(2018) 70:200
Miyagoshi et al. Earth, Planets and Space
https://doi.org/10.1186/s40623-018-0975-5

Open Access

EXPRESS LETTER

Effects of adiabatic compression
on thermal convection in super‑Earthsof various sizes
Takehiro Miyagoshi1* , Masanori Kameyama2 and Masaki Ogawa3

Abstract
We present two-dimensional numerical models of thermal convection of a compressible fluid in the mantles ofsuper-Earths calculated under the truncated anelastic liquid approximation to discuss how adiabatic compression
affects the thermal convection, depending on planetary mass. The convection is driven by basal heating, the viscosity
depends on temperature, and the thermal expansivity and the reference density depend on the depth. We varied allof the magnitude of adiabatic heating, the Rayleigh number, the depth profile of the thermal expansivity, and that of
the reference density in accordance with the planetary mass. The effects on thermal convection become substantial,
when the planetary mass normalized by the Earth’s mass Mp exceeds a threshold Mc, about 4. Hot plumes ascending
from the core–mantle boundary become thinner with increasing Mp; they become almost invisible except around thecore–mantle boundary, when Mp > Mc. The lithosphere that develops along the surface boundary due to the temperature dependence of viscosity becomes thicker with increasing Mp and is about twice as thick as that at Mp = 1 whenMp = 9.4. The convective velocity is almost independent of Mp. These results are in a striking contrast with earlier predictions that are made based on the models where the effects of adiabatic compression are neglected; it is importantto take account of the effects of adiabatic compression properly in the exploration of mantle dynamics such as plate
tectonics and hot spot volcanisms in massive super-Earths. Further researches are necessary to clarify the dependenceof Mc on the surface temperature and the material properties of the convecting mantle.
Keywords: Mantle convection, Super-Earths of various sizes, Adiabatic compression, Numerical simulations
Introduction
Motivated by recent detection of a large number ofsuper-Earths, i.e., extrasolar planets with a mass of up
to ten times the Earth’s (e.g., Borucki et al. 2011), many
researchers have explored dynamics of the mantle ofsuper-Earths, because it is a key to understanding theirtectonics, thermal history, and surface environments(e.g., Valencia et al. 2007; Valencia and O’Connell 2009;van Heck and Tackley 2011; Foley et al. 2012; Lenardic
and Crowley 2012; Stein et al. 2013; Tackley et al. 2013;
Stamenković and Breuer 2014). In the terrestrial planetsof our solar system, two of the most important factors*Correspondence: miyagoshi@jamstec.go.jp
1
Department of Deep Earth Structure and Dynamics Research, Japan
Agency for Marine-Earth Science and Technology, 3173‑25 Showa‑machi,
Kanazawa‑ku, Yokohama, Kanagawa 236‑0001, Japan
Full list of author information is available at the end of the article

that dominate mantle dynamics are the lithosphere andhot ascending plumes (e.g., Schubert et al. 2001; Davies2011). Whether the lithosphere on super-Earths is rifted
into mobile plates as observed on the Earth or remains
stagnant as observed on Venus and Mars has been acentral issue, and the predictions made in the literature
range from the stagnant lithosphere (e.g., O’Neill andLenardic 2007; Stein et al. 2013) to active plate tectonics(e.g., Valencia et al. 2007; Valencia and O’Connell 2009;van Heck and Tackley 2011; Foley et al. 2012; Tackley
et al. 2013). In their review of parameterized convection
models for super-Earths, Stamenković and Breuer (2014)conclude that whether plate tectonics can operate or notsensitively depends on the detail of the parameterization.
The vigor of hot ascending plumes, although it has not
drawn so much attention, is also an important issue, since
hot plumes affect the activity of hot spot volcanism and© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license,
and indicate if changes were made.

