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Temperature vs. Depth vs. time—Erf
For Plates (rocks), cooling skin thickness L=10km x (Age[m.y.])1/2
What about Seafloor Depth due to cooling?
Cooling rocks makes them denser:
is the coefficient of thermal expansion (units of inverse temperature)
Cooling rocks makes them contract (which is why they become denser):
For rocks, typically ~3x10-5 °C-1 (e.g. ~1% volume change per 300°C temperature change)
Δρ=−ρΔT
or fractional density change Δρρ
=−ΔT
ΔLL
=ΔT3
(fractional contraction in each direction)
Plate contraction: average temperature change ~600°C between plate & asthenosphere
Fractional density change between lith (cold plate) & asth:
Contraction:
Fractional density change Δρρ
=−ΔT =−3x10−5
°C−600°C( )=1.8%
ΔLL
=ΔT3
= 1.8%3
= 0.6% in each direction
Thus, a 100 m.y. old ~100km thick plate will have contracted about 600m vertically due to its cooling
Isostasy- concept of ‘floating’Lithosphere ‘floats’ on underlying mantle, surface relief is compensated by deeper root…
Plate contraction of 100 m.y. old lithosphere
Fractional density change Δρρ
=−ΔT =−3x10−5
°C−600°C( )=1.8%
ΔLL
=ΔT3
= 1.8%3
= 0.6% in each direction
Thus, a 100 m.y. old ~100km thick plate will have contracted about 600m vertically due to its cooling
But we see roughly 3km of deepening, not 600m. What gives?
Balance mass in columns…
ΔHtherm
L
ρm L+ ΔHtherm⎛⎝⎜
⎞⎠⎟= 1.02ρm⎛⎝⎜
⎞⎠⎟L
ΔHtherm
LΔHtherm = 0.02L
Predicts 2km subsidencefor 100km-thick lithosphere -- closer, but we see 3km! What’s Missing?
ρm1.02ρm
Mass in each column is the same
Can Find Isostatic Effects in Several Ways
(1) Mass added = mass displaced
(2) Same mass in each column of mantle + lithosphere + water
(3) Pressure at depth of compensation is uniform (similar idea to idea that mass of each column is the same – same overburden implies same pressure at the ‘depth of compensation’)
Mass added = mass displaced
Hot
2.5 km
Cool
Htherm
1= w Htherm / m
1
Maybe easiest conceptually, but hardest mathematically
ΔHtotal = ρwΔHthermal 1+ρwρm
+ρwρm
⎛
⎝⎜
⎞
⎠⎟
2
+ρwρm
⎛
⎝⎜
⎞
⎠⎟
3
+ ...⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
= ρwΔHthermal1
1− ρwρm
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
= ΔHthermalρm
ρm − ρw
⎡
⎣⎢
⎤
⎦⎥~
3.32.3
ΔHthermal
ρwΔHthermal = ρmε1 ⇔ extra subsidence ε1 =ρwρm
ΔHthermal
Fractional density change Δρρ
=−ΔT =−3x10−5
°C−600°C( )=1.8%
The ΔHthermal will be filled up by water, which has extra mass...
Model consistent with observations…
(2) Same mass in each column
dw
L
ρm L+dw⎛⎝⎜
⎞⎠⎟= 1.02ρm⎛⎝⎜
⎞⎠⎟L+ρwdw
L
ΔHtherm = 0.02L
ρm1.02ρmρm dw⎛
⎝⎜⎞⎠⎟−ρwdw = 1.02L−L⎛
⎝⎜⎞⎠⎟ρm = ΔHthermρm
dw = ΔHthermρm
ρm−ρw= ΔHtherm
3.32.3
dwρw
Usually leads to easier math
(3) Same mass displaced...
dw
L
ρm −ρw( )dw = − ρm −1.02ρm⎛⎝⎜
⎞⎠⎟L
= ρm0.02L= ρmΔHthermL
ρm1.02ρm
dw=ΔHthermρm
ρm−ρw=ΔHtherm
3.32.3
dwρw
If possible to do, is shortest math
mass deficitbalancesmass excess
(4) Same pressure at depth of compensation
dw
L
ρmg L+dw⎛⎝⎜
⎞⎠⎟= 1.02ρm⎛⎝⎜
⎞⎠⎟gL+ρwgdw
Lρm
1.02ρm
dwρw
(same math as for equal mass in columns,except for extra g in all terms)
Equal pressure at the ‘depth of isostatic compensation’
Base of Lithosphere is Compensation depth
Pressure = weight of overburden
Rheologic implications of isostasy
(1) What does the existence of isostasy imply about mechanical behaviour of lithosphere & underlying mantle?
(2) If oceanic lithosphere is denser than underlying mantle, why doesn’t it just sink???
Heat Flow q
q=-kdTdz
≈-kTm
2 κt
k : 3
W
m−°C=3
mWkm−°CFor rocks
Thermal conductivity k
Typical heatflow ~50mW/m2 [between 30-100mW/m2]
Old unit: 1 heat flow unit = 1cal/cm2-s ~ 42 mW/m2
(still in fairly common use, perhaps because Earth surface heatflow is typically of order 1 HFU)
Predicted heat flow scales with (age)-1/2
Lord Kelvin’s estimate for age of Earth…
q=-kdTdz
≈-kTm
2 κt~
470Wm2
age Ma[ ]
If continental heatflow is roughly 0.070W/m2 (70mW/m2),
Then this expression would suggest an age of the continents (=Earth) of ~45Ma.
We now know Earth is ~4.55Ga old (1000 times older)…
Because Kelvin neglected ?
(As a sidenote, Kelvin independently determined the age of the sun by assuming its energy source was the energy released by gravitational collapse – and also came up with ~40 Ma. The agreement between these two independent (mis-) estimates of the age of the solar system is what made him so certain he was right…)
45Ma =470W
m2
70Wm2
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
Ma[ ]