Download Geotherms One layer model

Geotherms
Reading: Fowler Ch 7
EPS 122: Lecture 19 – Geotherms
Equilibrium geotherms
One layer model
(a) Standard model:
k = 2.5 W m-1 °C-1
A = 1.25 x 10-6 W m-3
Qmoho = 21 x 10-3 W m-2
shallow T-gradient: 30 °C km-1
deep T-gradient: 15 °C km-1
Conductivity
reduce T-grad increases (b)
Heat generation
increase T-grad increases (c)
Basal heat flow
increase T-grad increases (d)
EPS 122: Lecture 19 – Geotherms
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Timescales
…long
Increase basal heat from
(a) Qmoho = 21 x 10-3 W m-2
to (d) Qmoho = 42 x 10-3 W m-2
Consider rock at 20 km depth
t=0
567 °C
t = 20 Ma
580 °C
t = 100 Ma
700 °C
t = 734 °C
melting and intrusion are
important heat transfer
mechanisms in the lithosphere
EPS 122: Lecture 19 – Geotherms
Timescales
From the diffusion equation
we can define the
characteristic timescale
the amount of time
necessary for a
temperature change to
propagate a distance l
thermal
diffusivity
characteristic thermal diffusion distance
the distance a change in temperature
will propagate in time thermal diffusivity of granite: 8.5 x 10-7 m2 s-1
l = 10 m = 4 years
l = 1 km = 37,000 years
l = 100 km = 370 Ma
EPS 122: Lecture 19 – Geotherms
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Instantaneous cooling
T=0
Semi-infinite half-space at temperature T0
Allow to cool at surface where T = 0
T = T0
No internal heating, use diffusion equation
The solution is the error function
time t1 calc error func T = 0.9T0
time t2 calc error func T = 0.6T0
time
EPS 122: Lecture 19 – Geotherms
Oceanic heat flow
– observations
• Higher for younger crust (mostly)
• Greater variability for younger crust
Stein & Stein, 1994
hydrothermal circulation
at mid-ocean ridges
EPS 122: Lecture 19 – Geotherms
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Oceanic heat flow
– observations
MidAtlantic
Ridge
Black Smokers
400°C water
The Blue Lagoon
EPS 122: Lecture 19 – Geotherms
Ocean basins
Sediment thickness
cuts off hydrothermal circulation
0
0.5
10
10
10
1
1
1
0.5
0.5
0
Thickest
sediments found
at the base of the
continental slope
– landslides
Thinnest at the
ridge – no time
for deposition
EPS 122: Lecture 19 – Geotherms
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Depth distribution
The ocean basins
Depth distribution is related to age
ie the time available for cooling
Good approximation to
observation out
to ~70 Ma
squares: North Atlantic
circles: North Pacific
EPS 122: Lecture 19 – Geotherms
– observations
Depth
…works best till ~70 Ma
for greater ages depth decreases
more slowly
Stein & Stein, 1994
Depth and heat flow
Heat flux
initially…
for greater ages Q decreases
more slowly
EPS 122: Lecture 19 – Geotherms
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A simple half-space model
ridge
x
T=0
T = Ta
3D convection and advection equation
z
Assume:
• temperature field is in equilibrium
• advection of heat horizontally is
greater than conduction
Also, t = x/u
i.e. distance and time related
by the spreading rate
We have already seen the solution….
EPS 122: Lecture 19 – Geotherms
A simple half-space model
ridge
x
Temperature gradient
T = Ta
T=0
z
Surface heat flow
…differentiate T-gradient
The observed heat flux
was:
this simple model provides
the t1/2 relation
EPS 122: Lecture 19 – Geotherms
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A simple half-space model
ridge
x
Temperature gradient
T = Ta
T=0
z
Estimate the lithospheric thickness…
T at base of lithosphere: 1100 °C
and Ta = 1300 °C
look up inverse error function
if = 10-6 m2 s-1
L in km, t in Ma
10 Ma L = 35 km
80 Ma L = 98 km
reasonable?
