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from a meridional ¬eld BM. For a successful dynamo there has to be a converse
loop that creates BM from Bφ, and as demonstrated by Cowling in 1934 no such
process exists if the system is totally axisymmetric.
The eddies in a rotating ¬‚uid undergoing turbulent convection are expected to
resemble the patterns of atmospheric circulation. These cyclonic (or anticyclonic)
disturbances will deform, and may amplify the magnetic ¬eld in a conducting ¬‚uid.
Consider then a decomposition of the magnetic ¬eld into a large-scale component
B and a much more rapidly varying small-scale ¬eld b , with a similar form for
the ¬‚ow ¬eld v. There will be an electromotive force (emf) = v — b that will
have a large-scale component
= v —b . (15.2.26)
When the turbulence is suf¬ciently complex the average over the small scales in
(15.2.26) does not vanish and the net emf is approximately proportional to the
large-scale component of the magnetic ¬eld,
=± B . (15.2.27)
In general, ± would be a tensor. If the small-scale motions are statistically
independent of direction, but do not have mirror symmetry, then ± is approximately
φ = ± B φ is able to generate a meridional
a scalar. The azimuthal component
¬eld BM ; this is known as the ±-effect.
Dominantly axisymmetric large-scale ¬elds can therefore be sustained by a
simple feedback loops. In the ±ω dynamo, the ±-effect creates BM from
Bφ and in turn the ω-effect produces Bφ from BM to give a self-sustaining
magnetic ¬eld. The size of the zonal ¬‚ows deduced from secular variation suggests
that the ±ω dynamo is a plausible candidate for the Earth™s core. An alternative, the
±2 dynamo, relies on the ±-effect to generate both BM from Bφ and Bφ from
BM . Such a dynamo model is more likely to lead to steady solutions. Depending
on the choice of ± and the differential rotation in the ¬‚ow the dominant solutions
can have either dipolar or quadrupolar symmetry. Reversal in the magnetic ¬eld
would occur when the form of internal motion changes, leading to a modi¬cation
of the preferred symmetry of the ¬eld con¬guration. Both ±- and ω-effects are
likely to be active, and the classi¬cation as ±ω or ±2 dynamos will depend on
which effect is more signi¬cant in generating BM .

15.2.4 Deviations from the reference state
The reference state introduced in (15.2.16) has purely radial variations of pressure
and temperature, whereas in the ¬‚ow ¬eld there will be additional components with
spatial and temporal variations; we therefore write
P = P0(r) + P (r, θ, •, t), T = T0(r) + ˜(r, θ, •, t), (15.2.28)
390 The Core and the Earth™s Dynamo

To ¬rst order in the pressure and temperature deviations (P , ˜) the equation of
motion takes the form
‚v ∇P 1
’ 2© — v + ν∇2v +
+ (v · ∇)v = ’ (∇ — B) — B; (15.2.29)
‚t ρ0 μ0ρ0
and the equation for the temperature evolution is
‚˜
+ v · ∇˜ = h + κH ∇2˜ ’ v · ∇T0 + dq, (15.2.30)
‚t
where dq represents the combined effect of Joule and viscous heating.
In many numerical simulations a somewhat simpli¬ed scenario is investigated.
The variation in density across the outer core is ignored and a Boussinesq
approximation is adopted in which just the density effects due to temperature
(and perhaps composition) are included (cf. the treatment of convection in Section
7.5.2). The resulting equations are similar to (15.2.29), (15.2.30), but the energy
dissipation dq cannot be included in a fully consistent fashion.


15.2.5 Non-dimensional treatment
In our discussion of ¬‚uid ¬‚ows in Chapter 7, we saw how considerable insight
into the physical character of the system can be obtained by working with
non-dimensional quantities, so that different con¬gurations can be compared
through a set of non-dimensional numbers. A similar approach can be made for
the magnetic ¬‚uid dynamics of the core. A wide variety of different choices
have been made in studies of the core dynamo and a convenient summary and
inter-comparison is provided by Kono & Roberts (2002).
The choice of non-dimensionalisation is designed to bring the main terms to
comparable size. There are two plausible choices for the length scale, either the
radius of the outer core, rc , or the span of the outer core from the mantle to the
core-mantle boundary Ls = rc ’ ri , i.e., 2260 km. There are more choices for the
time scale based on viscous, thermal or magnetic diffusion times. We will use the
magnetic diffusion time Ts = L2/», where » = 1/(μ0σ) is the magnetic diffusivity;
˜
s
Ts is of the order of 65,000 years. The corresponding ¬‚ow velocity Us = Ls/Ts
is around 10’6 ms’1 , and the effective ¬‚ow and diffusion time scales are then
comparable. A scaling for the magnetic ¬eld is provided by Bs = (©ρμ0»)1/2,
which ensures that the Lorentz and Coriolis force terms have a similar size.
The main scaling factors are thus:

L2 Ls
Ts = s = L2μ0σ,
L s = rc ’ ri , Us = ,
˜
s
» Ts
1/2
©ρ
1/2
Bs = (©ρμ0») = . (15.2.31)
σ˜
15.2 Convection and dynamo action 391

