Download Heat Conduction

Geodynamics
Lecture 7
Heat conduction and production
Lecturer: David Whipp
[email protected]
!
23.9.2014
Geodynamics
www.helsinki.fi/yliopisto
1
Goals of this lecture
•
Gain a conceptual and mathematical understanding of heat
conduction
!
•
Learn the concepts of radiogenic heat production in the Earth
!
•
View the effects of heat conduction and production on the
thermal field in the crust
2
Why talk about heat transfer?
Grand Prismatic Spring,Yellowstone National Park, U.S.
Image: http://en.wikipedia.org
The dynamics of the Earth are dominantly controlled by gravitational and
thermal processes. In fact, you could argue that thermal processes have the
largest impact. Why?
3
Why now?
www.usgs.gov
Faulting
Folding
Many rock properties are largely a function of temperature.
Faulting and brittle deformation may occur near the surface, but folding and
ductile flow are dominant at depth.
4
The effects of partial melting on rock strength
udio L. Rosenberg, Sergei Medvedev, and Mark R. Handy
•
Partial melting dramatically decreases rock strength
!
•
Only about 7% melt is required for to decrease rock
strength by 80-90%
Rosenberg et al., 2007
5
Heat transfer in the lithosphere
•
Conduction: The diffusive transfer of heat by kinetic atomic or
molecular interactions within the material. Also known as
thermal diffusion.
!
•
Advection: The transfer of heat by physical movement of
molecules or atoms within a material. A type of convection,
mostly applied to heat transfer in solid materials.
!
•
Production: Not really a heat transfer process, but rather a
source of heat. Sources in the lithosphere include radioactive
decay, friction in deforming rock or chemical reactions such as
phase transitions.
6
Heat transfer in the lithosphere
•
Conduction: The diffusive transfer of heat by kinetic atomic or
molecular interactions within the material. Also known as
thermal diffusion.
!
Focus in
this lecture
•
Advection: The transfer of heat by physical movement of
molecules or atoms within a material. A type of convection,
mostly applied to heat transfer in solid materials.
!
•
Production: Not really a heat transfer process, but rather a
source of heat. Sources in the lithosphere include radioactive
decay, friction in deforming rock or chemical reactions such as
phase transitions.
7
Basic ideas of heat conduction
•
The conduction of heat in solids is a
diffusion process, and well described by
Fourier’s laws, the basic mathematical
relationships describing diffusion
!
•
Fourier’s first law states that the flux of
heat in a material 𝑞 is directly proportional to the temperature gradient
!
•
What would this relationship look as an equation?
8
Basic ideas of heat conduction
•
The conduction of heat in solids is a
diffusion process, and well described by
Fourier’s laws, the basic mathematical
relationships describing diffusion
!
•
Fourier’s first law states that the flux of
heat in a material 𝑞 is directly proportional to the temperature gradient
!
•
What would this relationship look as an equation?
9
Fourier’s first law
•
In 1D, the mathematical translation of “Heat flux 𝑞 is directly
proportional to the thermal gradient in a material” is
!
q=
dT
k
dy
•
Here, 𝑇 represents temperature and 𝑦 represents spatial
position, depth in the Earth in this case
•
Thus, 𝑑𝑇/𝑑𝑦 is the change in temperature with distance, the
thermal gradient
•
The proportionality constant 𝑘 is known as the thermal
conductivity
10
Fourier’s first law
•
In 1D, the mathematical translation of “Heat flux 𝑞 is directly
proportional to the thermal gradient in a material” is
!
•
q=
dT
k
dy
Why is there a negative sign?
11
What is thermal conductivity?
