CHAPTER 5

Earth’s Magnetic Field

5.1 Introduction . Geomagnetism or the study of Earth’s magnetic field has a long history and hasrevealed much about the way the Earth works. As we shall see, the existence and characteristics of thefield 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 inferinformation about the age of the rocks, track the past motions of continents, and leads directly to theidea of sea-floor spreading. Variations in the external part of the geomagnetic field induce secondaryvariations 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 betweenthe behavior observed for a compass when moved around a sphere of magnetic material and the behaviorof a compass needle on Earth’s surface. His findings were published in 1600, predating Newton’sgravitational Principia by about 87 years. The magnetic compass had been in use, beginning with theChinese, since about the second century B.C., but temporal variations in the magnetic field were notwell documented until the seventeenth century by Henry Gellibrand. In 1680 Edmund Halley publishedthe first contour map of the geomagnetic variation as the declination was then known: he envisioned thesecular variation of the field as being caused by a collection of magnetic dipoles deep within the earthdrifting westward with time with about a 700 year period, a model not dissimilar to many put forwardthis century, although he did not know of the existence of the fluid outer core. A formal separationof the geomagnetic field into parts originating inside and outside the earth was first achieved by theGerman mathematician Karl Friedrich Gauss in the nineteenth century. Gauss deduced that by far thelargest contributions to the magnetic field measured at Earth’s surface are generated by internal ratherthan external magnetic sources, thus confirming Gilbert’s earlier speculation. He was also responsiblefor beginning the measurement of the geomagnetic field at globally distributed observatories, some ofwhich are still running today. The external contributions to the magnetic field arise from the fact thatEarth sits in a (small) interplanetary magnetic field and is in continual interaction with the temporallychanging solar wind. These time-varying magnetic fields also induce secondary magnetic fields in therocks in Earth’s crust and mantle. The internal magnetic field plays an important role in protecting usfrom cosmic ray particle radiation, because incoming ionized particles can get trapped along magneticfield lines, preventing them from reaching Earth. One consequence of this is that rates of production ofradiogenic nuclides such as 14C and 10Be are inversely correlated with fluctuations in geomagnetic fieldintensity.

The internal magnetic field can be divided into contributions from the crust and those originating inthe fluid outer part of Earth’s core. At Earth’s surface the crustal part is orders of magnitude weaker thanthat from the core, but remanent magnetization carried by crustal rocks has proved very important inestablishing 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 occurrenceof geomagnetic jerks is one notable exception); shorter period variations are usually attenuated by theirpassage 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 asnormal, the opposite polarity is reversed.

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The present and historical magnetic field is measured at observatories, by surveys on land and atsea, and from aircraft. Since the late 1950s a number of satellites, each carrying a magnetometer inorbit around Earth for months at a time, have provided more uniform coverage than previously possible.Early satellites only measured the magnitude of the field: however, it was shown in the late 1960sthat measurements of the field’s direction are also required to specify the field accurately. Prehistoricmagnetic field records can also be obtained through paleomagnetic studies of fossil magnetism recordedin rocks and archeological materials.

The magnetic field is a vector quantity, possessing both magnitude and direction; at any point on Eartha compass needle will point along the local direction of the field. Although we conventionally think ofcompass needles as pointing north, it is the horizontal component of the magnetic field that is directedapproximately in the direction of the North Geographic Pole. The three parameters conventionally usedto describe the magnetic field are intensity, B measured in Teslas, declination, D, and inclination, I(both in degrees). Figure 5.1 shows how they are related to conventional Cartesian coordinates centeredon a measurement location on the surface of the Earth, and oriented North (N), East (E) and down (V):

normal, the opposite polarity is reversed.

The present and historical magnetic field is measured at observatories, by surveys on land and at

sea, and from aircraft. Since the late 1950s a number of satellites, each carrying a magnetometer in

orbit around Earth for months at a time, have provided more uniform coverage than previously possible.

Early satellites only measured the magnitude of the field: however, it was shown in the late 1960s

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

is directed approximately in the direction of the North Geographic Pole. The difference in azimuth

between magnetic north and true or geographic north is known as declination (positive eastward) and

may be as much as several tens of degrees. The field also has a vertical contribution; the angle between

the horizontal and the magnetic field direction is known as the inclination and is by convention positive

downward (see Figure 5.1). At Earth’s surface today the field 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 thisdipole penetrates Earth’s surface at a longitude of of 288.9! E. The magnitude of the field, the magneticflux density passing through Earth’s surface, is measured in microTeslas, abbreviated as µT , with 1µT= 10"6T, and 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. Contoursof declination, inclination, and magnitude of the geomagnetic field for 1990 are shown on Figure 5.2.

The three parameters conventionally used to describe the magnetic field are intensity,B, declination,D, and inclination, I . Figure 5.1 shows how they are related to conventional Cartesian coordinatescentered on a measurement location on the surface of the Earth, and oriented North (N), East (E) and

down (V):

Figure 5.1

!Bh is the projection of the field vector onto the horizontal plane and Bz is the projection onto the

vertical axis. D is measured clockwise from North and can range from 0 ! 360!, or equivalently from"180 ! 180!. Note that except at high latitudes the range ofD is fairly small (# ±20! ). I is measuredpositive down from the horizontal and ranges from"90 ! +90! (because field lines can also point outof the Earth). From the diagram we have

Bh = B cos I and Bz = B sin I (5.1)

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Figure 5.1

Bh is the projection of the field vector onto the horizontal plane and Bz is the projection onto thevertical axis. D is measured clockwise from North and can range from 0→ 360, or equivalently from−180→ 180. Note that except at high latitudes the range of D is fairly small ( ∼ ±20 ). I is measuredpositive down from the horizontal and ranges from −90→ +90 (because field lines can also point outof the Earth). From the diagram we have

Bh = B cos I and Bz = B sin I (5.1)

The horizontal projection can also be projected onto the North (x) and East (y) axes (the directions inwhich measurements are usually made), i.e.,

Bx = B cos I cosD and By = B cos I sinD (5.2)

Maps of the present day values of B, D and I (Fig. 5.2) show that the field is a complicated functionof position on the surface of the Earth although it is approximately that of a dipole located at the centerof the Earth, with its axis tilted about 11 relative to the geographic axis; the axis of this dipole penetratesEarth’s surface at a longitude of of 288.9 E. The magnitude of the field is about twice as great at thepoles (about 60 µT) as at the equator (about 30 µT). Inclination ranges from +90 at the north magneticpole to -90 at the south magnetic pole. Declination may be a few tens of degrees.

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Oersted initial field model 2000, intensity, microTesla

510152025303540455055606570

Oersted initial field model 2000, inclination, degrees

-80-70-60-50-40-30-20-1001020304050607080

Oersted initial field model 2000, declination, degrees

-160-140-120-100-80-60-40-20020406080100120140160

Figure 5.2: Geomagnetic declination, inclination, and intensity for the year 2000

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IGRF 1990 Z and its time deriv-200-180-160-140-120-100-80-60-40-20020406080100120140160180

IGRF 1990, X and its time deriv-200-180-160-140-120-100-80-60-40-20020406080100120140160180

IGRF 1990, Y and its time deriv-200-180-160-140-120-100-80-60-40-20020406080100120140160180

Figure 5.3: X,Y, Z (contours in nT) for the International Geomagnetic Reference Field for 1990, andtheir time derviatives (color, nT/year).

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5.2 Some Physics of Magnetism . Magnetic fields are caused by electrical charge in motion. In 1819Oersted discovered that electric currents produce a magnetic force when he observed that a magneticneedle is deflected at right angles to a conductor carrying a current. Magnetic fields are also produced bypermanent magnets: although there is no conventional electric current in them, there are orbital motionsand spins of electrons (sometimes called “Amperian currents") which lead to a magnetization within thematerial and a magnetic field outside. The cooperative behavior that leads to permanent magnetizationis a quantum mechanical effect: however for our purposes we can describe the macroscopic effectsof 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 andmagnetization. The distinction is often not clearly made and units are used interchangeably, especiallybetween 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 H, measured in A/m.

(2) Magnetization M; also known as the intensity of magnetization or magnetic moment per unitvoume (M = m

V with m the magnetic moment). It is also measured in A/m.

(3) Magnetic induction B, also called magnetic flux density, also just called the field, measured inTesla, although the practical units of nT, µT, and mT are also used.

These quantities are related through the equation

B = µ0(H + M)

where µ0 is the permeability of free space (µ0 = 4π × 10−7NA−2, or equivalently H/m, or equivalentlyTm/A). The use of B and H in magnetics can be confusing, but in the absence of magnetization therelationship is always B = µ0H.In older books and publications you may come across gammas (γ) where 1 gamma = 10−5 Gauss, thecgs unit of magnetic field H. Through the vagaries of µo in cgs units, 1 gamma can be equated to 1nanoTesla (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 equalto a Coulomb of charge per second. The volume equivalent is current density J, which has units ofA/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 aproperty of the space in which the force acts, and the pattern of a field is portrayed by field lines. At anypoint in a field the direction of the force is tangential to the field line, and the intensity of the force isproportional to the density of field lines. Unlike gravitational and electrostatic forces, both of which actcentrally to the source of the force and follow an inverse square law, magnetic fields vary with azimuthand even in the simplest case of a magnetic dipole the field strength falls off as the cube of distance.Although magnetic monopoles do not really exist, they provide a useful mechanism for describing theeffects of magnetic fields and the concept of a magnetic potential analogous to gravitational potential.

