Heat Flow in Oceanic Crust

DefinitionsDefinitionsMeasuring heat flow

Kelvin and the age of the EarthRadioactivityy

Continental heat flow (1)Oceanic heat flow (1)

Global budget (1)Continental vs oceanic heat flow

Plate model for the oceansContinental heat flow (2)

Gl b l b d t (2)Global budget (2)Constraints on the temperature regime

ObjectivesHow do we determine temperature deep in the Earth? What is the Earth’s energy budget?What is the source of energy for geological processes? How does temperature controls physical properties and

i ?tectonic processes?

Thermodynamics1st law: heat is a form of energy dU = dQ -PdV ΔQ = C ρ ΔT 2nd law (simple form): heat goes from hot to cold (Entropy can only increase unless work is done. Work can be extracted from system only if there are a cold and a hotextracted from system only if there are a cold and a hot source). Conduction of heat (Fourier): q = -λ grad TConduction of heat (Fourier): q λ grad T Other mechanisms of heat transport (radiation, convection).

Three mechanisms of heat transport Conduction. Transport of energy in a medium (solid, fluid, or gas) without transport of matter. Convection. Energy is transported by movement of matter. Radiation. Electromagnetic waves transport energy in vacuum or in solid or fluid at very high temperaturevacuum, or in solid or fluid at very high temperature.All three mechanisms are found in the Earth. Near the surface, i.e. in the lithosphere conduction dominates.surface, i.e. in the lithosphere conduction dominates.

How do we measure heat flux?

Some numbersk (thermal conductivity) 3 W/m/K (average for most rocks)Temperature gradient 20-30 K/km Heat flux ~ 60 mW m-2

If di d d i h d hIf gradient does not decrease with depth, temperature at 100km >2000K and temperature at CMB > 60,000K

More numbersMean heat flux 80 mW m-2

Total energy loss 44 1012 W 44 TW or 1.3 1021J/yrTotal energy in quakes 1019 J/yrTotal tidal friction < 1017 J/yrEnergy from Sun 1.8 1017 W or 5 1024 J/yr2005 World energy consumption 15 TW or 5 1020 J/yr

Thermal diffusivity κ = λ / ρ C It gives scaling between time and length for heat transportIt gives scaling between time and length for heat transportτ = L2 / κNote κ ~ 10-6 m2 /s or 31 6 m2 / yrNote κ ~ 10 6 m2 /s or 31.6 m2 / yr

Kelvin and the age of the EarthJ.W Thomson (Lord Kelvin) tried to use the present temperaturepresent temperature gradient to calculate the age of the EarthgAssumed the Earth hot initiallyCooling by conduction

Kelvin calculated 25 Myr as the age of the EarthKelvin liked that number because it was consistent withKelvin liked that number because it was consistent with his calculation of the energy budget for the sun (assuming that the sun was radiating the gravitational energy g g gyaccumulated at formation. Kelvin calculation was flawed because

He ignored radio-activity He ignored convection

Kelvin assumptions were in all the following debatesKelvin assumptions were in all the following debates about thermal evolution of the Earth. Note that his model would work to estimate the age of the gsea floor.

C d i liConductive cooling modelNote that cooling remains gsuperficial. Even after 1Gyr, there is almost no cooling deeper than 600km.

As t ~l2/κ, it would take 100Gyr for cooling to reach 6000km!6000km!

RadioactivityR.J. Strutt (4th Lord Rayleigh) 1906

fl i l bHeat flux can entirely be accounted for by radio-activityactivity. “Crust” can not be thicker than 60km!!!(Before Mohorovicic discovered the Moho)

Radioactivity

First continental heat flux measurements by Bullard (1939).

Oceanic heat flow

Surprise !!!Continental crust is radioactive and thickOceanic crust is thin with almost no heat generationOceanic heat flow < Continental heat flow ?Apparently NO difference?

Energy Budget of the earth (1)Birch (1951)Total energy loss = 30 TWH t d ti i h d it 5 W/kHeat production in chondrites = 5 pW/kgMass of earth = 6 1024 kgCoincidence?Coincidence?

