thermal convection in earth’s inner core with phase change at its

8/10/2019 Thermal Convection in Earths Inner Core With Phase Change at Its

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arXiv:1306.248

2v1

[physics.geo-ph]11Jun2013

Geophys. J. Int.(0000)000, 000000

Thermal convection in Earths inner core with phase change at its

boundary

Renaud Deguen1

, Thierry Alboussiere2

and Philippe Cardin3

1Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218, USA.2 LGL, Laboratoire de G eologie de Lyon, CNRS, Universite Lyon 1, ENS-Lyon, Geode,

2 rue Raphael Dubois, 69622 Villeurbanne, France.3 ISTerre, Universite de Grenoble 1, CNRS, B.P. 53, 38041 Grenoble, France.

12 June 2013

SUMMARY

Inner core translation, with solidification on one hemisphere and melting on the other, providesa promising basis for understanding the hemispherical dichotomy of the inner core, as well as

the anomalous stable layer observed at the base of the outer core - the so-called F-layer - whichmight be sustained by continuous melting of inner core material. In this paper, we study indetails the dynamics of inner core thermal convection when dynamically induced melting andfreezing of the inner core boundary (ICB) are taken into account.If the inner core is unstably stratified, linear stability analysis and numerical simulations consis-tently show that the translation mode dominates only if the viscosity is large enough, with acritical viscosity value, of order3 1018 Pa s, depending on the ability of outer core convectionto supply or remove the latent heat of melting or solidification. If is smaller, the dynamicaleffect of melting and freezing is small. Convection takes a more classical form, with a one-cellaxisymmetric mode at the onset and chaotic plume convection at large Rayleigh number. be-ing poorly known, either mode seems equally possible. We derive analytical expressions for therates of translation and melting for the translation mode, and a scaling theory for high Rayleighnumber plume convection. Coupling our dynamical models with a model of inner core ther-mal evolution, we predict the convection mode and melting rate as functions of inner core age,

thermal conductivity, and viscosity. If the inner core is indeed in the translation regime, thepredicted melting rate is high enough, according toAlboussiere et al.(2010)s experiments, toallow the formation of a stratified layer above the ICB. In the plume convection regime, themelting rate, although smaller than in the translation regime, can still be significant if is nottoo small.Thermal convection requires that a superadiabatic temperature profile is maintained in the in-ner core, which depends on a competition between extraction of the inner core internal heatby conduction and cooling at the ICB. Inner core thermal convection appears very likely withthe low thermal conductivity value proposed byStacey & Loper(2007), but nearly impossi-ble with the much higher thermal conductivity recently put forward by Sha & Cohen(2011),de Koker et al.(2012) andPozzo et al.(2012). We argue however that the formation of an iron-rich layer above the ICB may have a positive feedback on inner core convection : it impliesthat the inner core crystallized from an increasingly iron-rich liquid, resulting in an unstablecompositional stratification which could drive inner core convection, perhaps even if the inner

core is subadiabatic.Key words: Instability analysis; Numerical solutions; Heat generation and transport; Seismicanisotropy.

1 INTRODUCTION

In the classical model of convection and dynamo action in Earths

outer core, convection is thought to be driven by a combination of

cooling from the core-mantle boundary (CMB) and light elements

(O, Si, S, ...) and latent heat release at the inner core boundary

(ICB). Convection is expected to be vigorous, and the core must

therefore be very close to adiabatic, with only minute lateral tem-

perature variations(Stevenson 1987), except in very thin, unstable

boundary layers at the ICB and CMB. To a large extent, seismo-

logical models are consistent with the bulk of the core being well-
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2 R. Deguen , T. Alboussiere and P. Cardin

mixed and adiabatic, which supports the standard model of outer

core convection. Yet seismological observations indicate the exis-

tence of significant deviations from adiabaticity in the lowermost

200km of the outer core(Souriau & Poupinet 1991). This layer,sometimes called F-layer for historical reasons, exhibits an anoma-

lously lowVPgradient which is most probably indicative of stablecompositional stratification (Gubbins et al. 2008), implying that the

lowermost 200 km of the outer core are depleted in light elements

compared to the bulk of the core. This is in stark contrast with theclassical model of outer core convection sketched above: in place

of the expected thin unstable boundary layer, seismological models

argues for a very thick and stable layer. Note also that the thickness

of the layer, 200 km, is much larger than any diffusion lengthscales, even on a Gy timescale, which means that if real this layer

must have been created, and be sustained, by a mechanism involv-

ing advective transport.

Because light elements are partitioned preferentially into the

liquid during solidification, iron-rich melt can be produced through

a two-stage purification process involving solidification followed

by melting (Gubbins et al. 2008). Based on this idea,Gubbins et al.

(2008) have proposed a model for the formation of the F-layer in

which iron-rich crystals nucleate at the top of the layer and melt

back as they sink toward the ICB, thus implying a net inward trans-port of iron which results in a stable stratification. In contrast,

Alboussiere et al. (2010) proposed that melting occurs directly at

the ICB in response to inner core internal dynamics, in spite of the

fact that the inner core must be crystallizing on average. Assuming

that the inner core is melting in some regions while it is crystalliz-

ing in others, the conceptual model proposed by Alboussiere et al.

(2010) works as follow : melting inner core material produces a

dense iron-rich liquid which spreads at the surface of the inner

core, while crystallization produces a buoyant liquid which mixes

with and carries along part of the dense melt as it rises. The strat-

ified layer results from a dynamic equilibrium between production

of iron-rich melt and entrainment and mixing associated with the

release of buoyant liquid. Analogue fluid dynamics experiments

demonstrate the viability of the mechanism, and show that a strati-fied layer indeed develops if the buoyancy flux associated with the

dense melt is larger (in magnitude) than a critical fraction ( 80%) of the buoyancy flux associated with the light liquid. This num-

ber is not definitive because possibly important factors were absent

in Alboussiere et al. (2010)s experiments (Coriolis and Lorentz

force, entrainment by thermal convection from above, ...) but it

seems likely that a high rate of melt production will still be re-

quired.

A plausible way to melt the inner core is to sustain dynam-

ically a topography that will bring locally the ICB at a potential

temperature lower than that of the adjacent liquid core, which al-

lows heat to flow from the outer core to the inner core. The melt-

ing rate is then limited by the ability of outer core convection to

provide the latent heat absorbed by melting, and only a significant

ICB topography can lead to a non-negligible melting rate. More

recently,Gubbins et al. (2011) andSreenivasan & Gubbins (2011)

have proposed that localized melting of the inner core might be

induced by outer core convection, but the predicted rate of melt

production is too small to produce a stratified layer according to

Alboussiere et al. (2010)s experiments. Furthermore, it is not clear

that the behavior observed in numerical simulations at slightly su-

percritical conditions would persist at Earths core conditions.

Among the different models of inner core dynamics proposed

so far (Jeanloz & Wenk 1988; Yoshida et al. 1996; Karato 1999;

Buffett & Wenk 2001; Deguen et al. 2011), only thermal con-

meltingcrystallization

dense liquidlayer

plumeslight

V

Figure 1. A schematic representation of the translation mode of the inner

core, with the grey shading showing the potential temperature distribution

(or equivalently the density perturbation) in a cross-section including the

translation direction (adapted fromAlboussiere et al.(2010)).

vection (Jeanloz & Wenk 1988; Weber & Machetel 1992; Buffett

2009; Deguen & Cardin 2011; Cottaar & Buffett 2012) is poten-

tially able to produce a large dynamic topography and associ-ated melting. Thermal convection in the inner core is possible if

the growth rate of the inner core is large enough and its ther-

mal conductivity low enough (Sumita et al. 1995; Buffett 2009;

Deguen & Cardin 2011). One possible mode of inner core ther-

mal convection consists in a global translation with solidification

on one hemisphere and melting on the other (Monnereau et al.

2010; Alboussiere et al. 2010; Mizzon & Monnereau 2013). The

translation rate can be such that the rate of melt produc-

tion is high enough to explain the formation of the F-layer

(Alboussiere et al. 2010). In addition, inner core translation pro-

vides a promising basis for understanding the hemispherical di-

chotomy of the inner core observed in its seismological prop-

erties (Tanaka & Hamaguchi 1997;Niu & Wen 2001;Irving et al.

2009; Tanaka 2012). Textural change of the iron aggregate dur-ing the translation (Monnereau et al. 2010; Bergman et al. 2010;

Geballe et al. 2013) may explain the hemispherical structure of the

inner core. Inner core translation, by imposing a highly asymmet-

ric buoyancy flux at the base of the outer core, is also a promis-

ing candidate(Davies et al. 2013; Aubert 2013) for explaining the

existence of the planetary scale eccentric gyre which has been

inferred from quasi-geostrophic core flow inversions (Pais et al.

