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Cross- refere nces

Geocentric Axial Dipole Hypothesis

Pole, Paleomagnetic

Time-averaged Paleomagnetic Field

GEODYNAMO

Intro ductio n

The discovery of the magnetic compass (q.v.) by the Chinese dates

back at least to the

A.D. 1st century A.D., and it was known in Europe

during the

A.D. 12th century. An early idea was that the magnetic nee-

dle was attracted to the pole star, but by the year 1600, William Gilbert

(q.v.) had realized that the source of the Earth ’s magnetic field came

from within the Earth rather than outside it. Final confirmation of this

was not made until 1838, when Carl Friedrich Gauss (q.v.) used a

spherical harmonic (q.v.) decomposition of the geomagnetic field to

establish that the main field is internal. The important discovery that

the field changes with time was made in 1634 by Henry Gellibrand

(q.v.). This showed that the Earth ’ s field cannot be a permanent mag-

net, as William Gilbert had envisaged. Edmond Halley (q.v.) investi-

gated the changes in the Earth ’s field and showed that some

magnetic features were drifting westward, and on this basis suggest ed

that the interior of the Earth might be liquid.

Evidence from the study of rock magnetism shows that the Earth ’s

magnetic field is certainly not static, but varies dramatically over long

periods of time. Indeed, the Earth has undergone many complete mag-

netic reversals throughout its history. The most widely accepted theory

is that the magnetic field is continually being created and destroyed

by fluid motions in the interior of the Earth, as suggested by Joseph

Larmor ( q.v.) (1919). Since electric ity and magnetism are commonly

generated by means of dynamos, the mechanism by which the Earth ’s

magnetic field is created is known as the geodynamo. Permanent mag-

netism does occur in the crustal magnetic field (q.v.) of the Earth, and

contributes a small and relatively static contribution to the main intern-

ally generated magnetic field, the core field.

There are also external components of the magnetic field measured

at the Earth ’ s surface. They can be distinguished from the internal core

field partly because they increase upward rather than decrease upward,

but also because they vary on a much shorter timescale. The origin of

these external fields is in the Earth ’s ionosphere (q.v.), where charged

particles in the solar wind interact with the upper atmosphere. Since

solar magnetic activity changes on a timescale of a few days, short

bursts of activity known as magnetic storms and substorms ( q.v. ) can

be detected in magnetic observatories. The external components of

the geomagnetic field will not be discussed further here as they are

not part of the geodynamo.

Magnet ic fields on other planets an d stars

The Earth is by no means the only planet to have a strong internal

magnetic field. Jupiter has a surface field 10 times larger than the

geomagnetic field, and Saturn, Uranus, and Neptune have surface

fields at least as strong. Internal fields exist on Mercury (though this

field is quite weak) and on Ganymede, one of Jupiter ’s moons. There

are planetary and satellite dynamos (q.v.) as well as a dynamo in the

Earth. Planetary magnetic fields have been reviewed by Stevenson

(1982, 2003) and Jones (2003).

Not all the planetary magnetic fields are dominated by an axial

dipole, as is the Earth. Uranus and Neptune have more complex fields.

Our moon, and the planet Mars, have crustal magnetic fields ( q.v. )

which are very likely to have been created by internal dynamos, which

operated early in their history, but have now ceased to function. The

issue of whether a planet or satellite has an internally generated mag-

netic field depends on the physical conditions in the interiors of the

planets and satellites (q.v.). The magnetic field of Sun ( q.v. ) is much

stronger than the geomagnetic field, and some stars have fields that

are stronger still. The solar magnetic field is believed to be generated

by dynamo processes occurring in the Sun ’s deep interior, (see the

Solar dynamo ).

The dynamo process

According to dynamo theory, the magnetic field in the Earth is main-

tained by a system of electrical currents flowing through its liquid

metal core. There are four key laws from electromagnetic theory that

describe how the magnetic field and the currents behave in terms of

concepts from elementary vector calculus. These are discussed in

detail in books on magnetohydrodynamics (q.v. ), or MHD for short,

such as Roberts (1967), Moffatt (1978), and Davidson (2001). The

first is Ampere ’s law, which can be written

rB ¼ mj ; (Eq. 1)

where B is the magnetic field, j is the current density (the current flow-

ing through a wire is j times its cross-sectional area), and m is the per-

meability of free space (the high core temperature makes the free space

value 4p  10

7

Tm A

1

appropriate). Maxwell showed that a dis-

placement c urrent term should be added t o t his e quation, but for

the Earth ’s core this extra term can be ignored, so Eq. (1) is some-

times called the pre-Maxwell equation. The second equation is

Faraday ’s law of electromagneti c induction, which is

] B

] t

¼rE ; (Eq. 2)

and says that if a magnetic field is varied in time, an electric field E is

created. The third result from electromagnetic theory is Ohm ’s law in a

moving conductor, which can be written

j ¼ s ðE þ u  B Þ: (Eq. 3)

Here the conductor (which may be fluid or solid) is moving with velo-

city u . s is the electrical conductivity of the core (q.v.), a physical

quantity of fundamental importance for the geodynamo. It is estimated

to be around 5  10

1

. Dividing by s and taking the curl of

Eq. (3) allows us to eliminate the electric field E, and dividing

Eq. (1) by m and taking the curl allows us to eliminate j . The

fourth law is simply

r B ¼ 0 ; (Eq. 4)

which is equivalent to saying there are no magnetic monopoles. If m

and s are constants, we obtain (using the vector identity curl curl ¼

grad div r

) the induction equation

] B

] t

¼rðu  B Þþr

B ; (Eq. 5)

where ¼ 1 =ms is called the magnetic diffusivity, and its typical value

in the core is  2m

 1

. The last diffusive term in Eq. (5) arises

because all metals have some electrical resistance.

GEODYNAMO 287

Seismology has shown that the Earth consists of a solid inner core,

the inner core radius being approximately 1100 km (see Core proper-

ties, physical ), surrounded by a fluid outer core of radius approxi-

mately 3500 km, surrounded in turn by the mantle. The electrical

conductivity of the mantle is very low compared with that of the outer

core, so the liquid metal outer core is the most natural seat of the

geodynamo.

We first consider what happens if there is no motion, u ¼ 0, so that

Eq. (5) becomes the diffusion equation. When the power flowing into

a laboratory electromagnet is switched off, the magnetic field usually

collapses quite quickly, but the Earth’s core is very large and this

slows down the decay. Equation (5) with u ¼ 0 can be solved for a

spherical conducting core surrounded by an insulating mantle (see

e.g., Moffatt, 1978), and the field is found to decay in time as

exp ðp

 t = a

Þ, where a is the core radius. Putting in the estimates

given above, the field reduces by a factor e (the e-folding time) in

20 ka. The core of the Earth is so large that even if the source of the

driving were suddenly removed, the field would decay only on this

very long timescale. However, the age of the Earth is around

4:5  10

y, and paleomagnetic studies show that the Earth’s magnetic

field has existed for at least 3  10

y. This is much longer than the

20 ka magnetic decay time, so an energy source for the geodynamo

(q.v.) is required to maintain the field.