5.3 Magnetic potential . It is useful to have a compact mathematical representation of the magneticfield for a particular time. Under certain assumptions, the magnetic field can be written as the gradientof a scalar magnetic potential (just as in the case of gravity). The assumptions required are basically thatthere are no electromagnetic sources in the measurement region and no time variations in the magneticfield. This is a good approximation in the atmosphere (since the conductivity close to the ground is10−13 S/m there are effectively no electric currents there). If we ignore time variations in the field then

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outside the Earth where the magnetic field is being generated B can be derived from a scalar magneticpotential 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 = ∇2Vm = 0 (5.4)

This is Laplace’s equation, which is also used to describe the behavior of the gravitational potential, andis the starting point for what is known as a spherical harmonic analysis of the field. Of course, we don’tmake measurements of the potential itself but of its gradient, the magnetic field (equation 5.3), which,is related to the potential by

Bx = −1r

∂Vm∂θ

and By = − 1r sin θ

∂Vm∂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=1

n∑m=0

Pmn ( cos θ)

[((ge)mn

( ra

)n+ (gi)mn

(ar

)n+1)

cosmφ

+(

(he)mn( ra

)n+ (hi)mn

(ar

)n+1)

sinmφ

](5.6)

where gmn and hmn are known as Gauss coefficients. Note that there is no n = 0 term because there areno magnetic monopoles. The e and i subscripts indicate external or internal origin for the sources, and ais the radius of the Earth (6.371× 106 m). Pmn are proportional to the associated Legendre polynomials,pnm. The normalization in geomagnetic work is usually the Schmidt normalization i.e.,

Pmn =(

2(n−m)!(n+m)!

) 12pnm for m 6= 0,

P 0n =pn0

Most importantly for our purposes

P 01 ( cos θ) = cos θP 1

1 ( cos θ) = sin θP 0

2 ( cos θ) = 12 (3 cos 2θ − 1)

P 03 ( cos θ) =

cos θ2

(5 cos 2θ − 3)

The P 0n 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 sphericalharmonic expansion to describe arbitrarily complicated magnetic fields.

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where we used the binomial expansion. This expression for g is correct to first order in the smallquantities !gr and !gt. To this accuracy, we see that we can neglect the transverse component, !gt, and

we only have to worry about !gr, i.e., to first order g still points towards the center of the body.!gr is a function of latitude. The full solution for the latitude dependence requires the use of

spherical harmonic analysis and we only outline the method here. It also turns out to be easiest to use the

gravitational potential, V , (see equation 4.5) because it is a scalar. We can always get the accelerationdue to gravity by differentiating the gravitational potential. From equation 4.5, we have that, outside a

spherical earth, the gravitational potential is just given by

V = !GM

r(4.18)

(Note that equation 4.15 for g outside a spherical earth can be gotten by differentiating the aboveequation). It is also easy to show that, outside a spherical Earth,

1r2

d

dr

!r2 dV

dr

"= "2V = 0 (4.19)

(in spherical symmetry.) The equation "2V = 0 is called Laplace’s equation and also holds outside anon-spherical Earth. In the case of a rotationally distorted Earth, the potentialV depends upon both r and" (the colatitude). In this case, the solution to Laplace’s equation can be written as a series expansion:

V (r, ") = !GM

r[J0P0( cos ") !

a

rJ1P1( cos ") !

a2

r2J2P2( cos ") !

a3

r3J3P3( cos ") · · ·] (4.20)

(a is the equatorial radius of the Earth). In this expansion, the J’s are empirically determined coefficientsand the P ’s are “Legendre polynomials.” The first few are

P0( cos ") = 1

P1( cos ") = cos "

P2( cos ") = 12 (3 cos 2" ! 1)

P3( cos ") =cos "

2(5 cos 2" ! 3)

Fig. 4.7

These functions, plotted in Figure 4.7, are progressively wigglier functions of the colatitude " and wecan choose the J’s so that the sum can approximate very accurately any observed dependence of V on

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Fig 5.4The Gauss coefficients are determined by fitting the gradient components of equation 5.6 to observa-

tions from magnetic observatories or satellite data for a particular time epoch. A maximum value of nto be used in the fitting procedure is usually assigned beforehand so that a finite number of coefficientscan be resolved. (A better technique is to ask for smooth models so that the resulting maps do not showunnecessary structure in locations with little data.) Perhaps the most important result of this kind ofanalysis is that 99% of the field is of internal origin. The coefficients for the internal field and their ratesof 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 isrepresented by the terms where n=1 (check out g0

1). The functions associated with coefficients of degree1 have wavelengths of one Earth circumference and can be thought of as representing geocentric dipolesalong three different axes: the spin axis (g0

1) and two equatorial axes intersecting at the Greenwichmeridian (g1

1) 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 isindicated by white and inward is denoted by black.

Fig 5.5. Note that g01 multiplies the shape denoted Y 0

1 . g11 multiplies the shape denoted Re(Y 1

1 ) andh1

1 multiplies the shape denoted Im(Y 11 ). g0

2 multiplies the shape denoted Y 02 and so on. The sum gives

the total potential and the observations are given by the appropriate derivatives of the potential.

One really important consequence of the spherical harmonic representation is that it is not only validat the place where the measurements are made. The model can be evaluated anywhere where Laplace’sequation holds: that is in the region where there are no sources. So far we have only assumed that thisis the case in the atmosphere, but a useful approximation is to suppose that we can neglect sources inboth the crust and mantle – effectively we assume that they can be considered electrical insulators with

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Table 5.1Gauss coefficients for the 1990 reference field (nT)

n m gmn hmn∂gm

n

∂t∂hm

n

∂t

1 0 -29775.4 0. 18.0208 01 1 -1850.99 5410.86 10.56840 -16.0712 0 -2135.81 0. -12.9179 02 1 3058.23 -2277.66 2.39650 -15.7802 2 1693.22 -380.030 -2.89000E-02 -13.7893 0 1314.58 0. 3.32890 03 1 -2240.19 -286.500 -6.66770 4.42103 2 1245.57 293.270 6.19000E-02 1.57653 3 806.540 -348.470 -5.86330 -10.55544 0 938.870 0. 0.480900 04 1 782.280 248.080 0.611800 2.55954 2 323.870 -239.530 -7.01810 1.81734 3 -422.730 87.0300 0.544800 3.09724 4 141.660 -299.380 -5.53540 -1.37855 0 -211.030 0. 0.630900 05 1 352.510 47.1700 -0.137700 -0.119505 2 243.790 153.470 -1.63150 0.461005 3 -110.780 -154.450 -3.11570 0.449105 4 -165.580 -69.2300 -6.65000E-02 1.65995 5 -37.0400 97.6700 2.31650 0.408406 0 60.6900 0. 1.28690 06 1 63.9400 -15.7800 -0.182100 0.246406 2 60.3600 82.7300 1.81150 -1.34756 3 -177.510 68.2900 1.31210 -3.80000E-06 4 2.04000 -52.4800 -0.171900 -0.881206 5 16.7100 1.79000 0.127200 0.452206 6 -96.2600 26.8500 1.15840 1.22447 0 76.5600 0. 0.589300 07 1 -64.1900 -81.0800 -0.506800 0.616307 2 3.71000 -27.3000 -0.307200 0.191207 3 27.5500 0.590000 0.626700 0.772307 4 0.940000 20.4300 1.58880 -0.520607 5 5.74000 16.3800 0.173200 -0.222107 6 9.77000 -22.6300 0.170700 4.41000E-07 7 -0.460000 -4.96000 0.292900 -3.43000E-08 0 22.4100 0. 0.165600 08 1 5.14000 9.74000 -0.676300 0.512508 2 -0.880000 -19.9300 -0.171600 -0.208208 3 -10.7600 7.09000 0.142600 0.328308 4 -12.3700 -22.1000 -1.12770 0.285708 5 3.79000 11.8700 -3.94000E-02 0.374208 6 3.78000 11.0000 -5.32000E-02 -0.458508 7 2.64000 -16.0100 -0.484300 -0.315408 8 -6.02000 -10.6900 -0.605300 0.60310

135

no permanent magnetization. Then it is possible to downward continue the field representation to thesurface of Earth’s core (r = .547a). Because the sources are much closer the higher degree (spatiallymore 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 thesurface 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 fieldevaluated 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 fromthe core – higher harmonics are dominated by crustal contributions so it is inappropriate to downwardcontinue these to the core. Clearly, the field from the core is masked by crustal contributions oncethe harmonic degree gets larger than about 14. Also note that the downward-continued field is stilldominated 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 pre-dominantly dipolar and internal in origin, to a first approximation we can neglect all but the internaldipole contributions to equation (5.6), writing

VD(r, θ, φ) = a(ar

)2

[g01P

01 ( cos θ) + P 1

1 ( cos θ)(g11 cosφ+ h1

1 sinφ)]

= a(ar

)2

[g01 cos θ + sin θ(g1

1 cosφ+ h11 sinφ)]

(5.7)

The dipole moment M = 4πa3

µ0(g1

1 , h11, g

01) in a geocentric cartesian coordinate system with rotation axis

along z, Greenwich meridian defining x and 90 E defining y.An even more drastic simplification known as the geocentric axial dipole (GAD) approximation for

the field supposes that the equatorial dipole contributions g11 and h1

1 can also be neglected, and the fieldrepresented as that from a dipole aligned with Earth’s rotation axis. Then we can write

136

VGAD = ag01

(ar

)2

P 01 ( cos θ) = ag0

1

(ar

)2

cos θ =µ0M cos θ

4πr2(5.8)

where M is 4πµ0g01a

3. Thus, from 5.5 (taking the three derivatives of the potential to get the three fieldcomponents

Bx =µ0M sin θ

4πr3and By = 0 and Bz =

2µ0M cos θ4πr3

(5.9)

Figure 5.7 is a cross section of the Earth, rotated so that the z-axis coincides with magnetic northand if the field were truly dipolar it clearly would not matter which cross section we choose because adipole field is rotationally symmetric about the axis going through the poles; in other words, By = 0.

