Transcript Slide 1

Heat Transfer as a Key
Process in Earth’s Mantle:
New Measurements, New
Theory
Anne M. Hofmeister
Collaborators
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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)
Outline
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Background
Heat transfer via vibrations
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Heat transfer via radiation
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A model for klat
Laser – flash data on klat (T)
Implications for magma genesis, Transition Zone
A model incorporating grain size
Does radiation or rheology have more impact?
Implications for the Lower Mantle
Merging Geological & Geophysical
Constraints
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Mantle convection is multiply layered.
The global power is low with no secular delay
Important Principles
Heat = Light
Macedonio Melloni (1843)
Photo Credit: www.Corbis.com
dk
dT
< 0 is destabilizing
dk
dT
> 0 is stabilizing
Dubuffett et al.(2002)
What drives convection?
buoyancy
heat diffusion viscous damping
vs.
&
which is more important?
Baal
Jehovah
Rheology
Thermal Conductivity
Momentum Eq.
Elliptic Eq.
Temperature Equation
Quasi hyperbolic nonlinear
Parabolic Equation
Credit: D. Yuen
Thermal conductivity is most important
property because it controls the
temperature, which then determines
the other physical properties.
k(T)
temperature
heat capacity
thermal expansivity
viscosity
density
To model convection we need:
k0 (ambient temperature and pressure)
k lat
P
k lat
T
k rad
T
Why is a model for k needed?
olivine (001)
and polycrystals
6
Chai et al.
(1996)
5
Kanamori et al.
(1968)
Beck
et al. (1978)
Scharmeli (1982)
k
4
tot
W/m-K
Kobayashi (1974)
Fa 8
3
Fa 13
Katsura (1995)
peridotite
Tommassi et al. (2001)
2
Schatz & Simmons (1972)
polycrystalline forsterite
400
600
800
1000
1200
Temperature, K
1400
1600
because of
crummy data!
Heat Transfer via Vibrations (phonons)
Debye (1914) used Claussius’
kinetic theory of gases to
relate the thermal conductivity
of a solid to the collisions
within its phonon gas:
1
2
k
ci ui  i

3 vib. modes
where ci is the heat capacity of the ith mode
ui is the group velocity
i is the mean free lifetime between collisions
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
X = Ae(-t)cos (wt)
Amplitude
0
A damped
harmonic oscillator
has a lifetime:
1


-A
Time (t)
Examples of vibrations
underdamped
damped
Heat Transfer via Vibrations (phonons)
+
mean free gas theory
damped harmonic oscillator model
gives
k0 

3MZ
CV u
2
1

(Hofmeister, 1999; 2001)
where  is obtained from IR reflectivity data
IR Spectrometer
Lifetimes ( = 1/)
are obtained from IR peak widths
1
-Mg SiO
2
100
4
0.8
80
0.6
60
Reflectivity
0.4