The generation mechanism for the geodynamo is that the field is

maintained by core motions (q.v.) inside the liquid metal outer core

of the Earth, so u is nonzero and the electromagnetic induction term

in Eq. (5), rðu  BÞ , is important. This mechanism has now been

demonstrated to work in laboratory experimental dynamos (q.v.). The

same process is used in power stations to generate virtually all our

electricity, though there the conducting material is an array of copper

wires rather than a core of liquid metal.

The velocity required to generate the magnetic field can be esti-

mated from the size of the induction term relative to the diffusion term.

This ratio is approximately U



a= ¼ R

, the magnetic Reynolds num-

ber, where U



is a typical value of the velocity of the fluid inside the

core relative to the frame rotating with the mantle. This can be esti-

mated by observing the westward drift (q.v.) velocity of the magnetic

field itself. Since the inhomogeneities in the field are to a large extent

carried along by the fluid motion in the core, this provides a rough

estimate of approximately U



 2  10

4

1

for the core flow,

giving R

300 (note though that in some places in the core the velo-

city can be as much as 8  10

4

1

, Bloxham and Jackson, 1991).

Unfortunately, it is not possible to get a complete picture of the flow

below the core-mantle boundary (CMB), but core-based inversions

(q.v.) can extract much useful information about core velocities. Since

is significantly larger than unity, this ensures that the induction

term, which generates the magnetic field, makes good the losses aris-

ing from magnetic diffusion. This large value of R

arising from these

natural estimates provides strong support for the dynamo hypothesis in

the Earth’s core. It is of interest to seek the minimum value of R

required to overcome diffusion. A number of bounding theorems have

been proved to answer this question.

The actual flow u inside the core that is driving the dynamo will be

time-dependent and have a complex structure. However, some idea

of the type of flow inside the core suggested by the theory of

the dynamics of the core (see section below on “Core dynamics”)is

shown in Figure G11.

Nondynamo generation mechanisms

Although the dynamo theory is a very reasonable hypothesis, it is

necessary to consider the other nondynamo theories (q.v.) which have

been suggested for driving the Earth’s magnetic field. Most of

these are capable of generating some magnetic field, but are not up

to the task of generating the rather strong magnetic field observed. Per-

manent magnets are demagnetized when raised to temperatures above

the Curie point temperature, which for the materials which constitute

the core is well below any reasonable estimate of core temperature.

Admittedly, the Curie point might be affected by the high pressure in

the Earth’s core (see Depth to the Curie temperature). The strong

variability of the geomagnetic field is the best argument against a per-

manent magnetism explanation. Thermoelectric effects can create cur-

rents and hence magnetic fields; indeed Stevenson (1987) has

proposed this mechanism for Mercury’s field, but it cannot produce

enough field to explain geomagnetism. Other possible mechanisms,

with references, are listed by Merrill et al. (1996), but none has gained

any widespread acceptance.

How is the geodynamo driven?

A more controversial issue is the energy source of the fluid flow inside

the core. The diffusion of the magnetic field is accompanied by ohmic

heating, sometimes called Joule dissipation. This loss of energy requires

a source to replace it. Core convection (q.v.) is the mostly widely

accepted source, and the two most developed driving mechanisms are

thermal convection and compositional convection. Other possibilities

are precession and tidal interaction. For a more detailed discussion on

these mechanisms (see Energy source for the geodynamo).

The fundamental source of energy for precession and tidal interac-

tions is the rotational energy of the Earth. If these contribute to driving

the geodynamo, they would lead to a slowing down of the Earth’s

rotation, i.e., to a lengthening of the day (see Decadal length of day

variations). Earth tides and oceanic tides are also slowing down the

rotation rate of the Earth, so a precession-driven dynamo would give

additional slowing.

Thermal convection derives its energy from the cooling down of the

Earth, although if there is radioactive heating in the core, this could

contribute too. The issue of whether there is radioactivity in the core

has a long history and remains highly controversial (see Radioactive

isotopes and their decay). There is no doubt that radioactivity is impor-

tant in the mantle, but whether radioactive elements were carried into

the core during formation is a very challenging problem for geoche-

mists. The primordial heating of the Earth occurred as part of the core

formation process through gravitational collapse, so the thermal energy

of the core may be thought of as originating from gravitational energy.

Figure G11 A possible configuration for the fluid flow and the

field inside the core (after Bloxham and Gubbins, 1989).

288 GEODYNAMO

Another source of energy for driving the dynamo is compositional or

chemical convection (q.v.) (Braginsky, 1963; J. Verhoogen (q.v.), 1961).

The solid inner core of the Earth contains a higher fraction of iron than

the fluid outer core (see Inner core composition), and so is more dense

than the fluid outer core. As the inner core grows, due to core cooling,

iron is deposited on the inner core, and light material is released at the

moment of freezing. This light material rises due to buoyancy, and stirs

up the outer core. Since the net effect of this process is to move heavier

material toward the center, gravitational energy is liberated. It is slightly

surprising that the core first freezes near the center of the Earth, where

the temperature is highest. This happens because the melting tempera-

ture of iron in the core increases strongly with pressure.

The question of whether tidal forcing or precessional forcing (see

Precession and core dynamics) is large enough to drive the geody-

namo has been discussed by Malkus (1994). The typical height of an

Earth tide inside the core is expected to be about h  0:15 m, and pre-

cession gives a similar displacement. The typical radial velocity is then

Oh 10

5

1

. The azimuthal velocity might be a little larger, so a

value not that far off the convective velocity U



might be achieved.

There is a difficulty because this velocity is fluctuating on a time-

scale of a day, whereas a flow varying only on a much longer time-

scale (a 1 ka) is needed to sustain a dynamo. Some mechanism

for converting oscillatory flows into steady flows is required, and

some suggestions for overcoming this difficulty have been proposed

(Kerswell, 2002; Aldridge, 2003). A recent example of a precession-

driven dynamo has been given by Tilgner (2005), though the con-

ditions under which these dynamos operate are still very far from

Earth-like.

Energy balance for a convective dynamo

This naturally leads on to the question of how efficient the geodynamo

actually is. In the solar dynamo (q.v.), only a tiny fraction of the heat

energy pouring out of the Sun is converted into the solar magnetic

energy: is the Earth more efficient in this respect? To investigate this

we need to examine the overall energy and entropy budget in the

Earth’s core, and to do this we need a model for the Core temperature

of the Earth (q.v.).

