(i) Structure and composition of Earth’s core
Lecture 3:
Thermal structure of the core and power
sources for the geodynamo
Lecture 3: Thermal structure of the core
and power sources for the geodynamo
3.1 Earth as a heat engine and geotherms
3.2 The Adiabatic gradient
3.3 Phase changes and the Clapeyron slope
3.4 Growth of the inner core
3.5 Power sources for the geodynamo
3.6 Age of the inner core: a paradox?
3.7 Summary
3.1.1 Earth as a heat engine
• Earth is a hot body
trying to cool down.
• This is the ultimate
origin of:
- Major phase changes
- Mantle convection
& plate tectonics
- Motions of the core
& the geomagnetic field
• So, worth trying to understand
Earth’s thermal structure & evolution....
3.1.2 Geothermal structure of Earth
• Geotherm: Change in temperature with depth in the Earth.
• Difficult to measure in Earth’s deep interior: assumptions
must be made concerning composition and mineralogy.
new estimates
(From Fowler, 2005)
3.2 The Adiabatic gradient
• In vigorously convecting regions of the Earth, where
material is well mixed, entropy S is constant so it is useful
to consider how T changes with P at constant S.
• Using the reciprocity theorem,
!
∂T
∂P
"
=−
S
!
∂T
∂S
" !
P
∂S
∂P
"
(1)
T
• But Maxwell’s relation from the 2nd derivative of G along
with the definition of thermal expansivity means that,
!
"
!
"
∂S
∂V
_= V α (2)
=−
=
∂P T
∂T P
• While the definition of specific heat per unit mass (cP) is,
T
cp =
m
!
∂S
∂T
"
(3)
P
3.2 The Adiabatic gradient
• Substituting from (2) and (3) into (1) and using ρ = m/V
gives,
"
!
∂T
Tα
=
∂P S
ρcp
• Then, assuming a hydrostatic pressure balance,
∂P
= −gρ
∂r
• Finally, using
T with r:
!
∂T
∂r
"
!
S
!
∂T
∂r
"
=
∂T
∂P
S
"
S
∂P
we get the variation of
∂r
=−
T αg
cp
Adiabatic
Gradient
3.3 Phase change: The Clapeyron slope
• Recall that the Gibbs potential is unchanged during a
phase transition i.e Ga1 = Ga2 and Gb1 = Gb2
( From
Adkins, 1983)
3.3 Phase change: The Clapeyron slope
• Considering the formula for change in the Gibbs Free Energy
dG1 =
dG2 =
!
!
∂G1
∂P
"
T
!
∂G1
∂T
"
∂G2
∂P
"
T
!
∂G2
∂T
"
dP +
dP +
dT = V1 dP − S1 dT
P
dT = V2 dP − S2 dT
P
• Then using the lack of change in the Gibbs potential,
V1 dP − S1 dT = V2 dP − S2 dT
• Collecting terms,
L= Latent Heat
∆S
L
dP
=
=
dT
∆V
T ∆V
Clausius-Clapeyron Eqn
• For substances that expand on phase change (e.g.
melting), the Clapeyron slope (dP/dT) is positive.
3.4 Growth of the inner core
• Geotherm and melting T
in the deep Earth:
(a) For inner and outer
core same composition
(b) For inner core with less
light element.
• In both cases the inner
core is solid while the
outer core is liquid.
From Fowler,
2005, (a) After
Jacobs, 1953
and (b) after
Stacey 1972.
• As Earth cools
geotherm shifts to
lower T => inner core
solidifies
3.4 Growth of the inner core
Temperature
T m(r = 0)
∆T c
Tc
T a(r )
T cen
Ti
Tm ðPÞ ¼
dT a
dP
δT c
adopt a slightly di
Stevenson et al. (
melting temperatu
Center
ICB
CMB
where Tm1 and T
rates the reduction
From
Nimmo,
light
element(s).
2007, Fig 2.