2
0.2
0
200
400
Dielectric
Function


2
FWHM
600
800
-1
Frequency, cm
1000
40
20
0
Let’s test the model against reliable
data
Compositional dependence of klat
60
MgO
Oxides
50
40
Calculated k
(W/m-K) 30
stishovite
SiO
Al O
2
2
Osako &
Kobayashi
1979
3
Yutatake &
Shimada
1976
20
CaO
10
TiO
2
MgSiO -PV
0
0
3
10
20
30
40
50
Measured k (W/m-K)
60
Compositional dependence of klat
10
spinel
Silicates
9
YAG
8
Calculated k,
W/(m-K) 7
old
new
TiO
-Fe SiO
grossular andradite
6
95
2
k0= C
2
4
5
MgSiO perovskite
3
5
magnetite forsterite
gahnite
Fo Fa
4
Gr An
30
70
Fo Fa
50
teph
3
2
90
monticellite
10
50
fayalite
solid solution aluminous garnets
2
3
4
5
6
7
8
9
Measured k, W/(m-K)
mostly from Horai (1971) and Fujisawa et al. (1968)
10
Pressure dependence of klat
0.3
NaCl
Calculated
d (ln k)/dP,
(ave. of 6
studies)
-1
GPa
NaClO
0.2
3
Pressure de
1  4 Th
 ln klat 
 3
P
KT
d (ln k)/dP =
olivine (ptgs)
Chai et al. (1996)
0.1
MgO
quartz
0
0
opx
stishovite
coesite
0.1
olivine and forsterite
0.2
0.3
-1
Measured d (ln k)/dP, GPa
predict :
 ln klat 
 2.5% /GP a
P
for MgSi perovskit e
To understand Earth processes,
we need to make measurements at high T
k (T )
D(T ) 
 CP
http://www.math.montana.edu
A laser-flash apparatus
near-IR
detector
furnace
Sample
under
cap
cap
support
CO2 laser
cabinet
How a laser flash apparatus works
fayalite at 1000o C
CO2 laser
pulse
Signal
fit
t half
detector
output
D  0.139
L2
thalf
detector
emissions
Sample
in furnace
CO2 laser
Time, ms
Advantages of LFA
6
Basalt
5
Signal
4
Heat transfer
by phonons
Signal/V
3
2
1
500oC
0
10
-1
9-1000
-500
0
500
1000
1500
2000
2500
3000
3500
Time /ms
8
7
Signal/V
6
signal
• Rapid and
accurate
• Contact free:
no power
losses from
cracks
• Phonon
component is
separated
from radiative
transfer
effects
Obsidian
5
4
phonons
3
2
photons
1
500oC
0
-1
-200
-100
0
100
200
300
Time /ms
time
400
500
600
700
Thermal diffusivity from lattice vibrations only
SrTiO 3
2.5
MgO ceramic
perovskite
D
2
(Mg,Fe)Al O
mm /sec
2
4
spinel
1.5
Diopside
0.5
400
800
1200
1600
2000
Mantle
Olivine
Mantle
garnet
Temperature, K
Once Dlat/T = 0, Dlat no longer effects convection.
More laser-flash results:
glass has low thermal diffusivity
1.4
A lb ite
1.2
D
1
2
m icro clin e
m m /sec
0.8
0.6
san id ine
0.4
O bsid ian
A lb ite glas s
2 00
400
600
800
1000
Tem perature, K
12 00
14 00
Transient
melting
experiments
1
Anthill garnet
0.9
D
2
mm /s
Change
upon
melting
0.8
0.7
Hawaii basalt
0.6
0.5
Iceland
part glass
glassy Hawaii
0.4
400
600
800
1000
Temperature, K
1200
1400
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 Mantle
Velocities in the Transition Zone
cannot be explained by adiabatic gradients or
by steep conductive temperature gradients
(super-adiabatic).
Velocity (km/sec)
Lower
Mantle
Transition
Zone
Upper
Mantle
Depth (km)
0
0
2
4
6
8
12
14
Deepest
Samples
Upper
Mantle
500
10
Transition Zone
Lower Mantle
1000
V
s
V
p
1500
2000
PREM (Anderson, 1989)
k (W/m-K)
o
0
0
2
4
U.M.
6
8
10
Temperature (K)
Olivine or Opx or Cpx
500 Majorite
Garnet
T.Z.
Silicate
Spinel
L.M.
1000
Silcate
Perovskite
Depth (km)
Depth (km)
0
500
1000
1500
T.Z.
2500
3000
adiabatic
U.M.
500
2000
metastable
extension
sub-adiabatic
L.M.
1000
1500
1500
Temp. is equivocal
because the phase
trans. has dT/dP~0
adiabatic
2000
k0 for mantle minerals
2000
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 superadiabatic T gradient, which is not supported by seismic velocities.
Temperature (K)
Depth (km)
0
500
1000
1500
T.Z.
2500
3000
adiabatic
U.M.
500
2000
metastable
extension
sub-adiabatic
L.M.
Also, nearly constant
temperatures suggest
buoyancy/ instability of
the Transition Zone:
1000
1500
Temp. is equivocal
because the phase
trans. has dT/dP~0
adiabatic
Mantle avalanche ???
2000
Alternatively, a chemical gradient exists across
the transition zone (Sinogeikin and Bass, 2002).
Then, the temperature gradient is unconstrained.
Layered mantle convection is implied.