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

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 (spatiallymore complex) sources appear much more important there and the field looks a lot more complicated.

5.5 The Dipole and Geocentric Axial Dipole Approximations . Since the geomagnetic field is predom-

inantly dipolar and internal in origin, to a first approximation we can neglect all but the internal dipole

contributions to equation (5.6), writing

VD(r, !, ") = a!a

r

"2[g0

1P 01 ( cos !) + P 1

1 ( cos !)(g11 cos" + h1

1 sin")]

= a!a

r

"2[g0

1 cos ! + sin !(g11 cos" + h1

1 sin")](5.7)

The dipole moment #M = 4!a3

µ0(g1

1 , h11, g

01) in a geocentric cartesian coordinate system with rotation

axis along z, Greenwich meridian defining x and 90! E defining y.An even more drastic simplification known as the geocentric axial dipole (GAD) approximation for

the field supposes that the equatorial dipole contributions g11 and h1

1 can also be neglected, and the field

represented as that from a dipole aligned with Earth’s rotation axis. Then we can write

VGAD = ag01

!a

r

"2P 0

1 ( cos !) = ag01

!a

r

"2cos ! =

µ0M cos !

4$r2(5.8)

whereM is 4!µ0

g01a3. Thus, from 5.5

Bx =µ0M sin !

4$r3and By = 0 and Bz =

2µ0M cos !

4$r3(5.9)

Figure 5.7 is a cross section of the Earth, rotated so that the z-axis coincides with magnetic north and

if the field were truly dipolar it clearly would not matter which cross section we choose because a dipole

field is rotationally symmetric about the axis going through the poles; in other words, By = 0.

Fig 5.7

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

Bx= 2 cot !m (5.10)

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Fig 5.7

Consider the measurement position on the surface of the Earth shown as a dot in Fig. 5.8.

Fig 5.8

Thus the inclination is directly related to the colatitude for a field produced by a geocentric axial dipole

(or g01). This allows us to calculate the distance away from the magnetic pole from the inclination of

the magnetic field, a result which will be useful in plate tectonic reconstructions. The intensity is also

related to !m because

B = (B2z + B2

x)12 =

µ0M

4"r3( sin 2!m + 4 cos 2!m)

12 =

µ0M

4"r3(1 + 3 cos 2!m)

12 (5.11)

5.6 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 by paleomagnetists as

#MI = $ #H

describes the induced magnetization acquired when a material is exposed to a magnetic field #H . $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 + $) #H = µ #H.

µ is the magnetic permeability. In practice $ may be dependent on field intensity, negative or need tobe 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

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Fig 5.8

Using the equations for Bz and Bx, we find that

tan I =BzBx

= 2 cot θm (5.10)

Thus the inclination is directly related to the colatitude for a field produced by a geocentric axial dipole(or g0

1). This allows us to calculate the distance away from the magnetic pole from the inclination ofthe magnetic field, a result which will be useful in plate tectonic reconstructions. The intensity is alsorelated to θm because

B = (B2z +B2

x)12 =

µ0M

4πr3( sin 2θm + 4 cos 2θm)

12 =

µ0M

4πr3(1 + 3 cos 2θm)

12 (5.11)

Because of difficulties in measuring the intensity of the field (see below) this equation is less useful thanequation 5.10.

137

5.5 Magnetic Poles and Dipoles and their Potentials . This section is intended to show some of theparallels between the magnetic potential and the gravitational potential by using two monopoles todescribe a dipole. In 1785 Coulomb showed that the force between the ends (or magnetic poles) of longthin magnetized steel needles obeyed an inverse square law. Thus supposing that monopoles existedand we have two with strengths p1 and p2 we can write an inverse square law for the force betweenthem, a Coulomb’s Law for magnetic poles

F (r) = Kp1p2

r2.

By analogy with the gravitational case we can define the magnetic field associated with a monopole p as

B(r) = Kp

r2

In SI units K = µ04π where µ0 = 4π × 10−7NA−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

V =∫ r

∞Bdr =

µ0p

4πr

In contrast to electrostatic charges, magnetic poles cannot exist in isolation: each positive pole must bepaired with a corresponding negative pole to form a magnetic dipole. We can construct the potential fora 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 pointof 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 pairof poles in a direction that makes an angle θ to the axis passing through both poles p and −p. if thedistances to the respective poles are r+ and r− respectively the net potential at (r, θ) will be

138

V =µ0p

4π( 1r+− 1r−

)=µ0p

4π(r− − r+r+r−

)Now we suppose that d r and use the approximations

r+ ≈ r −d

2cos θ

r− ≈ r +d

2cos θ′

Since d r we can write θ ≈ θ′ and on negelcting terms of order (d/r)2 we find

r− − r+ ≈ d cos θ

r+r− ≈ r2 −d2

4cos 2θ ≈ r2

and that the dipole potential at the point (r, θ) is

V (r, θ) =µ0

4π(dp) cos θ

r2=µ0m cos θ

4πr2

We call the quantity m = (dp) the magnetic moment of the dipole. Note that it is a vector quantity: ourcoordinate system is defined relative to the axis of the dipole. This is the same result as using the GAD(equation 5.8).

5.6 The origin of the magnetic field . From historical field measurements and paleomagnetic data (seebelow) it is clear that the field morphology changes quickly in geological terms. The implication is thatwhatever causes these fast changes is associated with rapid movement somewhere in the Earth. The onlyreasonable place for this to happen is the outer core which is known to be fluid (from seismology) andis inferred to be dominantly iron (from meteorites) and so is electrically conducting. Below a few 10’sof kilometers, the Earth is too hot to allow permanent magnetization and the field would decay rapidlyaway unless some regeneration mechanism is acting. We therefore must appeal to some convectivemotion of the outer core material resulting in dynamo action and so generating the field. To see this weconsider 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∂t

= ∇× (v ×B) + νm∇2B (5.12)

where B is the magnetic field, v is the velocity field, νm is the magnetic diffusivity and × indicatesvector cross product. The detailed analysis of this equation is beyond the scope of this course but weshall consider the physical meaning of the terms. The first term on the right hand side represents theinteraction of the magnetic field with the velocity field, the second term represents the diffusion ofmagnetic field through the material. In fact if the core fluid is at rest, i.e., ~v = 0, we have

∂B∂t

= νm∇2B (5.13)

This is a vector form of the diffusion equation. As the field diffuses through the material, electriccurrents are induced and the magnetic energy is converted to heat via ohmic dissipation. As magneticenergy is lost, the field strength decays. We can estimate this decay rate for a dipole field and find thatthe field strength decays to 1/e of its initial value in ∼15,000 years. i.e., the field strength varies as

B = B0e−t/τ (5.14)

139

where τ= 15,000 years.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 hasunits of m2/s. All diffusion equations have diffusivities with these kinds of dimensions. νm controls therate at which the field diffuses through the material and is related to the electrical conductivity, σ, by

νm =1µ0σ

(5.15)

where σ is estimated from shock wave experiments and theoretical calculations to be 3 × 105 Sm−1.Using these numbers gives

νm ≈ 2.65m2s−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 distanceL 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 arestuck to the material and move as the material moves – this is called the frozen-flux approximation. Witha finite νm, the field can move through the material causing electric currents to be induced and so givingohmic dissipation. On short time scales we can show by a dimensional analysis of 5.12 that for largescale changes in the magnetic field it is probably a reasonable approximation to neglect the diffusionterm and adopt the frozen flux approximation. The time scale for diffusion is

τd ≈L2

νm≈ 1010m2

3m2/s≈ 1010 s

while for convection it is

τv ≈L

U≈ 105m

10−3m2/s≈ 108 s

where we have taken L ≈ 105m, and U ≈ 10−3m/s to get diffusion times that are one or two ordersof magnitude larger than convection time scales. This is used as an argument in favor of neglectingdiffusion in large scale fields over short time intervals, and invoking the frozen-flux approximation. Itsvalidity on longer time intervals and for short length scales is more questionable. In the absence ofconvection, 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 itspresent value for billions of years.

