SEDSolid Earth DiscussionsSEDSolid Earth Discuss.1869-9537Copernicus GmbHGöttingen, Germany10.5194/sed-7-3817-2015A simple 3-D numerical model of thermal convection in Earth's
growing inner core: on the possibility of the formation of the
degree-one structure with lateral viscosity variations YoshidaM.myoshida@jamstec.go.jphttps://orcid.org/0000-0001-6069-5818Department of Deep Earth Structure and Dynamics Research, Japan
Agency for Marine-Earth Science and Technology (JAMSTEC), 2–15
Natsushima-cho, Yokosuka, Kanagawa 237-0061, JapanM. Yoshida (myoshida@jamstec.go.jp)15December201574381738414December20156December2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://se.copernicus.org/preprints/7/3817/2015/sed-7-3817-2015.htmlThe full text article is available as a PDF file from https://se.copernicus.org/preprints/7/3817/2015/sed-7-3817-2015.pdf
An east-west hemispherically asymmetric structure for Earth's inner
core has been suggested by various seismological evidence, but its
origin is not clearly understood. Here, to investigate the
possibility of an “endogenic origin” for the degree-one
thermal/mechanical structure of the inner core, I performed new
numerical simulations of thermal convection in the growing inner
core. A setup value that controls the viscosity contrast between the
inner core boundary and the interior of the inner core, ΔηT, was taken as a free parameter. Results show that the
degree-one structure only appeared for a limited range of ΔηT; such a scenario may be possible but is not considered
probable for the real Earth. The degree-one structure may have been
realized by an “exogenous factor” due to the planetary-scale
thermal coupling among the lower mantle, the outer core, and the
inner core, not by an endogenic factor due to the internal
rheological heterogeneity.
Introduction
After the segregation of the rocky mantle and molten iron core in the
early stage of Earth's formation (e.g., Stevenson, 1981), the inner
core was formed by gradual solidification of the molten iron core and
the size increased with age (Jacobs, 1953; Buffett et al., 1992) (for
example, see reviews by Buffett, 2000 and Sumita and
Yoshida, 2003). The solidification of the solid core could affect the
vigor of outer-core convection owing to the release of latent heat and
the passage of light elements toward the liquid outer core (e.g.,
Sumita and Yoshida, 2003). The resulting growth of the inner core may
have changed the convection style in the outer core and affected the
intensity of Earth's magnetic field throughout Earth's history.
Although the structure of the present inner core cannot be inferred
from surface geophysical observations, various seismological evidence
suggests that the inner core has an east-west, hemispherically
asymmetric structure in terms of seismic velocity, anisotropy, and
attenuation (Tanaka and Hamaguchi, 1997; Creager, 1999; Niu and Wen,
2001; Cao and Romanowicz, 2004; Deuss et al., 2010; Irving and Deuss,
2011; Waszek et al., 2011; Lythgoe et al., 2014) (see also a recent
review by Tkalčić, 2015).
Previous numerical simulations of Earth's mantle convection clarified
that for convecting rocky materials confined in a spherical shell, the
spherical harmonic degree-one structure was observed for a relatively
wide range of the parameter that controls the lateral viscosity
variations due to temperature variations (McNamara and Zhong, 2005;
Yoshida and Kageyama, 2006). This is because when the
temperature-dependence of viscosity is moderate, the highly viscous
lid that develops at the surface of the convecting mantle has the
longest-wavelength scale, and the dynamic instability at the bottom of
the lid concentrates in one area. This scenario characterizes the
degree-one thermal structure of mantle convection that lies between
the “mobile-lid regime” with weakly temperature-dependent viscosity
and the “stagnant-lid regime” with strongly temperature-dependent
viscosity (Yoshida and Kageyama, 2006).
It is possible that the degree-one seismic structure in the present
inner core originated from lateral temperature variations, which in
turn originated from lateral viscosity variations. Even given the
uncertainties in the rheological properties and composition of the
inner core materials, lateral viscosity variation offers considerable
potential as an “endogenic factor” that may explain the formation of
the degree-one seismic structure. It is therefore worth examining
whether a degree-one thermal/mechanical structure generated from the
lateral viscosity variations can be realized for solid materials
confined in a sphere. This topic had not been investigated in previous
numerical simulation models of inner core convection (Deguen and
Cardin, 2011; Cottaar and Buffett, 2012; Deguen, 2013; Deguen
et al., 2013).