Miyagoshi et al. Earth, Planets and Space(2018) 70:200

large igneous provinces that have been suggested to rift
the lithosphere on the Earth (e.g., Richards et al. 1989).
Before challenging these issues on the real super-Earths,
however, it is important to explore thermal convection in
the mantle of super-Earths as a problem of fluid dynamics. Recently, Miyagoshi et al. (2014, 2015, 2017) havefocused on the effects of adiabatic compression in superEarths of ten times the Earth’s mass, and found that theeffects are substantial, as summarized below. Here, weapply our earlier numerical models to super-Earths ofvarious mass to clarify the threshold in planetary mass
above which the effects of adiabatic heating we foundbecome important.
An important finding of Miyagoshi et al. (2014, 2015)is that adiabatic compression substantially reduces thevigor of thermal convection in a massive super-Earth incontrast to the predictions of many of earlier studies thatare based on the Boussinesq approximation where adiabatic compression is neglected (e.g., Valencia et al. 2007;
Valencia and O’Connell, 2009; van Heck and Tackley
2011; Foley et al. 2012). In their models, Miyagoshi et al.
(2014, 2015) found that the lithosphere becomes muchthicker and hot ascending plumes become thinner than
expected from Boussinesq models in a planet of ten times
the Earth’s mass. These results suggest that it is necessary
to revisit the issue of plate tectonics and hot spot volcanism in super-Earths, taking account of the effects ofadiabatic heating (e.g., Valencia et al. 2007; Valencia andO’Connell 2009; van Heck and Tackley 2011; Foley et al.
2012). In this paper, we apply our earlier models to superEarths of various mass Mp (the planetary mass divided
by the Earth’s mass) 1, 2, 4, 6, and 9.4 to clarify the minimum planetary mass above which the effects of adiabatic
compression on mantle convection become prominent.
To highlight the effects of adiabatic compression, we
keep our models as simple as possible and neglect otherfactors like the possible effects of complicated rheology
of mantle materials, the process of planetary formation,
the boundary condition on the core–mantle boundary,and internal heating.

Model
We only briefly describe the model and basic equations
here; the readers are referred to our previous studies formore detail (Miyagoshi et al. 2014, 2015, 2017).
We calculated a thermal convection of an infinite Prandtl
number fluid with a temperature-dependent viscosity
in a two-dimensional rectangular box of the aspect ratio
four under the truncated anelastic liquid approximation(TALA). All of the boundaries are shear stress free andimpermeable. The surface temperature Ts* and the bottom
temperature Tb* are both fixed, and the normalized surface

Page 2 of 8

temperature Ts*/(Tb* − Ts*) is fixed at 0.1. (The asterisks standfor dimensional quantities). The sidewalls are insulating.
The employed non-dimensional momentum equation is−∇p + ∇ · τ + Ra · ρ̄(z) · α(z) · T · ez = 0,




2
t
τij = η(T ) ∇u + ∇u ij − ∇ · uδij ,3(1)∇ · [ρ(z)u] = 0,
and the energy equationDi ∂uidT+ Di · α(z)wT = ∇ 2 T +
τij
.
ρ(z)
dt
Ra ∂xj(3)(2)

the mass conservation equation,(4)

The length scale for the normalization is the mantle
depth d*; the temperature scale is the temperature difference ∆T* ≡ Tb* − T*s; the time scale is d*2/κ* where κ* is thethermal diffusivity. ρ (z) and α(z) are the depth-dependent
reference density and thermal expansivity, respectively;
η(T) is the viscosity; δij is the Kronecker’s delta; z is theheight measured from the bottom boundary; p is nonhydrostatic pressure; u and ez are the fluid velocity vector
and the vertical unit vector; w is the vertical component of
fluid velocity; t is time; Ra is the Rayleigh number definedby Ra = ρ*0 α*0 g*ΔT*d*3/η*0 κ*; Di is the dissipation number
defined by Di = α*0 g*d*/Cp* . Here, ρ*0 and α*0 are the density
and thermal expansivity at the surface, respectively, g* is thegravity, Cp* is the specific heat, and η*0 is the viscosity on thebottom boundary that is chosen as the reference viscosity.
The viscosity η depends on the temperature asη = η∗ /η0∗ exp[ln (r)(Tb − T )] where r is the viscosity contrast between the top and bottom boundaries; we fixed r
at ­105 regardless of Mp, in order to focus on the effects ofadiabatic compression. The value of r is chosen so that thelithosphere develops along the cold surface of the planet.
We do not take account for the pressure-dependent viscosity, because the physical properties under very highpressure expected in massive super-Earths are still not wellknown. For example, Karato (2011) suggests that such ahigh pressure may reduce rather than increase the viscosity. As shown in Table 1, the thermal expansivity decreases
with the hydrostatic pressure that is calculated from thedepth asp∗ (z) = (127.4Mp + 7.241)(1 − z)(5)(Tachinami et al. 2014; Miyagoshi et al. 2014, 2015, 2017).
The reference density depends on z as[GPa]ρ(z) = 1 + [1.75(Mp )0.258 − 1](1 − z).