•
The mathematical translation of “Heat flux 𝑞 is directly
proportional to the thermal gradient in a material” is
Sandstone
q=
dT
k
dy
•
•
Thermal conductivity is a proportionality factor
•
Thermal conductivity of most crustal rocks is 2-3 W m-­‐1 K-­‐1
As you can easily see, rocks with a “high” thermal
conductivity will produce a large heat flow, whereas rocks
with a “low” thermal conductivity will have near zero heat
flow
Salt
Stüwe, 2007
12
Global heat flow map
http://www.cbe.cornell.edu
13
Global heat flow map
Global average: 87 mW m-2
Continents: 65 ± 1.6 mW m-2
Oceans: 101 ± 2.2 mW m-2
http://www.cbe.cornell.edu
Here we can clearly see the connection between geodynamic setting and heat flow
14
Radiogenic heat production
•
Radiogenic heat production, 𝐴 or 𝐻, results from the decay
of radioactive elements in the Earth, mainly 238U, 235U, 232Th
and 40K. 𝐴 is generally used for volumetric heat production and
𝐻 for heat production by mass.
•
These elements occur in the mantle, but are concentrated in
248
Heat Transfer
the crust, where radiogenic
heating can be significant
•
Table
Typicalheat
Concentrations
of the Heat-Producing
Elements
The4.3
surface
flow in continental
regions
is ~65inmW m-2
Several Rock Types and
Average Concentrations in Chondritic
and ~37 mW m-2 isthefrom
radiogenic heat production (57%)
Meteorites
Type
Rock
Reference
undepleted (fertile) mantle
“Depleted” peridotites
Tholeiitic basalt
Granite
Shale
Average continental crust
Chondritic meteorites
U (ppm)
Concentration
Th (ppm)
K (%)
0.031
0.001
0.07
4.7
3.7
1.42
0.008
0.124
0.004
0.19
20
12
5.6
0.029
0.031
0.003
0.088
4.2
2.7
1.43
0.056
Turcotte and Schubert, 2014
15
Radiogenic heat production
•
Radiogenic heat production, 𝐴 or 𝐻, results from the decay
of radioactive elements in the Earth, mainly 238U, 235U, 232Th
and 40K. 𝐴 is generally used for volumetric heat production and
𝐻 for heat production by mass.
•
These elements occur in the mantle, but are concentrated in
248
Heat Transfer
the crust, where radiogenic
heating can be significant
•
Table
Typicalheat
Concentrations
of the Heat-Producing
Elements
The4.3
surface
flow in continental
regions
is ~65inmW m-2
Several Rock Types and
Average Concentrations in Chondritic
and ~37 mW m-2 isthefrom
radiogenic heat production (57%)
Meteorites
Type
Rock
Reference
undepleted (fertile) mantle
“Depleted” peridotites
Tholeiitic basalt
Granite
Shale
Average continental crust
Chondritic meteorites
U (ppm)
Concentration
Th (ppm)
K (%)
0.031
0.001
0.07
4.7
3.7
1.42
0.008
0.124
0.004
0.19
20
12
5.6
0.029
0.031
0.003
0.088
4.2
2.7
1.43
0.056
Turcotte and Schubert, 2014
16
Radiogenic heat production
Heat production
Rock type
By mass, 𝐻 [W kg
Reference undepleated (fer/le) mantle
7.39E-12
2.44E-08
0.024
"Depleated" perido/tes
3.08E-13
1.02E-09
0.001
Tholeii/c basalt
1.49E-11
4.41E-08
0.044
Granite
1.14E-09
3.01E-06
3.008
Shale
7.74E-10
1.86E-06
1.857
Average con/nental crust
3.37E-10
9.26E-07
0.927
Chondri/c meteorites
3.50E-12
1.15E-08
0.012
By volume, 𝐴
By volume, 𝐴
[W m
[µW m
Calculated from Turcotte and Schubert, 2014
17
Radiogenic heat production
Heat production
Rock type
By mass, 𝐻 [W kg
Reference undepleated (fer/le) mantle
7.39E-12
2.44E-08
0.024
"Depleated" perido/tes
3.08E-13
1.02E-09
0.001
Tholeii/c basalt
1.49E-11
4.41E-08
0.044
Granite
1.14E-09
3.01E-06
3.008
Shale
7.74E-10
1.86E-06
1.857
Average con/nental crust
3.37E-10
9.26E-07
0.927
Chondri/c meteorites
3.50E-12
1.15E-08
0.012
By volume, 𝐴
By volume, 𝐴
[W m
[µW m
Calculated from Turcotte and Schubert, 2014
•
Typical heat production values for upper crustal rocks are
2-3 µW m-3, but the crustal average is <1 µW m-3
•
What does this suggest, and why might this occur?