Tm2 ¼ 0.
T m(r )
dT m
With the varia
dP
δr i
described by eqns
analytical express
Pressure
terms are given be
• Ta (r)
is adiabatic
temperature
profile,
Tm
(r) is melting curve
Figure
2 Schematic
of melting and
adiabatic
temperature
profiles. The IC (temperature T ) is defined by the
i
• Small
cooling
in
CMB
T
by
leadscurve
to increase
IC
c
intersection of the adiabat Ta and!T
thecmelting
Tm. The in8.02.3.4.1
Spe
radius
as defined
by crossing
of melting
curvebyand adiabat.
corresponding
temperature
at the CMB,
Tc, is obtained
Table 1 and eqn [
following the adiabat. Changing this CMB temperature by a
Qs and Es invo
small amount $Tc results in a change in IC radius, $ri, which
3.5.1 Power sources for the geodynamo
• Geodynamo: Mechanism by which core motions
generate the geomagnetic field.
Energy must be supplied to drive the core motions that produce the geomagnetic
field. Possible sources can be broadly categorized as:
(1) Thermal energy of initially hot planet as a consequence of its origin in
accretion (also known as the secular cooling `source’).
(2) Gravitational (buoyancy) energy from release of light elements during
freezing of liquid outer core into solid inner core.
(3) Latent heat released in freezing of liquid outer core into solid inner core.
(4) Energy released by decay of adioactive elements (eg Potassium-40) may
contribute to the core energy budget.
(5) Tidal/Precessional forces Gravitational influence of the moon and sun acting
on a non-spherical core will drive (tidal) flows, and cause the slow precession
of Earth’s rotation axis producing addition (precessional) flows. (WEAK?)
3.5.2 Energy budget in Earth’s core
• From conservation of energy on long time-scales &
assuming changes in KE and ME is negligible:
Change of
Internal Energy
d
dt
!
Internal (Radioactive)
heating
Flux of heat
ρedV = −
"
P u · dS −
"
q · dS +
!
ρhdV +
Work done by
surface pressures
!
ρu · ∇ψdV
Work done against
gravitational forces
• In Earth’s core where the inner core is growing and
solidifying, this balance can be rearranged & simplified
(see Nimmo (2007) for details) as:
Qcmb = QS + QL + QR + QG
Heat flux
through CMB
Core secular Latent heat
cooling
release
Radioactive
heating
Gravitational
energy release
may be thought of as a heat flow, multiplied by some
ows:
for the
entropy
terms is by
(Hewitt
al., 1975; Gubbins
efficiency factor
(<1)
and divided
someetcharacter2004)
istic operatinget al.,
temperature.
Higher heat flows result
"h dV
3.5.3
Entropy
budget
in Earth’s core
2.2–
in higher
rates
of entropy
Ds production.
r ? q $r ? i "h !
"
¼ –
þ
þ þ
½32#
Zore,
The
corresponding
eqnfields
T
T to T
T[12] in
nogeneral
term forequation
cost ofDtmaintaining
magnetic
DP • But
#T
equation.
theenergy/heat
entropy
terms
(Hewitt
al., 1975;
Dt forthis
where
the isheat
flux q etdepends
on Gubbins
the solute flux i
we must
consider
equation
al., 2004)
(eqn [9])
and the
entropyofs ‘entropy
depends balance’:
on P, T, and c
Dcet• Instead
" #c
dV
(eqn [3]). This equation summarizes the changes in
Ds
r ? q $r ? i "h
!
" entropy
¼ – arisingþfrom both
þ thermal
þ and compositional
½32#
due
to Ohmic
Change in
Dt
T ! is T
T T viscous and
effects. Here,
the combined
Ohmic
heating
entropy
due to internal
dissipation.