2008;Gillet et al. 2009).

However, inner core translation induces horizontal tempera-

ture gradients (see Figure1), and Alboussiere et al. (2010) noted

that finite deformation associated with these density gradients is

expected to weaken the translation mode if the inner core viscosity

is too small. They estimated from an order of magnitude analysis

that the threshold would be at 1018 Pa s. Below this threshold,thermal convection is expected to take a more classical form, with

cold plumes falling down from the ICB and warmer upwellings

(Deguen & Cardin 2011). Published estimates of inner core vis-

cosity range from 1011 Pa s to 1022 Pa s (Yoshida et al.1996; Buffett 1997; Van Orman 2004; Koot & Dumberry 2011;

Reaman et al. 2011, 2012) implying that both convection regime

seem possible.

The purpose of this paper is twofold : (i) to precise under what

conditions the translation mode can be active, and (ii) to estimate

the rate of melt production associated with convection, in particu-

lar when the effect of finite viscosity becomes important. To this


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Thermal convection in Earths inner core with phase change at its boundary 3

aim, we develop a set of equations for thermal convection in the

inner core with phase change associated with a dynamically sus-

tained topography at the inner core boundary (section3). The ki-

netics of phase change is described by a non-dimensional number,

noted Pfor phase change number, which is the ratio of a phasechange timescale (introduced in section2) to a viscous relaxation

timescale. The linear stability analysis of the set of equations (sec-

tion4) demonstrates that the first unstable mode of thermal con-

vection consists in a global translation when Pis small. When Pis large, the first unstable mode is the classical one cell convec-

tive mode of thermal convection in a sphere with an impermeable

boundary (Chandrasekhar 1961). An analytical expression for the

rate of translation is derived in section5.We then describe numeri-

cal simulation of thermal convection, from which we derive scaling

laws for the rate of melt production (section 6). The results of the

previous sections are then applied to the inner core, and used to

predict the convection regime of the inner core and the rate of melt

production as functions of the inner core growth rate and thermo-

physical parameters (section7).

2 PHASE CHANGE AT THE ICBAny phase change at the ICB will release or absorb latent heat,

with the rate of phase change v being determined by the Stefancondition,

sLv= qicb, (1)which equates the rate of latent heat release or absorption associ-

ated with solidification or melting with the difference of heat fluxqicbacross the inner core boundary. Heres is the density of thesolid inner core just below the ICB, and L is the latent heat of melt-ing. The heat conducted along the adiabatic gradient on the outer

core side is to a large extent balanced by the heat flow conducted

on the inner core side, the difference between the two making a

very small contribution toqicb. Convective heat transport in theinner core is small as well. Convection in the liquid outer core is amuch more efficient way of providing or removing latent heat and

qicbis dominated by the contribution of the advective heat flux(, )on the liquid side, which scales as

lcplu, (2)

where is the difference of potential temperature between the in-ner core boundary and the bulk of the core (Figure2), u is a typicalvelocity scale in the outer core, and l and cpl are the density andspecific heat capacity of the liquid outer core in the vicinity of the

inner core boundary (Alboussiere et al. 2010).

We choose as a reference radius the intersection of the mean

outer core adiabat with the solidification temperature curve (Fig-

ure 2), and note h(, ) the distance from this reference to theinner core boundary. At a given location on the ICB, the differ-

ence of potential temperature between the ICB and the outer core is

(, ) = (mp mad)p(, ), wherep(, )is the pressuredifference between the ICB and the reference surface (see Figure

2),mp =dTs/dpis the Clapeyron slope, andmad =dTad/dpisthe adiabatic gradient in the outer core. Taking into account the lo-

cal anomaly of the gravitational potential (due to the ICB topog-raphy and internal density perturbations), we have from hydrostatic

equilibriump = l(gicbh + ), which gives

= (mp mad)lgicb

h+

gicb

, (3)

Ts

Tad

pp

T

p

1

2

Figure 2. Temperature profiles (thick black lines) in the vicinity of the inner

core boundary. Profile1 corresponds to a crystallizing region, while profile

2 corresponds to a melting region. The thin black line is the outer core

adiabatTadand the thin grey line is the solidification temperature profile.

wheregicb is the average gravity level on the surface of the innercore. The surfaceheq(, ) =

/gicb is the equipotential sur-face which on average coincides with the ICB.

If the inner core is convecting, with a velocity field

u(r,,,t) = (ur, u, u), then the total rate of phase changeis

v= ric+ h

t ur, (4)

where ricis the mean inner core growth rate,ric(t)being the inner

core radius. Using Eqs.(2), (3)and (4), the heat balance (1) at theinner core boundary can be written as

ur ric h

t

h+ /gicb

, (5)

where the time scale is

= s L

2l cpl(mp mad) gicbu

. (6)

Withu 104m s1 and typical values for the other parameters(Table1), the phase change timescale is found to be of the orderof103 years, which will turn out to be short compared to the dy-namical time-scale of thermal convection in the inner core ( 1 Myor more). Noting = s l, the viscous relaxation timescale = /( gicb ric) is at most 0.1 My (for = 10

22 Pa s),

small as well compared to the inner core dynamical timescale. We

therefore adopt the hypothesis of isostasy and neglect h/t in (5),the heat transfer boundary condition finally adopted being written

ur ric = h+ /gicb

, (7)

where the unknown proportionality constant in Equation (2) has

been absorbed in, and will be treated as an additional source ofuncertainty.


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3.3 Set of equations

Introducing two new variables,

h= h +

gicb, (37)

p= p + s, (38)

one can write the momentum and entropy equation, together with

the boundary conditions relevant when phase change is allowed be-

tween solid inner core and liquid outer core:

u= 0, (39)

0= p+ s gicb

ric r er+

2u, (40)

D

Dt =2 +S(t), (41)

with boundary conditions atr = ricfrom (31), (32),(33), (36)and(7) :

= 0, (42)

r =

r

r

ur

+

1

r

ur

= 0, (43)

r =

r

ru

r

+ 1

r sin

ur

= 0, (44)

gicbh 2urr

+ p= 0, (45)

ur ric =h

. (46)

It can be seen from (45) and(46), that h is not necessary for theresolution of the equations, although it can be recovered once the

problem is solved, and can be eliminated between these two equa-

tions, leaving only one boundary condition:

gicb(ur ric) 2urr

+ p= 0, (47)

Incidently, it can also be seen that there is no need to explicitly

solve the gravitational equation (15), since

has been absorbedin the modified pressure(38).

The governing equations and boundary conditions are now

made dimensionless using the age of the inner core ic, its time de-pendent radiusric(t),/ric(t),/r

2ic(t)and S(t)r

2ic(t)/(6)as

scales for time, length, velocity, pressure and potential temperature

respectively. Using the same symbols for dimensionless quantities,

dimensionless equations can be written as

u= 0, (48)

0= p +Ra(t) r +2u, (49)

(t)

t = 2 (u P e(t)r)

+ 6

(t)

S(t)ic

S(t) + 2 P e(t)

,

(50)

where r = r er and S(t) = dS(t)/dt. The last terms in (50) aredue to dependency of the temperature scale on time, when used to

make the equations dimensionless. Three dimensionless parameters

are needed

(t) =r2ic(t)

ic, (51)

P e(t) =ric(t)ric(t)

, (52)

Ra(t) =sgicb(t)S(t)r

5ic(t)

62 . (53)

The dimensionless boundary conditions, atr = 1, can be written

= 0, (54)

r = r

r

ur

+

1

r

ur

= 0, (55)

r = r

r

ur

+

1

r sin

ur

= 0, (56)

P(t)(ur ric) 2ur

r + p= 0, (57)

where we have introduced the phase change number, P, charac-terizing the resistance to phase change :

P(t) = gicb(t) ric(t) (t)

. (58)

Pis the ratio between the phase change timescale and the vis-cous relaxation timescale =/( gicb ric) (equivalent to post-glacial rebound timescale). P = 0 corresponds to instantaneousmelting or freezing, while P corresponds to infinitely slowmelting or freezing. In the limit of infinite P, the boundary condi-tion (57) reduces to the condition ur = 0, which corresponds toimpermeable conditions. In contrast, when P 0, Equation (57)implies that the normal stress tends toward 0 at the boundary, which

corresponds to fully permeable boundary conditions. The generalcase of finite P gives boundary conditions for which the rate ofphase change at the boundary (equal to ur) is proportional to thenormal stress induced by convection within the spherical shell.