18
Radiogenic heat production
Stüwe, 2007
•
𝐻
Because radiogenic heat production is the result
of radioactive decay, the parent isotope
concentrations have continually decreased
throughout Earth’s history
•
Today, the total amount of radiogenic heat is
about half of what was produced in the
ArcheanS at ~3 Ga
Crustal heat production over time
19
Radiogenic heat production
Stüwe, 2007
•
𝐻
Because radiogenic heat production is the result
of radioactive decay, the parent isotope
concentrations have continually decreased
throughout Earth’s history
•
Today, the total amount of radiogenic heat is
about half of what was produced in the
ArcheanS at ~3 Ga
Crustal heat production over time
20
LETTER
RESEARCH LETTER
doi:10.1038/nature13728
Spreading continents kick-started plate tectonics
Patrice F. Rey1, Nicolas Coltice2,3 & Nicolas Flament1
Temperature (K)
Continent
c
d
e
Depth (km)
b
Depth (km)
Depth (km)
Aa
that of present-day tectonic forces driving orogenesis1. To explore the
Stresses acting on cold, thick and negatively buoyant oceanic
litho- 2,000
0 1,000
0
sphere are thought to be crucial to the initiation of subduction
and tectonic impact of a thick and buoyant continent surrounded by a stag1,2
the operation of plate tectonics , which characterizes the present- nant lithospheric lid, we produced a series of two-dimensional thermo200 was hotter mechanical
1,820 K numerical models of the top 700 km of the Earth, using
day geodynamics of the Earth. Because the Earth’s interior
293 K
in the Archaean eon, the oceanic crust may have been thicker, thereby temperature-dependent densities and visco-plastic rheologies that depend
making the oceanic lithosphere more buoyant than 400
at present3, and on temperature, melt fraction and depletion, stress and strain rate (see
whether subduction and plate tectonics occurred during this time is Methods). The initial temperature field is the horizontally averaged tem400 km
600
ambiguous, both in the geological record and in geodynamic models4. perature profile of a stagnant-lid convection calculation for a mantle
Here we show that because the oceanic crust was thick and buoyant5, ,200 K hotter than at present (Fig. 1A, a and Extended Data Fig. 2). The
lateral
temperature gradients
ensures that
no convective
stresses(g cm–3)
early continents may have produced intra-lithospheric gravitational absence ofB
a Deviatoric
Reference
density
stress (MPa)
stresses large enough to drive their gravitational spreading, to initi- act on the lid, allowing100
us to isolate
dynamic effects2.8
of the3.0
continent.
300 the
500
3.2 3.4
0
0
ate subduction at their margins and to trigger episodes of subduc- A buoyant and stiff continent 225 km thick (strongly depleted mantle
Basaltic crust
tion. Our model predicts the co-occurrence of deep to progressively root 170 km thick overlain by felsic crust 40 km thick; see Fig. 1B, a) is
shallower mafic volcanics and arc magmatism within continents in inserted within the lid, on the left side of the domain to exploit
the symCrust
40
40
a self-consistent geodynamic framework, explaining the enigmatic metry of the problem (Fig. 1A, a). A mafic crust 15 km thick covers the
multimodal volcanism and tectonic record of Archaean cratons6. More- whole system (Fig. 1A, a), consistent with the common occurrence of
Strongly
over, our model predicts a petrological stratification and tectonic struc- thick greenstone80
covers on continents, as well as80
thick basaltic
crust on
3
depleted
ture of the sub-continental lithospheric mantle, two predictions that the oceanic lid .
are consistent with xenolith5 and seismic studies, respectively, and conOur numerical solutions show that the presence oflithospheric
a buoyant conDepletion
sistent with the existence of a mid-lithospheric seismic discontinuity7. tinent imparts 120
a horizontal force large enough120
to induce mantle
a long period
The slow gravitational collapse of early continents could have kick- (,50–150 Myr) of slow collapse of the whole continental lithosphere
Strain rate
started transient episodes of plate tectonics until, as the Earth’s inte- (Fig. 1 and Extended
Fig.