In this study, to explore the time-dependent behavior of the
convection regime in Earth's growing inner core and the possibility of
generating the degree-one thermal/mechanical structure from the
internal rheological heterogeneity, a new simple numerical model of
the growing inner core is constructed and a series of numerical
simulations of thermal convection are performed, assuming that the
solidification of the liquid core started at 4.5 Ga and that
the radius of the inner core gradually increased with the square root
of age.
Model
Convection in the inner core is computed numerically using a staggered
grid-based, finite-volume code, ConvGS (e.g., Yoshida, 2008). The
material of the inner core is modeled as a Boussinesq fluid with an
infinite Prandtl number confined in a sphere and modeled in spherical
coordinates (r, θ, φ). Impermeable,
shear-stress-free, isothermal conditions are imposed on the inner core
boundary (ICB) with a fixed dimensionless radius of rc′=1 (Fig. 1a). The driving force of convection is primordial heat alone
because the radiogenic heat production is negligibly small in the
inner core (e.g., Karato, 2003). The number of computational grids is
taken as 64(r)×64(θ)×192(φ)×2
(for Yin and Yang grids) (Yoshida and Kageyama, 2004). To avoid the
mathematical singularity at the Earth's center, an extremely small
sphere with a dimensionless radius of rδ′=10-6 is
imposed at the center of the model sphere.
Following standard techniques for mantle convection simulations (e.g.,
Schubert et al., 2001), the length L, velocity vector v,
stress tensor (or pressure) σ, viscosity η, time t, and
temperature T are non-dimensionalized as follows:
L=rc(t)L′,v=κ0rc(t)v′,σ=κ0η0rc(t)2σ′,η=η0η′,t=rc(t)2κ0t′,T=ΔT0⋅T′
where rc(t) denotes the time-dependence of the radius of
the inner core, which depends on time (age); κ0 denotes the
reference thermal diffusivity; η0, the reference viscosity;
ΔT0, the reference temperature difference; and the
subscript “0” refers the reference values for the inner core
(Table 1). In these equations, symbols with primes represent
dimensionless quantities.
Using these dimensionless factors, the dimensionless conservation
equations for mass, momentum, and energy, which govern inner core
convection, are expressed as
∇⋅v=0,-∇p+∇⋅τ+Ra(t)ζ(t)-3Ter=0,τ=η∇v+∇vtr∂T∂t+v⋅∇T=∇2T+H(t)ζ(t)-2,
respectively, where p represents the pressure; τ, the
deviatoric stress tensor; and er, the unit vector
in the radial direction. Primes representing dimensionless quantities
are omitted in Eq. (–).
In the numerical simulation for this study, instead of fixing the
dimensionless radius of the ICB, the radius of the inner core in the
thermal Rayleigh number, Ra, the internal heat-source number, H, and
the “spherical-shell ratio”, ζ, depend on age. They are given
by
Ra(t)≡ρ02α0ΔT0g0cp0rct-rδ3k0η0,H(t)≡Ωρ0rct-rδ2MmΔT0k0,ζ(t)≡rct-rδrct,
where ρ0 is the reference density; α0, the reference
thermal expansion coefficient; ΔT0, the reference
temperature difference across the inner core; g, the reference
gravitational acceleration; cp0, the reference specific heat at
constant pressure; k0, the reference thermal conductivity; and
Mm, the mass of the mantle (Table 1). Thermal
conductivity has received a lot of attention in recent mineral physics
studies, and the main finding is that it may be much larger than the
value of 36 Wm-1K-1 from Stacey and Davis (2008),
i.e., 100–200 Wm-1K-1 (de Koker et al., 2012; Pozzo
et al., 2012, 2014). Therefore, two end-member models with k0=36 and 200 Wm-1K-1 are investigated in this study.