Since we believe the fluid outer core is stirred it is reasonable to

assume it is approximately adiabatically stratified, so the temperature

gradient is given by the adiabatic gradient in the core (q.v.). This is

dependent on a quantity called the Grüneisen parameter (q.v.) which

itself varies somewhat between the inner core boundary (ICB) and

the Core-Mantle boundary but if we adopt the currently accepted esti-

mates, we find that the temperature drop from the ICB to the CMB is

about 1100 K (Braginsky and Roberts, 1995; Roberts et al., 2003). In

principle, the melting temperature of iron at high pressure (see melting

temperature of iron in the core), should determine the ICB tempera-

ture, but unfortunately, the impurities expected in the core significantly

depress the melting temperature, so this is somewhat uncertain. A

reasonable guess at the present time (Roberts et al., 2003) is

ICB

¼ 5100 K, giving T

CMB

¼ 4000 K. An important issue is the

amount of thermal conduction in the core (q.v.). Again there is uncer-

tainty due to the difficulty of the very high core pressures but if we

take 40 W m

1

as our estimate, the conducted flux down the

adiabat near the CMB is Q

cond

 6 TW, and only 0:25 TW near

the ICB. There are two reasons for the big difference; first the adia-

batic temperature gradient decreases somewhat with depth, so the con-

ducted heat flux per square meter decreases with depth, but secondly

the geometry of the much larger surface area near the CMB means that

a much larger number of terawatts is conducted there. The critical

question is whether the actual heat flux out of the core is greater or less

than this conducted flux. If the actual flux exceeds the conducted flux

cond

at any point in the fluid outer core, the excess flux is carried by

convection. If however the actual heat flux is less than Q

cond

then there

will be a region where there is a subadiabatic temperature gradient.

This is most likely to occur just below the CMB. Here there is either

a nonadiabatic stably stratified region where the temperature gradient

is significantly subadiabatic, thus reducing the conducted flux, or if

compositional convection stirs this subadiabatic region the heat flux

produced by compositional stirring is negative (back into the interior).

A further possibility is that light material released at the ICB as the

inner core forms accumulates just below the CMB. This possibility

has been called by Braginsky (1993) the “inverted ocean,” since it

would consist of a sea of relatively light material floating up against

the CMB.

Unfortunately, our knowledge of the heat flow across the Core-Mantle

Boundary is imprecise, and we cannot yet be certain whether an inverted

ocean exists or not. We know that at the present time there are 44 TW of

heat coming through the Earth’s surface, and that most of that originates

in the mantle. Estimates of the CMB heat flux range from 3 TW (Sleep,

1990) to 15 TW (Roberts et al., 2003), and are highly dependent on

whether there is core radioactivity or not. In principle, mantle convection

simulations could tell us the heat flux across the CMB, since this is one of

the bottom boundary conditions to such simulations, and so will affect the

structure of mantle convection. However, to date the uncertainties sur-

rounding mantle convection modeling (see e.g., Schubert et al.,2001)

preclude this possibility. Numerical simulations of the geodynamo

(q.v.), such as those of Glatzmaier and Roberts (1995, 1997), have typi-

cally adopted a compromise value of around 7 TW for the CMB heat

flux, which just makes the core fully convective.

The overall energy balance in the core, ignoring any contributions

from precession or tides, can be written

¼Q

CMB

Q

ICB

Q

ðQ

Þ: (Eq. 6)

where Q

is the rate of cooling, Q

CMB

is the heat flux passing through

the CMB, Q

ICB

is the heat flux passing through the ICB, Q

is the

latent heat released by the freezing process on the inner core, Verhoogen

(q.v.), 1961 Q

is the gravitational energy released by the same process,

and Q

is the heat produced by radioactivity (if any). For the sake of

definiteness, we give reasonable estimates for each of these quantities

(see e.g., Braginsky and Roberts, 1995; Roberts et al., 2003), but it can-

not be emphasized too strongly that all of these estimates are uncertain,

and it would be very surprising if improvements in our understanding of

high pressure physics did not lead to radical revision of these estimates

over the next decades. Discussions of some of the physics that goes into

these estimates can be found in the articles on core composition (q.v.),

core density (q.v.), and core properties, physical (q.v.). With this pro-

viso, we take the cooling rate of the core to be Q

 2:3 TW, the latent

heat released at the ICB to be Q

 4:0 TW, the gravitational energy

released at the ICB Q

 0:5 TW, and Q

ICB

 0:25 TW. This gives

CMB

7 TW, the value used by Glatzmaier and Roberts (1997). Q

the heat released by core radioactivity is extremely uncertain, as men-

tioned above. Perhaps the best hope is that an improved understanding

of the geodynamo might enable us to constrain Q

, as might a better

understanding of the thermal history of planetary interiors.

It might seem surprising that the ohmic dissipation generated by the

magnetic field does not appear in the energy balance. This is because it

is balanced by the work done by buoyant convection. Some of the heat

energy flowing through the core is extracted to drive the fluid motions,

only to be returned in full by the ohmic and viscous dissipation.

Entropy balance in the core

To get an estimate which involves the magnetic field we need to con-

sider the rate of entropy production (Hewitt et al., 1975; Gubbins,

1977; Gubbins et al., 1979; Braginsky and Roberts, 1995; Roberts

et al., 2003). As magnetic field diffuses, it generates heat through

ohmic dissipation,

¼ m

dv (Eq. 7)

GEODYNAMO 289

Viscous dissipation is believed to be negligible in comparison with

ohmic dissipation, because the magnetic energy is much larger than

the kinetic energy in the core, and we obtain (Roberts et al., 2003)

CMB

ICB

þQ



1 

CMB

ICB



þQ





1 

CMB





C MB



(Eq. 8)

where T

is the average temperature where dissipation occurs, and T

is the average temperature of the fluid outer core. As in a Carnot heat

engine, there are “efficiency factors ” which mean that thermal convec-

tion cannot extract more than ð1  T

CMB

= T

ICB

Þ20% of the heat flux

passing through the core, and for sources which are distributed through-

out the core, such as cooling and radioactivity, the efficiency factor may

be as low as 10%. Using the estimates above, we find that in the absence

of radioactivity the dissipation is around 1 TW, with about 0.5 TW being

from compositional convection Q

, and 0.5 TW being from thermal

convection (see also Christensen and Tilgner, 2004).

Most current numerical simulations of the geodynamo (q.v. ) have an

ohmic dissipation that is less than 1 TW, but this is most probably

because they are not yet in the right parameter regime for the geody-

namo. Roberts and Glatzmaier (2000) have shown that as the resolu-

tion is increased the dissipation rises (see also Kono and Robert s,

2002). Again some geodynamo simulations show a viscous dissipation

rate as large as the ohmic dissipation rate, but this again is likely to

be a consequence of the numerical difficulties in achieving the right

parameter regime.

