Miyagoshi et al. Earth, Planets and Space (2017) 69:46
DOI 10.1186/s40623-017-0630-6

Open Access

FULL PAPER

Extremely long transition phaseof thermal convection in the mantle of massive
super‑Earths
Takehiro Miyagoshi1* , Masanori Kameyama2 and Masaki Ogawa3

Abstract
Adiabatic compression is a key factor that exerts control over thermal convection in the compressible solid mantle ofsuper-Earths. To discuss the effects of adiabatic compression, we present a numerical model of transient convectionin the cooling mantle of a super-Earth that is ten times larger in size than the Earth. The calculations started with theshallow mantle that was hotter than expected by the extrapolation from the deep mantle conditions. This type of
initial thermal state of the mantle is expected to naturally occur in real super-Earths due to heating by giant impacts
at the time of their formation. With our initial setup conditions, the convection temporarily occurs as a layered convection for the first several to ten billion years of the calculation and then changes its style into a whole layer convection. The long duration of the transient stage suggests that mantle convection currently occurs as a temporal layered
convection in many of the super-Earths. A temporal layered convection, if it occurs, can exert control over the tectonic
activities of super-Earths. Future studies should clarify how internal heating and complicated rheological properties ofmantle materials including their pressure dependence affect the duration of the temporal layered convection.
Keywords: Super-Earths, Thermal convection, Mantle, Thermal evolution
Introduction
Dynamics of the mantle in super-Earths is a key issue
in studying their thermal history and surface environments. Stimulated by the detection of a large number ofsuper-Earths, which are extrasolar terrestrial planets witha mass of up to ten times the Earth’s (e.g., Borucki et al.
2011), many theoretical and numerical studies have been
conducted to explain the structure and dynamics of theirinterior. In many of these studies, mantle convection ismodeled by the parameterized convection method (e.g.,
Valencia and O’connell 2009; Stamenković et al. 2011)or by a steady convection of a Boussinesq fluid where
the effect of adiabatic compression is neglected (e.g.,van Heck and Tackley 2011; Foley et al. 2012; Stein et al.
2013). Our studies (Miyagoshi et al. 2014, 2015; Tachinami et al. 2014; Kameyama et al. 2015), however, report*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 adiabatic compression significantly affects thermal
convection in massive super-Earths by greatly reducing
the efficiency of convective heat transport and also by
reducing the activity of hot plumes ascending from thebottom boundary of the mantle. A simple extrapolationof mantle convection in the Earth and other planets ofour solar system to super-Earths is unreliable.
In the course of these numerical studies of mantle convection in super-Earths, we noticed that the nature of
the thermal convection substantially changes with time,as the convection approaches the statistical steady state,
and that the period of the transitional stage becomes aslong as ten billion years based on the parameter values ofmassive super-Earths; the length of the transitional stage
is comparable to the age of the universe. This finding suggests that the evolution of super-Earths may be strongly
dependent on their initial conditions determined by theprocess of planetary formation. Therefore, caution may
be necessary when numerically simulating mantle evolution in super-Earths. In their numerical models of statistically steady thermal convection of a temperature- and© The Author(s) 2017. 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 (2017) 69:46

pressure-dependent viscosity fluid that yields at highstress, Tackley et al. (2013) found that plume-like hot
regions arise from the mid-mantle and that subducting
slabs stop sinking there. Noack and Breuer (2014) conclude from their calculation of similar models of convection that evolves with time over a 2.5-Gyr time span thatthe pressure dependence of viscosity critically affects theoperation of plate tectonics in super-Earths. Given thelong time span needed for the effects of the initial conditions to disappear, caution is necessary to apply thesenumerical results to super-Earths.
The initial temperature distribution in super-Earths
is determined by the formation process of the planets.
In the first runaway growth stage of the formation process of a planet (e.g., Wetherill and Stewart 1989), planetesimals collide into the surface of the protoplanet andthe surface and the shallow mantle are strongly heatedby released gravitational energy. At the end of the lateroligarchic growth stage (e.g., Chambers 2001; Kokubo
and Ida 1998), giant impacts induce strong shock waves
within the planets to raise the temperature most prominently at the surface and the core-mantle boundaries
where there is a strong density increase that causes shock
wave reflection (e.g., Canup 2004). In both stages, thetemperature in the shallower mantle is likely to become
higher than that in the deep mantle, and the excess temperature of the shallow mantle can be as high as several
thousands of degrees [Fig. 4 of Canup (2004)]. The excess
temperature may survive the temperature change in the
mantle subsequently caused by core formation and heattransport in the magma ocean.
In this study, we address the issue of the potential
excess temperature in the shallow mantle exerting control over the subsequent thermal evolution of massive
super-Earths. For this purpose, we present an exampleof transient mantle convection we encountered in our
numerical studies of a massive super-Earth to discuss
how the nature of thermal convection can change inthe course of its approach to the statistical steady stateand on what timescale the change occurs (note that
Miyagoshi et al. (2015) reports only the convections intheir statistical steady states).

