Timescale of overturn in a magma ocean cumulate

Abstract

The formation and differentiation of planetary bodies are thought to involve magma oceans stages. We study the case of a planetary mantle crystallizing upwards from a global magma ocean. In this scenario, it is often considered that the magma ocean crystallizes more rapidly than the time required for convection to develop in the solid cumulate. This assumption is appealing since the temperature and composition profiles resulting from the crystallization of the magma ocean can be used as an initial condition for convection in the solid part. We test here this assumption with a linear stability analysis of the density profile in the solid cumulate as crystallization proceeds. The interface between the magma ocean and the solid is a phase change interface. Convecting matter arriving near the interface can therefore cross this boundary via melting or freezing. We use a semi-permeable condition at the boundary between the magma ocean and the solid to account for that phenomenon. The timescale with which convection develops in the solid is found to be several orders of magnitude smaller than the time needed to crystallize the magma ocean as soon as a few hundreds kilometers of cumulate are formed on a Mars-to Earth-size planet. The phase change boundary condition is found to decrease this timescale by several orders of magnitude. For a Moon-size object, the possibility of melting and freezing at the top of the cumulate allows the overturn to happen before complete crystallization. The convective patterns are also affected by melting and freezing at the boundary: the linearly most-unstable mode is a degree-1 translation mode instead of the approximately aspect-ratio-one convection rolls found with classical non-penetrative boundary conditions. The first overturn of the crystallizing cumulate on Mars and the Moon could therefore be at the origin of their observed degree-1 features.

Full text

Timescale of overturn in a magma ocean cumulate
A. Morisona,∗, S. Labrossea, R. Deguena, T. Alboussièrea
aUniversité de Lyon, ENSL, UCBL, CNRS, LGL-TPE, 46 allée d’Italie, F-69364 Lyon,
France
Abstract
The formation and dierentiation of planetary bodies are thought to involve
magma oceans stages. We study the case of a planetary mantle crystallizing
upwards from a global magma ocean. In this scenario, it is often considered
that the magma ocean crystallizes more rapidly than the time required for
convection to develop in the solid cumulate. This assumption is appealing since
the temperature and composition proles resulting from the crystallization of
the magma ocean can be used as an initial condition for convection in the
solid part. We test here this assumption with a linear stability analysis of the
density prole in the solid cumulate as crystallization proceeds. The interface
between the magma ocean and the solid is a phase change interface. Convecting
matter arriving near the interface can therefore cross this boundary via melting
or freezing. We use a semi-permeable condition at the boundary between the
magma ocean and the solid to account for that phenomenon. The timescale with
which convection develops in the solid is found to be several orders of magnitude
smaller than the time needed to crystallize the magma ocean as soon as a few
hundreds kilometers of cumulate are formed on a Mars- to Earth-size planet.
The phase change boundary condition is found to decrease this timescale by
several orders of magnitude. For a Moon-size object, the possibility of melting
and freezing at the top of the cumulate allows the overturn to happen before
complete crystallization. The convective patterns are also aected by melting
and freezing at the boundary: the linearly most-unstable mode is a degree-1
translation mode instead of the approximately aspect-ratio-one convection rolls
found with classical non-penetrative boundary conditions. The rst overturn
of the crystallizing cumulate on Mars and the Moon could therefore be at the
origin of their observed degree-1 features.
Keywords: magma ocean, overturn, mantle dynamics, linear stability
1. Introduction1
A common scenario considered for the formation of terrestrial planets is the2
crystallization of a global magma ocean from the bottom-up, because the liq-3
∗Corresponding author adrien.morison@ens-lyon.fr
Preprint submitted to Elsevier April 8, 2019

uidus of silicate magmas increases with pressure more steeply than the isentropic4
temperature, at least at low to moderate mantle pressure (Andrault et al., 2011;5
Fiquet et al., 2010; Thomas and Asimow, 2013; Boukaré et al., 2015). The6
crystallization of the surface magma ocean is expected to be rapid, around 17
Myr (e.g. Abe, 1997; Lebrun et al., 2013). This has led several authors to as-8
sume convection in the solid part of the crystallizing mantle does not start until9
the mantle is entirely crystallized (e.g. Hess and Parmentier, 1995; Abe, 1997;10
Parmentier et al., 2002; Elkins-Tanton et al., 2003, 2005; Zhang et al., 2013).11
However, this assumption deserves scrutiny since the compositional and thermal12
structure of the mantle after complete crystallization could be widely dierent13
if solid-state convection does set in during its crystallization.14
Two processes might lead to the destabilization of the solid mantle during its15
crystallization. First, assumming fractional crystallization, the surface magma16
ocean gets enriched in incompatible elements. As a secondary result, the new17
solid formed at the solid/liquid boundary gets richer and richer in these elements18
as crystallization progresses. Iron is such an element and its abundance is such19
that it aects signicantly the density of both the solid and the liquid. The solid20
formed at the end of the crystallization is richer in iron than the solid formed at21
the beginning of the crystallization, leading to an unstable setup with material22
denser at the top than at the bottom of the solid mantle.23
The second process that can further destabilize the solid mantle is the tem-24
perature gradient in the solid. The solidus temperature increases with pressure,25
and is steeper than the isentropic temperature prole. Assuming the tempera-26
ture in the solid stays close to the solidus, the resulting prole is hence unstable.27
This eect is enhanced by fractional crystallization and the associated enrich-28
ment of the solid in incompatible elements: their presence further decreases the29
solidus temperature and the compositional gradient discussed above induces an30
even steeper solidus.31
Numerical simulations including these processes suggest it is possible for32
solid-state convection to set in prior to the entire crystallization of the surface33
magma ocean (e.g. Maurice et al., 2017; Boukaré et al., 2018). Whether con-34
vection in the mantle starts during or after the crystallization of the surface35
magma ocean is found to have profound implications on the preservation of36
compositional heterogeneities as well as the dynamics of the mantle (Ballmer37
et al., 2017; Tosi et al., 2013). These results further conrm the need to assess38
the parameters controlling the onset of convection in the primitive mantle.39
A dynamical feature of the solid cumulate in contact with a magma ocean40
that has not been accounted for in the past studies is the possibility of exchange41
of matter at the boundary between the solid and the ocean via melting and42
freezing. We use a boundary condition developed for the inner core boundary43
(Deguen et al., 2013) to take this eect into account. This boundary condition44
is expected to have important eects on the convection pattern and heat ux45
as well as the timescale with which convection sets in (Deguen, 2013; Labrosse46
et al., 2018).47
Our aim is to assess how the timescale at which convection starts in the48
solid cumulate compares with the time needed to crystallize a surface magma49
2

