Heat Transfer as a Key Process in Earth’s Mantle: New Measurements, New

Theory

Anne M. Hofmeister

Collaborators• Janet Bowey (U. College London) • Bob Criss (Washington U.)• Paul Giesting (Notre Dame)• Gabriel Gwanmesia (U. Delaware)• Brad Jolliff (Washington U.)• Andrew Locock (Notre Dame)• Angela Speck (U. Missouri Columbia)• Brigitte Wopenka (Washington U.)• Tomo Yanagawa (Kyushu U.)• Dave Yuen (U. Minnesota)

OutlineOutlinen BackgroundBackgroundn Heat transfer via vibrationsHeat transfer via vibrations

n A model for kA model for klatlat

n Laser – flash data on kLaser – flash data on klat lat (T)(T)n Implications for magma genesis, Transition ZoneImplications for magma genesis, Transition Zone

n Heat transfer via radiationHeat transfer via radiationn A model incorporating grain sizeA model incorporating grain sizen Does radiation or rheology have more impact? Does radiation or rheology have more impact? n Implications for the Lower MantleImplications for the Lower Mantle

n Merging Geological & Geophysical Merging Geological & Geophysical ConstraintsConstraints

n Mantle convection is multiply layered.Mantle convection is multiply layered.n The global power is low with no secular delayThe global power is low with no secular delay

Important Principles

Heat = Light

< 0 is destabilizing

> 0 is stabilizing

Dubuffett et al.(2002)Macedonio Melloni (1843)

T

k

d

d

T

k

d

d

Photo Credit: www.Corbis.com

What drives convection?

buoyancyvs.

heat diffusion&

viscous damping

which is more important?

Momentum Eq. Elliptic Eq.

Temperature EquationQuasi hyperbolic nonlinear

Parabolic Equation

Thermal Conductivity

JehovahBaal

Rheology

Credit: D. Yuen

Thermal conductivity is most important property because it controls the

temperature, which then determines the other physical properties.

k(T) temperature

thermal expansivity

densityviscosity

heat capacity

To model convection we need:To model convection we need:

lat

T

k

∂∂

k0 (ambient temperature and pressure)

lat

P

k

∂∂

rad

T

k

∂∂

because ofcrummy data!

Why is a model for k needed?

2

3

4

5

6

400 600 800 1000 1200 1400 1600

ktot

W/m-K

Temperature, K

Katsura (1995)

Kanamori et al. (1968)

Schatz & Simmons (1972)polycrystalline forsterite

Kobayashi (1974)Fa 8

peridotiteTommassi et al. (2001)

Scharmeli (1982)

olivine (001) and polycrystals

Beck et al. (1978)

Fa 13

Chai et al. (1996)

Debye (1914) used Claussius’ kinetic theory of gases to relate the thermal conductivity of a solid to the collisions within its phonon gas:

∑=modes vib.

2

3

1iiiuck τ

where ci is the heat capacity of the ith modeui is the group velocity i is the mean free lifetime between collisions

Heat Transfer via Vibrations (phonons)

The formula was not very useful because the vibrations were treated as harmonic oscillators (i.e., non-interacting).

Instead the vibrations interact through damping !

The Lorentz Model

A damped harmonic oscillator has a lifetime:

X = Ae(-t)cos (t)

=

1τ

Amplitude

Time (t)

0

A

-A

Examples of vibrations

underdamped damped

Heat Transfer via Vibrations (phonons)

damped harmonic oscillator model

+

mean free gas theory

gives

(Hofmeister, 1999; 2001)

where is obtained from IR reflectivity data

=

1

3

2

0 uCMZ

k V

ρ

IR Spectrometer

Lifetimes ( = 1/)

are obtained from IR peak widths

0

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

200 400 600 800 1000

γ-Mg2SiO

4

Reflectivity DielectricFunction

,Frequency cm-1

/2π

1/=/2π=

FWHMπ2

Γ

FWHM2

=π

Let’s test the model against reliable Let’s test the model against reliable datadata

Compositional dependence of klat

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Calculated k(W/m-K)

Measured k (W/m-K)

Al2O

3

MgO

CaO

Oxides

MgSiO3-PV

TiO2

stishovite SiO

2

Osako &Kobayashi1979

Yutatake &Shimada1976

Compositional dependence of klat

Pressure dependence of klat

0 0.1 0.2 0.30

0.1

0.2

0.3

Measured d (ln k)/dP, GPa -1

Calculated d (ln k)/dP,

GPa-1

NaCl

NaClO3

quartz

olivine and forsterite

MgO

stishovitecoesite

opx

olivine (ptgs) Chai et al. (1996)

(ave. of 6 studies)

d (ln k)/dP = (1/3 + 4 γTh

)/KT

Pressure derivative of the thermal conductivity

( )