The Ekman number (7.6.9), the ratio of the viscous and Coriolis forces, is given by
ν
≈ 10’12 ,
Ek = (15.2.32)
2
©L
based on molecular diffusivity in the core. If an eddy diffusivity is employed,
representing the effects of small scale ¬‚ow within the core, then the Ekman number
is increased by a factor of around 103 . The viscosity of the ¬‚uid outer core is not
well constrained, but is assumed to be comparable to that of water.
The magnetic Rossby number, representing the balance between magnetic and
Coriolis forces, is de¬ned as
» 1
≈ 10’8 .
RoM = = (15.2.33)
2 2
©L μ0σ©L
˜
The Coriolis forces from the rapid rotation overwhelm both the viscous and
magnetic forces.
The ratio of the thermal to magnetic diffusion times de¬nes the Roberts number
κH
q= , (15.2.34)
»
which plays an analogous role to the Prandtl number Pr = ν/κH that describes the
relative signi¬cance of momentum and heat transport.
For the rapidly rotating ¬‚uid it is convenient to use a modi¬ed Rayleigh number to
include the angular velocity ©

g±thβL2
s
Ra = , (15.2.35)
©κH
where g is the acceleration due to gravity, ±th the thermal expansion coef¬cient
and β is the radial temperature gradient.
Following scaling by the non-dimensional quantities we can write the equations
governing the behaviour of the magnetic ¬‚uid, in the absence of internal heating,
in the form

B = ∇2B + ∇ — (v — B), (15.2.36)
‚t

+ v · ∇ v + 2^z — v = ’∇P + Ek ∇2v + (∇ — B) — B + Ra qθx,
RoM e
‚t
(15.2.37)

+ v · ∇ ˜ = q∇2˜, (15.2.38)
‚t
∇ · B = 0, (15.2.39)
∇ · v = 0, (15.2.40)

where ˜ is the non-dimensional temperature perturbation.
392 The Core and the Earth™s Dynamo

Induction equation
Consider the equation for the magnetic ¬eld (15.2.36), but neglect initially the
diffusion term in ∇2B. Then

B = ∇ — (v — B). (15.2.41)
‚t
This equation is analogous to the vorticity equation of ¬‚uid dynamics

= ∇ — (v — ), (15.2.42)
‚t
which states that vortex lines evolve as material lines. Thus, just as vortex lines
can be stretched and thereby be ampli¬ed, so may the magnetic ¬eld lines. The
presence of the neglected term from (15.2.36) will ultimately lead to diffusion of
the ¬eld behaviour.

Geostrophy and Taylor Columns
The inertial term in (15.2.37) appears with the very small magnetic Rossby number
RoM and the viscous term with the not much larger Ekman number Ek so we can
anticipate that as in the treatment of a rotating ¬‚uid in Section 7.6 we could make a
geostrophic approximation to yield a reduced equation
2^ — v = ’∇P + (∇ — B) — B + Ra qθx.
x (15.2.43)
The appropriate boundary condition on the ¬‚ow at the core“mantle boundary is
then the inviscid condition v · n = 0. In the absence of the magnetic ¬eld there
would be no variation in the velocity ¬eld in the direction parallel to the rotation
axis. However, such a free geostrophic solution is not possible for (15.2.43) unless
the Lorentz force obeys the very restrictive Taylor condition

(∇ — B — B)φdzdφ ≡ T (s) = 0, (15.2.44)
C(s)

where (s, φ, z) are cylindrical polar coordinates and C(s) is the cylinder of radius
s aligned with the rotation axis within the core.
The problem can be overcome by working with a nearly geostrophic state
including viscous terms, either through the reinstatement of Ek∇2v in (15.2.43)
or by inclusion of viscous effects in thin Ekman boundary layers at the top
and bottom of the ¬‚uid outer core. Nevertheless, the forms of ¬‚ows displayed
by many numerical dynamos with signi¬cant magnetic ¬elds have a strong
columnar component (Taylor columns) as suggested by the simple geostrophic
approximation.


15.3 Numerical dynamos
The advent of fast computers has meant that it is feasible to attempt full
three-dimensiona; calculations of the dynamo with coupled velocity and magnetic
15.3 Numerical dynamos 393

¬elds. Viscosity is retained in the calculations and the Ekman number Ek made
as small as possible. However, the parameter values are still a long way from
being geophysically reasonable even though computed ¬eld strengths are of the
right order for the Earth and irregular oscillatory behaviour is obtained.
The role of the inner core is particularly important. Firstly, the tangent cylinder
that is parallel to the rotation axis and touches the inner core at the equator separates
two quite different types of ¬‚ow with the most vigorous motion lying within the
tangent cylinder. Secondly and more importantly, because the inner core is a
conductor with similar conductivity to that of the outer core any magnetic ¬eld
lines that penetrate into the solid inner core can only evolve on the diffusion time
scale. This pinning of the ¬eld lines helps to stabilise what would otherwise be
wild oscillatory behaviour and to prevent reversals of the main dipole ¬eld on very
short time scales.
The numerical solution of the coupled magnetic, ¬‚ow and thermal equations
(15.2.36)“(15.2.38) is commonly undertaken using a spectral technique coupled
to a toroidal and poloidal decomposition. As in (15.1.4) the magnetic ¬eld is
expressed as
B(x, t) = ∇ — [T (r, t)x] + ∇ — ∇ — [S(r, t)x], (15.3.1)
and so satis¬es the condition ∇ · B = 0 (15.2.39). A similar form can be applied
for an incompressible ¬‚uid so that
v(x, t) = ∇ — [V(r, t)x] + ∇ — ∇ — [U(r, t)x], (15.3.2)
and ∇ · v = 0 (15.2.40).
The de¬ning scalars for the toroidal and poloidal vectors are then expanded in
m
terms of the Yl spherical harmonics (11.3.38), e.g.,
LB l
m m
T (r, θ, φ, t) = Tl (r, t)Yl (θ, φ), (15.3.3)
l=0 m=’l
LV l
m m
V(r, θ, φ, t) = Vl (r, t)Yl (θ, φ); (15.3.4)
l=0 m=’l

a comparable expansion can be made for the non-dimensional temperature
LT l
˜m(r, t)Yl (θ, φ).
m
˜(r, θ, φ, t) = (15.3.5)
l
l=0 m=’l

