Gubbins D., Herrero-Bervera E. Encyclopedia of Geomagnetism and Paleomagnetism

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where

total

¼ r

ðE

þ T

Þ; (Eq. 67)

total

¼ r

ðE

þE

ÞV þ I

þ I

¼ r

1

þ T

(Eq. 68)

Here E

follows from (Eq. 30) by thermodynamic linearization.

The expressions Eqs. (67) and (68) for u

total

and I

total

are different

from Eqs. (13) and (14). To clarify, we shall temporarily call the latter

total

and

total

, and introduce the next term in the e

– expansion of V,

writing V ¼ V

þ V

þ... , where V

has until now been written as

V. Equation (1) gives

H ðr

Þ¼ 0; (Eq. 69)

]

þ H ðr

þ r

Þ¼0 : (Eq. 70)

It can be readily shown from Eqs. (13) and (14) that

total

¼ u

total

þðE

þ U Þ r

; (Eq. 71)

total

¼ I

total

þðE

þ U Þ½ r

þðr

þ r

Þ: (Eq. 72)

According to Eqs. (49), (69), and (70), the additional terms cancel out

when (Eq. 71) is substituted into Eqs. (12), and (66) – (68) are recovered.

As was noted in “The adiabatic (isentropic) reference state, ” convec-

tion is driven by the superadiabatic temperature difference across the

layer rather than by the full temperature difference (adiabatic plus

superadiabatic), as was used in crudely estimating the Rayleigh num-

ber, Ra, for the Earth ’s core at the end of “The Boussinesq approxima-

tion (BA).” This means that a more realistic Ra is smaller by a factor

of order e

so that Ra  10

. This is still so large that the core must be

in a well mixed, turbulent state, and this confirms a posteriori the suit-

ability of the AA in describing core motions. It might seem that when,

thermodynamically, the convective state is a small perturbation away

from the adiabatic state, the heat flux, must also be a small perturba-

tion away from the adiabatic heat flux, I

. This however is not neces-

sarily the case. When Ra is large, the same motions that mix the fluid

and lead to (Eq. 48) also transport a large amount of heat. Even with a

small T

, this may be of the same order as, or even larger than, the heat

flux, K

G, down the adiabat. This question and the role of turbulence

is re-opened in the section “ The role of turbulence. ”

The AA generalizes easily to convection in multicomponent

fluids, such as the binary alloy often used to describe the dynamics of

Earth ’s core. As for S , the mass fraction xð¼ x

þ x

Þ of the light com-

ponent of the alloy is almost homogenized by convection when S is,

so that Hx

¼ 0 supplements (Eq. 48). Compositional differences create

buoyancy forces, so that  a

g in (Eq. 57) is replaced by Cg ,

where C ¼a

 a

is the co-density and a

¼r

 1

ð ] r= ] xÞ

P ; S

;

cf. (Eq. 26). Also, x

is governed by an equation similar to (Eq. 58),

although S

obeys a different boundary condition. Full details of

this generalized AA are given by Braginsky and Roberts (1995,

2003). It was basic in the geodynamo simulation of Glatzmaier and

Roberts (1996).

Altern ative forms of the AA and BA

The tem perature- based anelas tic approxi mation (TAA)

The central thermodynamic variable describing convection in the

AA is S

; see Eqs. (56)– (58). This is convenient when studying core

convection, because the nonadiabatic heat flux, I

, is mainly due to

turbulence and is therefore best expressed in terms of the inhomogene-

ities in the entropy (see “The role of turbulence .” ). For systems such as

the Earth ’s mantle and oceans however, the temperature, T

, is a more

convenient variable to employ. One is then led to a “ temperature-based

anelastic approximation ” or “ TA A ” in which S

has been replaced by

in Eqs. (56), (57), and (63):

H ðr

VÞ¼ 0 ; (Eq. 73)

V ¼HP

 a T

g þ F

= r

; (Eq. 74)

ðC

ÞþH  I

 r

a T

V  g ¼ Q

þ Q

; (Eq. 75)

where

¼K

H T

 K

H T

; and P

¼ P

= r

: (Eq. 76)

As before, the presence of r

a T

V  g in (Eq. 75) assures conservation

of total energy.

The derivation of Eqs. (73) – (76) requires only

¼ C

ða T

Þ P

; (Eq. 77)

which follows from (Eq. 23) and thermodynamic linearization. If we

use (Eq. 77) in full to remove reference to S

in Eqs. (56) – (58),

we obtain a theory in every way equivalent to the AA, though compli-

cated by P

terms from (Eq. 77). In order of magnitude, however,

= r

H  P

= H  a T

g , so that (Eq. 77) implies

¼ a T

½1 þ Oð e

gÞ; (Eq. 78)

¼ C

½1 þ Oðe

g Þ: (Eq. 79)

Whenever aT

 1, as is the case for fluids and solids (but not

for gases, where a T

 1), the product e

a T

g in Eqs. (78) and (79)

is negligibly small, even when e

 1. We may then replace a

g in

(Eq. 57) by a T

g and d

ð T

Þ in (Eq. 63) by d

ðC

Þ. [Variations in

may be small, and d

ð C

Þ in (Eq. 75) may then be replaced sim-

ply by C

, as is often done. We may also recall (see “Introduc-

tion ”) that, since aT

 1, the differences between v

and v

and

between C

and C

may be ignored.] Since the density change, Dr

across the system is not small when e

 1, it is necessary to retain

within the divergence in (Eq. 73).

Slow, creep ing motions occur in the Earth ’ s mantle through a pro-

cess called “subsolidus convection. ” This is well described by the

basic equations of “ Introduction,” the viscosity defining F

being of

order 10

Pa s ð n  2  10

1

Þ. The resulting motions are so

slow ðV  1 cm year

 1

Þ that the inertial term on the left-hand side

of (Eq. 74) is utterly insignificant. Nevertheless, Ra  10

 10

through much the mantle’ s vertical extent, in which therefore the tem-

perature is close to an abiabat. The AA is then appropriate but, because

a T

 3  10

 2

, the approximate form Eqs. (73)– (75) of the AA

applies almost equally well; see Schubert et al. (2001).

The generali zed Bous sinesq ap proximat ion (G BA)

The approximation just described becomes even simpler when

 gH =v

 1. For example, for the oceans, H  5 km and

 1 : 5kms

1

, gives e

 2  10

 2

. In such a case, we may

substitute

¼ r

þ r

; T

¼ T

þ T

; ...... (Eq. 80)

into Eqs. (73) to (76), where r

; T

; ... are constants. Because

 T

= T

 e

 1, we may omit r

and T

everywhere

except where Hr

and H T

are required, as in (Eq. 76). It follows that

H  V ¼ 0; (Eq. 81)

16 ANELASTIC AND BOUSSINESQ APPROXIMATIONS

V ¼HP

 aT

g þ F

= r

; (Eq. 82)

þ H  I

 r

a T

V  g ¼ Q

þ Q

; (Eq. 83)

where C

and the other coefficients are assumed to be constants and

¼K

H T

 K

; and P

¼ P

: (Eq. 84)

Since e

 1, the term  r

V  g required to maintain energy

conservation is small, but it has been retained in (Eq. 83), as it was

for the AA and TAA.