(6)
The basic equations are discretized by a finite difference method. We employed the ACuTEMAN numericalcode (Kameyama 2005; Kameyama et al. 2005) for theMiyagoshi et al. Earth, Planets and Space(2018) 70:200

Table 1 The pressure dependence
expansivity assumed in the modelofPage 3 of 8

thethermalPressure (GPa)Thermalexpansivity­(10−5K−1)

0

4.0

10

3.5

30

2.6

60

1.86

100

1.38

150

1.10

300

0.701

600

0.37641000

0.30341500

0.2587

calculation. The employed mesh is uniform and contains
1024 times 256 grid points.
We carried out numerical calculations for planetswith Mp= 1, 2, 4, 6, and 9.4. The corresponding values of(Ra, Di) are (5 × 107, 0.87), (1.3 × 108, 1.49), (3.2 × 108,2.57), (5.5 × 108, 3.52), and (1.0 × 109, 5.00), respectively. In these estimates, we assume that the modeled planet has a similar chemical composition with the
Earth’s and that g* and d* depend on the planetary mass∗ = g M 0.5
gnormalized by
theEarth’s
mass
M
as
⊕
p
p


0.28
and d ∗ = d⊕ Mp, respectively (Valencia et al. 2006,2007), where the subscript ⊕ stands for the Earth’s value; 1.34,
Ra thus calculated depends on Mp as Ra = Ra⊕ Mp
 0.78. Here,while Di depends on Mp as Di = Di⊕ Mp

Di⊕ = 0.87 and Ra⊕ = 5 × 107, as estimated with
Cp = 1.3 × 103 J/kg/K, α0 = 4 × 10−5 ­K−1, g⊕ = 9.8 m/
s2, and d⊕ = 2900 km. We fixed the temperature contrast
across the mantle ΔT* at 3000 K in the estimates of Ra;
the effects of variation in Ra due to a variation in ΔT*
on numerical results can be readily estimated from theparametrized relationships presented in Tachinami et al.
(2014) and Miyagoshi et al. (2015). We also fixed the nondimensional surface temperature Ts at 0.1, since Tachinami et al. (2014) has already investigated its effects on
numerical results for iso-viscous models.
We shortly note the validity of TALA in the numerical model of mantle convection in the presence of strong
compression. Compared with its untruncated counterpart (ALA), the effect on buoyancy of dynamic pressure
is neglected in TALA. Such a truncation can be safely
done in the mantles of terrestrial planets particularly
with large mass, because the dynamic pressure is muchsmaller than the static one in the planetary interiors. Thishas been already confirmed from numerical results with
Mp= 10 in Kameyama and Yamamoto (2018).

Results
The left column of Fig. 1 shows snapshots of the distribution of potential temperature Tp(x,z) (color) and fluid
velocity (arrows) we obtained at various Mp at the statistically steady state, while the right column shows the horizontal average Tp of the potential temperature Tp(x,z).
The potential temperature is calculated from the temperature T(x,z) as shown in Fig. 21α(ξ )dξ ,Tp (x, z) = T (x, z) exp −Diz(7)(Miyagoshi et al. 2014, 2015, 2017). We present Tp rather
than T in Fig. 1, because Tp is more directly related to thebuoyancy force that drives thermal convection.
At Mp= 1, hot plumes ascending from the core–mantle boundary are as prominent as the cold plumes thatdescend from the top cold thermal boundary layer (TBL),
and the convective velocity around the cold plumes iscomparable to that around the hot plumes. The heads ofthe hot plumes reach the base of the top TBL.
Hot ascending plumes become, however, less prominent as Mp increases: At Mp= 2, the convective flow
induced by hot plumes is still an important element of
the convection in the mantle. The excess temperature of
the heads of hot plumes with respect to the surrounding
mantle is, however, smaller than that calculated at Mp= 1
shown in Fig. 1a. At Mp= 4 and 6, the excess temperatureof hot plume heads becomes even smaller. Hot plumes
become faint except around the core–mantle boundary,and the heads are often detached from the stems of hot
plumes before they ascend to the base of the top TBL.
The excess temperature of hot plumes rapidly decreases
as they ascend, while cold descending plumes are prominent at all depths. At Mp= 9.4, hot plumes become
almost invisible, while cold descending plumes are stillconspicuous. The effects of adiabatic compression ontemperature contrast between plumes and the surrounding mantle are more conspicuous for hot plumes thanfor cold plumes, because the effect is proportional to thetemperature (Eq. 4).
Figure 2 shows snapshots of the temperature distribution from which the potential temperature shownin Fig. 1 is calculated. Cold plumes are conspicuous at
all Mp. Hot plumes, in contrast, become fainter as Mp
increases and are almost invisible at Mp= 4 or larger,
although the structures can be slightly observed in thepotential temperature distribution.
Figure 3a shows the normalized thickness of the top
TBL, or the lithosphere, plotted against Mp. The baseof the lithosphere is located at the depth level where
Tp − Ts becomes 98% of Tpm − Ts: Tpm is the local maximum of Tp in the shallow mantle (Miyagoshi et al. 2015).