Hollerbach (2000) recommends that the truncations of the magnetic, velocity
and temperature spherical harmonic expansions are distinct, because the various
¬elds exhibit structure in rather different length scales. Thus, the velocity ¬eld
v is likely to exhibit ¬ne-scale structure when the Ekman number Ek is small,
whereas the temperature ¬eld ˜ will have ¬ne structure when the Roberts number
394 The Core and the Earth™s Dynamo

q is small. Independent truncation allows ¬‚exibility in the representation, with
consequent savings in computational effort.
With the expansion in spherical harmonics we need to turn the original
magnetohydrodynamic equations and boundary conditions for v, B, and ˜ into
equivalent differential equations and associated boundary conditions for Um, Tlm
l
etc. that are suitable for numerical computation. Application of the curl (∇ — )
and the curl of the curl (∇ — ∇ — ) to (15.2.37) removes the pressure term. The r
components of these equations yield
Ł2 ‚
RoM ’ EkL2 Vl Yl = ^ · ∇ — F,
mm
x (15.3.6)
r ‚t
m
l
2

Ł
’ EkL2 L2UmYl = ’^ · ∇ — ∇ — F,
m
RoM x (15.3.7)
l
r ‚t
m
l

where, as in (15.1.8), Ł2 = l(l + 1) and L2 is the angular momentum operator
(15.1.6). The forcing vector for the ¬‚ow equations
F = ’2^z — v ’ RoM (v · ∇)v + (∇ — B) — B + Ra qθx.
e (15.3.8)
A comparable development can be made for the induction equation (15.2.36),
Ł2 ‚
’ L2 SmYl = x · ∇ — (v — B),
m
^ (15.3.9)
l
r ‚t
m
l
Ł2 ‚
’ L2 Tl Yl = x · ∇ — ∇ — (v — B).
mm
^ (15.3.10)
r ‚t
m
l
For the temperature equation we can also make a spectral development
Ł2 ‚ 2‚
’ q L2 + ˜mYl = ’v · ∇˜.
m
(15.3.11)
l
r ‚t r ‚r
m
l
The orthogonality of the spherical harmonics allows us to extract equations for
the individual spectral components so that, e.g., from (15.3.6) we ¬nd
2π π
Ł2 ‚
RoM ’ EkL2 Vl =
m
dθ sin θ [Yl ]— x · ∇ — F.
m
^
dφ (15.3.12)
r ‚t 0 0
The linear boundary conditions for the different ¬elds apply directly to the
individual spectral components.
The radial dependence of the various spectral components can be expanded in
terms of Chebyshev polynomials, such as,
Nu +2
2r ’ rc ’ ri
Um(r, t) Um (t)Tn’1(x) with x =
= , (15.3.13)
l l,n
rc ’ ri
n=1
and the variable x varies between ’1 and 1. Radial derivatives can be evaluated
using a recursion on n for the Um coef¬cients. The Chebyshev coef¬cients,
l,n
15.3 Numerical dynamos 395




1 2 3 4




5 6 7 8




Figure 15.4. Evolution of the magnetic lines of force in a meridional plane averaged over
longitude during a simulated polarity reversal. The simulated time sequence starts at the
top left and ends at the bottom right after approximately 5200 years, which is consistent
with the time scale of the geomagnetic ¬eld. Clockwise ¬elds are shown in black (with a
plus sign) and counterclockwise in grey (with a minus sign). [Courtesy of F. Takahashi.]



such as Um , are to be found by evaluating (15.3.12) at the Nu collocation points
l,n
representing the zeros of TNu (x), together with the two boundary conditions
to provide Nu + 2 equations. The Chebyshev representation automatically
concentrates resolution close to the boundaries, which is of considerable value for
resolving boundary layer features.
The differential equations for the spectral components are linear on the left-hand
side of each equation, such as (15.3.12), but on the right-hand side we have
spherical transforms of non-linear functions. Once the right-hand side terms are
known, Runge“Kutta or similar systems can be used to solve the differential
equations (see Hollerbach, 2000, for details of the computational scheme). The
evaluation of the non-linear terms becomes rapidly infeasible as the number of
spectral components included increases, because the computation cost is O(L4)
where L is the truncation level of the spectral expansion. In consequence a
˜pseudo-spectral™ method is employed with repeated switches between real and
spectral space. The ¬eld variables v, B, etc. are evaluated at appropriate points
396 The Core and the Earth™s Dynamo




90

60

30
Pole Latitude




0

-30

-60

-90
0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82
Time




Figure 15.5. Evolution of the morphology of the radial component of the magnetic ¬eld at
the core surface during a simulated polarity reversal. Positive signs indicate outward ¬elds
from the core and negative signs inward ¬elds. The time scale is non-dimensionalised by
the magnetic diffusion time of approximately 200,000 years. The maximum and minimum
contour values are ±0.5mT. The results are presented for spherical harmonic order less
than 12 to allow comparison with the inferences from the observed geomagnetic ¬eld in
Figure 15.1. [Courtesy of F. Takahashi.]