3),
in agreement with the dynamics of
160 Data
–15
–1
10
19 s
rior cooled and oceanic lithosphere became heavier, plate tectonics spreading for gravity currents . Hence, a continent of larger volume leads
225
became self-sustaining.
to larger gravitational power and faster collapse. BecauseAsthenosphere
of lateral spreadPresent-day plate tectonics is primarily driven by the negative buoy- ing of the b
continent, the100
adjacent
3.2 3.4
2.8pushed
3.0 under
300lithospheric
500 lid is slowly
0 b and Extended Data Fig. 3A,0a). For gravitational
ancy of cold subducting plates. Petrological and geochemical proxies of its margin (Fig. 1A,
Basaltic crust
subduction preserved in early continents point to subduction-like pro- stress lower than the yield stress of the oceanic lid, thickening of the margin
cesses already operating before 3 billion years (Gyr) ago8,9 and perhaps of the lid is slow, and viscous drips (that is, Rayleigh–Taylor instabilities)
40
40
Lithospheric
as early as 4.1 Gyr ago10. However, they are not unequivocal, and geo- detach from its base (Extended Data Fig. 3A, a and b). These
instabilities,
20
mantle of
dynamic modelling suggests that the thicker basaltic crust produced by typical of stagnant-lid convection , mitigate the thermal thickening
partial melting of a hotter Archaean or Hadean mantle would have had the lid.
80
80
3,4
increased lithospheric buoyancy and inhibited subduction . Mantle conWhen gravitational stresses overcome the yield stress of the lithospheric
vection under a stagnant lid with extensive volcanism could therefore lid, subduction is initiated (Fig. 1A, b and c). Depending on the half-width
have preceded the onset of subduction11. In this scenario, it is classically of the continent120
and its density contrast with the120
adjacent oceanic lid (that
Strain
rate
assumed that the transition from stagnant-lid regime to mobile-lid regime is, its gravitational power)
three
situations
can
arise: first,
subduction
Asthenosphere
–15 s–1
10
and the onset of plate tectonics require that convective stresses overcame initiates and stalls
(Extended Data Fig. 3b); second,
160the slab detaches and
160
21
the strength of the stagnant lid12 at some stage in the Archaean.
the lid stabilizes (Fig. 1A, d and e and Extended Data Fig. 3c); or third,
On the modern Earth, gravitational stresses due to continental buoyancy recurrent detachment of the slab continues until recycling of the oceanic
can contribute to the initiation of subduction2,13. The role of continental lid is completed, followed by stabilization (Extended Data Fig. 3d). When
LETTER
doi:10.1038/nature13728
Spreading continents kick-started plate tectonics
Patrice F. Rey1, Nicolas Coltice2,3 & Nicolas Flament1
Temperature (K)
Continent
Depth (km)
Aa
that of present-day tectonic forces driving orogenesis1. To explore the
Stresses acting on cold, thick and negatively buoyant oceanic
litho- 2,000
0 1,000
0
sphere are thought to be crucial to the initiation of subduction
and tectonic impact of a thick and buoyant continent surrounded by a stag1,2
the operation of plate tectonics , which characterizes the present- nant lithospheric lid, we produced a series of two-dimensional thermo200 was hotter mechanical
1,820 K numerical models of the top 700 km of the Earth, using
day geodynamics of the Earth. Because the Earth’s interior
293 K
in the Archaean eon, the oceanic crust may have been thicker, thereby temperature-dependent densities and visco-plastic rheologies that depend
making the oceanic lithosphere more buoyant than 400
at present3, and on temperature, melt fraction and depletion, stress and strain rate (see
whether subduction and plate tectonics occurred during this time is Methods). The initial temperature field is the horizontally averaged tem400 km
600
ambiguous, both in the geological record and in geodynamic models4. perature profile of a stagnant-lid convection calculation for a mantle
Here we show that because the oceanic crust was thick and buoyant5, ,200 K hotter than at present (Fig. 1A, a and Extended Data Fig. 2). The
lateral
temperature gradients
ensures that
no convective
stresses(g cm–3)
early continents may have produced intra-lithospheric gravitational absence ofB
a Deviatoric
Reference
density
stress (MPa)
stresses large enough to drive their gravitational spreading, to initi- act on the lid, allowing100
us to isolate
dynamic effects2.8
of the3.0
continent.
