GEO 5/6690 Geodynamics 10 Sep 2014

© A.R. Lowry 2014Read for Wed 10 Sep: T&S 150-171

Last Time: Radiogenic Heating; Topography

Radioactive decay of crustal K, Th, U adds to surface heat flow; heat production A (W/m3) is a source such that:

Material Properties of Rocks:• Heat Production A: - Only composition-dependent - A(granites) >> A(mafics) >> A(mantle rocks) - Normal crustal fractionation processes ~ exponential decay of A with depth

Topography can be handled in Fourier domain!

€

r∇ •

rq = A

€

∂2T

∂z 2= −

A

k

€

A = A0 exp −lrad z( )

€

qs = q0 + lrad A0

Next Journal Article Reading:For Monday Sep 15: Karlstrom et al. (2013)Geology 41(4), 459-498.

The Geotherm Cont’d:

Up to now, we’ve focused on the geotherm insteady-state…

However temperature is not always constant!

€

q = −kdT

dz

Recall the 1Dsteady-state heatequation in theabsence of radiogenicheating:

So, negativeconductivity anyone?

If boundary conditions of temperature or heat flow change with time, need to generalize Fourier’s lawto include time. This leads to the Heat Equation:

where is thermal diffusivity (m2/s), is mass density (kg/m3), c is specific heat (J kg-1 °K-1).

In 1D:

€

r∇ 2T −

1

κ

∂T

∂t= −

A

k

€

κ =k

ρc

€

∂2T

∂z 2−

1

κ

∂T

∂t= −

A

k

Example I: Subduction Initiation

t = 0

t = 3 t = 2

t = 1

Example II: Subduction Cessation?

Example III: “Instantaneous” Rifting (McKenzie EPSL 1978)

Basically a cooling model:

crust

mantlelithosphere

crustmantle

lithosphere

crust

mantlelithosphere

Elevation h decays ~ exponentially as

where is stretching factor; characteristic time is

€

=l2

π 2κ

€

h(t) = h0

β

πsin

π

β

⎛

⎝ ⎜

⎞

⎠ ⎟exp −

t

τ

⎛

⎝ ⎜

⎞

⎠ ⎟

l

How would we measure this?

• ~1-2 km over 10-100 Myrs 0.1 mm/yr.You’ll never see this with GPS!

• Need measurements of elevation change on geologicaltime scales!!!

• One approach is to look at the sedimentary record, e.g.,backstripping: Isostatic balance of water + sediment+ crust + mantle. After correcting for sedimentcompaction (can predict from load) and sea level change, get elevation change!

The Adiabat (the “other” part of the geotherm): approximately describes mean temperature in convecting systems

Dep

th

Temperature00 1200-1400 ºC

50-300 km

ConductiveC

onvective

Adiabat: Temperatureincreases with pressurebut energy doesn’t change

The Adiabat:

What happens when mantle material moves up or downwithin a vertical column of flow?

Assume:

• No transfer of heat energy into or out of volumetric elements (constant entropy s)

• Compressibility , coefficient of thermal expansion are independent of pressure P, temperature T

• Acceleration of gravity, g, is independent of depth

• No phase changes in the material

s

Definition of compressibility:

€

≡1

ρ

∂ρ

∂P⇒ ρ = ρ 0 exp βP( )

Change in pressure:

€

dP

dz= ρg⇒

P = −1

βln 1− ρ 0gβz( )

ρ =ρ 0

1− ρ 0gzβ

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

Change in entropy:

€

ds =c

TdT −

α

ρdP∴ ds = 0⇒

dT

dP

⎛

⎝ ⎜

⎞

⎠ ⎟s

=αT

ρc

Then a change in pressure (i.e. depth) yields a change intemperature:

€

dT

dz

⎛

⎝ ⎜

⎞

⎠ ⎟s

=αgT

c

(No heat energy transfer This is not convection!!!)