If we reconsider equation 5.12, the first term describing field changes derived from convective motionsmust be able to counteract the second diffusion term sometimes giving growth of the field. We thereforeneed a mechanism for converting mechanical energy into magnetic energy and this is exactly what adynamo does.

The complete solution of the geodynamo problem turns out to be remarkably difficult. In 1919Larmor first proposed that a dynamo in a conductive body might be a possibility in the context ofsunspot magnetic fields, but this idea received a significant setback in 1934 when in a famous theoremCowling showed that axially symmetric flows in the core cannot generate a dynamo. This raisedfears that a general antidynamo theorem might exist, but in the 1950’s it was shown mathematicallythat dynamo action could in principle occur. That is specific kinds of velocity structures specified inequation 5.12 could result in self-sustaining magnetic field. The study of such velocity fields is known as

140

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

!d ! L2

"m

while for convection it is

!v ! L

U

If we take L ! 105m, and U ! 10!3m.s!1 we get diffusion times that 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 frompaleomagnetism and show that the field has existedwith 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 convectivemotions

must be able to counteract the second diffusion term sometimes giving growth of the field. We therefore

need a mechanism for converting mechanical energy into magnetic energy and this is exactly what a

dynamo does.

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

sunspot magnetic fields, but this idea received a significant setback in 1934 when in a famous theorem

Cowling showed that axially symmetric flows in the core cannot generate a dynamo. This raised fears that

a general antidynamo theorem might exist, but in the 1950’s it was shown mathematically that dynamo

action could in principle occur. That is specific kinds of velocity structures specified in equation 5.12

could result in self-sustaining magnetic field. The study of such velocity fields is known as kinematic

dynamo theory. Attention has subsequently turned to the identification of plausible motions and forces

that will sustain realistic kinds of magnetic fields.

Fig 5.19 A simple disk dynamo: a) a potential difference is generated in the rotating disc, b) the

external circuit enables current to flow and c) the current supplies a field which reinforces the original

field

To see how dynamo action could occur we first 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

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

150

Fig 5.10 A simple disk dynamo: a) a potential difference is generated in the rotating disc, b) theexternal circuit enables current to flow and c) the current supplies a field which reinforces the originalfield

kinematic dynamo theory. Attention has subsequently turned to the identification of plausible motionsand forces that will sustain realistic kinds of magnetic fields.

To see how dynamo action could occur we first 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 5.10. As thedisc rotates in a magnetic field an electromotive force is established, i.e., there is a potential differencebetween the axis and the periphery of the disc. We can complete an electric circuit between the axis andthe periphery and an electric current will flow. We do this in a special way – we take a wire around thedisc to complete a loop and attach it with brushes to the axis and the edge of the disc. The wire behavesas 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 thefield. Indeed if we make the machine more complicated (by putting in resistors to model dissipationand coupling dynamos together) we can make the field reverse polarity and behave in an apparentlychaotic manner with oscillations in intensity and reversals at random intervals. This is similar to thebehavior we observe of the Earth’s magnetic field and it is interesting that steady motions (in this caseconstant rotation of the disc) can produce such complicated field behavior as a function of time. Whilethis model is undoubtedly very different from the Earth’s core, it does demonstrate that dynamo actionis 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 fluidouter core. The solid inner core is roughly the size of the moon but at the temperature of the surface ofthe sun. The convection in the fluid outer core is thought to be driven by both thermal and compositionalbuoyancy sources at the inner core boundary that are produced as the Earth slowly cools and iron in theiron-rich fluid alloy solidifies onto the inner core giving off latent heat and the light constituent of thealloy. These buoyancy forces cause fluid to rise and the Coriolis forces, due to the Earth’s rotation, causethe fluid flows to be helical. Presumably this fluid motion twists and shears magnetic field, generatingnew magnetic field to replace that which diffuses away. Figure 5.11 provides a heuristic illustration ofthis process.

Until relatively recently, no detailed dynamically self-consistent model existed that demonstrated thiscould actually work or explained why the geomagnetic field has the intensity it does, has a stronglydipole-dominated structure with a dipole axis nearly aligned with the Earth’s rotation axis, has non-dipolar field structure that varies on the time scale of tens to thousands of years and why the fieldoccasionally undergoes dipole reversals. In order to test the convective dynamo hypothesis and attemptto answer these longstanding questions, the first self-consistent numerical model, the Glatzmaier-Robertsmodel, was developed in the mid 1990s. It simulates convection and magnetic field generation in afluid outer core surrounding a solid inner core with the dimensions, rotation rate, heat flow and (asmuch as possible) the material properties of the Earth’s core. The magnetohydrodynamic equations thatdescribe this problem are solved using a spectral method (spherical harmonic and Chebyshev polynomial

141

! " #

$ % &

'(%)!"#)$*+!,-),%#(!+./,0))1-+2%+3.-+!4)5%-$*+!,-)3(%-6*)76%/877-/%/)9!:)!+).+.3.!4;)76.,!6.4*)$.7-4!6;)7-4-.$!4),!5+%3.#)&.%4$0)))'(%)#<%&&%#3)#-+/./3/)-&)$.&&%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(%)!<%&&%#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

expansions) that treats all linear terms implicitly and nonlinear terms explicitly. These equations aresolved over and over, advancing the time dependent solution 20 days at a time. See Glatzmaier’s webpage for more on this at http://www.es.ucsc.edu/ glatz/geodynamo.html This numerical dynamo (alongwith a suite of others that have been created over the past decade) is a great step forward in studyingthe dynamics of Earth’s core, but it remains far from realistic. One reason is that the very low viscosityof the fluid in the outer core would require that the numerical simulations be able to resolve very smallscale changes in both space and time: such resolution remains a significant computational challenge.

There are historical measurements of the magnetic field going back about 300 years (mostly sailingship navigation records) allowing time-dependent models of the magnetic field to be constructed (seemovie). 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 showsthat rather abrupt changes in the field are occasionally observed. Using the frozen flux approximationand some further assumptions, the time variation of the observed field can be converted into a model offluid flow at the top of the outer core (see ppt).

If we want to look further back we must look at the remanent magnetization acquired by crustal rocksas they form in a magnetic field: this is the subject of paleomagnetism.

142

Fig 5.12 The time derivation of the Y component at two magnetic obervatories in Europe. Note thevery rapid variations that can occur. The sudden changes in the time derivative are called "jerks" andthe fact that we see them puts constraints on the electrical conductivity in the mantle.

5.7 Induced and Remanent Magnetization . The magnetic properties of rocks are dependent on thestate of charged particles associated with the crystal structure – motions of and interactions amongthe electrons generate the magnetic moment. There are two basic kinds of magnetization: inducedmagnetization and remanent magnetization. In the presence of an externally applied magnetic field themagnetization 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 therewould be no possibility of a rock recording the paleomagnetic field. The constitutive relation, which iscommonly written as

MI = χH

describes the induced magnetization acquired when a material is exposed to a magnetic field H. χ

is the magnetic susceptibility and describes the response of electronic motions within the material tothe applied field. Materials can acquire a component of magnetization in the presence of an externalmagnetic field (such as that generated in Earth’s core). This induced magnetization is often consideredto 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.

143

µ is the magnetic permeability. In practice χ may be dependent on field intensity, negative or need tobe represented by a tensor (magnetically anisotropic materials). However, we will not consider suchcomplications 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 orbitaland spin properties of the electons in the material. Diamagnetism typically occurs in materials whereall the electrons spins are paired and is caused by precession of the electron orbits in the presence of anapplied field. The associated magnetic moment opposes the applied field so diamagnetic susceptibilityis reversible, weak, and negative. Paramagnetism occurs when one or more of the electron spinsare unpaired so the net magnetic moment of the atom (or ion) is non zero. These can then alignwith an applied field but the effect is opposed by thermal energy. The corresponding susceptibility issmall, positive and reversible. Both paramagnetism and diamagnetism are insignificant contributors tothe geomagnetic field. The important contributions come from materials (some metals) with atomicmoments that interact strongly with each other as a result of quantum mechanical exchange interactionsresulting in spontaneous exact alignment of magnetic moments (fig5.13). These are ferrimagnetic andferromagnetic materials and they can carry either an induced or remanent magnetization. The totalmagnetization of a rock will result from the sum of these two contributions

M = Mi + Mr = χH + Mr

is reversible, weak, and negative. Paramagnetism occurs when one or more of the electron spins are

unpaired so the netmagneticmoment of the atom (or ion) is non zero. These can then alignwith 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 frommaterials (somemetals) with atomicmoments that interact

stronglywith each other as a result of quantummechanical exchange interactions resulting in spontaneous

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 = !Mi + !Mr = " !H + !Mr

Figure 5.9a: Schematic representation of arrangement of magnetic moments in various materials

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 5.9b: Magnetization hysteresis loop for a ferromagnetic material

135

Figure 5.13: Schematic representation of arrangement of magnetic moments in various materialsThe behavior of the magnetization as a function of an applied field is shown in fig 5.14. As the

magnetising field increases, the induced magnetization will reach some saturation level then, when thefield is removed, a magnetization remains – this is called an isothermal remanent magnetization (IRM)as shown in fig 5.14. Repeated exposure to positve and negative fields result in a hysteresis loop butthe important point is that ferromagnetic materials retain a record of an applied magnetic field after it isremoved.

is reversible, weak, and negative. Paramagnetism occurs when one or more of the electron spins are

unpaired so the netmagneticmoment of the atom (or ion) is non zero. These can then alignwith 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 frommaterials (somemetals) with atomicmoments that interact

stronglywith each other as a result of quantummechanical exchange interactions resulting in spontaneous

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 = !Mi + !Mr = " !H + !Mr

Figure 5.9a: Schematic representation of arrangement of magnetic moments in various materials

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 5.9b: Magnetization hysteresis loop for a ferromagnetic material

135

Figure 5.14: Magnetization hysteresis loop for a ferromagnetic materialThe saturation magnetization of a ferromagnetic material depends on temperature, and above the

"Curie" temperature thermal perturbations destroy the remanent magnetization so that the only remaining

144

magnetization is from diamagnetic or paramagnetic effects.