The amount of approximated primordial heat that has existed since
Earth's formation, Ω, is taken as a free parameter estimated
from the heat released by mantle cooling in the present Earth
(Turcotte and Schubert, 2014):
Ω=-43πre3ρecpe0dTdtt=t0,
where re is the radius of Earth; ρe,
the average density of Earth; and cpe0, the average
specific heat at constant pressure for Earth. Here
dT/dtt=t0 is the present
cooling rate of the mantle (Turcotte and Schubert, 2014):
dTdtt=t0=-3λRTm2Ea,
where λ is the average radiogenic decay constant in the
mantle; R, the gas constant, Tm, the mean mantle
temperature; and Ea, the activation energy of dry
olivine. Using the values in Table 1, Ω is 11.3 TW, which is
a reasonable value compared with the total heat release by mantle
cooling estimated from a global heat-flow balance (Lay et al., 2008).
Following the work of Buffett et al. (1992), who studied an analytical
model for solidification of the inner core, the global heat balance of
the outer core is
4π3rb3-rc3ρcpdTsrdt=4πrc2fit-4πrb2fmt,
where Ts(r) is the “solidification temperature”, and
fi(t)(≡Fi(t)/(4πrc2)) and
fm(t)(≡Fm(t)/(4πrb2)) are the heat fluxes across the ICB and the
core-mantle boundary (CMB), respectively (Fig. 1b). The potential
temperature in the well-mixed liquid outer core is assumed to be
spatially uniform and slowly decreases with time, and the ICB is
assumed to be in thermodynamic equilibrium with the surrounding
liquid. Under these assumptions, the temperature through the outer
core is uniquely defined by the solidification temperature,
Ts(r), as a function of pressure or depth (see the
explanation in Buffett et al., 1992 for details). Here, the heat
sources associated with solidification of the inner core (i.e., the
release of latent heat and gravitational energy) are ignored because
these effects play a secondary role in the growth of the inner core
(Buffett et al., 1992).
According to Eq. () in Buffett et al. (1992), the radial
dependence of Ts is expressed as
Ts(r)=Ts(0)-2π3Gρ02r2∂Ts∂p,
where G is the gravitational constant, and ∂Ts/∂p is the solidification profile (Table 1).
Substituting Eq. () into Eq. (), an expression for
the radius of the inner core is obtained:
rc(t)=rb14πrb2N∫0tFm(t)dt12,
where the model constant N is expressed as
N=2π9rb3cp0ρ02G∂Ts∂p.
Assuming that the heat flux across the CMB is constant throughout
Earth's history (Buffett et al., 1992), a model constant,
Fm′, is obtained:
Fm′=4πNtrc-rci2,
where rci is an arbitrary value that represents the
initial radius of the inner core at the beginning of the
simulation. When the age of Earth's core is assumed to be
4.5 Ga (Lister and Buffett, 1998), the solidification of the
inner core is assumed to have begun at this age, and
rci is taken to be 21.5 km so that
rc-rci is exactly 1200 km,
Fm′ is 2.56×1012W, which
would be a lower limit value for the real Earth considering an even
relationship between the total plume buoyancy flux observed at the
Earth' surface and the inferred total CMB heat flow (e.g., Davies,
1988; Sleep, 1990; Davies and Richards, 1992). Equation ()
indicates that the relationship between the total CMB heat flow and
the radius of the growing inner core is Fm′∝rc2, which means that the radius of the inner
core is proportional to the square root of
Fm′. It should be noted that there is
a trade-off between the choices of Fm′ and
rci, which are critical for identifying the age of
the present-day inner core in the real Earth. In the present model, I
set rci to a significantly small value to see the
behavior of inner core convection over the longest geological time,
i.e., 4.5 Gyr (see Sect. 4 for discussion).
Finally, the time-dependent radius of the growing inner core used in
the present simulation is expressed as
rc(t)=Fm′4πNt‾12+rci,
where the dimensional time is scaled as
t‾=(r‾c(t)2/κ0)⋅t′ using the radius of the inner core at the previous time
step of the simulation, r‾c(t).
Equation () means that rc increases with the
square root of age, and eventually reaches rc=1221.5km after 4.5 Gyr, which matches the present
radius of Earth's inner core.
