Download Chapter 5: Geomagnetism

CHAPTER 5
Earth’s Magnetic Field
5.1 Introduction . Geomagnetism or the study of Earth’s magnetic field has a long history and has
revealed much about the way the Earth works. As we shall see, the existence and characteristics of the
field essentially demand that the core be made of electrically conducting material, that is convecting,
and acts as a self sustaining dynamo. The study of the field as it is recorded in rocks allows us to infer
information about the age of the rocks, track the past motions of continents, and leads directly to the
idea of sea-floor spreading. Variations in the external part of the geomagnetic field induce secondary
variations in Earth’s crust and mantle which are used to study the electrical properties of the Earth,
giving insight into temperature and composition in these regions.
The magnetic field was the first property attributed to the Earth as a whole, aside from its roundness.
William Gilbert, physician to Queen Elizabeth I of England, inferred this by making an analogy between
the behavior observed for a compass when moved around a sphere of magnetic material and the behavior
of a compass needle on Earth’s surface. His findings were published in 1600, predating Newton’s
gravitational Principia by about 87 years. The magnetic compass had been in use, beginning with the
Chinese, since about the second century B.C., but temporal variations in the magnetic field were not
well documented until the seventeenth century by Henry Gellibrand. In 1680 Edmund Halley published
the first contour map of the geomagnetic variation as the declination was then known: he envisioned the
secular variation of the field as being caused by a collection of magnetic dipoles deep within the earth
drifting westward with time with about a 700 year period, a model not dissimilar to many put forward
this century, although he did not know of the existence of the fluid outer core. A formal separation
of the geomagnetic field into parts originating inside and outside the earth was first achieved by the
German mathematician Karl Friedrich Gauss in the nineteenth century. Gauss deduced that by far the
largest contributions to the magnetic field measured at Earth’s surface are generated by internal rather
than external magnetic sources, thus confirming Gilbert’s earlier speculation. He was also responsible
for beginning the measurement of the geomagnetic field at globally distributed observatories, some of
which are still running today. The external contributions to the magnetic field arise from the fact that
Earth sits in a (small) interplanetary magnetic field and is in continual interaction with the temporally
changing solar wind. These time-varying magnetic fields also induce secondary magnetic fields in the
rocks in Earth’s crust and mantle. The internal magnetic field plays an important role in protecting us
from cosmic ray particle radiation, because incoming ionized particles can get trapped along magnetic
field lines, preventing them from reaching Earth. One consequence of this is that rates of production of
radiogenic nuclides such as 14 C and 10 Be are inversely correlated with fluctuations in geomagnetic field
intensity.
The internal magnetic field can be divided into contributions from the crust and those originating in
the fluid outer part of Earth’s core. At Earth’s surface the crustal part is orders of magnitude weaker than
that from the core, but remanent magnetization carried by crustal rocks has proved very important in
establishing seafloor spreading and plate tectonics, as well as a global magnetostratigraphic timescale.
The much larger part generated in Earth’s core exhibits secular or temporal variations (see Figure 5.3),
generally only considered to be observable on timescales greater than a year (the occasional occurrence
of geomagnetic jerks is one notable exception); shorter period variations are usually attenuated by their
passage through Earth’s moderately electrically conducting mantle. On very long timescales (about 106
years) the field in the core reverses direction, so that a compass needle points south instead of north,
and inclination reverses sign relative to today’s field. The present orientation of the field is known as
normal, the opposite polarity is reversed.
128
that measurements of the field’s direction are also required to specify the field accurately. Prehistoric
magnetic field records can also be obtained through paleomagnetic studies of fossil magnetism recorded
in rocks and archeological materials.
The magnetic field is a vector quantity, possessing both magnitude and direction; at any point on
Earth a compass needle will point along the local direction of the field. Although we conventionally
think of compass needles as pointing north, it is the horizontal component of the magnetic field that
present andinhistorical
magnetic
field
is measured
at observatories,
by surveys
on land and at
is directedThe
approximately
the direction
of the
North
Geographic
Pole. The difference
in azimuth
sea,
and
from
aircraft.
Since
the
late
1950s
a
number
of
satellites,
each
carrying
a
magnetometer
in
between magnetic north and true or geographic north is known as declination (positive eastward) and
around
Earth for
at a time,
uniform
coveragethe
than
previously
may be orbit
as much
as several
tensmonths
of degrees.
The have
field provided
also has amore
vertical
contribution;
angle
betweenpossible.
Early satellites
only measured
the magnitude
however,
was
shown inpositive
the late 1960s
the horizontal
and the magnetic
field direction
is knownofasthe
the field:
inclination
and isit by
convention
that
measurements
of
the
field’s
direction
are
also
required
to
specify
the
field
accurately.
Prehistoric
downward (see Figure 5.1). At Earth’s surface today the field is approximately that of a dipole located
◦
magnetic
field
records
can
also
be
obtained
through
paleomagnetic
studies
of
fossil
magnetism
at the center of the Earth, with its axis tilted about 11 relative to the geographic axis; the axis of thisrecorded
in rocks and
archeological
dipole penetrates
Earth’s
surface atmaterials.
a longitude of of 288.9◦ E. The magnitude of the field, the magnetic
The
magnetic
field Earth’s
is a vector
quantity,
possessinginboth
magnitudeabbreviated
and direction;
at any
point
flux density
passing
through
surface,
is measured
microTeslas,
as µT
, with
1µon Earth
−6
a
compass
needle
will
point
along
the
local
direction
of
the
field.
Although
we
conventionally
T= 10 T, and is about twice as great at the poles (about 60 µT) as at the equator (about 30 µT).think of
◦
◦
compass
as pointing
north,magnetic
it is the horizontal
of the
magnetic
field
that is directed
Inclination
rangesneedles
from +90
at the north
pole to -90component
at the south
magnetic
pole.
Contours
approximately
in the
direction
of the
Geographicfield
Pole.forThe
three
conventionally
used
of declination,
inclination,
and
magnitude
ofNorth
the geomagnetic
1990
areparameters
shown on Figure
5.2.
to
describe
the
magnetic
field
are
intensity,
B
measured
in
Teslas,
declination,
D
,
and
inclination,
I
The three parameters conventionally used to describe the magnetic field are intensity, B, declination,
(both
in
degrees).
Figure
5.1
shows
how
they
are
related
to
conventional
Cartesian
coordinates
centered
D, and inclination, I. Figure 5.1 shows how they are related to conventional Cartesian coordinates
thethe
surface
of of
thethe
Earth,
andand
oriented
North
(N),(N),
East
(E)(E)
andand
down (V):
centeredonona ameasurement
measurementlocation
locationonon
surface
Earth,
oriented
North
East
down (V):
Figure 5.1
Figure 5.1
Bh is the projection of the field vector onto the horizontal plane and Bz is the projection onto the
! h is the
vertical
axis. D
measured
clockwise
from
North andplane
can range
0 →projection
360◦ , or equivalently
projection
of isthe
field vector
onto the
horizontal
and Bfrom
onto the from
B
z is the
◦
◦
180
. Note that
except at
high
latitudes
the range from
of D is
fairly
( ∼ ±20◦ ). Ifrom
is measured
vertical −180
axis. →
D is
measured
clockwise
from
North
and can
0→
360small
, or equivalently
◦ down from the horizontal and ranges from −90 → +90◦ (because field
◦ lines can also point out
positive
−180 → 180 . Note that except at high latitudes the range of D is fairly small ( ∼ ±20 ). I is measured
the Earth).
the diagram
we have
positiveof
down
from theFrom
horizontal
and ranges
from −90 → +90◦ (because field lines can also point out
of the Earth). From the diagram we have
Bh = B cos I and Bz = B sin I
(5.1)
(5.1)
Bh = B cos I and Bz = B sin I
The horizontal projection can also be projected onto the North (x) and East (y ) axes (the directions in
which measurements are usually made), i.e.,
124
Bx = B cos I cos D
and By = B cos I sin D
(5.2)
Maps of the present day values of B, D and I (Fig. 5.2) show that the field is a complicated function
of position on the surface of the Earth although it is approximately that of a dipole located at the center
of the Earth, with its axis tilted about 11◦ relative to the geographic axis; the axis of this dipole penetrates
Earth’s surface at a longitude of of 288.9◦ E. The magnitude of the field is about twice as great at the
poles (about 60 µT) as at the equator (about 30 µT). Inclination ranges from +90◦ at the north magnetic
pole to -90◦ at the south magnetic pole. Declination may be a few tens of degrees.
129
160
140
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
Oersted initial field model 2000, declination, degrees
80
70
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
Oersted initial field model 2000, inclination, degrees
70
65
60
55
50
45
40
35
30
25
20
15
10
5
Oersted initial field model 2000, intensity, microTesla
Figure 5.2: Geomagnetic declination, inclination, and intensity for the year 2000
130
180
160
140
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
IGRF 1990 Z and its time deriv
180
160
140
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
IGRF 1990, X and its time deriv
180
160
140
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200
IGRF 1990, Y and its time deriv
Figure 5.3: X,Y, Z (contours in nT) for the International Geomagnetic Reference Field for 1990, and
their time derviatives (color, nT/year).
131
5.2 Some Physics of Magnetism . Magnetic fields are caused by electrical charge in motion. In 1819
Oersted discovered that electric currents produce a magnetic force when he observed that a magnetic
needle is deflected at right angles to a conductor carrying a current. Magnetic fields are also produced by
permanent magnets: although there is no conventional electric current in them, there are orbital motions
and spins of electrons (sometimes called “Amperian currents") which lead to a magnetization within the
material and a magnetic field outside. The cooperative behavior that leads to permanent magnetization
is a quantum mechanical effect: however for our purposes we can describe the macroscopic effects
of the observed magnetization and associated magnetic fields using classical electromagnetic theory.
Magnetic fields exert a force on both current-carrying conductors and permanent magnets.
So far we have not been very explicit about distinguishing magnetic field from magnetic induction and
magnetization. The distinction is often not clearly made and units are used interchangeably, especially
between H and B, so it is important to be careful. To simplify things we will only consider SI units.
There are three kinds of magnetic vectors:
(1) Magnetic field B, also called magnetic flux density, also called magnetic induction, measured in
T esla, although the practical units of nT, µT, and mT are also used.
(2) Magnetizing field H, measured in A/m. This is also often called the magnetic field, although it
cannot be measured directly.
(3) Magnetization M; also known as the intensity of magnetization or magnetic moment per unit
voume (M = m/V with m the magnetic moment). It is also measured in A/m.
These quantities are related through the equation
B = µ0 (H + M)
where µ0 is the permeability of free space (µ0 = 4π × 10−7 N A−2 , or equivalently H/m, or equivalently
Tm/A). The use of B and H in magnetics can be confusing, but in the absence of magnetization the
relationship is always B = µ0 H.
In older books and publications you may come across gammas (γ ) where 1 gamma = 10−5 Gauss, the
cgs unit of magnetic field H. Through the vagaries of µo in cgs units, 1 gamma can be equated to 1
nanoTesla (nT), even though one is a unit of H and one a unit of B.
While we are talking about units, let us recall that the Ampere (A) is a unit of current, and is equal
to a Coulomb of charge per second. The volume equivalent is current density J, which has units of
A/m2 . The Ohm (Ω) is a unit of resistance, and the volume equivalent is resistivity, with units of Ωm.
A reciprocal Ohm is conductance, with units of Siemens (S), and the volume equivalent is conductivity,
S/m.
Gravitational, electrostatic and magnetic forces all have fields associated with them. The field is a
property of the space in which the force acts, and the pattern of a field is portrayed by field lines. At
any point in a field the direction of the force is tangential to the field line, and the intensity of the force
is proportional to the density of field lines. Although magnetic monopoles do not exist, they provide a
useful mechanism for describing the effects of magnetic fields and the concept of a magnetic potential
analogous to gravitational and electrostatic potential. Like gravitational and electrostatic forces, force
due to a monopole follows an inverse square law. But, in the real world only magnetic dipoles exist
(which can be thought of as pairs of monopoles, although this is a fiction), and magnetic fields vary with
azimuth and dipole field strength falls off as the cube of distance.