Magnetic Mineralogy

The oxides of iron and titanium are by far the most important terrestrial magnetic minerals. Mag-netite (Fe3O4, Curie temperature 580 C) and its solid solutions with ulvospinel (Fe2TiO4) are the mostimportant magnetic minerals in crustal rocks, although hematite, pyrrhotite also play a role in paleo-magnetic studies. A ternary composition diagram showing the solid solutions of the important mineralsis shown in fig 5.15. Typical values of magnetization for rocks cover a wide range, 10−1 → 1 A/m forbasalts, around 10−3A/m for red sediments to 10−4 → 10−5A/m for limestones. Curie temperatures alsovary widely with composition.

The saturation magnetization of a ferromagnetic material depends on temperature, and above the

”Curie” temperature thermal perturbations destroy the remanentmagnetization so that the only remaining

magnetization is from diamagnetic or paramagnetic effects.

Magnetic Mineralogy

The oxides of iron and titanium are by far the most important terrestrial magnetic minerals. Mag-

netite (Fe3O4, Curie temperature 580! C) and its solid solutions with ulvospinel (Fe2TiO4) are the

most important magnetic minerals in crustal rocks, although hematite, pyrrhotite also play a role in pale-

omagnetic studies. A ternary composition diagram showing the solid solutions of the important minerals

is shown in fig 5.9c. Typical values of magnetization for rocks cover a wide range, 10"1 ! 1A/m for

basalts, around 10"3A/m for red sediments to 10"4 ! 10"5A/m for limestones. Curie temperatures

also vary widely with composition.

Figure 5.9c: Ternary composition diagram of the iron-titanium oxide system

How long can a rock preserve a record of the magnetic field?

Rocks are not pure magnetic minerals, but assemblages of fine-grained ferromagnetic materials

dispersed in a diamagnetic and paramagnetic matrix. The net magnetization will correspond to the

minimum energy state, a complicated tradeoff among energy effects from magnetostatic effects, shape,

and magnetocrystalline anisotropies. 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 as whether it

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

remanent magnetization carried by a number of magnetic minerals (see fig 5.9c) 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

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 allmagnetic order is lost, varieswidelywith chemical composition and structure. Thermoremanent

magnetization (TRM) is acquired whenmagnetic 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 5.9d

Once the rock cools the remanence is locked in (blocked) and provides a permanent (albeit somewhat

noisy) record of the ambient magnetic field direction. Thermoremanent magnetization is commonly

136

Figure 5.15: Ternary composition diagram of the iron-titanium oxide system

How long can a rock preserve a record of the magnetic field?

Rocks are not pure magnetic minerals, but assemblages of fine-grained ferromagnetic materialsdispersed in a diamagnetic and paramagnetic matrix. The net magnetization will correspond to theminimum energy state, a complicated tradeoff among energy effects from magnetostatic effects, shape,and magnetocrystalline anisotropies. Factors that affect how long a rock can retain a record of themagnetic field are composition, temperature, and volume of the magnetic particle as well as whether itis divided into one or more magnetic domains of more or less uniform magnetization.

5.8 The Crustal Field and Paleomagnetism . The crustal magnetic field arises from induced andremanent magnetization carried by a number of magnetic minerals (see fig 5.15) that occur naturallyin the rocks that make up the crust. Although some minerals are magnetically viscous (i.e., theirmagnetization changes rapidly to follow the direction of any ambient field; this corresponds to aninduced magnetization), many rocks carry an imprint of the ambient magnetic field at the time of theirformation, which remains stable over geological timescales. Two important mechanisms for acquiringthis 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 andstructure. Thermoremanent magnetization (TRM) is acquired when magnetic material cools from aboveits Curie point in a magnetic field and records its direction at the time of cooling. This process is shownschematically in fig 5.16

Once the rock cools the remanence is locked in (blocked) and provides a permanent (albeit somewhatnoisy) record of the ambient magnetic field direction. Thermoremanent magnetization is commonlyfound in basalts and lava flows, and for low intensity field similar to that of the Earth, its magnitude isproportional to the intensity of the magnetic field in which it is acquired.

145

Figure 5.9d: Schematic depiction of cooling through the Curie temperature when

magnetization changes from paramagnetic to ferromagnetic. Subsequent cooling

results in the magnetization in the magnetic grains becoming ”blocked” along

easy magnetization directions close to that of the applied field.

found in basalts and lava flows, and for low intensity field similar to that of the Earth, its magnitude is

proportional to the intensity of the magnetic field in which it is acquired.

Remanent magnetization can also be acquired by sediments, as small magnetic particles are incorpo-

rated into the sediment; on average these will align with the ambient field during deposition and become

locked into position during subsequent dewatering and compaction of the sediment (fig 5.9e). This is

called depositional or post-depositional remanent magnetization, DRM or PDRM, depending on when

it is acquired. Note that magnetic grains may have elongate shapes that means that their final orientation

as they settle to the bottom may change due to the action of gravity thus giving an inclination error.

Chemical, mineralogical, and metamorphic processes after rock formation can also influence the

magnetization, sometimes causing the signal to be partially or completely overprinted by a latermagnetic

field.

Paleomagnetism, the study of the fossil magnetic record, has important applications in the earth

sciences. Taking oriented samples of crustal rocks and measuring their directions of magnetization in

a magnetometer allows the determination of the local paleofield direction when the rock acquired its

magnetization. During the late 1950s and 1960s it became increasingly apparent that these observations

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

137

Figure 5.16: Schematic depiction of cooling through the Curie temperature when magnetizationchanges from paramagnetic to ferromagnetic. Subsequent cooling results in the magnetization in themagnetic grains becoming "blocked" along easy magnetization directions close to that of the appliedfield.

Remanent magnetization can also be acquired by sediments, as small magnetic particles are incor-porated into the sediment; on average these will align with the ambient field during deposition andbecome locked into position during subsequent dewatering and compaction of the sediment (fig 5.17).This is called depositional or post-depositional remanent magnetization, DRM or PDRM, dependingon when it is acquired. Note that magnetic grains may have elongate shapes that means that their finalorientation as they settle to the bottom may change due to the action of gravity thus giving an inclinationerror. Chemical, mineralogical, and metamorphic processes after rock formation can also influencethe magnetization, sometimes causing the signal to be partially or completely overprinted by a latermagnetic field.

Figure 5.17: Schematic of the aquisition of a DRM

Paleomagnetism, the study of the fossil magnetic record, has important applications in the earth

146

sciences. Taking oriented samples of crustal rocks and measuring their directions of magnetization ina magnetometer allows the determination of the local paleofield direction when the rock acquired itsmagnetization. During the late 1950s and 1960s it became increasingly apparent that these observationsdid 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 morecompelling evidence was found in the patterns of seafloor magnetic anomalies in the crustal field. Oceanfloor 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 theridge axis. Successive normal and reverse polarity epochs of the magnetic field will have normally andreversely magnetized basalt associated with them. The associated magnetic field can be measured witha magnetometer towed behind a ship; anomalies in the magnitude of the magnetic field are observedwith distinctive lineated or striped anomaly patterns approximately parallel to ridge axes. The pattern ofpolarity changes in the geomagnetic field as a function of distance from the ridge axis can be correlatedwith land-based sections dated by radiometric methods; distance is transformed to age providing thebasis 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 directionsobserved approximate those of a dipole aligned along Earth’s rotation axis, paleomagnetic observationsand seafloor magnetic anomalies provide constraints used in continental reconstructions over geologictime. 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 rockformations a record of the direction of past magnetic fields. These are, as we have seen, directly relatedto the position of the Earth’s spin axis in the past. In the continent’s frame of reference, then, it is theNorth pole which appears to move. Tracks of these past pole positions are called apparent polar wanderpaths (APWP). An example of a apparent polar wander path is shown in Fig. 5.18.

