heat transfer as a key process in earth’s mantle: new measurements, new theory anne m. hofmeister
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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
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