EPS 122: Lecture 19 – Geotherms
A simple half-space model
Ocean depth
…apply isostasy
Column of lithosphere
at the ridge
=
Rearrange
Need (z)
…density as a
function of T
coefficient of
thermal expansion
…and T as a function
of age
Substitute…
EPS 122: Lecture 19 – Geotherms
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A simple half-space model
Ocean depth
…apply isostasy
Approximate L Rearrange…
Appropriate values:
w = 1.0 x 103 km m-3
a = 3.3 x 103 km m-3
= 3 x 10-5 °C-1
= 10-6 m2 s-1
Ta = 1300 °C
Observed…
t in Ma and d in km
The simple half-space cooling model
matches ocean depths out to ~70 Ma
i.e. lithosphere cools, contracts and subsides
EPS 122: Lecture 19 – Geotherms
The “plate” model
The lithosphere has a fixed thickness at
the ridge and cools with time
The asthenosphere below is constant
temperature
ridge
ridge
x
z
x
T = Ta
T=0
T = Ta
T=0
Simple half-space model
T = Ta
…asymptotic values of Q, depth etc.
z
…cools and thickens for ever
EPS 122: Lecture 19 – Geotherms
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Depth and heat flow
– observations
Which model(s) fit the data?
HS
– Half-space model
GDH1 – plate model
PSM
– plate model
Stein & Stein, 1994
The GDH1 “plate” model does
a better job of fitting the
depth data
(which is better constrained)
All fit the heat flow data
(within error)
EPS 122: Lecture 19 – Geotherms
Depth distribution
The ocean basins
Depth distribution is related to age
ie the time available for cooling
Good approximation to
observation out
to ~70 Ma
Plate model:
There is a limit to the lithospheric
thickness available for cooling
squares: North Atlantic
circles: North Pacific
EPS 122: Lecture 19 – Geotherms
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crust
crust
lithosphere
lithosphere
A hybrid?
Continents
• thicker crust
• similar
lithosphere
“Plate” model fits depth and Q best
(cratons?)
but there is other geophysical evidence for a
thickening lithosphere
• increasing elastic thickness
• increasing depth to low velocity asthenosphere
thermal boundary layer with small-scale convection
EPS 122: Lecture 19 – Geotherms
The mantle geotherm
convection
rather than conduction
more rapid heat transfer
Adiabatic
temperature
gradient
Raise a parcel of rock…
If constant entropy:
lower P expands
larger volume reduced T
This is an adiabatic gradient
Convecting system close to adiabatic
EPS 122: Lecture 19 – Geotherms
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The adiabatic temperature gradient
Need the change of temperature with pressure at constant entropy, S
using reciprocal theory
Some thermodynamics…
Maxwell’s thermodynamic
relation
coefficient of
thermal expansion
specific heat
Substitute…
…adiabatic gradient as a
function of pressure
EPS 122: Lecture 19 – Geotherms
The adiabatic temperature gradient
…adiabatic gradient as a
function of pressure
…but we want it as a function of depth
For the Earth
Substitute…
…adiabatic gradient as a
function of radius
Temperature gradient for the
uppermost mantle
0.4 °C km-1
at greater depth
using
0.3 °C km-1
due to reduced T = 1700 K
= 3 x 10-5 °C-1
g = 9.8 m s-2
cp = 1.25 x 103 J kg °C-1
EPS 122: Lecture 19 – Geotherms
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Adiabatic temperature gradients
Models agree that gradient is close to adiabatic,
particularly in upper mantle
…why would it not be
adiabatic?
greater uncertainty for the lowest
500-1000 km of the mantle
big range of estimated T for CMB
2500K to ~4000K
This is the
work of
Jeanloz
and
Bukowinski
in our dept
EPS 122: Lecture 19 – Geotherms
Melting in the mantle
100 km
200 km
Different
adiabatic
gradient
for fluids:
~ 1 °C km-1
Potential temperature: T of rock at surface if rises
along the adiabat
EPS 122: Lecture 19 – Geotherms
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