in real space, and the necessary operations performed to construct F and v — B in
the spatial domain. The required derivatives are then constructed in the spectral
domain. Hollerbach (2000) show how to make ef¬cient choices for the spatial
sampling to achieve accurate results without excessive computational effort.
In the presence of an inner core with ¬nite conductivity, the evolution of the
magnetic ¬eld is determined not only by changes at the boundary between the
inner and outer core, but also by prior variations in the inner core. The associated
magnetic diffusion time for the inner core imposes a new time scale on the whole
dynamo system that has a signi¬cant effect on the behaviour (see, e.g., Glatzmeier
& Roberts, 1996).
The complications in solving the coupled magnetohydrodynamic equations
come from the very small values of the magnetic Rossby number RoM (∼10’8 )
and the Ekman number Ek (<10’9 ) in equations such as (15.3.12). These terms
15.4 Evolution of the Earth™s core 397

are associated with very ¬ne length scales (of the order of metres) and rapid time
variations. Available computational resources have limited calculations to values
several orders of magnitude larger than expected core conditions. Nevertheless
dynamic behaviour with Earth-like states has been simulated.
Takahashi et al. (2005) have carried out a spectral approach using all spherical
harmonics of degree and order up to 255, using the Earth Simulator computer in
Japan, and have been able to reduce the Ekman number to 4—10’7 . Their dynamo
models employ electrically insulating, no-slip and isothermal boundary conditions
at the core“mantle boundary and the inner core boundary. Distinct reversals of
the magnetic ¬eld are simulated with a relatively rapid transition between ¬eld
states (Figure 15.4) In addition to full reversals this high resolution model shows
excursions of the magnetic ¬eld that do not lead to a sustained polarity change
(Figure 15.5).
Almost all numerical dynamo models produce a signi¬cant axial dipole
component above the trend of the power law decay of the higher ¬eld harmonics,
whereas the quadrupole term lies somewhat below the trend. A major limitation
arises from limited knowledge of the physical properties of the core. Some aspects
can be expected to be addressed by ab initio calculations as discussed in Chapter 9,
but parameters such as the viscosity of the core are likely to remain elusive.
15.4 Evolution of the Earth™s core
One of the intriguing questions about the Earth is the way that the core has evolved
in time. When did the inner core form and how might it be growing? As we shall
see the answers to such questions depend critically on the Earth™s heat budget, and
in particular the rate at which heat leaves the metallic core and enters the silicate
mantle.
It is tempting to suggest that the onset of observable magnetic ¬elds about 3.5 Ga
ago marks the point at which the inner core started to form and the dynamo began
to evolve towards its current con¬guration. However, dynamo action does appear
to be possible in a ¬‚uid sphere and the presence of a small inner core should not
have a profound effect on the ¬‚ow ¬eld. As we shall see, it is likely that the growth
of the inner core, once solidi¬cation occurs, is quite rapid. This change could well
affect the amplitude of the magnetic ¬eld that could be sustained by dynamo action.
15.4.1 Energy balance
Following Buffet et al. (1996)we can use the energy balance equations to
understand the evolution of the core. We assume that there is no mass transport
across the core“mantle boundary Sc and then look at the energy budget in the core.
We can integrate the expressions (15.2.24) and (15.2.24) over the whole volume of
the core Vc to describe the rate of change of the kinetic and magnetic ¬elds. The
rate of change of mechanical energy for the core is thus
d 12
2 ρv dV = v · (j — B)dV ’ v · (ρ∇• + ∇P)dV. (15.4.1)
dt Vc Vc Vc
398 The Core and the Earth™s Dynamo

With the magnetohydrodynamic assumption of the neglect of displacement
currents, the rate of change of magnetic energy for the whole core is
2 J2
1B
d
dV = ’ v · (j — B)dV ’ dV. (15.4.2)
2
Vc μ0 σ˜
dt Vc Vc

We can express the internal heat budget in the form
J2
d
ρU dV = ’ q · dS + ρh dV + dV ’ P(∇ · v)dV, (15.4.3)
Vc σ˜
dt Vc Sc Vc Vc

where U is the speci¬c internal energy, h is the rate of radioactive heating per unit
mass and q is the heat ¬‚ux into the base of the mantle.
The contributions to the speci¬c internal energy can be built up from
thermodynamic considerations and so
P
dU = T dS + dρ + μζ dζ, (15.4.4)
ρ2
where ζ is the concentration of light elements in the core, and μζ is the
corresponding chemical potential. We also have the conservation of mass in terms
of the continuity equation (8.1.1),

= ’ρ(∇ · v), (15.4.5)
Dt
and can describe the diffusion of the lighter elements by

ρ = ’∇ · i, (15.4.6)
Dt
with i the ¬‚ux of the lighter component.
With the aid of (15.4.4)“(15.4.6) we can rewrite the internal energy budget
equation (15.4.3) as
J2
DS
ρT dV = ’ q · dS + dV + ∇ · i dV. (15.4.7)
Dt Vc σ˜
Vc Sc Vc

The changes in the heat content of the core are due to the heat ¬‚ow from the core,
in the interior, and diffusive segregation of lighter elements. Gravitational energy
is released by thermal contraction and compositional fractionation; there will be a
modi¬cation of the compression, with a small amount of adiabatic heating, as the
net radial mass distribution changes.
With respect to the thermal evolution of the core, the changes in the gravitational
and internal energies will dominate those from the mechanical and magnetic
energies so that from (15.4.1)“(15.4.2):
2 J2
1B
d 12
2 ρv + 2 μ dV = ’ v · (ρ∇• + ∇P)dV ’ dV ≈ 0.(15.4.8)
σ˜
dt 0
Vc Vc Vc
15.4 Evolution of the Earth™s core 399

Thus, we can equate the total ohmic dissipation ¦ to the difference between the
change in gravitational energy and the work done against pressure gradients
J2
¦=’ dV = v · (ρ∇• + ∇P)dV. (15.4.9)
Vc σ˜ Vc

The second integral in (15.4.9) would vanish if the state of the core were
hydrostatic, and so ohmic heating is maintained by the departures of the core from
a hydrostatic state due to convection.
Under the same approximation of neglect of changes in magnetic, kinetic and
nuclear energies, the rate of change dΣ/dt of the total energy within the core
obtained by summing (15.4.1)“(15.4.3) is
d d
Σ= ρU dV + ρv · ∇• dV,
dt dt Vc Vc

