cooling plate depth vs. age

Cooling Plate Depth vs. Age~120 km

Temperature vs. Depth vs. timeErfFor 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 300C temperature change)

Plate contraction: average temperature change ~600C between plate & asthenosphereFractional density change between lith (cold plate) & asth:

Contraction:Thus, a 100 m.y. old ~100km thick plate will have contracted about 600m vertically due to its cooling

Isostasy- concept of floatingLithosphere floats on underlying mantle, surface relief is compensated by deeper root

Plate cooling & depth cartoon

Plate contraction of 100 m.y. old lithosphereThus, a 100 m.y. old ~100km thick plate will have contracted about 600m vertically due to its coolingBut we see roughly 3km of deepening, not 600m. What gives?

Balance mass in columnsPredicts 2km subsidencefor 100km-thick lithosphere -- closer, but we see 3km! Whats Missing?Mass in each column is the same

Can Find Isostatic Effects in Several WaysMass added = mass displacedSame mass in each column of mantle + lithosphere + waterPressure 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 displacedMaybe easiest conceptually, but hardest mathematicallyModel consistent with observations

(2) Same mass in each columnUsually leads to easier math

(3) Same mass displaced...If possible to do, is shortest mathmass deficitbalancesmass excess

(4) Same pressure at depth of compensation(same math as for equal mass in columns,except for extra g in all terms)Equal pressure at the depth of isostatic compensationBase of Lithosphere is Compensation depthPressure = weight of overburden

Rheologic implications of isostasyWhat does the existence of isostasy imply about mechanical behaviour of lithosphere & underlying mantle?

If oceanic lithosphere is denser than underlying mantle, why doesnt it just sink???

Heat Flow qFor rocksThermal conductivity kTypical 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 Kelvins estimate for age of EarthIf 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)