Equations (81) –(84) define what Braginsky (1964) and Braginsky

and Roberts (1995 ) termed the “ homogeneous model. ” The density

appearing in Eqs. (82) and (83) is the constant reference density r

and, because of this similarity with the BA, we call the present theory

the “generalized Boussinesq approximation ” or “ GBA,” but often it is

simply called the “ Boussinesq approximation. ” As its name suggest s,

the GBA is more widely applicable than the BA. The BA requires that

(Eq. 36) holds, but the GBA may be used when

 e

 1 : (Eq. 85)

This extension in the range of validity comes about because, whereas

the BA involves the departure T

from a constant reference tempera-

ture T

, the GBA employs the deviation T

from an adiabatic T

The GBA is especially appropriate for the oceans . The fractional

changes in density created by the pressure and temperature are of order

and e

, respectively and, as in “The Boussinesq approximation

(BA), ” they are small. The pressure in the ocean changes from 1 atm

at its surface to several times 10

atm at a depth of several kilometers.

Hence P

cannot be treated as a small correction to a constant P

;

the changes in pressure and temperature are of the same order so that

(Eq. 85) holds and not (Eq. 36). Nevertheless, as in the TAA above, it

is only the temperature-created change in density, r

g , that drives

convection. Although the GBA is relevant to oceanic convection,

compositional buoyancy, arising from differences in salinity, is also a

significant factor that requires Eqs. (81) –(84) to be appropriately gen-

eralized; see Kamenkovich (1977).

Despite the fact that e

 0: 2 in the Earth ’s core, so that r

 r

and T

 T

are poor approximations, the GBA is often used in study-

ing core convection. Since aT

 5  10

 2

however, the approxima-

tions T

 C

and a

 a T

are well satisfied, with an error

only of order 10

 3

, so that the TAA is applicable.

The ro le of turbul ence

It was observed in “The anelastic approximation (AA) ” that, when Ra

is large, there is every likelihood that the fluid motions are turbulent

and therefore inaccessible to accurate numerical simulation either

now or in the foreseeable future. This formidable difficulty besets

the natural sciences. Progress can be made only by abandoning all

hopes of describing V; S

;... in detail, and by seeking instead the

evolution of their averages,

V ; S

;...; these may be smooth enough

to be numerically resolvable. The remaining “turbulent ” fields,

; S

;... , are often termed the “sub-grid scale ” fields, and are

numerically unobtainable. Since however small-scale motions may

diffuse the mean fields far more effectively than the molecular diffu-

sivities, their effect on the mean fields cannot be ignored. It is abso-

lutely necessary to incorporate a description of the small-scale fields

into the mean field theory.

The fields are separated into mean and turbulent parts:

V ¼

V þ V

; S

¼ S

þ S

;......: (Eq. 86)

(We omit the

from S

because turbulence automatically implies con-

vection.) Consider first the transport of entropy. The ensemble avera ge

of the entropy equation, in the form (Eq. 58), is

þ H  I

¼ s

; (Eq. 87)

where

¼ ]

þ V  H and

¼ I

þ I

; s

¼ T

1

ðQ

þ Q

ÞI

 HT

(Eq. 88)

Here

¼ r

(Eq. 89)

is the “ turbulent entropy flux,” from which the “ turbulent heat flux,”

¼ T

can be obtained. This may be comparable with the molecu-

lar heat flux down the adiabat, I

¼K

H T

, and is large compared

with

¼K

, which is of order e

where e

 1; the super-

fix

stands for “molecular.” The viscous dissipation rate has a mole-

cular part,

v m

¼ p

v m

, and a turbulent part, Q

¼ p

, that

is usually much larger.

Similar arguments apply to (Eq. 57), the average of which may be

written as

V  F

¼ r

 r

g þ F

; (Eq. 90)

where

¼ H  p

╯

─

╰ nt

; and p

¼r

(Eq. 91)

is the “Reynolds stress tensor ” ; F

n t

may be combined with the

viscous force,

, of molecular origin created by V, previously

denoted by H  p

╯

─

╰n

. It is however generally much greater (except

perhaps, in the case of the Earth ’ s core, in Ekman layers). The rate

of working of the turbulent buoyancy force is

¼r

 g ¼I

 H T

; (Eq. 92)

and may be comparable with the rate of working,  r

V  g ,of

the mean buoyancy force. By using (Eq. 90) to remove the averaged

fields from (Eq. 57), an equation governing V

is derived. Taking the

scalar produ ct of this with r

and averaging, it is found that, in sta-

tistically steady turbulence ðE

ðV

¼ constant Þ,

¼ Q

n t

; (Eq. 93)

i.e., the buoyant energy released on the turbulent scales supplies on

average the kinetic energy necessary to maintain the turbulent motions

against viscous decay. It should be emphasized however that, in mag-

netohydrodynamic systems such as the Earth ’s core, the right-hand

side of (Eq. 93) is replaced by the sum of the viscous and ohmic dis-

sipations, and the ohmic dissipation may dominate; see Braginsky and

Roberts (1995, 2003).

In summary, the mean fields are governed by

H ðr

VÞ¼0; (Eq. 94)

V ¼HP

 a

g þ F

ntotal

; (Eq. 95)

þ H  I

qtotal

¼ Q

þ Q

ntotal

; (Eq. 96)

where

¼ P

and

ntotal

¼ F

þ F

; Q

ntotal

¼ Q

þ Q

;

qtotal

¼ I

þ I

: (Eq. 97)

ANELASTIC AND BOUSSINESQ APPROXIMATIONS 17

Equations (94) – (96) have the same form as Eqs. (56) – (58), but

involve mean quantities,

; Q

n t

; p

╯

─

╰n t

and I

, thathaveto besup-

plied by experimental measurement or by turbulence theory. Until

this is done, the turbulent entropy and momentum fluxes must be

parameterized.

“ Local turbulence theory ” provides one possible parameterization;

this rests on the assumption of “scale separation, ” i.e., that the length

and time scales on which the mean fields evolve are so widely sepa-

rated from turbulent scales that the effects of the latter are, as far as

the mean fields are concerned, only “ local.” The diffusive action

of local turbulence on the averaged motion can be parameterized

by turbulent diffusivities, K

and v

, that are much greater than

their molecular counterparts, e.g.,

¼K

. [This parameteriza-

tion is also commo nly used in describing astrophysical convection.]

Such forms are appropriate when the turbulence is isotropic but when,

as for the Earth ’ s core, this is probably not the case, anisotropic (ten-

sor) diffusivities are more realistic, e.g.,

¼K

╯

─

╰ t

 HS

. In either

case, it is H

and not HT

, that is significant. For this reason “the ane-

lastic approximation (AA) ” is more convenien t for core convection

than the alternative “ temperature-based anelastic approximation

(TAA),” even though the conditions required for the validity of the lat-

ter are fairly well satisfied. For the compositionally generalized AA of

“the anelastic approximation (AA), ” the turbulent flux

¼ r

x analogous to (Eq. 89) may be parameterized by

¼K

╯

─

╰ t

 Hx

with the same turbulent diffusivity K

╯

─

╰ t

as in I

¼K

╯

─

╰ t

 HS

; see

Braginsky and Roberts (1995, 2003).

More detailed discussions about the role of turbulence may be found

in Braginsky (1964), Braginsky and Meytlis (1990), and Bragin sky

and Roberts (1995, 2003). See also Core turbulence .