A steady state version of the set of equation (48)-(58)is found

by usingr2ic/as a timescale instead ofic, and keepingricand Sconstant. All the equation remain unchanged except the heat equa-

tion which now writes

t = 2 u + 6. (59)

This will be used in section 6where numerical simulations with

constant inner core radius and thermal forcing will be used to derive

scaling laws.

With the assumptions made so far, the velocity field is known

to be purely poloidal (Ribe 2007), and we introduce the poloidal

scalarPdefined such that

u= (Pr) . (60)

Taking the curl of the momentum equation(49) gives

Ra(t)L2 =

22

L2P, (61)

where the angular momentum operatorL2 is

L2 = 1

sin

sin

1

sin2

2

2. (62)

Horizontal integration of the momentum equation (see

Forte & Peltier (1987);Ribe (2007), where this is done component-

wise in spherical harmonics) shows that, on r = 1

p +

r

r2P

=Cst. (63)

This expression can be used to eliminate pin the boundary condi-tion (57). Noting that

ur =1

rL2P, (64)

continuity of the normal stress at the ICB (equation (57)) gives the

following boundary condition atr = 1:

r

r2P

2

rL2P

P(t)

1

rL2P ric

=Cst, (65)


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while the stress-free conditions (55)and(56) take the form

r

r

1

r2

r(rP)

+

1

r2L2P =Cst, (66)

which can be rewritten as

2P

r2 +

L2 2

Pr2

=Cst. (67)

At this stage, there are two unknown scalar field variables,andP. They are expanded as

= tml (r, t) Ym

l , (68)

P = pml (r, t) Ym

l , l 1, (69)

whereYml (, ), forl 0,m [l; l] are surface spherical har-monics, which satisfy L2Yml =l(l+1)Y

ml . The momentum equa-

tion (61)takes the form

Ra(t)tml = D2lp

ml , (70)

where

Dl = d2

dr2+

2

r

d

dr

l(l+ 1)

r2 . (71)

The stress-free boundary condition(67) can be written as

d2pmldr2

+ [l(l+ 1) 2]pml

r2 = 0, l 1, (72)

and the boundary condition (65), derived from normal stress bal-

ance, as

d

dr

rDlp

ml 2l(l+ 1)

pmlr

=l(l+ 1)P(t)

pmlr

, l 1. (73)

With(72), the equation above can also be written:

r2d3pmldr3

3l(l+ 1)dpml

dr =

l(l+ 1)P(t)

6

r

pml , l 1.

(74)

The thermal equation is also written in spherical harmonic expan-sion but cannot be solved independently for each degree and order

due to the non-linearity of the advection term, which is evaluated

in the physical space and expanded back in spherical harmonics.

4 LINEAR STABILITY ANALYSIS

We investigate here the linear stability of the system of equa-

tions describing thermal convection in the inner core with phase

change at the ICB, as derived in section 3. The calculation given

here is a generalization of the linear stability analysis of thermal

convection in an internally heated sphere given by Chandrasekhar

(1961). The case considered by Chandrasekhar (1961), where a

non-deformable, impermeable outer boundary is assumed, corre-sponds to the limit P of the problem considered here.

We assume constant Ra and P (and = 1, P e = S =0), thus ignoring that the base diffusive solution itself is time-dependent. This assumption is essentially correct when the growth

rate of the fastest unstable disturbance is much larger than the

growth rate of the radius of the inner core. The basic state of the

problem is then given by

= 1 r2, (75)

u= 0, (76)

which is the steady conductive solution of the system of equation

developed in section3. We investigate the stability of this conduc-

tive state against infinitesimal perturbations of the temperature and

velocity fields. The temperature field is written as the sum of the

conductive temperature profile given by equation(75) and infinites-

imal disturbances ,(r,,,t) = (r) +(r,,,t). The ve-locity field perturbation is noted u(r,,,t), and has an associ-ated poloidal scalar P(r,,,t). We expand the temperature andpoloidal disturbances in spherical harmonics,

=

l=0

lm=l

tml (r)Ym

l (, ) elt, (77)

P =

l=1

lm=l

pml (r)Ym

l (, ) elt, (78)

wherel is the growth rate of the degreel perturbations (note thatsince mdoes not appear in the system of equations, the growth rateis function ofl only, notm).

The only non-linear term in the system of equations is the ad-

vection of heatu in Equation (59), which is linearized as

ur

r = 2rur = 2L

2P . (79)

The resulting linearized transport equation for the potential temper-

ature disturbance is

t 2

= 2L2P+ 6. (80)

Using the decompositions (77) and(78), the linearized system of

equations is then, forl 1,

Ra tml = D2lp

ml , (81)

(l Dl) tml = 2l(l+ 1)p

ml , (82)

with the boundary conditions given by Equations (72) and (73),

withtml (r= 1) = 0.Developing thetml in series of spherical Bessel functions and

solving for pm

l , we obtain an infinite set of linear equations in per-turbation quantities, which admits a non trivial solution only if its

determinant is equal to zero (see AppendixA for the details of the

calculation). This provides the following dispersion equation,

ql3(P)2l,i+q

l4(P)

1

4l+ 6

2l,j

ql1(P)2l,i+q

l2(P)

+

l

4l,i+

6l,i

2l(l+ 1)Ra 1

1

2ij

= 0, with i, j = 1, 2,...

(83)

where ||...|| denotes the determinant. Herel,idenotes theith zeroof the spherical Bessel function of degreel. The functionsql1(P)to

q

l

4(P) are given in AppendixAby equations (A.24),(A.25),(A.27)and (A.28).Solving equation(83) for a given value ofl and l = 0gives

the critical valueRac of the Rayleigh number for instability of thedegreel mode, as a function of P. The resulting marginal stabilitycurves forl = 1to 4 are shown in figure3.The first unstable modeis always thel = 1mode, for which equation(83) reduces to

Ra 61,i+ 1

41,i

4

ij +

21,i

Ra

P +

20

3 21,jRa

= 0.

(84)

A useful first approximation is obtained by keeping only the i =


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Figure 4. Temperature field (left, red=hot, blue=cold) and vorticity field

at O(P) (right, blue=negative, red=positive) in a meridional cross-section(the direction of translation is arbitrary).

5.2 Translation velocity at O(P)

The translation velocity at O(P) can be obtained by calculating thetemperature field at O(P)and the velocity field at O(P2), whichallows to determine the constantA in the expression of the poloidalscalar at O(P) (equation (101)). The procedure, detailed in Ap-

pendixB,is complicated by the non-linearity of the heat equation :coupling of higher order harmonics component of the temperature

and velocity fields contribute to the l= 1component of the temper-ature field. Taking into account the effect of the non-linear coupling

of thel = 2components of the temperature and velocity fields, weobtain

V =

6

5

Ra

P

1 0.0216 P+ O(P2)

. (110)

which suggests that the effect of deformation becomes important

when Pis a significant fraction of1/0.0216 46, in agreementwith the prediction of the linear stability analysis.

The temperature field and the -component of the vorticityfield at O(P), as calculated in Appendix B, are shown in figure

(4).

5.3 The effect of the boundary layer

Let us finally discuss the influence of the thermal boundary layer

that must develop in the solid inner core near the melting side when

a convective translation exists. From the thermal equation (50),

and with the boundary condition (54), a thermal boundary layer

of thicknessV1 results from the balance between convective anddiffusive terms, so that the degree one temperature component (97)

may be approximated by

t01 6

V

r eV(r1)

. (111)

We now note that

D1

eV(r1)

=

1 +

2

V r

2

V2r2

V2eV(r1), (112)

so that to a good approximation,

D1

eV(r1)

V2eV(r1) and D21

eV r

V4eV(r1)

(113)

when V 1. Under this assumption, the resulting general solutionfor the velocity poloidal component(101)becomes

p01 V0

2

r+

Ar+Br3 +Cr5 10

V20V6

eV(r1)

P+ O(P2)

,

(114)

Following the same path as above, in the limit of infinite viscosity,

the translation velocityVis found to be

V V0

1

5

V0

5

V20+

30

V30

30

V40

(115)

when the effect of the boundary layer is taken into account.

5.4 Melt production

We define the rate of melt production Mas the volume of meltproduced at the surface of the inner core by unit area and unit of

time, averaged over the ICB. In the case of a pure translation, the

volume of melt produced by unit of time is simply given by the

translation velocity V multiplied by the cross-section r2ic of theinner core, so Mis given by

M= V r2ic

4r2ic=

V

4. (116)

For a more general inner core flow, Mcan be calculated fromthe radial velocity at the ICB as

M= 1

2|ur(ric) ric| =

1

8

,

|ur(ric) ric| sin dd,

(117)

where the overbar. . .denotes the average over a spherical surface.In the case of al = 1,m = 0flow, this reduces to

M= 1

4

,

|p01cos | sin dd=1

2|p01|. (118)

and gives M=V /4for a pure translation, which has p01= V /2.