431ln lat

T

Th

KP

k γ+=

∂

∂

( )

perovskite MgSifor

/GPa%5.2ln

:predict

lat =∂

∂

P

k

To understand Earth processes, we need to make measurements at high T

PC

TkTD

)()(

ρ=

http://www.math.montana.edu

A laser-flash apparatus

CO2 lasercabinet

near-IR detector

cap

support

Sampleundercap

furnace

How a laser flash apparatus works

CO2 laserpulse

fit

detector output

Sig

nal

Time, ms

t half

half

2

139.0t

LD =fayalite at 1000o C

detector

emissions

Samplein furnace

CO2 laser

Advantages of LFA• Rapid and

accurate• Contact free:

no power losses from cracks

• Phonon component is separated from radiative transfer effects -200 -100 0 100 200 300 400 500 600 700

Time /ms

-1

0

1

2

3

4

5

6

7

8

9

10

Signal/V

-1000 -500 0 500 1000 1500 2000 2500 3000 3500

Time /ms

-1

0

1

2

3

4

5

6

Signal/V

Heat transferby phonons

photons

phonons

Sig

nal

time

Obsidian

Basalt

500oC

500oC

sign

al

Once Dlat/T = 0, Dlat no longer effects convection.

0.5

1.5

2.5

400 800 1200 1600 2000

D

mm2/sec

Temperature, K

Thermal diffusivity from lattice vibrations only

MgO ceramicSrTiO

3-

perovskite

(Mg,Fe)Al2O

4

spinel

MantleOlivine

Mantlegarnet

Diopside

More laser-flash results:glass has low thermal diffusivity

0.4

0.6

0.8

1

1.2

1.4

200 400 600 800 1000 1200 1400

D

mm2/sec

Temperature, K

Albite

Albite glass Obsidian

microcline

sanidine

Transient melting experiments

0.4

0.5

0.6

0.7

0.8

0.9

1

400 600 800 1000 1200 1400

D

mm2/s

Temperature, K

Hawaii basalt

Iceland part glass

glassy Hawaii

Anthill garnet

Change upon melting

Results from laser-flash measurements

• The thermal diffusivity of melts or glasses is lower than that of minerals or rocks

• Thus, runaway melting is a possible mechanism for magma generation in the upper mantle

• D and klat (of minerals, rocks, and glasses) are independent of T at high T

• Thus, radiative transfer is the key process inside Earths’ mantle

Implications for Earth’s MantleImplications for Earth’s Mantle

PREM (Anderson, 1989)

Velocities in the Transition Zone cannot be explained by adiabatic gradients or by steep conductive temperature gradients (super-adiabatic).

Lower Mantle

UpperMantle

TransitionZone

0 2 4 6 8 10 12 140

500

1000

1500

2000

Velocity (km/sec)

Depth (km)

VpV

s

Lower Mantle

UpperMantle

Transition Zone

Deepest Samples

k0 for mantle minerals

Low thermal conductivity is expected for the TZ, as it is rich in garnet (e.g., Vacher et al. 1998). But, low k suggests a super-adiabatic T gradient, which is not supported by seismic velocities.

0 2 4 6 8 100

500

1000

1500

2000

ko (W/m-K)

Depth (km)

Olivine or Opx or Cpx

MajoriteGarnet

SilicateSpinel

Silcate Perovskite

L.M.

T.Z.

U.M.500 1000 1500 2000 2500 3000

0

500

1000

1500

2000

Temperature (K)

Depth (km)

adiabatic

sub-adiabatic

adiabatic

L.M.

T.Z.

U.M.

Temp. is equivocal because the phase trans. has dT/dP~0

metastableextension

Also, nearly constant temperatures suggest buoyancy/ instability of the Transition Zone:

Alternatively, a chemical gradient exists across the transition zone (Sinogeikin and Bass, 2002). Then, the temperature gradient is unconstrained.Layered mantle convection is implied.

Mantle avalanche ???

500 1000 1500 2000 2500 30000

500

1000

1500

2000

Temperature (K)

Depth (km)

adiabatic

sub-adiabatic

adiabatic

L.M.

T.Z.

U.M.

Temp. is equivocal because the phase trans. has dT/dP~0

metastableextension

Shells in the Egg Nebula

Credit: R. Thompson (U. Arizona) et al., NICMOS, HST, NASA

Hot Gas

Cool Dust

Radiative Transfer

The two types of radiative transfer

diffusive direct

Earth: diffusive Laboratory: direct

990 K ~1 km 1000 K recorder heater

hot

cold

298 K ~ 5 mm 800K

Diffusive Radiative Transfer

• Earth’s mantle is internally heated and consists of grains which scatter and partially absorb light

• Because the grains cannot be opaque,

they cannot be blackbodies• The light emitted =

the emissivity x the blackbody spectrum• Emissivity = absorptivity (Kirchhoff, ca. 1869).

We measure absorption with a spectrometer.

