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The Earth’s magnetic field
The Earth’s magnetic field
The Earth's magnetic field crudely resembles that of
a central dipole. On the Earth's surface the field
varies from being horizontal and of magnitude
about 30 000 nT near the equator to vertical and
about 60 000 nT near the poles; the root mean
square (rms) magnitude of the vector over the
surface is about 45 000 nT.
The internal geomagnetic field also varies in time,
on a time-scale of months and longer, in an as yet
unpredictable manner. This so-called secular
variation (SV) has a complicated spatial pattern,
with a global rms magnitude of about 80 nT/year.
Secular Variation
Variation of the dipole axis represented by the location of the
North Geomagnetic Pole.
(After Fraser-Smith, 1987)
Secular Variation
Variation of the dipole moment from successive spherical
harmonic analyses.
(After Fraser-Smith, 1987)
Secular Variation
Play Shockwave Movies
The Earth’s magnetic field
Consequently, any numerical model of the
geomagnetic field has to have coefficients which
vary with time. These coefficients are computed
from observations from geomagnetic observatories
that are distributed throughout the world, and from
satellite observations (Magsat which flew in 197980, and the current Ørsted mission). They are
usually updated every 5 years to produce the
International Geomagnetic Reference Field (IGRF).
The IGRF is a series of mathematical models
describing the Earth’s main field and its secular
variation.
The Earth’s magnetic field
Each model comprises a set of spherical harmonic coefficients
(called Gauss coefficients in recognition of Gauss’s development
of this technique for geomagnetism), in a truncated series
expansion of a geomagnetic potential function:

i 1
a
m
m
m




P
cos

g
cos
m


h
  l

l
l sin m 
0 l 1 m0  r 
a
l
where  is the geomagnetic potential, a is the mean radius of
the Earth (6371.2 km) and r, , , are the geocentric spherical
coordinates (r is the distance from the centre of the Earth,  is
the longitude eastward from Greenwich and  is the colatitude
(90° minus the latitude). 0 is the permeability of free space.
The Earth’s magnetic field
m
P
The
l cos   terms are Schmidt quasi-
normalized associated Legendre functions of degree
n and order m (n1 and mn). This gives values for
the coefficients in nano-Tesla (nT).
The maximum spherical harmonic degree of the
expansion is N.
The Earth’s magnetic field
The IGRF models for the main field are truncated at
N=10 (120 coefficients) which represents a
compromise adopted to produce well-determined
main-field models while avoiding most of the
contamination resulting from crustal sources.
The coefficients of the main field are rounded to the
nearest nanoTesla (nT) to reflect the limit of the
resolution of the available data. The IGRF models
for the secular variation are truncated at N=8 (80
coefficients). This time the coefficients are rounded
at the nearest 0.1 nT/year, this time to reduce the
effect of accumulated rounding error.
The Earth’s magnetic field
The main terms of interest in the above are the
harmonics of order 0, which are zonal harmonics.
Coefficents g10, g 20, g 30 , are the coefficients of the
geocentric axial dipole, the geocentric axial
quadrupole, and the geocentric axial octupole
respectively. The other coefficients are non-zonal,
the major one of interest being the equatorial
1
1
dipole ( g 1, h1), which causes the main dipole to be
inclined to the axis of rotation by about 10.5°.
The Earth’s magnetic field
Some low-degree Legendre functions. Functions P0() to
P6() are shown in the interval –1<  < 1.
The Earth’s magnetic field
Surface projection of the geocentric axial dipole term. Yellow region indicates a
downward pointing field and green indicates an upward pointing field.
The Earth’s magnetic field
The main terms of interest in the above are the
harmonics of order 0, which are zonal harmonics.
Coefficents g10, g 20, g 30 , are the coefficients of the
geocentric axial dipole, the geocentric axial
quadrupole, and the geocentric axial octupole
respectively. The other coefficients are non-zonal,
the major one of interest being the equatorial
1
1
dipole ( g 1, h1), which causes the main dipole to be
inclined to the axis of rotation by about 10.5°.