6 NUMERICAL RESULTS AND SCALING LAWS

6.1 Method

The code is an extension of the one used in Deguen & Cardin

(2011), with the boundary condition derived in section 3 now im-

plemented. The system of equations derived in section3 is solved

in 3-D, using a spherical harmonic expansion for the horizontal de-

pendence and a finite difference scheme in the radial direction. The

radial grid can be refined below the ICB if needed. The non-linear

part of the advection term in the temperature equation is evalu-

ated in the physical space at each time step. A semi-implicit Crank-

Nickolson scheme is implemented for the time evolution of the lin-

ear terms and an Adams-Bashforth procedure is used for the non-

linear advection term in the heat equation. The temperature field is

initialized with a random noise covering the full spectrum. We use

up to 256 radial points and 128 spherical harmonics degree. Care

has been taken that the ICB thermal boundary layer, which can be

very thin in the translation mode, is always well resolved.

The code has the ability to take into account the growth of

the inner core and the evolution of the internal heating rate S(t),which is calculated from the thermal evolution of the outer core

(Deguen & Cardin 2011). In this section, we will first focus on sim-

ulations with a constant inner core radius and steady thermal forc-

ing (internal heating rate Sconstant). Simulations with an evolvinginner core will be presented in section7.

Each numerical simulation was run for at least 10 overturn

timesric/Urms, whereUrmsis the RMS velocity in the inner core.


11/25


12/25


1

10

100

101

102

103

104

104

105

106

107

P

u

Ra

Figure 6. Mean degree u of the kinetic energy (as defined in Equation119), as a function ofP, for simulations with Ra = 104,105, 106, and107. The grey scale of the markers give the Rayleigh number of the simu-lation. u is close to 1 for

P29 for allRa, although the departure from

1 increases withRa when Papproaches 29 from below.

components, as in the translation mode, and increases as the char-

acteristic lengthscale of the flow decreases.

Figure6 shows the calculated value of for Ra = 104,105,106 and107 as a function ofP.remains very close to 1 as long asP is smaller than a transitional value Pt 29. There is a rapid in-crease ofuabove Pt, showing the emergence of smaller scale con-vective modes at the transition between the translation mode and

the high-Pregime. We interpret this sharp transition as being dueto the negative feedback that the secondary flow and smaller scale

convection have on the translation mode : advection of the poten-

tial temperature field by the secondary flow decreases the strengthof its degree one component and therefore weakens the translation

mode, which in turn give more time for smaller scale convection

to develop, weakening further the degree one heterogeneity. The

value of Pt does not seem to depend on Ra in the range exploredhere. Figure6 further shows that ICB phase change has a strong

influence on the flow up to P 300, which is confirmed by directvisualization of the flow structure.

Figure7a shows the translation rate V(circles) and time aver-aged RMS velocity (triangles) as a function of P for various valuesofRa. Here bothV andUrmsare multiplied byRa

1/2. The grey

dashed line shows the analytical prediction for the translation rate

in the rigid inner core limit. Below Pt, there is a good quantitativeagreement between the numerical results and the analytical model.

The fact thatUrms V for P < Pt indicates that there is, as ex-pected, negligible deformation in this regime. V andUrms divergefor P > Pt, the translation rate becoming rapidly much smallerthan the RMS velocity. As already suggested by the evolution of

, phase change at the ICB has still an effect on the convectionforPup to 300. Phase change at the ICB has a positive feed-back on the vigor of the convection: melting occurs preferentially

above upwelling, where the dynamic topography is positive, which

enhances upward motion. Conversely, solidification occurs prefer-

entially above downwellings, thus enhancing downward motions.

This effect becomes increasingly small as Pis increased, and theRMS velocity reaches a plateau whenP 103, at which the ef-

104

103

102

101

100

101

102

101

100

101

102

103

104

105

106

104

105

106

107

P

(urms,V

)

Ra1/2

Translation rateurms

1/2

Ra

106

105

104

103

102

101

100

102 101 100 101 102 103 104 105 106

104 105 106 107

P

M

Ra1

/2

1

1/2

Ra

Figure 7. a) RMS velocity (triangles) and translation velocity (circles) as

a function ofP, for Rayleigh numbers between 3 103

and 107

(greyscale). The inner core translation rate is found by first calculating the net

translation rate Vi=x,y,z of the inner core in the directions x,y,z of a

cartesian frame, given by the average over the volume of the inner core

of the velocity componentui=x,y,z(which can be written as functions of

the degree 1 components of the poloidal scalar at the ICB, see equation

B.42in AppendixB). We then write the global translation velocity as V =V2x +V

2y +V

2z. The grey dashed line shows the prediction of the rigid

inner core model.b) MRa1/2 as a function ofP, the grey scale of themarkers giving the value ofRa. The grey dashed line shows the predictionof the rigid inner core model, showing excellent agreement between the

theory and the numerical calculations for P small.

fect of phase change at the ICB on the internal dynamics becomes

negligible.

Figure7b shows the rate of melt production (defined in Equa-

tion(117)), multiplied by Ra1/2, as a function of P for variousvalues ofRa. Again, the prediction of the rigid inner core model(Equation (116), grey dashed line in Figure 7b) is in very good

agreement with the numerical results as long as P < Pt. ForP > Pt, the rate of melt production appears to be inversally pro-portional to P.


13/25


0.0

0.2

0.4

0.6

0.8

1.0

1.2

101

102

103

104

105

106

Ra/P

Vtr/V0

Translation rateM

Figure 8. Translation rate and melt production, normalized by the low Plimit estimate given by Equation (107), as a function ofRa/P, for P =102.

6.3 Scaling of translation rate, convective velocity, and melt

production

We now turn to a more quantitative description of the small- Pandlarge-Pregimes. We first compare the results of numerical simula-tions at P< Pt with the analytical models developed in section5.We then focus on the large-Pregime, and develop a scaling theoryfor inner core thermal convection in this regime, including a scaling

law for the rate of melt production.

6.3.1 Translation mode

Figure 8shows the translation rate (circles) and the rate of melt pro-

duction M(diamonds), normalized by the rigid inner core estimategiven by Equation (107), as a function ofRa/P, forP = 102.The translation rate increases from zero whenRa/P is higher thana critical value (Ra/P)c which is found to be in excellent agree-ment with the prediction of the linear stability analysis. Increas-

ing Ra/P above (Ra/P)c, the translation rate increases beforeasymptoting toward the prediction of the rigid inner core model

(dashed line). The prediction of our model including a boundary

layer correction (Equation(115), black line in Figure 8)is in good

agreement with the numerical results for Ra/P 103, demon-strating that our analytical model captures fairly well the effect of

the thermal boundary layer. As expected (see section5.4), the rate

of melt production is equal to1/4of the translation rate.Figure9 shows the effect of increasing Pon the translation

rate. In this figure, we have kept only simulations with Ra/Plargerthan105 to minimize the effect of the boundary layer, and furthercorrected the translation velocity with the boundary layer correc-

tion (Equation (115)) found in section 5.3, in order to isolate as

much as possible the effect ofPon the translation mode. The O(P)model developed in section5.2(Equation(110), black line) agrees

with the numerical simulations within 1% for P up to 3, butfails to explain the outputs of the numerical simulations when Pislarger, which indicates that higher order terms in Pbecome impor-tant.

Overall, our analytical results (stability analysis and finite am-

plitude models) are in very good agreement with our numerical

0.8

0.9

1.0

102

101

100

101

102

105

106

P

Vtr

/V0

Ra/P

Figure 9.Translation rate (normalized by the low Plimit estimate given byEquation (107)) as a function ofP, for different values ofRa/P. The thickblack line show the prediction of the O(P)model given by equation (110).

simulations when Pis small, which gives support to both our the-ory and to the validity of the numerical code.

6.3.2 Plume convection

IfPis large, the translation rate of the inner core becomes vanish-ingly small, but, as long as Pis finite, there is still a finite rate ofmelt production associated with the smaller scale topography aris-

ing from plume convection. A scaling for the melt production in

the limit of large Pand largeRa can be derived from scaling rela-tionship for infinite Prandtl number convection with impermeable

boundaries.Parmentier & Sotin (2000) derived a set of scaling laws

h

a

rIC

B

Figure 10. A schematic of inner core plume convection, and definition of

the length scales used in the scaling analysis. Streamlines of the flow are

shown with thin arrowed grey lines.