ocean. Dierent scenarios are explored to determine the parameters controlling50
the onset of convection in the magma ocean cumulate. We consider the case51
where fractional crystallization happens during the entire cooling history of the52
magma oceans as well as the case where no compositional fractionation occurs.53
We explore the classical case for which no matter crosses the boundary between54
the magma ocean and the solid cumulate, and also the case with a boundary55
that allows matter transfer accross it. The study is applied to the Earth, Mars,56
and the Moon.57
2. Methods58
We consider a mantle that is initially fully molten and crystallizes from the59
bottom or some intermediate depth upward. The goal of the present study is to60
determine the timescale for convection to start in the solid part of the mantle61
as the magma ocean crystallizes.62
For the sake of simplicity, we assume the compaction length to be small and63
neglect the thickness of a mush layer at the phase change interface. Matter on64
one side of the boundary is entirely liquid while matter on the other side is65
entirely solid. We nonetheless allow for compositional fractionation to occur66
as the mantle crystallizes. The temperature at the solid/liquid boundary is67
denoted Tmand referred to as the melting temperature.68
Depending on how the temperature prole in the magma ocean compares69
with the prole of the melting temperature, two situations can occur. Either the70
solidication of the ocean progresses from the bottom up, or the solidication71
starts from an intermediate depth leading to a setup in which the solid part of72
the mantle is surrounded by two magma oceans. In this second scenario, the73
crystallization of the surface magma ocean (SMO) is thought to be a lot faster74
than the crystallization of the basal magma ocean (BMO) (Labrosse et al., 2007).75
We assume the solid mantle is a spherical shell of internal radius R−and76
external radius R+. Since the crystallization of the BMO is much slower than the77
crystallization of the SMO, we assume R−to be constant even for the case where78
the solid shell is surrounded by two magma oceans. The presence or absence of a79
BMO however aects the boundary condition applied at the bottom boundary80
of the solid mantle (see section 2.4).81
As the magma ocean cools down, R+increases to reach the total radius of the82
planetary body, denoted by RT. The temperature at the top boundary of the83
solid follows the melting temperature. The composition of the solid changes as84
well with the radius if we assume fractional crystallization occurs. For the sake85
of simplicity, we only consider fractionation of iron. The mass fraction of FeO,86
denoted by C, varies between 0 (e.g. Forsterite) and 1 (e.g. Fayalite). Although87
simplistic, such a model allows us to study the eect of the density gradient88
due to fractional crystallization on the dynamics of the solid. Figure 1 shows89
the composition and temperature proles at two dierent times. We assume90
the velocity of the freezing front ˙
R+does not vary laterally and that the SMO91
is well mixed, the temperature and compositional elds in the resulting solid92
3

Figure 1: Temperature and composition reference proles. Solid lines are the proles at time
t, dashed lines the proles at time t+δt. The green area is the solid mantle at time t. The
yellow area represents the part of the surface magma ocean (SMO, in red) that has crystallized
during δt. All the annotations on the axes are written at time t(see table 1 for the meaning of
symbols). Notice how the melting temperature decreases between the two instants owing to
the enrichment in iron of the surface magma ocean. The slopes of the curves are exaggerated
for readability purpose.
hence only vary with the radial position (as long as no solid-state convection93
operates).94
In this section, we introduce the simple phase diagram we use to compute the95
resulting temperature and composition proles in the solid under the assumption96
that no convection occurs in the solid (section 2.1). This serves as base state97
which stability against overturning motion is studied. We don’t treat the full98
dynamics of the overturn but compute, using a linear stability analysis, the99
growth rate of an overturning instability to compare it to the crystallization100
rate of the magma ocean. The latter is computed using a magma ocean cooling101
model which gives R+as a function of time, as described in section 2.2.102
2.1. Composition and temperature reference proles103
Under the assumption that no convection occurs during crystallization, one104
can determine the resulting temperature and compositional proles in the cu-105
mulate. These proles are used as reference proles in order to perform the106
linear stability analysis (section 2.5).107
4

Symbol Description Earth Moon Mars
Input parameters
R−Internal radius of the solid shell 3871 km∗737 km 2090 km
RTTotal radius of the planet 6371 km 1737 km 3390 km
T−Temperature at the bottom boundary†4500 K 1500 K 2400 K
T∞Black body equilibrium temperature 255 K 255 K 212 K
εEmissivity¶10−4110−3
gGravity acceleration 9.81 m/s21.62 m/s23.71 m/s2
RaSRayleigh number of SMO 1030 1028 5×1028
αThermal expansion coecient 10−5K−1
CpHeat capacity 103J K−1
κThermal diusivity 10−6m2/s
LhLatent heat 4×105J kg−1
σStefan-Boltzmann constant 5.67 ×10−8Wm−2K−4
ρReference density 4×103kg/m3
∆ρmSolid/liquid density contrast 2×102kg/m3
ηViscosity in the solid 1018 Pa s
Cl0Iron content of the primitive SMO†0.1
DSolid/liquid partition coecient of iron‡0.6
βCompositional expansion coecient -0.33
∂Tm/∂P Clapeyron slope 2×10−8K Pa−1
∂Tm/∂C Dependence of Tmon iron content −700 K
Computed dimensional variables
LMFinal thickness of solid mantle RT−R−2500 km 1000 km 1300 km
TmMelting temperature Tm(P, C )described by eq. (2.4)
T+Temperature at the top boundary T+(t)with eq. (2.5)
TpPotential temperature at the surface Tp(t)with eq. (2.8)
TsTemperature at the surface of the planet Ts(t)with eq. (2.9)
R+External radius of the solid shell R+(t)with eq. (2.10)
LThickness of the solid shell L=R+−R−
C0Iron content of the rst solid KCl0= 0.06
ClIron content of the SMO KCl(t) = C+(t)with eq. (2.2)
τStokes Stokes time ηL2/(∆ρgL3
M)
Dimensionless numbers
Ra(t)Thermal Rayleigh number ρgα∆T L3/(ηκ)
Rc(t)Compositional Rayleigh number ρgβL3/(ηκ)
W(t)Freezing front velocity (Peclet number) L˙
R+/κ
Γ(t)Thickness of the solid part L/LM
ΓS(t)Thickness of the SMO (RT−R+)/LM
Φ±Phase change number§10−2;∞
Table 1: Symbols used in this paper. All quantities with a +superscript are evaluated at
the top boundary (R+), while quantities with a −superscript are evaluated at the bottom
boundary (R−). ¶The emissivity values for the Earth and Mars are chosen so that the
crystallization time scale of the SMO is of the order of 1 Myr (Lebrun et al., 2013). For
the Moon, we neglect the eects of the atmosphere and assume a black body cooling. ∗This
choice assumes a 400 km thick basal magma ocean. Using R−= 3471 km does not change
signicantly the results. †From Andrault et al. (2011), ‡from Andrault et al. (2012). §10−2:
ow-through, ∞: non-penetrative. For the Moon and Mars, the possibility of a BMO is not
considered and Φ−=∞(see section 2.4 for details).
5