5.3 Magnetic potential . It is useful to have a compact mathematical representation of the magnetic
field for a particular time. Under certain assumptions, the magnetic field can be written as the gradient
of a scalar magnetic potential (just as in the case of gravity). The assumptions required are basically that
132
there are no electromagnetic sources in the measurement region and no time variations in the magnetic
field. This is a good approximation in the atmosphere (since the conductivity close to the ground is
10−13 S/m there are effectively no electric currents there). If we ignore time variations in the field then
outside the Earth where the magnetic field is being generated B can be derived from a scalar magnetic
potential as
B = −∇Vm
(5.3)
Furthermore, another of Maxwell’s equations is ∇ · B = 0 (a consequence of the absence of monopoles)
and we have
∇ · ∇Vm = ∇2 Vm = 0
(5.4)
This is Laplace’s equation, which is also used to describe the behavior of the gravitational potential, and
is the starting point for what is known as a spherical harmonic analysis of the field. Of course, we don’t
make measurements of the potential itself but of its gradient, the magnetic field (equation 5.3), which,
is related to the potential by
Bx = −
1 ∂Vm
r ∂θ
and
By = −
1
∂Vm
r sin θ ∂φ
and
Bz = −
∂Vm
∂r
(5.5)
where r, θ, φ are radius, colatitude and longitude respectively.
Laplace’s equation in spherical coordinates can be solved by separation of variables and the general
solution for the magnetic potential is an infinite series of the form:
Vm (r, θ, φ) = a
n
∞ X
X
"
Pnm ( cos θ)
(ge )m
n
r n
n=1 m=0
+
(he )m
n
r n
a
+
(hi )m
n
a
+
a n+1 r
(gi )m
n
a n+1 r
cos mφ
#
sin mφ
(5.6)
where gnm and hm
n are known as Gauss coefficients. Note that there is no n = 0 term because there are
no magnetic monopoles. The e and i subscripts indicate external or internal origin for the sources, and a
is the radius of the Earth (6.371 × 106 m). Pnm are proportional to the associated Legendre polynomials,
pnm . The normalization in geomagnetic work is usually the Schmidt normalization i.e.,
Pnm
=
2(n − m)!
(n + m)!
1
2
pnm
for m 6= 0,
Pn0 =pn0
Most importantly for our purposes
P10 ( cos θ) = cos θ
P11 ( cos θ) = sin θ
(3 cos 2 θ − 1)
cos θ
P30 ( cos θ) =
(5 cos 2 θ − 3)
2
P20 ( cos θ) =
1
2
The Pn0 functions (plotted below – also, see chapter 4) are progressively wigglier functions of the
colatitude, θ, and by including higher and higher degree and order terms we can use a spherical
harmonic expansion to describe arbitrarily complicated magnetic fields.
133
P1 ( cos θ) = cos θ
P2 ( cos θ) =
P3 ( cos θ) =
1
2
(3 cos 2 θ − 1)
cos θ
(5 cos 2 θ − 3)
2
Fig. 4.7
Fig 5.4
The
Gauss
coefficients
are
determined
by
fitting
gradient
components
of equation
5.6we
to observaThese functions, plotted in Figure 4.7, are progressivelythe
wigglier
functions
of the colatitude
θ and
tions
from
magnetic
observatories
or
satellite
data
for
a
particular
time
epoch.
A
maximum
can choose the J’s so that the sum can approximate very accurately any observed dependence of V onvalue of n
to be used in the fitting procedure is usually assigned beforehand so that a finite number of coefficients
can be resolved. (A better technique is to ask 106
for smooth models so that the resulting maps do not show
unnecessary structure in locations with little data.) Perhaps the most important result of this kind of
analysis is that 99% of the field is of internal origin. The coefficients for the internal field and their rates
of change with time are given for a five-year time interval about 1990 in Table 5.1.
The most striking feature of the values for coefficients listed in Table 5.1 is that 90% of the field is
represented by the terms where n=1 (check out g10 ). The functions associated with coefficients of degree
1 have wavelengths of one Earth circumference and can be thought of as representing geocentric dipoles
along three different axes: the spin axis (g10 ) and two equatorial axes intersecting at the Greenwich
meridian (g11 ) and 90◦ East (h11 ). Fields produced by higher order coefficients are more difficult to
visualize. We sketch below the geometry of fields produced by a few coefficients where outward flux is
indicated by white and inward is denoted by black.
Fig 5.5. Note that g10 multiplies the shape denoted Y10 . g11 multiplies the shape denoted Re(Y11 ) and
multiplies the shape denoted Im(Y11 ). g20 multiplies the shape denoted Y20 and so on. The sum gives
the total potential and the observations are given by the appropriate derivatives of the potential.
h11
One really important consequence of the spherical harmonic representation is that it is not only valid
at the place where the measurements are made. The model can be evaluated anywhere where Laplace’s
equation holds: that is in the region where there are no sources. So far we have only assumed that this
is the case in the atmosphere, but a useful approximation is to suppose that we can neglect sources in
both the crust and mantle – effectively we assume that they can be considered electrical insulators with
134
Table 5.1
Gauss coefficients for the 1990 reference field (nT)
n
m
gnm
hm
n
m
∂gn
∂t
1
1
2
2
2
3
3
3
3
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
0
1
0
1
2
0
1
2
3
0
1
2
3
4
0
1
2
3
4
5
0
1
2
3
4
5
6
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
8
-29775.4
-1850.99
-2135.81
3058.23
1693.22
1314.58
-2240.19
1245.57
806.540
938.870
782.280
323.870
-422.730
141.660
-211.030
352.510
243.790
-110.780
-165.580
-37.0400
60.6900
63.9400
60.3600
-177.510
2.04000
16.7100
-96.2600
76.5600
-64.1900
3.71000
27.5500
0.940000
5.74000
9.77000
-0.460000
22.4100
5.14000
-0.880000
-10.7600
-12.3700
3.79000
3.78000
2.64000
-6.02000
0.
5410.86
0.
-2277.66
-380.030
0.
-286.500
293.270
-348.470
0.
248.080
-239.530
87.0300
-299.380
0.
47.1700
153.470
-154.450
-69.2300
97.6700
0.
-15.7800
82.7300
68.2900
-52.4800
1.79000
26.8500
0.
-81.0800
-27.3000
0.590000
20.4300
16.3800
-22.6300
-4.96000
0.
9.74000
-19.9300
7.09000
-22.1000
11.8700
11.0000
-16.0100
-10.6900
18.0208
10.56840
-12.9179
2.39650
-2.89000E-02
3.32890
-6.66770
6.19000E-02
-5.86330
0.480900
0.611800
-7.01810
0.544800
-5.53540
0.630900
-0.137700
-1.63150
-3.11570
-6.65000E-02
2.31650
1.28690
-0.182100
1.81150
1.31210
-0.171900
0.127200
1.15840
0.589300
-0.506800
-0.307200
0.626700
1.58880
0.173200
0.170700
0.292900
0.165600
-0.676300
-0.171600
0.142600
-1.12770
-3.94000E-02
-5.32000E-02
-0.484300
-0.605300
135
∂hm
n
∂t
0
-16.071
0
-15.780
-13.789
0
4.4210
1.5765
-10.5554
0
2.5595
1.8173
3.0972
-1.3785
0
-0.11950
0.46100
0.44910
1.6599
0.40840
0
0.24640
-1.3475
-3.80000E-0
-0.88120
0.45220
1.2244
0
0.61630
0.19120
0.77230
-0.52060
-0.22210
4.41000E-0
-3.43000E-0
0
0.51250
-0.20820
0.32830
0.28570
0.37420
-0.45850
-0.31540
0.60310
no permanent magnetization. Then it is possible to downward continue the field representation to the
surface of Earth’s core (r = .547a). Because the sources are much closer the higher degree (spatially
more complex) sources appear much more important there and the field looks a lot more complicated.
The spherical harmonic expansions allows us to plot the power in each harmonic degree – both at the
surface and at the core-mantle boundary (here, harmonic degree is designated by n – see figure 5.6).
Fig 5.6. Power as a function of spherical harmonic degree plotted up to degree 21 for the field
evaluated at the surface and the field evaluated at the core mantle boundary (using equation 5.6).
The usual interpretation of this figure is that harmonic degrees up to about 14 are due to the field from
the core – higher harmonics are dominated by crustal contributions so it is inappropriate to downward
continue these to the core. Clearly, the field from the core is masked by crustal contributions once
the harmonic degree gets larger than about 14. Also note that the downward-continued field is still
dominated by the dipole field (n = 1) once the crustal field is discounted.
5.4 The Dipole and Geocentric Axial Dipole Approximations . Since the geomagnetic field is predominantly dipolar and internal in origin, to a first approximation we can neglect all but the internal
dipole contributions to equation (5.6), writing
a 2
[g10 P10 ( cos θ) + P11 ( cos θ)(g11 cos φ + h11 sin φ)]
r
a 2
=a
[g10 cos θ + sin θ(g11 cos φ + h11 sin φ)]
r
VD (r, θ, φ) = a
3
(5.7)
1
1 0
The dipole moment M = 4πa
µ0 (g1 , h1 , g1 ) in a geocentric cartesian coordinate system with rotation axis
along ẑ , Greenwich meridian defining x̂ and 90◦ E defining ŷ .
An even more drastic simplification known as the geocentric axial dipole (GAD) approximation for
the field supposes that the equatorial dipole contributions g11 and h11 can also be neglected, and the field
represented as that from a dipole aligned with Earth’s rotation axis. Then we can write
136
r
1
1
1
3
# = 4πa (g 1 , h1 , g 0 ) in a geocentric cartesian coordinate system with rotation
The dipole moment M
1
1 1
µ0
axis along ẑ, Greenwich meridian defining x̂ and 90◦ E defining ŷ.
An even more drastic simplification known as the geocentric axial dipole (GAD) approximation for
the field supposes that the equatorial dipole contributions g11 and h11 can also be neglected, and the field
represented as that from a dipole aligned with Earth’s rotation axis. Then we can write
2
cos θ
!
"2 P 0 ( cos θ) = ag 0 !aa "22cos θ = µ0µMM
0 a
cos θ
VGAD = ag
(5.8)
0
01 a
01
10
cos θ = 4πr2 2
(5.8)
VGAD = ag1
r P1 ( cos θ) = ag1 r
r
r
4πr
0 3
where M is 4π
µ0 g1 a . Thus, from 5.5 (taking the three derivatives of the potential to get the three field
4π 0 3
where
M is µ0 g1 a . Thus, from 5.5
components
sinθθ
2µ
Mcos
cosθ θ
µµ
2µ00M
0 Msin
0M
and BByy =
= 00 and Bzz =
(5.9)
and
=
(5.9)
BxBx==
3
3
3
4πr
4πr
4πr
4πr3
Figure 5.7
5.7 isisaacross
crosssection
sectionofofthethe
Earth,
rotated
z-axis
coincides
magnetic
Figure
Earth,
rotated
so so
thatthat
the the
z-axis
coincides
withwith
magnetic
northnorth
and
and
if
the
field
were
truly
dipolar
it
clearly
would
not
matter
which
cross
section
we
choose
if the field were truly dipolar it clearly would not matter which cross section we choose becausebecause
a dipolea
dipole
is rotationally
symmetric
about
axisthrough
going through
thein
poles;
other words,
field
is field
rotationally
symmetric
about the
axisthe
going
the poles;
otherinwords,
By = 0.By = 0.
Fig 5.7
5.7
Fig
Consider the measurement position on the surface of the Earth shown as a dot in Fig. 5.8.