14/25


for high Rayleigh number internally heated thermal convection in a

cartesian box, in the limit of infinite Prandtl number, but we found

significant deviations from their model in our numerical simula-

tions, which we ascribe to geometrical effects due to the spherical

geometry. We therefore propose a set of new scaling laws for con-

vection in a full sphere with internal heating.

Quantities of interest are the horizontal and vertical velocities

uand w, the mean inner core potential temperature , the ther-

mal boundary layer thickness , the thermal radius of the plumesa, the average plume spacingh, and a length scale for radial vari-ations of the velocity, which we note r (see Figure10). The hor-izontal length scaleh is related to the number Nof plumes perunit area byN 1/2h.

Outputs of numerical simulations (, , RMS velocityUrms, RMS radial velocity wrms, RMS horizontal velocity urms,N) are shown in Fig.11a-d for Ra between 105 and 3 108.The boundary layer thickness is estimated as the ratio of themean potential temperature in the inner core, , over the timeand space averaged potential temperature gradient at the ICB :

= / /ricb. The time-averaged numberNof plumeper unit area is estimated by counting plumes on horizontal sur-

faces on typically 50 different snapshots. Both and follow

well-defined power law behaviors over this range ofRa. In con-trast, the RMS velocities and plume density Nseem to indicate achange of behavior atRa close to107. ForRa 0 meansmelting, andur ric < 0 means solidification) which, accordingto Equation57, can be written as

ur ric = P12

ur

r

+ p . (129)As discussed above, we have

urr

icb

w

r Ra1+2. (130)

The dynamic pressure is given by the horizontal component of the

Stokes equation,

0 = Hp+ (u)H (131)

which implies that

p u

h Ra1+2. (132)


15/25


101

105 106 107 108

Ra

102

101

105

106

107

108

Ra

102

105

106

107

108

Urms,wrms,urms

Ra

101

102

105

106

107

108

Ra

N

Figure 11. a) Mean potential temperature as a function ofRa for Plarger than103.b) Boundary layer thickness . c) RMS velocity Urms(squares),RMS vertical velocity wrms(diamonds), and RMS horizontal velocityurms (circles). d) Number of plumes Nper unit surface. In figuresa) to d), the thickred lines show the predictions of the scaling theory developed in section 6.3.2with =0.238. The dashed lines show the result of the individual leastsquare inversion for each quantity forRa 105.

Both terms follow the same scaling, which implies that the global

rate of melt production scales as

M Ra1+2P1. (133)

With= 0.238 0.003, we obtain M Ra0.5240.006P1.Figure 12b shows PM as a function ofRa, for P 103

corresponding to the plume convection regime. There is an almost

perfect collapse of the data points, which supports the fact that

M P in this regime. The kink in the curve at Ra 3 104

corresponds to the transition from steady convection to unsteady

convection. Above this transition, the data points are well fitted by

a power-law of the form M=aP1Rab. Least square regressionfor Ra 3105 givesa= 0.460.04and b= 0.5540.006, inreasonable agreement with the value found above. In dimensional

terms, M a(/ric)P1

Rab

and the mass flux of molten mate-rial isic M ak/(cpric)P

1Rab.

7 APPLICATION

7.1 Evolutive models

The analytical model for the translation mode and the scaling laws

for large-Pconvection derived in the previous sections strictly ap-ply only to convection withric andS constant. We therefore firstcheck that our models correctly describe inner core convection

when ricand Sare time-dependent, by comparing their predictions

101

102

103

104

102

103

104

105

106

107

108

109

P

M

Ra

Rac= 1545.6

Figure 12.Rate of melt production (multiplied by P) as a function ofRa,for numerical simulations with P 103 . The value of the critical Rayleighnumber as predicted by the linear stability analysis in the limit of infinite P(Equation89,Rac = 1545.6) is indicated by the arrow.

with the outcome of numerical simulations with inner core growth

and thermal history determined from the core energy balance.

To account for the inner core secular evolution, we follow the


16/25


a)

101

100

101

102

103

104

105

106

107

108

109

102

101

100

101

102

103

104

105

0.8

1

0.8

1

P

Ra

A

B

C

D

1020

Pas

1017

Pas

Translation

Plume

Convection

b) A - Tic = 0.8, = 1020 Pa s B - Tic= 1, = 1020 Pa s

1011

1010

109

0.7 0 .6 0 .5 0 .4 0 .3 0. 2 0. 1 0 .0

time (Gy)

V,

M,

Urms,

ric

(m/s)

1011

1010

109

0.7 0.6 0 .5 0 .4 0 .3 0. 2 0. 1 0 .0

time (Gy)

C - Tic= 0.8, = 1017 Pa s D - Tic= 1, = 1017 Pa s

1011

1010

109

0.7 0 .6 0 .5 0 .4 0 .3 0. 2 0. 1 0 .0

time (Gy)

V,

M,

Urms,

ric

(m/s)

1011

1010

109

0.7 0.6 0 .5 0 .4 0 .3 0. 2 0. 1 0 .0

time (Gy)

Figure 13. a)Trajectories of the inner core state in a Ra Pspace, for the four cases A,B,C, and D discussed in the text. The line annotations give thevalue ofTic for each case. The dashed lines shows the future trajectory of the inner core. b) Time evolution ofV, M,Urmsand ric for cases A to D. Redline : inner core growth rate ric. Black line : translation rateV. Orange line : RMS velocityUrms. Blue line : dimensional melting rate(/ric) M. Predictionsfor the RMS velocity (or translation velocity in the translation regime) and melting rate Mare shown with thick dashed and dash-dotted lines, respectively. In

the = 1020 Pa.s cases, the translation model (Equation115) is used to predict V and M. In the = 1017 Pa.s case, the high-Pscaling is used for Urmsand M. In the = 1020 Pa.s, Tic = 0.8and Tic= 1cases, the translation rate and the RMS velocity are equal. For these simulations, the Rayleigh numberwas calculated assuming a thermal conductivityk = 79W.m1.K1 and a phase change timescale = 1000 yr. Values of other physical parameters used

for these runs are summarized in Table1.

procedure explained inDeguen & Cardin(2011), where the growth

of the inner core and its cooling rate are determined from the core

energy balance. In this framework, a convenient way to write S(t)is

S=sg

T

KS3

f(ric) T

1ic 1

, (134)

where f(ric) is a decreasing order one function ofr ic defined inDeguen & Cardin (2011) (Equation 19, page 1104), g = dg/dr,KS is the isentropic bulk modulus, the Gruneisen parameter, and

Tic=

dTsdTad

1

1ic

, (135)

whereic is the age of the inner core, = r2ic/(6) is the cur-

rent inner core thermal diffusion time, and dTs/dTad is the ratioof the Clapeyron slopedTs/dP to the adiabatdTad/dP. The non-dimensional inner core age Ticis a convenient indicator of the ther-mal state of the inner core, with Tic < 1 implying unstable stratifi-cation for most of inner core history (see Deguen & Cardin(2011),

Figure 3a). We give for reference in table 2the values of the age of

the inner coreiccorresponding to Tic = 0.8, 1, and 1.2, for a ther-mal conductivity equal to 36, 79, and 150 W.m1.K1. With the

inner core growth history determined from the core energy balance

and S(t) calculated from Equation (134), the evolution ofRa(t)and P(t)can then be calculated.

Figure 13a shows the trajectories of the inner core state in

a Ra Pspace for four different scenarios, superimposed on aregime diagram for inner core thermal convection. According to

Equation(6), the ICB phase change timescale scales as r1ic ,

and therefore P ricalways increases during inner core history. In

contrast, the evolution ofRa(t) is non monotonic, with the effect ofthe increasing inner core radius and gravity opposing the decrease

with time of the effective heating rate S(t). BecauseSeventuallybecomes negative at some time in inner core history, Ra reachesa maximum before decreasing and eventually becoming negative,

resulting in a bell shaped trajectory of the inner core in the Ra Pspace. The maximum in Ra may or may not have been reached yet,depending on the value ofTic.

The scenarios A-Dshown in Figure13a have been chosen to

illustrate four different possible dynamic histories of the inner core.

In casesA and C, which have Tic = 0.8,Ra remains positive andsupercritical up to today, thus always permitting thermal convec-

tion. In casesB andD, which have Tic= 1,Ra has reached a max-imum early in inner core history, before decreasing below supercrit-

icality, at which point convection is expected to stop. In these twocases, only an early convective episode is expected(Buffett 2009;

Deguen & Cardin 2011). In casesA and B, which have = 1020

Pa s, P(t) is always smaller than the transitional Pt and thermalconvection therefore should be in the translation regime; Cases C

and D, which have = 1017 Pa s, have P(t) > 102 > Pt andthermal convection should be in the plume regime.