Several problems (K/U ratio)Can not be in equilibrium with present heat production is heat is conducted to the surfaceconducted to the surface Note this buget is obsolete (Current estimate of energy loss is 44TW)

Q i C li H d iQuestion: Cooling vs Heat production

Cooling half space or plate

CQ = 490 ± 20 QmWm-2 Myr1/2

based on petrology andpetrology and physical properties

Where hydrothermal circulation is shut off, heat flux datadata fitshut off, heat flux datadata fit model

N i f d t fit li h lf d lNoise free data fit cooling half space model at young ages

Heat flux reflects age of oceanicHeat flux reflects age of oceanic lithosphere

Predicted heat flux

Isostatic balance

Bathymetry fits model for agesBathymetry fits model for ages <80Myr

For ages > 80 Myr, heat flux at base of plate balances heat flux at surface: no more cooling

Hotspots perturb bathymetryHotspots perturb bathymetry profiles

CONTINENTAL HEAT FLUX

Steady state thermal model forSteady state thermal model for stable continents

Heat flux and temperature gradient decrease with depth because of h pdepth because of h.p.The higher the crustal h.p., the lower Moho andthe lower Moho and mantle temperature

Heat flux variations in stableHeat flux variations in stable continents (e.g. Canadian Shield)

Qs = Qm + ∫ A dz Qm can not vary by more than +/- 3 mWm-2

Qm < 20 mWm-2 (lowest heat flux measured) Best estimates 13 < Qm < 15 mWm-2

Kapuskasing crustal sectionGrenville provinceG it d h t fl d t i iGravity and heat flux data inversion

Moho heat flux in stable continentsMoho heat flux in stable continentsLocation Heat flux (mWm‐2)Location Heat flux (mWm )

Baltic Shield (Archean) 7‐15

Vredefort (South Africa) 18

Slave (Archean, Canada) 12‐24

Kapuskasing (Superior, Canada) 11‐13

Abitibi (Superior, Canada) 10‐14p

Siberian craton (Archean) 10‐12

Dharwar (Archean, India) 11

Norwegian Shield (Proterozoic) 11Norwegian Shield (Proterozoic) 11

Trans‐Hudson‐orogen (Proterozoic, Canada) 11‐16

Grenville (Canada) 13

Kalahari (Proterozoic, South Africa) 17‐25

Calculating temperature in theCalculating temperature in the lithosphere: 1-D heat equation

Lithospheric temperaturetemperature profiles depend on crustal heat production (surface heat flow)flow)

When differences in surfaceWhen differences in surface heat flux are only due to crustal heat production , Moho temperature varies by 150 d150 degrees

Profiles are very i i hsensitive to Moho

heat flow

Uncertainty of +/- 3 mWm-2 gives +/-50km on depth to 1350 adiabatp

Mantle convection

Adiabatic temperature gradient

Rayleigh number

Boundary layers and temerature profileBoundary layers and temerature profile in convecting fluid

Balancing the b dgetBalancing the budgetOceanic heat loss Crustal radio-activityOceanic heat lossHotspotsContinental heat loss

Crustal radio activityMantle radio-activityCore heat flowSecular cooling of mantle

Oceanic heat floOceanic heat flowRaw average of all heatRaw average of all heat flux data 80 mWm-2

Noisy data at young ages y y g gbecause of hydrothermal circulation Better to rely on models

Age distribution of sea floor

Total energy loss of cooling oceanic lithosphere

Age < 80Ma, use halfAge 80Ma, use half space cooling and age distribution ~24 TWAge > 80 Ma, use constant flux 48 mWm-2

~5TW Depends very much on age distribution of sea floor.

Hot spotsHot spotsWeak heat flow anomalyWeak heat flow anomaly on hot spotsUse sea floor bathymetry y yto estimate heat input from buoyancy ~2-4TWPlate may be subducted before heat flows out

Continental heat loss: eliminating the bias in the data

Method 1. Determine average heat flux for each geological age and

eight according toweight according to areal distribution 65mWm-2

Method 2. Determine area weighted averages 63 W 263 mWm-2

Total continents (210 106

km2 = 14TW)km = 14TW)

Heat flux in CanadaBlack triangles represent heat flux sites. Note distribution.