Consider the measurement position on the surface of the Earth shown as a dot in Fig. 5.8.
Using the equations for Bz and Bx , we find that
tan I =
Bz
= 2 cot θm
Bx
(5.10)
133
Fig 5.8Fig 5.8
Using the equations for Bz and Bx , we find that
Thus the inclination is directly related to the colatitude for a field produced by a geocentric axial dipole
from
the magnetic pole from the inclination of
(or g10 ). This allows us to calculate the distance away B
z
= 2 cot
θm
taninI plate
= tectonic
the magnetic field, a result which will be useful
reconstructions.
The intensity is also
Bx
related to θm because
(5.10)
Thus the inclination is directly related to the colatitude for a field produced by a geocentric axial dipole
1 to µ
1 from
1 from the inclination of
µ0 Mthe magnetic
(or g10 ). This 2allows2 us
calculate
the distance away
pole
0M
( sin 2 θm + 4 cos 2 θm ) 2 =
(1 + 3 cos 2 θm ) 2
(5.11)
B = (B + B ) 2 =
3
3
the magnetic zfield, xa result4πr
which
will be useful in plate 4πr
tectonic
reconstructions (on a practical note,
when doing an inverse cotangent calculation, it is useful to know that cot −1 (x) = π/2 − tan −1 (x)). The
5.6 Induced and Remanent Magnetization . The magnetic properties of rocks are dependent on the
intensity is also related to θm because
state of charged particles associated with the crystal structure – motions of and interactions among
the electrons generate the magnetic
are two basic kinds
magnetization: induced
1 moment.
1
1
µ0 M There
µof0 M
2
2
2 =
= (Bz2 +magnetization.
Bx2 ) 2 =
(
sin
θ
+
4
cos
θ
)
(1 +magnetic
3 cos 2 θmfield
) 2 the
(5.11)
m
m
magnetization andBremanent
In
the
presence
of
an
externally
applied
4πr3
4πr3
magnetization of a substance may be written as the sum of an induced and a remanent (or spontaneous)
Because of difficulties in measuring the intensity of the field (see below) this equation is less useful than
component.
equation 5.10.
#R
# =M
#I +M
M
# R is generated by the material
The remanent component, M
137 itself, and without this contribution there
would be no possibility of a rock recording the paleomagnetic field. The constitutive relation, which is
commonly written by paleomagnetists as
#
# I = χH
M
5.5 Magnetic Poles and Dipoles and their Potentials . This section is intended to show some of the
parallels between the magnetic potential and the gravitational potential by using two monopoles to
describe a dipole. In 1785 Coulomb showed that the force between the ends (or magnetic poles) of long
thin magnetized steel needles obeyed an inverse square law. Thus supposing that monopoles existed
and we have two with strengths p1 and p2 we can write an inverse square law for the force between
them, a Coulomb’s Law for magnetic poles
F (r) = K
p1 p2
.
r2
By analogy with the gravitational case we can define the magnetic field associated with a monopole p as
B(r) = K
p
r2
µ0
In SI units K = 4π
where µ0 = 4π × 10−7 N A−2 (or equivalently henry/meter) is the permeability
constant.
The magnetic potential V at a distance r from a pole of strength p is defined in terms of the work
required to move a pole of unit strength from infinity to position r
Z
r
V =
Bdr =
∞
µ0 p
4πr
In contrast to electrostatic charges, magnetic poles cannot exist in isolation: each positive pole must be
paired with a corresponding negative pole to form a magnetic dipole. We can construct the potential for
a dipole by summing the potential for two equal but opposite poles p and −p located a distance d apart,
and then letting their separation become infinitesimally small compared with the distance to the point
of observation.
Figure 5.9: Building a magnetic dipole from two fictitious monopoles
Referring to figure 5.9 we can calculate the potential V at a distance r from the mid-point of a pair
of poles in a direction that makes an angle θ to the axis passing through both poles p and −p. if the
distances to the respective poles are r+ and r− respectively the net potential at (r, θ) will be
138
V =
µ0 p 1
1 µ0 p r− − r+ −
=
4π r+
r−
4π
r+ r−
Now we suppose that d r and use the approximations
d
cos θ
2
d
r− ≈ r + cos θ0
2
r+ ≈ r −
Since d r we can write θ ≈ θ0 and on negelcting terms of order (d/r)2 we find
r− − r+ ≈ d cos θ
r+ r− ≈ r2 −
d2
cos 2 θ ≈ r2
4
and that the dipole potential at the point (r, θ) is
V (r, θ) =
µ0 m cos θ
µ0 (dp) cos θ
=
2
4π
r
4πr2
We call the quantity m = (dp) the magnetic moment of the dipole. Note that it is a vector quantity: our
coordinate system is defined relative to the axis of the dipole. This is the same result as using the GAD
(equation 5.8).
From this we can derive the radial, Br , and azimuthal, Bθ , components of the field:
Br = −
∂V
µo m2 cos θ
=
∂r
4πr3
Bθ = −
∂V
µo m sin θ
=
∂θ
4πr3
5.6 The origin of the magnetic field . From historical field measurements and paleomagnetic data (see
below) it is clear that the field morphology changes quickly in geological terms. The implication is that
whatever causes these fast changes is associated with rapid movement somewhere in the Earth. The only
reasonable place for this to happen is the outer core which is known to be fluid (from seismology) and
is inferred to be dominantly iron (from meteorites) and so is electrically conducting. Below a few 10’s
of kilometers, the Earth is too hot to allow permanent magnetization and the field would decay rapidly
away unless some regeneration mechanism is acting. We therefore must appeal to some convective
motion of the outer core material resulting in dynamo action and so generating the field. To see this we
consider the equation governing the time variation of the magnetic field in a moving, conductive fluid.
The equation can be straightforwardly derived from Maxwell’s equations and is
∂B
= ∇ × (v × B) + νm ∇2 B
∂t
(5.12)
where B is the magnetic field, v is the velocity field, νm is the magnetic diffusivity and × indicates
vector cross product. The detailed analysis of this equation is beyond the scope of this course but we
shall consider the physical meaning of the terms. The first term on the right hand side represents the
interaction of the magnetic field with the velocity field, the second term represents the diffusion of
magnetic field through the material. In fact if the core fluid is at rest, i.e., ~v = 0, we have
∂B
= νm ∇ 2 B
∂t
139
(5.13)
This is a vector form of the diffusion equation. As the field diffuses through the material, electric
currents are induced and the magnetic energy is converted to heat via ohmic dissipation. As magnetic
energy is lost, the field strength decays. We can estimate this decay rate for a dipole field and find that
the field strength decays to 1/e of its initial value in ∼15,000 years. i.e., the field strength varies as
B = B0 e−t/τ
(5.14)
where τ = 15,000 years. The total energy budget for the dynamo is 15–20 TW (compare this with 44 TW
for heat loss).
The value of τ depends upon the magnetic diffusivity νm . ∇2 has units of m−2 (every time you
differentiate something with respect to a length scale, L, you introduce a dimension of L−1 ) so νm has
units of m2 /s. All diffusion equations have diffusivities with these kinds of dimensions. νm controls the
rate at which the field diffuses through the material and is related to the electrical conductivity, σ , by
νm =
1
µ0 σ
(5.15)
where σ was originally estimated from shock wave experiments and theoretical calculations to be 3 × 105
Sm−1 , but now is considered to be 1.3 − −1.6 × 105 Sm−1 based on computer modeling and diamond
anvil cell measurements. Using these numbers gives
νm ≈ 0.530m2 s−1
(5.16)
A way to give a physical interpretation to νm is to say that in a time t, the field will diffuse a distance
L through the material where
L≈
√
tνm
(5.17)
If the core were perfectly conducting, (σ = ∞) then L would be zero. In this case the field lines are
stuck to the material and move as the material moves – this is called the frozen-flux approximation. With
a finite νm , the field can move through the material causing electric currents to be induced and so giving
ohmic dissipation. On short time scales we can show by a dimensional analysis of 5.12 that for large
scale changes in the magnetic field it is probably a reasonable approximation to neglect the diffusion
term and adopt the frozen flux approximation. The time scale for diffusion is
τd ≈
L2
1010 m2
≈ 3 × 109 s
≈
νm
0.5m2 /s
or about 100 years, while for convection it is
τc ≈
105 m
L
≈ −3 2 ≈ 108 s
U
10 m /s
or about 3 years, where we have taken L ≈ 105 m from the size of the core and U ≈ 10−3 m/s from
the secular variation of Earth’s field. Thus diffusion times are one or two orders of magnitude larger
than convection time scales. This is used as an argument in favor of neglecting diffusion in large scale
fields over short time intervals, and invoking the frozen-flux approximation. Its validity on longer
time intervals and for short length scales is more questionable. In the absence of convection, the
estimated decay rate of the field is simply too fast to be consistent with observations. These come from
paleomagnetism and show that the field has existed with an intensity similar to its present value for
billions of years.
If we reconsider equation 5.12, the first term describing field changes derived from convective motions
must be able to counteract the second diffusion term sometimes giving growth of the field. We therefore
140
diffusion in large scale fields over short time intervals, and invoking the frozen-flux approximation. Its
validity on longer time intervals and for short length scales is more questionable.
In the absence of convection, the estimated decay rate of the field is simply too fast to be consistent
with observations. These come from paleomagnetism and show that the field has existed with an intensity
similar to its present value for billions of years.
If we reconsider equation 5.21, the first term describing field changes derived from convective motions
must
be able to counteract
the second
diffusion term
sometimes
giving growth
of the
field.
need a mechanism
for converting
mechanical
energy
into magnetic
energy
and
thisWe
is therefore
exactly what a
need
a
mechanism
for
converting
mechanical
energy
into
magnetic
energy
and
this
is
exactly
what a
dynamo does.
dynamo does.
The complete solution of the geodynamo problem turns out to be remarkably difficult. In 1919
The complete solution of the geodynamo problem turns out to be remarkably difficult. In 1919
Larmor first proposed that a dynamo in a conductive body might be a possibility in the context of
Larmor first proposed that a dynamo in a conductive body might be a possibility in the context of
sunspot
magnetic
receiveda asignificant
significant
setback
in 1934
in a famous
theorem
sunspot
magneticfields,
fields,but
but this
this idea
idea received
setback
in 1934
whenwhen
in a famous
theorem
Cowling
showed
that
axially
symmetric
flows
in
the
core
cannot
generate
a
dynamo.
This
Cowling showed that axially symmetric flows in the core cannot generate a dynamo. This raised fears that raised
fears
that a antidynamo
general antidynamo
theorem
exist,
butitinwas
theshown
1950’s
it was shown
a general
theorem might
exist, might
but in the
1950’s
mathematically
thatmathematically
dynamo
thataction
dynamo
could occur.
in principle
of velocity
in
couldaction
in principle
That is occur.
specificThat
kindsisofspecific
velocity kinds
structures
specified structures
in equationspecified
5.12
could result
in self-sustaining
magnetic field.magnetic
The study
of such
fields
known asfields
kinematic
equation
5.12 could
result in self-sustaining
field.
Thevelocity
study of
suchisvelocity
is known as
dynamodynamo
theory. Attention
has subsequently
turned to the turned
identification
plausible motions
and forcesmotions
kinematic
theory. Attention
has subsequently
to theofidentification
of plausible
that
will
sustain
realistic
kinds
of
magnetic
fields.
and forces that will sustain realistic kinds of magnetic fields.
5.19
simpledisk
diskdynamo:
dynamo: a)
is generated
in theinrotating
disc, b)disc,
the b) the
FigFig
5.10
AA
simple
a) aa potential
potentialdifference
difference
is generated
the rotating
external
circuit
enablescurrent
current to flow
supplies
a field
which
reinforces
the original
external
circuit
enables
flow and
andc)c)the
thecurrent
current
supplies
a field
which
reinforces
the original
fieldfield
To see
how
dynamoaction
actioncould
could occur
a simple
machine
– the–Faraday
disc dynamo.