Figure13b shows outputs from numerical simulations corre-

sponding to the inner core histories shown in Figure 13a. The nu-

merical results are compared to the predictions for the RMS veloc-

ity Urms(equal to the translation rate Vtr in the translation regime)and melting rate Mfrom the analytical translation model (Equation


17/25


Table 2. Correspondence between ic andTic for three values of in-ner core thermal conductivity, assuming dTs/dTad = 1.65 0.11(Deguen & Cardin 2011).

k(W.m1.K1)

36 79 150

0.8 1.18 0.23Gy 0.54 0.11Gy 0.28 0.06Gy

Tic = 1.0 1.48

0.29Gy 0.68

0.13Gy 0.36

0.07Gy

1.2 1.77 0.35Gy 0.81 0.16Gy 0.43 0.08Gy

(116)) and the large-Pscaling laws (Equation (133)). The agree-ment is good in both the translation and plume convection regimes,

except at the times of initiation and cessation of convection.

There is always a lag between when conditions become su-

percritical and when the amplitude of convective motions become

significant, due to the finite growth rate of the instability. From

Equation (88), the timescale for instability growth, = 1/, isapproximately (in dimensional form)

5r2ic

Ra

P

Ra

P c1

ric

600km2 90My

Ra

PRa

P

c 1

(136)

in the translation regime, and

77r2ic

(Ra Rac)1

ric600km

2 80MyRa/Rac 1

(137)

in the large-Pregime. In both cases, the timescale for the growth ofthe instability will typically be a few tens of My, thus explaining the

delayed initiation of convection seen in the numerical simulations.

In cases B and D, the flow occurring after t 0.46 Gy,at a time where the models predict no motion (because S < 0),corresponds to a slow relaxation of the thermal heterogeneities left

behind by the convective episode.

Apart during the initiation and cessation periods of convec-tion, the models developed for steady internal heating and constant

inner core radius agree very well with the full numerical calcula-

tions, and can therefore be used to predict the dynamic state of the

inner core and key quantities including RMS velocity and melt pro-

duction rate.

7.2 Melt production

Experiments by Alboussiere et al. (2010) have shown that the de-

velopment of a stably stratified layer above the ICB by inner core

melting is controlled by the ratioBof the buoyancy fluxes arisingfrom the melting and freezing regions of the ICB. By using the ana-

lytical translation model and the scaling laws for plume convection

developed in the last two sections, we can now estimate todays

value ofBas a function of the state and physical properties of theinner core, and assess the likelihood of the origin of the F-layer by

inner core melting.

With M being the non-dimensional rate of melt produc-tion defined in Equation (117), the mean solidification rate is

(/ric) M+ ricfrom conservation of mass. The buoyancy flux as-sociated with the release of dense fluid by melting can be written as

gicb(/ric) M, while the buoyancy flux associated with thesolidification is gicb[(/ric) M+ ric], whereis the frac-tion of the ICB density jump due to the compositional difference.

According to Alboussiere et al. (2010)s experiments, a stratified

0.8

0.6

0.4

0.2

0.8

1014

1015

1016

1017

1018

1019

1020

1021

0.5 0.6 0.7 0.8 0.9 1.0

100

101

102

103

104

105

1060.0

0.2

0.4

0.6

0.8

1.0

Tic

(Pa.s

)

B

P

translation regime

plume convection

noconvection

Figure 14. Inner core regime diagram and map of the buoyancy ratio

B, as functions ofP andTic. The corresponding values of assuming = 1000 yr are given on the right hand size vertical axis. According

toAlboussiere et al.(2010)s experiments, inner core melting can produce

a stably stratified layer at the base of the outer core ifB > 0.8 (white

contour).

layer is expected to form above the ICB if the magnitude of the

buoyancy flux associated with melting is more than 80 % of the

buoyancy flux associated with solidification,i.e.if

B = gicb(/ric) M

gicb[(/ric) M+ ric]=

M

M+ ric ric/>0.8,

(138)

which requires that

ricM >4 ric. (139)

In the translation regime, in which M=V /4, this requires that the

rate of translation is at least 16 times larger than the mean solidifi-cation rate of the inner core.

The current inner core growth rate can be expressed as

ric = 3

ric

f(ric) dTsdTad

1

Tic(140)

where the function f(ric) 0.8 at the current inner core radius(Deguen & Cardin 2011). Using this expression, the buoyancy ratio

B is

B = 1

1 +

dTsdTad

1

Tic3f(ric)

V

4

1(141)

in the translation regime, with the translation velocity V given by

Equation(115), and

B = 1

1 +

dTsdTad

1

Tic3f(ric)

aP1Rab1

(142)

in the high-Pregime.

7.3 Todays inner core regime and rate of melt production

The inner core dynamic regime depends mostly on the value of

its non-dimensional age Tic and of P, both parameters being verypoorly constrained. The value of Tic dictates whether the innercore has a stable or unstable temperature profile, and the parameter


18/25


P determines the convection regime if the inner core is unstableagainst thermal convection. Other parameters have a comparatively

small influence on the inner core dynamics, and on the value ofB .With this idea in mind, it is useful to rewrite the Rayleigh number

as a function of Pand Tic:

Ra=sgicbSr

5ic

62

= A

f(ric) T1ic 1

P, with A = 2

2

sgicbr3

icT2 k . (143)

The exact value of the pre-factor A affects the value ofB , butnot the inner core regime (stably stratified inner core, translation,

or plume convection) which is determined by P and Tic. The un-certainty on A comes mostly from the uncertainty on , which isdifficult to estimate without a better understanding of the dynamics

of the F-layer. IfPand Ticare kept constants, changing A by an or-der of magnitude would change the translation velocity and melting

rates (in both regimes) by a factor of 3. Figure14showsB asa function ofTicand P, calculated from Equation(141)and Equa-tion(142)withRa given by Equation (143) and A = 3 105 (cor-responding to parameters values given in table1). Figure14serves

both as a regime diagram for the inner core, and as a predictive map

forB and the likelihood of the development of a stratified layerat the base of the outer core.

The inner core has currently an unstable thermal profile only

ifTicis smaller than 0.87. The mode of thermal convection thendepends on P, with the translation regime (small P) being the mostefficient at producing melt. Plume convection generates less melt,

but the rate of melt production still remains significant as long as

P is not too large ( not too small). The critical value ofB =0.8 (white contour in Figure14) suggested by the experiments ofAlboussiere et al.(2010) is almost always reached in the translation

regime, but only in a small part of the parameter space in the plume

convection regime.

8 COMPOSITIONAL EFFECTS

We have so far left aside the possible effects of the compositional

evolution of the outer and inner core on the inner core dynamics.

We will argue here that the development of an iron rich layer above

the inner core can have a possibly important positive feedback on

inner core convection: irrespectively of the exact mechanism at the

origin of the F-layer(Gubbins et al. 2008; Alboussiere et al. 2010;

Gubbins et al. 2011), its interpretation as an iron rich layer implies

a decrease with time of light elements concentration in the liquid

just above the ICB. This in turn implies that the newly crystal-

lized solid is increasingly depleted in light elements, and intrin-

sically denser, which may drive compositional convection in the

inner core. The reciprocal coupling between the inner core and the

F-layer may create a positive feedback loop which can make thesystem (inner core + F-layer) unstable. The mechanism releases

more gravitational energy than purely radial inner core growth with

no melting, and should therefore be energetically favored.

We note cs and cl the light element concentration in the in-ner and outer core, respectively, cs,licb their values at the ICB, and

cs,licb = dcs,licb/dt their time derivatives at the ICB. The concen-

tration in the liquid and solid sides of the ICB are linked by the

partition coefficientk,csicb=k clicb. Introducingc= c c

sicb, the

equation of transport of light element can be written as

Dc

Dt =c

2c +Sc, c(ric) = 0, (144)

with

Sc= csicb= k c

licb c

licb

dk

dt, (145)

which is an exact analogue of the potential temperature transport

equation (29). The only - but important - difference is that the

source termSc is a dynamical quantity which depends on the con-vective state of the inner core and on the dynamics of the F-layer

rather than being externally imposed like the effective heating rateS, which means that the dynamics of the inner core and F-layermust be considered simultaneously.

