Definitions
Measuring heat flow
Kelvin and the age of the Earth
Radioactivity
y
Continental heat flow (1)
Oceanic heat flow (1)
Global budget (1)
Continental vs oceanic heat flow
Plate model for the oceans
Continental heat flow (2)
Gl b l budget
Global
b d t (2)
Constraints on the temperature regime
Objectives
y How do we determine temperature deep in the Earth?
y What is the Earth’s energy budget?
y What is the source of energy for geological processes?
y How does temperature controls physical properties and
tectonic
i processes??
Thermodynamics
y 1st law: heat is a form of energy dU = dQ -PdV
y ΔQ = C ρ ΔT
y 2nd law (simple form): heat goes from hot to cold (Entropy
can only increase unless work is done. Work can be
extracted from system only if there are a cold and a hot
source).
y Conduction of heat (Fourier): q = -λ
λ grad T
y Other mechanisms of heat transport (radiation,
convection).
Three mechanisms of heat transport
y Conduction. Transport of energy in a medium (solid, fluid,
or gas) without transport of matter.
y Convection. Energy is transported by movement of matter.
y Radiation. Electromagnetic waves transport energy in
vacuum or in solid or fluid at very high temperature.
vacuum,
temperature
y All three mechanisms are found in the Earth. Near the
surface, i.e. in the lithosphere conduction dominates.
How do we measure heat flux?
Some numbers
y k (thermal conductivity) 3 W/m/K (average for most
rocks)
y Temperature gradient 20-30 K/km
y Heat flux ~ 60 mW m-2
y If gradient
di does
d
not decrease
d
with
i h depth,
d h temperature at
100km >2000K and temperature at CMB > 60,000K
More numbers
y Mean heat flux 80 mW m-2
y Total energy loss 44 1012 W 44 TW or 1.3 1021J/yr
y Total energy in quakes 1019 J/yr
y Total tidal friction < 1017 J/yr
y Energy from Sun 1.8 1017 W or 5 1024 J/yr
y 2005 World energy consumption 15 TW or 5 1020 J/yr
y Thermal diffusivity κ = λ / ρ C
y It gives scaling between time and length for heat transport
y τ = L2 / κ
y Note κ ~ 10-66 m2 /s or 31.6
31 6 m2 / yr
Kelvin and the age of the Earth
y J.W Thomson (Lord
Kelvin) tried to use the
present temperature
gradient to calculate the
age
g of the Earth
y Assumed the Earth hot
initially
y Cooling by conduction
y Kelvin calculated 25 Myr as the age of the Earth
y Kelvin liked that number because it was consistent with
his calculation of the energy budget for the sun (assuming
g the ggravitational energy
gy
that the sun was radiating
accumulated at formation.
y Kelvin calculation was flawed because
y
y
He ignored radio-activity
He ignored convection
y Kelvin assumptions were in all the following debates
about thermal evolution of the Earth.
y Note that his model would work to estimate the age
g of the
sea floor.
C d i cooling
li
Conductive
model
g remains
Note that cooling
superficial. Even after 1Gyr,
there is almost no cooling
deeper than 600km.
As t ~l2/κ, it would take
100Gyr for cooling to reach
6000km!
Radioactivity
y R.J. Strutt (4th Lord
Rayleigh) 1906
y Heat flux
fl can entirely
i l be
b
accounted for by radioactivity.
activity
y “Crust” can not be thicker
than 60km!!!
y (Before Mohorovicic
discovered the Moho)
Radioactivity
First continental heat flux
measurements by Bullard
(1939).
Oceanic heat flow
Surprise !!!
y Continental crust is radioactive and thick
y Oceanic crust is thin with almost no heat generation
y Oceanic heat flow < Continental heat flow ?
y Apparently NO difference?
Energy Budget of the earth (1)
y
y
y
y
y
Birch (1951)
Total energy loss = 30 TW
H t production
Heat
d ti in
i chondrites
h d it = 5 pW/kg
W/k
Mass of earth = 6 1024 kg
Coincidence?
y
y
y
Several problems (K/U ratio)
Can not be in equilibrium with present heat production is heat is
conducted to the surface
Note this buget is obsolete (Current estimate of energy loss is
44TW)
y Question:
Q
i
Cooling
C li vs Heat
H production
d i
Cooling half space or plate
CQ = 490 ± 20
mWm-2 Myr1/2
based on
petrology and
physical
properties
Where hydrothermal circulation is
shut off, heat flux datadata fit
model
N i free
f data
d t fit cooling
li half
h lf space model
d l
Noise
at young ages
Heat flux reflects age of oceanic
lithosphere
Predicted heat flux
Isostatic balance
Bathymetry fits model for ages
<80Myr
For ages > 80 Myr, heat flux at base of plate
balances heat flux at surface: no more cooling
Hotspots perturb bathymetry
profiles
CONTINENTAL HEAT FLUX
Steady state thermal model for
stable continents
y Heat flux and temperature
gradient decrease with
depth because of h.p.