To see
how
dynamo
occurwe
wefirst
firstconsider
consider
a simple
machine
the Faraday
disc dynamo.
This
consists
of
a
disc
on
an
axle
rotating
in
a
magnetic
field
and
is
illustrated
in
Figure
5.19. As
the As the
This consists of a disc on an axle rotating in a magnetic field and is illustrated in Figure
5.10.
disc rotates in a magnetic field an electromotive force is established, i.e., there is a potential difference
disc rotates in a magnetic field an electromotive force is established, i.e., there is a potential difference
between the axis and the periphery of the disc. We can complete an electric circuit between the axis and
between the axis and the periphery of the disc. We can complete an electric circuit between the axis and
the periphery and an electric current will flow. We
150do this in a special way – we take a wire around the
disc to complete a loop and attach it with brushes to the axis and the edge of the disc. The wire behaves
as an electromagnet when the current flows and the original field is reinforced.
The interesting thing about this machine is that it does not depend upon the initial polarity of the
field. Indeed if we make the machine more complicated (by putting in resistors to model dissipation
and coupling dynamos together) we can make the field reverse polarity and behave in an apparently
chaotic manner with oscillations in intensity and reversals at random intervals. This is similar to the
behavior we observe of the Earth’s magnetic field and it is interesting that steady motions (in this case
constant rotation of the disc) can produce such complicated field behavior as a function of time. While
this model is undoubtedly very different from the Earth’s core, it does demonstrate that dynamo action
is capable of producing many of the observed properties of the magnetic field.
A more realistic mechanism for generating Earth’s field is a convective dynamo operating in the fluid
outer core. The solid inner core is roughly the size of the moon but at the temperature of the surface of
the sun. The convection in the fluid outer core is thought to be driven by both thermal and compositional
buoyancy sources at the inner core boundary that are produced as the Earth slowly cools and iron in the
iron-rich fluid alloy solidifies onto the inner core giving off latent heat and the light constituent of the
alloy. These buoyancy forces cause fluid to rise and the Coriolis forces, due to the Earth’s rotation, cause
the fluid flows to be helical. Presumably this fluid motion twists and shears magnetic field, generating
new magnetic field to replace that which diffuses away. Figure 5.11 provides a heuristic illustration of
this process.
141
!
"
#
$
%
&
'(%)α−ω)$*+!,-),%#(!+./,0))1-+2%+3.-+!4)5%-$*+!,-)3(%-6*)76%/877-/%/)9!:)!+).+.3.!4;)
76.,!6.4*)$.7-4!6;)7-4-.$!4),!5+%3.#)&.%4$0)))'(%)ω<%&&%#3)#-+/./3/)-&)$.&&%6%+3.!4)6-3!3.-+;)
9":)!+$)9#:;)=6!77.+5)3(%),!5+%3.#)&.%4$)!6-8+$)3(%)6-3!3.-+!4)!>./;)3(%6%"*)9$:)#6%!3.+5)!)
?8!$687-4!6)3-6-.$!4),!5+%3.#)&.%4$0)@*,,%36*)./)"6-A%+;)!+$)$*+!,-)!#3.-+),!.+3!.+%$;)"*)
3(%)α<%&&%#3;)=(%6%"*)(%4.#!4)87=%44.+5)9%:)#6%!3%/)4--7/)-&),!5+%3.#)&.%4$0)'(%/%)4--7/)
#-!4%/#%)9&:)3-)6%.+&-6#%)3(%)-6.5.+!4)$.7-4!6)&.%4$;)3(8/)#4-/.+5)3(%)$*+!,-)#*#4%0
Figure 5.11
Until relatively recently, no detailed dynamically self-consistent model existed that demonstrated this
could actually work or explained why the geomagnetic field has the intensity it does, has a strongly
dipole-dominated structure with a dipole axis nearly aligned with the Earth’s rotation axis, has nondipolar field structure that varies on the time scale of tens to thousands of years and why the field
occasionally undergoes dipole reversals. In order to test the convective dynamo hypothesis and attempt
to answer these longstanding questions, the first self-consistent numerical model, the Glatzmaier-Roberts
model, was developed in the mid 1990s. It simulates convection and magnetic field generation in a
fluid outer core surrounding a solid inner core with the dimensions, rotation rate, heat flow and (as
much as possible) the material properties of the Earth’s core. The magnetohydrodynamic equations that
describe this problem are solved using a spectral method (spherical harmonic and Chebyshev polynomial
expansions) that treats all linear terms implicitly and nonlinear terms explicitly. These equations are
solved over and over, advancing the time dependent solution 20 days at a time. See Glatzmaier’s web
page for more on this at http://www.es.ucsc.edu/ glatz/geodynamo.html This numerical dynamo (along
with a suite of others that have been created over the past decade) is a great step forward in studying
the dynamics of Earth’s core, but it remains far from realistic. One reason is that the very low viscosity
of the fluid in the outer core would require that the numerical simulations be able to resolve very small
scale changes in both space and time: such resolution remains a significant computational challenge.
142
There are historical measurements of the magnetic field going back about 300 years (mostly sailing
ship navigation records) allowing time-dependent models of the magnetic field to be constructed (see
movie). Some magnetic observatories have been in operation for a very long time (about 150 years).
Fig 5.12 shows the time derivations of the Y component at two observatories in Europe which shows
that rather abrupt changes in the field are occasionally observed. Using the frozen flux approximation
and some further assumptions, the time variation of the observed field can be converted into a model of
fluid flow at the top of the outer core (see ppt).
Fig 5.12 The time derivation of the Y component at two magnetic obervatories in Europe. Note the
very rapid variations that can occur. The sudden changes in the time derivative are called "jerks" and
the fact that we see them puts constraints on the electrical conductivity in the mantle.
If we want to look further back we must look at the remanent magnetization acquired by crustal rocks
as they form in a magnetic field: this is the subject of paleomagnetism.
5.7 Induced and Remanent Magnetization . The magnetic properties of rocks are dependent on the
state of charged particles associated with the crystal structure – motions of and interactions among
the electrons generate the magnetic moment. There are two basic kinds of magnetization: induced
magnetization and remanent magnetization. In the presence of an externally applied magnetic field the
magnetization of a substance may be written as the sum of an induced and a remanent (or spontaneous)
component.
M = MI + MR
The remanent component, MR is generated by the material itself, and without this contribution there
would be no possibility of a rock recording the paleomagnetic field. The constitutive relation, which is
commonly written as
143
MI = χH
describes the induced magnetization acquired when a material is exposed to a magnetizing field H. χ
is the magnetic susceptibility and describes the response of electronic motions within the material to
the applied field. Materials can acquire a component of magnetization in the presence of an external
magnetic field (such as that generated in Earth’s core). This induced magnetization is often considered
to be proportional in magnitude to and along the direction of the external field. Thus we can write
B = µ0 (H + M) = µ0 (1 + χ)H = µH.
µ is the magnetic permeability. In practice χ may be dependent on field intensity, negative or need to
be represented by a tensor (magnetically anisotropic materials). However, we will not consider such
complications here.
Three main classes of magnetic behavior can be distinguished on the basis of magnetic susceptibility:
diamagnetism, paramagnetism, and ferromagnetism. All of these are fundamentally due to the orbital
and spin properties of the electons in the material. Diamagnetism typically occurs in materials where
all the electrons spins are paired and is caused by precession of the electron orbits in the presence of an
applied field. The associated magnetic moment opposes the applied field so diamagnetic susceptibility
is reversible, weak, and negative. Paramagnetism occurs when one or more of the electron spins
are unpaired so the net magnetic moment of the atom (or ion) is non zero. These can then align
with an applied field but the effect is opposed by thermal energy. The corresponding susceptibility is
small, positive and reversible. Both paramagnetism and diamagnetism are insignificant contributors to
the geomagnetic field. The important contributions come from materials (some metals) with atomic
is reversible, weak, and negative. Paramagnetism occurs when one or more of the electron spins are
moments that interactunpaired
strongly
with
each
other
result
ofzero.
quantum
mechanical
exchange interactions
so the net
magnetic
moment
of the as
atoma(or
ion) is non
These can then
align with an applied
field but the effect is opposed by thermal energy. The corresponding susceptibility is small, positive
resulting in spontaneous
exact
alignment
of
magnetic
moments
(fig5.13).
These
are
ferrimagnetic and
and reversible. Both paramagnetism and diamagnetism are insignificant contributors to the geomagnetic
field. The
come either
from materials
metals) with
moments that
interact
ferromagnetic materials
andimportant
theycontributions
can carry
an (some
induced
oratomic
remanent
magnetization.
The total
strongly with each other as a result of quantum mechanical exchange interactions resulting in spontaneous
magnetization of a rock
will
result
from
the
sum
of
these
two
contributions
exact alignment of magnetic moments (fig5.9a). These are ferrimagnetic and ferromagnetic materials
and they can carry either an induced or remanent magnetization. The total magnetization of a rock will
result from the sum of these two contributions
! =+
! +M
!r =
! +M
!+
M
H
r Mr
M=M
i Mi M
r = χχH
Figure 5.9a: Schematic representation of arrangement of magnetic moments in various materials
Figure 5.13: Schematic representation of arrangement of magnetic moments in various materials
behavior of the magnetization
as a function of
of anan
applied
field is shown
As the in fig 5.14. As the
The behavior of theThe
magnetization
as a function
applied
fieldin fig
is 5.9b.
shown
magnetising field increases, the induced magnetization will reach some saturation level then, when the
magnetising field increases,
the ainduced
reach
somemagnetization
saturation
field is removed,
magnetizationmagnetization
remains – this is called will
an isothermal
remanent
(IRM)level then, when the
as show in fig 5.9b. Repeated exposure to positve and negative fields result in a hysteresis loop but the
field is removed, a magnetization
remains
–
this
is
called
an
isothermal
remanent
magnetization (IRM)
important point is that ferromagnetic materials retain a record of an applied magnetic field after it is
as shown in fig 5.14.removed.
Repeated exposure to positve and negative fields result in a hysteresis loop but
the important point is that ferromagnetic materials retain a record of an applied magnetic field after it is
removed.
144
Figure 5.9b: Magnetization hysteresis loop for a ferromagnetic material
The behavior of the magnetization as a function of an applied field is shown in fig 5.9b. As the
magnetising field increases, the induced magnetization will reach some saturation level then, when the
field is removed, a magnetization remains – this is called an isothermal remanent magnetization (IRM)
as show in fig 5.9b. Repeated exposure to positve and negative fields result in a hysteresis loop but the
important point is that ferromagnetic materials retain a record of an applied magnetic field after it is
removed.
Figure hysteresis
5.9b: Magnetization
loop formaterial
a ferromagnetic material
Figure 5.14: Magnetization
loop for hysteresis
a ferromagnetic
The saturation magnetization of a ferromagnetic material depends on temperature, and above the
135
"Curie" temperature thermal
perturbations
destroy the
magnetization
so that the
remaining
The saturation
magnetization
ofremanent
a ferromagnetic
material depends
ononly
temperature,
and above the
temperature
thermal perturbations
magnetization is from”Curie”
diamagnetic
or paramagnetic
effects.destroy the remanent magnetization so that the only remaining
magnetization is from diamagnetic or paramagnetic effects.
Magnetic MineralogyMagnetic Mineralogy
Thetitanium
oxides ofare
ironbyand
areimportant
by far the terrestrial
most important
terrestrial
magnetic
minerals. MagThe oxides of iron and
fartitanium
the most
magnetic
minerals.