=’ q · dS ’ Pv · dS. (15.4.10)
Sc Sc

The work done against the Lorentz forces and the work associated with P∇ · v do
not appear in (15.4.10) because they represent transfer of energy between different
forms within the core.
15.4.2 Thermal and compositional effects
Figure 15.6 illustrates the processes that are likely to contribute to the evolution of
the core through the thermal and compositional release of gravitational energy. The
processes are shown separately, but all four will in fact operate at the same time.
Because of the extraction of heat ¬‚ux q from the core into the mantle above, there
will be cooling of the core and a contraction in the core radius, rc ’ rc’δrc , with the
generation of a thermal boundary layer below the cooler mantle. The heat ¬‚ux can
be expected to vary over the surface of the core because of the strong variation in
the properties of the mantle just above the core“mantle boundary. Over time parts
of the cold and dense thermal boundary layer will become unstable and sink to mix
into the bulk of the outer core. At the inner core boundary, the lighter components
are expelled as the inner core solidi¬es and the inner core radius will tend to grow,
ri ’ ri + δri . There will also be release of latent heat as the inner core material
freezes. The light compositional boundary layer will be unstable and there will be
overturn and mixing of the lighter components into the outer core.
The long term effect of inner core growth will be a redistribution of mass, with
a concentration of the heavier iron in the inner core and a consequent reduction in
density in the outer core, since the mass of the core must remain constant.
Because motions are rapid the mixing of cooler and lighter components will be
ef¬cient and the outer core temperature pro¬le will be approximately adiabatic and
the composition nearly uniform. Even though the ¬‚uctuations about the adiabatic,
isentropic, well-mixed and hydrostatic state will be small, they are needed to
produce the convective ¬‚ows and buoyancy ¬‚uxes that sustain the dynamo.
400 The Core and the Earth™s Dynamo




Mantle
thermal
q
boundary layer
mixing of
Outer Core
contraction colder material



ρ ρ + δρ


(a) (b)
Inner Core

ρi


(c) (d)
growth mixing of
lighter material
expelled
ρ ρ ’ δρ
light component




Figure 15.6. Core processes contributing to thermal and compositional release of gravita-
tional energy: (a) heat ¬‚ux extracted from the core leads to local contraction near the core
mantle boundary, (b) the cold, dense thermal boundary layer becomes unstable and mixes
deeper into the core, (c) lighter elements are expelled from the inner core in growth by
solidi¬cation, (d) the light compositional boundary layer becomes unstable and mixes into
the overlying ¬‚uid.


15.4.3 Inner core growth in a well-mixed core
The cooling of the core is controlled, in large part, by the much more massive
mantle with relatively sluggish ¬‚ows. The mantle regulates the total heat ¬‚ux
Q(t) = Sc (q) · dS across the core-mantle boundary, and the magnitude and time
dependence of Q(t) will depend on the details of mantle convection.
The part of the thermal heat ¬‚ux that is signi¬cant is that part that is not
associated with thermal conduction along the adiabat, i.e. Q(t)/Nu, where the
Nusselt number Nu was introduced in (7.5.2). The effective heat ¬‚ux

1
q— · dS = 1’ Q(t). (15.4.11)
Nu
Sc
15.4 Evolution of the Earth™s core 401

Diffusive transport of light elements along the pressure gradient with radius is
expected to be slower and less signi¬cant than thermal diffusion. So all the
gravitational energy released by compositional segregation will contribute to ¦
(15.4.9), but only the convective heat ¬‚ux q— contributes to the thermal part.
There is a density jump (”ρ)ζ across the boundary between the inner and outer
cores r = ri associated with the compositional changes, which will be less than
the seismological estimates that include the phase change to the solid state. The
solidi¬cation at this boundary will release latent heat L per unit mass. It is also
convenient to introduce the mass-averaged gravitational potential of the outer core,
1
•= ρ• dV.
¯ (15.4.12)
MVoc oc

Buffet et al. (1996) show that the rate of ohmic dissipation ¦ under the
assumption of ef¬cient convective mixing of compositional and thermal anomalies
can be expressed as
±th ρ0L
dri
¦ = 4πr2 (”ρ)ζ ’ • ’ •(ri )
¯
i
Cp
dt
±th
q— · dS • ’ •(rc ) ,
’ ¯ (15.4.13)
Cp Sc

where, as in (15.2.18), ±th is the coef¬cient of thermal expansion. If mixing is not
ef¬cient, diffusion would smooth out the effects before the motion had a chance
to produce Joule heating. We can think of (15.4.13) as representing the effect of
the redistribution of a compositional mass de¬cit from the inner core and a mass
excess from the thermal boundary layer at the top of the core. The contribution
¦ζ = 4πr2(”ρ)ζ dri /dt is purely compositional, and the remainder ¦T = ¦ ’ ¦ζ
i
is thermal.
The contributions of compositional changes to the internal energy will be
dominated by the other terms in the internal heat budget of the core, so that to
an good approximation we can write (15.4.7) as
J2
DS
ρT dV = ’Q(t) + dV, (15.4.14)
Dt σ˜
Vc Vc
DT Dp drc
= ρCp dV ’ ±th T dV ’ ρL dS, (15.4.15)
Dt Dt dt
Vc Vc Si

where Si is the surface of the inner core. Small effects, such as heat generated by
mixing, are neglected in (15.4.15).
Buffet et al. (1996 - Appendix B) demonstrate that the advective term Vc ρT (v ·
∇S)dV in (15.4.15) cancels out the thermal contribution ¦T to the convective
energy release. As a result (15.4.15) can be rearranged to the form
‚T ‚p dri
Q(t) = ’ ρCp dV + ±th T dV + ρL dS + ¦C. (15.4.16)
‚t ‚t dt
Vc Vc Si
402 The Core and the Earth™s Dynamo