hp
y The higher the crustal h.p.,
the lower Moho and
mantle temperature
Heat flux variations in stable
continents (e.g. Canadian Shield)
y Qs = Qm + ∫ A dz
y Qm can not vary by more than +/- 3 mWm-2
y Qm < 20 mWm-2 (lowest heat flux measured)
y Best estimates 13 < Qm < 15 mWm-2
y Kapuskasing crustal section
y Grenville province
y Gravity
G it andd heat
h t flux
fl data
d t inversion
i
i
Moho heat flux in stable continents
Location
Heat flux (mWm‐2)
Baltic Shield (Archean)
7‐15
Vredefort (South Africa)
18
Slave (Archean, Canada)
12‐24
Kapuskasing (Superior, Canada)
11‐13
Abitibi (Superior, Canada)
p
10‐14
Siberian craton (Archean)
10‐12
Dharwar (Archean, India)
11
Norwegian Shield (Proterozoic)
11
Trans‐Hudson‐orogen (Proterozoic, Canada)
11‐16
Grenville (Canada)
13
Kalahari (Proterozoic, South Africa)
17‐25
Calculating temperature in the
lithosphere: 1-D heat equation
Lithospheric
temperature
profiles depend
on crustal heat
production
(surface heat
flow)
When differences in surface
heat flux are only due to
crustal heat production ,
Moho temperature varies by
150 degrees
d
Profiles are very
i i to Moho
h
sensitive
heat flow
Uncertainty of +/- 3
mWm-2 gives +/-50km on
depth
p to 1350 adiabat
Mantle convection
Adiabatic temperature gradient
Rayleigh number
Boundary layers and temerature profile
in convecting fluid
Balancing the budget
b dget
y Oceanic heat loss
activity
y Crustal radio
radio-activity
y Hotspots
y Mantle radio-activity
y Continental heat loss
y Core heat flow
y Secular cooling of mantle
Oceanic heat flo
flow
y Raw average of all heat
flux data 80 mWm-2
y Noisy
y data at young
y
g ages
g
because of hydrothermal
circulation
y Better to rely on models
Age distribution of sea floor
Total energy loss of cooling oceanic
lithosphere
y Age < 80Ma, use half
space cooling and age
distribution ~24 TW
y Age > 80 Ma, use
constant flux 48 mWm-2
~5TW
y Depends very much on
age distribution of sea
floor.
Hot spots
y Weak heat flow anomaly
on hot spots
y Use sea floor bathymetry
y
y
to estimate heat input
from buoyancy
y ~2-4TW
y Plate may be subducted
before heat flows out
Continental heat loss: eliminating the bias in
the data
y Method 1. Determine
average heat flux for
each geological age and
weight
eight according to
areal distribution
65mWm-2
y Method 2. Determine
area weighted averages
63 mWm
W -22
y Total continents (210 106
km2 = 14TW)
Heat flux in Canada
Black triangles represent heat flux sites. Note distribution.
prod ction of BSE,
BSE cr
st and mantle
Heat production
crust,
U(ppm)
Th(ppm)
K(ppm)
A(pW/kg)
Hart & Zindler (1986)
0.021
0.079
264
4.9
McDonough & Sun (1985)
0.020
0.079
240
4.8
Palme & O’Neil (2003)
0.022
0.083
261
5.1
Lyubetskaya & Korenaga (2007)
0.017
0.063
190
3.9
0.013
0.040
160
2.8
11.3
5.6
6
15000
330
BSE MORB mantle source Langmuir et al. (2005)
Continental crust
R d i k d G (2003)
Rudnick and Gao
(
)
Jaupart and Mareschal (2003)
Total BSE ~ 20 TW
Continental crust 7TW Mantle ~ 13 TW
293‐352
Core heat loss?
y > 4 TW (hot spots)
y Conductive heat flux on
adiabat -> 4TW
y Thermodynamic
efficiencyy of dynamo
y
~10%
y Ohmic dissipation of
dynamo (0.1 -> 1 TW)
y 10 TW ?
F
From
Ni
Nimmo (2007)
Mantle cooling
y Mantle heat loss 39TW
y Core flux 9TW
y Mantle radioactivity 13 TW
y Secular cooling must provide
17 TW (52 1019 J/yr)
J/ )
y Rate ~ 110 K/Gyr
From Abbott et al.
al
(1994)
Summary budget
y Mantle heat loss: 39TW
y Mantle heat production:
13 +// 4 TW
y Urey number: 0.33 +/0 11
0.11
Total heat loss 46+/-2TW
other (differentiation, (differentiation
tidal, .. ) < 1TW
core heat flow (9+/‐5TW)
Secular cooling mantle
(18+/‐8 TW)
Radio‐activity crust (7+/
7+/‐1TW)
1TW)
Radioactivity mantle (3 /4
(13+/‐4 TW)
)
Summary budget
Mantle heat loss:
39TW
Mantle heat
production: 13 +/4 TW
Urey number:
0.33 +/- 0.11
Flux de chaleur et régime thermique
8
Appendice E: Calcul de la température en fonction de la profondeur en
régime stationnaire
Dans le cas d’un régime stationnaire, l’équation de la chaleur à une dimension prend la forme:
dq
= −A(z)
dz
(E1)
où z est la profondeur, A(z) le production de chaleur, et le signe du flux q a été changé implicitement (z
est positif vers le bas et q est défini positif vers le haut). On obtient pour le flux
z
q(z) = q0 −
A(z )dz (E2)
0
avec q0 le flux de surface
surface. Donc le flux décroit
decroit avec la profondeur d’autant
d autant plus que les sources de chaleur
sont concentrées près de la surface. Et pour la température:
1
q0 z
−
T (z) = T0 +
K
K
0
z
dz
z
0
A(z”)dz”
(E3)
Si A(z) = A0 pour z < h0 , on obtient:
T (z) = T0 +
A0 z 2
q0 z
−
K
2K
(E4)
On peut ainsi itérativement calculer la température pour un modèle avec plusieurs couches dans lequel
les sources de chaleur sont constantes.
Si les sources décroissent exponentiellement avec la profondeur, A(z) = A0 exp(−z/D):
q(z) = q0 − A0 D (1 − exp(−z/D))
et
(E5)
qr z A0 D2
+
(1 − exp(−z/D))
(E6)
K
K
où qr = q0 − A0 D est le flux réduit. Notez que pour q0 et qr fixés, la température à une profondeur
donnée diminue avec D.
T (z) =