P P

is the Gr¨neisen parameter, for which estimates are available (Merkel et

u

al., 2000). Gravity g and the sound speed uP are given in the preliminary

reference Earth model PREM (Dziewonski & Anderson, 1981). Here ± is

the coe¬cient of thermal expansion and cp is the speci¬c heat at constant

pressure. When solving equation (11.1), the temperature at the CMB must

be supplied; in principle it can be found from mantle convection studies.

The pressure is found from the hydrostatic equation ∇Pa = ρa g, the density

ρa being given by PREM. The liquidus temperature at which freezing occurs

increases with pressure as we go deeper into the core at a rate faster than the

adiabatic temperature increases, so the inner core forms ¬rst at the centre

of the planet. In principle, when the temperature and pressure are known,

the freezing point and hence the location of the inner core is determined.

In practice, the location of the Earth™s inner core is known from seismology.

This is fortunate, because the freezing point of iron is signi¬cantly depressed

by the impurities in the outer core, and the exact amount of the depression

is hard to estimate; for other planets, where seismology is not yet available,

we have to rely on theoretical estimates to determine where the inner core

lies. For the Earth, it is generally believed that the CMB temperature is at

about 4 000 K and equation (11.1) then gives the ICB temperature at about

5 100 K.

When the temperature structure is known, the next step is to use high

pressure physics estimates of the thermal conductivity (45 W m’1 K’1 is a

typical value) to ¬nd the heat ¬‚ux conducted down the adiabat. This is

comparable to the convected ¬‚ux. If the Nusselt number N u is de¬ned as

the ratio of conducted to convected ¬‚ux, then N u ∼ 1, very di¬erent from

Dynamos in planets 165

solar convection. Conduction down the adiabat generates entropy at a rate

Σ ∼ 170 MW K’1 .

The energy balance is

QCMB = QICB + QL + QS + QG + QR , (11.2)

which relates the heat ¬‚ux QCMB coming out of the CMB to the small

amount (0.3 TW) coming out of the inner core, QICB . QS is the rate of

core cooling, known from the time evolution of equation (11.1), provided

TCMB can be found from mantle convection studies. QL is the latent heat

released at the ICB and QG is the gravitational energy liberated by the

central condensation as the inner core grows. These can both be estimated

in terms of the rate of growth of the inner core, which in turn depends on

how the temperature structure given by equation (11.1) evolves with time.

QG involves the density jump at the ICB. The Earth™s inner core density

= 12730 kg m’3 and the ¬‚uid outer core density = 12160 kg m’3 and the

di¬erence is the density jump across the core. However, not all this jump

releases useful energy for the dynamo. There are two parts contributing (i)

due to release of light material, and (ii) due to contraction on solidi¬cation.

The estimates of Roberts et al. (2002), with the age of the inner core taken as

1.2 Gyrs, suggest that the useful part (i) gives QG ∼ 0.5 TW. This age for the

inner core is consistent with the Labrosse et al. (2001) value of 1 ± 0.5 Gyr.

The cooling QS ∼ 2.3 TW, and the latent heat QL ∼ 4.0 TW. If QR , the

radioactive term, is zero then we have QCMB ∼ 7.1 TW. However, if there

is radioactivity in the core, this estimate of the CMB heat ¬‚ux could be a

serious underestimate. The value of 7.1 TW for CMB heat ¬‚ux apparently

causes no great di¬culty for mantle convection models, but the same is true

for larger values, too.

The ¬‚ux conducted down the adiabat near the CMB is around 6 TW

using the above estimates, and because the latent heat is released at the

ICB, the total heat ¬‚ux exceeds the conducted ¬‚ux everywhere, so there

is convection throughout the core on this model. However, a rather small

reduction in CMB heat ¬‚ux would change this. If the CMB heat ¬‚ux is

less than 6 TW, the top of the core is subadiabatic. It would still convect

through compositional convection, and would still be close to adiabatic, but

one would expect convection there to be much less vigorous; this thermally

stable layer is Braginsky™s ˜inverted ocean™ (Braginsky, 1993).

Energy balance does not allow us to investigate the amount of dissipation

QD , since the work done by buoyancy cancels the dissipation. We need

to consider the entropy balance; following the discussion of Roberts et al.

Jones

166

(2002).