We consider a magma ocean crystallizing from some depth R−up to its top108
radius RT. The mass fraction of the heavy component (FeO) is C(r)in the solid109
and Cl(t)in the liquid, assuming that no diusion (nor convection) operates in110
the solid (therefore Cdoes not depend on time) and convection mixes the liquid111
eciently (therefore Cldepends only on time). At the freezing front, the phase112
relation is113
C(R+(t)) = DCl(t)(2.1)
with Dthe partition coecient (considered constant) and R+(t)the time-114
evolving radius of the freezing interface.115
Assuming the magma ocean undergoes fractional crystallization, the com-116
position prole in the cumulate is exponential. At the radial position rit is117
C(r) = 


C0RT3−(R−)3
RT3−r31−Dif r < Rs
1if r > Rs,
(2.2)
with118
Rs=(R−)3C
1
1−D
0+RT31−C
1
1−D
01/3
(2.3)
the value of R+at which Clreaches 1(see appendix A for more details).119
Since the diusion timescale is much larger than the other time scales con-120
sidered here, we assume the temperature prole in the cumulate stays close to121
the melting temperature. We take into account variations of the melting tem-122
perature Tmdue to both the pressure and the composition. A higher concentra-123
tion in iron leading to a lower melting temperature, the resulting temperature124
prole in the solid is steeper than a constant-concentration solidus when frac-125
tional crystallization is accounted for (Figure 1). The melting temperature Tm
126
veries:127
dTm
dr=∂Tm
∂P
∂P
∂r +∂Tm
∂C
∂C
∂r .(2.4)
With ∂P
∂r =−ρg and eq. (2.2), one obtains128
dTm
dr=−ρg ∂Tm
∂P + 3C(1 −D)r2
RT3−r3
∂Tm
∂C .(2.5)
For the sake of simplicity, we assume ∂Tm
∂P and ∂Tm
∂C to be constant (see table 1129
for values).130
We denote T=T − Tisen the superisentropic temperature in the solid, with131
Tisen =T−exp αg(R−−r)
Cp(2.6)
the isentropic temperature prole in the solid, with αthe coecient of ther-132
mal expansion, gthe acceleration of gravity and Cpthe heat capacity. We133
6

assume the variations of α,Cpand gwith depth to be negligible. The reference134
superisentropic temperature (denoted ¯
T) gradient is then:135
d¯
T
dr=−ρg ∂Tm
∂P +3C(1−D)r2
RT3−r3
∂Tm
∂C +αg
Cp
T−exp αg(R−−r)
Cp.(2.7)
2.2. Crystallization time scale136
Assuming the temperature prole in the SMO to be isentropic and neglecting137
variations of α,gand Cpwith depth, the potential temperature at the surface is:138
Tp=T+exp −αg(RT−R+)
Cp.(2.8)
Note that we are neglecting the temperature drop across the boundary layer at139
the bottom of the magma ocean. This is justied by the very small viscosity of140
the magma and the main buoyancy force coming from cooling to the atmosphere141
at the top surface.142
King et al. (2012) showed that the scaling law for the heat ux in a ro-143
tating uid (such as the surface magma ocean) depends on how the quantity144
RaSE3/2
S=αg∆T ν 1/2
κ(2Ω)3/2compares to 1, with ESthe Ekman number and RaS
145
the Rayleigh number in the SMO. A conservative lower bound with the ther-146
mal expansivity α∼10−5K−1, the gravity g∼10 m/s2, the super-isentropic147
temperature dierence ∆T∼1 K, the kinematic viscosity ν∼10−5m2/s,148
the thermal diusivity κ∼10−6m2/sand the rotation rate Ω∼10−4s−1is149
RaSE3/2
S∼1051. We then consider the heat ux is not controlled by rota-150
tion and scales as Nu = 0.16Ra2/7
SΓ6/7
Swith ΓS= (RT−R+)/L the dimension-151
less thickness of the SMO (King et al., 2012). Note that this scaling does not152
depend on the Prandtl number in the range of values explored by King et al.153
(2012), i.e. 16Pr 6100. Since Pr ∼10 is a reasonable value for a magma154
ocean, we assume this scaling holds for our study. We neglect variations of RaS
155
with time and assume the magma ocean behaves like a gray body at its upper156
surface. Heat ow conservation at the surface gives the following equation for157
the surface temperature Ts:158
k(Tp− Ts)
LM
0.16Ra2/7
SΓ−1/7
S=εσ(T4
s− T 4
∞)(2.9)
where T∞is the black body equilibrium temperature, σis the Stefan-Boltzmann159
constant and εthe emissivity. The emissivity should depend on the atmosphere160
dynamics and composition (particularly its water content) and vary with time.161
Taking this eet into account would require an atmosphere model (e.g. Abe,162
1997; Lebrun et al., 2013). For the sake of simplicity, we assume the emissivity163
to be constant, tuning its value to obtain a crystallization timescale that matches164
the ones of Lebrun et al. (2013) (see table 1 for values).165
As the SMO crystallizes (i.e. R+increases with time), we assume the tem-166
perature at the top of the solid mantle T+follows the solidus (eq. (2.5)), and167
7

the temperature prole in the SMO follows an isentropic prole. As R+grows,168
two phenomena produce heat that should be evacuated: the crystallization it-169
self with an associated latent heat Lh, and the cooling of the magma ocean.170
Assuming this heat is entirely evacuated through radiation in the atmosphere171
modeled as a gray body, one obtains the following equation:172
εσR2
T(T4
s−T 4
∞) = ρLhR+2dR+
dt−ρCp
d
dt ZRT
R+
T+exp αg(R+−r)
Cpr2dr!.
(2.10)
The last term of this equation can be developed (keeping in mind that the lower173
bound of the integral R+depends on time). This yields the time derivative174
of T+, which is written as a derivative with respect to R+using the chain175
rule. One obtains an ordinary dierential equation on R+(t)whose numerical176
integration gives the position of the interface between the solid and the surface177
magma ocean as a function of time.178
2.3. Set of dimensionless equations179
L=R+−R−,L2
M/κ,κ/L,ηL3/κ,∆T=T−−T+are used as scales for180
length, time, velocity, mass and temperature respectively. Note that R+and T+
181
vary with time as the surface magma ocean crystallizes. LM=RT−R−is the182
thickness of the solid mantle once the SMO is entirely crystallized. Note that183
all scales depend on time except the one for time itself, which is why Γ = L/LM
184
appears in the following equations. The dimensionless radial position is built185
as 1 + (r−R−)/L so that it is between 1 and 2 at all times. Similarly, the186
dimensionless temperature is chosen as (T−T+)/∆Tso that it is between 0187
and 1 at all times.188
Using the same symbols for dimensionless quantities, dimensionless conser-
vation equations of mass, momentum, heat and iron fraction are written as:
∇ · u= 0 (2.11)
0=−∇p+∇2u+ Ra (Θ − hΘi)ˆ
r+ Rc (c− hci)ˆ
r(2.12)
Γ2∂Θ
∂t +u·∇(Θ + ¯
T)− ∇2Θ = W (r−1)∂Θ
∂r +∂¯
T
∂r +
Θ!(2.13)
Γ2∂c
∂t +u·∇(c+¯
C) = W(r−1) ∂c
∂r .(2.14)
uis the velocity eld, pthe dynamic pressure, Θthe temperature perturbation189
with respect to the reference prole ¯
Tand cthe composition perturbation with190
respect to the reference prole ¯
C.hxidenotes the lateral average of the quantity191
x.Ra is the thermal Rayleigh number, Rc is the compositional Rayleigh number.192
The terms on the right hand side of eqs. (2.13) and (2.14) are due to the time193
dependence of the scales Land ∆T, which brings new advection terms associated194
with the change of frame, with W=L˙
R+/κ the dimensionless velocity of the195
freezing front. See table 1 for the denition and values of the various symbols.196
8