300 the
500
3.2 3.4
0
0
ate subduction at their margins and to trigger episodes of subduc- A buoyant and stiff continent 225 km thick (strongly depleted mantle
Basaltic crust
tion. Our model predicts the co-occurrence of deep to progressively root 170 km thick overlain by felsic crust 40 km thick; see Fig. 1B, a) is
shallower mafic volcanics and arc magmatism within continents in inserted within the lid, on the left side of the domain to exploit
the symCrust
40
40
a self-consistent geodynamic framework, explaining the enigmatic metry of the problem (Fig. 1A, a). A mafic crust 15 km thick covers the
multimodal volcanism and tectonic record of Archaean cratons6. More- whole system (Fig. 1A, a), consistent with the common occurrence of
Strongly
over, our model predicts a petrological stratification and tectonic struc- thick greenstone80
covers on continents, as well as80
thick basaltic
crust on
3
depleted
ture of the sub-continental lithospheric mantle, two predictions that the oceanic lid .
are consistent with xenolith5 and seismic studies, respectively, and conOur numerical solutions show that the presence oflithospheric
a buoyant conDepletion
sistent with the existence of a mid-lithospheric seismic discontinuity7. tinent imparts 120
a horizontal force large enough120
to induce mantle
a long period
The slow gravitational collapse of early continents could have kick- (,50–150 Myr) of slow collapse of the whole continental lithosphere
Strain rate
started transient episodes of plate tectonics until, as the Earth’s inte- (Fig. 1 and Extended
Fig.
3),
in agreement with the dynamics of
160 Data
–15
–1
10
19 s
rior cooled and oceanic lithosphere became heavier, plate tectonics spreading for gravity currents . Hence, a continent of larger volume leads
225
became self-sustaining.
to larger gravitational power and faster collapse. BecauseAsthenosphere
of lateral spreadPresent-day plate tectonics is primarily driven by the negative buoy- ing of the b
continent, the100
adjacent
3.2 3.4
2.8pushed
3.0 under
300lithospheric
500 lid is slowly
0 b and Extended Data Fig. 3A,0a). For gravitational
ancy of cold subducting plates. Petrological and geochemical proxies of its margin (Fig. 1A,
Basaltic crust
subduction preserved in early continents point to subduction-like pro- stress lower than the yield stress of the oceanic lid, thickening of the margin
cesses already operating before 3 billion years (Gyr) ago8,9 and perhaps of the lid is slow, and viscous drips (that is, Rayleigh–Taylor instabilities)
40
40
Lithospheric
as early as 4.1 Gyr ago10. However, they are not unequivocal, and geo- detach from its base (Extended Data Fig. 3A, a and b). These
instabilities,
20
mantle of
dynamic modelling suggests that the thicker basaltic crust produced by typical of stagnant-lid convection , mitigate the thermal thickening
partial melting of a hotter Archaean or Hadean mantle would have had the lid.
80
80
3,4
increased lithospheric buoyancy and inhibited subduction . Mantle conWhen gravitational stresses overcome the yield stress of the lithospheric
vection under a stagnant lid with extensive volcanism could therefore lid, subduction is initiated (Fig. 1A, b and c). Depending on the half-width
have preceded the onset of subduction11. In this scenario, it is classically of the continent120
and its density contrast with the120
adjacent oceanic lid (that
Strain
rate
assumed that the transition from stagnant-lid regime to mobile-lid regime is, its gravitational power)
three
situations
can
arise: first,
subduction
Asthenosphere
–15 s–1
10
and the onset of plate tectonics require that convective stresses overcame initiates and stalls
(Extended Data Fig. 3b); second,
160the slab detaches and
160
22
the strength of the stagnant lid12 at some stage in the Archaean.
