GEOMAGNETISM: a dynamo at the centre of the Earth Lecture 1 How the dynamo is powered Lecture 2 How...
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Transcript of GEOMAGNETISM: a dynamo at the centre of the Earth Lecture 1 How the dynamo is powered Lecture 2 How...
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