Note that these equations are written under the assumption that ˙
R−= 0.197
Other terms would appear on the right hand side of eqs. (2.13) and (2.14) in the198
general case involving the crystallization of a basal magma ocean. For Earth’s199
case, we assume the basal ocean crystallizes much slower than the surface ocean,200
and as such we neglect ˙
R−(Labrosse et al., 2007). We assume the diusion of the201
compositional eld is negligible since the diusion coecient of composition is202
much smaller than that of heat. Moreover, diusion of ¯
Tis neglected while that203
of Θis retained in order to ease the linear stability analysis. This is justied a204
posteriori by the fact that the diusion timescale is much longer than the other205
timescales considered in this study.206
2.4. Phase change boundary condition207
In the classical Rayleigh-Bénard setup, convecting matter arriving near an208
horizontal boundary forms a topography whose height is limited by the weight209
viscous forces can sustain. This topography is often neglected and a non-210
penetrative boundary condition is assumed at the interface (ur(R+) = 0). How-211
ever, in the system studied here, the boundary between the magma ocean and212
the cumulate is a phase change interface. A topography of the solid with respect213
to the equilibrium position of the solid/liquid interface can then be eroded by214
melting or freezing. Provided that the melting/freezing time is short compared215
to the time needed to build the topography by viscous forces, it is thus possible216
to have a non-zero normal velocity accross the interface. This is taken into ac-217
count with the help of the boundary condition introduced for the inner core by218
Deguen et al. (2013). This boundary condition, which translates the continuity219
of normal stress across the interface, is written in dimensional form as:220
∆ρmgτφur+ 2η∂ ur
∂r −p= 0.(2.15)
where ∆ρmis the density dierence between the solid and liquid phases and τφ
221
is the phase change timescale. Note that our denition of the dynamic pressure222
(dened here as p=P−hPi) diers from that of ˆpused by Deguen et al. (2013).223
The laterally constant term ∆ρmgτφ˙
Ris thus included in pinstead of explicitly224
appearing in the boundary condition. The dimensionless form of the boundary225
condition is226
±Φ±ur+ 2∂ ur
∂r −p= 0 (2.16)
where Φis the phase change number dened as:227
Φ±=|∆ρm|±gLτφ
η(2.17)
(the superscript +denotes the interface between the SMO and the solid at R+
228
while the superscript −denotes the interface between the BMO and the solid at229
R−). Moreover, the continuity of tangential stress is simply written as a classic230
free-slip boundary condition.231
9

The phase change timescale τφis related to the time needed to transport232
latent heat in the magma ocean from the areas that freeze to the areas that233
melt (Deguen et al., 2013):234
τφ=ρLh
(ρ−∆ρm)2Cp(∂PTm−∂PTisen)gu0(2.18)
where u0is the velocity scale in the magma ocean. A reasonable value for the235
latter is u0∼1 m s−1(Lebrun et al., 2013). Using nominal values for the other236
parameters, we nd that τφ∼104s. Plugging this in the expression of the phase237
change parameter eq. (2.17) yields Φ∼10−5.238
The phase change number Φcompares the phase change timescale τφ(i.e.239
the time needed to erode topography via melting and freezing) to the viscous240
timescale (i.e. the time needed to build topography with viscous forces). The241
value of Φallows to tune continuously the boundary condition between a non-242
penetrative classical condition (Φ→ ∞) and a fully permeable condition (Φ→243
0). Although this number should depend on time since Ldepends on time and244
τφdepends also on time but in a non trivial way, it is kept constant in this245
study. Two extreme values are tested for the SMO/solid interface: Φ = ∞246
which leads to the classical non-penetrative boundary condition and Φ = 10−2
247
which leads to a ow-through boundary (we use this value rather than 10−5
248
because the resolution of radial modes is more computationally demanding as Φ249
decreases, while the overturn timescale is not aected as shown in the results).250
For the Earth, these two values are also considered at the bottom of the solid,251
accounting for the possible presence of a basal magma ocean (BMO, Labrosse252
et al., 2007). For Mars and the Moon, we do not consider the possibility of a253
BMO and the bottom interface is hence non-penetrative, ur(R−) = 0. Rather254
than being realistic, these extreme constant values are used to study how the255
possibility of melting and freezing at the interface aects the stability of the256
solid, both in terms of onset time of overturn and preferred mode of motion.257
The estimated nominal value being Φ∼10−5, we expect the real system should258
be closer to the ow-through case than to the classical non-penetrative case.259
2.5. Determination of overturn timescale260
We start from a completely molten primitive mantle (R+=R−and T+=261
T−). We numerically integrate eq. (2.10) to obtain R+as a function of time (the262
potential surface temperature Tpand the surface temperature Tsare computed263
using eq. (2.8) and eq. (2.9)).264
At each timestep of this integration, we compute the reference tempera-265
ture and composition proles in the solid as shown in section 2.1 as well as266
the dimensionless numbers Ra(t),Rc(t),W(t)and Γ(t). Using a Chebyshev-267
collocation approach (e.g. Guo et al., 2012; Canuto et al., 1985), the set of lin-268
earized equations around the reference state is written as an eigenvalue problem269
(see appendix B). Solving numerically this problem yields the growth rate and270
shape of the most unstable mode of overturn. The inverse of that growth rate271
is the timescale for convection to set in in the solid shell. We compute this272
10

