that turbulent values of these di¬usion coe¬cients are more appropriate (and

then the question of whether isotropic or anisotropic di¬usion is appropriate

Jones

168

Ra Spatially Chaotic

GR

6

10

Ordered

4

10

2 No Convection BZ

10

E

-6 -5 -4 -3

10 10 10 10

Hyperdiffusivity

Used

GR

Ra

6

10 High Rm

4

10

Low Rm BZ

2

10

q

0.1 1 10 100

Hyperdiffusivity

Used

Fig. 11.1. The di¬erent regimes of parameter space explored by numerical models.

Dynamo action is only possible with high Rm convection. BZ and GR locate typical

solutions of the ˜Busse-Zhang™ and ˜Glatzmaier-Roberts™ type.

arises; Braginsky & Meytlis, 1990), E ∼ 10’9 which is still too small to deal

with numerically.

In Figure 11.1 (Sarson, 2000), we show a schematic diagram indicating

which parts of the parameter space have been explored (see also Busse,

2000). It is not possible to reduce the Ekman number much below 10’4 in

spherical codes (for plane layer codes we can do better, see below). Hyper-

di¬usivity (see e.g. Zhang & Jones (1997) for an explanation of what this in-

volves), which enhances di¬usion in the latitudinal and azimuthal directions,

but not the radial direction, has to be used to explore the low E regime,

and this introduces further uncertainties. At, for example, E ∼ 10’3 , dy-

namos are found at mildly supercritical Rayleigh number provided q 1,

the ˜Busse-Zhang™ regime. If q ∼ 1 the magnetic Reynolds number is too

small to give dynamo action. To correct this, we must increase the Rayleigh

number, in order to increase the ¬‚ow velocity. In principle, this should al-

Dynamos in planets 169

low us to achieve dynamo action at lower q, but in practice raising Ra at

¬xed E makes the ¬‚ow more chaotic and small scale, and no large scale

dipole ¬eld results. We need to lower E as well as raise Ra in order to keep

the ¬‚ow su¬ciently coherent to generate a dipole dominated ¬eld. This is

the ˜Glatzmaier-Roberts™ regime (see also Kuang & Bloxham, 1997), but as

noted above it can only be found by introducing hyperdi¬usion, with its

concomitant uncertainties.

There is, therefore, much less freedom to choose the parameters E and q

than one would like. We still have to decide on what values of P r and Ra

to choose. P r only a¬ects the inertial term. For behaviour on a timescale of

tens of years and greater, the inertial term is rather small, and its neglect can

be formally justi¬ed by letting P r be large. However, on molecular values at

least, P r is small not large. If the dynamo is in the correct low E regime, the

inertial term will become less important as E is reduced and so the solutions

will become independent of Prandtl number. Since dynamo codes are not yet

run in the low E regime, it is not surprising that authors report signi¬cant

Prandtl number dependence in their results. Dormy et al. (2001) note that

since dynamo codes are not run in the correct regime, great care must be

taken in interpreting the results, and in how the dimensionless variables are

to be translated back into physical variables. Finally, how is the Rayleigh

number to be chosen? This measures the ratio of the superadiabaticity

to the di¬usion coe¬cients, and there is no direct method of determining

this. Instead, we choose the Rayleigh number so that the heat ¬‚ux gives

the correct value. We show below that this criterion gives us the the typical

velocity of the ¬‚ow, and this is similar to that of the ˜westward drift™ velocity.

Ohmic Dissipation

Since essentially all the magnetic energy generated by the dynamo ends

up as ohmic dissipation, we can test whether our dynamo solutions have

a total dissipation comparable with the available energy estimates given in

the previous section.

Gubbins (1977) showed that if the ¬eld inside the core minimises the dis-

sipation subject to the constraint that the ¬eld at the CMB is the observed

¬eld, this minimum dissipation is

∞

·rcmb (2n + 1)(2n + 3)

Qmin = qn , qn = Rn , (11.5)

µ n

n=1

Jones

170

where qn is the dissipation from the spherical harmonics of order n, and

n

rearth 2n+4 2 2

gn + hm

m

Rn = (n + 1) (11.6)

n

rcmb

m=0

is the Mauersberger-Lowes spectrum extrapolated to the core surface (see

e.g., Langel, 1987), and gn and hm are the usual Gauss coe¬cients (see e.g.

m

n

Backus et al., 1996). The Mauersberger-Lowes spectrum Rn at the CMB is

well-approximated for n ≥ 3 by

Rn = 1.51 — 10’8 exp(’0.1n) T2 , (11.7)

which leads to Qmin ∼ 44 MW, with the peak dissipation occurring at around

n ∼ 12. This value of n is coincidentally at about the limit of what can be

observed, as higher harmonics are obscured by crustal magnetism.

Dynamo models suggest that the actual dissipation QD Qmin , so 44 MW

is a gross underestimate. The dynamo is very ine¬cient, in the sense that

the actual dissipation is orders of magnitude greater than the minimum

necessary dissipation. The reasons are (i) most of the ¬‚ux generated in

dynamo models never leaves the core. The toroidal ¬eld generated is nec-

essarily trapped in the core, but models show that only a small fraction of

the poloidal ¬eld leaves the core to form the observed potential ¬eld. (ii)

although the ¬eld escaping from the core is mostly dipolar in the models,

the internal ¬eld has a much more complex structure than the very sim-

ple structure of the minimising ¬eld. So not only is there far more ¬eld in

the core than is strictly necessary to generate the observed dipole ¬eld, its

structure is also rather complex.

The upshot is that dynamo models do suggest that the dissipation is of

the order of 1 TW, in agreement with the estimates of section 3. Any energy

source which falls signi¬cantly short of this ¬gure is insu¬cient. However, it

is not yet possible to make very reliable estimates of the dissipation with the

current generation of dynamo models, because the dissipation occurs mainly

in the range n ∼ 10 ’ 40 and this range is a¬ected by hyperdi¬usivity. An

interesting theoretical question is what is the nature of the power spectrum

in dynamo models. Formula (11.7) is empirical, and indeed power law spec-

tra ¬t just as well. However, a recent simulation (Roberts & Glatzmaier,

2000) was ¬tted well by the formula

Rn = 1.51 — 10’8 exp(’0.055n) T2 , (11.8)

suggesting that an exponential law may have some as yet unknown theoret-

ical basis. The form of this power spectrum has important implications for

dynamo theory, because it connects directly with an outstanding problem at

Dynamos in planets 171