Miyagoshi et al. Earth, Planets and Space

Mp=1

a

(2018) 70:200

Page 4 of 8

Tp(x,z) 0.6651

0.6955

1

b

z
0
0.1

0.6

Tp

d

c

Mp=2

0.58900.6194e

Mp=40.52240.5528fg

Mp=60.49450.5249hi

Mp=9.40.47320.5036j(|V| = 3000)
Fig. 1 The left column shows snapshots of the distribution of the non-dimensional potential temperature Tp(x,z) (color) and fluid velocity (arrows)
at various Mp. Tp is normalized with the temperature unit of ΔT* = 3000 K. The color scale is changed from panel to panel so that Tp of the isothermalcore of the convection is represented by green color. See the sample arrow beneath the panel (i) for the non-dimensional magnitude of thevelocity. The right column shows the horizontally averaged Tp(x,z), Tp (the red lines). The dashed black lines show the location of the base of thelithosphere (for details, see the text)

The height of the base of the lithosphere thus defined isshown in the right column in Fig. 1 by the black dashedlines. The figure shows that the normalized thickness of
the lithosphere is almost independent of Mp. This behavior is different from that found in the earlier models

where the Boussinesq approximation is employed: Under
this approximation, the flux of convective heat transport
q, which is approximately proportional to the inverse of
the normalized thickness of the lithosphere, is known
to depend on the Rayleigh number as q ∝ Ra0.258 (e.g.,Miyagoshi et al. Earth, Planets and Space

a

Mp=1

b

Mp=2

c

Mp=4

(2018) 70:200

Page 5 of 8

d Mp=6e Mp=9.4

0.1

T(x,z)

1.1

Fig. 2 Snapshots of the temperature distribution (color) from which the potential temperature presented in Fig. 1 is calculatedMiyagoshi et al. Earth, Planets and Space

Non-dimensional thicknessof the lithospherea(2018) 70:200

Page 6 of 8

◀ Fig. 3 a The plot of L (the thickness of the lithosphere normalized

by the mantle depth) against Mp. The dotted line shows the plot thatis expected from the Boussinesq approximation L(Mp = 1) · Mp−0.346
where L(Mp = 1) is the normalized thickness at Mp= 1 (for details,
see the main text). b The dependence of the thickness of thelithosphere h = Ld* on Mp, whered* is the mantle depth that 0.28
depends on Mp as d ∗ = d⊕ Mp; the thickness is normalized
by its value at Mp= 1. The dashed line shows the dependence
suggested earlier from the Boussinesq approximation (Valencia
et al., 2007); the thickness depends on Mp as Mp−0.45 in their models.
c The dimensional root-mean-square average velocity V*rms of theconvection plotted against Mp. The dashed line shows the relationship∗ (M = 1) · M1.19 suggested in Valencia et al. (2007). (Here,of Vrmspp
V*rms(Mp= 1) is the value of V*rms at Mp= 1.)

0.6

0.4

0.2

0

1

2

4

6

8

10

Mp

Thickness of the lithosphere(normalized by that at Mp=1)

b
2.0

1.0

0

1

2

4

6

8

10

Mp

c

*
Vrms
(×10−1[cm/yr])
42effect of adiabatic compression is taken into account.
(In the figure, the lithosphere at Mp= 1 is much thicker
than that we usually observe for the Earth, because
we neglected internal heating and plate motion in ourmodels.)When converted
to dimensional quantity by using
0.28
d ∗ = d⊕ Mp, which is described in “Model” section, the thickness of the lithosphere (= d* times thenormalized thickness shown in Fig. 3a) increases withincreasing Mp, as shown in Fig. 3b (again, the thickness
is normalized by the value at Mp= 1), despite that a larger
Mp implies a thicker mantle, and hence a higher Rayleighnumber. The lithosphere becomes about twice thicker, as
the planetary mass becomes 9.4 times larger. For comparison, we show the dependence of the dimensional
thickness on Mp expected in an earlier parameterized
convection model where the Boussinesq approximation
is employed (Valencia et al. 2007); here the thickness isexpressed as M−0.45(the dashed line). When the effectspof adiabatic compression are considered, the lithospherebecomes thicker rather than thinner with increasing Mp.
We also present a plot of the dimensional root-mean*square velocity of the convection Vrms
= Vrms κ*/d* (here*−62
κ = 10 ­[m /s]) against Mp. The
non-dimensionalaverage
Vx2 + Vz2 avevelocity Vrms is given as Vrms =,time_ave