Fig 5.18

The fundamental assumption in constructing apparent polar wander paths for continental blocks is that

147

when averaged over sufficient time, the earth’s magnetic field reduces to that of a dipole aligned along therotational axis and located at the earth’s center. That is, all terms in the spherical harmonic representationexcept for g0

1 will cancel out in a time average. This is the geocentric axial dipole hypothesis. In order totrack the movement of the spin axis with respect to the continent, we must sample rocks of known ageand orientation, obtain a magnetization which represents the ancient magnetic field, and then calculatethe 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 widelyused, but highly optimistic, rule of thumb is to use 10 separately oriented samples from 10 separaterock units, which span at least 10,000 years. Data from the past 5 million years suggest that hundredsof sites spanning hundreds of thousands of years may be a more realistic requirement. The positionon the Earth where this average dipole would pierce the surface is the paleomagnetic pole. This thenis the position of the geographic rotation axis at the time the rocks were magnetized in the continentalreference frame. The pole calculated from a single measurement is called a virtual geomagnetic pole, orVGP. We may calculate this pole position (specified by its colatitude and longitude : θp, ψp) if we knowthe declination and inclination (D and I) of the ancient field and the site of the rocks sampled (θs, ψs).This is done using the sine and cosine rules for triangles on the surface of a sphere.

Fig 5.10

of the geographic rotation axis at the time the rocks were magnetized in the continental reference frame.

The pole calculated from a single measurement is called a virtual geomagnetic pole, or VGP. We may

calculate this pole position (specified by its colatitude and longitude : !p, "p) if we know the declination

and inclination (D and I) of the ancient field and the site of the rocks sampled (!s, "s). This is done

using the sine and cosine rules for triangles on the surface of a sphere.

Fig 5.10

Remember that the lengths of the sides of the spherical triangle are also measured as angles. Two

formulae come in handy:

sinA

sin a=sinB

sin b=sinC

sin c(5.12)

and

cos a = cos b cos c + sin b sin c cosA (5.13)

Let us now sketch the geometry of our problem:

S is ourmeasurement site,P is the location of the virtual geomagnetic pole (whichwewant to determine)andN is the presentNorth Pole. Themagnetic colatitude of the samplemagnetization is the distance from

the measurement location to the paleomagnetic pole, !m. This can be determined from the inclination,

I , of the sample, using arguments introduced in the first section of this chapter,

cot !m = 12 tan I (5.14)

139

Fig 5.19

Remember that the lengths of the sides of the spherical triangle are also measured as angles. Twoformulae come in handy:

sinAsin a

=sinBsin b

=sinCsin c

(5.18)

and

cos a = cos b cos c+ sin b sin c cosA (5.19)

Let us now sketch the geometry of our problem:S is our measurement site, P is the location of the virtual geomagnetic pole (which we want to determine)and N is the present North Pole. The magnetic colatitude of the sample magnetization is the distancefrom the measurement location to the paleomagnetic pole, θm. This can be determined from theinclination, I, of the sample, using arguments introduced in the first section of this chapter,

cot θm = 12 tan I (5.20)

Now the declination, D, is just the angle from the present North pole to the line joining S and P so

cos θp = cos θs cos θm + sin θs sin θm cosD (5.21)

which allows us to calculate θp. To determine ψp we could use the sine formula:

148

Fig 5.11

Now the declination,D, is just the angle from the present North pole to the line joiningM and P so

cos (!p) = cos (!s) cos !m + sin (!s) sin !m cosD (5.15)

which allows us to calculate !p.

To determine "p we use

sin ("p ! "s)sin !m

=sinD

sin (!p)(5.16)

or

sin ("p ! "s) =sin !m sinD

sin !p(5.17)

Because !p has been determined and "s, D and !m are known, we can determine "p. For certain

orientations this formula will give the longitude "p incorrectly. The formula is valid if cos !m >cos !s cos !p. If cos !m < cos !s cos !p then use

sin (180 + " ! "p) =sin !m sinD

sin !p(5.18)

NOTE: " is measured eastward from the Greenwich meridian and goes from 0 " 360!. ! is measuredfrom the North pole and goes from 0 " 180!. Of course ! relates to latitude, # by ! = 90 ! #. !m 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

140

Fig 5.20

sin (ψp − ψs)sin θm

=sinDsin θp

(5.22)

or

sin (ψp − ψs) =sin θm sinD

sin θp(5.23)

However, there can be an ambiguity between the angle you get and 180 degrees minus that angle (whichboth give the same sine value). It is easiest to use the cosine formula again:

cos θm = cos θs cos θp + sin θs sin θp cos (ψp − ψs)

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 measuredfrom the North pole and goes from 0→ 180. Of course θ relates to latitude, λ by θ = 90− λ. θm is themagnetic colatitude. Be sure not to confuse latitudes and colatitudes.

These formulae allow us to compute the location of the paleomagnetic pole assuming that the fieldhas remained dipolar. By hypothesis, the paleomagnetic pole averaged to the geographic North polewhen the rock formed. (If measurements are taken from a time period when the field was reversed thepaleomagnetic south pole was at the North pole.) By using measurements from a continental block ofmany different dates, the apparant polar wander path as a function of time can be constructed. The polarwandering path will be the same for all measurements from a single plate and can be used to reconstructthe absolute location of the plate in the past (relative to the geographic poles) though you should notethat longitudinal motions of continents will not be detectable because of the assumed axial symmetryof the field. The polar wandering paths of two adjacent plates can be used to compute the minimiumrelative velocities between plates.

149

5.10 Secular and Paleosecular Variation . The magnetic field varies through time as we all knowfrom 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 beof internal origin. However, it should be kept in mind that time scale is not an entirely reliable meansof determining the source of a particular field. Geomagnetic jerks are a notable example of rapid fieldvariations of internal origin. At present, the field is mainly that of a dipole inclined at about 11 to therotation 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 westwardat a rate of about 0.2/year. This westward drift accounts for about a quarter of the variation in the recentnon-dipole field. Perhaps the most dramatic variation in the magnetic field is that the axial geocentricdipole coefficient changes sign occasionally, meaning that compasses would point South! The processtakes from four to ten thousand years to complete and the last so-called reversal was about 780,000years ago.

Grand Spectrum

Frequency, Hz

Am

plitu

de, T

/√H

z

Reversals

Cryptochrons?

Secular variation

Annual and semi-annual

Daily variation

Solar rotation (27 days)

Storm activityQuiet days

Schumann resonances

Powerline noiseRadio

??

1 se

cond

10 k

Hz

1 ho

ur

1 da

y

1 th

ousa

nd y

ears

10 m

illio

n ye

ars

1 m

onth

1 ye

ar

1 m

inut

e

Internal origin External origin

The full spectrum of behavior exhibited by the geomagnetic field can only be determined from pa-leomagnetic and associated geochronological studies of material from the geological and archeologicalrecord. Typically only observations of field direction are possible but when the magnetic remanence ac-quisition process can be simulated in the laboratory the magnitude may also be estimated. Although suchrecords can never have the spatial or temporal resolution of historic measurements of the geomagneticfield, 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 ageocentric dipole). Was the paleofield always as dominantly dipolar as it appears now?

150

(2) It was noted very early in geomagnetic studies that large features of the geomagnetic non-dipolefield appeared to move westward through time with a drift rate of approximately 0.2 per year. Thisstriking behavior became known as westward drift. Is westward drift an intrinsic feature of thepaleofield?

(3) The present dipole moment of the geomagnetic field is roughly 8 × 1022Am2. What is the averagevalue and how much does it change?

(4) What is the long-term nature of secular variation? Is the present or historic variability representativeof the field in general?

(5) Magnetohydrodynamical equations describing the conditions in the Earth’s core do not care whatsign the field is (i.e., whether it is “normal" or “reversed"). Are there any detectable differencesbetween the two polarity states (for example, does the field spend more time in one polarity state thanthe 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 polaritystate to another?

(7) How long has the core been producing a magnetic field and is there any observable effect of thegrowth 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 mediaincluding both sedimentary and igneous sequences. Each of these materials has specific advantages anddisadvantages 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 nodefinitive answers yet. For example, in order to investigate whether westward drift is intrinsic to thefield, 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 whatdistance can prominent features of the directional data be recognized? Existing data sets obtainedfrom lake sediment data, appear to show that similar geomagnetic field features can be observed inrecords 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 secularvariation, then we should see the same curves all over the world, offset slightly in time. What we seeinstead are quite different records with apparent spatial scales on the order of a few thousand kilometers.

During times of stable magnetic polarity, secular variation causes the orientation of Earth’s dipoleaxis to change by some 10− 15 on timescales of a few hundreds to thousands of years; correspondinglocal fluctuations in field intensity may be as much as factor of two or three, but the global dipolemoment is somewhat less variable. A large departure from the approximately geocentric axial dipolarstate for the field is known as a geomagnetic excursion and is usually accompanied by strong fluctuationsin intensity of the field (see ppt for some examples). For a long time, it was conventional wisdom thatexcursions were artifacts of the recording process and were not characteristic of the real geomagneticfield. However, these misgivings were laid to rest by reproducibility arguments. The most convincingaspect of well studied excursions is that they are recorded in several separate sections. There are only afew 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 andthe position of the dipole axis (that is the VGP as inferred from a single measurement location) traces apath to more than 45 away from the spin axis. Because they are of short duration (a few thousand yearsor less) it is difficult to find geological records of an excursion that confirm it as a globally synchronousevent; the most recent global geomagnetic excursion, known as the Laschamp event, appears to havetaken place about 40 thousand years ago. Excursions are often thought of as aborted reversals, andthe evidence available so far suggests that a continuum of behavior exists ranging from normal secularvariation through excursions to full polarity reversals.