Heat prod ction of BSE cr st and mantleHeat production of BSE, crust, and mantleU(ppm) Th(ppm) K(ppm) A(pW/kg)

BSE BSE 

Hart & Zindler (1986) 0.021 0.079 264 4.9

McDonough & Sun (1985) 0.020 0.079 240 4.8

Palme & O’Neil (2003) 0.022 0.083 261 5.1

Lyubetskaya & Korenaga (2007) 0.017 0.063 190 3.9

MORB mantle source MORB mantle source 

Langmuir et al. (2005) 0.013 0.040 160 2.8

Continental crust

R d i k  d G ( ) 6

Total BSE 20 TW

Rudnick and Gao (2003) 11.3 5.6 15000 330

Jaupart and Mareschal (2003) 293‐352

Total BSE ~ 20 TW

Continental crust 7TW Mantle ~ 13 TW

Core heat loss?Core heat loss? > 4 TW (hot spots) 4 TW (hot spots)Conductive heat flux on adiabat -> 4TWThermodynamic efficiency of dynamo y y~10% Ohmic dissipation of dynamo (0.1 -> 1 TW)10 TW ?

F Ni (2007)From Nimmo (2007)

Mantle coolingMantle heat loss 39TWCore flux 9TWMantle radioactivity 13 TWSecular cooling must provide 17 TW (52 1019 J/ )17 TW (52 1019 J/yr)Rate ~ 110 K/Gyr

From Abbott et alFrom Abbott et al. (1994)

Summary budgetMantle heat loss: 39TWMantle heat production: 13 / 4 TW

Total heat loss 46+/-2TWother 

(differentiation  13 +/- 4 TW Urey number: 0.33 +/-0 11

core heat flow (9+/‐5TW)

(differentiation, tidal, .. ) < 1TW

0.11

Radio‐activity crust 

(7+/‐1TW)

Secular cooling mantle

(18+/‐8 TW)

(7+/ 1TW)

Radioactivity mantle 

(13+/‐4 TW)( 3 / 4 )

Summary budgetMantle heat loss: 39TWMantle heat production: 13 +/-4 TW4 TW Urey number: 0.33 +/- 0.11

Flux de chaleur et regime thermique 8

Appendice E: Calcul de la temperature en fonction de la profondeur enregime stationnaire

Dans le cas d’un regime stationnaire, l’equation de la chaleur a une dimension prend la forme:

dq

dz= −A(z) (E1)

ou z est la profondeur, A(z) le production de chaleur, et le signe du flux q a ete change implicitement (zest positif vers le bas et q est defini positif vers le haut). On obtient pour le flux

q(z) = q0 −∫ z

0

A(z′)dz′ (E2)

avec q0 le flux de surface Donc le flux decroit avec la profondeur d’autant plus que les sources de chaleuravec q0 le flux de surface. Donc le flux decroit avec la profondeur d autant plus que les sources de chaleursont concentrees pres de la surface. Et pour la temperature:

T (z) = T0 +q0z

K− 1

K

∫ z

0

dz′∫ z′

0

A(z”)dz” (E3)

Si A(z) = A0 pour z < h0, on obtient:

T (z) = T0 +q0z

K− A0z

2

2K(E4)

On peut ainsi iterativement calculer la temperature pour un modele avec plusieurs couches dans lequelles sources de chaleur sont constantes.

Si les sources decroissent exponentiellement avec la profondeur, A(z) = A0 exp(−z/D):

q(z) = q0 − A0D (1 − exp(−z/D)) (E5)

et

T (z) =qrz

K+

A0D2

K(1 − exp(−z/D)) (E6)

ou qr = q0 − A0D est le flux reduit. Notez que pour q0 et qr fixes, la temperature a une profondeurdonnee diminue avec D.