timescale at each timestep of the evolution of the SMO. By comparing this273
timescale with the corresponding time in the evolution of the SMO, we can as-274
sess whether convection is able to take place before the entire magma ocean is275
crystallized. Three dierent models are considered for the bulk of the solid:276
1. full model: compositional, thermal, and moving frame terms are taken277
into account;278
2. thermal model: compositional terms are left out, modeling the ideal case279
where no fractional crystallization occurs and the sources of instability280
are purely thermal (eq. (2.14) and the corresponding buoyancy term in281
eq. (2.12) are ignored);282
3. frozen-time model: moving frame terms (right-hand-side of eqs. (2.13)283
and (2.14)) are left out, resulting in a frozen-time approach where all284
long term evolution terms are ignored when studying the stability of the285
system at a given instant.286
We also compare the timescale obtained by linear stability analysis with the287
Stokes time τStokes =ηL2/(∆ρgL3
M)computed at each time to check whether288
this time is a relevant proxy of the stability of the solid mantle.289
3. Results290
The destabilization timescales for the Earth, Mars, and the Moon with var-291
ious boundary conditions along with the time needed to crystallize the remain-292
ing SMO are shown on Figure 2. Comparison of the destabilization timescales293
obtained for various bulk setups and boundary conditions yields information re-294
garding their contribution to the destabilization of the solid.295
The simplest cases are the one neglecting the compositional eects on den-296
sity. For such cases, the destabilization timescale tends to innity for a given297
non-zero thickness of crystallized mantle. This thickness corresponds to the one298
needed for instabilities to overcome diusion of perturbations of the reference299
state. In other words, it corresponds to the thickness above which the Rayleigh300
number in the solid part is above the critical Rayleigh number. For the Moon,301
this thickness is never reached and the Moon’s mantle stays stable with respect302
to purely thermal convection. For the Earth and Mars, this thickness is reached303
rather early, after ∼500 km of solid mantle is formed. As crystallization pro-304
gresses, the thickness and the temperature contrast between the top and the305
bottom of the solid mantle increase. The available buoyancy in the system there-306
fore increases. This leads to a strong decrease of the destabilization timescale,307
which becomes much shorter than the time needed to crystallize the remaining308
surface magma ocean (up to 6 orders of magnitude, depending on which bound-309
ary conditions are considered). This suggests that even in the purely thermal310
case, solid-state convection sets in before the mantle is completely crystallized311
for planets larger than Mars.312
The cases taking compositional eects on density into account are always313
unstable. This contrasts with the purely thermal cases and is due to the fact314
11

Figure 2: Growth time of the most unstable mode as a function of the crystallized mantle
thickness for the Earth, Mars, and the Moon. The solid black line is the time necessary to
crystallize the remaining surface magma ocean. Colors represent dierent boundary condi-
tions: both horizontal boundaries non-penetrative (blue); ow-through boundary condition
between the solid and the surface magma ocean to model the possibility of melting and freez-
ing (see section 2.4 for details) (green); and ow-through boundary conditions for both hori-
zontal boundaries assuming the presence of a basal magma ocean (red). Linestyles represent
dierent approximations regarding compositional eects (fractional crystallization and eect
on density) and moving frame contributions: both are taken into account (solid lines), compo-
sitional eects are neglected (dash-dotted lines), or moving frame terms are neglected (dotted
lines). The black dashed line is the Stokes time for each thickness, given for comparison.
that diusion of the composition eld is neglected. There is no mechanism315
to damp perturbations around the reference state, the latter is hence always316
unstable. Similarly to what is observed for the thermal cases, the destabilization317
timescale drops dramatically as the solid mantle thickens. For the Earth and318
Mars, the destabilization timescale ends up being shorter than the crystallization319
time of the remaining SMO by several orders of magnitude. The case where320
moving frame terms are neglected exhibits a shorter destabilization time scale321
at small thickness. The moving frame terms play a stabilising role only at the322
begining of mantle crystallization for the Earth and Mars but are signicant323
through the entire Moon’s mantle crystallization. The stabilising eect of the324
moving terms can be understood from the energy conservation eq. (2.13). Taking325
a temperature perturbation θ > 0and the associated velocity perturbation326
ur>0, one can notice there is a competition between the advection term327
ur∂r¯
T < 0and the moving frame term W(r−1)∂rθwhose average is negative.328
The same reasoning can be made with a negative perturbation and on the iron329
conservation eq. (2.14).330
For the Moon, the destabilization timescale is always greater than the time331
needed to crystallize the SMO. However, it should be noted that in this study332
the time to crystallize the SMO is computed assuming a well-mixed SMO with a333
surface behaving like a black body. The formation of a light solid crust enriched334
in plagioclase when around 80% of the SMO is crystallized is expected to slow335
down the solidication of the SMO by a few million years (e.g. Elkins-Tanton336
et al., 2011). This would leave enough time for convection to set in in the solid337
12

Φ+=∞
Φ−=∞
(a) classical case
Φ−=∞
Φ+= 10−2
(b) ow-through at top
Φ+= 10−2
Φ−= 10−2
(c) ow-through at top and
bottom
Figure 3: Most unstable convection modes for the Earth when a 1700 km thick mantle has
crystallized, for dierent boundary conditions represented by the values of the Φparameters
at the top and the bottom, as indicated. The dark zones represent negative temperature
anomalies while the bright zones represent positive temperature anomalies. The stream-
lines are superimposed. Note that the linear stability analysis oers no constraint on the
orientation and amplitude of these modes, only their harmonic degree and radial shape. (a):
both boundaries non-penetrative, the convection rolls have an aspect ratio approximatively
equal to 1; (b): ow-through top boundary, the ow pattern is of spherical harmonic degree
one, the streamlines go through the top boundary but go around the central part; (c): ow-
through conditions at both boundaries, the ow pattern is of spherical harmonic degree one,
the streamlines go through both boundaries, resulting in a translation mode of convection.
Similar behavior is obtained for the other bodies.
since the destabilization timescale we nd is much shorter than that.338
The three boundary conditions exhibits dierent destabilization timescales.339
The case where both boundaries are non-penetrative (which is the case classi-340
cally considered) needs more time to destabilize than the case where the bound-341
ary between the surface magma ocean and the solid allows melting and freezing.342
Convective patterns obtained with a ow-through boundary are substantially343
dierent than the classical ones (Figure 3). Aspect-ratio-1 rolls are obtained344
with classical boundary conditions. However, when the top boundary allows345
phase change, a spherical-harmonic-degree-1 near-translation mode develops.346
Matter freezes on one side of the spherical shell, goes around the core or basal347
magma ocean, and melts on the other side. In the case with a basal magma348
ocean and its boundary with the solid of ow-through type, matter also crosses349
the inner boundary of the spherical shell, resulting in a true translation mode.350
These two translation modes involve very little or no deformation of the solid351
compared to the classical case, and therefore less viscous forces acting against352
convection. This explains the smaller destabilization timescale associated with353
these modes as well as the lower critical thickness in the purely thermal case.354
Figure 4 shows the transition between the non-penetrative and the ow-355
through regime occurs over a rather short range of values of the phase change356
number. Φ+.1leads to near-translation while Φ+&100 leads to classical357
aspect-ratio-one rolls.358
A notable feature on Figure 2 is the steep decrease of the destabilization359
13