Radiative Transfer
Hot Gas
Cool Dust
Shells in the Egg Nebula
Credit: R. Thompson (U. Arizona) et al., NICMOS, HST, NASA
The two types of radiative transfer
diffusive
cold
direct
hot
Earth: diffusive
990 K
~1 km
1000 K
Laboratory: direct
recorder
298 K
heater
~ 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
d
• Emissivity = absorptivity (Kirchhoff, ca. 1869).
We measure absorption with a spectrometer.
2 
4dn
krad ,dif (T ) 
3
 dA
(1  e ) [ I BB ( ν, T )]
0 (1  dA) ν 2 T dν
UV
visible
50
2.5
Fa
10
40
2
2000
30
1.5
2500 K
1400
20
1
1500 K
10
6
BB
Blackbody intensity, 10 W/cm /cm
2000 K
0.5
2
True absorption coefficient, 1/cm
Diffusive radiative transfer is calculated from
spectra from the near-IR through the ultraviolet,
accounting for scattering losses at grain boundaries:
0
0
5000
15000
Wavenumbers, cm
-1
Visible region from Taran and Langer (2001)
Ullrich et al. (2002)
25000
interface reflectivity
krad depends strongly and non-linearly
on grain-size (d) due to competing
effects:
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
UM
Lithosphere
Lower Mantle
TZ
5
50
perovksite (using k for MgSiO )
k
0
lat
silicates
3
4
d = 0.1 cm
(lower mantle)
3
40
30
olivine
k
lat
MgO
rad,dif
W/m-K
k
2
1 cm
5 cm
20
0.5 cm
1 10 cm
MgO
10
0.01 cm
0
500
1000
1500
2000
2500
0
3000
Temperature, K
for ~0.1% interface reflectivity
Radiative transfer is large in
the lower mantle, which
promotes stability
But in the transition zone, the
negative T gradient of radiative
transfer is destabilizing for large
grain sizes
Does radiative transfer or viscosity
affect convection more?
work in progress by Tomo Yanagawa,
Dave Yuen, and Masao Nakada
Vertical viscosity contrast is e ~ 107
k=1
represents upper mantle
k = 1 + 4T3
credit: Tomo Yanagawa
Vertical viscosity contrast is e ~ 103
k contrast is 5
credit: Tomo Yanagawa
represents Lower Mantle
Implications
Radiative transport exerts greater control
over convection than viscosity
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Blob-like convection in Upper Mantle
An almost stagnant Lower Mantle
Is there evidence ?
Tomography show
that
the middle of
the lower mantle
is less
heterogeneous
than the rest
Masters et al.
(2000)
Possible stratigraphies for layered
convection (categorized by different
modes of heat transport)
Upper Mantle
Transition Zone
Lower
Mantle
slab
Equatorial Section
N
Lower mantle
L= 2 flow
Polar
Section
Does the Earth’s engine lack
sufficient vigor to produce whole
mantle convection?
Strong radiative transfer in the lower mantle
limits strong convection.
The current model for the global heat
flux assumed constant k and thus
overestimated power:
Global Power
70
140
Binned heat flux, mW/m 2
Oceanic Flux
120
60
Half-space cooling model
-1/2
J = 501 t
-1/2
= k T (t)
0 m
100
50
Global
Power,
32 TW from
Curve Fitting
80
J = 134 t
40
-0.19
TW
30
60
Binned Data
Pollack et al. (1993)
40
0
40
80
6
120
20
160
31 TW at mid-ocean
Age, 10 yr
Half-space cooling model with constant k gives 44 TW.
Analysis of the raw data gives 31 TW
Geologic evidence for weak
convection
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A global power of 31 TW is consistent with
an enstatite chondrite model of the Earth,
which also explains its O isotopes and
huge Fe core (Lodders, Javoy).
The long-standing existence of basaltic
volcanism of the oceanic crust implies
near steady-state heat expulsion.
MORB and hot-spot melting is runaway,
requires little excess heating.
Layered (weak) convection may address
different styles of upper and lower mantle
Conclusions
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Variable thermal conductivity exerts great
control over convection, more than viscosity
Mantle convection is multiply layered
Global power and estimated bulk
compositions agreeing implies that Earth
cools as radioactivity decreases
Radiative transport is a key process in the
Earth, as is surmised for the Universe