The former is assumed
to be negligible
due to heat flux due to flux of
production
(radioactive)
where ½31#
the heat
q depends
on the
soluteheating
i by
solute
and flux
the volumetric
Ohmic
dissipation
isflux
given
H þ Qg þ QL
Dt
(eqn [9]) and the entropy s depends on P, T, and c
! 2 "
2
hat it makes each of
J
B changes in
(eqn
[3]).
Thisdissipation
equationissummarizes
the
where
Ohmic
%
!
¼
½33#
ard the core energy
2 %l 2
%
$
0
entropy
nt of energy
beingarising from both thermal and compositional
•effects.
Entropy
balance
the influence
ofBmagnetic
Here,
! isexplicitly
the
and Ohmic
where
J is combined
the includes
electric viscous
current
density,
and l are
on radioactive
heat
fields on heat
flow,
this
isiswhy
it must
be
when
former
to considered
be
negligible
typical
values
forassumed
the
magnetic
field
and the
lengthar coolingdissipation.
and
con- The
studying the power used to generate the geodynamo.
scale atOhmic
which dissipation
dissipation occurs
l, gravitational,
andvolumetric
½31#
and the
is given(e.g.,
by Labrosse,
2003), respectively, and % is the electrical conductivws. With the excep! 2 "
2
hseofterms turn out to
ity. It is because
of Bthe appearance of this term that
J
! ¼ %balance
the entropy
mayEarth’s
be used core
to ½33#
determine
rgy
ate of core cooling
3.5.3 Entropy
%budget
$20 %l 2in
whether or not a dynamo can operate.
This equation also
eing
Making
use
of eqnsdensity,
[9] can
and B
[10]
andl employing
diabatic where
flow,
•heat
By expand
terms
and rearranging
this
beand
written
in the
J is the
electric
current
are
heat
divergence theorem, we have
y role in typical
the
global
form
(Nimmo,
2007).
valuesthefor
the magnetic field and the lengthonZ
Z
Z
however, do figure
#q$
r?q
rT
scale
at
which
dissipation
occurs
(e.g.,
Labrosse,
and
dV By
þ compositional
q ? 2 dV
dV ¼ Byrinternal
?
balance, which is
T and % is the
T
T
By Latent
2003),
conductivepheating Z electrical
(gravitational)
work
By core respectively,
!
"2
heat
cooling
Q
rT
cmb
ity. It is because of the
t to
–
k of this
dV term that
¼ appearance
+ EL +may
ETZRc +beEG
=TE + EΦ
the entropyESbalance
ling
!used kto" determine
1
$#D
whether or not a dynamo
– i2 Þ dV
1
þ
ð – #D r$By? iOhmic
þ can operate.
also
By
thermal
&T
#D T
Heating
eating
(dissipation)
diffusion
Making
use
of
eqns
[9]
and
[10]
and
employing
ow,
½34#
ot enter the
the divergence
global
theorem,
we
have
obal
• Note that here E refers to rate of entropy production and
Z
Z
Z notation
ure
# q $ has been
notr?q
energy! This
used
rT to allow easy
dVofþNimmo,
q ? 2007.
dV
dV ¼withr
? paper
is
comparison
the
2
T
T
T
"
Z !
Qcmb
rT 2
–
k
dV
¼
Tc
T
!
"
Z
must have formed more recently.
For a present-day estimated CMB heat flow of
6–14 TW, the net entropy production rate available
to drive the dynamo is 300–1000 MW K!1, sufficient
to generate roughly 1.5–5 TW of Ohmic dissipation.
is small. Again, however, the existence of radioactive
potassium in the core has a significant effect on the
cooling history of the core, and the growth of the IC.
The role of potassium in controlling the age of the IC
is discussed further in Section 8.02.5.3.