í)],í( [

í )1(

)e1(

3

4)(

02

2

, dT

TI

dA

dnTk BB

dA

difrad ∂∂

+−

= ∫∞ −

d

Diffusive radiative transfer is calculated from spectra from the near-IR through the ultraviolet,

accounting for scattering losses at grain boundaries:

0

10

20

30

40

50

5000 15000 25000

True absorption coefficient, 1/cm

Blackbody intensity, 10

6 W/cm

2/cm

Wavenumbers, cm -1

0

1

2

0.5

1.5

2.5

2000

2000 K1500 K

BB1400

Fa10

visible UV

2500 K

interface reflectivity Visible region from Taran and Langer (2001)

Ullrich et al. (2002)

1) small grains scatter light repeatedly, providing a short mean free path, which suppresses krad

2) small grains absorb light weakly, providing a large mean free path, which inhances krad

3) small grains emit weakly which suppresses krad

krad depends strongly and non-linearly on grain-size (d) due to competing

effects:

0

1

2

3

4

5

0

10

20

30

40

50

500 1000 1500 2000 2500 3000

klat

silicates

krad,dif

W/m-K

klat

MgO

Temperature, K

MgO

perovksite (using k0 for MgSiO

3)

olivine

d = 0.1 cm(lower mantle)

0.01 cm

0.5 cm1 cm

5 cm

10 cm

LithosphereUM

TZ Lower Mantle

for ~0.1% interface reflectivity

Radiative transfer is large in Radiative transfer is large in the lower mantle, which the lower mantle, which

promotes stabilitypromotes stability

But in the transition zone, the But in the transition zone, the negative T gradient of radiative negative T gradient of radiative transfer is destabilizing for large transfer is destabilizing for large

grain sizesgrain sizes

Does radiative transfer or viscosity affect convection more?

work in progress by Tomo Yanagawa, Dave Yuen, and Masao Nakada

k=1 k = 1 + 4T3

Vertical viscosity contrast is eγ ~ 107

represents upper mantlecredit: Tomo Yanagawa

Vertical viscosity contrast is eγ ~ 103

represents Lower Mantlecredit: Tomo Yanagawa

k contrast is 5

ImplicationsImplications

Radiative transport exerts greater Radiative transport exerts greater control over convection than control over convection than viscosityviscosity

n Blob-like convection in Upper MantleBlob-like convection in Upper Mantlen An almost stagnant Lower MantleAn almost stagnant Lower Mantle

Is there evidence ?

Masters et al. (2000)

Tomography shows that the middle of the lower mantleis less heterogeneous than the rest

Possible stratigraphies for layered convection (categorized by different

modes of heat transport)

Upper Mantle

Transition Zone

Lower

Mantle

slab

Equatorial Section

Lower mantleL= 2 flow

N

PolarSection

Does the Earth’s engine lack Does the Earth’s engine lack sufficient vigor to produce whole sufficient vigor to produce whole

mantle convection?mantle convection?

The current model for the global The current model for the global heat flux assumed constant heat flux assumed constant kk

and thus overestimated power:and thus overestimated power:

Strong radiative transfer in the lower Strong radiative transfer in the lower mantle limits strong convection.mantle limits strong convection.

40

60

80

100

120

140

0 40 80 120 160

Binned heat flux, mW/m

2

Age, 106 yr

J = 501 t-1/2 = k0T

m(πκ )t -1/2

Oceanic Flux

=134J t-0.19

Binned Data .(1993)Pollack et al

Curve Fitting

- Half space cooling model

32 TW from

20

30

40

60

50

70

Global,Power

TW

Global Power

31 TW at mid-ocean

Half-space cooling model with constant k gives 44 TW.Analysis of the raw data gives 31 TW

Geologic evidence for weak Geologic evidence for weak convectionconvection

n A global power of 31 TW is consistent with A global power of 31 TW is consistent with an enstatite chondrite model of the Earth, an enstatite chondrite model of the Earth, which also explains its O isotopes and which also explains its O isotopes and huge Fe core (Lodders, Javoy). huge Fe core (Lodders, Javoy).

n The long-standing existence of basaltic The long-standing existence of basaltic volcanism of the oceanic crust implies volcanism of the oceanic crust implies near steady-state heat expulsion.near steady-state heat expulsion.

n MORB and hot-spot melting is runaway, MORB and hot-spot melting is runaway, requires little excess heating.requires little excess heating.

n Layered (weak) convection may address Layered (weak) convection may address different styles of upper and lower mantledifferent styles of upper and lower mantle

ConclusionsConclusions

n Variable thermal conductivity exerts great Variable thermal conductivity exerts great control over convection, more than viscositycontrol over convection, more than viscosity

n Mantle convection is multiply layeredMantle convection is multiply layeredn Global power and estimated bulk Global power and estimated bulk

compositions agreeing implies that Earth compositions agreeing implies that Earth cools as radioactivity decreasescools as radioactivity decreases

n Radiative transport is a key process in the Radiative transport is a key process in the Earth, as is surmised for the UniverseEarth, as is surmised for the Universe