the lid stabilizes (Fig. 1A, d and e and Extended Data Fig. 3c); or third,
On the modern Earth, gravitational stresses due to continental buoyancy recurrent detachment of the slab continues until recycling of the oceanic
can contribute to the initiation of subduction2,13. The role of continental lid is completed, followed by stabilization (Extended Data Fig. 3d). When
b
d
Larger horizontal stresses force subduction
of buoyant oceanic lithosphere
Slab detaches, margin stabilizes
e
Depth (km)
c
Depth (km)
Continent weakens, spreads laterally
1D steady-state heat conduction + production
•
Consider the heat flux across a slab of
thickness 𝛿𝑦
•
The net heat flux is simply the heat flux out
minus the heat flux in, or
•
Fig. 4.5, Turcotte and Schubert, 2014
q(y + y)
q(y)
Using Taylor series expansion and Fourier’s
first law, it can be shown that
 ✓ ◆
dq
d
dT
q(y +
k
y) q(y) = y dy = y dy
dy
 ✓ 2 ◆
d T
= y
k
2
dy
for constant thermal conductivity 𝑘
23
1D steady-state heat conduction + production
Fig. 4.5, Turcotte and Schubert, 2014
•
If we assume the only source of changing the
heat flux in the slab is radiogenic heat, we can
say
 ✓ 2 ◆
d T
⇢H y = y
k
dy 2
or
2
d
T
k 2 + ⇢H = 0
dy
where 𝜌 is rock density
•
Assuming 𝑇 = 𝑇0 and 𝑞 = 𝑞0 at 𝑦 = 0, the
solution is
q0
T = T0 + y
k
⇢H 2
y
2k
With this equation, we can plot a geotherm, or change in temperature with depth
24
Heat conduction in the mantle?
!
!
q0
T = T0 + y
k
⇢H 2
y
2k
•
70 mW m-2 is a reasonable average heat flow value observed
at the surface
•
If we assume a surface temperature 𝑇0 = 0°C, rock density
𝜌 = 3300 kg m-3, 𝐻 = 7.38 ⨉ 10-12 W kg-1 and 𝑘 = 4 W m-1 K-1,
what is the predicted temperature at 100 km depth?
•
•
Does this seem reasonable?
What about at 200 km depth?
25
Heat conduction in the mantle?
Fig. 4.8, Turcotte and Schubert, 2014
Clearly, heat flow in the mantle is not strictly conductive!
26
Geotherms in the continental crust
With a 40-km-thick crust,
you can see that heat
production can
increase the Moho
temperature considerably
!
!
Note, that this model has a
constant heat flow basal
boundary condition
𝑘 = 2.75 W m-­‐1 K-­‐1 𝑞m = 20 mW m-­‐2 ymax = 40 km
27
Geotherms in the continental crust
Why might this matter?
The approximate depth at
which crustal rocks
transition from brittle
deformation to ductile
deformation corresponds
roughly to the 300°C
isotherm in the crust
!
Based on the models here,
that depth could vary from
~15-40 km (!), which has
serious implications for
how the crust deforms
𝑘 = 2.75 W m-­‐1 K-­‐1 𝑞m = 20 mW m-­‐2 ymax = 40 km
28
Distribution of heat producing elements
Heat production
Rock type
By mass, 𝐻 [W kg
Reference undepleated (fer/le) mantle
7.39E-12
2.44E-08
0.024
"Depleated" perido/tes
3.08E-13
1.02E-09
0.001
Tholeii/c basalt
1.49E-11
4.41E-08
0.044
Granite
1.14E-09
3.01E-06
3.008
Shale
7.74E-10
1.86E-06
1.857
Average con/nental crust
3.37E-10
9.26E-07
0.927
Chondri/c meteorites
3.50E-12
1.15E-08
0.012
•
By volume, 𝐴
By volume, 𝐴
[W m
[µW m
As we’ve seen, however, the concentration of heat-producing
elements is not constant in the crust, but decreases with depth
29
Geotherms in the continental crust
!
melting of crust
𝐴 = 2.5 𝜇W m-­‐3 𝑘 = 2.75 W m-­‐1 K-­‐1 𝑞m = 20 mW m-­‐2 ymax = 40 km
Onset of partial
As before, changing the
concentration of heatproducing elements shows
us something important
about how we might expect
the crust to deform
!