151

5.11 Geomagnetic field reversals . The Earth’s dipole field flips polarity at irregular intervals andwhen the polarity is the same as the present day the polarity is said to be normal. On average, the fieldspends about half its time in each state. When reversed polarities were first discovered, there was somediscussion about whether or not magnetic minerals could become magnetized in the opposite directionto the applied field. While this is true for some minerals under special conditions, the same patternof reversals is found in rocks at different geographical sites. Furthermore, the same pattern is foundin lavas and in various sediments so the only reasonable hypothesis is that the geomagnetic field itselfis reversing. The latest transition, from the Matuyama reversed polarity epoch to the present normalpolarity epoch (the Brunhes epoch), took place at about 0.78 Ma. The time between successive reversalsis 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 therewas 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 consistentlyreversed in polarity; other intervals may also exist. The transition to such states may reflect changingphysical 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 aswe have about its intensity during Archean and Early Proterozoic times suggests that it was roughly thesame order of magnitude as it is today. Reversals are observed from Precambrian times to the presentthough the frequency of reversals seems to change considerably through time.

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 reversal 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 the foundation for the hypothesis of sea-floor spread-

ing and was the key to the plate tectonic revolution. Consider Figure 5.15, a sketch of an oceanic ridge.

145

Figure 5.21: Geomagnetic reversal rate for the past 160Myr.

What happens during a reversal? Usually the intensity decreases by about an order of magnitudefor several thousand years while the field maintains its direction. The field then undergoes complicateddirectional changes over a period of 1000 – 4000 years and finally the intensity grows with the fieldhaving reversed polarity. The total time span of a reversal is up to 10,000 years. Some reversals seemto show concurrent changes in intensity and direction and the complete reversal occurs in about 5000years. One explanation of the behavior during a reversal is that the dipole field decays away and thenondipole field stays roughly at the same level giving rise to the direction changes when the dipole fieldis small. The dipole field then grows with opposite polarity.

152

Fig. 5.14

Fig. 5.15

As the oceanic plates diverge, molten mantle material flows upward to fill the gap, this magma then

cools by conductive heat loss to the surface. The ridge is elevated because the rocks are hotter and

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

polarity of the field when the rock formed. The blocks of normal and reversed magnetization produces

lineated variations in the geomagnetic field aligned parallel to the ridge crest. These stripes then provide

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

seamount in Figure 5.16:

This seamount was extruded during a period of normal polarity and has a magnetization parallel to

the Earth’s magnetic field. To the South (right), the field generated by the seamount adds to the Earth’s

field and to the North (left) it opposes the Earth’s field. The intensity of the net field is measured by a

146

Fig. 5.22

The reversal sequence of the magnetic field has been calibrated for the last five million years bydating basalts of known polarity. A recent version is shown in Fig 5.22. Portions of the time scale whichare of one dominant polarity, lasting from one to two million years are designated Chrons and the mostrecent four chrons are named after great scientists who contributed significantly to our understanding ofthe geomagnetic field. The Chrons older than about five million years have a somewhat cumbersomenumbering scheme.

Knowledge that the Earth’s magnetic field reversed polarity frequently in the past helped explainsome of the mysterious features of the geophysical observations made in the oceans. One such mysterywas that of the curious magnetic “stripes" observed in the geomagnetic field data over much of the oceanfloor. The notion that oceanic crust forms at the great mid-ocean ridges (another mystery of the seafloor) while the magnetic field reverses polarity served as the foundation for the hypothesis of sea-floorspreading and was the key to the plate tectonic revolution. Consider Figure 5.23, a sketch of an oceanicridge.

Fig. 5.14

Fig. 5.15

As the oceanic plates diverge, molten mantle material flows upward to fill the gap, this magma then

cools by conductive heat loss to the surface. The ridge is elevated because the rocks are hotter and

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

polarity of the field when the rock formed. The blocks of normal and reversed magnetization produces

lineated variations in the geomagnetic field aligned parallel to the ridge crest. These stripes then provide

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

seamount in Figure 5.16:

This seamount was extruded during a period of normal polarity and has a magnetization parallel to

the Earth’s magnetic field. To the South (right), the field generated by the seamount adds to the Earth’s

field and to the North (left) it opposes the Earth’s field. The intensity of the net field is measured by a

146

Fig. 5.23

As the oceanic plates diverge, molten mantle material flows upward to fill the gap, this magma thencools by conductive heat loss to the surface. The ridge is elevated because the rocks are hotter andare therefore more buoyant. We have considered these features of seafloor spreading in detail in otherchapters. As the rock cools it gains a TRM and the direction of the magnetization depends upon thepolarity of the field when the rock formed. The blocks of normal and reversed magnetization produceslineated variations in the geomagnetic field aligned parallel to the ridge crest. These stripes then providea record of the direction and rate of sea-floor spreading between two plates and may be used to helpreconstruct plate motions and so unravel the evolution of the ocean basins. Fig 5.24 shows a pictureof the magnetic anomalies off the coast of California which allows detailed mapping of the age of the

153

ocean floor (though see the next section where some of the complexities associated with this processare discussed). This leads to a quantitative estimate of spreading rates. If we make the assumption thatspreading rates are relatively constant over time, the pattern of magnetic anomalies can be used to extendthe time scale back much further – all the way to the age of the oldest oceanic crust – about 140Ma (seeppt for examples). Where possible, radiometric dates have been tied into the reversal pattern and theresulting time scale has a wide range of applications from marine geophysics to stratigraphy.

Fig. 5.24

5.12 Controls on oceanic magnetic anomalies . A magnetic anomaly is the magnetic field remainingafter the reference field (produced by Gauss coefficients such as those listed in Table 5.1) has beenremoved. Anomalies usually result from the magnetization of nearby geologic features. Consider theseamount in Figure 5.25:

This seamount was extruded during a period of normal polarity and has a magnetization parallel tothe Earth’s magnetic field. To the South (right), the field generated by the seamount adds to the Earth’sfield and to the North (left) it opposes the Earth’s field. The intensity of the net field is measured by amagnetometer towed behind a ship. A magnetic anomaly is calculated by subtracting a reference fieldfrom the measured field. The magnitude of these anomalies is only a few hundred nanoTeslas or about1% of the dipole field. Referring to our seamount, there would be a negative anomaly to the left and apositive one to the right.

Magnetic anomalies with respect to normal oceanic crust are similarly controlled by the magnetizationof the sea-floor. Using an argument similar to that made for the measurement of gravity, it can be

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 anomalieswith respect to normal oceanic crust are similarly controlled by themagnetization

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

field and are insensitive to slight variations perpendicular to it. It also interesting to point out that

normally magnetized blocks do not always produce positive anomalies as shown below:

Fig. 5.17

When both the crustal magnetization and the geomagnetic field are steep, the normal blocks produce

positive anomalies. At the equator, however, blocksmagnetized 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

Fig. 5.25

easily shown that magnetometers generally only measure the anomalous field intensity parallel to thereference field and are insensitive to slight variations perpendicular to it. It also interesting to point outthat normally magnetized blocks do not always produce positive anomalies as shown below (fig 5.26):

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 anomalieswith respect to normal oceanic crust are similarly controlled by themagnetization

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

field and are insensitive to slight variations perpendicular to it. It also interesting to point out that

normally magnetized blocks do not always produce positive anomalies as shown below:

Fig. 5.17

When both the crustal magnetization and the geomagnetic field are steep, the normal blocks produce

positive anomalies. At the equator, however, blocksmagnetized 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

Fig. 5.26When 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 geomagneticfield produce negative anomalies. In the equatorial case, when the lengths of the blocks is long comparedto the depth of the ocean, as when the spreading center is oriented North-South, no anomaly at all willbe 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 theEarth’s magnetic field. It is possible to determine the intensity of ancient magnetic fields, because theprinciple mechanisms by which rocks get magnetized (i.e., thermal, chemical, and detrital) produceremanent 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. Forexample, since TRM is proportional to B, all we need do, in principle, is measure the natural remanentmagnetization (NRM), and then give the rock a TRM in the laboratory in a known field. Since

TRMlab

Blab=

NRM

Bancient

the ambient field responsible for the NRM (Bancient) is easily calculated. Similarly, for a DRM, onewould 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

155

are different for the different magnetization mechanisms; we consider TRM first. The principal failingof the simple method outlined above is that most rocks alter when heated in the laboratory and so theTRM acquired by heating the rock to above its Curie Temperature and cooling in a known field may haveno relationship to the TRM acquired originally. Minerals such as maghemite are unstable at elevatedtemperatures and convert to other magnetic phases. Magnetite itself oxidizes. Such changes in themagnetic mineralogy have a drastic effect on the magnetization and careful lab work and consistencychecks are needed to detect such changes and compensate for them. Most data quoted for absolutepaleofield intensity values are determined in this or similar ways. Types of materials that carry TRMare igneous bodies and archeological material such as baked hearths or pottery. Recently rapidlycooled submarine basaltic glass has been used very successfully to recover a large number of absolutepaleointensity measurements.

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, bottommostcurve is the polarity interval length.

