Heat Transfer as a Key Process in Earths Mantle: New Measurements, New Theory

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Heat Transfer as a Key Process in Earths Mantle
New Measurements, New Theory
Anne M. Hofmeister
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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)

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Outline

• Background
• Heat transfer via vibrations
• A model for klat
• Laser flash data on klat (T)
• Implications for magma genesis, Transition Zone
• Heat transfer via radiation
• A model incorporating grain size
• Does radiation or rheology have more impact?
• Implications for the Lower Mantle
• Merging Geological Geophysical Constraints
• Mantle convection is multiply layered.
• The global power is low with no secular delay

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Important Principles
Heat Light
lt 0 is destabilizing
gt 0 is stabilizing
Dubuffett et al.(2002)
Macedonio Melloni (1843)
Photo Credit www.Corbis.com
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What drives convection?
buoyancy
heat diffusion
viscous damping
vs.

which is more important?
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Jehovah
Baal
Thermal Conductivity
Rheology
Temperature Equation Quasi hyperbolic
nonlinear Parabolic Equation
Momentum Eq. Elliptic Eq.
Credit D. Yuen
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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
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To model convection we need
k0 (ambient temperature and pressure)
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Why is a model for k needed?
because of crummy data!
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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
where ci is the heat capacity of the ith mode ui
is the group velocity ti is the mean free
lifetime between collisions
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The formula was not very useful because the
vibrations were treated as harmonic oscillators
(i.e., non-interacting).
Instead the vibrations interact through damping !
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The Lorentz Model A damped harmonic
oscillator has a lifetime
X Ae(-?t)cos (wt)
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Examples of vibrations
damped
underdamped
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Heat Transfer via Vibrations (phonons)

damped harmonic oscillator model
mean free gas theory
gives
(Hofmeister, 1999 2001)
where G is obtained from IR reflectivity data
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IR Spectrometer
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• Lifetimes (t 1/G)
• are obtained from IR peak widths

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Lets test the model against reliable data
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Compositional dependence of klat
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Compositional dependence of klat
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Pressure dependence of klat

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To understand Earth processes, we need to make
measurements at high T
http//www.math.montana.edu
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A laser-flash apparatus
near-IR detector
furnace
Sample under cap
cap support
CO2 laser cabinet
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How a laser flash apparatus works
fayalite at 1000o C
CO2 laser pulse
fit
detector
Signal
emissions
t half
detector output
Sample in furnace
CO2 laser
Time, ms
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Advantages of LFA
Basalt

• Rapid and accurate
• Contact free no power losses from cracks
• Phonon component is separated from radiative
  transfer effects

Heat transfer by phonons
Signal
500oC
Obsidian
signal
phonons
photons
500oC
time
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Once ?Dlat/?T 0, Dlat no longer effects
convection.
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More laser-flash results glass has low thermal
diffusivity
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Transient melting experiments
Change upon melting
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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

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Implications for Earths Mantle
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Velocities in the Transition Zone cannot
be explained by adiabatic gradients or by steep
conductive temperature gradients
(super-adiabatic).
PREM (Anderson, 1989)
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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.
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Also, nearly constant temperatures suggest
buoyancy/ instability of the Transition Zone
Mantle avalanche ???
Alternatively, a chemical gradient exists across
the transition zone (Sinogeikin and Bass, 2002).
Then, the temperature gradient is
unconstrained. Layered mantle convection is
implied.
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Radiative Transfer
Hot Gas
Cool Dust
Shells in the Egg Nebula Credit R. Thompson (U.
Arizona) et al., NICMOS, HST, NASA
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The two types of radiative transfer

• diffusive

cold
direct
hot
Earth diffusive
Laboratory direct

990 K 1 km 1000 K
recorder heater
298 K 5 mm 800K
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Diffusive Radiative Transfer

• Earths 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.

d
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Diffusive radiative transfer is calculated
from spectra from the near-IR through the
ultraviolet,accounting for scattering losses at
grain boundaries
Visible region from Taran and Langer
(2001) Ullrich et al. (2002)
interface reflectivity
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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
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for 0.1 interface reflectivity
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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

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Does radiative transfer or viscosity affect
convection more?
work in progress by Tomo Yanagawa, Dave Yuen,
and Masao Nakada
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Vertical viscosity contrast is eg 107
k1
k 1 4T3
represents upper mantle
credit Tomo Yanagawa
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Vertical viscosity contrast is eg 103
k contrast is 5
represents Lower Mantle
credit Tomo Yanagawa
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Implications

• Radiative transport exerts greater control
  over convection than viscosity
• Blob-like convection in Upper Mantle
• An almost stagnant Lower Mantle

Is there evidence ?
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Tomography shows that the middle of the lower
mantle is less heterogeneous than the rest
Masters et al. (2000)
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Possible stratigraphies for layered convection
(categorized by different modes of heat transport)
Upper Mantle
Transition Zone
slab
Lower
Mantle
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Equatorial Section
N
Lower mantle L 2 flow
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Polar Section
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Does the Earths 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

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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
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Geologic evidence for weak convection

• 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

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Conclusions

• 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