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GJI Geomagnetism, rock magnetism and palaeomagnetism

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Geophysical Journal International
Geophys. J. Int. (2009) 179, 1414–1428

doi: 10.1111/j.1365-246X.2009.04361.x

Modelling the palaeo-evolution of the geodynamo
Julien Aubert,1 Ste´phane Labrosse2 and Charles Poitou3
1Dynamique des Fluides Ge´ologiques, Institut de Physique du Globe de Paris, Universite´ Paris-Diderot, INSU/CNRS, 4 Place Jussieu, 75252, Paris cedex 05, France. E-mail: aubert@ipgp.jussieu.fr 2Laboratoire des sciences de la Terre, Ecole normale supe´rieure de Lyon, Universite´ de Lyon, CNRS UMR 5570, 46 Alle´e d’Italie, 69364 Lyon Cedex 07, France 3Pale´omagne´tisme, Institut de Physique du Globe de Paris, Universite´ Paris-Diderot, INSU/CNRS, 4 Place Jussieu, 75252, Paris cedex 05, France
Accepted 2009 August 11. Received 2009 May 22; in original form 2008 December 10
SUMMARY Although it is known that the geodynamo has been operating for at least 3.2 Ga, it remains difﬁcult to infer the intensity, dipolarity and stability (occurrence of reversals) of the Precambrian magnetic ﬁeld of the Earth. In order to assist the interpretation of palaeomagnetic data, we produce models for the long-term evolution of the geodynamo by combining core thermodynamics with a systematic scaling analysis of numerical dynamo simulations. We update earlier dynamo scaling results by exploring a parameter space, which has been extended in order to account for core aspect ratios and buoyancy source distributions relevant to Earth in the Precambrian. Our analysis highlights the central role of the convective power, which is an output of core thermodynamics and the main input of our updated scalings. As the thermal evolution of the Earth’s core is not well known, two end-member models of heat ﬂow evolution at the core–mantle boundary (CMB) are used, respectively, terminating at present heat ﬂows of 11 TW (high-power scenario) and 3 TW (low power scenario). The resulting models predict that until the appearance of the inner core, a thermal dynamo driven only by secular cooling, and without any need for radioactive heating, can produce a dipole moment of strength comparable to that of the present ﬁeld, thus precluding an interpretation of the oldest palaeomagnetic records as evidence of the inner core presence. The observed lack of strong long-term trends in palaeointensity data throughout the Earth’s history can be rationalized by the weakness of palaeointensity variations predicted by our models relatively to the data scatter. Speciﬁcally, the most signiﬁcant internal magnetic ﬁeld increase which we predict is associated to the sudden power increase resulting from inner core nucleation, but the dynamo becomes deeper-seated in the core, thus largely cancelling the increase at the core and Earth surface, and diminishing the prospect of observing this event in palaeointensity data. Our models additionally suggest that the geodynamo has lied close to the transition to polarity reversals throughout its history. In the Precambrian, we predict a dynamo with similar dipolarity and less frequent reversals than at present times, due to conditions of generally lower convective forcing. Quantifying the typical CMB heat ﬂow variation needed for the geodynamo to cross the transition from a reversing to a non-reversing state, we ﬁnd that it is unlikely that such a variation may have caused superchrons in the last 0.5 Ga without shutting down dynamo action altogether.
Key words: Dynamo: theories and simulations; Palaeointensity; Palaeomagnetic secular variation; Reversals: process, timescale, magnetostratigraphy.

1 INTRODUCTION
Palaeomagnetic records can be used in order to shed light onto the past of the Earth’s dynamo. However, with increasing age, rock sequences are likely to be affected by weathering, alteration and metamorphism, thus destroying the pristine information on the early magnetic ﬁeld. Although the situation constantly improves with newer samples and better techniques, especially regarding palaeoin-
1414

tensity determinations (see Tauxe & Yamazaki 2007, for a review), uncertainties remain concerning the key characteristics of the geodynamo in the Precambrian. A ﬁrst quantity of interest is the dipole moment. The oldest (3.2 Ga BP) reliable palaeointensity record to date (Tarduno et al. 2007) reveals that the virtual dipole moment (VDM) was possibly as large as its present value of 8 × 1022 A m2, although the consideration of the experimental cooling rate effect may lead to a twofold decrease in the determined VDM. As a
C 2009 The Authors Journal compilation C 2009 RAS

result, the debate concerning the long-term evolution of the VDM remains lively (Dunlop & Yu 2004), though there are suggestions (Macouin et al. 2004) of a long-term average monotonous increase from 3 × 1022 A m2 at 1000–2000 Myr to 8 × 1022 A m2 at present times. In addition, although it has been proposed (Stevenson et al. 1983; Hale 1987) that the power increase subsequent to inner core nucleation and onset of chemical convection in the core could cause a sudden increase in the dipole moment, it appears that until now, the palaeomagnetic data scatter has prevented a proper resolution of this feature. A second quantity of interest is the dipolarity of the ﬁeld, which can be assessed (McFadden et al. 1991) through the analysis of latitude dependence in palaeosecular variation (PSV). Following this line, Smirnov & Tarduno (2004) proposed that the dipolarity should have been higher some 2.5 Ga ago. However, the existence of antipodal directions, which would be a clear evidence of high dipolarity (Dunlop & Yu 2004), is lacking in their data, and more generally, there are considerable issues with this method (Hulot & Gallet 1996), including the difﬁculty to separate the dipolarity effect from the secular variation effect in the PSV. The PSV can indeed also be used to study the intrinsic variability of the geomagnetic pole, as recently done by Biggin et al. (2008), who found evidence of a more stable dynamo 2.4–2.8 Ga ago. This led to the suggestion that a third quantity of interest, the reversal frequency of the geodynamo, was lower at that time than at present. The same conclusion was reached for the last 500 Myr (Eide & Torsvik 1996) through direct magnetostratigraphic analysis, which is obviously a more robust method when data is available, but some contrary indications of high reversal frequency have also been reported for the same period (Pavlov & Gallet 2001). It should be noted however that the strongest signal on the reversal frequency curve is related to the irregular occurrence of superchrons (see for instance, Pavlov & Gallet 2005), where reversal frequency goes down to zero. A last important palaeomagnetic result is the timing of the oldest known reversal, which apparently occurred 2.7 Ga ago (Strik et al. 2003), though there are some indications of reversals occurring 3.2 Ga ago (Tarduno et al. 2007).
As none of the debate presented above is currently settled, the goal of this study is to gain insight from the comparison of palaeomagnetic results with synthetic time evolution models for the dipolarity, intensity and stability of the Precambrian Earth dynamo. Since a few years, a scaling theory (Christensen & Tilgner 2004; Christensen & Aubert 2006; Olson & Christensen 2006) is available to predict the main characteristic quantities of the present-day geodynamo and planetary dynamos. The central control variable in almost all scalings is the convective buoyancy ﬂux, which is equivalent to convective power available for the dynamo. Thus, in order to achieve scaling predictions relevant to the Precambrian Earth dynamo, one needs a time-series of the dynamo power, which can be evaluated from core thermodynamics and Earth cooling histories (e.g. Labrosse 2003; Lister 2003). Furthermore, the results of Christensen & Tilgner (2004), Christensen & Aubert (2006) and Olson & Christensen (2006) need to be extended to cases where the outer core aspect ratio varies, and where the partitioning between the inner- and outer-boundary originated buoyancies also varies. In these respects, this paper takes advantage of advances made in both ﬁelds of numerical dynamo modelling and core thermodynamics to update a previous analytical treatment (Olson 1981). In Section 2, we introduce the numerical dynamo model, which we use for our systematic parameter space study. The numerical results are presented in Section 3.1. Then we turn to the investigation of various Earth cooling histories, and how they relate to the dynamo power (Section 3.2). Finally, we produce our time evolution model
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Modelling the palaeo-evolution of the geodynamo 1415
for palaeomagnetic observables (Section 3.3) and discuss the results in light of the palaeomagnetic observations and geophysical constraints (Section 4).

2 MODEL

2.1 Outline

We consider an electrically conducting, incompressible ﬂuid in a self-gravitating spherical shell between radii r i and r o. The shell is rotating about an axis ez with an angular velocity , and convecting thermally and chemically. We study the Earth’s core at various stages of its existence, hence its rotation rate, aspect ratio χ = r i/r o and thermochemical buoyancy partitioning are variable over geological times, but can be assumed to be constant control parameters over timescales relevant for core dynamics. We deﬁne the deviation temperature ﬁeld T and light element mass fraction ﬁeld ξ with respect to the isentropic temperature and well-mixed mass fraction, and, within the Boussinesq approximation, both buoyancy effects are grouped into a codensity (or density anomaly) ﬁeld (Braginsky & Roberts 1995) C such that

C = αρT + ρξ .