The problem with sedimentary paleointensity data is that, laboratory conditions cannot duplicate thenatural environment. First of all, most sediments carry a post-depositional remanence as opposed to adepositional one. Secondly, it seems that the intensity of remanence is a function of not only field butmagnetic mineralogy and even chemistry of the water column. Since mineralogy and chemistry can vary

156

with climate conditions considerable care is required to ensure that these variations can be adequatelycompensated for. The best we can presently hope for from sediments is relative paleointensity variationswith time. An example of a such a record is provided in Figure 5.27. This particular record seemsto show higher field intensities correlating with lower reversal rate, but this result has not so far beenconfirmed elsewhere.

Many such records are available covering the past few hundred thousand years, and these have beenaveraged to provide an estimate of geomagnetic dipole moment variations. The field varies substantiallyover this time interval. Data from the last 5 million years shows that the present field is somewhat higherthan 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 ofthe Earth, paleomagnetists have often found it convenient to express paleointensity values in termsof the equivalent geocentric dipole moment which would have produced the observed intensity atthat paleolatitude. This moment is called the virtual dipole moment or VDM. First, the magneticpaleocolatitude, θm, is calculated as before from the observed inclination and using the dipole formulatan (I) = 2 cot (θm) (see eqn 5.10 and 5.11). Then

V DM =4πr3

µoBancient(1 + 3 cos 2θm)−

12 (5.25)

5.14 The magnetospheric ring current. The largest component of the external variations inEarth’s magnetic field comes from a ring of current circulating westward at a distance of 2-9 Earth radii.To understand why this might be we need to consider the nature of charged particles moving in a plasma.A charged particle moving with velocity v in a magnetic field B will be acted on by the Lorentz force

f = ev ×B

where here we assume an electron with a charge of−e. If v is perpendicular to B then the particle movesin a circular path whose radius can be obtained by balancing the centripetal force with the Lorentz force:

mv2

r= evB

r =mv

eB.

Consider a particle thus spiraling around a magnetic field line near the equator. The main magnetic field,originating inside the core, has a radial gradient and is larger closer to Earth. The radius of curvature ofa particle will thus be smaller closer to Earth. Looking down from the North pole, a positive particlewill circle clockwise and drift westward; an electron will do the opposite, and together they make awestward directed current that circles Earth between 2 and 9 Earth radii. The strength of the ring currentis characterized by the ‘Dst’ (disturbance storm time) index, an index of magnetic activity derived froma network of low- to mid-latitude geomagnetic observatories that measures the intensity of the globallysymmetrical part of the equatorial ring current.

157

1982.53 1982.535 1982.54 1982.545 1982.55

-300

-250

-200

-150

-100

-50

0

Year

Dst

, nT

Magnetic storm of June 1982

sudden commencement

recovery

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 fieldassociated with the sudden increased pressure of the solar wind on Earth’s magnetosphere. Followingthe commencement the main phase of the storm is associated with a large decrease in the magnitude ofthe fields as the ring current is energized – the effect of the ring current is to cancel Earth’s main dipolefield slightly. Finally, there is a recovery phase in which the field returns quasi-exponentially back tonormal. 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.

5.15 Electromagnetic induction. One of the key concepts in geomagnetic induction is that of theskin depth, the characteristic length over which electromagnetic fields attenuate. We can derive the skindepth starting with Faraday’s Law:

∇×E = −∂B∂t

and Ampere’s Law:∇×H = J

where J is current density (A/m2), E is electric field (V/m), B is magnetic flux density or induction (T),H is magnetic field intensity (A/m). We can use 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 are related by magnetic permeabilityµ 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 = −µ∂H∂t

∇×H = σE .

If we take the curl of these equations and use ∇× (∇×A) = ∇(∇ •A)−∇2A

∇2E = µ∂

∂t(∇×H) = µσ

∂E∂t

(5.27)

158

∇2H = σ(∇×E) = µσ∂H∂t

. (5.28)

You will recognize these as diffusion equations (1/µσ acts as a diffusivity, and if you plug in the unitsit does indeed come out with units of m2/s). Now if we consider sinusoidally varying fields of angularfrequency ω and a uniform conductivity σ

E(t) = Eoeiωt∂E∂t

= iωE

H(t) = Hoeiωt ∂H

∂t= iωH

and so∇2E = iωµσE (5.29)

∇2H = iωµσH . (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 propagatingvertically into a half-space, these equations decouple to

∂2E∂z2

+ k2E = 0

∂2H∂z2

+ k2H = 0

with solutionsE = Eoe−ikz = Eoe−iαze−βz (5.31)

H = Hoe−ikz = Hoe

−iαze−βz (5.32)

where we have defined a complex wavenumber

k =√iωµσ = α− iβ (5.33)

and an attenuation factor, which is called a skin depth

zs = 1/α = 1/β =√

2σµoω

. (5.34)

The skin depth is the distance that field amplitudes are reduced by 1/e, or about 37%, and the phaseprogresses one radian, or about 57 . In practical units

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

159

material σ, S/m 1 year 1 month 1 day 1 hour 1 sec 1 ms

core 3×105 4 km 770 m 200 m 40 m 71 cm 23 mmlower mantle 10 900 km 170 km 46 km 9 km 160 m 5 mseawater 3 1600 km 470 km 85 km 17 km 280 m 9 msediments 0.1 9000 km 1700 km 460 km 95 km 1.6 km 50 mupper mantle 0.01 3×104 km 104 km 1500 km 300 km 5 km 158 migneous rock 1×10−5 106 km 2×105 km 4×104 km 9500 km 160 km 5 km

Skin depths for a variety of typical Earth environments and a large range of frequencies. One can seethat the core is effectively a perfect conductor into which even the longest period external magnetic fieldcannot penetrate. However, 1-year variations can penetrate into the lower mantle. The very large rangeof conductivities found in the crust indicates the need for a corresponding large range of frequencies inelectromagnetic sounding of that region.

than a meter for conductive rocks and kilohertz frequencies to thousands of kilometers in mantle rocksand periods of days (see table). Skin depth is a reliable indicator of maximum depth of penetration.

5.16 Geomagnetic depth sounding. Geomagnetic depth sounding, or GDS, is used to derive theelectrical conductivity inside Earth using measurements of the magnetic field. The geomagnetic ringcurrent generates a field that is largely dipolar in morphology (it is like being inside a big solenoid), andif one chooses a coordinate system defined by the dipole field (a geomagnetic coordinate system), thenthe field can be described as P 0

1 in morphology. Equation 5.6 reduces to

V (r, θ) = aP 01 ( cos θ)

[(ge)01

(r

a

)+ (gi)01

(a

r

)2](5.35)

(notice that we have kept the internal AND external Gauss coefficients, since we will be interested inthe 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 gidirectly. However, most of the time one just has horizontal (H) and vertical (Z) components of B asrecorded by a single observatory. We can obtain H and Z from the appropriate partial derivatives of Vto obtain:

−H = µo

(1r

∂V

∂θ

)r=a

= µ(ge + gi)∂

∂θP 0

1 ( cos θ) = µ(ge + gi)∂

∂θ( cos θ)

= µAH sin θ (5.37)

and−Z = µ

(∂V

∂r

)r=ao

= µ(ge − 2gi)P 01 ( cos θ)

= µAZ cos θ (5.38)

160

where we have defined new expansion coefficients

AH = ge + gi

AZ = ge − 2gi .

Thus we can define a new electromagnetic response

W (ω) =Z(ω) sin θH(ω) cos θ

=AZAH

(5.39)

where θ is the co-latitude of the observatory and we have included the ω dependence to remind us thatthis is all for a particular frequency. W is related to Q by

Q(ω) =gi(ω)ge(ω)

=1−AZ/AH2 +AZ/AH

.

The admittance or inductive scale length c provides a practical unit for interpretation, and is given by

c(ω) =a

2W (ω). (5.40)

Note that for causal systems, the real part of c is positive, and the imaginary part is always negative. Fora uniform earth of conductivity σa, the resistivity is given by

ρa = 1/σa = ωµo|ic|2 . (5.41)

3 4 5 6 7 8 9−1000

−500

0

500

1000

1500

2000

2500

3000

Log (Period, s)

C, k

m Real (C)

Imag (C)

1 day 1 month 1 year 11 years1 hour

Figure 5.29: The admittance, c, based on an analysis of geomagnetic observatory data. The red lineis a fit to the data for the model shown in Figure 5.30.

161

0 500 1000 1500 2000 2500 300010 −3

10 −2

10 −1

10 0

10 1

10 2

10 3

Depth, km

Con

duct

ivity

, S/m

Olivine at 1400°C

Bounds on all models

Bounds for monotonic models

Pero

vski

te a

t 200

0°C

Ringwoodite/Wadsleyite at 1500°C

UPP

ER M

AN

TLE

TRA

NSI

TIO

N Z

ON

E

LOW

ER M

AN

TLE C

ore

Figure 5.30: Electrical conductivity in Earth’s mantle derived from the GDS data shown in Figure5.29. The blue line shows one model that fits the data (the fit is shown in Figure 5.29), but theyellow regions give upper and lower bounds on average conductivity for the three mineralogical mantleregimes. Black dots are conductivities of representative mantle minerals at appropriate pressures andtemperatures.

162