Final rema rks

An alternati ve expres sion of the BA

Formulations of the BA, different from “the Boussinesq approximation

(BA)” and “the generalized Boussinesq approximation (GBA), ” are

often preferred; see for example Spiegel and Veronis (1960). In the

convecting state, (Eq. 31) is replaced by

r ¼ r

þ r

; T ¼ T

þ T

; P ¼ P

þ P

;...; (Eq. 98)

where the primed variables arise from convection and variables carry-

ing the suffix “ b ” refer to a “basic conductive state ” that is in mechan-

ical and thermal equilibrium: V

¼ 0. Evidently, unlike “The

Boussinesq approximation (BA), ” the basic state is nonuniform.

Mechanical equilibrium requires

H P

¼ r

g ¼r

H U : (Eq. 99)

This implies that r

, P

, and therefore T

are constant on equipotential

surfaces, i.e., surfaces on which U is constant. Thermal equilibrium

requires that T

satisfies the imposed thermal boundary conditions

together with

H ðK

Þþ Q

¼ 0 ; (Eq. 100)

obtained from Eqs. (3) and (19).

According to Eqs. (46) and (98), T

¼ T

þ T

 T

. On substituting

this into Eqs. (81) – (83) and assuming that T

 T

 T

 T

the equations governing the convective motions become

H  V ¼ 0 ; (Eq. 101)

V ¼HP

 a T

g þ F

; (Eq. 102)

þ V  Hð T

 T

Þ¼½H ðK

H T

ÞþQ

= r

: (Eq. 103)

The term in (Eq. 103) that is proportional to the superadiabatic gradi-

ent is included implicitly in the GBA and is significant when

 e

ð 1 Þ .

One of the disadvantag es of the present formulation is apparent in

the case of “free convection, ” in which boundary conditions prevent

and T

from being constant on equipotential surfaces. Simultaneous

solution of Eqs. (99) and (100) is then impossible, and the assumption

¼ 0 is untenable; convection occurs for all nonzero Ra. By adopt-

ing Eqs. (31) or (46) in preference to (Eq. 98), free convection motions

do not have to be treated separately; they are on the same footing as

convective instabilities and are included with them.

A more ge neral CA

In some contexts, convection is weak or nonexistent in some regions

of the system but is strong in other regions. For example, consider

the thermal boundary layer at a no-slip wall confining a strongly

convecting system, or consider a transition layer via which convection

in a superadiabatic region penetrates into an adjacent subadiabatic

region. In such situations it is useful to select the basic state in a more

general way than has so far been contemplated. The basic state

(denoted by subscript “b ” ) is again hydrostatic, but satisfies the heat

equation (5) in the form

þ H  I

¼ Q

; (Eq. 104)

which reduces to (Eq. 48) in a strongly convecting region but to (Eq.

100) in an adjacent weakly or nonconvecting region. Since

ð¼ V  HS

Þ involves V, (Eq. 104) imposes a consistency require-

ment on the solution of the convection equations. Despite this compli-

cation, the theory uses thermodynamic linearization and is therefore

easier to apply than the full panoply of “Basic equations.”

Denoting departures from the basic state by

, convection is gov-

erned by

H ðr

V Þ¼0 ; (Eq. 105)

V ¼HP

þðT

H S

 S

ÞþF

= r

; (Eq. 106)

ð T

Þr

V ðS

H T

 T

H S

ÞþH  I

¼ Q

; (Eq. 107)

where P

¼ P

= r

. In deriving (Eq. 106), the relations H r

2

H P

 r

and r

¼ v

2

 r

obtained from (Eq. 22)

have been used. It follows from these that r

H P

 P

H r

ð T

 S

H T

Þ , by (Eq. 27). The buoyancy term T

H S

 S

in (Eq. 106) can also be written as  a

g þ P

H S

or as

 a T

g þ P

a HT

; cf. Eqs. (62) and (74).

A generalized CA of the type considered here has been developed

by Rogers and Glatzmaier (2005).

Boundary con ditions for CA

Boundary conditions are specific to the system under consideration

and are imposed on the solutions of the basic equations of “Basic

equations.” They translate directly into corresponding conditions on

solutions of the CA equations. Their application to the AA, however,

raises questions that are briefly addressed here, using the Earth’s core

as an illustration.

The core and mantle form one, coupled system that must obey the

correct thermal conditions at the CMB. The first of these is

Core

¼ T

Mantle

; on the CMB; (Eq. 108)

18 ANELASTIC AND BOUSSINESQ APPROXIMATIONS

where each side is the sum of adiabatic and convective parts, e.g.,

Core

¼ T

Core ;a

þ T

Core; c

. The second expresses continuity of the nor-

mal component of the heat flux

Core;n

¼ I

Mantle ;n

; on the CMB; (Eq. 109)

where each side is the sum of adiabatic and convective parts, e.g.,

Core ;n

¼ I

Core;a

þ I

Core; cn

. In the case of the core, these were denoted

earlier simply by I

¼K

]

¼ K

G and the normal component

of I

It is clearly inconvenient to have to solve for core and mantle con-

vection simultaneously but, because V

mantle

 V

core

, the two systems

can be considered separately. Each of the four fluxes comprising

(Eq. 109) are a few TW and, since I

Core; cn

ðr C

Core

and

Mantle ; cn

ðr C

Mantle

, it follows that T

Core ;c

 T

Mantle ; c

.Toa

very good approximation therefore, (Eq. 108) reduces to

Mantle

¼ T

Core ;a

; on the CMB : (Eq. 110)

This is the thermal condition to apply to mantle convection. It would

be wrong from a physical point of view to assign T

CMB

when solving

core convection. Having solved the mantle convection equations with

the T

CMB

prescribed, I

Mantle ;n

on the CMB becomes known (in princi-

ple), and this determines the right-hand side of (Eq. 109), which

becomes the condition to be applied to core convection:

Core;cn

¼ I

CMB

; where I

CMB

¼ðNu  1Þ I

Core ;a

; on the CMB :

(Eq. 111)

Here Nu ¼ I

Mantle ;n

= I

Core ;a

is the Nusselt number, a nondimensional

measure of the importance of convective heat transport in the core

relative to the conductive; in general it is a function of horizontal posi-

tion on the CMB (see Core-mantle coupling, thermal ).

In the bulk of the core, turbulent motions are responsible for the flux

but V

, falls to zero as the CMB is approached, so that thermal con-

duction must increasingly carry the necessary heat flux until, on the

CMB itself, I

¼K

. Since I

is typically large, H

T must

be large too, both at the CMB and in a thin, nonadiabatic, boundary

layer abutting the CMB. We shall not discuss the structure of the

boundary layer here (see Core, boundary layers ).

A note on termino logy

All CAs filter out elastic waves (seismic/acoustic) and they could

therefore all be described as “ anelastic, ” i.e., not elastic. Paradoxically,

the AA and TAA are the only CAs that recognize elasticity at all, in

the sense that they include the slow elastic response of the density to

changes in pressure. If these are termed “anelastic, ” the BA is totally

anelastic! A CA is commonly called “Boussinesq ” if it uses

H  V ¼ 0 but “anelastic ” if it uses H ðr

V Þ¼0 instead.

Stanislav I. Braginsky and Paul H. Roberts

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cal Journal Supplement, 121: 247–

264.

Monin, A.S., 1990. Theoretical Geophysical Fluid Dynamics,

Dordrecht: Kluwer.