(1)

Here α is the thermal expansion coefﬁcient, ρ is the ﬂuid density and ρ is the density difference between the light components that contribute to chemical convection and pure iron. The temperature and molar fraction ﬁelds are assumed to have the same diffusivity κ, due to turbulent mixing in the outer core. This allows us to write a single transport equation for the codensity C, which is solved numerically in a dimensionless form, together with the magnetic induction equation for the solenoidal magnetic ﬁeld B in the magnetohydrodynamic approximation, and the Navier–Stokes and thermochemical transport equations for the incompressible velocity ﬁeld u, and pressure P

∂u

r

∂t + u · ∇u + 2ez × u + ∇ P = RaQ ro C

+ (∇ × B) × B + E∇2u (2)

∂B ∂t

=

∇

× (u × B) +

E ∇2B Pm

(3)

∂C ∂t

+ u · ∇C

=

E Pr

∇2C

+

ST /ξ

(4)

∇·u=0

(5)

∇ · B = 0.

(6)

Here r is the radius vector. Time is scaled with the inverse of the rota-
tion rate −1. Length is scaled with the shell gap D = r o − r i. Velocity is scaled with D. Magnetic induction is scaled by (ρμ)1/2 D,

where ρ is the ﬂuid density and μ the magnetic permeability of the

ﬂuid. The Ekman number E, magnetic Prandtl and Prandtl numbers

Pm and Pr are deﬁned as

E= ν

(7)

D2

Pm

=

ν λ

(8)

Pr

=

ν κ

.

(9)

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1416 J. Aubert, S. Labrosse and C. Poitou

Here ν, λ are, respectively, the viscous and magnetic diffusivities of the ﬂuid. In this study, the codensity boundary conditions relevant to thermochemical convection are treated in the following way: at the inner-core boundary, the release of latent heat and light elements correspond to a positive mass anomaly ﬂux Fi (expressed in kilograms per second), which we consider uniform and imposed on the long term by global core thermodynamics. The dimensional form of Fi writes

Fi = − κ∇C · dS,

(10)

Si

where Si is the inner boundary surface. Similarly, at the outer boundary, we consider that the mantle imposes a uniform mass anomaly

ﬂux (which in fact corresponds to a heat ﬂux without chemical

contributions). The dimensional form of Fo writes

Fo = − κ∇C · dS.

(11)

So

Here So is the outer boundary surface. Note that Fo is deﬁned with respect to the adiabatic heat ﬂux carried out at the outer boundary
(this is the reference state of the Boussinesq system). Hence, Fo can be either positive or negative, but in any case the total mass anomaly ﬂux F = Fi + Fo must be positive for convection to occur. The codensity is scaled with F/4π D3 . The Rayleigh number based on mass anomaly ﬂux, RaQ, which appears in (2) is therefore deﬁned as

RaQ

=

go F . 4πρ 3 D4

(12)

Here go is gravity at radius r = r o. The present formulation is

slightly different from, but equivalent to that in Christensen &

Aubert (2006), where a Rayleigh number based on advected buoy-

ancy

ﬂux

Ra

∗ Q

was

introduced.

For

sufﬁciently

supercritical

convec-

tion the conversion from their formalism to ours can be achieved

through

RaQ

≈

rori D2

Ra

∗ Q

.

(13)

The

reason

for

introducing

this

change

of

formulation

is

that

Ra

∗ Q

is singular in the case where the inner core is not present.

In Section 3.1, we carry out a systematic analysis of the Boussi-

nesq system and let the mass anomaly ﬂuxes Fi and Fo be independent. The two associated control parameters are the Rayleigh

number RaQ and the fraction of inner-boundary originated buoyancy f i = Fi/F. However, we emphasize that in the Earth’s core, Fi and Fo are related through a consideration of the complete core thermodynamics, such as done in Sections 3.2 and 3.3. A general

situation describing an Earth system which is slowly cooling on

geological timescales is Fi = Fo, in which case the basic state over which the Boussinesq system is considered has a decreasing

temperature and increasing light element mass fraction, while the

Boussinesq system itself is statistically stationary. This can be ac-

counted for within our framework by adding a volumetric correction

term ST/ξ in (4), such that the mass anomaly budget of the spherical shell vanishes (Braginsky & Roberts 1995; Kutzner & Christensen

2002). The correction term then has the following dimensionless

expression:

ST/ξ

=

3 (1 − 2 fi) . ro3 − ri3

(14)

The other boundary conditions at both boundaries are of rigid type for velocity, and insulating for the magnetic ﬁeld (the effect on inner-core conductivity on the long-term behaviour of dynamo

simulations has been found to be negligible, as shown by Wicht 2002). The numerical implementation PARODY-JA is used in this study (see Aubert et al. 2008, for details). The numerical scheme is of ﬁnite-differencing type in the radial direction with up to 120 grid points, and uses a spherical harmonic decomposition in the lateral directions up to degree and order 106. No particular symmetry along longitude was assumed.
Table 1 gives details on the 43 models which we have integrated for this study. The parameter range is E = 3 × 10−5 to 3 × 10−4 for the Ekman number, RaQ = 10−6 to 10−3 for the Rayleigh number, Pm = 1–10 for the magnetic Prandtl number, and the Prandtl number Pr is set to 1 in all simulations. It should be kept in mind that due to computational limitations, and just like all accessible numerical dynamo simulations (Christensen & Aubert 2006), our simulations operate in a parametric regime still very far from that of the Earth’s core, where E ≈ 3 × 10−15, RaQ ≈ 10−13 and Pm = 10−6. Scaling analyses attempt to overcome this intrinsic limitation by identifying robust trends supported by reasonable underlying physical considerations. Most of the scalings used here bear little residual inﬂuence of the various diffusivities, which gives credence to their applicability to core conditions, since diffusivities represent the main reason why models and Earth are distant in parameter space. The models presented here explore the new parameter space axes relative to χ and f i. The aspect ratio goes down to χ = 0.01, where the inner core is practically non-existent in the simulation. This situation is meant to simulate the core before or at the inner core nucleation time. The geophysically relevant buoyancy driving mode is therefore secular cooling ( f i = 0), which, in our formalism, corresponds to no buoyancy at the inner boundary, a positive volumetric source term, and a ﬁxed heat ﬂow at the outer boundary. Several models have been computed at χ = 0.05, where f i has been set to 0, 0.5 and 1. These cases correspond to a system where the inner core has just nucleated, and the buoyancy driving is therefore a mix of secular cooling and chemical convection. Finally, models with χ = 0.35 were also needed for comparison with present Earth. Core thermodynamics predict that the main buoyancy source of the present-day geodynamo (in terms of available power) is chemical convection (Lister & Buffett 1995). For that reason, cases with f i = 0, χ = 0.35 are left out of the present study. However, the uncertainties pertaining to the determination of core–mantle boundary (CMB) heat ﬂow (Lay et al. 2008) and core adiabat leave some room for the determination of f i. Here we use the published runs from Christensen & Tilgner (2004), Christensen & Aubert (2006) and Olson & Christensen (2006), which have been performed with ﬁxed temperature boundary conditions. In our formalism, this is roughly equivalent to f i = 0.5 (see Fig. 1). In addition, we have included models with f i = 1, and also with f i = 2, 10, describing situation of dominant chemical convection where the CMB heat ﬂow is, respectively, just adiabatic, or below the adiabat. In the latter case, a stably stratiﬁed layer exists at the top of the outer core.
We deﬁne several outputs, which are all averaged over times much longer than core ﬂow timescales (but shorter than geological timescales for Earth’s mantle variations), and over the full volume of the spherical shell or the surface of the outer boundary. Most of these bear the same deﬁnitions as in Christensen & Aubert (2006): the root-mean-squared velocity inside the shell Ro, rms magnetic ﬁeld amplitude inside the shell Lo, mean harmonic degree in the velocity ﬁeld l, ohmic dissipation fraction of the convective power f ohm, ratio bdip of the mean ﬁeld strength inside the shell Lo to the dipole strength on the outer boundary Bdip, and ratio f dip of Bdip to the rms amplitude of the magnetic ﬁeld at the outer boundary truncated at spherical harmonic degree 12. Parameters with deﬁnitions which are
C 2009 The Authors, GJI, 179, 1414–1428
Journal compilation C 2009 RAS

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Modelling the palaeo-evolution of the geodynamo 1417

Table 1. Numerical models and results (see text for details). Models C and T are visualized in Fig. 2.