104
105
106
107
Destabilization timescale τ(year)
Γ=1/3
10−410−310−210−1100101102103104
Phase change number Φ+
103
104
Destabilization timescale τ(year)
Γ=2/3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 4: Destabilization timescale of several harmonics degree (l= 1 to 15) as a function of
the phase change number value for the Earth. The bottom boundary is non-penetrative. Top:
833 km are crystallized (mid-radius ¯r∼4288 km), bottom: 1667 km are crystallized (mid-
radius ¯r∼4704 km). The most unstable mode is the one with the shortest destabilization
timescale. One can notice that in the non-penetrative case (Φ+→ ∞), the most unstable
mode corresponds to aspect-ratio-1 rolls. The typical roll size of the most unstable mode (¯rπ/l)
is roughly 900 km for the top case (l= 15) and 1850 km for the bottom case (l= 8). However,
with a ow-through boundary (Φ+→0), the most unstable mode is the near-translation
mode for both cases.
14

Figure 5: Ratio between the destabilization timescale obtained for the purely compositional
case τC(thermal terms are left out) and the timescale obtained for the purely thermal case τT
(compositional terms are left out). When this ratio is above one, it means the thermal reference
prole is more unstable than the compositional reference prole. The Moon is not shown
here since the purely thermal case is never unstable (τT→ ∞). The colors are the same as in
Figure 2, blue: non-penetrative condition for both horizontal boundaries (Φ±=∞); green:
ow-through condition at the boundary between the solid and the surface magma ocean; and
red: ow-through condition at both horizontal boundaries.
timescales at the end of the crystallization when compositional terms are taken360
into account. That decrease is due to the strong (i.e. very unstable) compo-361
sitional gradient appearing at the end of the crystallization. It does not aect362
the destabilization timescale obtained with non-penetrative boundary condi-363
tions; this can be explained by the fact that the strong compositional gradient364
is in a very thin layer at the top of the domain where vertical velocities vanish,365
and therefore does not contribute to the driving of the down- and up-welling366
currents.367
A comparison between the purely thermal and purely compositional cases for368
the Earth and Mars is shown on Figure 5. The ratio between the destabilization369
timescales for theses two cases is 0before the critical thickness for the purely370
thermal case is reached. For Mars, the compositional prole is always more371
unstable than the thermal prole and controls the destabilization timescale of372
the system. For the Earth, however, the ratio between the two cases is fairly373
close to 1for a large part of the crystallization history: neither the thermal nor374
the compositional prole dominates the destabilization timescale of the system.375
Figure 6 shows that the destabilization timescale τLSA is proportional to the376
Stokes time τStokes =ηL2/(∆ρgL3
M). Two eects alter this relation: moving377
frame terms whose eects are not included in the Stokes time, and the strong378
compositional gradient at the end of the crystallization whose eects depend379
15

Figure 6: Growth time of the most unstable mode versus the Stokes time for the Earth,
Mars, and the Moon. The solid line is the destabilization timescale obtained with the linear
stability analysis τLSA (case with all terms accounted for). The dashed lines correspond to
τLSA ∝τStokes. Composition, temperature and moving frame terms are all taken into account.
The colors are the same as in Figure 2, blue: non-penetrative condition for both horizontal
boundaries (Φ±=∞); green: ow-through condition at the boundary between the solid and
the surface magma ocean; and red: ow-through condition at both horizontal boundaries.
on the boundary condition. It should be noted that the ratio τLSA/τStokes de-380
pends on the body and the boundary conditions considered. Notably, permeable381
boundary conditions lead to a decrease of τLSA.382
4. Discussion383
We showed for the Earth and Mars that the growth timescale of convective384
instabilities in a crystallizing mantle from the bottom up is several orders of385
magnitude smaller than the time needed to fully crystallize that mantle. This386
holds even without taking into account fractional crystallization and the un-387
stable density gradient it induces. This contrasts with the assumptions made388
in several studies (Hess and Parmentier, 1995; Elkins-Tanton et al., 2003; Tosi389
et al., 2013) where the overturn is assumed to take place because of the compo-390
sitionally induced density gradient after the entire mantle is crystallized. The391
numerical simulations performed by Ballmer et al. (2017) for Earth-like objects392
lead to a destabilization of the solid after a few Myr, and those performed by393
Maurice et al. (2017) for Mars-like objects lead to a destabilization after roughly394
1 Myr. These times are not easily comparable to the timescales we compute via395
linear stability analysis since the physical problems are dierent in non-trivial396
ways: the simulations of Ballmer et al. (2017) are in a 2D aspect-ratio-1 carte-397
sian box, those of Maurice et al. (2017) are in cylindrical geometry with a vari-398
able viscosity, a melt extraction mechanism and a solidus temperature that de-399
pends only on pressure. However, despite these dierences, the destabilization400
time uncovered by these simulations are rather similar to the one we predict for401
the non-penetrative cases: of the order of 1 Myr for the Earth and 0.5 Myr for402
16

Mars. This conrms the linear growth rate of instabilities is a relevant proxy403
for the timescale at which convection sets in.404
Moreover, allowing transfer of matter via melting and freezing at the inter-405
face between the solid and the surface magma ocean reduces dramatically the406
timescale with which solid-state convection can set in. It also changes the shape407
and harmonic degree of the most unstable mode: a degree-one translation mode408
is preferred. Therefore, the possibility of melting and freezing at the interface409
should be accounted for when studying the overturn of the primitive mantle of410
planetary bodies. For example, the case of the Moon is an interesting potential411
application. This body has a strong dichotomy: the near-side presents more412
mare basalts, more KREEP material, and a thinner crust than the far-side.413
Wasson and Warren (1980) already proposed that such features could be due to414
a slower cooling of the lunar magma ocean on the near side than on the far-side.415
A permeable boundary would allow the solid mantle to overturn with a domi-416
nant degree-one before the entire crystallization of the mantle (keeping in mind417
that the end of the crystallization is much slower than what we predict with418
our simple model, see Elkins-Tanton et al. (2011)). The mechanisms proposed419
to build a degree 1 at the scale of the Moon involve the dynamics of an entirely420
crystallized lunar mantle (e.g. Parmentier et al., 2002; Zhong et al., 2000). The421
possibility to form a degree one while the crystallization of the magma ocean422
is still ongoing is therefore worth exploring with more complete models to test423
whether this dominant degree-one can be conserved after crystallization of the424
magma ocean and/or helps the development of degree-one instabilities such as425
the ones predicted in the aforementioned studies. It is also tempting to asso-426
ciate the degree-one feature of Mars (the Marsian dichotomy) to the same pro-427
cess but, as explained above, the rst degree-one overturn of the solid mantle428
is expected to happen long before its complete crystallization. Secondary over-429
turning instabilities are possible after the rst one that we cannot investigate430
with the tools presented above. A more complete study investigating the nite431
amplitude dynamics is necessary to understand the implications of this work to432
planets larger than the Moon.433
It should be noted that several parameters involved in the problem are badly434
constrained. The viscosity of the solid mantle and even its rheology is such a pa-435
rameter. It is highly dependent on how close the temperature in the solid is from436
the solidus and could easily vary by a few orders of magnitude (e.g. Solomatov,437
2015). Since the destabilization timescale scales as the Stokes time (Figure 6),438
it is directly proportional to the viscosity and could therefore vary by a few or-439
ders of magnitude. The strong relation between the viscosity of the cumulate440
and the overturn scaling has been investigated by Ballmer et al. (2017): their441
numerical experiments conrm the overturn onset scales as the Stokes time. It442
should be noted that our ow-through boundary conditions does not aect this443
result, it only reduces the proportionality factor between the Stokes time and444
the growth time of instabilities (Figure 6). This validates the general approach445
proposed by Boukaré et al. (2018) to assess whether solid-state convection sets446
in before the magma ocean is entirely crystallized: they compare the Stokes447
time with the time needed to crystallize the magma ocean and their numeri-448
17

