GEOMAGNETISM: a dynamo at the centre of the Earth

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GEOMAGNETISM a dynamo at the centre of the Earth

• Lecture 1 How the dynamo is powered
• Lecture 2 How the dynamo works
• Lecture 3 Interpreting the observations
• Lecture 4 Thermal core-mantle interactions

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Lecture 1How the dynamo is powered

• Gubbins, D., D. Alfe, G. Masters, D Price M.J.
  Gillan Can the Earths dynamo run on heat
  alone?
• Gross thermodynamics of 2-component core
  convection
• - both under review for Geophysical Journal
  International

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ENERGY LOST THROUGH ELECTRICAL RESISTANCE

• Magnetic field decays in 15,000 years
• Energy loss is 1011 - 1012 W

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THE MODEL

• Core cooling drives convection
• Perhaps some radioactive heating
• Inner core freezes -gt more latent heat
• and releases light material that drives
  convection through
• Release of gravitational energy

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Mantle
K40
O
inner core
H
latent heat
Fe
S
Si
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THE BASIC STATE

• Pressure is nearly hydrostatic
• Convective velocity gtgt diffusion
• means core is well mixed
• including entropy
• Temperature is adiabatic

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GRUNEISSENS PARAMETER
Thermodynamic definition
Hydrostatic pressure
Seismic parameter
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Temperature in the core is found by integrating
up from the inner core boundary, where T is the
known melting temperature
Time evolution of the (logarithm) of temperature
is then the same everywhere
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INNER CORE FREEZING
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THE FIRST TWIST...

• Conservation of energy does not equate the energy
  required with the heat lost by the magnetic
  field, in fact it does not involve the magnetic
  field at all!

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ENERGY FLOW CHART
dynamo
conduction convection
electrical heating
buoyancy
expansion
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ENTROPY BALANCE

• Dissipation gives entropy gains
• thermal conduction
• electrical conduction
• Offset by entropy losses if TingtTout

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BACKUS IDEAL DYNAMO
Efficiency

• Can be greater than unity.
• This is because the output of the heat engine,
  the electrical heating, is used again in powering
  the convection.
• A Carnot engine driving a disk dynamo achieves
  the ideal bound

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THE SECOND TWIST...

• Cooling and contraction releases a significant
  amount of Earths gravitational energy
• Freezing also releases gravitational energy
• Is this available to the dynamo? Some think so
• But only about 5 is available

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GRAVITATIONAL ENERGY

• Is calculated from the work done in assembling
  all the mass from infinity
• The gravitational force is conservative, so we
  can do this however we like
• Assemble the mass of the Earth slowly,
  maintaining hydrostatic pressure
• Then all of the gravitational energy goes into
  compaction, except for.
• a small amount caused by pressure heating

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PRESSURE HEATING
Drop temperature for change in volume
From the Maxwell relation
Heat released
Divide by specific heat released
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PRESSURE EFFECT ON FREEZING
The change in volume on freezing also releases
gravitational energy The change in volume on
freezing is related to the latent heat (L)
through the Clausius-Clapeyron equation
Again the only part of this gravitational energy
that is available to drive convection is a small
amount of pressure heating
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The increase in melting temperature caused by the
higher causes the inner core to grow a little
more The latent heat released is identically
equal to the gravitational energy change, because
of the Clausius-Clapeyron equation
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SUMMARY - HEAT ONLY
Entropy balance choose LHS and find cooling rate
and radioactive heating h
Energy balance find heat flux from cooling rate
and radioactive heating h
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ADIABATIC GRADIENTS
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HEAT BUDGET
Earths heat budget Crustal radioactivity 9 TW
mostly lower crust mantle radioactivity
25 TW chondritic composition core
radioactivity 0 TW iron meteorites,
chemistry cooling 10 TW includes
core, mantle TOTAL 44 TW Surface heat
flux Cooling rate 36 K/Gyr From core 3 TW
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RESULTS FOR THERMAL CONVECTION
MODEL dTc/dt dri/dt ICage QL QS Q K/Gyr
km/Gyr Ma TW TW TW GAMP02 214
1414 288 10.2 10.9 21.6 LPL97 234
1550 263 8.4 14.2 23.0 NOIC 565
0.0 28.8 28.8
Comparison between 3 models of Gubbins et al
2002 LaBrosse et al 1997 (modified) and a model
with no inner core (L0)
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COMPOSITIONAL CONVECTION

• Light material released at the inner core
  boundary on freezing rises to stir the core
• Energy source is Earths gravitational energy
• This changes as light material rises, heavy iron
  sinks
• Compositional convection stirs the core directly,
  there is no thermal efficiency factor

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Mantle
K40
O
inner core
H
latent heat
Fe
S
Si
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THE STORY SO FAR...

• Thermal convection cannot drive the dynamo
  because too much heat is needed
• This means we have no means of generating a
  magnetic field before the inner core formed, the
  inner core must be as old as the magnetic field
• Compositional convection can help drive the
  dynamo
• The solid inner core can include 8 S or Si to
  explain the density. When this mixture freezes,
  it all freezes.
• A liquid Fe8S8 O can explain the density of
  the liquid outer core
• When Fe8O mixture freezes, the O is left in the
  liquid
• This provides the source of buoyancy for
  compositional convection

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NEXT...

• We see if compositional plus thermal convection
  can drive the dynamo
• We estimate the cooling rates and radioactive
  heating needed by balancing the entropy
• Then we use the cooling rate and radioactive
  heating to calculate the heat flux across the
  core-mantle boundary and the inner core age.

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CORE COMPOSITION OF PRICE, ALFE GILLAN (2001)
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DENSITY REDUCTIONS FROM PURE IRON AT ICB PRESSURE
AND TEMPERATURE
r Dr Solid iron 13.16 8
S/Si 12.76 3.0 0.40 Melting 12.52 1.8 0.24 8
O 12.17 2.8 0.37
Ideal solutions theory predicts densities well,
but not diffusion constants or free energies
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DISSIPATION ENTROPY

• Thermal conduction 200-500 MW/K
• Ohmic heating 50-500 MW/K
• Molecular diffusion 1 MW/K
• Round up 1000 MW/K

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FINAL EQUATIONS
Find the cooling rate and radioactive heating
from the entropy balance
And find the heat flux from cooling rate and
radioactive heating
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THE MODELS

• E 1000 MW/K rounding up
• E 546 MW/K, heat conducted down by compositional
  convection
• E 262 MW/K Dynamo fails
• Repeat with enough radioactive heating to make
  the inner core last 3.5 Gyr

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RESULTS FOR COMPOSITIONAL CONVECTION
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CONCLUSIONS

• Compositional convection only doubles the
  efficiency of the dynamo
• With present estimates and no radioactivity in
  the core, the age of the inner core is less than
  1Ga
• The simplest way to alter this result is to
  increase the seismological estimate of the
  density jump at the inner core boundary
• At present it seems impossible to drive the
  dynamo without an inner core