E

RaQ

Pr Pm χ

fi

Ro

Lo

bdip

f dip

l

τ diss/τ mag

p

f ohm

3 × 10−5 1.80 × 10−6 1 1 0.05 0.5 5.03 × 10−3 3.22 × 10−3

9.56 0.74 10.0 7.90 × 10−4 8.32 × 10−7 0.23

3 × 10−5 4.50 × 10−6 1 2 0.01 0 3.83 × 10−3 1.19 × 10−2

4.61 0.67 16.3 9.56 × 10−4 1.79 × 10−6 0.62

3 × 10−5 4.50 × 10−6 1 1 0.01 0 4.28 × 10−3 1.18 × 10−2

3.56 0.81 16.6 1.88 × 10−3 2.09 × 10−6 0.58

3 × 10−5 9.00 × 10−6 1 1 0.01 0 5.98 × 10−3 1.17 × 10−2

4.03 0.78 20.1 9.46 × 10−4 4.20 × 10−6 0.52

3 × 10−5 9.00 × 10−7 1 2 0.05 1 4.99 × 10−3 3.91 × 10−3 16.0 0.66 7.0 6.94 × 10−4 6.24 × 10−7 0.27

3 × 10−5 1.80 × 10−6 1 2 0.05 1 6.81 × 10−3 4.24 × 10−3 24.0 0.48 8.1 4.01 × 10−4 1.36 × 10−6 0.25

3 × 10−5 9.00 × 10−7 1 2 0.05 0.5 3.56 × 10−3 2.80 × 10−3 13.7 0.63 7.8 7.31 × 10−4 3.26 × 10−7 0.25

10−4

6 × 10−5

1 5 0.01 0 1.21 × 10−2 2.24 × 10−2

7.19 0.53 13.3 4.23 × 10−4 2.63 × 10−5 0.45

10−4

6 × 10−5

1 2 0.01 0 1.20 × 10−2

0

n/a

n/a 14.5

0

2.65 × 10−5

0

10−4

3 × 10−5

1 5 0.05 1 2.22 × 10−2 1.27 × 10−2 36.0 0.32 6.0 3.34 × 10−4 2.59 × 10−5 0.18

10−4

3 × 10−5

1 5 0.05 0.5 1.32 × 10−2 1.95 × 10−2

9.23 0.53 8.1 6.09 × 10−4 1.66 × 10−5 0.38

T

10−4

1.5 × 10−5 1 10 0.05 0 5.19 × 10−3 1.88 × 10−2

7.17 0.49 9.2 8.17 × 10−4 3.98 × 10−6 0.54

C

10−4

1.5 × 10−5 1 10 0.05 1 1.20 × 10−2 1.76 × 10−2 13.3 0.47 7.4 3.98 × 10−4 1.21 × 10−5 0.32

10−4

1.5 × 10−5 1 10 0.05 0.5 8.44 × 10−3 1.99 × 10−2

9.36 0.49 7.6 6.03 × 10−4 7.59 × 10−6 0.43

10−4

3 × 10−5

1 2 0.05 0 8.64 × 10−3 1.64 × 10−2

4.66 0.62 12.6 1.51 × 10−3 1.02 × 10−5 0.43

10−4

3 × 10−5

1 2 0.05 1 2.38 × 10−2 5.63 × 10−3 40.4 0.32 6.1 4.08 × 10−4 2.54 × 10−5 0.07

10−4

3 × 10−5

1 2 0.05 0.5 1.60 × 10−2 1.29 × 10−2

7.93 0.68 7.1 9.56 × 10−4 1.68 × 10−5 0.25

10−4

6 × 10−5

1 2 0.05 0.5 2.06 × 10−2 1.71 × 10−2

7.23 0.64 8.1 6.65 × 10−4 3.56 × 10−5 0.30

10−4

6 × 10−5

1 2 0.05 0 1.54 × 10−2 8.53 × 10−3 10.4 0.53 14.1 4.63 × 10−4 2.43 × 10−5 0.11

10−4

6 × 10−5

1 2 0.05 1 3.22 × 10−2 5.93 × 10−3 44.1 0.28 6.2 2.81 × 10−4 5.09 × 10−5 0.06

10−4

3 × 10−5

1 5 0.01 0 8.36 × 10−3 1.95 × 10−2

6.64 0.50 11.5 6.93 × 10−4 1.12 × 10−5 0.49

10−4

3 × 10−5

1 5 0.05 0 8.00 × 10−3 1.99 × 10−2

6.55 0.52 11.7 7.66 × 10−4 1.02 × 10−5 0.50

10−4

1.5 × 10−5 1 10 0.01 0 5.33 × 10−3 1.89 × 10−2

7.20 0.44 9.1 7.60 × 10−4 4.40 × 10−6 0.53

10−4

3 × 10−5

1 10 0.01 0 7.83 × 10−3 2.27 × 10−2

7.68 0.48 11.5 4.57 × 10−4 1.13 × 10−5 0.50

10−4

3 × 10−5

1 10 0.05 0 7.58 × 10−3 2.23 × 10−2

7.39 0.48 11.9 4.80 × 10−4 1.04 × 10−5 0.50

10−4

3 × 10−5

1 10 0.05 1 1.83 × 10−2 1.90 × 10−2 19.0 0.38 7.0 2.71 × 10−4 2.59 × 10−5 0.25

10−4

3 × 10−5

1 10 0.05 0.5 1.20 × 10−2 2.23 × 10−2 10.2 0.50 8.8 3.99 × 10−4 1.68 × 10−5 0.37

10−4

4 × 10−4

1 5 0.01 0 3.63 × 10−2 2.40 × 10−2 47.8 0.17 13.7 1.05 × 10−4 2.09 × 10−4 0.26

10−4

×10−4

1 5 0.01 0 1.70 × 10−2 2.34 × 10−2 11.0 0.50 12.4 3.04 × 10−4 4.65 × 10−5 0.38

10−4

1.5 × 10−4 1 5 0.01 0 2.26 × 10−2 2.26 × 10−2 19.2 0.30 11.5 2.14 × 10−4 7.26 × 10−5 0.32

10−4

1.5 × 10−4 1 5 0.05 0 2.21 × 10−2 2.05 × 10−2 18.8 0.33 13.1 1.98 × 10−4 6.81 × 10−5 0.30

10−4

2.5 × 10−4 1 5 0.01 0 2.92 × 10−2 2.06 × 10−2 46.9 0.18 13.5 1.28 × 10−4 1.26 × 10−4 0.26

10−4

×10−4

1 2 0.05 0.5 2.80 × 10−2 1.74 × 10−2

8.31 0.54 7.6 4.72 × 10−4 6.17 × 10−5 0.26

×10−4

2 × 10−4

1 2 0.05 0.5 4.14 × 10−2 1.72 × 10−2 18.3 0.38 7.2 2.73 × 10−4 1.27 × 10−4 0.18

3 × 10−4 1.80 × 10−5 1 3 0.35 10 2.28 × 10−2 3.52 × 10−3 14.3 0.73 6.9 1.03 × 10−3 2.96 × 10−5 0.02

3 × 10−4 2.48 × 10−5 1 3 0.35 10 2.64 × 10−2 1.61 × 10−3 38.1 0.49 7.2 7.75 × 10−4 4.20 × 10−5 0.04

3 × 10−4 3.15 × 10−5 1 3 0.35 10 2.92 × 10−2 2.30 × 10−3 70.0 0.35 7.6 6.61 × 10−4 5.45 × 10−5 0.07

3 × 10−4 4.50 × 10−5 1 3 0.35 10 3.34 × 10−2 6.78 × 10−3 60.0 0.40 8.0 5.81 × 10−4 8.00 × 10−5 0.04

3 × 10−4 7.20 × 10−5 1 3 0.35 10 4.01 × 10−2 1.17 × 10−2 106.3 0.27 8.3 5.03 × 10−4 1.31 × 10−4 0.10

3 × 10−4 1.08 × 10−4 1 3 0.35 10 4.87 × 10−2 1.61 × 10−2 161.7 0.22 8.4 4.44 × 10−4 2.30 × 10−4 0.14

3 × 10−4 4.50 × 10−4 1 3 0.35 2 4.61 × 10−2 1.59 × 10−2 85.8 0.20 9.0 4.80 × 10−4 2.02 × 10−4 0.13

3 × 10−4 9.41 × 10−4 1 3 0.35 1 4.82 × 10−2 2.83 × 10−2 20.2 0.40 10.1 5.52 × 10−4 2.95 × 10−4 0.24

3 × 10−4 1.05 × 10−3 1 3 0.35 1 5.14 × 10−2 2.78 × 10−2 28.2 0.32 10.0 5.09 × 10−4 3.29 × 10−4 0.23

proper to this study are the following: the convective power density p

is the ratio of the convective power deﬁned in Christensen & Aubert

(2006)

to

the

shell

volume

V

=

4π

(r

3 o

−

r

3 i

)/3.

