enriched (De-
schamps et al., 2012) points to rather efficient equilibration
between the magma ocean(s) and much of the solid man-
tle during crystallisation, such as predicted by a subset of
our models. For the Earth, the subset of our models with 8+
smaller than 100 suggests that chemical (half-)equilibrium
between a solid mantle and a (100 to 1000 km thick) TMO
can be accomplished in less than 1 Myr, i.e. before the
TMO is fully solidified or becomes a mush (Fig. 7). Equi-
libration over such a short timescale makes the solid mantle
mostly homogeneous (with some heterogeneities as seen be-
fore), which could explain the pyrolitic nature of the man-
tle (Wang et al., 2015; Zhang et al., 2016; Kurnosov et al.,
2017). Therefore, the final-stage TMO and subsequent mush
may be efficiently equilibrated with most of the solid man-
tle. In this case, we expect solid compositions that are by far
not as enriched in FeO as predicted by fractional crystallisa-
tion models (e.g. Elkins-Tanton, 2012), in which strong en-
richment only occurs because the final-stage TMO is fully
separated from the solid mantle, with strong disequilibrium
between the two reservoirs.
Similarly, we expect moderate enrichment (in FeO and in-
compatible trace elements) and roughly basaltic to komati-
itic (i.e. the melting product of a hot pyrolitic mantle) major-
element compositions of the primary crust. As our models do
not explicitly address the final and mush stages of the TMO
and consider only a strongly simplified compositional model
with just two components, (Fe, Mg)O, more detailed studies
with a more complex compositional treatment are needed in
order to predict the composition of the early crust.
In our models, we consider a simplified initial condition
with bulk planetary TMO and BMO compositions (XBulk
FeO =
0.12 and XTMO, BMO
FeO =0.12). While this condition may be
realistic for the formation of the solid mantle due to equi-
librium crystallisation, the TMO and BMO would be signif-
icantly more FeO enriched initially if they were formed by
fractional crystallisation (Solomatov and Stevenson, 1993a;
Xie et al., 2020). In our models, the initial cumulate down-
wellings formed at the TMO–solid mantle boundary are de-
pleted in FeO and, hence, buoyant, resisting solid mantle
convection and delaying compositional equilibration, but this
effect would be strongly diminished (or even opposite) for a
more realistic initial condition. Conversely, the initially de-
pleted cumulate upwellings from the BMO–mantle bound-
ary in our models advance convection and equilibration. As
these effects that depend on our choice of the initial condition
scale with buoyancy number, B, we choose a conservative
value of B=1.0 (see Sect. 2.1). Using higher values of B
(B3) is expected to advance TMO–mantle equilibration
for fractional crystallisation of the solid mantle shell. Our
models predict that such an equilibration can occur swiftly
to avoid extreme enrichments of the TMO during progres-
sive crystallisation and, thus, to prevent a subsequent global-
scale overturn with deep-mantle stratification. In our set-
up, the impact of the value of Bis fairly limited due to
weak compositional contrasts. Indeed, Fig. 4 exhibits com-
positional contrasts of at most 1XS
FeO =0.08. With B3,
the buoyancy would still be dominated by the thermal term
of order 1T =1, rather than by compositional buoyancy
B1XS
FeO 0.3. The value of Bwill only have a significant
impact in the late stages of the crystallisation of the magma
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 433
ocean, when fractional crystallisation leads to strong compo-
sitional contrasts.
Smaller planets than Earth are less likely to be chemically
equilibrated for a given bulk composition. First, they tend to
cool faster, as they contain a smaller total reservoir of heat
and volatiles (i.e. stabilising a less massive atmosphere to
shield cooling). Moreover, the Ra number is lower for small
planets, such that equilibration is expected to take longer ac-
cording to our results. Thus, the Martian mantle might be
less equilibrated (more stratified) than that of Earth (Elkins-
Tanton et al., 2003; Maurice et al., 2017). On the other hand,
super-Earths are expected to be well equilibrated, particu-
larly as BMOs are likely to be stabilised in their interiors due
to high CMB pressures (Stixrude, 2014; Caracas et al., 2019)
this has a strong effect on equilibration timescales. Whether
or not chemical equilibration during the magma ocean stage
is efficient has important implications for the composition
of the primary crust, the propensity of overturn, and related
stabilisation of a deep dense layer, as well as the long-term
evolution of terrestrial planets.