Oberbeck, A., 1888. On the phenomena of motion in the atmosphere.

Sitz. König. Preuss. Akad. Wiss., 261–275. English translation in

Saltzman (1962).

Ogura, Y., and Phillips, N.A., 1962. Scale analysis of deep and shal-

low convection in the atmosphere. Journal of the Atmospheric

Sciences, 19: 173–179.

Rayleigh, Lord. 1916. On convection currents in a horizontal layer

of fluid, when the higher temperature is on the under side.

Philosophical Magazine (6), 32: 529–546. Reprinted in Saltzman

(1962).

Rogers, T.M., and Glatzmaier, G.A., 2005. Penetrative convection

within the anelastic approximation. Astrophysics Journal, 620:

432–441.

Saltzman, B., (ed.) 1962. Selected Papers on the Theory of Thermal

Convection with Special Application to the Earth’s Planetary

Atmosphere, New York: Dover.

Schubert, G., Turcotte, D.L., and Olson, P., 2001. Mantle Convection

in the Earth and Planets, Cambridge: University Press.

Spiegel, E.A., and Veronis, G., 1960. On the Boussinesq approxi-

mation for a compressible fluid. Astrophysics Journal, 131:

442–497.

Tackley, P.J., 1997. Effects of phase transition on three-dimensional

mantle convection. In Crossley, D.J., (ed.), Earth’s Deep Interior.

London: Gordon and Breach.

Cross-references

Convection, Chemical

Convection, Nonmagnetic Rotating

Core Convection

Core-mantle Coupling, Thermal

Core Turbulence

Core, Boundary Layers

Geodynamo, Energy Sources

Magnetoconvection

Magnetohydrodynamics

ANELASTIC AND BOUSSINESQ APPROXIMATIONS 19

ANISOTROPY, ELECTRICAL

In the geosciences, the term “electrical anisotropy” describes the direc-

tional dependence of the electrical conductivity of Earth materials.

In the case of intrinsic anisotropy, this directional dependence occurs

on all scales, while in the case of macroscopic anisotropy composite

media are formed from isotropic components in such a way that the bulk

conductivity is anisotropic. Intrinsic anisotropy can be described with

numerical modeling of the electromagnetic (see EM ) induction process

(e.g., Pek and Verner, 1997). An example of macroscopic anisotropy is

presented in FigureA2.

Field observations: magnetotellurics

Fournier (1963) first realized that differences between amplitudes of

orthogonal electromagnetic fields, measured with magnetotellurics

(MT) (q.v.), can be associated with electrical anisotropy. In modern

MT, anisotropic Earth models are derived from differences between

the phases (phase lags between electric and magnetic fields) of two

polarizations of the electromagnetic field (Kellett et al., 1992)—see

FigureA2. In contrast, amplitude differences are rarely evaluated

now due to galvanic distortion (q.v.) (Simpson and Bahr, 2005). Tests

are available which allow to distinguish between anisotropic Earth

models and models of conductivity anomalies that only generate aniso-

tropic data (Bahr et al., 2000). In these tests the phase differences

obtained from many MT sites in a target area are compared, and

anisotropic Earth models are considered if all sites have similar phase

differences.

Laboratory measurements and scaling

The ratio between the highest and the lowest conductivity is referred to

as anisotropy factor. The electrical anisotropy factors found in labora-

tory studies of crustal rocks (e.g., Rauen and Lastovickova, 1995) are

smaller than those found in MT earth models, and this discrepancy has

been explained with scaling techniques in composite media (Bahr,

1997) (FigureA3). In contrast, the anisotropy factor of hydrogen diffu-

sity in olivine can be as large as 40 (Kohlstedt and Mackwell, 1998)

and this can result in a large electrical anisotropy of “wet” olivine

(Lizzaralde et al., 1995).

Geodynamic implications

MT data from Northern Bavaria have been interpreted with electrical

anisotropy in the middle crust with the direction of highest conduc-

tance (electromagnetic strike) aligned parallel to paleoterrane bound-

aries (Bahr et al., 2000). Anisotropic conductive structures in the

continental crust can therefore be fossil remainings of terrane accre-

tion. Marine-controlled electromagnetic data from the Pacific show

that the oceanic crust can be electrically anisotropic, with the direction

of highest conductance aligned parallel to the fossil spreading direction

(Everett and Constable, 1999). Strain-aligned mineralogical fabric

with connected accumulations of trace conductors such as graphite

or sulfides can account for this anisotropy. Graphite is also the most

likely cause of electrical anisotropy in the North American lithosphere

(Mareschal et al., 1995).

With long period MT, electrical anisotropy has been found at the

base of the Australian lithosphere with the direction of highest conduc-

tance rotated 30



relative to the direction of the present day absolute

Figure A2 Isotropic and anisotropic MT Earth models and the

corresponding transfer functions, displayed as apparent

resistivity r

(amplitude of the E/B transfer function) and phase j

(phase lag between the electric field E and the magnetic field B),

as a function of period T. Due to the skin effect, electromagnetic

fields with longer periods penetrate deeper into the Earth than

shorter periods. The conductive (black) intermediate layer of the

isotropic model generates a minimum in the apparent resistivity

curve at intermediate periods as well as a variation in the phase

curve, regardless of the polarization of the electromagnetic field,

or E

. In the anisotropic model, this intermediate layer

is conductive only in the x direction and therefore affects only

the transfer function for the E

polarization, resulting in a

phase difference between the E

and the E

polarization in

the intermediate period band.

Figure A3 Electrical connectivity g of a random resistor network

representing a two-component medium as function of the fraction

p of the high conductive component. At the percolation threshold,

a small increase of this fraction results in a relatively large

increase of the connectivity. Suppose the high conductive

component is distributed in microcracks: If these cracks occur

slightly more often in the y direction than in the x direction, then a

large difference between the connectivities g

and g

in the y and x

direction is created. This difference results in anisotropic bulk

conductivity.

20 ANISOTROPY, ELECTRICAL

plate motion (APM) that is determined relativ e to the hotspot reference

frame (Simpson, 2001). But the APM direction that is determined rela-

tive to a reference frame defined by requiring no net rotation of the

lithosphere matches the electrical anisotropy direction better (Simpson,

2002). Mantle flow models suggest that shear of the mantle imparted

by the drag of an overriding plate favors alignment of olivine [100]

axes parallel to the direction of plate motion (McKenzie, 1979), and

this has been confirmed by seismological studies with surface waves

(e.g., Debayle and Kennett, 2000). Hydrogen diffusivities enhance

the conductivity of olivine crystals anisotropically (e.g., Lizzaralde

et al., 1 99 5) , p ro vi di n g a p os si bl e e xp la na ti on f o r t he M T a ni so t r o py i f

hotspots are not stationary relative to the deep mantle (Simpson, 2002).

However, electrical anisotropy has also been found below the European

lithosphere where current plate motion is slow, and therefore paleoflow

can also explain the electrical anisotropy of the sublithospheric mantle

(Leibecker et al., 2002; Gatzemeier and Moorkamp, 2005).

Karsten Bahr

Bibliogr aph y

Bahr, K., 1997. Electrical anisotropy and conductivity distribution

functions of fractal random networks and of the crust: the scale

effect of connectivity. Geophysical Journal International , 30:

649– 660.

Bahr, K., Bantin, M., Jantos, Chr., Schneider, E., and Storz, W., 2000.