Figure 7: Thickness of the solid cumulate at which the destabilization timescale equals the
time needed to crystallize the rest of the SMO for several values of the partition coecient,
D∈[0.01,0.99]. The Moon is not shown here since the destabilization timescale is greater
than the time needed to crystallize the SMO. The colors are the same as in Figure 2, blue:
non-penetrative condition for both horizontal boundaries (Φ±=∞); green: ow-through
condition at the boundary between the solid and the surface magma ocean; and red: ow-
through condition at both horizontal boundaries.
18

cal experiments yields that syn-crystallization convection is possible when the449
ratio between these two times exceeds ∼5×104. This value however was de-450
termined with non-penetrative boundary conditions, the actual threshold should451
be a few orders of magnitude higher (meaning syn-crystallization convection452
is allowed for shorter solidication timescales) since the ow-through bound-453
ary condition leads to a faster destabilization of the cumulate for the same454
Stokes time. Another aspect that deserves care is that for Earth-sized bodies,455
the Stokes time should incorporate both the thermal and compositional density456
constrasts. Boukaré et al. (2018) compare the “compositional” Stokes time with457
the solidication timescale; while this is perfectly valid for the Moon and Mars458
for which the thermal density contrast is much smaller than the compositional459
one, this does not hold for the Earth where both terms have similar magnitudes460
(Figure 5). The tremendous importance of the viscosity is why a viscosity of461
1018 Pa s is assumed in this study since it is a higher bound for the near-solidus462
viscosity (see Solomatov, 2015, and references therein) and gives the most con-463
servative estimate for the destabilizing time. The viscosity could be signicantly464
lower if the melt fraction is important in the cumulate, Solomatov (2015) sug-465
gests 1014 Pa s as a lower bound at 40% melt fraction (roughly the rheological466
transition). Another potential eect of viscosity that is neglected in the study is467
dynamical: since solid state convection occurs during the crystallization of the468
magma ocean, the temperature in the solid departs from the solidus tempera-469
ture prole and as a result the viscosity increases. Moreover, the compositional470
prole becomes gravitationally stable with iron-enriched heavy material being471
transported from the top to the bottom of the solid. These two eects com-472
bined may lead to the stopping of the solid state convection (Solomatov, 2015).473
Depending on the size of the magma ocean considered, it could then be possible474
either that the magma ocean crystallizes completely before convection may start475
again in the solid, or that convection sets in again in the solid before it is en-476
tirely crystallized. Studying this scenario requires a more complex method that477
a simple linear stability analysis since it involves a non-linear feedback between478
the dynamics of the solid part and its viscosity, temperature, and compositional479
elds.480
Another unconstrained parameter is the partition coecient of iron between481
the solid and liquid. An exploration of this parameter shows that the eect482
of the partition coecient is rather limited for the Earth, and slightly more483
important for Mars (Figure 7). This is in agreement with Figures 2 and 5484
showing the dierence between the purely thermal case (corresponding to the485
extreme value D= 1) and the purely compositional case is rather small for the486
Earth but more important for Mars.487
Finally, our choice of a constant emissivity results in a roughly constant488
solidication rate, whereas more sophisticated cooling models including an at-489
mosphere predict most of the mantle crystallizes quickly, and the solidication490
slows down when only a shallow magma ocean remains. Although such eects491
are important to build realistic solidication models, they should not aect dra-492
matically our results. Indeed, a faster crystallization at the beginning would lead493
to a destabilization of the solid mantle at a larger thickness, but we expect this494
19

dierence to be small since the destabilization timescale is rapidly much lower495
than the solidication timescale.496
5. Conclusions497
Upward crystallization of the silicate mantle of planets within a magma498
ocean is expected to produce a unstably stratied situation, because of both499
temperature and composition. In this study, we have addressed the question500
of whether the overturning instability develops faster than the time it takes to501
crystallize the magma ocean. To that end, we have developed a linear stability502
analysis tool to compute the growth rate of the fastest overturning mode and503
studied systematically the eect of the most important parameters: the planet’s504
size (Moon to Earth size), the partition coecient and the type of boundary505
condition between the solid and the liquid. In particular, we have introduced a506
boundary condition that accounts for the possibility of melting and freezing at507
the interface between the solid mantle and the magma ocean.508
This study shows convection is likely to start in the solid mantle of the509
Earth, Mars and the Moon before the entire crystallization of the surface magma510
ocean. Evolution models of the primitive mantle of planetary bodies should511
therefore account for convection and the associated mixing in the solid part of512
the crystallizing mantle.513
This result holds for the Earth and Mars even without fractional crystalliza-514
tion and the unstable compositional gradient it creates in the cumulate. The515
value of the partition coecient is found to have little impact on the timing of516
mantle overturn.517
The timescale at which convection sets in scales as the Stokes time. Speci-518
cally, it is proportional to the viscosity of the solid. However, it should be kept519
in mind that these results are obtained assuming a newtonian rheology and a520
constant viscosity in the solid mantle. Given the central role of viscosity in this521
problem, better knowledge of the viscosity and rheology of the primitive solid522
mantle is of primary importance to study its dynamics.523
Finally, the possibility of exchange of matter between the solid mantle and524
the magma ocean(s) should be accounted for in dynamical models of the primi-525
tive mantle since it greatly alters the pattern of convection as well as the desta-526
bilization timescale. It could even be a way of producing degree-one structures527
such as the ones observed on the Moon and Mars.528
Acknowledgements529
We thank the editor Bruce Buet and an anonymous reviewer for their useful530
remarks that helped to improve the clarity of our paper. This study is funded531
by the French Agence Nationale de la Recherche (grant number ANR-15-CE31-532
0018-01, MaCoMaOc).533
20