The Earth’s magnetic field
Some low-degree Legendre functions. Functions P0() to
P6() are shown in the interval –1<  < 1.
The Earth’s magnetic field
Surface projection of the geocentric axial quadrupole term. Yellow region indicates a
downward pointing field and green indicates an upward pointing field.
The Earth’s magnetic field
The main terms of interest in the above are the
harmonics of order 0, which are zonal harmonics.
Coefficents g10, g 20, g 30 , are the coefficients of the
geocentric axial dipole, the geocentric axial
quadrupole, and the geocentric axial octupole
respectively. The other coefficients are non-zonal,
the major one of interest being the equatorial
1
1
dipole ( g 1, h1), which causes the main dipole to be
inclined to the axis of rotation by about 10.5°.
The Earth’s magnetic field
Some low-degree Legendre functions. Functions P0() to
P6() are shown in the interval –1<  < 1.
The Earth’s magnetic field
Surface projection of the geocentric axial octupole term. Yellow region indicates a
downward pointing field and green indicates an upward pointing field.
The Earth’s magnetic field
The main terms of interest in the above are the
harmonics of order 0, which are zonal harmonics.
Coefficents g10, g 20, g 30 , are the coefficients of the
geocentric axial dipole, the geocentric axial
quadrupole, and the geocentric axial octupole
respectively. The other coefficients are non-zonal,
the major one of interest being the equatorial
1
1
dipole ( g 1, h1), which causes the main dipole to be
inclined to the axis of rotation by about 10.5°.
The Earth’s magnetic field
Surface projection of the equatorial dipole term. Yellow region indicates a downward
pointing field and green indicates an upward pointing field.
The Earth’s magnetic field
The Earth’s magnetic field
I = +90o
I = 0o
I = 0o
I = -90o
This variation on timescales
of decades to hundreds or
years is known as secular
variation.
When averaged over time
scaled of several thousands
of years the magnetic field
resembles that of a dipole,
where there is a simple
relationship between
latitude and the inclination
of the Earth’s magnetic
field:
Tan I = 2 Tan l
Generation of the Earth’s magnetic field
The dynamo theory of the Earth’s magnetic field
originates from papers by Elasser and Bullard in the
1940’s that the electrically conducting core of the Earth
acts like a self-exciting dynamo, and produces the
electrical currents needed to sustain the field.
The solution of this idea in an Earth-like scenario has,
however, proven to be very difficult. This is because
the Earth has a homogenous, highly electrically
conductive, rapidly rotating, convecting fluid that forms
the dynamo. Thus the equations needed to provide a
solution to the generation of the field are, of necessity,
fluid dynamical ones. With the advent of more
powerful supercomputers major advances have been
made in recent years.
Why does the Earth have a magnetic field?
The Earth has, at its centre, a dense liquid
core, of about half the radius of the Earth,
with a solid inner core. This core is though
to be mostly made of molten iron, perhaps
mixed with some lighter elements. The
Earth's magnetic field is generated by fluid
motions in the Earth's core, from circulating
flows that help get rid of heat produced
there. The source of this heat is poorly
understood: it might come from some of the
iron becoming solid and joining the inner
core, releasing latent heat, or perhaps it is
generated by radioactivity, like the heat of
the Earth's crust. The circulation of the
molten iron in the outer core is very slow,
and the energy involved is just a tiny part of
the total heat energy contained in the core.
By moving through the existing magnetic
field, the molten iron creates a system of
electric currents, spread out through the
core. These currents create the magnetic
field.
Generation of the Earth’s magnetic field
A number of concepts are central to the
understanding of the geodynamo:
Frozen-in-field theorem: If a magnetic field exists in
a perfectly conducting medium, the magnetic field
lines be carried along with the fluid medium. This is
a central concept because it means that the
differential motions of the fluid stretch the field lines
and add energy to the field. In the case of the Earth,
however, the fluid is not a perfect conductor and the
magnetic field will therefore diffuse away with time.
