source.

11.5 Dynamical regimes in planetary cores

Rotvig & Jones (2002) and Jones & Roberts (2000) have considered plane

layer models to gain some understanding of the low E dynamical regime. In

this geometry we can no longer compare results with geomagnetic studies,

but there are signi¬cant computational advantages in Cartesian geometry

(the non-existence of useful fast Legendre transforms is the root of the prob-

lem for spectral spherical codes). We can get E small enough to get into

the correct dynamical regime, where the basic balance of terms is correct.

This is signalled by the magnetic ¬eld satisfying Taylor™s (1963) constraint.

When this is achieved, the terms in the equation of motion are in MAC

balance (Braginsky, 1967), that is viscous forces and inertial acceleration

are negligible, while pressure force, Lorentz force and buoyancy force are all

comparable with the Coriolis acceleration.

We therefore have (Starchenko & Jones, 2002)

2|„¦ — u| ∼ |∇p|/ρ ∼ |j — B|/ρ ∼ g(±Ta S/cp + ±ξ ξ) , (11.9)

where S is the entropy ¬‚uctuation, ξ is the composition ¬‚uctuation, ρ is

the density and ±ξ is the compositional expansion coe¬cient, with ±ξ ≈ 0.6

being typical for terrestrial cores.

This is a completely di¬erent balance from that in the solar convection

zone, where mixing length balance occurs,

|u · ∇u| ∼ U— /l ∼ g±Ta S/cp ,

2

(11.10)

l being the mixing length and U— being a typical velocity. In non-magnetic

planetary atmospheres a geostrophic balance is common,

2ρ„¦ — u ∼ ’∇p , (11.11)

with either the thermal and viscous terms being much smaller. In laboratory

convection the motion is on short length scales (tall thin rolls) so that viscous

forces can be signi¬cant through the particular geometry of the motion.

Taking S— as a typical entropy ¬‚uctuation, and ignoring compositional

terms as appropriate for Jupiter,

2„¦U— ∼ g±Ta S— /cp . (11.12)

Jones

172

The heat ¬‚ux equation gives

Q

F ∼ ρTa U— S— ∼ . (11.13)

2

4πrcmb

Eliminating the entropy ¬‚uctuation S— ,

1

g±rcmb Q 2

U— ∼ . (11.14)

M „¦cp

Putting in the standard estimates for thermodynamic quantities, we obtain

for Jupiter, U— ∼ 2 — 10’3 m s’1 . For the Earth, we can form the mass ¬‚ux

equation analogous to the heat ¬‚ux equation, and we obtain (see Starchenko

& Jones, 2002 for details) U— ∼ 2 — 10’4 m s’1 .

These estimates are in good agreement with velocities inferred from mea-

surements of the secular variation (Bloxham & Jackson, 1991) for the Earth,

Russell et al. (2001) for Jupiter. This agreement provides useful evidence

that MAC balance does operate in the cores of these two planets. For ex-

ample, if the mixing length balance, equation (11.10), is used in place of

equation (11.9), the typical velocity is orders of magnitude too large.

The typical magnetic ¬eld can be estimated from

2ρ„¦U— ∼ |j — B| ∼ B— /µr— ,

2

(11.15)

where r— is lengthscale of the variation of the ¬eld, |B|/|∇ — B|. Eliminating

U— ,

g±ρ2 µ2 r— rcmb Q„¦ 1

2

— 4

B— ∼ . (11.16)

M cp

How do we choose r— ? Unfortunately, this is not at all clear. Studies of ¬‚ux

’1/2

ropes (Galloway, Proctor & Weiss, 1978) suggest r— ∼ Rm . Numerical

simulations at Rm ∼ O(102 ) suggest r— ∼ d/50, where d = rcmb ’ ricb . Mag-

netic ¬eld saturates when the stretching properties of the ¬‚ow are inhibited

by Lorentz force. Dynamic simulations suggest that ¬‚ux ropes thicken in

the fully nonlinear regime: it might therefore be that r— eventually becomes

independent of Rm at large Rm . For the Earth, it is reasonable to take

r— ∼ d/50 as suggested by the simulations with Rm close to its terrestrial

value, and we obtain B— ∼ 0.5 — 10’2 T, about ten times the dipole ¬eld

extrapolated to the CMB (Table 11.1), a reasonable value consistent with

numerical models, which indicate that about 10% of the core ¬eld escapes

through the CMB.