At the beginning of the simulation, the initial condition for
dimensionless temperature with significant small-scale lateral
perturbations is given as
T′(r,θ,ϕ)=0.5+ω⋅Y3417θ,ϕ⋅sinπr1′-r′r1′,
where Yℓm(θ, ϕ) is the fully normalized
spherical harmonic function of degree ℓ and order m and ω(=0.1) is the amplitude of perturbation.
The viscosity of the inner core materials, ηT′, in
this model depends on temperature according to a dimensionless
formulation (Yoshida, 2014):
ηT′=η0′⋅exp2Tave′(t)E′T′+Tave′(t)-E′2Tave′(t),
where Tave′(t) is the dimensionless average
temperature of the entire sphere at each time step. A model parameter,
E′, controls the viscosity contrast between the ICB with
T′=0 and the interior of the inner core. In the present
model, E′=ln(ΔηT) varies from ln(100)=1 (i.e., no laterally variable viscosity) to ln(105)=11.51.
Resutls
Figure 2 shows the time evolution of the convection pattern in the
inner core for models with η0=1017Pas,
k0=36Wm-1K-1, and with ΔηT=100 (Fig. 2a–d), ΔηT=103 (Fig. 2e–h), ΔηT=103.5 (Fig. 2i–l), and ΔηT=104
(Fig. 2m–p). When ΔηT=100 (i.e., the viscosity of
the inner core is homogeneous) the convection pattern kept
a short-wavelength structure over almost all of the simulation time
and the “present” inner core at 0.0 Ga has numerous
downwellings uniformly distributed in the sphere (Fig. 2d). On the
other hand, when ΔηT is large enough to make the surface
thermal boundary layer stagnant (ΔηT≥104), the
convection pattern tends towards a short-wavelength structure with
age, and secondary cold plumes from the bottom of the highly viscous
lid are evenly distributed in the sphere (Fig. 2p). A remarkable
change in the convection pattern is found for moderate values of
ΔηT: when ΔηT=103 and 103.5,
the convection patterns shift towards a long-wavelength structure with
increasing age, and eventually the longest-wavelength thermal
structures develop for the inner core at 0.0 Ga (Fig. 2h
and l)
To quantitatively assess the variations in thermal and mechanical
heterogeneities in the inner core with time, Fig. 3 shows the power
spectra of the temperature and root-mean-square velocity fields
throughout the modelled inner core at each time step. On the other
hand, when ΔηT is ≤102.5, it is found that the
scales of the thermal and mechanical heterogeneities generally tend to
shorten with increasing age, although long-wavelength structures
develop just after the beginning of the simulation (“A” in
Fig. 3a and c) in spite of an initial temperature condition with
a short-wavelength structure (Eq. ). When ΔηT
is moderate (i.e., ΔηT=103 and 103.5) it
appears that the scales of the thermal and mechanical heterogeneities
generally tend to shorten with increasing age before
c. 2.0 Ga, but the degree-one structure begins to develop
after 1.0 Ga (see “B” in Fig. 3e–h).
Even when η0=1016Pas and k0=36–200 Wm-1K-1, these conclusions remain
essentially unchanged: the degree-one thermal/mechanical structure
only appeared for a limited range of parameter values for lateral
viscosity variations, i.e., ΔηT=104 and 104.5
for the model with k0=36Wm-1K-1 (“A” in
Fig. 4) and ΔηT=103 and 103.5 for the model
with k0=200Wm-1K-1 (“A” in Fig. 5). These
results imply that the degree-one structure is found in the models
with a wide range for the Rayleigh number, as also shown in mantle
convection simulations (McNamara and Zhong, 2005; Yoshida and
Kageyama, 2006).
This degree-one convection pattern is similar to the familiar
“sluggish-lid regime” or the “transitional regime” in thermal
convection of the mantle that has already been found in numerical
simulations of mantle convection (e.g., Solomatov, 1995) (Fig. 6b). In
this regime, the flow velocities of downwelling plumes from the
sluggish-lid are large compared with the interior of the inner core,
and the global flow pattern in the sphere corresponds to the
temperature distribution. This is because temporal changes in thermal
heterogeneity roughly correlate with those in mechanical
heterogeneity, as shown in Figs. 3, 4 and 5. Also, once the degree-one
structure is formed, the sluggish-lid regime generates the largest
magnitude of flow velocity and the most laterally heterogeneous
velocity field in the inner convecting region (Figs. 6b and 7a) when
compared with other regimes such as the mobile-lid regime (Fig. 6a)
and the stagnant-lid regime (Fig. 7b).