5 Conclusions
In this work we use a numerical model to investigate the
thermo-chemical evolution of the convecting solid mantle
bounded at the top and/or bottom by magma oceans. We
parameterise fractional crystallisation and melting processes
of dynamic topography at one or both solid mantle bound-
aries, and determine the timescales to reach chemical half-
equilibrium between solid mantle and magma ocean(s).
We show that these fractional crystallisation and dynamic
melting processes at one or both boundaries play an impor-
tant role in the chemical evolution of the solid mantle. Effi-
cient transfer of FeO across the TMO–mantle and/or BMO–
mantle boundary can prevent strong enrichment of the last-
stage magma ocean and, therefore, any strong chemical strat-
ification of the early fully solid mantle. Moreover, this ef-
ficient transfer of FeO renders the timescales of chemical
(half-)equilibration between the solid mantle and magma
ocean(s) shorter than (or of the order of) 1 Myr. As magma
ocean crystallisation occurs in few million years (Abe, 1997;
Lebrun et al., 2013; Salvador et al., 2017; Nikolaou et al.,
2019), our study suggests that chemical equilibrium between
solid and liquid reservoirs can be reached before the end of
magma ocean crystallisation. Therefore, a strong chemical
stratification of the solid mantle is unlikely to occur, and the
first crust is not expected to be extremely enriched in FeO.
This prediction fundamentally contrasts with that of classi-
cal models of fractional crystallisation of the magma ocean
(e.g. Elkins-Tanton, 2012).
However, more studies are needed to better constrain
chemical equilibration timescales. This could be achieved,
for instance, as more realistic compositional models and
phase diagrams are accounted for, and/or a moving bound-
ary approach is applied to explicitly model the evolution of
one or both crystallisation fronts.
https://doi.org/10.5194/se-12-421-2021 Solid Earth, 12, 421–437, 2021
434 D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs
Appendix A: Regression method
The best fitting coefficients of all regressions are obtained
using a simple algorithm. Each free parameter has an initial
possible minimum and maximum, chosen here between 1
and 1. All eight parameters aiin Eq. (15) are scanned be-
tween these minimum and maximum boundaries using ho-
mogeneous steps. For each point in that 8-D space, we com-
pute the misfit between predicted and observed timescale as
logPcasestS,Eq/2
pred tS,Eq/2
case 2
. The set of best fitting param-
eters are found by selecting the lowest misfit between the
analytical formulation and the data.
When the best fitting coefficients are found after a first
search, new iterations of the algorithm are requested using
more refined windows in the parameter space located around
the previous best fitting parameters. When a best parameter is
found at the boundary of the parameter space, the parameter
space is widened such that the best fitting coefficients are
independent from the initial boundaries in parameter space.
Iterations of the search are performed until the solution is
converged below fourth digit precision.
Figure A1 shows the regression for all cases that reached
chemical half-equilibrium.
Figure A1. Regression for all data with Eq. (15) for the solid mantle bounded by (a) a TMO of 1000, 500, and 100 km (VS=4.7×1020,6.7×
1020, and 8.6×1020 m3respectively); (b) a BMO of 1000, 500, and 100 km (VS=7.1×1020,8.2×1020, and 8.9×1020 m3respectively);
and (c) a TMO and BMO of 500–500, 500–100, 100–500, and 100–100 km (VS=5.8×1020 ,6.6×1020 ,7.7×1020, and 8.4×1020 m3
respectively). Colours indicate the corresponding Rayleigh number, Ra, and coloured outlines represent the phase change number, 8.
Solid Earth, 12, 421–437, 2021 https://doi.org/10.5194/se-12-421-2021
D. P. Bolrão et al.: Timescales of chemical equilibrium between the solid mantle and MOs 435
Code availability. The code is available for collaborative studies
upon request from the corresponding author.
Data availability. The data that support the findings of this study
are available from the corresponding author upon reasonable re-
quest.