Model description
Since the model and basic equations adopted here are
the same as those implemented in our previous studies(Miyagoshi et al. 2014, 2015), we only briefly summarize
them here. The compressible Navier–Stokes equation
with the infinite Prandtl number, the mass conservation
equation, and the heat transport equation are solved ina two-dimensional rectangular box with an aspect ratioof four under the truncated anelastic liquid approximation using the numerical simulation code ACuTEMan

Page 2 of 10(Kameyama 2005; Kameyama et al. 2005). The absolutetemperature is fixed at Ts and Tb at the top and bottom boundaries, respectively, while there is no heat
flux on the sidewalls. We neglected the possible effects
of decreasing Tb with time due to core cooling on the
dynamics of the compressible mantle, as further discussed in "Discussion" section. All boundaries are shear
stress-free and impermeable.
The viscosity strongly depends on temperature:(1)η = ηb exp [E(Tb − T )]where ηb is the viscosity at the bottom boundary and E isa constant, called the Frank-Kamenetskii approximation
parameter. We consider the temperature dependence ofviscosity in the calculations because it is the agent thatinduces the lithosphere, a prominent feature of mantle
convection. This approximation gives a fairly good estimate of the temperature-dependent viscosity beneaththe stagnant lithosphere, as discussed in Appendix Aof Reese et al. (1999). We ignore, however, the pressure
dependence of viscosity, considered in earlier models(e.g., Tackley et al. 2013), to make the effects of adiabatic
compression conspicuous (see "Discussion" section forfurther discussion). We set the ratio in viscosity between
the top and bottom boundaries(2)�η = exp [E(Tb − Ts )]6

at 10 .
The basic equations in their non-dimensional forms are   
∇ ∗ · ρ ∗ z ∗ u∗ = 0,(3)   −∇ ∗ p∗ + ∇ ∗ · τ ∗ + Ra∗ ρ ∗ z ∗ α ∗ z ∗ T ∗ ez = 0,(4)

τij∗ = η∗ T ∗



 


2 ∗ ∗
∇ u + ∇ u ij − ∇ · u δij ,
3
∗ ∗

t

∗ ∗





  ∗ ∗
 ∗  dT ∗∗∗ ∗+ Di · α z w Tρ zdt ∗Di∗ ∂u∗= ∇ ∗2 T ∗ + ∗ τij∗ ∗i .
Ra
∂xj

(5)

∗

(6)

The quantities with asterisks are non-dimensional.
Here, u and ez are the fluid velocity and the vertical unit
vector, p is non-hydrostatic pressure, w is the verticalcomponent of fluid velocity, and δij is Kronecker’s delta.
These equations are non-dimensionalized by the lengthscale of d (the depth of the mantle), the timescale of d2/κ(where κ = 10−6 m2/s is the thermal diffusivity), and thetemperature scale of ∆T ≡ Tb − Ts. Here, T* = T/ΔT is
the normalized absolute temperature rather than therelative temperature (T − Ts)/ΔT that is often adopted
in Boussinesq models. The viscosity η, the thermalMiyagoshi et al. Earth, Planets and Space (2017) 69:46

