GE 11a 2014, Lecture 6 Conduction, the lithosphere and isostacy.
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Transcript of GE 11a 2014, Lecture 6 Conduction, the lithosphere and isostacy.
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GE 11a 2014, Lecture 6Conduction, the lithosphere and isostacy
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To zero’th order, the earth’s surface is bimodal in height with respect to sea level
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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
MoonMars
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The Catastrophists view of the North Atlantic
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Cartoon of crust and lithosphere on the board…
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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 box
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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.
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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
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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
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Case 2: Sun’s accretion, continued:
Age ≤ 0.5MsxV2
3.8x1026 J/s
Age ≤ 1015 s ~ 30 Million years
Potential energy = -GMimj
Rji
Total mass M at center-of-masslocation, i
Component particle mass mat location j
Rji
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 equalto the escape velocity from the Sun today:
(plus any contained in rotationor other motion of cloud)
V = (2GMs/R)0.5 = 618 km/s
i0.5miv2 = 0.5Ms(6.18x105)2
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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
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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 balanceof surface
t0t1t2
0
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
Melting point of rock
Jheat = k(dT/dx)
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Note that conduction also leads to a change in rheology between interior and outer shell
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Rayleigh number = Buoyancy
Viscous drag XMomentum diffusivity
Thermal diffusivity
acceleration Thermal expansion
Kinematic viscosity Thermal diffusivity
Length scale
Temperature contrast
If > ~1000, convection ensues. The mantle is ~106
What are the dynamics of the hot, viscous (fluid like) interior?
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A numerical model of whole-mantle convection in a2-D earth