Kinem atic dynamo mod els

A major difficulty in constructing geodynamo models was noted by

Cowling (1934). He showed that it is impossible to maintain an axisym-

metric magnetic field by means of dynamo action, a result now known as

Cowling ’s theorem (q.v.). This was the first of a class of antid ynamo the-

orems (q.v.) showing that fields with too much symmetry cannot be cre-

ated by dynamo action. The earliest geodynamo models based on

electromagnetic induction appeared in the 1950s (Bullard (q.v.) and

Gellman, 1954), thus showing that although Cowling ’s theorem (q.v.)

is a mathematical inconvenience (because symmetric solutions of partial

differential equations are much easier to find) it is not a fatal objection to

the dynamo idea. These models only considered the induction equation

(5), the velocity field of the fluid flow being imposed. This is known

as the kinematic dynamo problem, the significance of the word kine-

matic being that the velocity field is simply prescribed, rather than taking

the flow to be a solution of an equation of motion. In practice fairly sim-

ple flows are chosen, but despite the induction equation (5) being linear

in B , the nonaxisymmetric three-dimensional nature of the solutions

makes it hard to solve, even with today ’s fast comput ers. Because of

the linearity, the induction equation (5) has either exponentially growing

or decaying solutions. We say that a flow is a kinematic dynamo if the

solutions grow rather than decay. Fast dynamos, that is dynamos whose

growth rate remains finite in the limit of small magnetic diffusion

!1) are usually studied in the kinematic dynamo approximation.

It was noticed early on in the study of kinematic dynamos that quad-

rupolar dynamos, that is dynamos with a dominant quadrupole field

(see symmetry and the geodynamo) rather than a dipole field can also

be obtained, depending on the chosen flow. The issue of what makes

some flows give dipolar dynamos and others quadrupolar dynamos

is not yet fully resolved, so we cannot yet give a definite answer to

the question why is the Earth dipole dominated rather than quadrupole

dominated. Numerical simulations of the geodynamo ( q.v. ) usually, but

not always, give dipolar fields.

Another issue to arise out of kinematic dynamo studies was the

role of meridional circulation, which is the axisymmetric component

of flow in the radial and latitudinal direction. (The axisymmetric

component of flow in the longitudinal direction is called the azimuthal

flow.) The time dependence of a kinematic dynamo is proportional to

exp ð st Þ , and s is in general complex. If s happens to real, we say that

the dynamo is steady, but if s has an imaginary part we say it is oscil-

latory. The solar dynamo (q.v.) reverses continually in an approxi-

mately periodic 22-year cycle, and is therefore naturally modeled as

an oscillatory dynamo, but the geodynamo is comparatively steady,

althoug h it does reverse (irregularly) occasionally (see Reversals, the-

ory ). Kinematic dynamo calculations (Roberts, 1972) showed that mer-

idional circulation helps to make a dynamo steady, indicating that

meridional circulation could play an important role in the geodynamo.

Recent work on kinematic dynamos relevant to the Earth can be found

in e.g., Gubbins et al. (2000a,b) and Sarson (2003).

Mean field dynamo model s

In the 1960s, mean field dynamos (q.v.) were developed (Steenbeck

et al., 1966) see also Moffatt (1978). The basic idea is that small-scale

core turbulence (q.v.) could help to generate the large-scale field. If the

magnetic field is B þ b, and the velocity is U þ u where b and u are

the small-scale turbulent parts, and B and U are the mean parts aver-

aged over the small scales, then the induction equation averaged over

the small scales becomes

¼rðU  BÞþrð

u  bÞþr

B: (Eq. 9)

Provided certain criteria are satisfied the new term can be written

rð

u  bÞ¼raB (Eq. 10)

where a is in general a function of position in the core. More generally,

a is a tensor, but it is usually taken to be isotropic so that the compara-

tively simple form of Eq. (10) can be used. This term in the induction

equation is known as the a-effect. The types of turbulent flow which

give a nonzero a are those with nonzero mean helicity. The helicity

of a flow is u  v where v ¼ru is the vorticity. Flows with a

“screw-type” motion have helicity. Parker (1955) pointed out that a

blob of hot fluid, rising because of its buoyancy, would twist as it rises

due to the action of Coriolis force in a rotating body like the Earth or

the Sun. This gives a natural source of helicity. The great advantage of

mean field models is that axisymmetric mean fields are now possible

solutions of Eqs. (9) and (10). The nonaxisymmetric components of

the field which must be there because of Cowling’s theorem (q.v.),

do not need to be calculated explicitly. This simplification made it

feasible to include dynamics into dynamo models, to obtain what are

now known as intermediate dynamo models (see section below on

“Intermediate dynamo models”).

Mean field models have been extensively used in the solar dynamo

(q.v.) problem, because they give rise to dynamo waves (q.v.), which

can explain many features of the solar dynamo (q.v.). The main disad-

vantage of mean field models is that the distribution of a over the core

depends on the form of the turbulence in the Earth’s core. This cannot

be observed, and indeed it is not even known whether such turbulence

satisfies the criteria required for a mean field dynamo to be valid. If it

were the case that the spatial form of a was not critical, this would be

less important, but unfortunately numerical calculations indicated that

the a-distribution can make a very big difference to the types of

dynamo generated (Hollerbach et al., 1992).

Core dynamics

Some of the most interesting properties of the geodynamo are con-

cerned with the dynamics that is the force balance in the core. This

is described by the Navier-Stokes equation with additional forces

which are important in the geodynamo. A simplification that is often

290 GEODYNAMO

used is the Boussinesq approximation (see the article on Boussinesq

and anelastic approximations ), in which density variati ons are ignored

except where they are multiplied by g , so that buoyancy forces can

be included. This approximation is not strictly valid in the core, as

density variations of order 20% can occur, but it is nevertheless a use-

ful simplif ication of equations which are difficult to solve, and we

adopt it here.

The equation of motion for a rotating fluid in which Lorentz (mag-

netic) forces, buoyancy, and viscosity act can be written

D t

þ 2 V  u



¼rp þ j  B þ rn r

u þ r g a T

(Eq. 11)

Here r is the density (assumed constant), u is the velocity, p is the

pressure, j is the current density, B is the magnetic field, n is the kine-

matic viscosity, g is gravity, a is the coefficient of expansion, T is the

temperature, and

r is the unit vector in the direction of gravity, the

radial direction. Not surprisingly, such a complex equation with so

many different forces leads to many different physical effects. Note

that the Coriolis acceleration, 2V  u is often thought of as a force

(when multiplied by r ) in a rotating frame. D/D t is the convective deri-

vative, ] =] t þ u r.

Since the temperature occurs in this equation, another equation

determining T is needed,

] T

] t

¼ k r

T  u rT þ Q: (Eq. 12)

Here Q is the heat source and k is the thermal diffusivity.