To overcome this diffusion dynamo action is
necessary to add energy back into the system.
Generation of the Earth’s magnetic field
Poloidal and Toroidal fields: Toroidal fields have no
radial component and cannot, therefore, be
observed at the earth’s surface whereas poloidal
fields do have a radial component. The
geomagnetic field at the Earth’s surface is therefore
poloidal.
A central issue in geodynamo models is how it’s
possible to generate a toroidal field from a poloidal
one and vice-versa as the feed-backs between the
two provide the necessary energy to maintain the
geodynamo.
Generation of the Earth’s magnetic field
Production of a toroidal
magnetic field in the
core: w-model
a) an initial poloidal field
passing through the
Earth’s core is subjected
to an initial cylindrical
shear motion of the fluid.
b) The fluid motion
‘drags’ out the magnetic
field lines, and after one
complete circuit we have
generated 2 new toroidal
loops of opposite sign.
(After Parker, 1955)
Generation of the Earth’s magnetic field
Poloidal and Toroidal fields: Toroidal fields have no
radial component and cannot, therefore, be
observed at the earth’s surface whereas poloidal
fields do have a radial component. The
geomagnetic field at the Earth’s surface is therefore
poloidal.
A central issue in geodynamo models is how it’s
possible to generate a toroidal field from a poloidal
one and vice-versa as the feed-backs between the
two provide the necessary energy to maintain the
geodynamo.
Generation of the Earth’s magnetic field
Rotation due to Coriolis force
(anticlockwise in Northern Hemisphere) =
Helicity
Production of a poloidal magnetic field in the northern
hemisphere: a-model
A region of fluid upwelling interacts with the field line.
Because of the Coriolis force the fluid exhibits helicity
(rotating as it moves upwards). The magnetic field line is
carried along and twisted to produce a poloidal loop.
(After Parker, 1955)
The Alpha – Omega Dynamo Cycle
Consider an initial dipolar
poloidal field, such as in
(a). The omega-effect
consists of (b,c)
differential rotation,
wrapping the magnetic
field around the rotational
axis, thereby creating (d)
a quadrupolar toroidal field
magnetic field inside the
core. A closure of the
dynamo cycle requires a
bit of symmetry breaking,
brought about by the
alpha-effect, whereby (e)
helical upwelling and
downwelling creates loops
of magnetic field. These
loops coalesce (f) to
reinforce the original
dipolar field.
Why does the Earth have a magnetic field?
Because the Earth is spinning the
convection in the outer core is influenced
by the motion of the planet. The picture
on the right depicts region where fluid
flow is greatest within the outer core
(the core-mantle boundary is the blue
mesh; and the inner-outer core
boundary is the red mesh).
The flows form an imaginary “tangent
cylinder” due to the effects of large
rotation, small fluid viscosity and the
presence of a solid inner core within the
spherical shell of the outer core.
Why does the Earth have a magnetic field?
Because the flow lines within the outer
core form a ‘tangent cylinder’ the
magnetic field lines generated also
tend to wrap around the inner core.
On the left is depicted a snapshot of
the 3D magnetic field structure
simulated with a computer generated
field model. Magnetic field lines are
blue where the field is directed inward
and yellow where directed outward.
The rotation axis of the model Earth is
vertical and through the centre. A
transition occurs at the core-mantle
boundary from the intense,
complicated field structure in the fluid
core, where the field is generated, to
the smooth, potential field structure
outside the core. The field lines are
drawn out to two Earth radii. The
magnetic field is wrapped around the
“tangent cylinder” due to the shear of
the zonal fluid flow.
Reversals of the field
The Earths magnetic field is known to have reversed its polarity in the past:
that is the magnetic field has “flipped” to flow from North to South. These
reversals are not periodic and episodes of constant polarity vary greatly in
length. The history of reversals of the Earths magnetic field is very well
known for the past 200 million years. Magnetostratigraphy involves
measuring the pattern of reversals in a sequence of rocks. This yields a
unique fingerprint, or “barcode”, of reversals that can be matched to known
reversal records elsewhere. This fingerprint can be used to date rocks and in
correlating different rock sequences.