Mag◦
◦
netite
(F
e
O
,
Curie
temperature
580
C)
and
its
solid
solutions
with
ulvospinel
(F
e
3
4
2 T iO4 ) are the
netite (F e3 O4 , Curie temperature 580 C ) and its solid solutions with ulvospinel (F e2 T iO4 ) are the most
most
important
magnetic
minerals
in
crustal
rocks,
although
hematite,
pyrrhotite
also
play
important magnetic minerals in crustal rocks, although hematite, pyrrhotite also play a role in paleo-a role in paleomagnetic studies. A ternary composition diagram showing the solid solutions of the important minerals
magnetic studies. A ternary composition diagram showing the solid solutions of the important minerals
−1
− 1A/m for
is shown in fig 5.9c. Typical values of magnetization for rocks cover a wide
−1 range, 10
is shown in fig 5.15. basalts,
Typicalaround
values10of−3magnetization
for
rocks
cover
→
1
A/m
fortemperatures
−4 a wide
−5 range, 10
A/m
for
red
sediments
to
10
−
10
A/m
for
limestones.
Curie
basalts, around 10−3 A/m
sediments
to 10−4 → 10−5 A/m for limestones. Curie temperatures also
also for
varyred
widely
with composition.
vary widely with composition.
5.9c: Ternary
composition
diagram of theoxide
iron-titanium
Figure 5.15: TernaryFigure
composition
diagram
of the iron-titanium
system oxide system
How long can a rock preserve a record of the magnetic field?
How long can a rock preserve a record of the magnetic field?
Rocks are not pure Rocks
magnetic
minerals,
but assemblages
of assemblages
fine-grainedofferromagnetic
materials materials
are not
pure magnetic
minerals, but
fine-grained ferromagnetic
dispersed
in paramagnetic
a diamagnetic and
paramagnetic
The net magnetization
will to
correspond
to the
dispersed in a diamagnetic
and
matrix.
The netmatrix.
magnetization
will correspond
the
minimum
energy state,
a complicated
tradeoff
amongfrom
energy
effects from magnetostatic
effects, shape,
minimum energy state,
a complicated
tradeoff
among energy
effects
magnetostatic
effects, shape,
and anisotropies.
magnetocrystalline
anisotropies.
Factors
that affect
how
long
a rock
can retain
a record of the
and magnetocrystalline
Factors
that affect
how long
a rock
can
retain
a record
of the
magnetic
field
are
composition,
temperature,
and
volume
of
the
magnetic
particle
as
well
magnetic field are composition, temperature, and volume of the magnetic particle as well as whether itas whether it
is divided into one or more magnetic domains of more or less uniform magnetization.
is divided into one or more magnetic domains of more or less uniform magnetization.
5.7 The Crustal Field and Paleomagnetism . The crustal magnetic field arises from induced and
5.8 The Crustal Field and Paleomagnetism . The crustal magnetic field arises from induced and
remanent magnetization carried by a number of magnetic minerals (see fig 5.9c) that occur naturally
remanent magnetization
carried by a number of magnetic minerals (see fig 5.15) that occur naturally
in the rocks that make up the crust. Although some minerals are magnetically viscous (i.e., their
magnetization changes rapidly to follow the direction of any ambient field; this corresponds to an induced
145
magnetization), many rocks carry
an imprint of the ambient magnetic field at the time of their formation,
which remains stable over geological timescales. Two important mechanisms for acquiring this fossil
magnetization or magnetic remanence are temperature changes and depositional processes. When a rock
is heated it gradually loses its magnetization. The Curie temperature of a mineral, the temperature above
which all magnetic order is lost, varies widely with chemical composition and structure. Thermoremanent
magnetization (TRM) is acquired when magnetic material cools from above its Curie point in a magnetic
in the rocks that make up the crust. Although some minerals are magnetically viscous (i.e., their
magnetization changes rapidly to follow the direction of any ambient field; this corresponds to an
induced magnetization), many rocks carry an imprint of the ambient magnetic field at the time of their
formation, which remains stable over geological timescales. Two important mechanisms for acquiring
this fossil magnetization or magnetic remanence are temperature changes and depositional processes.
When a rock is heated it gradually loses its magnetization. The Curie temperature of a mineral,
the temperature above which all magnetic order is lost, varies widely with chemical composition and
structure. Thermoremanent magnetization (TRM) is acquired when magnetic material cools from above
its Curie point in a magnetic field and records its direction at the time of cooling. This process is shown
schematically in fig 5.16
Figure 5.16: Schematic depiction
of cooling
through
the ofCurie
temperature
magnetization
Figure 5.9d:
Schematic
depiction
cooling
through the when
Curie temperature
when
changes from paramagnetic to magnetization
ferromagnetic.
Subsequent
cooling results
in the magnetization
the
changes
from paramagnetic
to ferromagnetic.
Subsequent in
cooling
resultsalong
in the easy
magnetization
in the magnetic
grains
becoming
along
magnetic grains becoming "blocked"
magnetization
directions
close
to that ”blocked”
of the applied
easy magnetization directions close to that of the applied field.
field.
in basalts and
lava flows,
and for low
intensity
fielda similar
to that
of thesomewhat
Earth, its magnitude is
Once the rock coolsfound
the remanence
is locked
in (blocked)
and
provides
permanent
(albeit
proportional
to
the
intensity
of
the
magnetic
field
in
which
it
is
acquired.
noisy) record of the ambient magnetic field direction. Thermoremanent magnetization is commonly
magnetization
can alsofield
be acquired
small magnetic
particles
found in basalts and lavaRemanent
flows, and
for low intensity
similarby
tosediments,
that of theasEarth,
its magnitude
is are incorporated
into
the
sediment;
on
average
these
will
align
with
the
ambient
field
during
deposition
and become
proportional to the intensity of the magnetic field in which it is acquired.
locked into position during subsequent dewatering and compaction of the sediment (fig 5.9e). This is
Remanent magnetization can also be acquired by sediments, as small magnetic particles are incorcalled depositional or post-depositional remanent magnetization, DRM or PDRM, depending on when
porated into the sediment;
on average these will align with the ambient field during deposition and
it is acquired. Note that magnetic grains may have elongate shapes that means that their final orientation
become locked into position
during
subsequent
dewatering
compaction
of the thus
sediment
5.17). error.
as they settle
to the
bottom may
change dueand
to the
action of gravity
giving (fig
an inclination
This is called depositional
or post-depositional
remanent
magnetization,
DRMrock
or PDRM,
Chemical,
mineralogical, and
metamorphic
processes after
formationdepending
can also influence the
on when it is acquired.
Note that magnetic
grains
may
shapes
that means
that theirby
final
magnetization,
sometimes
causing
the have
signalelongate
to be partially
or completely
overprinted
a later magnetic
orientation as they settle
to the bottom may change due to the action of gravity thus giving an inclination
field.
error. Chemical, mineralogical,
and metamorphic
after rock
formation
can alsoapplications
influence in the earth
Paleomagnetism,
the study of processes
the fossil magnetic
record,
has important
sciences.
Taking
oriented
samples
of
crustal
rocks
and
measuring
their
directions
magnetization in
the magnetization, sometimes causing the signal to be partially or completely overprinted by of
a later
a
magnetometer
allows
the
determination
of
the
local
paleofield
direction
when
the
rock
acquired its
magnetic field.
magnetization.
During
the
late
1950s
and
1960s
it
became
increasingly
apparent
that
these
observations
Paleomagnetism, the study of the fossil magnetic record, has important applications in the earth
did not support the existence of a predominantly dipolar field configuration like that existing today,
sciences. Taking oriented
samples of crustal rocks and measuring their directions of magnetization in
unless the continents had moved around on Earth’s surface since the time the rocks were formed.
a magnetometer allows the determination of the local paleofield direction when the rock acquired its
These observations provided important evidence supporting the theory of plate tectonics. Even more
magnetization. Duringcompelling
the late 1950s
andwas
1960s
it in
became
increasingly
apparent
these observations
evidence
found
the patterns
of seafloor
magneticthat
anomalies
in the crustal field. Ocean
floor basalts extruded at mid-ocean ridges rapidly cool and acquire a thermoremanent magnetization.
146the ridge by plate motion so that age increases with distance from the
The material is carried away from
ridge axis. Successive normal and reverse polarity epochs of the magnetic field will have normally and
reversely magnetized basalt associated with them. The associated magnetic field can be measured with
a magnetometer towed behind a ship; anomalies in the magnitude of the magnetic field are observed
137
Figure 5.17: Schematic of the aquisition of a DRM
did not support the existence of a predominantly dipolar field configuration like that existing today,
unless the continents had moved around on Earth’s surface since the time the rocks were formed.
These observations provided important evidence supporting the theory of plate tectonics. Even more
compelling evidence was found in the patterns of seafloor magnetic anomalies in the crustal field. Ocean
floor basalts extruded at mid-ocean ridges rapidly cool and acquire a thermoremanent magnetization.
The material is carried away from the ridge by plate motion so that age increases with distance from the
ridge axis. Successive normal and reverse polarity epochs of the magnetic field will have normally and
reversely magnetized basalt associated with them. The associated magnetic field can be measured with
a magnetometer towed behind a ship; anomalies in the magnitude of the magnetic field are observed
with distinctive lineated or striped anomaly patterns approximately parallel to ridge axes. The pattern of
polarity changes in the geomagnetic field as a function of distance from the ridge axis can be correlated
with land-based sections dated by radiometric methods; distance is transformed to age providing the
basis for the global magnetostratigraphic timescale that spans the approximate time interval 0-160 Ma,
the age of the oldest ocean basins.
Together with the assumption that averaged over sufficient time the geomagnetic field directions
observed approximate those of a dipole aligned along Earth’s rotation axis, paleomagnetic observations
and seafloor magnetic anomalies provide constraints used in continental reconstructions over geologic
time. Paleomagnetism is also useful for studying regional tectonic problems.
5.9 Paleomagnetic poles and apparent polar wander . As a continent moves, it carries in its rock
formations a record of the direction of past magnetic fields. These are, as we have seen, directly related
to the position of the Earth’s spin axis in the past. In the continent’s frame of reference, then, it is the
North pole which appears to move. Tracks of these past pole positions are called apparent polar wander
paths (APWP). An example of a apparent polar wander path is shown in Fig. 5.18.
147
Fig 5.18
The fundamental assumption in constructing apparent polar wander paths for continental blocks is that
when averaged over sufficient time, the earth’s magnetic field reduces to that of a dipole aligned along the
rotational axis and located at the earth’s center. That is, all terms in the spherical harmonic representation
except for g10 will cancel out in a time average. This is the geocentric axial dipole hypothesis. In order to
track the movement of the spin axis with respect to the continent, we must sample rocks of known age
and orientation, obtain a magnetization which represents the ancient magnetic field, and then calculate
the position of the pole. The instantaneous field deviates quite significantly from that of an axial dipole,
and directions need to represent some significant (geologically speaking) amount of time. A widely
used, but highly optimistic, rule of thumb is to use 10 separately oriented samples from 10 separate
rock units, which span at least 10,000 years. Data
Fig from
5.10 the past 5 million years suggest that hundreds
of sites spanning hundreds of thousands of years may be a more realistic requirement. The position
on the Earth where this average dipole would pierce the surface is the paleomagnetic pole. This then
ofthe
theposition
geographic
rotation
axis at rotation
the timeaxis
the rocks
in the
continentalinreference
frame.
is
of the
geographic
at thewere
timemagnetized
the rocks were
magnetized
the continental
The
pole
calculated
from
a
single
measurement
is
called
a
virtual
geomagnetic
pole,
or
VGP.
We
may
reference frame. The pole calculated from a single measurement is called a virtual geomagnetic pole,
or
declination
calculate
this pole
position
by its(specified
colatitudebyand
θp , ψ
VGP.