A number of dimensionless parameters can be formed from these

equations. The Ekman number, E ¼ n= 2O d

, measures the relative

importance of the viscous force to the Coriolis force. To estimate this

we need to know the core viscosity (q.v.), but E is certainly very

small, probably of order 10

15

in the core. There are two diffusion

coefficients, n and k , and a third,  , occurs in the induction equation

(5), so we have two dimensionless ratios, Pr ¼ n =k , known as the

Prandtl number, and Pm ¼ n= known as the magnetic Prandtl num-

ber. Another useful combination involving the magnetic field is the

Elsasser (q.v. ) number L ¼ B

= 2 Omr, where B

is a typical value

of the magnetic field. The Elsasser number at the CMB, where the

field is typically 0.5 mT, is about 0.25, but inside the core the field

is likely to be stronger, and the Elsasser number is believed to be of

order unity or a little more.

Rapidl y rotating co nvection

Much of our understanding of convection in rapidly rotating systems

comes from the problem of thermally driven nonmagnetic rotating

convection (q.v.), and the problem of convection in an imposed mag-

netic field (usually either a uniform magnetic field or one with a sim-

ple geometry) which is called magnetoconvection (q.v.). The articles

on these topics give a more detailed discussion, but there is a very

large literature on both these topics, and some understanding is essen-

tial for an appreciation of the geodynamo. Our knowledge has been

enhanced by fluid dynamic experiments (q.v. ) and experimental dyna-

mos (q.v.). It turns out that magnetic field affects the convection con-

siderably, but at least the nonmagnetic case gives us a framework on

which to build understanding of the magnetic case.

The onset of instability in a rapidly rotating sphere is now fairly

well understood (Roberts, 1968; Busse, 1970; Jones et al., 2000).

Rapid rotation means the Ekman number, E ¼ n= 2O d

is small, and

in this case convection occurs in a columnar form, somewhat as illu-

strated in Figure G11 . However, in nonmagnetic convection the col-

umns are tall but very thin, so that instead of the three columns

shown in Figure G11, a much larger number of order E

 1=3

fit into

the sphere. These columns are known as Busse columns (Busse, 1970;

Busse and Carrigan, 1976), and their basic structure can be understood

in terms of the Proudman-Taylor theorem (q.v.). The onset of convection

in a uniformly heated sphere with stress-free boundaries in the absence

of rotation occurs when the Rayleigh number Ra ¼ g abr

= kn exceeds

crit

¼ 1 546 (Chandrasekhar, 1961), where b ¼ Q =3 k, so the static

gradient is b r . Gravity is taken to be  gr , as in a unifor m sphere gravity

increases linearly from the center. In the small Ekman number limit, i.e.,

in rapid rotation, the critical Rayleigh number for the onset of convection

approaches 4 : 117E

4 =3

at Prandtl number Pr ¼ 1 (Jones et al., 2000),

which gets very large at small E. Although the onset of instability is

strongly delayed by the rotation as the Rayleigh number is increased

above critical, once the critical value for convection is achieved the flow

becomes time-dependent, and then aperiodic, rather quickly. Abdulrah-

man et al. (2000) showed that the bifurcations to chaotic convection

occur in a relatively small w i ndow i n R a yl e i gh num be r s pa ce

crit

< Ra < R

crit

þ O ðE

 2=3

Þ .

Convection continues to occur in tall thin columns in the fully non-

linear regime provided the Rossby number Ro ¼ U



= aO remains

small, although the thickness of the columns does increase somewhat

over its very thin linear value, and the columns tend to become rather

transient, and occur fairly randomly throughout the sphere. Experi-

ments suggest that in the nonlinear regime, inertial effects take over

from viscous effects, and the leading order balance is between inertia,

Coriolis force, and buoyancy (for details see Aubert et al., 2001).

When a magnetic field is applied, the Proudman-Taylor theorem

(q.v.) no longer controls the pattern of convection, as the Lorentz force

becomes important. An important effect of magnetic field is to

increase the thickness of the Busse columns (see e.g., Jones et al.,

2003), which occurs even at small Elsasser number L  OðE

1=3

Þ field

strengths. If the Elsasser number is further increased to O(1) values,

the columns are still visible, but the number fitting into the sphere

is much reduced. Because magnetic field can break the Proudman-

Taylor constraint, the critical Rayleigh number is reduced in the pre-

sence of a L  Oð1Þ field, so magnetic field can help the system

convect, see e.g., Fearn et al. (1988).

Dynamical regime in the core

In the magnetic case it is not unreasonable as a first approximation to

assume that the large-scale flows and fields have dominant length scales

of the size of the core radius. It is then possible to do a useful scale

analysis of the dynamics of the Earth ’ s core using Eq. (11) (see e.g.,

Braginsky and Roberts, 1995; Starchenko and Jones, 2002), although

we must keep in mind this can only give order of magnitude estimates,

and more exact results require that the geometry of the convection and

the spherical domain be taken properly into account.

Some of the physical quantities in the core are quite accurately

known, others somewhat less so, but here we only need order of

magnitude estimates, to see which terms are large and which small

in Eq. (11). We take as a typical value for the velocity in the core



¼ 2  10

4

1

, see section “The dynamo process.” For D/Dt

we take 1/300 years, or 10

10

1

. Significant changes to the field

occur on this timescale, though a full reversal of the field might take

ten times as long. The density r 10

kg m

3

in the core. The Earth ’s

angular velocity O ¼ 7  10

5

, and already we can see that the mag-

nitude of the inertial term jDu=Dtj is very small compared to the

Coriolis term j2V  uj.

We turn next to the Lorentz force term, j  B. The field at the CMB

is about 5  10

4

T. The current is rB=m, and since the curl is a

space derivative, we crudely estimate it at jBp=amj where a is the core

radius, 3:5  10

m. The Coriolis term j2rV  uj3  10

4

3

while the Lorentz term jj  Bj has magnitude 2:5  10

7

3

. This

is smaller than the Coriolis term, but it is generally believed that the

invisible field inside the core is at least 10 times larger than the visible

field, which leaves the core, increasing the magnitude of the Lorentz

term. Furthermore, the magnetic field pattern at the CMB suggests that

it varies on a shorter length scale than the core radius, which will

GEODYNAMO 291

enhance j . It is therefore likely that the Lorentz force will be compar-

able to the other forces at least in some places. The main argument for

a significan t Lorentz force is that the induction equation (5) is linear in

B and therefore cannot determine the field strength. This must be lim-

ited by the Lorentz force term, so j  B must be significant at least

somewhere in the core, and most dynamo models suggest that equili-

bration of the magnetic field takes place when the Lorentz and Coriolis

forces are comparable.