Electrical anisotropy from electromagnetic array data: implications

for the conduction mechanism and for distortion at long periods.

Physics of the Earth and Planetary Interiors , 11 9 : 237– 257.

Debayle, E., and Kennett, B.L.N., 2000. The Australian continental

upper mantle: structure and deformation inferred from surface

waves, Journal of Geophysical Research, 105: 25423 –25450.

Fournier, H., 1963. La spectrograph ic directionelle magnéto-tellurique

a Garchy (Niève). Annales Geophysicae , 19 : 138 –148.

Gatzemeier, A., and Moorkamp, M., 2005. 3D modelling of electrical

anisotropy from electromagnetic array data: hypothesis testing for

different upper mantle conduction mechanisms. Physics of the

Earth and Planetary Interiors , 149 : 225– 242.

Kellett, R.L., Mareschal, M., and Kurtz, R.D., 1992. A model of lower

crustal electrical anisotropy for the Pontiac Subprovince of

the Canadian Shield. Geophysical Journal International , 11 1 :

141– 150.

Kohlstedt, D.L., and Mackwell, S., 1998. Diffusion of hydrogen and

intrinsic point defects in Olivine. Z. Phys. Chemie, 207 : 147– 162.

Leibecker, J., Gatzemeier, A., Hönig, M., Kuras, O., and Soyer, W.,

2002. Evidence of electrical anisotropic structures in the lower

crust and the upper mantle beneath the Rhenish Shield. Earth

Planet Science Letter, 202 : 289– 302.

Lizzaralde, D., Chave, A., Hirth, G., and Schultz, A., 1995. North-

eastern Pacific mantle conductivity profile from long-period mag-

netotelluric sounding using Hawaii-to-California submarine cable

data. Journal of Geophysical Research, 100: 17837 –17854.

Mareschal, M., Kellett, R.L., Kurtz, R.D., Ludden, J.N., and Bailey, R.C.,

1995. Archean cratonic roots, mantle shear zones and deep electrical

anisotropy. Nature, 375:134–137.

McKenzie, D., 1979. Finite deformation during fluid flow. Geophysical

Journal of the Royal Astronomical Society, 58 : 689–715.

Pek, J., and Verner, T., 1997. Finite-difference modelling of magneto-

telluric fields in two-dimensional anisotropic media. Geophysical

Journal International ,

128: 505– 521.

Rauen, A., and Lastovickova, M., 1995. Investigation of electrical

anisotropy in the deep borehole KTB. Surveys in Geophysics , 16:

37– 46.

Simpson, F., 2001. Resistance to mantle flow inferred from the elec-

tromagnetic strike of the Australian upper mantle. Nature, 412:

632– 635.

Simpso n, F., 2002. Intensity and direction of lattice-preferred orien-

tation of olivine: are electrical and seismic anisotropies of the

Australian mantle reconcilable? Earth Planet Science Letter, 203:

535– 547.

Simpso n, F., and Bahr, K., 2005. Practical Magnetotellurics.

Cambridge: Cambridge University Press.

Cr oss-referenc es

Galvanic Distortion

Induction

Magnetotellurics

Mantle, Electrical Conductivity, Mineralogy

ANTIDYNAMO AND BOUNDING THEOREMS

Dynamos are devices that convert mechanical energy into electromag-

netic energy. While technical dynamos depend on a suitable arrange-

ment of multiply connected regions (usually wires) of high electrical

conductivity within an insulating space, the generation of magnetic

fields in planets and stars must occur in simply connected domains

of essentially uniform finite conductivity. For a long time after Larmor

(1919) (see Larmor, J.) first proposed this homogeneous dynamo process

as the origin of magnetic fields in sunspots it has been doubtful whether it

is possible. Mathematicians and geophysicists have proved antidynamo

theorems in order to determine the conditions under which homogeneous

dynamos are possible. The first and most famous theorem has been formu-

lated and proved by Cowling (1934) (see Cowling, T.G.). He demon-

strated that axisymmetric or two-dimensional magnetic fields cannot

be generated by the homogeneous dynamo process (see Cowling’s

Theorem). Since the Earth’s magnetic field is nearly axisymmetric the

dynamo hypothesis of the origin of geomagnetism remained in doubt for

a quarter of a century until Backus (1958) and Herzenberg (1958) could

show in a mathematically convincing way that nonaxisymmetric fields

could indeed be generated (see Dynamo, Herzenberg and Dynamo,

Backus).

Antidynamo and bounding theorems can be divided into the follow-

ing categories:

1. Lower bounds on magnetic Reynolds numbers or related quantities

that are necessary for amplifying magnetic fields

2. Structures and symmetries of velocity fields that cannot be dyna-

mos (e.g., Toroidal theorem)

3. Structures and symmetries of magnetic fields that can not be gener-

ated by the dynamo process (e.g., Cowling’s theorem)

Lower bounds for magnet ic Reynolds numbers

Nearly all theorems that have appeared in the literature are based on

the equation of magnetic induction for the magnetic flux density B,

þ v r



B þrðlrBÞ¼B rv; (Eq. 1)

which can be derived from Maxwell’s equations in the magnetohydro-

dynamic approximation in which the displacement current is neglected

and from Ohm’s law for a moving conductor,

lrB ¼ v  B þ E (Eq. 2)

where E is the electric field, v is the velocity vector field, and l is the

magnetic diffusivity. The latter is defined as the inverse of the product

of magnetic permeability and electrical conductivity, l ¼ðs  mÞ

 1

In writing equation 1, a solenoidal velocity field v has also been assumed.

ANTIDYNAMO AND BOUNDING THEOREMS 21

Antidynamo theorems of type (i) can be obtained by multiplying equation

1byB and integrating it over a finite domain V o f f i ni te co nd uc ti vi ty

d t

V þ V

jB j

V ¼

B ðB rÞvd

V 

l jr B j

(E q. 3 )

Here the integral on the left hand side is extended over the entire space

V þ V

where the integral over the electrically insulating complement

of V arises from the partial integration of the term rðlr B Þ (for

details see Backus, 1958). Using the maximum m(t) of the rate of strain

tensor and the minimum decay rate s

, which are defined by

m ðt Þ¼ ma x

ð q

þ q



; s

 mi n

ljr B j

V þ V

jB j

(E q. 4 )

the inequality

d t

V þ V

jB j

V ðmð t Þs

V þV

j Bj

V (Eq. 5)

can be obtained (Backus, 1958). This relationship indicates the rate of

stretching of the magnetic field by the velocity field must be suffi-

ciently strong in order to overcome the effect of Ohmic dissipation.

In the special case when V is a sphere of constant l with radius r

and when the velocity field is steady such that m ðt Þ¼ v

= r

can be

written with a constant v

of the dimension of a velocity, inequality

(Eq. 5) can be obtained in the form

d t

V þ V

jB j

V ðRm  p

V þ V

jB j

V (Eq. 6)

where the magnetic Reynolds number Rm  v

= l has been intro-

duced and the expression s

¼ p

l =r

for the smallest rate of decay

of a magnetic field in a conducting motionless sphere has been used.

This inequality states that unless the magnetic Reynolds number Rm

exceeds the value p

a dynamo cannot exist in a conducting sphere.