Page 3 of 10

expansivity α, and the density ρ are normalized by theirvalues at the surface boundary ηs = ηbΔη, α0, and ρ0,
respectively. The time derivative in Eq. (6) is Lagrangian.
Hereinafter, the values are non-dimensional unless otherwise stated and we drop the asterisks. The units of normalization are summarized in Table 1. The mantle depth
d is calculated from the scaling
law suggested by Valen 0.28
cia et al. (2007) as d = d⊕ Mp
where d⊕ is the mantle depth of the Earth (~2900 km) and Mp (=10 in thispaper) is the planetary mass normalized by the Earth’smass.
In the basic equations, we assume a simple linear
dependence of the reference density ρ on the depth 1 − z(z is the height from the bottom), 0.258
ρ(z) = 1 + 1.75 Mp− 1 (1 − z),(7)based on the earlier models of mantle structure (Valencia et al. 2006). On the other hand, the thermal expansivity α depends on the depth as shown in Fig. 1a. The
dependence is the same as the one in our previous studies (Tachinami et al. 2014; Miyagoshi et al. 2014; 2015); αdecreases by a factor of 10 from the surface to the bottom
boundary. This profile is estimated for MgO in a former
ab initio calculation (Tsuchiya and Tsuchiya 2011) and isconsistent with the previous laboratory measurements
under high pressure (Chopelas and Boehler 1992).
There are three free parameters in the formulation:
the Rayleigh number Ra = ρ0α0gΔTd3/ηsκ where g is thegravitational acceleration; Ts/∆T is the normalized surface temperature; Di = α0gd/Cp is the dissipation number where Cp is the specific heat. The numerical results
depend on the surface temperature Ts/ΔT in our models
in contrast to Boussinesq models since the magnitude ofthe effects of adiabatic compression is proportional tothe absolute temperature [see the second term of the lefthand side of Eq. (6)]. The normalized bottom temperature is calculated from Ts/ΔT as Tb/ΔT = Ts/ΔT + 1 andhence is not an independent free parameter.
We adopt Ra = 104, Ts/∆T = 0.1, and Di = 5, whichcorrespond to a planet of ten times the Earth’s mass.

Table 1 The scaling unit for non-dimensionalization
The scaling unit

SymbolValue

Length (mantle depth)d5500 (km)Thermal diffusivityκ

10−6 (m2/s)

Time

d2/κ

3.0 × 1019 (s) ~ 950 Gyr

Velocity

κ/dThermal expansivity

α0

Temperature∆T−4

5.7 × 10(cm/year)4 × 10−5 (K−1)
3000 (K)

a

z 1.0
0.8
0.6
0.4
0.2
0

0

1.0

0.5

α

b

z 1.0
0.8
0.6
0.4
0.2
0

0.2

0.4

0.6

0.8

1.0

Tave

Fig. 1 a The plot of the normalized thermal expansion coefficient α(horizontal axis) against height z (vertical axis). b The solid line shows
the horizontally averaged initial temperature profile plotted against
height z. As for the dashed line, see the text

These values are the same as the values for Case 5 inTable 1 of Miyagoshi et al. (2015). Note that the freeparameters are all defined with the surface values ofmaterial properties. The Rayleigh number becomes
1010 when it is defined with the viscosity at the bottom
boundary, as calculated from the assumed value of Δη.
The magnitude of the effects of adiabatic compression
is proportional to α*(z*)Di rather than Di at depth z, as
shown in Eq. (6), and the effective dissipation number,i.e., the dissipation number averaged over the entire mantle becomes 0.79:
 1
Di = Diαdz.
(8)0This value is close to the one assumed in an earlier model(Tackley et al. 2013).
We started the calculation from the initial temperature profile illustrated by the solid line in Fig. 1b. We
obtained this initial temperature profile as follows: WeMiyagoshi et al. Earth, Planets and Space (2017) 69:46

Page 4 of 10

first assumed that the non-dimensional temperature
is constant at (Ts + Tb)/2 = 0.6 over the entire mantle except along the boundaries (see the dotted line in
Fig. 1b) and started to calculate convection. Then, we
uniformly heated the mantle, until the average temperature of the whole mantle became 1.05 times that at theequilibrium state. Starting from this temperature profile,
we observed the evolution of thermal convection in thecooling mantle.

Results
In this section, we present the overall structure of thermal convection and its development with time in Figs. 2
and 3. Figure 2 shows the distribution of temperature andconvective flow (left column) and the plots of horizontally averaged temperature against height (right column)
at each time interval. Figure 3 shows the distribution ofpotential temperature Tp(x, z) calculated from Fig. 2 as
1α(ξ )dξ .