One of the difficulties of numerical simulations is that they require a

substantial amount of dissipation, in particular viscous dissipation, for

numerical stability. Thus the balance in the models is often between

Coriolis force and viscosity, with the Lorentz force playing only a

small role in limiting the field strength without greatly affecting the

flow. This is known as the weak field regime, in contrast to the strong

field regime in which the Lorentz force is comparable to the other

large forces in the equation of motion. This is the essential difference

between the strong and weak field dynamo regimes. We address this

issue further in the section “ Taylor’s constraint. ”

The buoyancy force drives the flow, and so we would expect this

force to be comparable to the Coriolis and Lorentz force terms. If we

estimate g at 10 ms

2

, and the coefficient of expansion a at

 5

 1

, we find that the typical temperature fluctuation is in the

range 10

3

– 10

 4

K . It is rather remark able that such tiny temperature

fluctuations in the core are driving the whole dynamo process, but we

should remember that core flows are very slow; even a lethargic snail

could go faster! We also have a consistency check here, because the

convective heat flux

conv

¼ r c

T d S ; (Eq. 13)

can be estimated, and when we remember that T here is the very small

superadiabatic temperature, which is 10

3

–10

4

K only, we reassur-

ingly discover that this flux is comparable to that sugge sted by the

energy arguments of the section above “Energy balance for a convective

dynamo. ”

The viscous term in the equation of motion turns out to be very

small, and so is not in the primary force balance, except in thin bound-

ary layers in the core ( q.v. ). It is however necessary to include the vis-

cous term in the force balance when doing numerical simulations, to

ensure numerical stability. Pressure forces are always important in con-

vecting fluids; when buoyant fluid rises, the pressure forces push the

ambient fluid aside to make room for the hot rising fluid.

The main balance of forces in the core is therefore between Lorentz

force, Coriolis force, buoyancy force, and pressure force. This is

known as MAC balance, M standing for magnetic, A for Archimedean

force (Buoyancy), and C for Coriolis force (Braginsky, 1967). The

word “magnetostrophic ” is sometimes used as a synonym for MAC

balance, though more commonly magnetostrophic balance refers to

situations where buoyancy is absent. Pressure can usually be elimi-

nated by taking the curl of the equation of motion, which converts it

into a vorticity equation.

If compositional convection is present, an additional equation of

similar form to the temperature equation (12) is required for x, the

fraction of light materi al, and the resulting buoyancy must be included

in the equation of motion (see e.g., Braginsky and Roberts, 1995).

Equations (5), (11), and (12) can be solved numerically in dynamo

simulations, but an important issue is whether laminar or turbulent values

should be used for the diffusivities. Laminar values of k an d n are so small

it is h ard to b eliev e they c an be r elev ant t o the co re. Turb ule nt

values are therefore generally used, and this raises the interesting

qu est ion of th e n at ure o f t u rbu le nce i n t he Earth ’s co re. Th e ine rtial

terms, which are responsible for cascading energy down to small

scales in “normal” flu id s are o nl y eff ectiv e o n ti ny len gth scal es i n

the core, so core turbulence may be quite different from normal tur-

bulence. In particular, the diffusion processes may be quite anisotro-

pic, as suggested by Braginsky and Meytlis (1990).

W aves an d instabi lities

Calculating how the field and the flow evolves in the core can only be

done using sophisticated numerical dynamo models. However, insight

can be obtained into the rather complicated nature of the dynamics of

the core by considering simple equilibrium flows and fields, and ana-

lyzing the magnetohydrodynamic waves (q.v.) and instabilities that

these simple equilibria undergo when they are slightly perturbed. This

leads to linear problems which can be investigated mathematically.

Perhaps the most fundamental problem is the case of a uniform mag-

netic field with fluid at rest (or in uniform motion) with a uniform

temperature gradient imposed. Linearizing Eqs. (5), (11), and (12)

about a uniform magnetic field B

and constant temperature gradient

b , so that B ¼ B

þ b

, the velocity is u

and the temperature is

T ¼ bz þ T

. The primed quantities are assumed small so that squares

and products of primed quantities are neglected, and then wave solu-

tions with u

; b

, and T

proportional to exp ið k  x  o t Þ can be found.

Equations (5), (11), and (12) then give the dispersion relation between

k and o. If the temperature gradient is stabilizing (subadiabatic) then

real values of o are found if diffusion is neglected. If diffusion is

retained, o is complex with a negative imaginary part corresponding

to damped oscillations. If the temperature gradient is superadiabatic,

growing waves (instabilities) are found provided the diffusion is not

too large.

The dispersion relation is actually very complicated, but it can be

simplified when as in the Earth ’s core, different types of waves have

very different frequencies (Fearn et al ., 1988). The fastest waves are

inertial waves (see the article on gravity-inertio waves and inertial

oscillations ), which balance the Coriolis and inertial accelerations,

and the inertial wave frequency is given by o

¼ 2 ðV  k Þ=j k j. The

typical period is therefore of the order of a day. These waves have

not yet been observed in the core, but they are expected to be driven

by tidal forcing. Alfvén waves (q.v.) result from a balance of inertia and

Lorentz force in the equation of motion, when combined with the induc-

tion equation. These waves have frequency o

¼ðB

 k Þ=ð mrÞ

1 =2

and travel at the Alfvén speed, B

=ðmrÞ

1=2

, which for a moderate 1 mT

core field is around 10

 2

 1

, giving around 60 years for the

wave to travel round the core. An impo rtant class of Alfvén waves are

those corresponding to azimuthal motion constant on cylinders,

which are called torsional oscillations (q.v.). These are believed to be

important in the core; see “ Taylor’s constraint ” below.

Another timescale comes from the temperature gradient, from the

balance of buoyancy and inertia in the equation of motion, combined

with the temperature equation (12). The frequency of internal grav-

ity waves is o

¼ðg abÞ

1=2

= k j, where b is the subadiabatic tem-

perature gradient, and k

is the component of k perpendicular to

gravity. In a convectively unstable region, the temperature gradient is

superadiabatic and b is negative. Then o

is imaginary, which corre-

sponds to an exponentially growing unstable mode, with j o

j being

the growth rate. With the estimate of 10

4

K for a typical superadia-

batic temperature fluctuation, b  10

 4

= a, giving a typical growth rate

of about 1 year.

When all the terms in the equation of motion are present, the disper-

sion relation gives a fast inertial wave and a slow wave in which only

the time-derivative terms in the induction and temperature equations

are important, inertia being negligible. These slow waves are known

as MAC waves and have frequency

MAC

¼ o

ð1 þ o

1=2

; where o

¼ o

(Eq. 14)

When jo

j > jo

j, o

MAC

is imaginary, corresponding to a growing

convective mode. The rate of growth is slow, since t

¼ 2p=o

of the order of some thousands of years. It is comparable to the magnetic

diffusion time, since the ratio t

diff

¼ B

=2mrO ¼ L and the

Elsasser number L has a value of O(1) in the core. The MAC growth

292 GEODYNAMO

rate will be a little larger than 1 = t

because jo

j> o

, but these

slow growth times are consistent with the time taken for the typical con-

vective velocity to take fluid across the core, t

conv

10

s, so the dyna-

mical picture does seem to be self-consistent. The main role of

the Coriolis acceleration and the Lorentz force is to constrain the con-

vection and slow it down from an unimpeded growth rate of about a

year down to t

conv

There are, however, some difficulties. Although for general wave

vectors k the MAC wave timescale is slow, it is much faster if k and

V are perpen dicular. This is the case for motions which are indepen-

dent of z, the coordinate parallel to the rotation axis. Then o

is infi-

nite, which means we have to restore inertia, and we then get the much

faster torsional wave frequency. It is also possible for k to be perpen-

dicular to both V and B

. Such waves have motion only along

“plates, ” planes containing the rotation vector and the magnetic field

vector. For these waves Eq. (14) is degenerate, with both o

and

zero. Such motions are unaffected by the field and the rotation

and so will grow on the much shorter o

timescale. We therefore

expect these modes to domina te small scale convection in the core

(Braginsky and Meytlis, 1990).