Here, we might predict
partial (to complete)
melting of the crust and
very weak rocks if we don’t
consider decreasing heat
production
30
Geotherms in thick continental crust
!
melting of crust
𝐴 = 2.5 𝜇W m-­‐3 𝑘 = 2.75 W m-­‐1 K-­‐1 𝑞m = 20 mW m-­‐2 ymax = 80 km
Onset of partial
The crust beneath the
Tibetan Plateau is thought
to be 70-80 km thick
resulting from extensive
shortening during
orogenesis, and it is likely
that this over-thickened
crust is partially molten at
depth as a result of
radiogenic heat
!
How might this affect the
plateau topography?
31
Geodynamic implications
•
We’ve now see that radiogenic heat production can
significantly influence the subsurface thermal field, particularly
in the crust
•
•
This may influence geodynamic behavior of the deforming
crust by affecting the depth of the transition from brittle to
ductile deformation, or the potential for partial melting of
the crust
Another important influence on the crustal thermal field is
topography at the Earth’s surface
•
Why might topography matter?
32
Thermal field beneath periodic topography
•
In our calculations of temperatures in the Earth thus far, we’ve
assumed a known surface temperature 𝑇0 at 𝑦 = 0
•
Atmospheric temperatures decrease with elevation following
an atmospheric lapse rate 𝛽 (typical value: 𝛽 = 6.5°C km-1)
•
Combined, this suggests surface temperatures should vary with
elevation in mountainous regions, an effect we can simulate
•
Importantly, this is clearly a 2D problem
33
Thermal field beneath periodic topography
•
To do this, we compensate for the temperature variations with elevation by
scaling the temperature by 𝛥𝑇 at 𝑦 = 0, yielding
⌘
qm y ⇢H0 h2r ⇣
2⇡x 2⇡y/
y/hr
T (x, y) = T0 +
+
1 e
+ T cos
e
k
k
with
✓
◆
qm
⇢H0 hr
T =
h0
k
k
where 𝑇0 is the surface temperature a sea level in nature, 𝑞m is the mantle heat
flux, 𝑘 is thermal conductivity, 𝜌 is density, 𝐻0 is the surface heat production
value, ℎr is the e-folding depth for heat production, 𝜆 is the wavelength of the
34
topography and ℎ0 is the amplitude
Thermal field beneath periodic topography
𝐴 = 2.5 𝜇W m-­‐3 ℎr = 10 km 𝑘 = 2.75 W m-­‐1 K-­‐1 𝑞m = 20 mW m-­‐2 ℎ0 = 4 km 𝜆 = ??? km 𝛽 = 6.5°C km-­‐1
35
Thermal field beneath periodic topography
𝐴 = 2.5 𝜇W m-­‐3 ℎr = 10 km 𝑘 = 2.75 W m-­‐1 K-­‐1 𝑞m = 20 mW m-­‐2 ℎ0 = 4 km 𝜆 = ??? km 𝛽 = 6.5°C km-­‐1
36
Thermal field beneath topography
•
Topography can influence the geometry of the thermal field
beneath it, particularly as the relief and wavelength grow
•
The depth of the perturbation is related to the wavelength,
with longer wavelengths affecting deeper crustal levels
!
•
This has important implications for data that depend on the
thermal structure of mountainous regions, such as lowtemperature thermochronology data
37
Recap
•
Heat conduction, or diffusion, is the process by which
molecular or elemental interactions transfer energy to
neighboring particles
!
•
Heat conduction is dominantly controlled by rock material
properties, such as the thermal conductivity or radiogenic heat
production
!
•
The thermal field in the crust plays an important role in
determining how the crust deforms
38
References
Rey, P. F., Coltice, N., & Flament, N. (2015). Spreading continents kick-started plate tectonics. Nature,
513(7518), 405–408. doi:10.1038/nature13728
!
Rosenberg, C. L., Medvedev, S. & Handy, M. (2007) Effects of melting on faulting and continental deformation.
In: The Dynamics of Fault Zones, edited by M. R. Handy, G. Hirth, N. Hovius. Dahlem Workshop Report
95, The MIT Press, Cambridge, Mass., USA, 357-401.
39