Another version of this theorem was obtained by Childress (1969),

which assumes the same form (Eq. 6) except that v

in the definition

of Rm denotes the maximum of j vj , which is the more common defini-

tion of the magnetic Reynolds number. For an improvement of

Backus’ bound, see Proctor (1977). A related criterion has been

derived by Proctor (1979). It states for

jrvj

V 

16r

(Eq. 7)

no growing magnetic field can be obtained. The expression on the left

hand side denotes, of course, the viscous dissipation of the velocity

field v divided by the viscosity. It should be noted that all equations

and derivations used in this article remain valid in a rotating system,

which makes the results especially useful for geophysical and astro-

physical applications.

Veloci ty fiel ds that cann ot act as dyn amos

For the discussion of other dynamo theorems it is convenient to intro-

duce the general representation for solenoidal vector fields such as B

and v (in the case when rv ¼ 0 can be assumed),

B ¼rðrh  r Þþrg  r ;

v ¼rðrF  r ÞþrC  r  v

pol

þ v

tor

(Eq. 8)

where r is the position vector. The poloidal scalar functions h and

F and the toroidal scalar functions g and C are uniquely defined when

the condition is imposed that their averages over surfaces jr j¼const .

vanish (Backus, 1958). A famous theorem of type (ii) that was already

known to Elsasser (1946) (see Elsasser W.) and been discussed in

detail by Bullard and Gellman (1956) is the Toroidal theorem, which

states that a purely toroidal velocity field for which F  0 holds can-

not generate a magnetic field when l ¼ lð r Þ is a function of the radius

r ¼jr j only. The validity of this theorem can easily be seen when

equation 1, multiplied by r,

þ v r



B  r  lð r Þr

B  r ¼ B rv  r (Eq. 9)

is considered. Since the right hand side vanishes for a purely toroidal

velocity field, equation 9 becomes the heat equation for the “tempera-

ture ” B  r without heat sources for which only decaying solutions

exist. After the r-component of B has decayed it is easy to see that

the other components must decay as well. The question could be raised

whether a purely poloidal velocity field can generate a magnetic field.

The answer is “yes. ” Gailitis (1970) provides an example in that

he used an axisymmetric purely poloida l velocity field for his ring

dynamo (see Dynamo, Gailitis ).

Since a finite radial component of the velocity field is required for

dynamo action according to the Toroidal theorem, it is of interest to

see whether bounds on its magnitude can be obtained. The condition

maxjv  r j l M

pol

= M (Eq. 10)

derived on the basis of equation 9 for a sphere of constant diffusivity l

provides a necessary condition for a dynamo (Busse, 1975). Here M

denotes the total energy of the magnetic field, while M

pol

refers to

the energy of its poloidal component.

A related criterion that is independent of magnetic energies has

recently been derived by Proctor (2004). It states that the energies of

poloida l and toroidal components of the magnetic field decay expo-

nentially if

j v

pol

tor

ﬃﬃﬃ

pol

(Eq. 11)

is satisfied where j v

pol

and jv

tor

indicate the maximum values of

j v

pol

j and jv

tor

j in the conducting sphere of radius r

. For

j v

tor

> pl=r

this criterion is more severe than Childress ’ criterion

mentioned above in connection with relationship (Eq. 6).

Antidynamo theorems have also been derived for veloci ty fields

with only a radial component (Namikawa and Matsushita, 1970) in

which case the property rv ¼ 0 must be dropped, of course. For a

discussion of antidynamo theorems for a combination of a toroidal

velocity field and a purely radial velocity we refer to Ivers (1995)

and earlier papers mentioned therein.

Magnetic fields that cannot be generated

by the dynamo process

We have already mentioned the fact that equation 1 does not permit

growing axisymmetric magnetic fields as solutions as stated by Cowl-

ing’s theorem. But it is also generally believed that neither a purely

toroidal nor a purely poloidal magnetic field can be generated by the

dynamo process in a sphere. The assumption of spherical symmetry

is essential since the boundary conditions for poloidal and toroidal

components separate on a spherical surface. The only other configura-

tion with this property appears to be the planar layer, which may be

regarded as the limit of a thin spherical shell. Proofs in the spherical

and planar cases are thus essentially identical. A proof that any physi-

cally reasonable purely toroidal field must decay has been obtained by

22 ANTIDYNAMO AND BOUNDING THEOREMS

Kaiser et al. (1994). In the poloidal case a proof depending on a phy-

sical plausibility argument has been given by Kaiser (1995).

Concluding remarks

The search for antidynamo theorems has led to some interesting math-

ematical theorems in the past decades. Just as important are the physi-

cal insights into the dynamo process that have been gained. The

interactions of poloidal and toroidal components of the magnetic field

are evidently essential for the operation of a dynamo. To summarize

the main results of antidynamo theorems the Table A1 has been com-

posed. In it we have included the case of helical symmetry, which is

known to be dynamo friendly. Indeed Lortz (1968) was able to derive

a dynamo with helical symmetry.

Friedrich Busse and Michael Proctor

Bibliography

Backus, G., 1958. A class of self-sustaining dissipative spherical dyna-

mos, Annals of Physics, 4: 372–447.

Bullard, E., and Gellman, H., 1954. Homogeneous dynamos and ter-

restrial magnetism. Philosophical Transactions of the Royal

Society of London, A247: 213–278.

Busse, F.H., 1975. A necessary condition for the geodynamo. Journal

of Geophysical Research, 80: 278–280.

Childress, S., 1969. Théorie magnétohydrodynamique de l’effet

dynamo, Lecture Notes. Département Méchanique de la Faculté

des Sciences, Paris.

Cowling, T.G., 1934. The magnetic field of sunspots. Monthly Notices

of the Royal Astronomical Society, 34:39–48.

Elsasser, W.M., 1946. Induction effects in terrestrial magnetism. Phy-

sical Review, 69: 106–116.

Gailitis, A., 1970. Magnetic field excitation by a pair of ring vortices.

Magnetohydrodynamics (N.Y.), 6:14–17.

Herzenberg, A., 1958. Geomagnetic Dynamos. Philosophical Transac-

tions of the Royal Society of London , A250: 543–585.

Ivers, D.J., 1995. On the antidynamo theorem for partly symmetric

flows. Geophysical and Astrophysical Fluid Dynamics, 80: 121–128.

Kaiser, R., 1995. Towards a poloidal magnetic field theorem. Geophy-

sical and Astrophysical Fluid Dynamics, 80: 129–144.

Kaiser, R., Schmitt, B.J., and Busse, F.H., 1994. On the invisible

dynamo. Geophysical and Astrophysical Fluid Dynamics,77:91–109.

Larmor, J., 1919. How could a rotating body such as the sun become a

magnet? Reports of the British Association for the Advancement of

Science, 159–160.

Lortz, D., 1968. Exact solutions of the hydromagnetic dynamo pro-

blem. Plasma Physics, 10: 967–972.

Namikawa, T., and Matsushita, S., 1970. Kinematic dynamo problem.

Geophysical Journal of the Royal Astronomical Society, 19:319–415.

Proctor, M.R.E., 1977. On Backus’ necessary condition for dynamo

action in a conducting sphere. Geophysical and Astrophysical

Fluid Dynamics, 9:89–93.

Proctor, M.R.E., 1979. Necessary conditions for the magnetohydrody-

namic dynamo. Geophysical and Astrophysical Fluid Dynamics,

14: 127–145.