Author contributions. DPB, MDB, AM, ABR, SL, and PJT de-
signed the study. DPB, AM, ABR, PS, SL, and PJT developed the
code. MDB, AM, ABR, and SL supported DPB with respect to in-
vestigating the results. ABR fitted the data and obtained the empir-
ical scaling law. DPB made the figures and wrote the initial paper.
All co-authors provided input and suggestions on the paper.
Competing interests. The authors declare that they have no conflict
of interest.
Acknowledgements. We thank Antonio Manjón-Cabeza Córdoba,
the editor Julien Aubert, and the two anonymous reviewers for use-
ful comments that improved the first version of this paper. We grate-
fully acknowledge support from the Swiss National Science Foun-
dation (SNSF; grant nos. 200021E-164337 and ANR-15-CE31-
0018-01).
Financial support. This research has been supported by the Swiss
National Science Foundation (grant no. 200021E-164337).
Review statement. This paper was edited by Susanne Buiter and re-
viewed by two anonymous referees.
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... The depth at which the initial crystallization occurs in the magma ocean is determined by the intersection between the mantle adiabat and the pyrolytic liquidus (Solomatov, 2015). This initial point of crystallization may have ranged between the coremantle boundary (CMB) and mid-mantle depth because the shapes of the liquidus and solidus (melting curves) of the pyrolytic mantle are not well constrained at depth (Bolrão et al., 2021). Accordingly, the existing theoretical and numerical modeling studies on magma ocean solidification are classified into two types: (A) magma ocean solidification from the CMB upward (Elkins-Tanton et al., 2003;Elkins-Tanton, 2008) and (B) magma ocean solidification originating at a mid-mantle depth and progressing toward the top and the bottom of the mantle (Labrosse et al., 2007;Mosenfelder et al., 2009;Bolrão et al., 2021). ...
... This initial point of crystallization may have ranged between the coremantle boundary (CMB) and mid-mantle depth because the shapes of the liquidus and solidus (melting curves) of the pyrolytic mantle are not well constrained at depth (Bolrão et al., 2021). Accordingly, the existing theoretical and numerical modeling studies on magma ocean solidification are classified into two types: (A) magma ocean solidification from the CMB upward (Elkins-Tanton et al., 2003;Elkins-Tanton, 2008) and (B) magma ocean solidification originating at a mid-mantle depth and progressing toward the top and the bottom of the mantle (Labrosse et al., 2007;Mosenfelder et al., 2009;Bolrão et al., 2021). ...
... In contrast to the type-A model, the type-B theoretical and numerical works (Labrosse et al., 2007;Stixrude et al., 2009;Caracas et al., 2019;Bolrão et al., 2021), considered several melting curves of mantle materials deduced from highpressure melting experiments (Fiquet et al., 2010;Deng and Lee, 2017) , which suggest that solidification of magma ocean could begin from the mid-mantle region where the adiabat intersects the liquidus of the mantle, and crystals are neutrally buoyant (Mosenfelder et al., 2009;Caracas et al., 2019;Bolrão et al., 2021). ...
... Compositional changes associated with the phase change are also required. Both aspects have already been reported and the implications of such a boundary have been explored , Agrusta et al., 2019, Bolrão et al., 2021, Lebec et al., 2023. The possibility of melting and freezing at one of the horizontal boundaries helps convection in the solid. ...
... An important feature of the model, already included in StagYY for a few previous studies [Agrusta et al., 2019, Bolrão et al., 2021, Lebec et al., 2023, is the solid-liquid phase change boundary condition at the bottom of the solid shell [e.g. Labrosse et al., 2018], ...
... In the context of the present study, we use a unique compositional information, representing the FeO content of the mineral assemblage. A complexity added to this approach by the phase change boundary condition and already treated by Bolrão et al. [2021] comes from the fact that the flow crossing the phase change boundary implies exchange of FeO with the BMO. In practice, when solid material crosses the boundary by melting, the associated tracers are removed while new tracers are added to regions where crystallisation occurs. ...
... The timescale of the basal MO crystallization, which has large uncertainty because several parameters to determine the timescale cannot be well constrained, is estimated based on fractional crystallization (Labrosse et al.,