Tp (x, z) = T (x, z) exp −Diz(9)

We present the potential temperature as in Tachinami
et al. (2014) and Miyagoshi et al. (2015) because it is more
relevant to mantle convection than the temperature itself.
Temperature variation caused by adiabatic compression does not drive the mantle convection; nevertheless,it overshadows the potential temperature variation thatdrives the mantle convection at large Di values assumed
here (Fig. 2).
At the initial transient stage (panels a–d in Figs. 2, 3),
the convection is layered. In the upper layer, convection
starts at t ∼ 0.156 × 10−2 when the thermal boundary
layer (TBL) along the surface boundary becomes sufficiently thick owing to the conductive cooling from theboundary; cold plumes begin to descend from the TBL.
Plume heads take a time on the order of 0.9 × 10−2 tosink to the layering boundary (panels a–d in Figs. 2, 3).
In contrast, hot ascending plumes have been prominent
in the lower layer since the beginning of the calculation, as shown in Fig. 3 (the panels a–d); low viscositythat results from high temperature (Fig. 2) is the reason
for the higher plume activity. Indeed, the fluid velocity ishigher in the lower layer (see the arrows in Fig. 2). The
heads of the hot plumes ascend to the layering boundary at the height of z ~ 0.5 where the temperature of theplume head decreases to the ambient temperature owing
to adiabatic decompression. The boundary hardly moves
throughout the initial stage.
Figures 3 and 4 show why convection is layered in thisstage. Although the temperature in the upper layer islower than that in the lower layer (Fig. 2), the potential
temperature in the upper layer is higher than that in thelower layer. The difference in the potential temperature
between the two layers decreases with time owing to thecooling from the surface boundary and heating from thebottom boundary, but the upper layer is still slightly hotter until t ∼ 1.2 × 10−2.
The initial stage of convection ends at t ∼ 1.2 × 10−2
(about 11.4 billion years in dimensional quantity). After
that, cold plumes penetrate through the layering boundary and the convective structure changes to a whole layer
one (panel f in Figs. 2, 3). The distribution of horizontally
averaged temperature becomes close to the adiabatic
one in the entire convecting mantle (from panels g to l
in Fig. 2). Whole layer convection is the convective flow
pattern that we obtained as the statistical steady state in
Miyagoshi et al. (2015).
Effects of the transition from layered convection towhole layer convection are observed in the plot of theroot-mean-square (RMS) fluid velocity, VRMS, as shown in
Fig. 5b. The VRMS rapidly increases at t around 1.2 × 10−2
and remains high after that. In contrast, the transition
does not noticeably affect the heat flux at the surface(Fig. 5a). The surface heat flux gradually decreases withtime because of the thickening of the surface TBL. The
heat flux at the surface is larger than that at the bottom
boundary after t = 0.004, and the mantle cools duringthat period of time as a whole. After the transition, thegradual cooling from the surface continues, until theheat flux at the bottom increases and that at the surface
decreases to the common equilibrium value of 3.5.
Figure 6 presents the nature of the transition fromlayered convection to whole layer convection in
more detail. In this figure, we show −(dTave/dz) (redcurves), the adiabatic temperature gradient calculated
from −(dTave/dz)ad = α · Di · Tave (blue curves), andthe plots of the horizontally averaged RMS velocity
Vx2 (x, z) + Vz2 (x, z) ave_x , where Tave is
VRMS_h (z) =the horizontally averaged temperature, Vx and Vz are thex- and z-components of the velocity, and 〈〉ave_x means
the average in x (horizontal) direction. All of the quantities are plotted against height during the period when
the convection is layered. The left column shows thatthe two temperature gradients, −(dTave/dz) and −(dTave/dz)ad, coincide with each other in the lower layer whereplumes are prominent throughout the initial transient
stage. In the upper layer, however, the two gradients substantially deviate from each other. The black dashed lines
in the left column show the depth where the difference[−(dTave/dz)ad] − [−(dTave/dz)] takes its maximum value.
The descent of cold plume heads is most strongly impededat this depth, and VRMS_h takes its local minimum value.
Indeed, the dashed lines indicate the maximum depth ofcold descending plume heads. The transition from layered
convection to whole layer convection occurs when theMiyagoshi et al. Earth, Planets and Space (2017) 69:46

Page 5 of 10

t = 0.156E-2

a

z

1

0
0.1

g

1.1

Tave

b

t = 0.605E-2hct = 0.793E-2idt = 1.05E-2

jet = 1.20E-2

kf

t = 1.34E-2l= 40000.1

1.1

Fig. 2 Snapshots of the calculated temperature distribution (left column) and its horizontal average Tave plotted against height z (right column)
taken at time t = 0.156E-2 (panels a and g), 0.605E-2 (panels b and h), 0.793E-2 (panels c and i), 1.05E-2 (panels d and j), 1.20E-2 (panels e and k),
1.34E-2 (panels f and l)Miyagoshi et al. Earth, Planets and Space (2017) 69:46

Page 6 of 10

at = 0.156E-2bt = 0.605E-2ct = 0.793E-2

dt = 1.05E-2

et = 1.20E-2

f

t = 1.34E-2

0.47

0.50

Fig. 3 Snapshots of the potential temperature distributions that
correspond to the temperature distributions shown in Fig. 2, taken
at time t = 0.156E-2 (panel a), 0.605E-2 (panel b), 0.793E-2 (panel c),
1.05E-2 (panel d), 1.20E-2 (panel e), 1.34E-2 (panel f)

depth indicated by the dashed line reaches the boundary
at the height of z ~ 0.5 (Fig. 3).

