GE 11a 2014, Lecture 6
Conduction, the lithosphere and isostacy
To zero’th order, the earth’s 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
reflects something unique (and uniquely important) on Earth
Mars
Moon
The Catastrophists view of the North Atlantic
Cartoon of crust and lithosphere on the board…
A shaggy dog story about the first organized thought on this subject:
Lord Kelvin’s response to uniformitarianism+catastrophism
Lord Kelvin looking into a box
• 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’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.
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-1m-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 ≤ (2x1030 Kg)/(3.8x1026) x E
Age ≤ 5000 x E
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 ~ 5x107 J/Kg
If the sun is a ball of gasoline, it is ≤ 2.5x1011 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 inelastic
Case 2: Sun’s accretion, continued:
Total mass M at center-of-mass
location, i
-GMimj
Potential energy =  R
ji
(plus any contained in rotation
or other motion of cloud)
Rji
Component particle mass m
at location j
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:
V = (2GMs/R)0.5 = 618 km/s
i0.5miv2 = 0.5Ms(6.18x105)2
0.5MsxV2
Age ≤ 3.8x1026 J/s
Age ≤ 1015 s ~ 30 Million years
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 surface
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
Melting point of rock
1500
t2
t1
t0
T (˚C)
‘pinned’ by radiative balance
of surface
0
Radial distance
Jheat = k(dT/dx)
dT/dt = k d2T/dx2
k = thermal diffusivity ~ 5x10-3 cm2/s (= ‘conductivity’/(densityxCv))
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
Q.E.D.: Physicists rule; geologists drool
Note that conduction also leads to a change in rheology between interior and outer shell
What are the dynamics of the hot, viscous (fluid like) interior?
Rayleigh number =
Buoyancy
Viscous drag
acceleration
Momentum diffusivity
X Thermal diffusivity
Thermal expansion
Temperature contrast
Length scale
Kinematic viscosity
Thermal diffusivity
If > ~1000, convection ensues. The mantle is ~106
A numerical model of whole-mantle convection in a
2-D earth