The changes in the temperature T and the pressure P appearing in (15.4.16) can be
estimated by looking at the changes in the average conditions in the core over time
scales long compared with those of convective ¬‚uctuations, but short compared
with the time over which the Earth cools. Thus
‚p ‚T ρgγth T
= ’ρg, =’ , ri < r < rc , (15.4.17)
‚r ‚r KS
where γth is the Gr¨ neisen parameter (6.1.13). The equations (15.4.17) need to
u
be supplemented with the condition that the composition is uniform on these time
scales in ri < r < rc . The acceleration due to gravity g, given by (6.1.1), will
depend on P, T and composition through the density ρ. The pressure p must vanish
at the free surface, and thus takes the form (6.1.2). The core liquid is regarded as
an ideal mixture, with the light component present only in the ¬‚uid outer core. The
mass fraction ζ is given by
ζ(t) = Ml/Moc (t), ri < r < rc , (15.4.18)
where Ml is the total mass of light element and Moc (t) is the slowly decreasing
mass of the outer core as the heavy component freezes onto the inner core.
The temperature is controlled by the requirement that the surface of the inner
core is in thermodynamic equilibrium with the surrounding ¬‚uid, i.e., T (ri , t) is
equal to the liquidus temperature TL(p, ζ). An approximate linear expansion for TL
can be made about the conditions where the inner core just begins to form, (p0, ζ0),
‚TL ‚TL
TL(p, ζ) = TL(p0, ζ0) + (p ’ p0) + (ζ ’ ζ0). (15.4.19)
‚p ‚ζ
The material properties ρ, KS, γth will be controlled by the equation of state and
will vary in time. The full set of equations describing the evolution of the core will
therefore require numerical solution. However, as demonstrated by Buffet et al.
(1996), the main properties can be found through analytical approximations based
on power series that yield results within 8% of those from more detailed numerical
models including the effect of self-gravitation.
The density and thermodynamic parameters are approximated by their values ρ0,
K0, γ0 at the centre of the Earth so that there are simple solutions for p and T as a
function of radius:
p(r) = p0 + Ar2, with A = 2πGρ2/3, (15.4.20)
0

and
2 ’r2 )/r2
T (r) = TL(ri )e’φ(r with φ = Ar2γ0/K0.
, (15.4.21)
c
i
c

The liquidus temperature TL(ri ) as a function of inner core radius ri takes the form
r3
Ar2 ‚T ‚TL 2 ‚TL
c i
TL(ri ) = ’ Ari ’ ζ0, (15.4.22)
3 ’ r3
φ ‚p ‚p ‚ζ rc
0 i
where (‚T /‚p)0 is the adiabatic gradient evaluated at p = p0.
15.4 Evolution of the Earth™s core 403

Because we have made the assumption of constant density ρ0 the pressure does
not change with time, ‚p/‚t = 0, and the energy balance does not depend on
pressure. This is an oversimpli¬cation, but the associated error is small. This means
that once the total heat ¬‚ux Q(t) is prescribed, the energy balance only depends on
the rates of change of temperature ‚T/‚t and the rate of change of the inner-core
radius dri /dt. These quantities are related by
‚T (r, t) d 2 2 dri
22
TL(ri )eφ(ri /rc ) e’φ(r /rc )
= , (15.4.23)
‚t dri dt
where TL(ri ) is speci¬ed by (15.4.22). Because the temperature variation across
the core is not large, φ = 0.26, and we can make the approximation
2 ’r2 )/r2
e’φ(r = 1 ’ φ(r2 ’ r2)/r2 + O(φ2). (15.4.24)
c
i
i c

We can now evaluate the volume integral over ‚T/‚t that is required in (15.4.16)
as
‚T dTL(ri ) ri φ dri
ρ0Cp dV = H + 2 2 TL(ri ) , (15.4.25)
‚t rc
dri dt
Vc

where the scaling factor
3 r2
2π 3
’i
H= r ρ0Cp 1 ’ φ . (15.4.26)
3c 5 r2
c
The form (15.4.25) implicitly assumes that the inner core is adiabatic, which will
be a reasonable form until the core is almost totally solid.
The gravitational potential for the pressure distribution (15.4.20) is • =
Ar2/ρ0 + const and so the average potential in the outer core, which is needed
in (15.4.13) , can be found from
3A r5 ’ r5c
i
•= + const;
¯ (15.4.27)
3 ’ r3
5ρ0 ri c
the constant does not present any problem since we need differences in potential.
With the aid of the representation (15.4.25) we can express the heat balance
equation (15.4.16) in terms of the radius of the inner core ri (t) by eliminating the
dependence on temperature. In the level of approximation to which we are working
we can neglect the small effect of thermal contraction at the top of the core, and
introduce the fractional inner core radius
ri (t)
ξi = . (15.4.28)
rc
The heat ¬‚ux equation (15.4.16) can then be rewritten as an ordinary differential
equation for ξi of the form f(ξi)dξi/dt = Q(t), with the heat ¬‚ux Q(t) as a
forcing term. The terms appearing in f(ξi) are rational functions of ξi and so an
analytical integral for ξi(t) can be found once Q(t) is speci¬ed.
For the current inner core ξi = 0.349 and so ξ4 = 0.015, we can therefore make a
i
404 The Core and the Earth™s Dynamo