The

magnetic

dissipation time τ diss is deﬁned as in Christensen & Tilgner (2004),

by dividing the magnetic energy by the power dissipated through

ohmic losses. However, in that study τ diss was normalized by the dipole diffusion time. Here we choose to normalize this quantity

by the standard magnetic diffusion time τ mag = D2/λ, in order to

account for the variation in the shell gap D throughout geological

time. For the determination of the system behaviour regarding the

occurrence of reversals, a standard length of three to ﬁve magnetic

diffusion times was used.

2.2 Generalized relationship between convective power and mass anomaly ﬂux/Rayleigh number
Here we derive a general relation between the convective power density p and the Rayleigh number RaQ. Buffett et al. (1996)
C 2009 The Authors, GJI, 179, 1414–1428 Journal compilation C 2009 RAS

demonstrated that under the assumption of good mixing (sufﬁciently supercritical convection), the total dissipation of the dynamo is proportional to the sum of the inner- and outer- boundary originated mass anomaly ﬂuxes

= i + o = Fi(ψi − ψ) + Fo(ψ − ψo).

(15)

Here ψ is the gravitational potential such that the gravity vector is g = −∇ψ, and ψi, ψo, ψ are, respectively, the inner boundary, outer boundary, and mean values of the gravitational potential. The physical meaning of (15) is that the dissipation results from taking mass anomaly at a given gravitational potential, and redistributing it at the mean gravitational potential, which is where the good mixing assumption enters.
In the present context of radial gravity, the gravitational potential is expressed as ψ = r 2go/2r o + cst. The expression for ψ is (Buffett et al. 1996)

ψ = 3go ro5 − ri5 . 10ro ro3 − ri3

(16)

1418 J. Aubert, S. Labrosse and C. Poitou

2.5 2
1.5 1

Δ T heating (fi

fi replaced by 1)

"Thermochemical" (f =0.5)
i

"Chemical" (f =1)
i

"Thermal" (fi=0)

Subadiabatic (fi=2)

Subadiabatic (fi=10)

p/RaQ

0.5

00

0.1

0.2

0.3

0.4

aspect ratio ri/ro

Figure 1. Test of the perfect mixing theoretical p/RaQ relationship with numerical data. Symbols represent the numerical data, and lines represent
the theoretical prediction of eqs (18) and (17). Red colour: Christensen
& Aubert (2006) data points. Our theory applies to their choice of ﬁxed temperature boundary conditions provided f i and 1 − f i are replaced by 1 in eq. (18). White symbol ﬁlling and light grey line: cases of secular cooling with f i = 0. Medium-grey symbol ﬁlling and line: cases of thermochemical convection with f i = 0.5. Dark-grey symbol ﬁlling and black line: cases of purely chemical convection with f i = 1. Blue symbols and lines: cases where the shell is stably stratiﬁed at the outer boundary (subadiabatic system) with f i = 2. Green symbols: same as blue, for f i = 10. For symbol shape deﬁnitions see Fig. 3.

Expanding (15) with the help of (16), and making use of the fact that the conservation of energy, when averaged over times long relatively to core ﬂow timescales, yields (ρ 3 D2) pV = (recall that p is a dimensionless quantity), we obtain after some algebra the following proportionality relationship between the power per unit volume p and the total mass anomaly ﬂux F

p = γ RaQ

(17)

with

γ = 3(ro − ri)2 2 ro3 − ri3 ro

fi

3 ro5 5 ro3

− ri5 − ri3

− ri2

+ (1 − fi)

ro2

−

3 5

ro5 ro3

− −

ri5 ri3

.

(18)

Eqs (18) and (17) are the generalizations to arbitrary buoyancy distributions of the relationships obtained in appendix A of Christensen & Aubert (2006). They are tested versus our numerical data in Fig. 1. The agreement between theoretical and numerical values of p/RaQ becomes good as the supercriticality of convection increases. Cases with a strong stable density stratiﬁcation (i.e. f i 1) need a stronger level of mixing to approach the theoretical line. This is simply a consequence of the fact that convection does not ﬁll the entire shell in these cases. For a given level of convection supercriticality, the mass anomaly, while indeed produced at the inner boundary, fails to be redistributed at the mean gravitational potential of the shell to a greater extent.
For the scaling relationships to be presented in the next section, the use of p instead of RaQ as a basic scaling parameter is motivated by several important reasons. First, the theory underlying these scalings (Christensen & Aubert 2006) relies on convective power arguments, while RaQ is merely an approximate proxy for

convective power. In our present study, where we vary the buoyancy partition f i and the aspect ratio χ , the use of RaQ as a basic scaling parameter does not capture the geometrical and buoyancy distribution effects contained in (18), thus resulting in a large scatter that the use of p corrects for. Another advantage of using p instead of RaQ is that p is a geophysical parameter which is constrained by thermodynamic studies of the Earth’s core (e.g. Lister 2003, see Section 3.2). Finally, expressing the scalings with p leads to a useful internal consistency relationship to be detailed in Section 3.3.
2.3 Statistics of least-squares ﬁts
In Section 3.1, we obtain power laws of the form y = axb from numerical data. The problem reduces to getting a linear least squares ﬁt of the form ln y = ln a + b ln x. In this ﬁt, the slope b is usually fairly well constrained by the large number of data points available. The standard least-squares error σ b on b is therefore discussed only as we check the internal consistency of the scalings. For all other purposes, the vertical variance σ 2 of the errors ei = ln yˆi − ln yi is used, and as a standard practice, we present the best-ﬁtting laws y = axb together with their 3σ lines y = (a/e3σ )xb and y = ae3σ xb, which theoretically enclose 99.7 per cent of the data if the distribution of errors ei is normal.
3 RESULTS
3.1 Results from numerical dynamos
We ﬁrst present (Fig. 2) images from two models with a small inner core (χ = 0.05, see Table 1 for other parameters). Model C is driven by inner-boundary originated buoyancy (chemical convection, f i = 1), and model T is driven by volumetric heating and outer-boundary originated buoyancy (secular cooling, f i = 0). Convection sets up where the thermochemical gradients are most unstable: one single convection cell near the inner boundary for C, ﬁve cells extending from mid-shell to the outer boundary for T. DMFI visualization (Aubert et al. 2008) reveals that the magnetic ﬁeld is generated according to the classical macroscopic α2 mechanism (Olson et al. 1999), although model C additionally has an enhanced toroidal ﬁeld production by zonal ﬂow near the outer boundary. Both models are dipole-dominated, with magnetic dipoles of similar relative strengths (C and T, respectively, have f dip = 0.47, 0.49). They are typically less dipolar than similar models with χ = 0.35 (see Fig. 6). In model T, the lower dipolarity can be explained by shallow convection columns which enhance magnetic ﬂux expulsion and thus enrich the outer boundary power spectrum in multipolar content. In model C, the small inner core surface over which the buoyancy is distributed favours the occurrence of magnetic upwellings (Aubert et al. 2008) which reduce the dipolarity by frequently disrupting the magnetic dipole. In both models, the absence of the inner core favours a global axisymmetric poloidal circulation, which, at a given instant in time, concentrates magnetic ﬁeld at one pole and disperses it at the other pole (in the snapshots presented in Fig. 2, ﬂux concentration is occurring at the south pole for model C and the north pole for model T). It should ﬁnally be noted that in thermal models, the absence of buoyancy at the inner-core boundary decouples the inner core from the convection and dynamo processes, with two consequences: ﬁrst, thermal models with χ = 0.05 or 0.01 yield almost the same results, which are presumably those which would be obtained from a case where r i = 0. Second, the relationship
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Figure 2. Magnetic ﬁeld and ﬂow morphologies from a chemically driven model (C, f i = 1), and a model driven by secular cooling (T, f i = 0). See Table 1 for other parameters. From top to bottom: Hammer projections of the radial magnetic ﬁeld at the outer boundary, harmonic degree power spectra normalized by the total power, DMFI equatorial and polar visualizations. The DMFI images present magnetic ﬁeld tubes (grey) with thicknesses normalized by the local magnetic energies, as well as two isosurfaces of the axial vorticity ωz = (∇ ×u) · ez, with levels −0.54 (blue) and 0.54 (red) for model C, and ± 0.18 for T. The outer boundary is colour coded with the radial magnetic ﬁeld, with similar colour scheme as on the Hammer projections. The thick white line is the rotation axis. For other details on DMFI imaging see Aubert et al. (2008).

between reverse magnetic ﬂux patch locations and inner core size (Stanley et al. 2007) does not hold if secular cooling drives the dynamo.
Fig. 3 presents a plot of the Rossby number Ro (or dimensionless rms ﬂow velocity) versus the dimensionless convective power p.

The best ﬁt and 3σ lines have equations

Ro = (0.69, 1.31, 2.49) p0.42.