Interestingly, if we use r— ∼ d/50 for Jupiter, we get a core ¬eld of B— ∼

2 — 10’2 T which is reasonable if about 10% of the core ¬eld escapes to

’1/2

form the observed dipole. If we assume that r— ∼ Rm , this gives a smaller

Dynamos in planets 173

value of B— (smaller than the observed ¬eld), because Jupiter has a higher

magnetic Reynolds number.

For moderate Rm , geodynamo models typically give for the Elsasser num-

ber Λ

2

B—

∼ 4 ’ r— = 4dRm ’1 ∼ d/50 ,

Λ= (11.17)

2µρ„¦·

with a typically Earth-like value of Rm ∼ 200. For Jupiter™s metallic hy-

drogen core, we expect Rm ∼ 104 , and Λ ∼ 20. A possible problem with

this is that magnetic instabilities may occur when Λ is this big (e.g. Zhang,

1995). One possibility is that Jupiter™s dynamo is located not deep inside

the conducting core, but at the interface of the metallic hydrogen core and

the molecular atmosphere. Since it is likely that the electrical conductivity

goes smoothly to zero with distance from the centre (Kirk & Stevenson,

1987), there must be a zone where the Elsasser number and the magnetic

Reynolds number assume Earth-like values, and this may be a promising

location for the jovian dynamo.

Saturn may be driven by compositional as well as thermal convection,

Stevenson (1982), and the uncertainty in the core energy ¬‚uxes means that

typical velocities and ¬eld strengths are also uncertain. We can only com-

pare with the ¬eld strength, and this suggests that Saturn, like Jupiter, is

probably in MAC balance. A similar uncertainty holds for Ganymede; as

mentioned above, the thermal history, and consequently the current core

heat ¬‚ux, is unknown.

Mercury™s small size makes it likely that inner core nucleation started

early, so that by now a large solid inner core probably exists (Stevenson et

al., 1983). The sulphur (and other impurities) present depress the freezing

point, and as the inner core grows, the relative fraction of impurity rises in

the ¬‚uid left, so it is di¬cult to freeze the core entirely. The ¬‚uid outer core

is therefore probably only a thin shell; the thickness of this shell depends

on the (unknown) initial sulphur concentration, but values of ∼ 100 km

to ∼ 500 km are plausible. The thermal strati¬cation in the liquid iron

core is likely to be stable (Stevenson et al., 1983) but gravitational energy

QG ∼ 6 — 109 (d/100)2 W, where d is the outer core thickness in km, is

available from compositional convection (Stevenson, 1987). The rotation

rate of Mercury is slow, and the core magnetic ¬eld is much weaker than

that in other planets. These facts may well be related. If we take the

shell thickness d = rcmb ’ ricb ∼ 100 km, we obtain from MAC balance

U ∼ 3 — 10’3 m s’1 which implies a magnetic Reynolds number (based on

length d) of over 100. To obtain the observed ¬eld strength, however, we

Jones

174

need a rather small r— < 1 km. It may be that Mercury™s ¬eld has a relatively

smaller dipole component than the Earth, so the core ¬eld may be more than

ten times the escaping observed ¬eld. This could come about because the

slower rotation and relatively stronger driving may impose less order on the

¬‚ow (see Figure 11.1).

The planets Uranus and Neptune present a di¬erent problem. The heat

¬‚ux coming out of Uranus is about 3 — 1014 W and for Neptune 3 — 1015 W.

If we assume MAC balance, we have a typical velocity of 2 — 10’4 m s’1 for

Uranus, and about 3 times larger for Neptune. With the large di¬usivity

· ∼ 102 m2 s’1 , we have for Uranus a magnetic Reynolds number of only

about 40, which is probably too small to sustain dynamo action; Neptune is

marginal. MAC balance is therefore probably not possible in these planets.

Holme & Bloxham (1996) also point out that the ohmic dissipation associ-

ated with the observed ¬elds would be larger than the total heat ¬‚ux if the

core ¬eld is signi¬cantly larger than the observed ¬eld.

The absence of a dynamo in Venus is also of interest. Even with its slow