Discussion
Our numerical models show that in the mantle of manyof massive super-Earths, a layered convection can temporarily occur and continue for several billion years if
the shallow mantle is hotter than expected by the adiabatic extrapolation from deep mantle as a consequenceof the formation process of the planets. The thermaland dynamic state of the interior may be in the transient

stage, even today, in super-Earths. The transient-layered
convection, if it occurs, would substantially affect thestyle of tectonic activities on the planet. The occurrenceof plate tectonics on super-Earths may not be solely
determined by the planet’s size and surface environment(e.g., Valencia et al. 2007; Valencia and O’Connell 2009;
Korenaga 2010; van Heck and Tackley 2011; Foley et al.
2012; Stein et al. 2013; Stamenković and Breuer 2014). It
may also depend on whether or not the convection is in
its initial transient stage.
Because of its potential effects on tectonic evolution, it
is important to understand how long initial layered convection can continue in a super-Earth. In our numerical model, the transition to the whole layer convection
occurs when the cold plume’s head descends to the layering boundary (Fig. 3). The cold plume heads further
penetrate into the deep mantle because their potential
temperature is lower than the average potential temperature of the deep mantle and hence the cold plumeheads are denser than the average lower mantle materials
when they are brought into the lower mantle. Note thatthe average potential temperature in the shallower mantle is still slightly higher than that of the deep mantle andthat the mantle is still stably stratified as a whole at the
moment of the transition. Although the convective instability at the base of the lithosphere takes some time, on
the order of 10−3 (Fig. 3a) to induce cold plumes, most of
the duration of the layered convection (0.012) is spent forcold plumes to move downward to the layering boundary. Figure 6 suggests that the duration is scaled with theoverturn time of mantle convection in the shallow mantle: In the shallow mantle, the RMS average velocity VRMS
is about 100 (see Fig. 6). The overturn time of convection
τ defined by ds/VRMS where ds is the depth of the shallow mantle beneath the lithosphere (about 0.3, Fig. 6) is,
therefore, 0.003, and the duration of the layered convection is 4τ.
The duration would be shorter at a higher Rayleighnumber because of the shorter turnover time of convection. On the contrary, a higher surface temperaturesuggested for some super-Earths (Léger et al. 2009) may
extend the duration, because a higher surface temperature reduces the vigor of the convection driven by coolingfrom the surface boundary and hence makes the turnovertime of convection longer (Tachinami et al. 2014).
There are many other factors that can affect the duration but are neglected for simplicity in this study. Oneof such factors is internal heating. The duration would
be shorter if the deep mantle contains an internal heatsource that reduces the excess temperature of the shallow mantle (i.e., the potential temperature in theshallow mantle minus that in the deep mantle). One possible internal heat source is radioactive elements. If theMiyagoshi et al. Earth, Planets and Space (2017) 69:46

Page 7 of 10

z 1.0

0.5

0
0.1

0.3

0.5

Tp_aveFig. 4 Development of horizontally averaged potential temperature
Tp_ave against height z. The red, green, blue, purple, light blue, andorange lines show the plots taken at t = 0, 0.156 × 10−2, 0.605 × 10−2,
0.793 × 10−2, 1.01 × 10−2, and 1.34 × 10−2, respectively