(19)

When cast into a p − Ro space, the Christensen & Aubert (2006) scaling is virtually unchanged by the addition of the new data points

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1420 J. Aubert, S. Labrosse and C. Poitou
10 10

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Ro

10
10
10 10 10 10 10 10 10 10 p
Figure 3. Dimensionless rms velocity, or Rossby number Ro as a function of the dimensionless convective power p. Red crosses are the Christensen & Aubert (2006) data points. Other symbol colours are deﬁned in Fig. 1. Symbol shapes are as follows: small aspect ratio models (χ < 0.05) have circles for Pm = 10, squares for Pm = 5, diamonds for Pm = 2, ﬁlled stars for Pm = 1. Present aspect ratio (χ = 0.35) additional models (triangles and open stars) have Pm = 3. Our data set is ﬁltered to exclude the non-dynamo run.
10

Lo/f1o/h2m

10

10

10 10 10 10 10 10 10 p

Figure 4. Dimensionless magnetic ﬁeld, or Lorentz number Lo, corrected

with the ohmic dissipation fraction

f

0.5 ohm

,

as

a

function

of

the

dimensionless

convective power p. Symbols as in Fig. 3. Our data set is ﬁltered to exclude

dynamos with a dipole fraction f dip ≤ 0.35.

with variable inner core size and buoyancy distribution, with a scatter which is also unchanged. In the dipole-dominated regime ( f dip > 0.35), the same remarks hold for the Lorentz number Lo (or dimensionless rms magnetic ﬁeld) scaling (Fig. 4), which obeys

Lo/ fo0h.m5 = (0.62, 1.17, 2.22) p0.34.

(20)

We deduce that the convective power p is the primary scaling parameter for magnetic and velocity ﬁeld amplitude, and integrates the relevant dependencies on inner core size and buoyancy distributions for our present purposes. We further check (Fig. 5) the relation between the magnetic dissipation time τ diss and the magnetic Reynolds number Rm = RoPm/E (Christensen & Tilgner 2004)

τdiss/τmag = (0.11, 0.26, 0.65)Rm−1.0.

(21)

τdiss/τmag

10

10

102

Rm

103

Figure 5. Magnetic dissipation time τ diss, normalized by the standard magnetic diffusion time τ mag, as a function of the magnetic Reynolds number Rm. Symbols as in Fig. 3, except the red crosses which represent the Christensen & Tilgner (2004) models. Our data set is ﬁltered to exclude the non-dynamo run.

For this last scaling, the newer runs cause a signiﬁcantly larger scatter than that obtained by Christensen & Tilgner (2004). We attribute this to the rather small size (26 models) and limited parameter space extent of the data set used in that study, and conclude that the aspect ratio and buoyancy distribution have little inﬂuence on this scaling.
For an extrapolation of the three scalings presented above to Earth’s core conditions, the scaling prefactors need to be determined. Here we derive a useful internal consistency relationship tying these prefactors. We ﬁrst deﬁne the dimensional values Brms and U rms for Lo and Ro, expand them using (19) and (20) and also recall the deﬁnition of τ diss

Brms = Lo(ρμ)1/2 D = c1 fo0h.m5 p0.34(ρμ)1/2 D,

(22)

τdiss = c2 Rm−1 D2/λ with Rm = Urms D/λ

(23)

Urms = ( D)Ro = c3 p0.42 D.

(24)

Here the ci coefﬁcients are scaling prefactors, to be chosen within the 3σ range of each scaling. In the case of the geodynamo, where the magnetic diffusivity is much larger than the viscous diffusivity,
and the magnetic energy much larger than the kinetic energy, we expect most of the convective power ρ 3 D2 pV to be dissipated through ohmic losses (i.e. f ohm ≈ 1), hence

ρ

3 D2 p =

Br2ms 1 . 2μ τdiss

(25)

After expanding all quantities with the use of their scaling laws, the dimensionless version of (25) writes

1 = c12c3 p0.1.

(26)

2c2

This reveals a good consistency of the scalings: for instance, the dependency in λ vanishes as it should, thanks to the scaling exponent −1 in (23). However consistency is not perfect, as witnessed by the residual power 0.1 at which p appears. As in the next section, p will be assumed to vary over geological time, this will prevent (26) to be exactly satisﬁed with time-independent values for c1−3. This problem can be related to the standard error in the least-squares determination of each exponent. We obtained standard exponent

C 2009 The Authors, GJI, 179, 1414–1428 Journal compilation C 2009 RAS

1

dipolar dynamos

0.8

0.6

fdip

0.4

nondipolar dynamos

0.2

no reversals

reversals

0 10

10

10

100

Rol

Figure 6. Dipolar fraction f dip as a function of the local Rossby number Rol = Rol/π . Symbols as in Fig. 3. The dashed lines delineate the transition point from stable to reversing dynamos: red dashes for the χ = 0.35 models from Christensen & Aubert (2006), green for χ = 0.35 and f i > 1, light grey for χ = 0.01–0.05 and f i = 0, medium grey for χ = 0.01–0.05 and f i = 0.5, black for χ = 0.01–0.05 and f i = 1. Our data set is ﬁltered to
exclude the non-dynamo run.

errors of σ b1 = 0.0169, σ b2 = 0.0639 and σ b3 = 0.0178 for the Lo, τ diss and Ro scalings, respectively. The error on the exponent of the ohmic dissipation, in the right-hand side of (26), is therefore

σb = 2σb1 + 0.42σb2 + σb3 = 0.078.

(27)

As σ b is comparable with the residual power 0.1 present in (26), we conclude that this inconsistency reﬂects the inherent error introduced by data scatter in the least-squares ﬁtting procedure used to derive the three scalings. This error remains small however, and can be compensated (as will be done in Section 3.3) by choosing a set of c1−3 that minimizes the deviations caused by p0.1 (typically 20 per cent).
The control parameter Rol = Rol/π was identiﬁed in Christensen & Aubert (2006) as the main parameter governing the dipolarity and stability versus reversals of numerical dynamo models. Here we follow the same approach (Fig. 6). In the dipoledominated regime ( f dip > 0.35), Fig. 6 shows again that the absence of the inner core generally decreases the dipolarity, as already explained in Fig. 2. In contrast, the inﬂuence of buoyancy distribution on dipolarity is not clear-cut and not monotonic. The critical Rossby number for reversals Rolc is located in a narrow range Rolc = 0.04−0.12. The frequency of polarity reversals increases with increasing departures of Rol from Rolc.
We then analyse (Fig. 7) the ratio bdip, which determines the relative strength of the internal ﬁeld and the dipole ﬁeld at the outer boundary. Since, as shown by Fig. 6, the geodynamo is likely to have had a dipolarity lower or equal to the present-day value throughout its history, we restrict the data to the range 0.4 < f dip < 0.7. In this range, we ﬁnd little variability of bdip with the convective power (or Rol), whereas the main source of variability comes from f i and χ . This can be attributed to depth variations of the dynamo region, the dipole seen at the surface being reduced in the case of a deeper dynamo. Indeed chemical models (deeper dynamos) have a markedly larger bdip value than thermal models (shallow dynamos, see Fig. 2). The highest bdip values are obtained for models with f i > 1, where the extent of the dynamo region is restricted even further by a stably stratiﬁed layer at the top of the shell (see for

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Modelling the palaeo-evolution of the geodynamo 1421
102

bdip

101

100

10

10

Rol

Figure 7. Ratio bdip of the rms magnetic ﬁeld inside the shell to the dipole ﬁeld at the outer boundary, as a function of the local Rossby number Rol. Symbols as in Fig. 3. The dashed lines locate the predictions of model (28), with same colour conventions as in Fig. 6. Our data set is ﬁltered to retain dynamos with a dipole fraction 0.4 < f dip < 0.7.

instance, Christensen 2006). In Section 3.3, a model will be needed for bdip in order to estimate the dipole moment at any point in the geological history. We adopt the simplest possible linear dependency

bdip = 7.3(1 − χ )(1 + fi).

(28)

The last scaling which we check (Fig. 8) is the relationship (Olson & Christensen 2006) between the local Rossby number Rol and the dynamo control parameters p, Ek, Pm and Pr. As there does not currently exist a physical rationale to exclude some control parameters, the powers in this last scaling were obtained by an approach of empirical scatter minimization. We conﬁrm (Fig. 8a) the scaling relationship obtained by Olson & Christensen (2006)

Rol = (0.34, 0.68, 1.35) p0.48 E −0.32 Pr 0.19 Pm−0.19.

(29)

Unlike in the previous scalings, there appears an additional dependence on the aspect ratio χ , which we resolve by adopting the following relationship, yielding a smaller data scatter (Fig. 8b):

Rol (1 + χ )

=

(0.33, 0.54, 0.89) p0.48 E −0.32 Pr 0.19 Pm−0.19.

(30)

Note that factoring out a (1 + χ ) dependence also slightly reduces

the scatter in the critical Rolc values obtained from Fig. 6. We therefore subsequently replace the criterion for reversals Rolc = 0.04–0.12 with

Rolc = (0.33, 0.54, 0.89) p0.48 E −0.32 Pr 0.19 Pm−0.19 (1 + χ )

≈ 0.04−0.1.