We may
calculate
this(specified
pole position
its longitude
colatitude: and
longitude
: θp , ψthe
we know
p ) if we know
p ) if
iss ,done
anddeclination
inclination and
(D inclination
and I) of the
field
the site
the the
rocks
(θs , ψsampled
the
(Dancient
and I ) of
theand
ancient
fieldofand
sitesampled
of the rocks
ψs ).
s ). This(θ
usingisthe
sine
and the
cosine
for triangles
on the
surface
a sphere.
This
done
using
sinerules
and cosine
rules for
triangles
onofthe
surface of a sphere.
Fig
Fig 5.19
5.10
148
Remember that the lengths of the sides of the spherical triangle are also measured as angles. Two
formulae come in handy:
sin A
sin B
sin C
=
=
sin a
sin b
sin c
(5.12)
Remember that the lengths of the sides of the spherical triangle are also measured as angles. Two
formulae come in handy:
sin B
sin C
sin A
=
=
sin a
sin b
sin c
(5.18)
cos a = cos b cos c + sin b sin c cos A
(5.19)
and
Let us now sketch the geometry of our problem:
Fig5.11
5.20
Fig
S is ourNow
measurement
site, D,
P isisthe
the the
virtual
geomagnetic
we want
to determine)
the declination,
justlocation
the angleoffrom
present
North pole pole
to the(which
line joining
M and
P so
and N is the present North Pole. The magnetic colatitude of the sample magnetization is the distance
cos (θpto
) =the
cospaleomagnetic
(θs ) cos θm + sin
(θs ) sin
cos D can be determined
(5.15)
from the measurement location
pole,
θmθ. m This
from the
inclination,
I
,
of
the
sample,
using
equation
5.10
introduced
in
the
first
section
of
this
chapter,
which allows us to calculate θp .
To determine ψp we use
cot θm =
1
2
tan I
(5.20)
sin (ψp − ψs )
sin D
(5.16)
=
Now the declination, D, is just the angle from
the present
sin θm
sin North
(θp ) pole to the line joining S and P so
or
cos θp = cos θs cos θm + sin θs sin θm cos D
sin θm sin D
sin θp
sin (ψp − ψψsp) we
= could use the sine formula:
which allows us to calculate θp . To determine
(5.21)
(5.17)
sin D we can determine ψp . For certain
(ψp − θψms )are known,
Because θp has been determined and ψsin
s , D and
=
(5.22)
sin θmψp incorrectly.
sin θp The formula is valid if cos θm >
orientations this formula will give the longitude
or cos θs cos θp . If cos θm < cos θs cos θp then use
sin θm sin D
sin (180
) =sin θm sin D
sin (ψ+p ψ−−ψsψ)p=
sinθ θp
sin
p
(5.18) (5.23)
◦
However,
can be aneastward
ambiguity
between
the angle
you get
and
180
degrees
minus
angle (which
NOTE:there
ψ is measured
from
the Greenwich
meridian
and
goes
from
0 → 360
. θ that
is measured
◦ use the cosine formula again:
bothfrom
givethe
theNorth
samepole
sineand
value).
It
is
easiest
to
goes from 0 → 180 . Of course θ relates to latitude, λ by θ = 90 − λ. θm is
the magnetic colatitude. Be sure not to confuse latitudes and colatitudes.
θmcompute
= cos θthe
θp + sin
θs sin
θp cos (ψp −
ψs )assuming that the field
s cos
These formulae allow cos
us to
location
of the
paleomagnetic
pole
has remained dipolar. By hypothesis, the paleomagnetic pole averaged to the geographic North pole
when the rock formed. (If measurements are taken149
from a time period when the field was reversed the
paleomagnetic south pole was at the North pole.) By using measurements from a continental block of
many different dates, the apparant polar wander path as a function of time can be constructed. The polar
wandering path will be the same for all measurements from a single plate and can be used to reconstruct
the absolute location of the plate in the past (relative to the geographic poles) though you should note
so
cos (ψp − ψs ) =
cos θm − cos θs cos θp
sin θs sin θp
(5.24)
Because θp has been determined and ψs , D and θm are known, we can determine ψp .
NOTE: ψ is measured eastward from the Greenwich meridian and goes from 0 → 360◦ . θ is measured
from the North pole and goes from 0 → 180◦ . Of course θ relates to latitude, λ by θ = 90 − λ. θm is the
magnetic colatitude. Be sure not to confuse latitudes and colatitudes.
These formulae allow us to compute the location of the paleomagnetic pole assuming that the field
has remained dipolar. By hypothesis, the paleomagnetic pole averaged to the geographic North pole
when the rock formed. (If measurements are taken from a time period when the field was reversed the
paleomagnetic south pole was at the North pole.) By using measurements from a continental block of
many different dates, the apparant polar wander path as a function of time can be constructed. The polar
wandering path will be the same for all measurements from a single plate and can be used to reconstruct
the absolute location of the plate in the past (relative to the geographic poles) though you should note
that longitudinal motions of continents will not be detectable because of the assumed axial symmetry
of the field. The polar wandering paths of two adjacent plates can be used to compute the minimium
relative velocities between plates.
5.10 Secular and Paleosecular Variation . The magnetic field varies through time as we all know
from using navigational charts. Variations occur on time scales of milliseconds to millions of years.
Variations with time scales less than 1 – 10 years are mainly of external origin. Longer term variations
(on time scales of 10 to 10,000 years) are called secular variations and are typically considered to be
of internal origin. However, it should be kept in mind that time scale is not an entirely reliable means
of determining the source of a particular field. Geomagnetic jerks are a notable example of rapid field
variations of internal origin. At present, the field is mainly that of a dipole inclined at about 11◦ to the
rotation axis. The strength of the dipole (measured as a dipole moment), presently 7.94 × 1022 Am2
but is decreasing by ∼ .05%/year. If we subtract out the dipole contribution to the field, the remainder
(called the non-dipole field) has a complicated shape some of which appears to drift roughly westward
at a rate of about 0.2◦ /year (22 km/yr or ≈ 0.001 m/s, which we used in the diffusion calculation). This
westward drift accounts for about a quarter of the variation in the recent non-dipole field. Perhaps the
most dramatic variation in the magnetic field is that the axial geocentric dipole coefficient changes sign
occasionally, meaning that compasses would point South! The process takes from four to ten thousand
years to complete and the last so-called reversal was about 780,000 years ago.
The full spectrum of behavior exhibited by the geomagnetic field can only be determined from paleomagnetic and associated geochronological studies of material from the geological and archeological
record. Typically only observations of field direction are possible but when the magnetic remanence acquisition process can be simulated in the laboratory the magnitude may also be estimated. Although such
records can never have the spatial or temporal resolution of historic measurements of the geomagnetic
field, there are interesting questions which can be asked. These include the following:
(1) The present geomagnetic field is dominantly dipolar (roughly 90% of the field can be explained by a
geocentric dipole). Was the paleofield always as dominantly dipolar as it appears now?
(2) It was noted very early in geomagnetic studies that large features of the geomagnetic non-dipole
field appeared to move westward through time with a drift rate of approximately 0.2◦ per year. This
striking behavior became known as westward drift. Is westward drift an intrinsic feature of the
paleofield?
(3) The present dipole moment of the geomagnetic field is roughly 8 × 1022 Am2 . What is the average
value and how much does it change?
150
Grand Spectrum
Reversals
Cryptochrons?
Secular variation
Amplitude, T/√Hz
?
?
Annual and semi-annual
Solar rotation (27 days)
Daily variation
Internal origin
External origin
Storm activity
Quiet days
10 kHz
1 second
1 minute
1 hour
1 day
1 year
Schumann resonances
1 month
1 thousand years
10 million years
Powerline noise
Radio
Frequency, Hz
(4) What is the long-term nature of secular variation? Is the present or historic variability representative
of the field in general?
(5) Magnetohydrodynamical equations describing the conditions in the Earth’s core do not care what
sign the field is (i.e., whether it is “normal" or “reversed"). Are there any detectable differences
between the two polarity states (for example, does the field spend more time in one polarity state than
the other or do gauss coefficients have different mean values)?
(6) How often does the field reverse ? What is the behavior of the field during transition from one polarity
state to another?
(7) How long has the core been producing a magnetic field and is there any observable effect of the
growth of the inner core?
Paleomagnetic records used to address these and other geomagnetic questions are derived from
archeological materials such as pottery, baked hearths, and adobe bricks as well as geological media
including both sedimentary and igneous sequences. Each of these materials has specific advantages and
disadvantages and the most complete understanding comes from an integrated view of all the records.
The above questions are still very much subjects of investigation and for many of them there are no
definitive answers yet. For example, in order to investigate whether westward drift is intrinsic to the
field, one requires a global data set which is accurately dated; such a data set does not exist. However,
another question which could be addressed by existing paleosecular variation records is: Over what
distance can prominent features of the directional data be recognized? Existing data sets obtained
from lake sediment data, appear to show that similar geomagnetic field features can be observed in
records across the North American continent, but these records are quite different from those obtained,
for example, from European lakes. If westward drift of the non-dipole field were dominating secular
variation, then we should see the same curves all over the world, offset slightly in time. What we see
instead are quite different records with apparent spatial scales on the order of a few thousand kilometers.
151
During times of stable magnetic polarity, secular variation causes the orientation of Earth’s dipole
axis to change by some 10 − 15◦ on timescales of a few hundreds to thousands of years; corresponding
local fluctuations in field intensity may be as much as factor of two or three, but the global dipole
moment is somewhat less variable. A large departure from the approximately geocentric axial dipolar
state for the field is known as a geomagnetic excursion and is usually accompanied by strong fluctuations
in intensity of the field (see ppt for some examples). For a long time, it was conventional wisdom that
excursions were artifacts of the recording process and were not characteristic of the real geomagnetic
field. However, these misgivings were laid to rest by reproducibility arguments. The most convincing
aspect of well studied excursions is that they are recorded in several separate sections. There are only a
few excursions which have been sufficiently well documented to convince the most skeptical audience.
The characteristics of excursions are that the field decreases to about 30% or less of the normal value and
the position of the dipole axis (that is the VGP as inferred from a single measurement location) traces a
path to more than 45◦ away from the spin axis. Because they are of short duration (a few thousand years
or less) it is difficult to find geological records of an excursion that confirm it as a globally synchronous
event; the most recent global geomagnetic excursion, known as the Laschamp event, appears to have
taken place about 40 thousand years ago. Excursions are often thought of as aborted reversals, and
the evidence available so far suggests that a continuum of behavior exists ranging from normal secular
variation through excursions to full polarity reversals.
5.11 Geomagnetic field reversals . The Earth’s dipole field flips polarity at irregular intervals and
when the polarity is the same as the present day the polarity is said to be normal. On average, the field
spends about half its time in each state. When reversed polarities were first discovered, there was some
discussion about whether or not magnetic minerals could become magnetized in the opposite direction
to the applied field. While this is true for some minerals under special conditions, the same pattern
of reversals is found in rocks at different geographical sites. Furthermore, the same pattern is found
in lavas and in various sediments so the only reasonable hypothesis is that the geomagnetic field itself
is reversing. The latest transition, from the Matuyama reversed polarity epoch to the present normal
polarity epoch (the Brunhes epoch), took place at about 0.78 Ma. The time between successive reversals
is extremely irregular, but over long periods there are systematic changes in geomagnetic reversal rates.
Figure 5.21 shows a fairly steady rise in average reversal rate since about 84 Ma. Prior to that there
was a long period during which no known reversals occurred, the Cretaceous Long Normal Superchron.
Another such reversal-free interval occurs from about 320–250 Ma when the field was consistently
reversed in polarity; other intervals may also exist. The transition to such states may reflect changing
physical conditions for the geodynamo, related to Earth’s cooling history and growth of the inner core.
The existence of the geomagnetic field since about 3 Ga is well documented, and such information as
we have about its intensity during Archean and Early Proterozoic times suggests that it was roughly the
same order of magnitude as it is today. Reversals are observed from Precambrian times to the present
though the frequency of reversals seems to change considerably through time.