Proctor, M.R.E., 2004. An extension of the toroidal theorem. Geophy-

sical and Astrophysical Fluid Dynamics, 98: 235–240.

Cross-references

Cowling’s Theorem

Cowling, Thomas George (1906–1990)

Dynamo, Backus

Dynamo, Gailitis

Dynamo, Herzenberg

Elsasser, Walter M. (1904–1991)

Larmor, Joseph (1857–1942)

ARCHEOLOGY, MAGNETIC METHODS

Magnetism and archeology

Introduction

Magnetic methods have become important tools for the scientific

investigation of archeological sites, with magnetic prospection surveys

and archeomagnetic dating being the most prominent ones. The princi-

ples behind these techniques were initially applied to larger and older

features, for example prospecting for ore deposits (see Magnetic

anomalies for geology and resources) or paleomagnetic dating (see

Paleomagnetism). When these techniques were adapted for archeologi-

cal targets it was soon established that very different methodologies

were required. Archeological features are relatively small and buried

at shallow depth and the required dating accuracy is in the order of

tens of years. More importantly, the relationship between archeological

features and magnetism is often difficult to predict and the planning of

investigations can hence be complicated. Related is the problem of

interpretation. Geophysical results on their own are only of limited

use to resolve an archeological problem. It is the archeological inter-

pretation of the results using all possible background information (site

conditions, archeological background knowledge, results from other

investigations, etc.), which provides useful new insights. If the rela-

tionship between magnetic properties and their archeological forma-

tion is unknown, such interpretation may become speculative.

All magnetic investigations depend on the contrast in a magnetic

property between the feature of interest and its surrounding environ-

ment, for example the enclosing soil matrix. The most important mag-

netic properties for archeological studies are magnetization and

magnetic susceptibility.

Remanent magnetism

Thermoremanent magnetization is probably the best understood mag-

netic effect caused by past human habitation. If materials that are rich

in iron oxides are heated above their Curie temperature and then

allowed to cool in the ambient Earth’s magnetic field they have the

potential to acquire a considerable thermoremanence that is fixed in

the material until further heating. Typical archeological examples are

kilns and furnaces, often built of clay, which during their heating

cycles often exceed the Curie temperatures of magnetite (Fe

) and

maghaemite (g-Fe

) (578



C and 578–675



C, respectively). Such

iron oxides are commonly found in the clay deposits that were used

for the construction of these features. Even if the clays only contained

weakly magnetic haematite or goethite, the heating and cooling cycles

may have converted these into ferrimagnetic iron oxides (see Magnetic

mineralogy, changes due to heating). Similarly, fired bricks and

pottery can exhibit thermoremanence but when the finished bricks are

used as building material their individual vectors of magnetization will

point into many different directions producing an overall weakened

magnetic signature (Bevan, 1994). The same applies to heaps of

pottery shards. The strong magnetic remanence of kilns led to

their discovery with magnetometers in 1958 (Clark, 1990), which trig-

gered the widespread use of archeological magnetometer surveys today.

Table A1 Existence of homogeneous dynamos

Properties Of magnetic field Of velocity field

Axisymmetry No Yes

Purely toroidal No No

Purely poloidal No (?) Yes

Helical symmetry Yes Yes

ARCHEOLOGY, MAGNETIC METHODS 23

Kilns also helped to establish the archeomagnetic dating technique as a

new tool for chronological studies in archeology (Clark et al., 1988).

Detrital remanence is caused when the Earth ’s magnetic field aligns

magnetized particles that are suspended in solution and gradually settle

(see Magnetization, depositional remanent). An example is stagnant

water loaded with “magnetic sediments”. The resulting deposits can

exhibit a weak but noticeable remanence, which could be used for dat-

ing and prospection. It is suspected that similar effects have produced

a small remanence and hence a noticeable positive signal in Egyptian

mud-bricks, which were made from wet clay pushed into moulds and

dried in the sun (Herbich, 2003). However, other magnetometer sur-

veys over mud-brick structures (Becker and Fassbinder, 1999) have

shown a negative magnetic contrast of these features against the sur-

rounding soil. Although it is possible to consider burning events that

could lead to such results during the demolition of buildings, it is more

likely that these bricks were made from clay that has lower magnetic

susceptibility than the soil on the building site.

Induced magnetism

Any material with a magnetic susceptibility will acquire an induced mag-

netization in the Earth’s magnetic field (see Magnetic susceptibility).

Hence, if past human habitation has led to enhanced levels of magnetic

susceptibility in the soils, magnetic measurements can be used for their

detection. The relationship between human activities and the enhance-

ment of magnetic susceptibility was investigated by Le Borgne (1955,

1960), distinguishing thermal and bacterial enhancement. When soil is

heated in the presence of organic material (for example during bush- or

camp-fires), oxygen is excluded and the resulting reducing conditions

lead to a conversion of the soil’shematite(a-Fe

, antiferromagnetic)

to magnetite (Fe

, ferrimagnetic) with a strong increase of magnetic

susceptibility. On cooling in air, some of the magnetite may be re-

oxidized to maghemite (g-Fe

, ferrimagnetic), thereby preserving the

elevated magnetic susceptibility. In contrast, the “fermentation effect”

refers to the reduction of hematite to magnetite in the presence of anerobic

bacteria that grew in decomposing organic material left by human habita-

tion, either in the form of rubbish pits (“middens”)orwoodenbuilding

material. This latter effect requires further research but it is reported

that changes in pH/Eh conditions as well as the bacteria’s use of iron as

electron source are responsible for the increase of magnetic susceptibility

(Linford, 2004). The level of magnetic susceptibility that can be reached

through anthropogenic enhancement also depends on the amount of iron

oxides initially available in the soil for conversion. The level of enhance-

ment can hence be quantified by relating a soil’s current magnetic suscept-

ibility to the maximum achievable value. This ratio is referred to as

“fractional conversion” and is determined by heating a sample to about

700



C to enhance its magnetic susceptibility as far as possible (Crowther

and Barker, 1995; Graham and Scollar, 1976). Whether the initial mag-

netic susceptibility was enhanced by pedogenic or anthropogenic effects

can however not be distinguished with this method. It is also worth

remembering that magnetite and maghemite have the highest magnetic

susceptibility of the iron oxides commonly found in soils, and in the

absence of elemental iron, a sample’s magnetic susceptibility is hence a

measure for the concentration of these two minerals.

More recent investigations have indicated additional avenues for the

enhancement of magnetic susceptibility. One of the most interesting is

a magnetotactic bacterium that thrives in organic material and grows

magnetite crystals within its bodies (Fassbinder et al., 1990) (see also

Biomagnetism). Their accumulation in the decayed remains of wooden

postholes led to measurable magnetic anomalies and is probably

responsible for the detection of palisade walls in magnetometer

surveys (Fassbinder and Irlinger, 1994). Other causes include the

low-temperature thermal dehydration of lepidocrocite to maghemite

(e.g., Özdemir and Banerjee, 1984) and the physical alteration of the

constituent magnetic minerals, especially their grain size (Linford

and Canti, 2001; Weston, 2004). Magnetic susceptibility can also be

enhanced by the creation of iron sulfides in perimarine environments

with stagnant waters (Kattenberg and Aalbersberg, 2004). These may

fill geomorphological features, like creeks, that were used for settle-

ments and can therefore be indirect evidence for potential human

activity.