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A. Composition prole resulting from the fractional crystallization628
of the surface magma ocean629
Conservation of the heavy component implies that630
d
dt ZR+
R−
C(r)r2dr+1
3RT3−R+3Cl!= 0,(A.1)
where no compressibility eect on density is considered, allowing the bulk den-631
sity to drop out of the equation. Using eq. (2.1) and ˙
Cl=˙
R+dCl
dR+, assuming632
˙
R+>0at all time and R−constant:633
1
Cl
dCl
dR+= 3(1 −D)R+2
RT3−R+3.(A.2)
Using eq. (2.1), eq. (A.2) can be written for C:634
1
C
dC
dR+−1
D
dD
dR+= 3(1 −D)R+2
RT3−R+3.(A.3)
Since C(r)does not depend on time, this equation holds for any r6R+(t)(i.e.635
everywhere in the solid) and can be written as:636
1
C
dC
dr−1
D
dD
dr= 3(1 −D)r2
RT3−r3.(A.4)
23

Equation (A.4) is general and allows to take into account variations of D.637
However, it is useful to consider the limiting case of a constant partition coe-638
cient D. In that case, a solution to this equation is639
C=C0 RT3−R−3
RT3−r3!1−D
,(A.5)
with C0=DCl0the mass fraction of FeO in the rst solid formed.640
Note that eq. (A.5) diverges when r→RTbut is in fact only valid as long641
as C < 1and Cl<1. When Clreaches 1, the solid formed has the same642
composition as the liquid. The complete solution therefore is643
C=


C0RT3−R−3
RT3−r31−Dif r < Rs
1if r > Rs,
(A.6)
with644
Rs=(R−)3C
1
1−D
0+RT31−C
1
1−D
01/3
(A.7)
the value of R+(t)such that Cl(t) = 1.645
B. Linear Stability646
Since the solid is considered isoviscous and no source of toroidal ow is
imposed at the boundaries, the velocity eld can be expressed in terms of the
scalar poloidal potential P:u=∇×∇×(Pr)(e.g. Ricard and Vigny, 1989; Ribe,
2007). Linearizing eqs. (2.11) to (2.14) around the reference state (u=0;¯
T;¯
C)
gives:
Q=∇2P(B.1)
∇2Q= RaΘ− hΘi
r+λ+ Rcc− hci
r+λ(B.2)
Γ2∂Θ
∂t +∂¯
T
∂r
L2P
r+λ− ∇2Θ = W (r−1)∂Θ
∂r +∂¯
T
∂r +
Θ!(B.3)
Γ2∂c
∂t +∂¯
C
∂r
L2P
r+λ=W(r−1) ∂c
∂r .(B.4)
The boundary conditions on the temperature and composition perturbations
are trivial:
Θ±= 0,(B.5)
c±= 0.(B.6)
24

The boundary condition eq. (2.16) and the free-slip boundary condition are
written in term of the poloidal potential as:
±Φ±1
r+λL2P+∂
∂r 2
r+λL2P − (r+λ)Q= 0 (B.7)
∂2P
∂r2+ (L2−2) P
(r+λ)2= 0.(B.8)
λ=R−/L −1is a curvature term due to the denition of the dimensionless647
radius. L2is the horizontal laplacian: L2•=∂r((r+λ)2∂r•)−(r+λ)2∇2•. The648
quantity Qis introduced to ease the formulation of this system as an eigenvalue649
problem involving square matrices.650
The perturbations P,Q,Θand care developed using spherical harmonics,651
e.g.652
P=
∞
X
l=1
l
X
m=−l
Pl(r)Ym
l(θ, φ)eσlt(B.9)
where land mare the spherical harmonics degree and order and σlis the growth653
rate associated to the harmonic degree l. The system is laterally degenerated654
and mdoes not aect the growth rate of the perturbation nor the shape of the655
radial modes Pl(r),Ql(r),Θl(r)and cl(r). These radial modes are discretized656
using a Chebyshev collocation approach (e.g. Guo et al., 2012; Canuto et al.,657
1985). Each radial mode is expressed as a vector whose components are the658
values at the N+ 1 Chebyshev nodal points (respectively denoted P,Q,T659
and C). Radial derivatives evaluated at the nodal points ri=1
23 + cos iπ
N
660
can then be expressed with a dierentiation matrix d, e.g. ∂rP(ri) = (dP)i. We661
formulate the system of linearized equations along with the associated boundary662
conditions as663
LX =σlRX (B.10)
25

with
X=





P0 : N
Q0 : N
T1 : N−1
C1 : N−1
(B.11)
L=
0 : N0 : N1 : N−1 1 : N−1






















d2+ (l2−2)r−2
λ0 0 0 0
D2
−1 0 0 1 : N−1
d2+ (l2−2)r−2
λ0 0 0 N
l2(Φ+r−1
λ
−2r−2
λ+ 2r−1
λd)−(1+rλd)0 0 0
0 D2
−Rar−1
λ
−Rcr−1
λ1 : N−1
l2(−Φ−r−1
λ
−2r−2
λ+ 2r−1
λd)−(1+rλd)0 0 N
−(∂r¯
T)l2r−1
λ0 D2+W+(r−1)d+ (∂r¯
T)+101 : N−1
−(∂r¯
C)l2r−1
λ0 0 W+(r−1)d1 : N−1
(B.12)
R=
0 : N0 : N1 : N−1 1 : N−1






0 0 0 0 0 : N
0 0 0 0 0 : N
0 0 Γ21 0 1 : N−1
0 0 0 Γ211 : N−1
(B.13)
where 1is the identity matrix, rij =ri1ij,rλ=r+λ1,l2=l(l+ 1) and664
D2=d2+ 2r−1
λd−l2r−2
λThe extra row and column on top and right of the665
matrices are respectively the column and row indices of each of the submatrices.666
For example, the top left submatrix of the matrix Lis only the rst row (hence667
the 0on the extra column) of the matrix d2+ (l2−2)r−2
λ.668
At a given instant during the crystallization, all the dimensionless numbers669
W,λ,Γ,Ra and Rc appearing in the matrices Land Rare known. For any670
harmonic degree lof the perturbation, nding its growth rate σland associated671
vertical mode Xis an eigenvalue problem. The largest eigenvalue is the growth672
rate, and the associated eigenvector represent the vertical modes. At a given673
instant, we look for the harmonic degree lwith the highest growth rate σl, which674
is then used to compute the dimensional destabilization time scale L2
M/(κσ).675
26