3.5.5 Present-day Energy and Entropy Budget
Table 4
Individual contributions to energy/entropy budgets for a present-day core with parameters given in Table 2 and a
CMB heat flow of 9 TW
K¼0
K ¼ 300 ppm
Q
Qs, Es
QL, EL
Qg, Eg
QR, ER
Qk, Ek
QH, EH
Qcmb, !E
dTc/dt (K Gy!1)
dri/dt (km Gy!1)
IC age (My)
56
E
Q
E
(TW)
(%)
(MW K!1)
(%)
(TW)
(%)
(MW K!1)
(%)
2.2
4.2
2.5
0
4.9
0
9.0
25
47
28
0
73
268
618
0
!162
!219
537
8
28
64
0
1.7
3.3
1.9
2.1
4.9
0
9.0
19
37
21
24
56
205
474
65
!162
!168
431
7
26
59
8
!37
788
590
!30
605
780
Q and E refer to the energy and entropy contributions, respectively; !E is the entropy available to drive the dynamo, dTc/dt is the core
cooling rate and dri/dt the IC growth rate. Two cases are shown, one with no potassium (K) and one with 300 ppm potassium in the core. IC
age is calculatedof
assuming
a constant CMB heat flow.
Energetics
the Core
From Nimmo,
2007, Table 4.
(a)
Temperature (K)
• Compositional
driving is major contribution to entropy
6000
Temperatures
needed to
drive
the dynamo,
Core in the present day core.
5000
4000
3000
2000
Mantle
Heat flow (TW)
1000
3.5.7
Mantle control of geodynamo
(b)
entropy budget
100
80
Heat flow
Surface
60
Mantle he
at
40
20
production
Core
0
(c)
Entropy production
(MW K–1)
1000
Entropy production
800
600
400
From Nimmo,
2007, Figure 4.
Entropy
200
1000
2000
Time (My)
3000
4000
• Fluctuations in form of mantle convection (e.g. intermittent
instabilities of lower thermal BL) lead to variations in CMB
heat flux, and hence available entropy.
Figure 4 Example of 2-D numerical mantle thermal evolution calculations, based on the method described in Xie and Tackley
(2004). (a) Evolution of mantle potential temperature and core temperature at the CMB as a function of time. (b) Evolution of
surface heat flow, CMB heat flow, and radiogenic mantle heat production. (c) Rate of entropy production, calculated using the
methods of Section 8.02.3.4 and the parameters given in Section 8.02.4. Model output courtesy of Paul Tackley.
through time (e.g., Buffett, 2002; Labrosse, 2003).
Since the rate of entropy production depends on
rate of entropy production should therefore be appropriate to the situation just prior to the onset of IC
Temp
20
CMB heat flux
3000
2000
(a)
(b)
10 Core 57
Energetics of the
Radiogenic heat flux
3.5.8 Change in CMB heat flux
Entropy production rate
Temperature
(MW K–1) (K)
(b)
ppm
300
ppmpotassium
potassium
300
potassium
0 ppm
potassium
0 ppm
9TW CMB heat flux
Core temp
era
ture
800
4000
innerCMB Dimensionless
heat flux
core radius
40
0.8
2000
Time (My)
20
0.4
Radiogenic heat flux
Net entropy production rate
1000
50
1.0
30
0.6
3000
400
2000
0
0
3000
4000
10
0.2
0
0
Dimensionless
IC radius
Heat flux (TW)
7000
1600
6000
1200
5000
(TW)
4000
From Nimmo,
2007, Figures 5,6.
Entropy production rate
(MW K–1)
Dimensionless IC radius
1600
1.0 MW K!1,
Figure 6 Same
as for Figure 5, except that rate of entropy production prior to IC solidification is fixed at 135
300 ppm potassium
and CMB heat flow following
onset
of
solidification
is
fixed
at
9
TW.
20 TW CMB heat flux
0 ppm potassium
0.8
1200
drive the dynamo has a strong influence on the evodecreased, the numerical model shown0.6in Figure 4
800 temperature. Figure 6 shows an
lution of the core
demonstrates that the decrease may actually be rather
0.4
Net
entropy
production
rate
identical set of results to Figure 5, except assuming a
small and thus a scenario invoking a constant
CMB
400
constant (pre-IC)
entropy production rate of
heat flow mayDimensionless
not be unreasonable.