Why does it reverse?
Because the field is generated by
fluid flow the geometry of the field is
affected by any instabilities within
the flow. The field is time varying,
but polarity is stabilised by the
presence of an inner core. Any field
reversal must reverse the field in the
inner core and the inner core
therefore damps the effect of any
turbulence in flow in the outer core.
If instabilities build up for a long
enough period of time the flow can
induce the opposite polarity and
hence we have a reversal.
How does it reverse?
A full understanding of how the field
reverses has yet to be achieved. Two
main approaches are currently being
used to study how the field reverses.
The first is the study of the
behaviour of the field through the
magnetic records preserved in rock
sequences. However to get a full 3-d
picture of what happens multiple
studies of the same reversal are
required from multiple sites spread
out around the globe. This will take
some time to achieve. The second
approach is the use of
supercomputers to model the
behaviour of the field through time.
The model on the left depicts a
reversal in the computer model of
Glatzmaier & Roberts (1995). Note
that the field lines become chaotic
during the reversal but revert to a
dipole geometry once the reversal is
complete.
How does it reverse?
A snapshot of the 3D magnetic field
structure simulated with the
Glatzmaier-Roberts geodynamo
model. Magnetic field lines are blue
where the field is directed inward
and yellow where directed outward.
The rotation axis of the model Earth
is vertical and through the centre. A
transition occurs at the core-mantle
boundary from the intense,
complicated field structure in the
fluid core, where the field is
generated, to the smooth, potential
field structure outside the core. The
field lines are drawn out to two Earth
radii. Magnetic field is wrapped
around the “tangent cylinder” due to
the shear of the zonal fluid flow.
How does it reverse?
500 years before the middle of a
magnetic reversal.
How does it reverse?
During the middle of a magnetic
reversal
How does it reverse?
500 years after the middle of a
magnetic reversal
How does it reverse?
A snapshot of the simulated
magnetic field structure within
the core. Lines are blue where
outside the inner core and
orange within the inner core.
The rotation axis is vertical.
Preferred Reversal Paths?
Transitional VGP paths for the
upper Olduvai transition in the
Cristolo sediment.
362 transitional VGPs from 121
volcanic record of reversals
younger than 16 Ma.
After: Tric et al., 1991; Prevot &
Camps, 1993
Preferred Reversal Paths?
Equator crossings for sedimentary VGP transition paths up to
1995.
After: McFadden & Merrill, (1995)
What triggers reversals?
Polarity Lengths and Superchrons
Non-Dipole Fields in the Past?
A fundamental assumption in all plate reconstructions based on
magnetic data is that we are dealing with a time-averaged geocentric
axial dipole.
(Kent & Smethurst 1998)
Non-Dipole Fields in the Past?
The addition of varying proportions of non-dipole fields will have the effect of
changing the inclination vs latitude relationship:
Tan I = 2 Tan l.
Here the addition of quadrupole fields is modelled. (Kent & Smethurst 1998)
Non-Dipole Fields in the Past?
The addition of varying proportions of non-dipole fields will have the effect
of changing the inclination vs latitude relationship:
Tan I = 2 Tan l.
Here the addition of octupole fields is modelled. After Kent & Smethurst 1998.
Non-Dipole Fields in the Past?
The Frequency vs Inclination
distribution for Palaeozoic
Rocks is far from dipolar.
This could indicate:
•Non-dipole fields
•Shallowing of inclinations
in sedimentary sequences
•Inadequate global sampling
(i.e. – we just happen to
have sampled the lowlatitude continents)
After Kent & Smethurst 1998.
Non-Dipole Fields in the Past?
It is possible to model the distributions in terms of non-dipole
contributions to the main field.
After Kent & Smethurst 1998.