further approximation of neglecting terms higher than cubic in ξi in the differential
equation for the inner core radius, and then the solution can be expressed as
t
’1
Q(„)d„ = ξ2 + Gζ + L ’ Z ξ3 + O(ξ4).
M (15.4.29)
i i i
0
The time t is thus measured from the instant at which the temperature falls below
the liquidus and the inner core begins to grow.
The factor M in (15.4.29) serves to make the heat ¬‚ux dimensionless,
1φ ‚TL ‚T
ρ0CpAr5
M = 4π ’ ’ , (15.4.30)
c
3 5 ‚p ‚p 0
which is the heat that is required to cool the entire core to its solidi¬cation
temperature.
The set of parameters on the right hand side of (15.4.29) describe the relative
proportions of the various physical processes in the evolution of the core:
’1
ζ0 ‚TL ‚TL ‚T
Z= ’ ,
Ar2 ‚ζ ‚p ‚p 0
c
4π Ar5 (”ρ)ζ
c
Gζ = , (15.4.31)
5 M ρ0
4π ρLr3 c
L= .
3M
The quantity Z represents the effect of composition on the liquidus temperature,
Gζ the gravitational energy release and ohmic dissipation due to compositional
segregation and L the effect of latent heat release.
Many of the quantities needed for evaluating (15.4.29) are not well known, but
the greatest uncertainty is in the difference in temperature gradients appearing in
M, Z. This difference can be cast into a slightly more helpful form by using
Lindemann™s relation to describe the melting curve
‚TL TL
= 2(γth ’ 1 ) , (15.4.32)
3K
‚p T
where KT is the isothermal bulk modulus (see, e.g., Stacey, 1992). This relation
is based on the concept that melting will occur when the amplitude of atomic
vibrations about their equilibrium position exceeds a certain fraction of the
interatomic distance in the basic lattice.
The adiabatic temperature gradient takes the form γth T/KS in terms of the
isentropic modulus KS. The difference in the gradient at the centre of the Earth
is thus approximately
‚TL ‚T TL(0) KT 2
’ = 2’ γth ’ ; (15.4.33)
‚p ‚p KT KS 3
0
estimates of the central temperature show considerable variation in the range
4000“6000 K that will affect the size of the gradient difference.
15.4 Evolution of the Earth™s core 405


2000
Inner Core Radius [km]

1500



1000
2.5 TW


500
4.0 TW
6.0 TW

0
0 2 4 6
Time [Ga]

Figure 15.7. Evolution of the radius of the inner core to the present-day values and into
the future for three estimates of the net ¬‚ux across the core-mantle boundary (in TW =
1012 W) using the analytic approximation (15.4.34). The indicators mark the current Earth
conditions.

The analytic equation for the growth of the inner core (15.4.29) can be used to
develop an explicit expression for ξi to the same level of approximation as before:
ξi(t) = ξ0(t) ’ δξi(t). (15.4.34)
The leading order approximation is controlled just by the heat loss into the mantle
1/2
t
ξ0(t) = Q(„)d„/M , (15.4.35)
0
whereas the correction term
(Gζ + L ’ Z)ξ2
+ O(ξ4)
0
δξi(t) = (15.4.36)
0
2 + 3(Gζ + L ’ Z)ξ0
includes the in¬‚uence of latent heat and gravitational energy associated with
compositional change.
Figure 15.7 displays the results from (15.4.34) using the numerical values for
the parameters suggested by Buffet et al. (1996), for different assumptions about
the net heat ¬‚ux into the mantle. The growth curves for the inner core have been
adjusted so that they pass through the current Earth conditions. A low mantle heat
¬‚ux means that an inner core needs to form early in Earth history, whereas more
rapid cooling allows the process to be delayed. The solid curve for an average
heat ¬‚ux of 4.0 TW is representative of current estimates. A feature of all the
solutions is the relatively rapid initial growth of the inner core; once solidi¬cation
starts an inner core several hundred kilometres in radius is formed within 200 Ma.
406 The Core and the Earth™s Dynamo

This change can be expected to have a signi¬cant in¬‚uence on the evolution of the
magnetic ¬eld as the character of the dynamo changes.
The analytic solution has been obtained with the aid of a number of
simpli¬cations including neglect of self-gravitation and the variations of the density
with radius. Despite the limitations the results agree with more detailed numerical
solutions within 8%, which is less than the error in the density approximation! The
numerical solutions give slightly more rapid growth for the same average heat ¬‚ux.
From the analysis above, we see that the relative importance of thermal and
compositional convection in the outer core is primarily controlled by the heat ¬‚ux
from the core and the radius of the inner core. For current estimates of heat ¬‚uxes of
the order of 3“4 TW and the present inner core radius, about two thirds of the heat
available for ohmic dissipation can come from compositional convection. This heat
¬‚ux is close to the adiabatic value, and so thermal convection would be expected to
be of lesser importance than compositional convection. The thermal effects would
largely be associated with latent heat release. When the inner core was smaller, and
the heat ¬‚ux to the mantle most likely larger, thermal convection would have been
the dominant energy source for the geodynamo.
Appendix: Table of Notation




Stress and Strain
x - position vector
xi - position coordinates
ξ - initial position vector
ξi - initial position coordinates
u - displacement vector
v - velocity vector
f - acceleration vector
g - external force vector
ν - local body moment
n - normal vector, orthogonal vector
F - deformation gradient
J = det F - Jacobian, determinant of deformation gradient
A - displacement gradient
Σ
dS, dΣ - surface elements
dV, dm - volume, mass elements
σij - stress tensor
σr - principal stresses
σPK, σSK - ¬rst and second Piola“Kirchhoff tensors
σ, t - traction vectors
„ - traction vector at a surface
»r - stretches
U, V - left and right stretch tensors
R, Q, P - rotation matrices
N ≡ n— - skew matrix
E, e - Green strain tensor, Cauchy strain tensor
D/Dt - material derivative
p - pressure
Πij - momentum ¬‚ux density tensor



