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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
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 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!!!)