What happens during a reversal? Usually the intensity decreases by about an order of magnitude
for several thousand years while the field maintains its direction. The field then undergoes complicated
directional changes over a period of 1000 – 4000 years and finally the intensity grows with the field
having reversed polarity. The total time span of a reversal is up to 10,000 years. Some reversals seem
to show concurrent changes in intensity and direction and the complete reversal occurs in about 5000
years. One explanation of the behavior during a reversal is that the dipole field decays away and the
nondipole field stays roughly at the same level giving rise to the direction changes when the dipole field
is small. The dipole field then grows with opposite polarity.
152
The existence of the geomagnetic field since about 3 Ga is well documented, and such information as
we have about its intensity during Archean and Early Proterozoic times suggests that it was roughly the
same order of magnitude as it is today. Reversals are observed from Precambrian times to the present
though the frequency of reversals seems to change considerably through time.
Figure 5.13: Geomagnetic
rate for the reversal
past 160Myr.
Figure 5.21:reversal
Geomagnetic
rate
for the past 160Myr.
What happens during a reversal? Usually the intensity decreases by about an order of magnitude
for several thousand years while the field maintains its direction. The field then undergoes complicated
directional changes over a period of 1000 – 4000 years and finally the intensity grows with the field
having reversed polarity. The total time span of a reversal is up to 10,000 years. Some reversals seem
to show concurrent changes in intensity and direction and the complete reversal occurs in about 5000
years. One explanation of the behavior during a reversal is that the dipole field decays away and the
nondipole field stays roughly at the same level giving rise to the direction changes when the dipole field
is small. The dipole field then grows with opposite polarity.
The reversal sequence of the magnetic field has been calibrated for the last five million years by
dating basalts of known polarity. A recent version is shown in Fig 5.14. Portions of the time scale which
are of one dominant polarity, lasting from one to two million years are designated Chrons and the most
recent four chrons are named after great scientists who contributed significantly to our understanding of
the geomagnetic field. The Chrons older than about five million years have a somewhat cumbersome
numbering scheme.
Knowledge that the Earth’s magnetic field reversed polarity frequently in the past helped explain
some of the mysterious features of the geophysical observations made in the oceans. One such mystery
was that of the curious magnetic “stripes” observed in the geomagnetic field data over much of the ocean
floor.
The notion that oceanic crust forms at the great mid-ocean ridges (another mystery of the sea floor)
while the magnetic field reverses polarity served as
theFig.
foundation
Fig.
5.22
5.14 for the hypothesis of sea-floor spreading and was the key to the plate tectonic revolution. Consider Figure 5.15, a sketch of an oceanic ridge.
The reversal sequence of the magnetic field has been calibrated for the last five million years by
145
dating basalts of known polarity. A recent version
is shown in Fig 5.22. Portions of the time scale which
are of one dominant polarity, lasting from one to two million years are designated Chrons and the most
recent four chrons are named after great scientists who contributed significantly to our understanding of
the geomagnetic field. The Chrons older than about five million years have a somewhat cumbersome
numbering scheme.
Knowledge that the Earth’s magnetic field reversed polarity frequently in the past helped explain
some of the mysterious features of the geophysical observations made in the oceans. One such mystery
5.15geomagnetic field data over much of the ocean
was that of the curious magnetic “stripes" observedFig.
in the
floor. The notion that oceanic crust forms at the great mid-ocean ridges (another mystery of the sea
the oceanic
plates
diverge, polarity
molten mantle
material
flows
upward to fill
gap,
this magma of
then
floor) while theAs
magnetic
field
reverses
served
as the
foundation
forthethe
hypothesis
sea-floor
cools
by
conductive
heat
loss
to
the
surface.
The
ridge
is
elevated
because
the
rocks
are
hotter
and
spreading and was the key to the plate tectonic revolution. Consider Figure 5.23, a sketch of an oceanic
are therefore more buoyant. We have considered these features of seafloor spreading in detail in other
ridge.
chapters. As the rock cools it gains a TRM and the direction of the magnetization depends upon the
polarity of the field when the rock formed. The blocks of normal and reversed magnetization produces
153 parallel to the ridge crest. These stripes then provide
lineated variations in the geomagnetic field aligned
a record of the direction and rate of sea-floor spreading between two plates and may be used to help
reconstruct plate motions and so unravel the evolution of the ocean basins.
5.12 Controls on oceanic magnetic anomalies . A magnetic anomaly is the magnetic field remaining
after the reference field (produced by Gauss coefficients such as those listed in Table 5.1) has been
removed. Anomalies usually result from the magnetization of nearby geologic features. Consider the
Fig. 5.14
Fig. 5.15
Fig. 5.23
As the oceanic As
plates
diverge,
molten
mantle
flowsflows
upward
gap,this
this
magma
the oceanic
plates
diverge,
molten material
mantle material
upwardtotofill
fill the
the gap,
magma
then then
cools by
conductive
to the surface.
The ridge
is elevated
because the
the rocks
hotter
and and
cools by conductive
heat
loss toheat
theloss
surface.
The ridge
is elevated
because
rocksareare
hotter
are
therefore
more
buoyant.
We
have
considered
these
features
of
seafloor
spreading
in
detail
in
other
are therefore more buoyant. We have considered these features of seafloor spreading in detail in other
chapters. As the rock cools it gains a TRM and the direction of the magnetization depends upon the
chapters. As the
rock ofcools
it gains
a TRM
and the
of theandmagnetization
depends
upon the
polarity
the field
when the
rock formed.
Thedirection
blocks of normal
reversed magnetization
produces
polarity of the field
when
the rock
The
blocks
normal
and
reversed
magnetization
produces
lineated
variations
in theformed.
geomagnetic
field
alignedofparallel
to the
ridge
crest. These
stripes then provide
a record
of geomagnetic
the direction andfield
rate aligned
of sea-floor
spreading
between
platesThese
and may
be used
to help
lineated variations
in the
parallel
to the
ridgetwo
crest.
stripes
then
provide
reconstruct
plate
motions
and
so
unravel
the
evolution
of
the
ocean
basins.
a record of the direction and rate of sea-floor spreading between two plates and may be used to help
Controlsand
on oceanic
magnetic
anomaly
is the Fig
magnetic
remaining
reconstruct plate5.12
motions
so unravel
the anomalies
evolution. Aofmagnetic
the ocean
basins.
5.24field
shows
a picture
the reference
fieldcoast
(produced
by Gauss coefficients
such asdetailed
those listed
in Tableof
5.1)
hasage
beenof the
of the magneticafter
anomalies
off the
of California
which allows
mapping
the
removed. Anomalies usually result from the magnetization of nearby geologic features. Consider the
ocean floor (though
see the next section where some of the complexities associated with this process
seamount in Figure 5.16:
are discussed). This
to a was
quantitative
estimate
of of
spreading
rates.andIf has
weamake
the assumption
Thisleads
seamount
extruded during
a period
normal polarity
magnetization
parallel to that
the Earth’s
magnetic
field. To
thetime,
South the
(right),
the field
byanomalies
the seamountcan
addsbetoused
the Earth’s
spreading rates are
relatively
constant
over
pattern
of generated
magnetic
to extend
field much
and to the
North –(left)
it opposes
thethe
Earth’s
The
intensity
of the net
field–isabout
measured
by a (see
the time scale back
further
all the
way to
agefield.
of the
oldest
oceanic
crust
140Ma
ppt for examples). Where possible, radiometric dates have
been tied into the reversal pattern and the
146
resulting time scale has a wide range of applications from marine geophysics to stratigraphy.
5.12 Controls on oceanic magnetic anomalies . A magnetic anomaly is the magnetic field remaining
after the reference field (produced by Gauss coefficients such as those listed in Table 5.1) has been
removed. Anomalies usually result from the magnetization of nearby geologic features. Consider the
seamount in Figure 5.25:
Fig.Fig.
5.16
5.25
Thismagnetometer
seamount towed
was extruded
during
a period
of normal
polarity
and has aareference
magnetization
parallel to
behind a ship.
A magnetic
anomaly
is calculated
by subtracting
field
the Earth’s
magnetic
To magnitude
the Southof(right),
the field
generated
by thenanoTeslas
seamount
adds to the Earth’s
from the
measuredfield.
field. The
these anomalies
is only
a few hundred
or about
1% of
field.
Referring
to our seamount,
therefield.
wouldThe
be a negative
anomaly
the field
left and
field and
to the
thedipole
North
(left)
it opposes
the Earth’s
intensity
of thetonet
isameasured by a
positive onetowed
to the right.
magnetometer
behind a ship. A magnetic anomaly is calculated by subtracting a reference field
Magnetic anomalies with respect to normal oceanic crust are similarly controlled by the magnetization
from the
measured
field.anThe
magnitude
ofthat
these
is only aoffew
hundred
or about
of the
sea-floor. Using
argument
similar to
madeanomalies
for the measurement
gravity,
it can benanoTeslas
easily
1% ofshown
the dipole
field. Referring
toonly
ourmeasure
seamount,
there would
be a negative
to the left and a
that magnetometers
generally
the anomalous
field intensity
parallel to anomaly
the reference
field
andtoare
positive
one
theinsensitive
right. to slight variations perpendicular to it. It also interesting to point out that
normallyanomalies
magnetized with
blocksrespect
do not always
produce
positivecrust
anomalies
as shown below:
Magnetic
to normal
oceanic
are similarly
controlled by the magnetization
of the sea-floor. Using an argument similar to that made for the measurement of gravity, it can be
easily shown that magnetometers generally only measure the anomalous field intensity parallel to the
154
Fig. 5.16
magnetometer towed behind a ship. A magnetic anomaly is calculated by subtracting a reference field
from the measured field. The magnitude of these anomalies is only a few hundred nanoTeslas or about
1% of the dipole field. Referring to our seamount, there would be a negative anomaly to the left and a
positive one to the right.
Magnetic anomalies with respect to normal oceanic
crust are similarly controlled by the magnetization
Fig. 5.24
of the sea-floor. Using an argument similar to that made for the measurement of gravity, it can be easily
shown that magnetometers generally only measure the anomalous field intensity parallel to the reference
insensitive
to slight
variations
perpendicular
to it. It alsotointeresting
point out thatto point out
reference field
fieldand
andareare
insensitive
to slight
variations
perpendicular
it. It alsoto interesting
normally
magnetized
blocks
do
not
always
produce
positive
anomalies
as
shown
below:
that normally magnetized blocks do not always produce positive anomalies as shown below (fig 5.26):
Fig. 5.26
When both the crustal magnetization and theFig.
geomagnetic
field are steep, the normal blocks produce
5.17
155
When both the crustal magnetization and the geomagnetic field are steep, the normal blocks produce
positive anomalies. At the equator, however, blocks magnetized in the same direction as the geomagnetic
field produce negative anomalies. In the equatorial case, when the lengths of the blocks is long compared
147
positive anomalies. At the equator, however, blocks magnetized in the same direction as the geomagnetic
field produce negative anomalies. In the equatorial case, when the lengths of the blocks is long compared
to the depth of the ocean, as when the spreading center is oriented North-South, no anomaly at all will
be detected, because the observer is so far from the edge of the block.
5.13 Geomagnetic field intensity . We turn now to the question of the variation in intensity of the
Earth’s magnetic field. It is possible to determine the intensity of ancient magnetic fields, because the
principle mechanisms by which rocks get magnetized (i.e., thermal, chemical, and detrital) produce
remanent magnetizations that are linearly related to the ambient field (for low fields such as the Earth’s).
What is required, then, is to determine the proportionality factor relating magnetization and field. For
example, since TRM is proportional to B, all we need do, in principle, is measure the natural remanent
magnetization (NRM), and then give the rock a TRM in the laboratory in a known field. Since
N RM
T RMlab
=
Blab
Bancient
the ambient field responsible for the NRM (Bancient ) is easily calculated. Similarly, for a DRM, one
would just redeposit the sediment in a known field and thereby calculate the ratio of DRM to the field,
then calculate the ancient field from the NRM.