Archeological prospection

Archeological prospection refers to the noninvasive investigation of

archeological sites and landscapes for the discovery of buried archeo-

logical features. To understand past societies it is of great importance

to analyze the way people lived and interacted and the layout of arche-

ological sites gives vital clues; for example the structure of a Roman

villa’s foundations or the location of an Iron-aged ditched enclosure

within the wider landscape. Such information can often be revealed

without excavation by magnetic surveys. These techniques have there-

fore become a vital part of site investigation strategies. Buried archeo-

logical features with magnetic contrast will produce small anomalies in

measurements on the surface and detailed interpretation of recorded

data can often lead to meaningful archeological interpretations. The

techniques are not normally used to “treasure-hunt” for individual fer-

rous objects but rather for features like foundations, ditches, pits, or

kilns (Sutherland and Schmidt, 2003).

Magnetic susceptibility surveys

Since human habitation can lead to increased magnetic susceptibility

(see above), measurements of this soil property are used for the iden-

tification of areas of activity. Such surveys can either be carried out

in situ (i.e., with nonintrusive field measurements) or by collecting soil

samples for measurements in a laboratory. These two methods are

often distinguished as being volume- and mass-specific, respectively,

although such labeling only vaguely reflects the measured properties.

Most instruments available for the measurement of low field mag-

netic susceptibility internally measure the “total magnetic susceptibil-

ity” (k

with units of m

), which is proportional to the amount of

magnetic material within the sensitive volume of the detector: the more

material there is, the higher will be the reading. For field measure-

ments, the amount of investigated material is usually estimated by

identifying a “volume of sensitivity ” (V ) for the employed sensor

(e.g., a hemisphere with the sensor’s diameter for the Bartington

MS2D field coil). The “volume specific magnetic susceptibility” is

then defined as k ¼ k

/V (dimensionless). In contrast, laboratory mea-

surements normally relate instrument readings to the weight of a sam-

ple (m), which can be determined more accurately. The “mass specific

magnetic susceptibility” is then w ¼ k

/m (with units of m

1

Accordingly, it is possible to calculate one of these quantities from

the other using the material’s bulk density (r): w ¼ k/r.

The main difference between field and laboratory measurements,

however, is the treatment of samples. It is commo n practice (Linford,

1994) to dry and sieve soil samples prior to measuring their magnetic

susceptibility in the laboratory. Drying eliminates the dependency of

mass specific magnetic susceptibility on moisture content, which

affects the bulk density. Sieving removes coarse inclusions (e.g., peb-

bles) that are magnetically insignificant. In this way, laboratory mea-

surements represent the magnetic susceptibility of a sample’s soil

component and can therefore be compared to standard tables. For

field measurements, however, results can be influenced by nonsoil

inclusions and the conversion of volume-specific measurements to

mass-specific values is affected by changes in environmental factors

(e.g., soil moisture content).

The measured magnetic susceptibility depends on the amount of

iron oxides available prior to its alteration by humans (mostly related

to a soil’s parent geology), and also on the extent of conversion due

to the anthropogenic influences. As a consequence, the absolute value

24 ARCHEOLOGY, MAGNETIC METHODS

of magnetic susceptibility can vary widely between different sites and

“enhancement” can only be identified as a contrast between areas of

higher susceptibility compared to background measurements. There

is no predetermined threshold for this contrast and it is therefore justi-

fiable to use the more qualitative measurements of field-based mag-

netic volume susceptibility (see above) for archeological prospection.

The instrument most commonly used for field measurements of

volume specific magnetic susceptibility is the Bartington MS2D field

coil consisting of a 0.2 m diameter “loop,” which derives 95% of its

signal from the top 0.1 m of soil (Schmidt et al., 2005). Since most

archeological features are buried deeper than this sensitivity range,

the method relies on the mixing of soil throughout the profile, mostly

by ploughing.

Magnetic susceptibility surveys, either performed as in situ field

measurements or as laboratory measurements of soil samples, can be

used in three different ways: (i) as primary prospection method to

obtain information about individual buried features, (ii) to complement

magnetometer surveys and help with their interpretation by providing

data on underlying magnetic susceptibility variations, or (iii) for

a quick and coarse “reconnaissance” survey using large sampling

intervals to indicate areas of enhancement instead of outlining indivi-

dual features. Figure A4a shows the magnetic susceptibility survey

(MS2D) of a medieval charcoal burning area. Since a detailed study

of this feature was required, the data were recorded with a spatial reso-

lution of 1 m in both x- and y-directions. Corresponding magnetometer

data (FM36) are displayed in Figure A4b. The magnetic susceptibility

results outline the burnt area more clearly and are on this site well sui-

ted for the delineation of features. On former settlement sites where

ploughing has brought magnetically enhanced material to the surface

and spread it across the area, magnetic susceptibility measurements

at coarse intervals of 5, 10, or 20 m can be used to identify areas of

enhancement that can later be investigated with more detailed sam-

pling, for example with a magnetometer (see below). Even where

ploughing has mixed the soil, magnetic susceptibility measure-

ments can vary considerably over short distances of about 2 m. It is

hence not normally possible to interpolate coarsely sampled data and

settlement sites can only be identified if several adjacent measure-

ments have consistently high levels compared to the surrounding area.

Figure A5 shows data from a survey at Kirkby Overblow, North York-

shire, in search for a lost medieval village. The dots mark the

individual measurements and their size represents the strength of the

magnetic susceptibility, highlighting the singularity of each measure-

ment. Nevertheless, areas of overall enhancement can be identified

and these were later investigated with magnetometer surveys. The gray

shading in this diagram visualizes the data in a contiguous way,

coloring the Voronoi cell around each individual measurement accord-

ing to its magnetic susceptibility. The limitations of such diagrams for

widely separated measurements should be considered carefully.

Magnetometer surveys

In contrast to surveys that measure magnetic susceptibility directly,

magnetometer surveys record the magnetic fields produced by a con-

trast in magnetization, whether it is induced as a result of a magnetic

susceptibility contrast, or remanent, for example from thermoremanent

magnetization. If the shape and magnetic properties of a buried arche-

ological feature were known, the resulting magnetic anomaly could be

calculated (Schmidt, 2001) (see also Magnetic anomalies, modeling).

The inverse process, however, of reconstructing the archeological

feature from its measured anomaly, is usually not possible due to the

nonuniqueness of the magnetic problem and the complex shape

and heterogeneous composition of such features. Some successful

inversions were achieved when archeologically informed assumptions

were made about the expected feature shapes (e.g., the steepness of

ditches) and magnetic soil properties (for example from similar sites)

(Neubauer and Eder-Hinterleitner, 1997; Herwanger et al., 2000).

If surveys are conducted with sufficiently high spatial resolution, the

mapped data often already provide very clear outlines of the buried

features (Figure A6), allowing their archeological interpretation even

without data inversion. However, when interpreting measured data

directly, the typical characteristics of magnetic anomalies have to be

taken into consideration. For example, in the northern hemisphere

Figure A4 Medieval charcoal production site in Eskdale, Cumbria. (a) Magnetic susceptibility survey with Bartington MS2D field coil.

(b) Fluxgate gradiometer survey with Geoscan FM36. Both surveys were conducted over an area of 40 m  40 m with a spatial

resolution of 1 m  1m.

ARCHEOLOGY, MAGNETIC METHODS 25