In general, the fact that the thermal and compositional dif-

fusivities are different can be of importance, and would lead to

double-diffusive type convection. But this is not the case in the

translation regime, for which diffusion does not play any role

as long as the translation rate is large enough ( i.e. if the Peclet

number P e = V ric/ 1). Thanks to the potential tempera-ture/composition analogy noted above, the translation model de-

veloped for thermal convection can be extended to include compo-

sitional effects, the translation rate being given by

V = 1

5

s

(S+ cSc)

1/2

ric (146)

when compositional effects are accounted for. We therefore need

to compare the magnitudes of S and cSc = ck clicb

cclicb dk/dt. Assuming (Gubbins et al. 2008) that the light ele-

ment concentration at the base of the F-layer is currently about

twice smaller than the outer core mean concentration, coc 5wt.%, we obtain clicb 0.5 coc/ic 10

18 wt.%.s1 with

ic 1 Gy. With c 1, this gives ck clicb 10

19 s1 if

k 0.1and ck clicb 10

20 s1 ifk 0.01, which is simi-lar or larger than the thermal contribution S 105 1015 1020 s1. The term c c

licbdk/dt might be positive as well. Ac-

cording to calculations byGubbins et al. (2013), the variation with

temperature of the partitioning behavior of Oxygen can produce an

unstable compositional gradient. As discussed inAlboussiere et al.

(2010) andDeguen & Cardin (2011), the effective partition coef-ficient may also decrease with time because of dynamical reasons

(the efficiency of melt expulsion from the inner core increases with

inner core size), which would also imply that this term is positive.

There is an additional feedback, this time negative, which

comes from the effect of composition on the solidification tem-

perature, which increases with decreasing light element concen-

tration. The decreasing light element concentration at the base of

the F-layer implies that the ICB temperature decreases with time

at a slower rate than if the composition is fixed, which results in

a smaller effective heating rate S[Equation (30)]. For a fixed in-ner core growth rate, this decreases the ICB cooling rate by an

amount equal to mc clicb, where mc = Ts/c 10

4 K

(Alfe et al. 2002) is the liquidus slope at the inner core boundary

pressure and composition. This adds a term mc clicb in Equa-tion(146). If only one light element is considered, the ratio of the

stabilizing term mc clicbover the destabilizing term c k c

licbis

mc/(ck) 0.1/k. The two terms are of the same orderof magnitude ifk 0.1, but the negative feedback dominates ifkis smaller.

The above estimates are clearly uncertain, and a dynami-

cal model of the F-layer will be required for assessing in a self-

consistent way the effect of the development of the F-layer on inner

core convection. There are several feedbacks of the formation of an

F-layer on inner core convection, either positive or negative, and it

is not clear yet whether the net effect would be stabilizing or desta-


19/25


bilizing. Still, it does suggest that the effect could be important, and

worth considering in more details.

9 SUMMARY AND CONCLUSIONS

Inner core translation can potentially explain a significant part of

the inner core structure, but its existence depends critically on the

value of a number of poorly constrained parameters. In this paper,we have studied in details the conditions for and dynamics of inner

core thermal convection when melting and solidification at the ICB

are allowed. We summarize here the main results and implications

of our work :

(i) If the inner core is convectively unstable, linear stability anal-

ysis (section 4), asymptotic calculations (section 5), and direct

numerical simulations (section6) consistently show that the con-

vection regime depends mostly on a non-dimensional number, the

phase change numberP, characterizing the resistance to phasechange [Equation (58)]. The convective translation mode domi-

nates only ifP< 29, which requires that the inner core viscosity islarger than a critical value estimated to be 3 1018 Pa s. IfPislarger (smaller viscosity), melting and solidification at the ICB have

only a small dynamical effect, and convection takes the usual formof low Prandtl number internally heated convection, with a one-cell

axisymmetric mode at the onset, and chaotic plume convection if

the Rayleigh number is large.

(ii) With published estimates of the inner core viscosity ranging

from 1011 Pa s to 1022 Pa s (Yoshida et al. 1996; Buffett 1997;Van Orman 2004; Mound & Buffett 2006; Koot & Dumberry

2011; Reaman et al. 2011, 2012), the question of which mode

would be preferred is open (although we note that the latest esti-

mate from mineral physics,1020 1022 Pa s (Reaman et al. 2012),would put the inner core, if unstably stratified, well within the trans-

lation regime).

(iii) The two convection regimes have been characterized in de-

tails in sections4to7;a summary of theoretical results and scaling

laws for useful dynamical quantities (RMS velocity, rate of melt

production, mean potential temperature, number of plumes per unit

area, and strain rate) is given in table 3. If the inner core is unsta-

bly stratified, the rate of melt production predicted by our models

is always large enough to produce an iron-rich layer at the base of

the outer, according to Alboussiere et al. (2010)s experiments, if

the inner core is in the translation regime (Figure14). In the plume

convection regime, the rate of melt production can still be signifi-

cant ifPis not too large ( not too small).(iv) Being driven by buoyancy, a prerequisite for the existence

of convective translation is that an unstable density profile is main-

tained within the inner core. Thermal convection requires that

a superadiabatic temperature profile is maintained with the in-

ner core, which is highly dependent on the core thermal history

and inner core thermal conductivity. With k = 36 W.m1.K1

as proposed byStacey & Davis (2008), this would be very likely

(Buffett 2009;Deguen & Cardin 2011). However, several indepen-

dent groups (Sha & Cohen 2011;de Koker et al. 2012;Pozzo et al.

2012) have recently argued for a much higher core thermal conduc-

tivity, around 150 W.m1.K1 or higher. This would make thermal

convection in the inner core, whether in the translation mode or

in the plume convection mode, impossible unless the inner core is

very young ( 300 My or less, which would require a probablyexcessively high CMB heat flux).

(v) Compositional convection might be a viable alternative to

thermal convection, either because the temperature dependency of

the light elements partitioning behavior can produce an unstable

compositional profile(Gubbins et al. 2013), or because of a pos-

sibly positive feedback of the development of the F-layer on inner

core convection. As proposed in section8,the formation of an iron-

rich layer at the base of the outer core over the history of the inner

core implies that the inner core crystallizes from a source which is

increasingly depleted in light elements. This in turn implies that the

newly crystallized solid is increasingly depleted in light element,

which results in an unstable density profile. Whether this positivefeedback is strong enough to overcome the stabilizing effect of a

possibly subadiabatic temperature profile depends on the dynamics

of the F-layer, and further work is needed to test this idea.

ACKNOWLEDGMENTS

We would like to thank the two anonymous referees for many help-

ful comments and suggestions. All the computations presented in

this paper were performed at the Service Commun de Calcul Inten-

sif de lObservatoire de Grenoble (SCCI). R. D. was supported by

grant EAR-0909622 and Frontiers in Earth System Dynamics grant

EAR-1135382 from the National Science Foundation. T. A. was

supported by the ANR (Agence National de la Recherche) within

the CrysCore project ANR-08-BLAN-0234-01, and by the program

PNP of INSU/CNRS. P. C. was supported by the program PNP of

INSU/CNRS.

APPENDIX A: LINEAR STABILITY ANALYSIS

We investigate here the linear stability of the system of equations

describing thermal convection in the inner core with phase change

at the ICB, as derived in section 3. The calculation given here is a

generalization of the linear stability analysis of thermal convection

in an internally heated sphere given byChandrasekhar (1961).

We assume constantRa and P. The basic state of the problemis then

= 1 r2, (A.1)

u= 0, (A.2)

which is the steady conductive solution of the system of equation

developed in section 3. We investigate the stability of this con-

ductive state against infinitesimal perturbations of the temperature

and velocity fields. The temperature field is written as the sum of

the conductive temperature profile given by equation (A.1) and in-

finitesimal disturbances , (r,,,t) = (r) + (r,,,t).The velocity field perturbation is noted u(r,,,t), and has an as-sociated poloidal scalar P(r,,,t). We expand the temperatureand poloidal disturbances in spherical harmonics,

=

l=0

lm=l

tml (r)Ym

l (, ) elt, (A.3)

P =

l=1

lm=l

pml (r)Ym

l (, ) elt, (A.4)

wherel is the growth rate of the degreel perturbations.The only non-linear term in the system of equations is the ad-

vection of heatu T, which is linearized as

ur

r = 2rur = 2L

2P . (A.5)


20/25


21/25


which we rewrite as

Bl,i =

ql1(P) +

ql2(P)

2l,i

jl+1(l,i)

l,iRa, (A.23)

where

ql1(P) = l+ 2

l+ 1

4l(l+ 1) 2 + (2l+ 1)P

6

l

1, (A.24)

ql2(P) = 2(l2 + 3l 1) + P 6l

4l(l+ 1) 2 + (2l+ 1)P 6l

. (A.25)

With this expression forBl,i, the constantsCl,iare given by

Cl,i=

ql3(P) +

ql4(P)

2l,i

jl+1(l,i)

l,iRa, (A.26)

where

ql3(P) =(l2 1)

l(l+ 2)ql1(P), (A.27)

ql4(P) =(l2 1)ql2(P) + 1

l(l+ 2) . (A.28)

Now, using Equations(A.13)and(A.15)for pm

l and Equation(A.9)fortml , the heat equation(A.8) gives

i

Al,i

l+

2l,i

2l(l+ 1)

Ra

4l,i

jl(l,ir)

=

i

Al,i

Bl,irl + Cl,ir

l+2

.