0

1

2

4

6

8

10

Mp

Deschamps and Sotin 2000) where Ra depends on Mp as
M1.34
p , as derived in “Model” section. Thus, the normalized thickness depends on Mp as L(Mp = 1) · Mp−0.346
under the Boussinesq approximation, where L(Mp = 1)
is the normalized thickness at Mp= 1. This dependence isshown by the dotted line in Fig. 3a. The plot shows thatthe lithosphere becomes substantially thicker, when thewhere Vx and Vz are the velocity vector component of horizontal and vertical directions,  ave means spatially averaged, and  time_ave means time averaged. As shown in the*
figure, Vrms
is almost independent of Mp, when the effectof adiabatic compression is taken into account. Although
the non-dimensional Vrms increases by a factor of about 2
as Mp increases from 1 to 9.4 (see the arrows in Fig. 1), this*increase in Vrms is canceled out in Vrms
by the increase in d*due to increasing Mp. In contrast, in the Boussinesq*
approximation model (Valencia et al. 2007), the Vrms
rap∗
idly increases with increasing Mp as Vrms (Mp = 1) · Mp1.19**
where Vrms
(Mp= 1) is the value of Vrms
at Mp= 1 (see thedashed line in Fig. 3c).

Miyagoshi et al. Earth, Planets and Space(2018) 70:200

Page 7 of 8

Discussion
In our numerical models, the effects of adiabatic compression on thermal convection become substantial in
the mantle of super-Earths, when the mass Mp exceeds
a threshold value Mc about 4. Hot ascending plumes
become less prominent as Mp increases and cannot directly ascend to the base of the lithosphere in aplanet with Mp larger than Mc ~ 4. As a consequence,
the flows induced by cold descending plumes become
the major component of the convective flow at large
Mp. The thickness of the lithosphere normalized by the
mantle depth is also affected by adiabatic compression.
The normalized thickness becomes almost independent of Mp, and hence the dimensional thickness of thelithosphere increases with increasing Mp, as shown in
Fig. 3b.
The hot plumes lose their buoyancy as they ascend in
the mantle of planets with a large Mp, because the temperature contrast between a hot plume and the surrounding mantle δΤ decreases bydδT= −α(z)Di · δT ,dz

(8)owing to adiabatic decompression, as the plume ascends.
(See Eq. (7).) At Mp= 2 (Di = 1.49), hot plumes ascend
to the base of the lithosphere that is at the height z of ≈0.68; the coefficient α(z)Di is about 0.59 there, according
to Table 1 and Eq. (5). At Mp= 4 (Di = 2.57), however, thesame value of the coefficient is reached at z ≈ 0.58, which
is well below the base of the lithosphere. At larger Mp,
the same value of the coefficient α(z)Di is reached evenat lower z. The lower level of z is the reason why the hot
plumes stop ascending before they reach the base of thelithosphere at Mp equal to 4 or greater.
An estimate
 1  of the average dissipation number
Di = Di 0 α α0 dz also leads to the same conclusion.
Di is 0.44, 0.57, 0.69, 0.74, 0.79 at Mp= 1, 2, 4, 6, and 9.4,
respectively; Di at Mp= 4 is already about 90% of that at
Mp= 9.4. The rather large value of Di at Mp > 4 is also thereason why the lithosphere becomes thicker (Fig. 3b) andthe root-mean-square average velocity does not increase(Fig. 3c) with increasing Mp at Mp > 4 in spite of the larger
Ra at larger Mp. The stronger effects of adiabatic compression counteract the effects of higher Ra at large Mp.
The role of adiabatic compression in controlling
the thickness of lithosphere is also readily concluded
from the comparison with results in our earlier work
(Miyagoshi et al. 2015). When both r and Di are fixed, theNusselt number increases as Ra increases (see Figure 8 of
Miyagoshi et al. (2015)), indicating the decrease in nondimensional thickness of the lithosphere. (It also showsthat its non-dimensional thickness is not determinedby only r). In our present model where both Ra and Di

