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GEOMAGNETISM: a dynamo at the centre of the Earth

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

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

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

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

Mantle

K40

O

H

inner core

latent heat

Fe

S

Si

- Pressure is nearly hydrostatic:
- Convective velocity >> diffusion…
- …means core is well mixed
- …including entropy
- Temperature is adiabatic

Thermodynamic definition:

Hydrostatic pressure:

Seismic parameter:

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:

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!

dynamo

conduction +

convection

electricalheating

buoyancy

expansion

Dissipation gives entropy gains:

- thermal conduction
- electrical conduction
Offset by entropy losses if Tin>Tout

“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

- Cooling and contraction releases a significant amount of Earth’s gravitational energy
- Freezing also releases gravitational energy
- Is this available to the dynamo? Some think so
- But only about 5% is available

- 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

Drop temperature for change in volume

From the Maxwell relation

Heat released

Divide by specific heat released:

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

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

Entropy balance: choose LHS and find cooling rate and radioactive heating h

Energy balance: find heat flux from cooling rate and radioactive heating h

Earth’s heat budget:

Crustal radioactivity9 TW mostly lower crust

mantle radioactivity 25 TWchondritic composition

core radioactivity0 TWiron meteorites, chemistry

cooling 10 TW includes core, mantle

TOTAL 44 TW Surface heat flux

Cooling rate: 36 K/GyrFrom core 3 TW

MODELdTc/dt dri/dtICage QLQSQ

K/Gyr km/GyrMaTWTWTW

GAMP02214 141428810.210.921.6

LPL97234 15502638.414.223.0

NOIC565 0.028.828.8

Comparison between 3 models of Gubbins et al 2002;

LaBrosse et al 1997 (modified); and a model with no inner core (L=0)

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

Mantle

K40

O

H

inner core

latent heat

Fe

S

Si

- 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 Fe+8%S+8% O can explain the density of the liquid outer core
- When Fe+8%O mixture freezes, the O is left in the liquid
- This provides the source of buoyancy for compositional convection

- 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.

r%Dr

Solid iron13.16

8% S/Si12.763.0%0.40

Melting12.521.8%0.24

8%O12.172.8%0.37

Ideal solutions theory predicts densities well, but not diffusion

constants or free energies

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

Find the cooling rate and radioactive heating from the entropy balance

And find the heat flux from cooling rate and radioactive heating

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

- 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