(31)

The aspect ratio thus disappears from our condition for the onset of reversals.

3.2 Core thermodynamics and cooling models
The previous section shows how time-average properties of the past geodynamo can be estimated from the convective power. The assumed time scale for this time average is long (say a million years) compared to core ﬂow timescales, but short compared to any geological evolution timescale of the Earth. The geological evolution

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1422 J. Aubert, S. Labrosse and C. Poitou
a.
100

b.
100

10

10

Rol Rol/(1+χ)

10

10

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10 10 10

10

10 10

100

p E Pr2/5 Pm

10 10 10 10 10 10

100

p E Pr2/5 Pm

Figure 8. Local Rossby number Rol as a function of the combination p E−2/3 Pr 2/5 Pm−2/5 proposed in Olson & Christensen (2006). (a) Without a dependence in χ , (b) with a dependence in χ . Symbols as in Fig. 3. Our data set is ﬁltered to exclude the non-dynamo run.

of dynamo properties can be obtained from our scaling laws if we have a long-term evolution model for the convective power, which we now derive from core thermodynamics (see the recent studies from Labrosse 2003; Lister 2003, and references therein). The fundamental inputs which are needed is an history of core-mantle heat ﬂow Qcmb(t), of the radioactive core heating Qr(t) and the heat ﬂow down the isentropic temperature gradient at the CMB Qa (hereafter termed adiabatic heat ﬂow), which is assumed constant over time. In what follows, we use the simple parametrization of Lister (2003) which holds if the inner core volume is small relative to the outer core volume (χ 3 1).
The entropy budget of the system, when time-averaged over timescales much longer than core ﬂow timescales, but shorter than geological timescales, gives an expression for the total dynamo dissipation as a function of the fundamental inputs listed above. The part i of the dissipation that originates from mass anomaly ﬂux at the inner boundary is then

i = (Qcmb − Qr )( L + B ),

(32)

where L and B are, respectively, the thermodynamic efﬁciencies

of latent heat and light element release, which are given by

L

=

2

3L[1 − e(φ)]χ + 3(L + B − C)χ

(33)

B

=

2

+

3Bχ 3(L + B

−

C)χ

.

(34)

The values of the thermodynamic parameters L, B, C, e(φ) and φ are given in Table 2. The part o of the dissipation that originates from mass anomaly ﬂux at the outer boundary is

o = (Qcmb − Qa ) S,

(35)

where S is the thermodynamic efﬁciency of thermal convection, given by

S = eφe(φ) − 1.

(36)

The dimensionless, total volumetric power p can be obtained through the time average conservation of energy ρ 3 D2 pV =
i + o, where V is the shell volume. Once the dissipations i and o are known, the associated mass anomaly ﬂuxes Fi and Fo can be retrieved from eq. (15), to ﬁnally determine f i = Fi/ (Fi + Fo). Note that for strongly subadiabatic cases, eq. (15) may

Table 2. Parameters used in the thermodynamic, core cooling (Section 3.2, upper panel), and palaeomagnetic (Section 3.3, lower panel) models.

Parameter

Meaning

Value

Reference

QCMB(t ) Qa Qr M L B C φ e(φ)

Earth’s cooling model Adiabatic heat ﬂow at the CMB Radiogenic heating in the outer core Heat capacity for solidiﬁcation Latent heat effect Buoyancy effect Compositional effect Adiabatic decay parameter Adiabatic decay integral

Variable 6 TW 0W 9.2 × 1029 J 2.1 0.86 −0.8 0.256 0.8595

See Section 3.2 See Section 3.2 See Section 3.2 Lister (2003) Lister (2003) Lister (2003) Lister (2003) Lister (2003) Lister (2003)

ro (t )
λ ν
κ
ρ

Outer core radius Earth’s rotation rate Outer core magnetic diffusivity Outer core viscosity Outer core thermochemical diffusivity (assumed to be the same as thermal diffusivity) Outer core density

3480 km variable 1.3 m2 s−1 10−6 m2 s
5 × 10−6 m2 s−1
104 kg m−3

See Section 3.3 Secco & Shloessin (1989) de Wijs et al. (1998) Stacey & Loper (2007), Labrosse et al. (2007), Lay et al. (2008) Dziewonski & Anderson (1981)

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Heat flow or Power (TW) Heat flow or Power (TW)

a: high power

b: low power

12

12

10

10

8

Qcmb

8

Qa

6

Φi Φo

6

4

P=Φi+Φo

4

Qcmb Qa Φi Φo P=Φi+Φo

2

Inner core nucleation

2

Inner core nucleation

0

4

3

2

1

0

time before present (Gyr)

0

4

3

2

1

0

time before present (Gyr)

Figure 9. High- (a) and Low- (b) power models for: heat ﬂow from the core to the mantle Qcmb, adiabatic heat ﬂow Qa, inner- and outer- boundary originated dissipations i, o, and total power P = i + o as functions of the time before present. The greyed area represents the uncertainty range for the adiabatic heat ﬂow. The inner core nucleates at a ≈ 1.8 Ga before present in the low power scenario, and a ≈ 0.8 Ga before present in the high-power scenario.

not hold because of incomplete mixing, which in this case might lead to an underestimation of Fi. It should be mentioned here that breaking the total dissipation into boundary-originated terms, and equating these two terms separately to their equivalent convective ﬂuxes is not a trivial operation, since dissipation is a global nonlinear quantity. The derivation presented in appendix shows that this is legitimate if the accuracy of the Boussinesq approximation is tolerable.
Once the inner core is present, the evolution of the inner core aspect ratio χ is constrained by the heat capacity for solidiﬁcation M (value in Table 2) through the equation

Md dt

χ 2 + (L + B − C)χ 3

= −(Qcmb − Qr ).

(37)

Eq. (37) can be integrated backwards in time from present (Labrosse et al. 2001) until the inner core age a (Here time is measured before present, therefore a > 0).
The amount of radioactive heating Qr in planetary cores is debated. Experiments of potassium partitioning between iron and silicates suggest that present maximum potassium concentration amounts to values ranging from 30 p.p.m (Hirao et al. 2006) to 60–130 p.p.m (Rama Murthy et al. 2003), the maximum value being obtained for a sulphur rich (10 per cent wt.) core, which is not favoured for the Earth based on geochemical constraints (McDonough 2003) yielding a sulphur content of 3 per cent wt. An upper bound of 60 p.p.m for the Earth’s core seems therefore reasonable and contributes 0.4 TW of present radioactive power, which is quite low when compared to typical Qcmb values. Radioactivity was obviously stronger in the past (in the case of potassium, the power is double every 1.26 Gyr backwards), but this would amount to typically 1.5 TW at 3 Ga ago, which again is quite low compared to estimated Qcmb at that time. In the present study, we therefore neglect the radioactive heating throughout the Earth’s history, that is, Qr = 0. Its inclusion is straightforward but unnecessary at this point.
The value of the adiabatic heat ﬂow Qa is uncertain and debated. Following Stacey & Loper (2007), Labrosse et al. (2007) and Lay et al. (2008), we adopt Qa = 6 TW for a central value and allow for an uncertainty range of 1 TW above and below this value. This would correspond to a central value of the upper outer core thermal conductivity of about 50 W m−1 K at the top of the core (Labrosse 2003).

We now turn to the central unknown of our analysis, the history of the heat ﬂow at the CMB, Qcmb(t). Since there are many uncertainties involved in the determination of Earth cooling models, as well as in our present modelling effort, our goal is not to propose a deﬁnitive model for the geologic evolution of the geodynamo, which would be based on a deﬁnitive model for Qcmb(t). Rather, we focus on two end-member scenarios representing the variety of geophysical situations which can be expected based on the uncertainties (Fig. 9). The ﬁrst one, which we label as the high-power scenario, was proposed by Labrosse et al. (2007). It is motivated by the large (about 10 TW) present heat ﬂows at the CMB deduced from post-perovskite seismological studies (Hernlund et al. 2005; Lay et al. 2006), from geochemical constraints and from the suggested present crystallization of a basal magma ocean in the lower mantle. As indicated by Fig. 9(a), it yields a typical dynamo power of 2.7 TW at present. The second, low power scenario is motivated by the fact that the scaling of ohmic dissipation in numerical dynamos (Christensen & Tilgner 2004) favours a low present dissipation of about 0.2–0.5 TW. Using the thermodynamic analysis presented above, this implies that the top of the Earth’s core is presently subadiabatic (Qcmb < Qa, see Fig. 9b). A variety of idealized, constant rate cooling histories can be built, which cross the adiabat at an age b. Plausible models are such that b ≤ a (Labrosse et al. 1997), because if b > a then convection stops in the Earth’s core between the adiabat crossing and the nucleation of the inner core. This is not acceptable since a conducting core would not cool fast enough to subsequently nucleate an inner core before present. Our second scenario is built according to this constraint, taking an initial CMB heat ﬂow of Qcmb = 11 TW, and a present value Qcmb = 3 TW, corresponding to a present dynamo power of about 0.3 TW.
3.3 Time evolution models for palaeomagnetic observables
We now combine the dynamo scaling study from Section 3.1 with the thermodynamic elements from Section 3.2 in order to evaluate how the main properties of the geodynamo evolve over time. At any point in time, the power p can be accessed from the analysis of the previous section. The Prandtl and magnetic Prandtl numbers are set according to the diffusivity values listed in Table 2. In order to determine the Ekman number E, the rotation rate of the Earth is needed. We use the length-of-day model (LOD) of Varga