The above analysis assumes uniform magnetic fields. Actually the

field inside the core is likely to have a complicated structure. Such

fields often become unstable, and allow magnetic energy to be con-

verted into kinetic energy. In the Sun, this can happen on a very short

timescale, and solar flares are the result. Nothing quite as dramatic is

expected in the Earth ’s core, because magnetic diffusion is much more

important there. Nevertheless magnetic instabilities could be important

in driving shorter timescale motions and in controlling the strength of

the magnetic field. Magnetic instabilities generally become important

when the Elsasser number is O(1) (Fearn, 1998), (see also Core

magnetic instabilities ) so this may well be an important mechanism

in the core.

Taylor ’s con straint

One interesting consequence of the MAC balance in the core pointed

out by J.B. Taylor (1963) was that the geomagnetic field must satisfy

a special condition, known as Taylor ’s condition (q.v.). Azimu thal

flows that are constant on cylinders coaxial with the rotation axis

play a rather special role in rotating flows, because the Coriolis accel-

eration can be balanced entirely by the pressure force. Such flows

are called geostrophic flows. The special feature of the cylinders is

that a column of fluid moving along such a cylinder has constant

height, so no vortex stretching occurs. J.B. Taylor noted that if we inte-

grate the net azimuthal force over a cylinder, the contribution from

the Coriolis term is zero, as is the pressure term, and since buoyancy

acts radially, not azimu thally, the only force left is Lorentz force. It

follows that

ð j  B Þ

d s ¼ 0 ; (Eq. 15)

where S is any cylindrical surface not intersecting the inner core. This

is known as Taylor ’s constraint, and provided viscosity is negligible it

must be satisfied in the core. Since the current is related to the field by

Ampere ’s law, (1), Taylor’ s condition is a constraint on the magnetic

field. An interesting question is what happens if we solve an initial

value problem in which Taylor’ s constraint is not initially satisfied?

We must then restore part of the inertial term, r ] u= ] t , and the system

responds with a fast torsional oscillation (q.v. ), in which Lorentz

forces directly accelerate the fluid. These oscillations have a period

of some tens of years, and there is evidence that they occur in the core,

as they may explain the decadal variations of the length of day (q.v.)

(Jault et al., 1988; Jackson, 1997; Jault, 2003), and possibly shorter

period elements of the geomagnetic secular variation (q.v. ), such as

the so-called geomagnet ic jerks ( q.v. ). On the much longer dynamo

timescale, these oscillations would then decay away, restoring a field

satisfying Taylor’s condition. We can view the MAC balance state satis-

fying Taylor’s constraint as a slowly evolving equilibrium state, with the

magnetic field slowly changing as the convection moves the core fluid,

on a timescale of hundreds of years. Short-term variations of the field

on small length scales can be explained by convection, as variations on

a short length-scale ‘ would occur on short timescale ‘=U



, but the

short-term variations on long length scales, such as give rise to the dec-

adal variations of the length of day, could be explained if torsional oscil-

lations were continually being excited, so that the core is always

vibrating around its equilibrium MAC state. How these torsional

oscillations are excited is not yet understood.

Intermediate dynamo models

The full time-dependent three-dimensional equations (5), (11), and

(12) are difficult to solve numerically, so it is only recently that com-

puters have become fast enough to make it feasible to study them.

Numerical simulations can also be hard to interpret. A compromise

position has been to still retain the a-effect, thus allowing axisym-

metric solutions to be found, but to include some dynamics, normally

the meridional circulation and the azimuthal flow driven by the mag-

netic field.

Intermediate dynamo models have often been used to investigate

how Taylor’s constraint could be met. It was suggested by Malkus

and Proctor (1974) that the magnetic field would drive flows of

exactly the right type to generate fields satisfying Taylor’s constraint.

Soward and Jones (1983) looked at an intermediate plane layer model

which did indeed lead to solutions satisfying Taylor’s constraint, in

accord with the Malkus-Proctor scenario. With values of a close to cri-

tical, the magnetic field did not satisfy Taylor’s constraint. The

azimuthal force balance was therefore between the Lorentz force and

the Ekman friction in the boundary layer which is OðE

1=2

Þ, so that

the magnetic field was limited to a value with Elsasser number

OðE

1=2

Þ. This balance is known as an Ekman state. However, as a

was increased, a value was reached at which the field did satisfy

Eq. (15), and at that point the field dramatically increased, leaving

the Ekman state and having a strength with Elsasser number O(1).

However, Braginsky (1976, 1994) proposed an almost axisymmetric

model-Z dynamo (q.v.) in which the Taylor constraint was met in an

unusual way. In an axisymmetric configuration, Eq. (15) can be

written



¼ 0; (Eq. 16)

where B

is the component of magnetic field pointing away from the

rotation axis. Taylor’s constraint could be satisfied if the meridional

part of B was purely in the z-direction, so B

¼ 0, hence the name

model Z dynamo. Braginsky proposed a specific distribution of a con-

centrated near the equatorial region and the CMB, which led to a

steady rather than an oscillatory dynamo, as part of the model Z sce-

nario. A more detailed analysis by Jault (1995) suggested however that

if the viscosity is reduced to very low values, “Taylorization” (that is

the approach to a state in which Eq. (15) holds) occurs in the conven-

tional Malkus-Proctor way, with B

being small but finite.

Fully three-dimensional simulations usually have to be run with a

viscosity too large to achieve “Taylorization” in order to maintain

numerical stability. However, Rotvig and Jones (2002) have run fully

three-dimensional simulations in plane geometry, which is less

demanding computationally, and allows smaller E to be reached. They

found that “Taylorization” did occur at the lowest values of E they

could reach.

Another way of simplifying the full three-dimensional problem, is

the so-called “2

-dimensional” approach. Here full resolution is used

in the r and y directions, but in the azimuthal fdirection a severe trun-

cation is imposed; in the most draconian approximation only the axi-

symmetric and one nonaxisymmetric expðimfÞ modes are retained

GEODYNAMO 293

(e.g., Jones et al., 1995). This still allows nonaxisymmetric convection

to occur, so no a-mechanism is needed, and the results are generally

qualitatively similar to those obtained from fully three-d imensional

simulations, but at much reduced cost. The disadvantage is that unless

results are checked by fully three-dimensional simulations, one cannot

be certain that the truncation is not affecting the behavior; nevertheless

“2

-dimensional ” models are very useful for testing out new ideas in

dynamo theory.