407
408 Appendix: Table of Notation

Solids and Fluids
- strain
eij - strain tensor
Dij - rate of deformation tensor
^
Dij - deviatoric strain rate tensor
- spin tensor, vorticity tensor
ij
cijkl - elastic modulus tensor
Cijkl - anelastic relaxation tensor
κ, K - bulk modulus
μ, G - shear modulus
ρ - density
¦ - seismic parameter
» - Lam´ modulus
e
E - Young™s modulus
… - Poisson™s ratio
• - gravitational potential
p, p , pf - pressure, incremental pressure, ¬‚uid pressure
g - acceleration due to gravity
· - shear viscosity
ζ - bulk viscosity
Gk - elastic Green™s tensor
i
φ(t) - relaxation function
ψ(t) - creep function, stream function
R», Rμ - isotropic relaxation functions
„R, „C - relaxation times
© - angular velocity vector
© - angular velocity

Waves
ω - angular frequency
θ, φ - coordinate angles
± - P wavespeed
β - S wavespeed
a - P slowness (1/±)
b - S slowness (1/β)
φ, c0 - wavespeed of sound waves in a ¬‚uid
p - horizontal slowness, phase slowness
n - direction of travel
s - ray direction
p - slowness vector
Q’1 - loss factor
Appendix: Table of Notation 409

Thermodynamic Quantities
h - heat production
q - heat ¬‚ux vector
T - temperature
k - thermal conductivity
κH - thermal diffusivity
U - internal energy density
H - thermal energy, enthalpy
Y - work rate
W - work density, strain energy density
Q - thermal contribution to internal energy
U - total energy
F, F - Helmholtz free energy
G, G - Gibbs free energy
S, S - entropy
V - volume
R - gas constant
±th - volume thermal expansion coef¬cient
γth - thermodynamic Gr¨ neisen parameter
u
KT , KS - isothermal, adiabatic bulk modulus
CP, CV - speci¬c heat at constant pressure, volume
pH, eH - pressure and energy on Hugoniot
L - latent heat
γc - Clapeyron slope

Non-dimensional Quantities for Fluids
Ek - Ekman number
Nu - Nusselt number
Pe - P´ clet number
e
Pr - Prandtl number
Ra - Rayleigh number
Ra - modi¬ed Rayleigh number for rotating system
Re - Reynolds number
q - Roberts number
Ro - Rossby number
RoM - magnetic Rossby number
θ - non-dimensional temperature
410 Appendix: Table of Notation

Electromagnetic
E - electric vector
B - magnetic induction
D - electric displacement
H - magnetic vector
j - electric current density
ρ
˜ - electric charge density
^
j - surface current density
ρ
^ - surface charge density
σ
˜ - conductivity
μ0, 0 - magnetic permeability, permittivity of free space
μ, - magnetic permeability, permittivity
c - speed of light
γi,jk - piezoelectric tensor
» magnetic diffusivity = 1/μ0σ ˜
-
W - stored energy in magnetic ¬eld
Q - dissipation due to Joule heating
S - Poynting vector
Q - energy ¬‚ux vector
d - penetration depth
n - complex refractive index
φ - phase difference

Quantum/Atomic
ψ - electron state
H - Hamiltonian
V - potential
T - kinetic energy
D - dynamical matrix
G - force tensor
μ - chemical potential
ρij - interatomic interactions
ωq - frequency of quantum state
¯
E - energy in lattice modes
H— , E— , V — - activation quantities
c, J - species concentration, ¬‚ux
D - diffusivity
b - Burgers vector
˜D - Debye temperature
kB, R - Boltzmann constant, gas constant
Appendix: Table of Notation 411

Deformation
d - grain size
TM - melting temperature
n, m - grain size exponent, stress exponent in creep dependence
σN - normal stress
σS - shear stress
”σ - deviatoric stress
pf - pore ¬‚uid pressure
»f - ratio between ¬‚uid and lithostatic pressure
σ1, σ2, σ3 - prevailing principal stresses
δs - slab dip angle
Vs - slab convergence rate
ts - spreading time
R - thermal Reynolds number
Q0 - surface heat ¬‚ow
TM - mantle temperature
”g - gravity anomaly
Z - gravitational admittance
D - ¬‚exural rigidity of a plate
Te - effective elastic thickness
h, k, l - Love numbers
ζ - sea level function
b - bouyancy ¬‚ux

Seismology
v - wavespeed
mij - moment tensor density
Mij - moment tensor
Mij - moment rate tensor
r - radial variable
H - modal operator
ue - modal eigendisplacement
I
cI - modal coef¬cients
W; T - displacement; traction coef¬cients for toroidal modes
U, V; P, S - displacement; traction coef¬cients for spheroidal modes
¦, Ψ - gravitational coef¬cients for spheroidal modes
b - radial column vector for spheroidal modes
B - boundary term for modal estimate
T - kinetic energy for modal estimate
T - kinetic energy per unit volume
V - potential energy per unit volume
G - gravitational constant
412 Appendix: Table of Notation

Core dynamics
P - augmented pressure
ζ - mass fraction of light constituent in core
i - mass ¬‚ux of light constituent
˜ - temperature deviation
θ - non-dimensional temperature perturbation
ri, rc - radii of inner and outer cores
ξi - fractional inner core radius

Mathematical
a, b, c, d - constants
(1),(2)
jl(x), hl (x) - spherical Bessel functions
zl(x) - combination of spherical Bessel functions
m
Pl(x), Pl (x) - Legendre functions
(1),(2)
Ql - travelling wave form of Legendre function
Pm, Bm, Cm - vector surface harmonics (spherical)
l l l
m
Yl - surface harmonics on a sphere
l - angular order of spherical harmonics
[l(l + 1)]1/2
Ł -
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