GE 11a 2014, Lecture 6Conduction, 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

MoonMars

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

• 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

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:

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

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

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)

Note that conduction also leads to a change in rheology between interior and outer shell

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?

A numerical model of whole-mantle convection in a2-D earth