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Figure 10. (a) Dynamo power and ohmic dissipation (respectively, left- and right-hand side of the internal consistency relationship (25), both multiplied by the shell volume V ), using the prefactor set (c1, c2, c3) = (1.65, 0.11, 1.31) for both scenarios. (b) rms core velocity U rms, with indications of the equivalent magnetic Reynolds numbers Rm at present, and a rough delineation of the dynamo onset which would correspond to Rm ≈ 40 (Christensen & Aubert 2006). (c) rms core magnetic ﬁeld Brms. (d) Local magnetic Reynolds number Rol (corrected by 1 + χ ). For this last scaling the central value 0.54 from (30) is used, and the 3σ uncertainty range is propagated to the location of the critical value Rolc for reversals obtained from (31) (shaded zone, lighter shade of grey means higher likeliness for reversals/higher reversal frequency). In (b)–(d), the dashes represent the epoch with no available palaeomagnetic samples.

et al. (1998), according to which the LOD has piecewise linearly increased from 19 hr 2.5 Ga ago to 20.8 hr 0.64 Ga ago, and to 24 hr today. As there is no constraint on earlier length of day, we backward continue the 2.5–0.5 Ga trend, thus yielding an initial length of day of 17 hr. It should be mentioned that the length-of-day variation should not exceed a factor 2 in any case, which has a weak impact on the scalings where the Ekman number is present.
The determination of absolute values for dynamo properties is subject to a considerable amount of uncertainty, which is discussed in detail in the next section. For that reason, we focus on the trends, or absence thereof, rather than the absolute values. Here we therefore present time evolution models obtained with scaling prefactors from (22) and (24) as close to their central values as possible, while still satisfying the constraint (26) of internal consistency. Using the central values for the prefactors ci as a starting point, we obtain p0.1c21c3/2c2 ≈ 0.2 throughout time. We therefore need to adjust the prefactors ci within the 3σ error range, increasing c1,3 and decreasing c2. In order to keep c1,3 close to their central values, we ﬁrst decrease c2 from 0.26 to its minimal acceptable value 0.11. There subsequently remains some discrepancy in (26), which we cancel out by increasing c1 from 1.17 to 1.65 while keeping c3 = 1.31. Our predictions for U rms (or the magnetic Reynolds number Rm), Brms and the local Rossby number Rol are reported in Figs 10(b), (c) and (d). We note that the model for Brms implicitly assumes that f ohm ≈

1 throughout time, and that the dynamo has been dipole-dominated throughout Earth’s history. This last point is reasonable since our models show that Rol, the parameter controlling the breakdown of dipolarity (Fig. 6) has been below its present-day value throughout Earth’s history (Fig. 10d).
An illustrative indication of how the internal properties evolution previously computed may translate to surface observables can be obtained by computing the true dipole moment

M = 4√πro3 Brms .

(38)

2μ bdip

For the determination of bdip, we use the simple model (28), the time evolution of which is presented in Fig. 11(a). In order to put the results in perspective with the considerable scatter in palaeointensity data, the resulting true dipole moment time-series (Fig. 11b) are presented together with virtual dipole moment values (VDM) from the IAGA palaeointensity database (Perrin & Schnepp 2004; Biggin et al. 2009).

4 DISCUSSION
Fig. 11(b) shows that throughout the Earth’s history, the dipole moments predicted by our models agrees with the observed palaeointensities to better than an order of magnitude, a fact which can be
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Modelling the palaeo-evolution of the geodynamo 1425

TDM / VDM (1022 A.m2)

a.
4

2

High power

Low power

0 30

b.

15

High power

Low power

10

fi

bdip

20

10

05

4

3

2

1

0

Time before present (Gyr)

5

04

3

2

1

0

Time before present (Gyr)

Figure 11. (a) Time evolutions for f i, obtained from (15), and for bdip, obtained from (28). (b) Time evolution models for the true dipole moment (TDM) M, with the same choice of scaling prefactors as in Fig. 10. The black dots represent the virtual dipole moment (VDM) values from the IAGA palaeomagnetic database (Perrin & Schnepp 2004; Biggin et al. 2009). Only VDM data obtained from Thellier–Thellier-type experiments using pTRM consistency checks (Thellier & Thellier 1959) were selected for robustness. Samples with an age of less than 10 Myr, of unknown or transitional polarity, or with VDM standard deviation greater than 20 per cent of the mean were also excluded, thus reducing the data set to 224 reliable points, 24 of which corresponding to the Precambrian.

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seen as a success for the underlying theories of convective dynamos and core thermodynamics. However, it must be kept in mind that a more speciﬁc analysis, such as discriminating between the highand low-power scenarios based on comparison with palaeointensity data, is necessarily plagued by a considerable amount of uncertainty. First, we have used central prefactor values for our scaling predictions, bearing a typical uncertainty of a factor 2 above and below the central line. Similarly, the conversion from deep to surface properties is also suffering from a similar factor 2 uncertainty (Fig. 7) and from departures from the perfect mixing theory (though in our models, f i does not greatly exceed 1, thus limiting the impact of incomplete mixing). The determination of thermodynamic properties in the upper part of Table 2 is also uncertain to a typical factor 2 (Lister 2003), which mainly affects the determination of p from a given cooling scenario. The diffusivities in the lower part of Table 2 are even less precisely determined, but we note that most of our scalings draw their interest from the fact that they are diffusivity-independent. Finally, the palaeointensity data against which we compare our models has a large scatter which could result from either time variability of the ﬁeld or intrinsic determination uncertainty (Tauxe & Yamazaki 2007). In these respects, the most signiﬁcant conclusion that can be drawn from the use of the two end-member cooling scenarios is an evaluation of the sensitivity of predicted palaeointensities to uncertainties in the Earth’s cooling history.
Studying the time variations of dynamo properties, rather than their absolute values, makes however more sense because among all the uncertainties listed above, a number can be supposed to be time-independent: material and thermodynamic properties, and diffusivities which are the main source of scatter around the central line in dynamo scalings. We additionally focus on trends, which are independent on the chosen cooling scenario. Still, the dipole moment variations predicted by our models are in any case small, owing to the relatively small exponent 1/3 to which the power enters in the magnetic ﬁeld scaling, and smaller than the typical scatter in palaeointensity data, a fact which rationalizes the observed lack of strong long-term trends (Tauxe & Yamazaki 2007) in the palaeointensity time-series.
The outstanding events in the evolution of the internal ﬁeld amplitude, as presented in Fig. 10(c), are the onset of dynamo action,
C 2009 The Authors, GJI, 179, 1414–1428 Journal compilation C 2009 RAS

and the nucleation of the inner core. Fig. 10(b) shows that the dynamo has operated above the typical value of the critical magnetic Reynolds number Rmc ≈ 40 (Christensen & Aubert 2006) throughout the period 3.2 Ga to present which is documented by palaeomagnetic records. Our simulations show that prior to the nucleation of the inner core, a thermal dynamo driven by secular cooling alone, and without any need for radioactivity, has no difﬁculties to generate a magnetic ﬁeld with strength comparable to that of the present ﬁeld: both cooling scenarios predict a dipole moment of 5 × 1022 A m2 3 Ga ago, on par with present-day values, and compatible with the ﬁndings of Tarduno et al. (2007). Based on this result, the suggestion (Dunlop 2007) that old palaeomagnetic samples such as those analysed by Tarduno et al. (2007) provide evidence that the inner core was already present before 3.2 Ga ago should be discarded. While the inner-core nucleation itself certainly has a signiﬁcant impact on the internal magnetic ﬁeld amplitude, the signature is largely attenuated in the dipole moment time-series (Fig. 11a), due to the evolution of the conversion factor bdip. Indeed, as the inner core nucleates, the onset of chemical convection brings additional thermodynamic efﬁciency in the system, extracting more power from a given CMB heat ﬂow. At the same time, however, the location of the dynamo shifts from mid-shell to the shell centre, thus masking the ﬁeld amplitude increase seen from the surface. The prospect of observing the signature of inner core nucleation through a sudden increase of the palaeointensity at the surface of the Earth, following the suggestion of Hale (1987), is therefore reduced.
We now discuss the predicted evolution for the dipolarity and stability of the geodynamo. Our results suggest that for a given convective forcing (more precisely, for a given local Rossby number), the absence of an inner core reduces the dipolarity. All forcing conditions being equal, the earlier geodynamo could therefore have been less dipolar than at present. However, inferring the dipolarity of the past geodynamo implies to weigh the effects of the geometry at given forcing, and of the forcing variations, a smaller Rol implying a possibly more stable and more dipolar dynamo (see Fig. 6). As, in both models, Rol has been generally lower than its present value throughout Earth’s history, and also less than an order of magnitude away from the critical value Rolc for reversals (Fig. 10d), it is possible that 3–2.5 Ga ago, the effects related to the absence of the inner core and to a lower Rol might have compensated to yield

minimal CMB heat flow (TW) CMB heat flow fluctuations (%)

