Cooling Plate Depth vs. Age

~120 km

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 cooling & depth cartoon

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

Heat Flow vs. Distance (N. Atlantic & Pacific)

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[ ]