When ΔηT is 104 or larger, the scales of the
thermal and mechanical heterogeneities under the stagnant-lid, whose
thickness is approximately 100 km, are quite small (the
dominant degrees are below 16) even after 4.5 Gyr
(Fig. 3i–l). Although the relatively long-wavelength mode is
intermittently dominant during the simulation even when ΔηT is quite large (“C” in Fig. 3k–l), it is immediately
damped over a short time-scale of ≤ c. 0.5 Gyr. As
a result, the stagnant-lid regime is maintained for almost the entire
simulation time.
Conclusions and discussion
The systematic numerical simulations conducted in this study
investigated the possibility of the formation of a degree-one
structure with lateral viscosity variations in thermal convection
within the age of Earth for a simulated inner core confined in
a sphere. The degree-one thermal/mechanical structure, however, only
appeared for a limited range of parameter values for lateral viscosity
variations. Considering the uncertainties in the exact magnitude of
lateral temperature variations, the rheology, and the composition of
Earth's inner core materials, the formation of a degree-one structure
with lateral viscosity heterogeneity under the limited geophysical
conditions confirmed here would be considered possible but not
probable for the real Earth. Namely, the degree-one structure of the
inner core may have been realized by an “exogenous factor” from
outside the ICB, rather than by an “endogenic factor” due to the
internal rheological heterogeneity.
An exogenous origin for the hemispherically asymmetric structure of
the inner core and the related hemispherical difference in the degree
of crystallization, i.e., freezing and melting, of the inner core
materials is consistent with a previous suggestion that
planetary-scale thermal coupling among the lower mantle, the outer
core, and the inner core plays a primary role in the growth and
evolution of the inner core (Aubert et al., 2008; Gubbins
et al., 2011; Tkalčić, 2015). More recently, a new isotopic
geochemical analysis study (Iwamori and Nakamura, 2015; Iwamori
et al., 2015) revealed that such planetary-scale thermal coupling
operates in the whole-Earth system through so-called “top-down
hemispheric dynamics”, in which the hemispheric supercontinent-ocean
distribution at Earth's surface controls the thermal convection system
in the whole Earth via the mantle, outer core, and inner core over the
course of Earth's history (see Fig. 13 of Iwamori and Nakamura, 2015).
It should be noted that the style of convection envisioned in this
study is only one form of degree-one convection. Another form involves
melting and solidification processes at the ICB, as suggested by
Alboussière et al. (2010) and Monnereau et al. (2010), who
concluded that the inner core translates laterally as a rigid body and
the return flow effectively occurs in the fluid outer core. Because
the present study adopts impermeable boundary conditions at the ICB,
this second form of degree-one convection is not permitted. This
scenario provides a strong alternate candidate for the form of
degree-one convection through top-down hemispheric dynamics, if
degree-one convection due to internal rheological heterogeneity is not
possible.
The age of the inner core is one of the most controversial issues in
Earth Science. As mentioned in Sect. 2, the value of the total heat
flow at the CMB is a key parameter that controls the speed of growth
of the inner core. A previous study (Cottaar and Buffett, 2012) has
shown that the minimum heat flow for inner core convection is
4.1 TW, giving a maximum inner core age of
1.93 Ga. Heat flow greater than 6.3 TW leads to the
present convection regime, but the inner core age is then less than
1.26 Ga. Following their analysis, if the total CMB heat flow
is c. 10 TW (Lay et al., 2008 and references therein), the age
of inner core should be younger (i.e., less than 1 Ga). If the
value of the total CMB heat flow is significantly larger than that
used in this study, the possibility of the formation of degree-one
structure by an endogenic factor due to rheological heterogeneity
becomes less likely, because it takes c. 3.0 Gyr to form the
degree-one structure when η0=1017Pas,
k0=36Wm-1K-1, and ΔηT is 3.5
(Fig. 3g). At the very least, the conclusion that the convective
motion in the inner core is maintained even for the present Earth can
be drawn from the models studied here.