TD TCMB

QD = [(QICB + QL )(1 ’ )

TCMB TICB

TCMB

+ (QS + QR )(1 ’ ) + QG ’ ΣTCMB ] , (11.3)

T

recalling that Σ is the entropy production due to conduction down the adia-

bat. QD is almost entirely ohmic dissipation, viscous dissipation being orders

of magnitude smaller in the core. TD is the mean temperature at which the

dissipation occurs (e¬ectively where the dynamo operates most strongly),

which clearly lies between TCMB and TICB . T is the mean temperature of

the outer core. Putting in numerical estimates (Roberts et al., 2002) gives

TD

QD = (0.5 TW + QG + 0.12QR ) , (11.4)

TCMB

indicating that thermal and compositional convection both contribute

roughly 0.5 TW, giving a total of around 1 TW to drive the dynamo. We

are therefore aiming at ¬nding a dynamo with about 1 TW of ohmic dissi-

pation. This is consistent with the output of current dynamo models. There

is still some uncertainty in the above heat ¬‚ux estimates; for example Lister

& Bu¬ett (1995) estimated the conducted ¬‚ux at the CMB as only 2.7 TW.

Early Earth

There is, however, a serious problem with all the above theory (Roberts et

al., 2002). While it can explain the current geodynamo, what was happening

before the inner core formed ∼ 1.2 Gyr ago? According to the paleomagnetic

evidence, the magnetic ¬eld dates back to at least 3.5 Gyr. It was suggested

(Hale, 1987) that the ¬eld strength intensi¬ed 2.7 Gyr ago, possibly corre-

sponding to the formation of the inner core. Before inner core formation,

the latent heat and the gravitational energy sources are not available, only

cooling. The power available for the geodynamo is then much reduced. Even

more seriously, most, if not all, of the core would be stably strati¬ed with

the above estimate of the cooling term. It is not clear how a dynamo could

be sustained under these circumstances.

If we assume the Earth was formed from material with solar abundances,

there is a signi¬cant depletion of radioactive potassium, 40 K, in the mantle.

This could either have been lost to space during the Earth™s formation,

which is the view favoured by most geochemists, or it could be trapped in

the core. If it has been trapped in the core, then the CMB heat ¬‚ux would

be much greater, possibly even as much as 20 TW (Roberts et al., 2002),

removing the di¬culty with the early dynamo. Including radioactivity in the

Dynamos in planets 167

core also has the e¬ect of altering the time at which the inner core formed,

generally increasing the age of the inner core (Labrosse et al., 2001). Another

possibility is that the primordial heat at formation was very large, and so

the rate of cooling, QS , is much larger than our 2.3 TW estimate, especially

during the time before the inner core formed.

11.4 Physical nature of convective dynamo solutions

A number of research groups have produced three-dimensional numerical

solutions of the geodynamo equations, and these have been recently reviewed

by Dormy et al. (2001), Jones (2000) and Busse (2000). We shall therefore

focus on a few particular issues here.

The geodynamo equations are usually solved in the Boussinesq approxima-

tion, which are given in e.g. Jones (2000), although Glatzmaier & Roberts

have also solved the anelastic equations, which allow for variations in the

properties of the Earth™s core (see e.g. Glatzmaier & Roberts, 1997). To

avoid complications in what is already a formidable set of equations, only one

source of convection (either thermal or compositional) is usually assumed.

The dimensionless parameters that occur in the equations are the Ekman

number E = ν/2„¦d2 (d is the core radius), the Roberts number q = κ/·,

the Rayleigh number Ra and the Prandtl number P r = ν/κ. Here · is

the magnetic di¬usivity, ν is the kinematic viscosity, and κ is the thermal

di¬usivity.

In the inner core, the magnetic di¬usion equation is solved, and appro-

priate continuity conditions are applied across the ICB (see e.g. Jones et

al., 1995). For the mechanical boundary conditions at the ICB, Glatzmaier

& Roberts (1996) used no-slip, while Kuang & Bloxham (1997) used stress-

free, arguing that since viscosity is arti¬cially enhanced in the models (see

below), stress-free represents the physical situation better. The di¬erent

assumptions for this boundary condition appear to make a signi¬cant dif-

ference to the nature of the solutions, but the detailed reasons for this are

not yet apparent.

The main problem with geodynamo solutions is that it is not possible to

solve the equations in the desired parameter regime. The molecular di¬usion

coe¬cients κ ∼ 2—10’5 m2 s’1 and ν ∼ 10’6 m2 s’1 lead to very small values