Second,
the
pre0.2
inner135 MW K!1 ("0.7 TW dissipation), appropriate to
sence of radioactive potassium
has more of an effect on
core radius
0
a more reasonable
of 9 TW. As one
when the CMB heat
0 CMB heat flow
1000
2000 the IC age3000
4000flow, and total entropy
would expect, the lower heat flow results in a slowerTime production
rate, are smaller.
(My)
core cooling rate and an older IC (0.61–0.84 Gy,
Figure 7 summarizes these results. Figure 7(a)
Figure 5 (a) Evolution of core temperature Tc (left-hand scale) and heat flow Qcmb (right-hand scale) through time, assuming
the initial
coreICtemperature
a function
ofkept
core
depending
onof the
potassium
concentration).
Theto IC plots
solidification.
During
solidification,as
the
heat flow is
a constant rate
entropy
production
of 500 MW K!1 prior
potassium
and
entropy
production
rate.
It
demonresulting
changes
in
core
temperatures
are
also
smalconstant at 20 TW. Results were obtained by integrating backward in time using the present-day conditions and parameters
specified
in Table 2. Given
fixed heat
flow or rate
of entropy production,
and a radioactive
heat higher
production
rate,oftheentropy
change
strates that,
as expected,
rates
ler.
In particular,
if 300a ppm
potassium
is present,
in
core
temperature
with
time
may
be
calculated.
Thick
lines
denote
case
with
no
core
potassium;
thin
lines
denote
case
with
production require more core cooling and thus
then the core initially heats up, because the radio300 ppm potassium. In the latter case, the radioactive heat production is given by the dotted line. (b) Evolution of rate of
higher initial temperatures. Adding potassium to the
active
decay alone is sufficient to account for the
entropy production and growth of IC. Thick lines denote potassium-free case; thin lines show 300 ppm potassium in the core.
core counteracts this effect. These effects are essenrequired entropy production. A corollary is that the
tially identical to those found by Buffett (2002) and
IC may have been present, disappeared as the core
8.02.4.3 up,
andand
integrates
Tc backward
in time
that
are
high, but
the Unfortunately,
presence of potassium
reduces
the
Labrosse
(2003).
as discussed
below,
heated
then began
to resolidify
(cf.such
Buffett,
the entropy
rate prioris tounlikely
IC formation
initial
temperature,
as expected
[77], from
the initial
temperature
of the from
core eqn
is sufficiently
2002).
Such aproduction
scenario, however,
simply
It have
is assumed
the
stays constant
at 500
MW K!1.to
5863
to 5140
The
magnitude
of an
this
effect
is
uncertain
that K.
only
results
indicating
initial
tembecause
the core
is expected
been that
initially
hot
the subsequent
gravitationaltoenergy
releasedremains
during
peraturedue
cooler
the present
can be
coredue
heattoflow
IC formation
mainly
to than
the that
shortof half-life
of day
potassium,
accretion
the Earth
(see below).
excluded
with any15confidence.
constant, of
which
is reasonable
given the young IC
which
produces
times more heat at 3.5 Gy BP
There
are two other interesting consequences of the
The
initial
core day.
temperature depends on both the
ages
obtained.
than
at the
present
lower
entropy
production
rate
assumed.
First,
the
manner
in
which
the
accreted,ofand
Figure 5(a) shows the evolution of Tc and CMB
Figure 5(b) showsEarth
the evolution
thethe
ICprocess
radius
CMB
heatintegrated
flow stays
almost from
constant
over 4 Gy.
of core
differentiation
(Stevenson,
1989).
heat flow,
backward
the present.
An
and
entropy
production
with time.