Seismic observations provide evidence for a seismic velocity
discontinuity about 200 km below the ICB separating an
isotropic layer in the uppermost inner core from an underlying
anisotropic inner core (e.g., Song and Helmberger, 1998). The results
presented here may imply that this “inner core transition zone”
represents the boundary between a sluggish, highly viscous, cold layer
and an underlying hot convection region (Figs. 6b and 7a). Sumita and
Yoshida (2003) predicted that there may be a characteristic structure
in the topmost section of the inner core that is similar to the plate
tectonic mechanism at Earth's surface, which is explained by the
existence of a crust-like, thin, low-degree partial-melting layer and
an underlying asthenosphere-like, high-degree partial-melting
layer. The ICB is, by definition, at melting temperature, which means
that it is possible that the viscosity could be lowest at the top and
gradually increase with depth into the interior of the inner
core. However, because the temperature across the inner core is never
very far from the melting temperature (Stacey and Davis, 2008),
depending in detail on the contribution from light element impurities,
viscosity variations in the underlying hot convection region should be
small.
In future, the evolution of the inner core should be resolved by
numerical simulations of the whole-Earth thermal convection system
because it is highly possible that the growth rate of the inner core
is determined by heat flow at the CMB, which largely depends on the
behavior and style of mantle convection (Buffett et al., 1992; Sumita
and Yoshida, 2003). The combined effects of the thermal and
compositional buoyancies (Lythgoe et al., 2015) and the effects of
non-uniform heat flux boundary condition at the ICB, rather than
a fixed temperature condition on the style and regime of inner core
convection, should also be studied numerically in future.
Acknowledgements
Some figures were produced using the Generic Mapping Tools (Wessel
et al., 2013). The calculations presented herein were performed
using the supercomputer facilities (SGI ICE-X) at JAMSTEC. The
present study was supported by a Grant-in-Aid for Exploratory
Research from the Japan Society for the Promotion of Science (JSPS
KAKENHI Grant Number 26610144). All of the simulation data are
available from the corresponding author upon request.
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Physical parameters for the simulation used in this study.
SymbolDefinitionValueUnitRefs.α0Thermal expansion coefficient19.74×10-6K-1a, bcp0Specific heat at constant pressure17.03×102Jkg-1K-1ag0Gravitational acceleration14.4002ms-2aρ0Density11.27636×104kgm-3ak0Thermal conductivity136 or 200W m-1K-1a, c, d, eκ0Thermal diffusivity14.01×10-6m2s-1aη0Viscosity11017 or 1016Pa sf–Temperatures at the inner core boundary and Earth's center5000 and 5030KaΔT0Temperature difference across the inner core130KaMmMass of the mantle4.043×1024kggτcFormation age of the core4.5Gahcpe0Average specific heat at constant pressure for Earth9.2×102Jkg-1K-1gρe0Average density of Earth5.520×103kgm-3gTmMean mantle temperature2250KgEaActivation energy for dry olivine540×103Jmol-1g, iλAverage decay constant for the mixture of radioactive isotopes in the mantle2.77×10-10yr-1gdT/dtt=t0Present cooling rate of the mantle64.7KGyr-1Eq. ()GGravitational constant6.67384×10-11m3kg-1s-2gRGas constant8.3144621Jmol-1K-1g∂Ts/∂pSolidification profile7×10-9KPa-1j, kreRadius of Earth6.371×106mgrbRadius of the outer core3.48×106mlrc(t)Radius of the inner coreTime-dep.m–rcRadius of the present inner core1.2215×106mlrciInitial radius of the inner core2.15×104m2ΩModel constant (see text)1.13×1013WEq. ()NModel constant (see text)2.01×1016J m-2Eq. ()Fm′Model constant (see text)2.56×1012WEq. ()Dimensionless parameters Ra(t)Thermal Rayleigh numberTime-dep.–Eq. ()H(t)Internal heat-source numberTime-dep.–Eq. ()ζ(t)Spherical-shell ratio numberTime-dep.–Eq. ()
Definition: 1 indicates reference values for the inner core. References:
a Stacey and Davis (2008); b Vočadlo et al. (2003); c de Koker et al. (2012); d
Pozzo et al. (2012); e Pozzo et al. (2014); f Karato (2003); g Turcotte and Schubert
(2014); h Lister and Buffett (1998); i
Karato and Wu (1993); j Verhoogen (1980);
k Buffett et al. (1992); l Dziewonski and Anderson (1981). 2 indicates arbitrary values (see text).
