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

Lecture 1How the dynamo is powered

• 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

ENERGY LOST THROUGH ELECTRICAL RESISTANCE

• Magnetic field decays in 15,000 years

• Energy loss is 1011 - 1012 W

THE MODEL

• 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

inner core

O

H

SSi

K40

Fe

Mantle

latent heat

THE BASIC STATE

• Pressure is nearly hydrostatic:

• Convective velocity >> diffusion…

• …means core is well mixed

• …including entropy

• Temperature is adiabatic

gdr

dP

GRUNEISSEN’S PARAMETER

Thermodynamic definition: 5.1

S

S

p

S

P

T

T

K

C

K

Hydrostatic pressure: SS P

Tg

dr

dP

P

T

dr

dT

g

dr

dT

T

1Seismic parameter:

r

r

drg

TrTi

exp)( i

dt

dT

Tdt

dT

T

11 i

i

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 thenthe same everywhere:

INNER CORE FREEZING

dt

dT

drdTdrdTdt

dr i

am

i

//

1

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!

outin QQ

ENERGY FLOW CHART

dynamo

electrical heating

expansion

buoyancy

conduction +convection

ENTROPY BALANCE

Dissipation gives entropy gains:

• thermal conduction

• electrical conduction

Offset by entropy losses if Tin>Tout

Ok EET

Q

T

Q

out

out

in

in

BACKUS’ IDEAL DYNAMO

min

minmax

T

TT

Q

“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

THE SECOND TWIST...

• 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

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

PRESSURE HEATING

TT

KV

S

TVV

TTK

VTKQ TTa

2

05.0

TTC

K

Q

Q

p

T

s

a

Drop temperature for change in volume

From the Maxwell relation

Heat released

Divide by specific heat released:

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

V

V

L

TT

Pm

Again the only part of this gravitational energy that is availableto 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

SUMMARY - HEAT ONLYEntropy balance: choose LHS and find cooling rate and radioactive heating h

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

hqdt

dTqqQ r

csL

hedt

dTeeEEQ

TTEE r

csLrsLkO

maxmin

11

ADIABATIC GRADIENTS

HEAT BUDGET

Earth’s heat budget:Crustal radioactivity 9 TW mostly lower crustmantle radioactivity 25 TW chondritic compositioncore radioactivity 0 TW iron meteorites, chemistrycooling 10 TW includes core, mantle

TOTAL 44 TW Surface heat flux

Cooling rate: 36 K/Gyr From core 3 TW

RESULTS FOR THERMAL CONVECTION

MODEL dTc/dt dri/dt ICage QL QS QK/Gyr km/Gyr Ma TW TW TW

GAMP02 214 1414 288 10.2 10.9 21.6LPL97 234 1550 263 8.4 14.2 23.0NOIC 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 (L=0)

COMPOSITIONAL CONVECTION

• 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

inner core

O

H

SSi

K40

Fe

Mantle

latent heat

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 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

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.

CORE COMPOSITION OF PRICE, ALFE & GILLAN (2001)

DENSITY REDUCTIONS FROM PURE IRON AT ICB PRESSURE AND

TEMPERATURE

Solid iron 13.168% S/Si 12.76 3.0% 0.40Melting 12.52 1.8% 0.248%O 12.17 2.8% 0.37

Ideal solutions theory predicts densities well, but not diffusionconstants or free energies

DISSIPATION ENTROPY

• Thermal conduction 200-500 MW/K

• Ohmic heating 50-500 MW/K

• Molecular diffusion 1 MW/K

• Round up 1000 MW/K

hedt

dT

T

GeeEEE r

csLOk

max

hqdt

dTGqqQ r

csL

And find the heat flux from cooling rate and radioactive heating

Find the cooling rate and radioactive heating from the entropy balance

FINAL EQUATIONS

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

RESULTS FOR COMPOSITIONAL CONVECTION

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