amount of content of radioactive elements is equal to thatof the Earth, these elements would raise the temperature
by about 2000 K over 4.5 Gyr (e.g., Turcotte and Schubert
2014). This temperature increase is a substantial fractionof the excess temperature of the shallow mantle expected
to be induced by a giant impact on the Earth (e.g., Canup,2004) and may not be negligibly small even for a superEarth. Tidal heating due to the central star can be, however, more important. In an extrasolar terrestrial planet
GJ876d (Jackson et al. 2008) for which the semi-major
axis of the orbit is 0.0208 AU, for example, the tidal heating is suggested to be equivalent of 104–105 W m−2 surface heat flux, about 105 times the surface heat flux of the
Earth. In such a planet, the duration of the transient-layered mantle convection would be negligibly short in its
evolutionary history.
The potential influence of the bottom thermal boundary condition on convection is also worth discussing.
We assumed that the temperature is fixed at the bottomaboundary; however, the temperature may decrease withtime in super-Earths because of core cooling. Given the
low efficiency of convective cooling of the compressible
mantle of super-Earths found in Miyagoshi et al. (2015),
the amount of the temperature decrease is probably small
and is unlikely to substantially affect the duration of thetransient-layered convection.
It is difficult to predict the effects of the pressure- or
depth-dependent viscosity, which are neglected in thisstudy, on the duration. A previous one-dimensionalparameterized convection model (Stamenković et al.
2012) suggests that pressure-dependent viscosity makes
the duration of the transient stage longer. The mixing
length theory (Tachinami et al. 2011) and two-dimensional statistically steady models (Tackley et al. 2013)
suggest that high viscosity in the deep mantle due toits pressure dependence or the post-perovskite transition reduces the activity of convection in contrast to ourmodel suggesting that a vigorous convection occurs fromthe beginning of the calculated time span. The suggested
lower activity of the deep convection layer would affectthe duration of the transient-layered convection. Whether
the duration becomes longer or shorter, however, depends
on the internal heating rate and other factors.
Can a lateral thermal heterogeneity overlaid on theinitial temperature distribution affect the subsequenttransient-layered convection? When a giant impact
occurs, a large lateral contrast arises in the temperature
distribution of the mantle (e.g., Canup 2004). In the past
numerical studies of Mars (Roberts et al. 2009) and Mercury (Roberts and Barnouin 2012), an initial temperature
anomaly caused by a giant impact laterally spread in only
10–100 Myr. This characteristic time is shorter than theturnover time of mantle convection expected for superEarths, and the influence of initial lateral heterogeneities
is probably small.

bheat flux

t

VRMS

t

Fig. 5 a The average heat flux at the surface of the planet (red) and the core-mantle boundary (blue), both plotted against time. b The RMS-averaged velocity plotted against timeMiyagoshi et al. Earth, Planets and Space (2017) 69:46

Page 8 of 10

t = 0.156E-2
z 1.0

z 1.0

0.5

0.5

0
-1

2

8

5

0

0

400

800

-(dTave/dz), -(dTave/dz)ad

1200

VRMS_ht = 0.414E-2
1.0

1.0

0.5

0.5

0

0
0

2

4

6

400

800

1200

400

800

1200

300

600

900

t = 0.605E-2

1.0

1.0

0.5

0

0

0.5

0

2

4

6

0

0

t = 0.793E-2
1.0

1.0

0.5

0.5

0

0

2

4

6

0

0

Fig. 6 The gradient of horizontally averaged temperature −(dTave/dz) (red curves), the adiabatic temperature gradient calculated from −(dTave/dz)ad = α · Di · Tave (blue curves, both left column), and the horizontally averaged RMS velocity (right column), all plotted against height z at various
timesMiyagoshi et al. Earth, Planets and Space (2017) 69:46

Conclusions
The numerical example presented in this study showsthat the thermal state of the mantle can take geologic
time to approach its statistical steady state in massive
super-Earths because of the effects of adiabatic compression: When the shallow mantle is hotter than expected by
the adiabatic extrapolation from the deep mantle, mantle
convection temporarily occurs as a layered convection at
the beginning of the calculation. It takes approximately
ten Gyr in the model until this transient-layered convection yields to the whole mantle convection that occurs
as the statistically steady state (see Figs. 2, 4). The transition in the convective flow pattern occurs when the cold
plume’s head in which the potential temperature is lower
than the average potential temperature of the deep mantle descends to the layering boundary. The depth of thelayering boundary is determined by hot plumes ascending from the bottom boundary. Mantle convection thatcurrently occurs in many of massive super-Earths may
be transient that may significantly affect the style of tectonic activities on these planets; whether or not plate
tectonics operates, for example, is not solely determinedby the size of the planet or properties of its mantle
materials.
Authors’ contributions
TM performed numerical simulations and the analysis of simulation dataand prepared 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 would like to thank two anonymous reviewers and the editor 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. This work was supported by JSPS KAKENHI Grant
Numbers JP25287110, JP15H05834 and a joint research program at Geodynamics Research Center, Ehime University
Competing interests
The authors declare that they have no competing interests.
Received: 25 November 2016 Accepted: 18 March 2017

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