TheThe
highenergy
core
associated
the that
differentiation
be
While
(e.g.,toNimmo
et al., 2004)
entropyparametrized
production scalings
rate prior
IC formation
of
cooling
ratewith
means
the IC is aprocess
young may
feature
suggest
CMB heat
flow heat
should
have
calculated
a fairly
straightforward
by
requires
a CMB
flow
of declined
20 TW,
500 MWthat
K!1the
(0.6
Gy old).in As
expected,
the entropy manner,
production
over
history
the mantle
comparing
the(by
gravitational
potential
of a
and a Earth
present-day
coreascooling
rate of 84temperature
K Gy!1 in
rate
increases
a factor of #3)
when energies
IC formation
the absence of radioactive heating. This CMB heat flow
begins. While one might expect such an increase to
is a significant fraction of the total global surface heat
have significant effects on the observed magnetic
flow of 42 TW (Sclater et al., 1980), and exceeds the
field, in practice the observations are sufficiently
likely value deduced from present-day observations
sparse that no such effects have been detected
(Section 8.02.4.3.7). These high heat fluxes suggest
(Section 8.02.2.4). Because of the high CMB heat
that a dynamo requiring an entropy production rate
flow, the presence of potassium has little effect on
!1
of 500 MW K ("2.5 TW dissipation) is unrealistic.
the age of the IC, though it does affect the early core
Figure 5 shows two scenarios: one with no radiotemperature as noted above.
active heating, the other with 300 ppm potassium in
As noted by Buffett (2002) and Labrosse (2003),
the core. In either case, the initial core temperatures
and shown in eqn [76], the dissipation required to
• Appearance of inner core causes increase in entropy
production making it easier to drive a dynamo.
• Increase in CMB heat flux makes inner core grow faster.
3.5.9 Influences on core thermal evolution
• CMB heat flux: varying with time & driven by mantle, but
depends on poorly understood changes in plate-tectonics.
• Presence of growing inner core, with light element
expelled on freezing efficiently supplying latent heat
and gravitational energy.
• Rate of core cooling: depends on presence of radioactive
elements e.g. Potassium (K) in the core.
• Enough entropy must be produced to balance Ohmic
heating due to the operation of the geodynamo.
3.6 Age of the inner core: a paradox?
• Old inner core:- Inner core is important power source
-Observation of ~3.5 Gyr geomagnetic field
-Hard to power dynamo before IC formation
=> inner core is older than 3.5Gyr.
• Young inner core: - Large present CMB heat flux (~9TW)
=> rapid core cooling and IC ~1Gyr old
-Even with radioactive potassium (K) in
the core, hard to make inner core older.
• Possible resolutions
(a) Calculations correct and IC is young.
(b) Calculations wrong because
- Entropy requirements for dynamo actually smaller.
and/or present CMB heat flux estimates are wrong.
- Properties (e.g. thermal conductivity) for core are incorrect.
3.7 Summary: self-assessment questions
(1) What is the adiabatic gradient: can you derive its form?
(2) What is the Clapeyron slope?
(3) What are the power sources that may drive the
geodynamo?
(4) Why is it important to consider entropy when modelling
the thermal evolution of Earth’s core?
(5) What are the arguments suggesting (i) An old (>3.5Gyr),
inner core and (ii) A young (<1 Gyr) inner core?
Next time: Earth’s magnetic Field: Sources and measurements
Homework for Lecture 3
• Exercise 3.1: The Adiabatic gradient in Earth’s core.
• Exercise 3.2: Inner core growth and implications for
Earth’s moment of inertia.
References
- Adkins, C. J., (1983) Equilibrium thermodynamics, Cambridge University Press.
- Jacobs, J.A., (1992) Deep interior of the Earth, Chapman and Hall.
- Fowler, C.M.R. (2005) The Solid Earth, 2nd Edition, Cambridge University Press.
- Nimmo, F., (2007) Energetics of the core. In Treatise on Geophysics, Vol 8 Ed.
P. Olson, Chapter 8.02, pp.31-65.