(a) Illustration for the numerical model of inner
core convection. (b) Schematic profile of the potential
temperature in the cooling core of the Earth used in an analytical
model for solidification of the inner core (after Buffett et al.,
1992). The solid line represents the temperature in the upper part
of the thick solid inner core, the liquid outer core, and the lower
mantle at some instant. The thick dashed line represents the
subsequent evolution of temperature as heat is continuously
extracted from the core. Heat fluxes fm and fi
pertain to the core-mantle and inner-core boundaries (CMB and ICB),
respectively. The solidification temperature
Ts(rc) is defined at the ICB, and
Ts(r, t) is the temperature within the inner core.
Time evolution of the convection pattern in the inner core
for models with η0=1017Pas, k0=36Wm-1K-1, and (a–d)ΔηT=100, (e–h)ΔηT=103,
(i–l)ΔηT=103.5, and (m–p)ΔηT=104. Blue and copper isosurfaces indicate
regions with lower and higher than average temperatures at each
depth. (a–d and m–p) blue: -0.3 K;
copper: +0.3 K. (e–f and i–k) blue:
-0.6 K; copper: +0.6 K. (g, h
and l) blue: -1.5 K; copper: +1.5 K.
Temporal changes in the heterogeneity of temperature and
root-mean-square velocity fields in the inner core for the models
with η0=1017Pas, k0=36Wm-1K-1, and (a–b)ΔηT=100, (c–d)ΔηT=102,
(e–f)ΔηT=103, (g–h)ΔηT=103.5, (i–j), ΔηT=104,
and (k–l)ΔηT=105. The logarithmic
power spectra are normalized by the maximum values at each elapsed
time. The four black wedges on the panels (a),
(e), (g), and (i) represent the ages that
correspond to Fig. 2.
Temporal changes in the heterogeneity of temperature and
root-mean-square velocity fields in the inner core for the models
with η0=1016Pas, k0=36Wm-1K-1, and (a–b)ΔηT=100, (c–d)ΔηT=102,
(e–f)ΔηT=103, (g–h)ΔηT=103.5, (i–j), ΔηT=104,
and (k–l)ΔηT=105. The logarithmic
power spectra are normalized by the maximum values at each elapsed
time.
Temporal changes in the heterogeneity of temperature and
root-mean-square velocity fields in the inner core for the models
with η0=1016Pas, k0=200Wm-1K-1, and (a–b)ΔηT=100, (c–d)ΔηT=102,
(e–f)ΔηT=103, (g–h)ΔηT=103.5, (i–j), ΔηT=104,
and (k–l)ΔηT=105. The logarithmic
power spectra are normalized by the maximum values at each elapsed
time.
Cross sections of temperature and velocity fields for the
models with η0=1017Pas, k0=36Wm-1K-1, and (a)ΔηT=100 and (b)ΔηT=103 at
0.0 Ga. The top panels show the cross sections at the
central depth of the inner core (i.e., radius of 610.8 km)
and bottom panels show the cross sections cut along the great
circles shown by dashed lines in the top panels. The contour
interval in the temperature plots is 3 K. These figures
correspond to (a) Figs. 2d and 3a, and (b)
Figs. 2h and 3e.
Cross sections of temperature and velocity fields for the
models with η0=1017Pas, k0=36Wm-1K-1, and (a)ΔηT=103.5 and (b)ΔηT=104 at
0.0 Ga. The top panels show the cross sections at the
central depth of the inner core (i.e., radius of 610.8 km)
and bottom panels show the cross sections cut along the great
circles shown by dashed lines in the top panels. The contour
interval in the temperature plots is 3 K. These figures
correspond to (a) Figs. 2l and 3g, and (b)
Figs. 2p and 3k.