GEOMAGNETISM: a dynamo at
the centre of the Earth
Lecture 3
Interpreting the Observations
OVERVIEW
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Historical and paleomagnetic data
Historical record gives good spatial resolution
Paleomagnetic record covers a long time interval
Measurements are interpreted in terms of a core field
generated by a dynamo
• Field morphology interpreted in terms of the
dynamics
• Secular variation in terms of core flows
• Is the geodynamo unstable?
HISTORICAL RECORD
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Accuracy limited by crustal field mainly
Global coverage => good spatial resolution
Many components measured
Known dates
Short duration: 1500-2000AD (500 years)
Length of day measured and
predicted (Jackson et al. 1993)
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~1500
1586
1600
16??
1695
1715
1777
1839
1840
1840
1840
185?
1887
1900-1926
1955
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1966
1980
2001--
Start of navigational records
Robert Norman measures inclination in London
William Gilbert publishes de Magnete
Jacques l’Hermite’s voyage across Pacific
Edmund Halley measures D in Atlantic
Feuille measures I in Atlantic and Pacific
James Cook’s voyages; solves longitude problem
Gauss measures absolute intensity
Gottingen Union Observatories set up
Royal Navy exploration of Southern Oceans
James Ross expedition to Antarctica
Suez Canal built
Challenger expedition
First magnetic surveys, first permanent observatories
Carnegie burns out in Apia harbour
Proton magnetometer starts widespread aero- and
marine surveys
First total intensity satellites (POGO)
Magsat
Oersted, Champ, etc: decade of magnetic
Voyage of HMS Challenger
Halley and the “Paramour”
Historical data (after Jackson, Walker &
Jonkers 2000)
POTENTIAL THEORY: uniqueness requires
measurements on the boundary of:
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The potential V
Normal derivative of the potential =Z…
…but not F (Backus ambiguity)
North component X…
... but not East component Y
D and I (to within a multiplicative constant)
D and H on a line joining the poles
SPHERICAL HARMONIC EXPANSIONS
a
V  a 
l ,m  r 
l 1
g
m
l
cosm  hlm sin m Pl m cos 
Differentiating the potential gives the magnetic field
components
Setting r=a the Earth’s radius gives a standard inverse problem
for the geomagnetic coeficients in terms of surface
measurements
Setting r=c the Core radius gives the magnetic field on the core
surface
DATA KERNELS (Gubbins and Roberts 1983)
The magnetic potential V at radius r is an average over the whole core surface:
Br ( r, , )   Br ( r ' , ' , ' ) N (cos )d'
where is the angular distance between (r, , ) and (r’, ' , ' )
Then
 dN 
Br ( r  a )   Br ( r' , ' , ' )
 d'
 dr r a
where
c2 a 2  c2 
 dN 
N r    
 
3/ 2
 dr r a 4aa 2  2ac cos  c2 
This is the data kernel for the inverse problem of finding the vertical component
of magnetic field at the core surface from measurements of vertical component
of magnetic field at the Earth’s surface.
The data kernel for a horizontal component measurement, Nh, is found by
differentiating with respect to 
DATA KERNELS
Smoothing
constraint
Data plane
LEAST SQUARES
L1 NORM
(double exponential)
Declination AD1600
Declination AD1990
The Tngent Cylinder
PALEO/ARCHEOMAGNETIC
DATA
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Locations limited
10x less accurate than direct measurement
Rarely is the date known accurately
Hence rarely more than one location at a
time
• Record is very long duration (Gyr)
THE TIME-AVERAGED PALEOMAGNETIC
FIELD LAST 5Ma
Hawaiian data last 50 kyr
from borehole data and
surface flows
(Teanby 2001)
Critical Rayleigh number for magnetoconvection
E=10-9
AN IMPORTANT INSTABILITY?
• Nobody has yet found a dynamo working in a sphere in the
limit E  0 (Fearn & Proctor, Braginsky, Barenghi,
Jones, Hollerbach)
• Perhaps there is none because the limit is structurally
unstable
• Small magnetic fields lead to small scale convection and a
weak-field state, which then grows back into a strong-field
state
• This may manifest itself in erratic geomagnetic field
behaviour
STABILISING THE GEODYNAMO
Time scale to change B
in outer core: 500 yr
Time scale in inner
core (diffusion) 5 kyr
DYNAMO CATASTROPHE
• The Rayleigh number is fixed
• The critical Rayleigh number depends on field
strength
• Vigour of convection varies with supercritical
Ra…
• So does the dynamo action
• If the magnetic field drops, so does the vigour of
convection, so does the dynamo action
• The dynamo dies
NUMERICAL DIFFICULTIES
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At present we cannot go below E  105
The resulting convection is large scale
The large E prevents collapse to small scales…
…and therefore the weak field regime
Hyperdiffusivity suggests smaller E…
...but the relevant E for small scale flow is actually
larger
CONCLUSIONS
• We are still some way from modelling the
geodynamo, mainly because of small E
• The geodynamo may be unstable, explaining
the frequent excursions, reversals, and
fluctuations in intensity
• Is the geodynamo in a weak-field state during
an excursion?
• If not, what stabilises the geodynamo?