In practice, however, there are problems which limit the usefulness of each approach. The problems
are different for the different magnetization mechanisms; we consider TRM first. The principal failing
of the simple method outlined above is that most rocks alter when heated in the laboratory and so the
TRM acquired by heating the rock to above its Curie Temperature and cooling in a known field may have
no relationship to the TRM acquired originally. Minerals such as maghemite are unstable at elevated
temperatures and convert to other magnetic phases. Magnetite itself oxidizes. Such changes in the
magnetic mineralogy have a drastic effect on the magnetization and careful lab work and consistency
checks are needed to detect such changes and compensate for them. Most data quoted for absolute
paleofield intensity values are determined in this or similar ways. Types of materials that carry TRM
are igneous bodies and archeological material such as baked hearths or pottery. Recently rapidly
cooled submarine basaltic glass has been used very successfully to recover a large number of absolute
paleointensity measurements.
The problem with sedimentary paleointensity data is that, laboratory conditions cannot duplicate the
natural environment. First of all, most sediments carry a post-depositional remanence as opposed to a
depositional one. Secondly, it seems that the intensity of remanence is a function of not only field but
magnetic mineralogy and even chemistry of the water column. Since mineralogy and chemistry can vary
with climate conditions considerable care is required to ensure that these variations can be adequately
compensated for. The best we can presently hope for from sediments is relative paleointensity variations
with time. An example of a such a record is provided in Figure 5.27. This particular record seems
to show higher field intensities correlating with lower reversal rate, but this result has not so far been
confirmed elsewhere.
Many such records are available covering the past few hundred thousand years, and these have been
averaged to provide an estimate of geomagnetic dipole moment variations. The field varies substantially
over this time interval. Data from the last 5 million years shows that the present field is somewhat higher
than average and that it ranges from about 10% of the average to about twice that value.
Because the intensity of the geomagnetic field varies by about a factor of two over the surface of
the Earth, paleomagnetists have often found it convenient to express paleointensity values in terms
of the equivalent geocentric dipole moment which would have produced the observed intensity at
that paleolatitude. This moment is called the virtual dipole moment or VDM. First, the magnetic
paleocolatitude, θm , is calculated as before from the observed inclination and using the dipole formula
tan (I) = 2 cot (θm ) (see eqn 5.10 and 5.11). Then
156
Figure 5.27: A 9 million year record of relative geomagnetic paleointensity variations from ODP site 522
(middle curve, fit to data points). Top panel shows the reversal rate throughout the record, bottommost
curve is the polarity interval length.
V DM =
1
4πr3
Bancient (1 + 3 cos 2 θm )− 2
µo
(5.25)
5.14 The external magnetic field. The largest component of the external variations in Earth’s
magnetic field comes from a ring of current circulating westward at a distance of 2-9 Earth radii.
The strength of the ring current is characterized by the ‘Dst’ (disturbance storm time) index, an index
of magnetic activity derived from a network of low- to mid-latitude geomagnetic observatories that
measures the intensity of the globally symmetrical part of the equatorial ring current.
Magnetic storm of June 1982
sudden commencement
0
-50
Dst, nT
-100
-150
recovery
-200
-250
-300
1982.53
1982.535
1982.54
Year
157
1982.545
1982.55
Figure 5.28: A typical magnetic storm as recorded by the Dst index.
The solar wind injects charged particles into the ring current, mainly positively charged oxygen
ions. Sudden increases in the solar wind associated with coronal mass ejections cause magnetic storms
(Figure 5.28), characterized by a sudden commencement, a small but sharp increase in the magnetic field
associated with the sudden increased pressure of the solar wind on Earth’s magnetosphere. Following
the commencement the main phase of the storm is associated with a large decrease in the magnitude of
the fields as the ring current is energized – the effect of the ring current is to cancel Earth’s main dipole
field slightly. Finally, there is a recovery phase in which the field returns quasi-exponentially back to
normal. All this can happen in a couple of hours, or may last days for a large storm.
On the 13th March 1989 a 600 nT magnetic storm caused the entire power grid in Quebec, Canada,
to collapse.
There is also a daily variation in the magnetic field of about 30–50 nT as a result of the sun
heating the electrically conductive ionosphere on the day-time side. Finally, lightning strikes produce
high frequency impulses and also excite a resonance in the Earth-iononsphere cavity at 7–8 Hz and
harmonics, the so-called Schumann resonances.
5.15 Electromagnetic induction. One of the key concepts in geomagnetic induction is that of the
skin depth, the characteristic length over which electromagnetic fields attenuate.
We can derive the skin depth starting with Faraday’s Law:
∇×E=−
∂B
∂t
and Ampere’s Law:
∇ × B = µo J
2
where J is current density (A/m ), E is electric field (V/m), B is magnetic flux density or just magnetic
field (T). We can use the absence of magnetic monopoles and the identity ∇ • ∇ × A = 0 to show that
∇•B=0
and
∇•J=0
in regions free of sources of magnetic fields and currents. B and H (magnetizing field, with units of
A/m) are related by magnetic permeability µ and J and E by conductivity σ :
B = µH
J = σE
(the units of σ are S/m, where S = 1/Ω; the latter equation is Ohm’s law), and so
∇×E=−
∂B
∂t
∇ × B = µo σE
.
If we take the curl of these equations and use ∇ × (∇ × A) = ∇(∇ • A) − ∇2 A to get
∂E
∂
(∇ × H) = µσ
(5.27)
∂t
∂t
∂B
∇2 B = σ(∇ × E) = µσ
.
(5.28)
∂t
You will recognize these as diffusion equations (1/µσ acts as a diffusivity, and if you plug in the units
∇2 E = µ
it does indeed come out with units of m2 /s). Now if we consider sinusoidally varying fields of angular
frequency ω and a uniform conductivity σo
E(t) = Eo eiωt
158
∂E
= iωE
∂t
∂B
= iωB
∂t
B(t) = Bo eiωt
and so
∇2 E = iωµo σo E
∇2 B = iωµo σo B
(5.29)
.
(5.30)
These are the equations describing propagation of electric and magnetic fields in a conductive medium.
In air and very poor conductors where σ ≈ 0, or if ω = 0, the equations reduce to Laplace’s equation.
If we now consider fields that are horizontally polarized in the xy directions and are propagating
vertically into a half-space, these equations decouple to
∂2E
+ k2 E = 0
∂z 2
∂2B
+ k2 B = 0
∂z 2
with solutions
E = Eo e−ikz = Eo e−iαz e−βz
B = Bo e
−ikz
= Bo e
(5.31)
−iαz −βz
e
(5.32)
iωµσ = α − iβ
(5.33)
where we have defined a complex wavenumber
k=
p
and an attenuation factor, which is called a skin depth
r
zs = 1/α = 1/β =
2
σµo ω
.
(5.34)
The skin depth is the distance that field amplitudes are reduced by 1/e, or about 37%, and the phase
progresses one radian, or about 57◦ . In practical units
p
zs ≈ 500m 1/(σf )
where circular frequency f is defined by ω = 2πf .
The skin depth concept underlies all of inductive electromagnetism in geophysics. Substituting a few
numbers into the equation shows that skin depths cover all geophysically useful length scales from less
than a meter for conductive rocks and kilohertz frequencies to thousands of kilometers in mantle rocks
and periods of days (see table). Skin depth is a reliable indicator of maximum depth of penetration.
material
core
lower mantle
seawater
sediments
upper mantle
igneous rock
σ , S/m
5
3×10
10
3
0.1
0.01
1×10−5
1 year
1 month
1 day
1 hour
1 sec
1 ms
4 km
900 km
1600 km
9000 km
3×104 km
106 km
770 m
170 km
470 km
1700 km
104 km
2×105 km
200 m
46 km
85 km
460 km
1500 km
4×104 km
40 m
9 km
17 km
95 km
300 km
9500 km
71 cm
160 m
280 m
1.6 km
5 km
160 km
23 mm
5m
9m
50 m
158 m
5 km
Skin depths for a variety of typical Earth environments and a large range of frequencies. One can see
that the core is effectively a perfect conductor into which even the longest period external magnetic field
cannot penetrate. However, 1-year variations can penetrate into the lower mantle. The very large range
of conductivities found in the crust indicates the need for a corresponding large range of frequencies in
electromagnetic sounding of that region.
159
5.16 Geomagnetic depth sounding. Geomagnetic depth sounding, or GDS, is used to derive the
electrical conductivity inside Earth using measurements of the magnetic field. The geomagnetic ring
current generates a field that is largely dipolar in morphology (it is like being inside a big solenoid), and
if one chooses a coordinate system defined by the dipole field (a geomagnetic coordinate system), then
the field can be described as P10 in morphology. Equation 5.6 reduces to
V (r, θ) =
2 0 a
0 r
+ (gi )1
(ge )1
a
r
aP10 ( cos θ)
(5.35)
(notice that we have kept the internal AND external Gauss coefficients, since we will be interested in
the internally induced fields).
One can define a geomagnetic response at a given frequency ω as simply the ratio of induced (internal)
to external fields:
Q(ω) =
gi (ω)
ge (ω)
.
(5.36)
In certain circumstances, such as satellite observations, one has enough data to fit the ge and gi
directly. However, most of the time one just has horizontal (H ) and vertical (Z ) components of B as
recorded by a single observatory. We can obtain H and Z from the appropriate partial derivatives of V
to obtain:
1 ∂V
r ∂θ
−H = µo
r=a
∂
∂
= µ(ge + gi ) P10 ( cos θ) = µ(ge + gi ) ( cos θ)
∂θ
∂θ
= µAH sin θ
and
(5.37)
∂V
−Z = µ
∂r r=ao
= µ(ge − 2gi )P10 ( cos θ)
= µAZ cos θ
(5.38)
where we have defined new expansion coefficients
AH = ge + gi
AZ = ge − 2gi
.
Thus we can define a new electromagnetic response
W (ω) =
AZ
Z(ω) sin θ
=
H(ω) cos θ
AH
(5.39)
where θ is the co-latitude of the observatory and we have included the ω dependence to remind us that
this is all for a particular frequency. W is related to Q by
Q(ω) =
gi (ω)
1 − AZ /AH
=
ge (ω)
2 + AZ /AH
.
The admittance or inductive scale length c provides a practical unit for interpretation, and is given by
c(ω) =
a
W (ω).
2
160
(5.40)
Note that for causal systems, the real part of c is positive, and the imaginary part is always negative. For
a uniform earth of conductivity σa , the resistivity is given by
ρa = 1/σa = ωµo |ic|2
.
(5.41)
3000
2500
2000
C, km
1500
Real (C)
1000
500
0
−500
−1000
Imag (C)
1 hour
3
1 day
4
5
1 month
6
Log (Period, s)
1 year
7
11 years
8
9
Figure 5.29: The admittance, c, based on an analysis of geomagnetic observatory data. The red line
is a fit to the data for the model shown in Figure 5.30.
Conductivity, S/m
Core
LOWER MANTLE
10 1
Perovskite at 2000°C
UPPER MANTLE
10 2
TRANSITION ZONE
10 3
Bounds for monotonic models
10 0
Ringwoodite/
Wadsleyite at 1500°C
10
Bounds on all models
−1
Olivine at 1400°C
10 −2
10 −3
0
500
1000
1500
Depth, km
2000
2500
3000
Figure 5.30: Electrical conductivity in Earth’s mantle derived from the GDS data shown in Figure
5.29. The blue line shows one model that fits the data (the fit is shown in Figure 5.29), but the
yellow regions give upper and lower bounds on average conductivity for the three mineralogical mantle
regimes. Black dots are conductivities of representative mantle minerals at appropriate pressures and
temperatures.
161