1426 J. Aubert, S. Labrosse and C. Poitou 10 8

100

High power

80

Low power

6

60

4

40

2

High power Low power

20

04

3

2

1

0

Time before present (Gyr)

04

3

2

1

0

Time before present (Gyr)

Figure 12. Minimal CMB heat ﬂow (left-hand side: absolute, right-hand side: relative to actual CMB heat ﬂow) needed to get a dynamo (solid lines) and reversals (dashed lines).

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a dynamo with about present-day dipolarity, in line with the ﬁndings of Roberts & Glatzmaier (2001), Smirnov & Tarduno (2004), and with less frequent reversals than at present, a result which is also in good qualitative agreement with the conclusions of Coe & Glatzmaier (2006), Biggin et al. (2008).
Mantle dynamics can cause sizeable temporal ﬂuctuations of the CMB heat ﬂow (see for instance Nakagawa & Tackley 2005), which would superimpose to the long-term geological trend which we used to produce our models. To assess the impact of these ﬂuctuations, we compute in Fig. 12 the minimal CMB heat ﬂow required to get a dynamo. Here it is assumed that Rmc = 40, but this value has a negligible effect on the result, which can be determined by simply writing down the condition p ≥ 0. If the goal is to maintain a working dynamo throughout the Precambrian, then the low-power scenario appears to impose quite drastic restrictions on the amplitude of allowed ﬂuctuations, especially near the inner core nucleation time, when the dynamo is very close to shut down. In contrast, the highpower scenario allows for ﬂuctuations of 30 per cent and larger throughout the period 3.2 Ga ago to present. The same analysis can be repeated for the onset of reversals, assuming for instance that the critical local Rossby number for reversals is Rolc/(1 + χ ) = 0.07. The thermal dynamo, which pertained to the early Earth conditions would have needed about 1–2 TW superadiabatic CMB heat ﬂow to reverse. Here both scenarios are compatible with the oldest timing for reversals, which is in the range 2.7–3.2 Ga (Strik et al. 2003; Tarduno et al. 2007). At recent-times, the amount of ﬂuctuations needed to shut down reversals is very close to that needed to shut down the dynamo altogether, a fact which does not support the likeliness of mantle-induced superchrons in the last 500 Myr. Rather, our results would support the idea of superchrons being integral to the variety of timescales spontaneously produced by the dynamo, as suggested by Hulot & Gallet (2003).
The palaeomagnetic predictions derived from the low-power cooling scenario are in fair agreement with palaeomagnetic observables (order of magnitude of the dipole moment, occurrence of reversals). This result conﬁrms that from the perspective of scaling analyses, the power requirement on the geodynamo is fairly economical, as initially suggested by Christensen & Tilgner (2004): the scenario indeed terminates at a dynamo power of only 0.3 TW. Its main drawback is the extreme sensitivity to mantle-induced heat ﬂow ﬂuctuations, which can shut down the dynamo if the heat ﬂow is too close to the critical heat ﬂow required to maintain convection. It is also conﬂicting with estimates of the present-day CMB heat ﬂow derived from the double crossing of the post-perovskite

phase transition (Hernlund et al. 2005; Lay et al. 2006). These determinations involve an extrapolation of pointwise temperature gradient estimations to the entire lower mantle, and multiplication of the result by the mantle thermal conductivity, both steps being rather uncertain. Such attempts lead to a heat ﬂow lower bound of about 7 TW (van der Hilst et al. 2007). Selecting a cooling model, which agrees with this constraint, such as our high-power scenario, does still yield palaeomagnetic predictions in agreement with the observations, while being more permissive on the magnitude of allowed mantle heat ﬂow ﬂuctuations. Note however that such a model necessarily yields higher ﬂow velocities inside the core (Fig. 10b) which are not unrealistic [the conversion factor from surface to deep ﬂow velocities can attain a factor 4 in the study of Christensen & Aubert (2006)] but certainly harder to reconcile with core surface ﬂow velocities of about 5 × 10−4 m s−1. We thus expect that a cooling scenario contained within the bounds represented by our two end-member scenarios would present less consistency problems than either of the end-members.
A striking feature of the recent palaeomagnetic record is the large dispersion of virtual dipole moments (see Fig. 11b), with, for instance, ﬂuctuations from about 2 × 1022 to 15 × 1022 A m2 in the last 500 Myr, occurring over very short (million year) timescales. These are unlikely to be explained by magnetohydrodynamic processes taking place in dynamo models, where the magnetic ﬁeld has typically smaller (and faster) ﬂuctuations (see for instance Olson 2008), with epochs of low dipole moment representing only rare events associated with reversals. Moreover, these are also too short to represent a response of the dynamo to changing mantle conditions, and, even if this was the case, the CMB heat ﬂow would have to come very close to the minimal 2 TW required to get a dynamo (Fig. 12) in order to cause such large magnetic ﬁeld ﬂuctuations. Reﬁnements in the knowledge of CMB heat ﬂow variations, in dynamo theory, and in variability analyses of palaeomagnetic samples, will be needed in order to conclude regarding the physical nature of these variations.
ACKNOWLEDGMENTS
We thank U.R. Christensen for providing the numerical data relative to published dynamo models. JA was supported by CNRS and programs Programme National de Plane´tologie (PNP) and Structure et Dynamique de l’Inte´rieur de la Terre (SEDIT) of the French Institut National des Sciences de l’Univers (INSU). SL was supported by program ANR BEGDy from French Agence Nationale
C 2009 The Authors, GJI, 179, 1414–1428 Journal compilation C 2009 RAS

de la Recherche. Numerical calculations were performed at the Me´soCentre de Calcul et de donne´es (MCCD) of IPGP, and at IDRIS, France. This is IPGP contribution 2547.
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1428 J. Aubert, S. Labrosse and C. Poitou
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APPENDIX
Here we follow the analysis presented by Lister (2003) to derive an approximate equivalence between dissipation at each boundary and its corresponding mass anomaly ﬂux. Eq. (24) in Lister (2003) gives the original deﬁnitions of boundary-originated dissipations i and
o as they stand in (32) and (35):

i = fc · g dV + Q∗ICB[1 − T / To(ri)]
V

(A1)

o

=

Q

∗ CMB

[1

−

T

/ To(ri)].

(A2)

Here fc is the convective compositional mass anomaly ﬂux, T o is the
adiabatic temperature gradient, T is the mass-averaged temperature in the shell, Q∗ICB,CMB are the superadiabatic heat ﬂows at inner-core boundary and CMB. The last term in eq. (24) of Lister (2003) is

neglected because the average temperature of dissipation is assumed to be T . Denoting the compositional mass anomaly ﬂux at the inner core boundary as Fci, and assuming, as in Lister (2003) that compositional diffusion can be neglected and that the convective (not total) compositional ﬂux is well mixed throughout the shell, we have ∇ · fc = Fci/V and an integration by parts of the ﬁrst term in (A1) yields

fc · g dV ≈ Fci (ψi − ψ).
V

(A3)

Integrating the deﬁnition of the adiabatic temperature gradient and

neglecting the radial variations of α and the heat capacity Cp, we

have

α Cp

(ψ

(r

)

−

ψi)

=

ln[To(r

)/

To(ri)].

(A4)

Under the assumption of moderate departure of T o from the massaveraged temperature T and of moderate density variations, the

mass average of the previous equation yields

α Cp

(ψi

−

ψ

)

≈

[1

−

T

/ To(ri)]

(A5)

and we have therefore

Q∗ICB[1 − T / To(ri)] ≈ Fti (ψi − ψ ),

(A6)

where Fti is the thermal mass anomaly ﬂux at the inner boundary. From here it is apparent that i as it stands in (32) corresponds to
i as used in (15), the same result holding for o. As discussed in Lister (2003), the accuracy of the various approximations used in this derivation reduce to that of the Boussinesq approximation.

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