increase with Mp, in contrast, its non-dimensional thickness does not decrease for given r. The difference fromthe results of Miyagoshi et al. (2015) clearly shows thatthe almost constant non-dimensional thickness of thelithosphere for the values of Mp employed here (Fig. 3a)
is caused by the counteraction between the increase in
Ra and that in Di with Mp.
The thick lithosphere (Fig. 3b) and the inactive convection (Fig. 3c) observed at large Mp imply that it is necessary to revisit the issue of plate tectonics on super-Earths.
In the literature, the stress induced in the lithosphereby mantle convection has been estimated and has been
compared with the rupture strength of the lithosphere tosee if plate tectonics operates in super-Earths (e.g., Valencia et al. 2007; Valencia and O’Connell 2009). In theseestimates, the stress is proportional to V*rms/h* where h*is the thickness of the lithosphere. Since V*rms increases
and h* decreases with increasing Mp under the Boussinesq approximation, it has been concluded that the stressin the lithosphere exceeds its rupture strength, and thatplate tectonics takes place more readily in super-Earths.
When the adiabatic compression is taken into account,however, h* increases with increasing Mp, and V*rms isalmost independent of Mp, as shown in Fig. 3; the figuresuggests that the stress decreases as Mp increases, andthat plate tectonics may be more difficult to operate inmassive super-Earths.
The low activity of hot ascending plumes at large Mp
shown in Fig. 1 contains an implication for the volcanism on super-Earths, too. On the Earth, hot plumes are
known to cause hot spot volcanism like the one observed
at Hawaii and large igneous provinces (LIP) that causes
rifting of plates (e.g., Coffin and Eldholm 1994). However, the lowered activity of hot plumes due to the effectof adiabatic compression in our results suggests that suchvolcanism as the one observed on the Earth is not important in super-Earths, when Mp exceeds around 4.
Of course, we neglected many factors that may affectthermal convection to fix the model presented here. As
has been already discussed, the vigor of mantle convection measured by the Nusselt number significantly
depends on the surface temperature in massive superEarths Tachinami et al. (2014). The dependence of viscosity on pressure is also important (e.g., Tackley et al.,2013). Besides, we neglected internal heating and highpressure induced phase changes such as the post-perovskite transition. Further numerical studies with theseeffects are necessary to predict with more confidence thethreshold in Mp above which the effects of adiabatic compression becomes important.
In summary, our numerical models suggest that thestrong effects of adiabatic compression make both plate
tectonics and volcanism caused by hot ascending plumes

Miyagoshi et al. Earth, Planets and Space(2018) 70:200

difficult to take place in super-Earths with Mp larger thanabout 4 under our model settings. To confirm this conclusion, further numerical experiments are necessary in
the future, taking account of the complicated rheology
of mantle materials (e.g., pressure- or stress-dependent
viscosity), internal heating and the long-lasting effects of
the initial condition on the thermal state of the mantle(Miyagoshi et al. 2017).
Authors’ contributions
TM performed numerical simulations, analysis of simulation data, andprepared the manuscript. MK developed the numerical simulation code
ACuTEMan and prepared the manuscript. MO prepared the manuscript. All
authors discussed numerical simulation results. All authors read and approved
the final manuscript.
Author details
1
Department of Deep Earth Structure and Dynamics Research, Japan Agency
for Marine-Earth Science and Technology, 3173‑25 Showa‑machi, Kanazawa‑ku, Yokohama, Kanagawa 236‑0001, Japan. 2 Geodynamics ResearchCenter, Ehime University, 2‑5 Bunkyo‑cho, Matsuyama, Ehime 790‑8577,
Japan. 3 Department of Earth Sciences and Astronomy, University of Tokyoat Komaba, 3‑8‑1 Komaba, Meguro, Tokyo 153‑8902, Japan.
Acknowledgements
The authors thank two anonymous reviewers for their helpful comments.
Numerical simulations were performed on the Earth Simulator and the supercomputer system at Japan Agency for Marine-Earth Science and Technology.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Data supporting the figures in this article are available upon request to theauthors.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
This work was supported by JSPS KAKENHI Grant Numbers JP25287110,
JP15H05834, a joint research program at Geodynamics Research Center,
Ehime University, and “Exploratory Challenge on Post-K computer” (Elucidationof the Birth of Exoplanets [Second Earth] and the Environmental Variations ofPlanets in the Solar System).

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 16 July 2018 Accepted: 11 December 2018

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