(A.29)

Multiplying equation(A.29)by r2jl(l,jr) where j is an integerin[0; [, integrating in r over [0;1], and using the orthogonalityrelation(A.11), we obtain

Al,j

l+

2l,j

2l(l+ 1)

Ra

4l,j

1

2[jl+1(l,j)]

2

=

i

Al,iBl,i

1

0

rl+2jl(l,jr)dr

+

i

Al,iCl,i

10

rl+4jl(l,jr)dr, (j = 1, 2, . . . ).

(A.30)

Equation (A.30)forms a set of linear homogeneous equations for

the constants Al,j , which admits non-trivial solutions only if itssecular determinant is equal to zero.

Before calculating the secular determinant of the system of

equations, we evaluate the two integrals on the right hand side,

starting with the integral ofrl+2jl(l,jr). Using the formula

1x ddx

m xn+1jn(x)

=xnm+1jnm(x) (A.31)

(Abramovich & Stegun 1965) withm= 1andn = k + 1 gives

d

dx

xk+2jk+1(x)

=xk+2jk(z), (A.32)

which allows to write, withk = l, 10

rl+2jl(l,jr)dr= 1

l+3l,j

l,j0

xl+2jl(x)dx

= 1

l+3l,j

xl+2jl+1(x)

l,j0

= jl+1(l,j)

l,j.

(A.33)

Now, using the recurrence relation

jn1+jn+1=2n+ 1

r jn (A.34)

(Abramovich & Stegun 1965) withn = l + 1, we rewrite the inte-gral ofrl+4jl(l,j r)as

10

rl+4jl(l,j r)dr= 1l+5l,j

l,j0

xl+4jl(x)dx (A.35)

= 1

l+5l,j

l,j0

xl+4jl+2(x)dx

+2l+ 3

l+5l,j

l,j0

xl+3jl+1(x)dx

(A.36)

The two integrals on the RHS can be calculated using the relation

(A.32) with k= l + 2andk = l + 1, respectively. With further useof the recurrence relation(A.34), we finally obtain 10

rl+4jl(l,j r)dr=

1

4l+ 6

2l,j

jl+1(l,j )

l,j. (A.37)

With the integrals estimated above, the system of equations(A.30) can be rewritten as

Al,j

l+

2l,j

2l(l+ 1)Ra

1

4l,j

1

2[jl+1(l,j)]

2

=

i

Al,i

ql3(P) +

ql4(P)

2l,i

4l+ 6

2l,j 1

+ql1(P) +ql2(P)

2l,i

jl+1(l,i)

l,i

jl+1(l,j)

l,j, (j= 1, 2, . . . ).

(A.38)

Introducing Ai = (jl+1(l,i)/3l,i)Al,i, and dividing by

jl+1(l,j )/l,j , we finally obtaini

Ai

ql3(P)

2l,i+q

l4(P)

1

4l+ 6

2l,j

ql1(P)2l,i+q

l2(P)

+

l

4l,i+

6l,i

2l(l+ 1)Ra 1

1

2ij

= 0, (j= 1, 2, . . . ).

(A.39)

This forms an infinite set of linear equations, which admits a

non trivial solution only if its determinant is zero :

ql3(P)2l,i+ql4(P) 1 4l+ 62l,j ql1(P)2l,i+ql2(P)+

l

4l,i+

6l,i

2l(l+ 1)Ra 1

1

2ij

= 0,

(A.40)

with i, j = 1, 2,.... Solving equation (A.40) for a given value ofl and l = 0 gives the critical value Rac of the Rayleigh numberfor instability of thel mode as a function of P. When solving nu-merically equations (A.40), the precision onRac depends on themaximum value ofiandj retained in the calculation, but the valueofRac converges relatively fast withi, j.


22/25


23/25


From this, we can calculate the associated velocity field,

p2 = Ra

D221

t2 (B.15)

=Ra

V0

D22

1+k=1

kr2k

(B.16)

=5

6V0PD

221

+

k=1 kr2k

(B.17)

=V0P+k=1

5

6

k(2k+ 7)(2k+ 5)(2k+ 2)2k

r2k+4 (B.18)

The general solution for p2,1is

p2,1= A2r2 +B2r

4 ++k=1

kr2k+4, (B.19)

where

k =5

6

k(2k+ 7)(2k+ 5)(2k+ 2)2k

(B.20)

= 25

2(2k+ 7)(2k+ 5)(2k+ 3)(2k+ 2)(2k+ 1)(2k 1)

(B.21)The constantsA2andB2have to be determined from the boundaryconditions. The stress free condition gives

3A2+ 8B2++k=1

k[2 + (k+ 2)(2k+ 3)] = 0 (B.22)

and the continuity of normal stress gives, ignoring O(P)terms,

15A2 21B2++k=1

k(2k+ 7)(2k2 + 2k 3) = 0. (B.23)

From equations (B.22) and (B.23), we obtain

B2 = 1

19

+

k=1

(k+ 1)(4k2 + 24k+ 19)k 0.0211 (B.24)

and

A2= 1

57

+k=1

k(32k2 + 186k+ 211)k 0.0346. (B.25)

B2 l= 1temperature field at O(P)and velocity field atO(P2)

Inserting in equation (B.9) the expression found above for the l= 2component of the velocity field,t1,1is now given by

t1,1 = p1,13

5A2r+B2r3 +

+

k=1

kr2k+3

+

k=1

kr2k .

(B.26)

After some rearrangements, we obtain

t1,1 = p1,1 3

5

+k=0

(A2k+1+B2k+k 0k) r2k+3,

(B.27)

where

k =k

i=0

iki. (B.28)

We can now determine the l = 1flow field by integrating theStokes equation with the above temperature field. Noting

p1= V0

2

r+ p1,1P+ p1,2P

2 + O(P3)

(B.29)

the second order contribution is given by

V0

2

P2p1,2= 6

V0

PRa D211 t1,1, (B.30)or

p1,2 = 10

D211

t1,1+Dr+Er3. (B.31)

We obtain

p1,2= 1

28Ar5

5

756Br7

5

2376Cr9

30

5

+k=0

A2k+1+ (B2 0)k+k(2k+ 9)(2k+ 6)(2k+ 7)(2k+ 4)

r2k+7

+Dr+Er3

(B.32)

which gives

p1,2= 128

Ar5 + 2531752

r7 566528

r9

30

5

+k=0

A2k+1+ (B2 0)k+k(2k+ 9)(2k+ 6)(2k+ 7)(2k+ 4)

r2k+7

+Dr+Er3

(B.33)

The constants A and Ecan be determined from the boundary cond-tions. The no-stress condition gives

E= 5

42A

115

24948+

+k=0

A2k+1+ (B2 0)k+ k(2k+ 9)(2k+ 4)

(B.34)

and continuity of the normal stress gives

23

7A= 6E+

316

1173

30

5

+k=0

A2k+1+ (B2 0)k+k(2k+ 9)(2k+ 6)(2k+ 7)(2k+ 4)

2(k+ 3)(4k2 + 24k+ 29).

(B.35)

Using equations (B.34) and(B.35), we obtain

A= 131

1764

3

2

+k=0

A2k+1+ (B2 0)k+k2k+ 7

0.0617.

(B.36)

The average velocity uxin thexdirection, defined as

ux= 1

Vic

Vic

uxdV (B.37)

is less than the infinite viscosity limit (hereVicis the volume of theinner core). Indeed, noting thatux= urcos usin , and that

ur =l,m

l(l+ 1)pml

r Pl, (B.38)

u =l,m

1

r

d

dr(rpml )

Pl

, (B.39)


24/25


we find

ux= 3

4

10

0

2p1cos

2 + d

dr(r p1)sin

2

r sin drdd

(B.40)

=

10

4p1+ 2r

dp1dr

r dr=

10

2d

dr

r2p1

dr (B.41)

= 2p1(r= 1) (B.42)

=V0

1 + (A +B+C) P+ O(P2)

, (B.43)

which gives

ux

6

5

Ra

P

1 0.0216 P+ O(P2)

. (B.44)

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