Conduction the lithosphere and isostacy To zeroth order the earths surface is bimodal in height with respect to sea level Similar things are also true for the moon and Mars though we will end up deciding it ID: 416691
Download Presentation The PPT/PDF document "GE 11a 2014, Lecture 6" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
GE 11a 2014, Lecture 6
Conduction, the lithosphere and
isostacySlide2
To
zero’th order, the earth’s surface is bimodal in height with respect to sea level Slide3
Similar things are also true for the moon and Mars, though we will end up deciding it
re
flects something unique (and uniquely important) on Earth
Moon
MarsSlide4
The Catastrophists view of the North Atlantic Slide5
Cartoon of crust and lithosphere on the board…Slide6
A shaggy dog story about the first organized thought on this subject:
Lord
Kelvin
’
s response to
uniformitarianism+catastrophism
• First quantitative estimates of the ages of celestial objects based on
‘
modern
’
physical theory (I.e., Newtonian physics, thermodynamics, Fick’s laws and the kinetic theory of gases).• Engaged a mature scientific community and discredited
‘
lax
’
logic of Uniformitarian dating• Arguments of this kind are still made to date astrophysical events, processes on other planets, and poorly sampled geologic events
Lord Kelvin looking into a boxSlide7
Lord Kelvin
’
s measurement of the age of the earth
Take 1: a proof was presented in his Ph.D. thesis, but he burned his writings on this work
after his thesis defense. It has never been recovered or reproduced.Slide8
Lord Kelvin
’
s measurement of the age of the earth
Take 2: determine the age of the Sun using principles of gravitation and thermodynamics;
infer this to be the maximum age of the Earth.
I: Measure flux of energy at earth
’
s surface
(best above atmosphere directly facing sun)
=1340 Js
-1
m
-2
II: Integrate over area of a sphere with radius
equal to distance from earth to sun (assumes
sun emits energy isotropically)
area = 4π(1.5x1011)2; power = 3.8x1026 Js
-1
If dJ/dt is a constant:
(dJ/dt)xAge ≤ mass of sun x initial energy content (
‘
E
’
, in J/Kg))
Age ≤ (2x10
30 Kg)/(3.8x1026) x E Age ≤ 5000 x ESlide9
Lord Kelvin
’
s measurement of the age of the earth
Take 2, continued:
Age of sun ≤ 5000 x initial energy content of sun in J/Kg
Case 1: If sun
’
s radiance is driven by a chemical reaction, like combustion, then it
’
s highest
plausible
initial energy content is ~ 5x10
7
J/Kg
If the sun is a ball of gasoline, it is ≤ 2.5x10
11
s, or
8000 years
, old
Case 2: Sun
’
s radiance is dissipating heat derived from its initial accretion:
Potential energy of pre-accretion cloud…
converts to kinetic energy when cloud collapses…
turns into heat if collisions between accreting material are inelasticSlide10
Case 2: Sun
’
s accretion, continued:
Age ≤
0.5M
s
xV
2
3.8x10
26
J/s
Age ≤ 10
15
s ~
30 Million years
Potential energy =
-
GM
i
m
j
R
ji
Total mass M at center-of-mass
location, i
Component particle mass m
at location j
R
ji
Solution depends on the distribution of mass and velocity in the cloud before its collapse to form the sun
One simple solution supposes all constituent masses arrived at the sun with a velocity equal
to the escape velocity from the Sun today:
(plus any contained in rotation
or other motion of cloud)
V = (2GM
s
/R)
0.5
= 618 km/s
i
0.5m
i
v
2
= 0.5M
s
(6.18x10
5
)
2
Slide11
Lord Kelvin
’
s measurement of the age of the earth
Take 3: directly determine age of the Earth by inverting the conductive temperature profile
observed in its outer few km of crust
Measurements from a geothermal area in Iceland
The archetype for the outer 300 km of the Earth
dT/dz ~ 1˚/40 meters, on average, near Earth
’
s surfaceSlide12
Lord Kelvin
’
s measurement of the age of the earth
Take 3: directly determine age of the Earth by inverting the conductive temperature profile
observed in its outer few km of crust
Q.E.D.: Physicists rule; geologists drool
T (˚C)
Radial distance
1500
‘
pinned
’
by radiative balance
of surface
t
0
t
1
t
2
0
dT/dt = k d
2
T/dx
2
k = thermal diffusivity ~ 5x10
-3
cm
2
/s (=
‘
conductivity
’
/(densityxC
v
))
Solution not simple, but is approximated by x = (kt)
0.5
, where
x = distance from surface to mid-point in T profile.
x ~ 30 km; t ~ 20 million years
Melting point of rock
J
heat
= k(
dT
/dx)Slide13
Note that conduction also leads to a change in rheology between interior and outer shellSlide14
Rayleigh number =
Buoyancy
Viscous drag
X
Momentum diffusivity
Thermal diffusivity
acceleration
Thermal expansion
Kinematic viscosity
Thermal diffusivity
Length scale
Temperature contrast
If > ~1000, convection ensues. The mantle is ~10
6
What are the dynamics of the hot, viscous (fluid like) interior? Slide15
A numerical model of whole-mantle convection in a
2-D earth