Role of the inner core

As mentioned in the energy balance section, the inner core makes an

important contribution to the driving of convection in the core. It also

significantly affects the dynamics of core convection. The importance

of geostrophic motions in rapidly rotating fluids suggest s that the solid

inner core could play an important role in the dynamics of the Earth ’s

interior. The cylinder, which just touches the inner core, is called the

inner core tangent cylinder ( q.v.). Whereas the volume of the inner

core itself is only about 4% of the volume of the outer core, the

fraction inside the tangent cylinder is much larger.

The flow outside the tangent cylinder can convect heat out in tall

thin column s, which although not exactly geostrophic, because some

vortex stretching occurs as the columns rotate due to the sloping bound-

aries, nevertheless minimizes the effect of rotation. Inside the tangent

cylinder the convection needs to get the heat out in a direction almost

parallel to the rotation axis, and tall thin columns run into the inner core,

causing additional friction. In consequence, most models show that con-

vection is more efficient outside the tangent cylinder than inside it. This

difference produces a latitudinal temperature gradient, with the poles

being slightly hotter than the equator on a sphere of constant radius.

Neglecting magnetic forces, the azimuthal component of the curl of

Eq. (11), then gives the thermal wind (q.v.) equation,

2 O

] u

] z

g a

] T

] y

(Eq. 17)

where z is the coordinate parallel to the rotation axis and y is the mer-

idional direction in spherical polars ðr ; y ; f Þ. This shows that u

decreases with z in the northern hemisphere. A similar effect drives

the jet stream in the Earth ’s atmosphere, though the sign is different

(the jet stream velocities increase with height, because the pole is

colder than the equator, and so the jet stream goes from west to east

relative to the surface) and of course temperature gradients and veloci-

ties are much smaller in the core. Nevertheless, the thermal wind is

sufficient to give a rotation of the inner core (q.v.) relative to the man-

tle. The strength of the thermal wind can also be affected by magnetic

torques. It has been suggested (Buffett and Glatzmaier, 2000) that the

acceleration produced by the thermal wind may be counteracted by a

gravitational coupling of the inner core to the mantle, which may be

expected if there are slight departures from axisymmetry in the figure

of the mantle and the inner core. The rotation of the inner core relative

to the mantle can in principle be measured by seismologists, but the

actual value of the inner core rotation rate is still controversial

(e.g., Collier and Helffrich, 2001).

Reversa ls an d field morpho logy

Two obvious questions about the geodynamo are (i) why is the field

dominated by an axial dipole component? (ii) why does the field per-

iodically reverse its polarity? Our lack of a full understanding of the

geodynamo is perhaps highlighted by the fact that neither of these

questions can be answered unambiguously. The majority of dynamo

simulations do show a dipole dominated field, but by no means all

do (e.g., Christensen et al. , 1999; Busse, 2002), and the Karlsruhe

experimental dynamo was dominated by an equatorial rather than an

axial dipole. Quadrupolar dynamos, equatorial dipoles, and axial

dipoles can all be produced from flows, which are apparently not that

dissimilar. A number of trends are apparent from the simulations; for

example, a strong azimuthal flow makes an equatorial dipole less

likely, as the shearing motion associated with differential rotation tends

to disrupt an equatorial dipole field. Dipolar fields tend to be more

common when the convection is driven by a flux of heat from the

inner core, whereas quadrupolar dynamos are more common in uni-

formly heated models, and there is a tendency (Busse, 2002) for

dipoles to be preferred over quadrupoles at higher Rayleigh number.

However, all these observations are model dependent, and it should

not be forgotten that it is not currently possible to run geodynamo

models in the correct parameter regime.

A large number of papers have addressed the issue of the theory of

reversals (q.v.) (see e.g., Sarson, 2000). One simple point is that the

dynamo equations are invariant under a reversal of the field direction,

so if a solution with “normal” polarity exists, an exactly similar one

with reversed polarity also exists. To see this note that the transforma-

tion B !B changes the sign of both B and j, so the Lorentz force

j  B is unchanged, and the induction equation is linear in B so rever-

sing the sign of B leaves this equation unchanged. This observation

shows that it is reasonable to expect the geodynamo to reverse occa-

sionally, but it doesn’t explain why reversals are comparatively infre-

quent (there are only a few reversals per million years on average)

and why they happen relatively quickly (around 5 ka or less) when

they occur.

The magnetic field is continually fluctuating, and reversals appear to

be part of this process. Magnetic excursions, events where the axis of

the dipolar component moves rapidly away from the geographic poles

occur much more frequently than full reversals, so it is possible that

excursions are in some sense failed reversals, where the fluctuation

is large but not large enough to cross a threshold leading to full

reversal. Hollerbach and Jones (1993, 1995) suggested that the thresh-

old necessary was that the field in the solid inner core had to be

reversed. Since this can only occur by magnetic diffusion, rather than

on the somewhat more rapid convective turnover time, only the larger,

longer, excursions would allow the inner core field to reverse, and

hence establish a full reversal. Numerical simulations of the geody-

namo also suggest that the core is in a continually fluctuating state,

and that some of these fluctuations occasionally lead to reversals.

Glatzmaier et al. (1999) found that a spatially inhomogeneous heat

flux at the core-mantle boundary could affect reversal frequency (see

also Inhomogeneous boundary conditions and the dynamo). This is

an interesting possibility, as mantle convection models suggest that

such inhomogeneities are likely, and indeed seismic measurements of

the CMB region also suggest there is such inhomogeneity. This would

provide a link between mantle convection and the dynamo, and since

mantle convection evolves on a very long million year timescale, this

could explain why reversals are very infrequent compared to excur-

sions. For example, on this picture during the 60-million-year Cretac-

eous superchron, during which there were no significant reversals,

mantle convection was in a pattern, which affected the core flow in

such a way as to make reversals unlikely. Sarson and Jones (1999)

suggested that this might be connected with the meridional circulation,

which would be associated with an inhomogeneous CMB heat flux, as

in their model the strength of the meridional circulation played an

important role in the reversal process.

There is much that we do not yet understand about the dynamics of

the Earth’s core and the reversal process, but progress is being made

on a number of different fronts: better numerical simulations, and dee-

per understanding of the processes involved; more detailed and more

accurate paleomagnetic studies giving information about the past beha-

vior of the geomagnetic field; new results from seismology, probing

the structure of the Earth’s deep interior ever more thoroughly; a

new and better understanding of the physical properties of matter at

very high pressure. With all these stimuli, our understanding of the

geodynamo will surely radically improve over the coming decades.

Chris Jones

294 GEODYNAMO

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