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between grains) sample. The length of the well packed sample in

the tube should be between 2.5 and 3.5 cm (or a length appropri-

ate to the diameter of the tube being used, as suggested by the

manufacturer of the MSB).

7. Place the packed tube into the tube guide in the MSB, wait for

stable reading before recording this first reading, R

. If the reading

is off-scale, remove the packed tube, set the Range Knob to 10

scale and then rezero the MSB before putting back the packed

sample. Record your reading, but remember to multiply this

reading by 10.

8. Remove the packed tube from the MSB and tap it several

times to further compress the solid in the tube and remove any

trapped air.

9. Place the packed tube back into the tube guide in the MSB, and

record a second reading, R

. If this second reading (R

) is not

the same as the first reading (R

), then repeat further the tap-

ping/measuring sequence until two successive readings are the

same. Even packing gives constant reading hence tap the sample

tube gently till a constant reading is indicated.

10. Take the temperature reading, T, of the area around the MSB;

remove the packed tube from the MSB and measure the sample

length in the tube in centimeters, L.

11. Immediately, measure the mass of the packed tube on a balance

and record its mass in grams, m

t,s

; find the actual mass of the sam-

ple (m) in the packed tube in grams by subtracting the mass of the

empty tube (m

) from this value; i.e.,

m ¼ m

t;s

 m

12. Do another measurement of the same sample by repacking the

same tube and following the procedures above. (Note: data for

the same empty tube, i.e., R

and m

should remain the same.)

13. Clean the tube by tapping out the powder, followed by several

washings with a squirt bottle filled with soapy water. Finally rinse

and dry with acetone.

Calculations and worked examples

1. Calculation of the calibration constant, C

bal

(a) HgCo(SCN)

has a w

¼ 16:44  10

6

cgs at 20



at T



C ¼

4 981

ð283 þ T Þ

 10

6

cgs;

then at T ¼ 21:0



C, w

4 981

ð283 þ 21Þ

 10

6

cgs ¼ 16:38  10

6

cgs:

(b) Refer to Table S3, data No. 1 for HgCo(SCN)

;

bal

ðR  R

ðm  10

;

on rearranging;

bal

ðm  10

LðR  R

Hence for T ¼ 21:0



C, C

bal

becomes;

bal

16:38  10

6

ð0:5082  10

3:25ð2 120 ð31ÞÞ

¼ 1:19076:

for Ni(en)

refer to Table S3, data No. 2;

bal

ðR  R

ðm  10

3:30  1:19076ð750 ð31ÞÞ

ð0:2787  10

;

i.e., w

¼ 11:01  10

6

cgs at T ¼ 21:0



Compare this value of w

for Ni(en)

with the value given in

literature, i.e., w

at 20:0



C ¼ 11:04  10

6

cgs and since

at T



C ¼

2 759

ð230 þ T Þ

 10

6

cgs;

then

at 21:0



C ¼

2 759

ð230 þ 21Þ

 10

6

cgs ¼ 10:99  10

6

cgs:

2. Worked examples: refer to Table S3, data for a-MnS (data No. 4)

bal

ðR  R

ðm  10

3:15  1:19076ð8 840 ð27ÞÞ

ð0:5358  10

;

i.e., w

¼ 62:07  10

6

cgs at T ¼ 23:0



Hence w

for this compound can be calculated using (7),

i.e., w

¼ w

 MW=Z ¼ 62:07  10

6

 87:002 g mol

1

, since

Z ¼ 1 for a-MnS,

therefore, w

¼ 5:40  10

3

emu mol

1

To calculate the effective magnetic moment, m

eff

, we make use of

(14), i.e.,

eff

¼ 2:828ðemu mol

1

KÞ

1=2

½w

ðT  yÞ

1=2

¼ 2:828ðemu mol

1

KÞ

1=2

½5:40  10

3

emu mol

1

ð296 ð490ÞK

1=2

¼ 5:83 BM:

The students will study a series of first row transition metal com-

pounds to determine the oxidation state of the ion in each compound

and the number of unpaired electrons, from the measurement of the

magnetic susceptibility. The compounds will include various transition

metal complexes varying in different number of d-electrons. The stu-

dents will directly measure the mass susceptibility of each compound

using a Magnetic Susceptibility Balance (MSB, Johnson Matthey

model). Given the values of Curie constant (y) for each compound,

the students will be able to find the effective magnetic moment of

the manganese ion from the measured values of the molar susceptibil-

ity of each compound using (14), hence will be able to calculate the

number of unpaired electrons.

Z.S. Teweldemedhin, R.L. Fuller and M. Greenblatt

Table S3 The list of chemicals to be used in this experiment

No. Chemical MW (g mole

1

)

Standard HgCo(SCN)

491.85816

1 MnSO

2H

O 187.03221

2 MnO

86.93685

3 Cu(HCOO)

H

O 171.59676

4Mn

157.8743

5 Ni(en)

351.1206

6 KMnO

158.03395

Fe(CN)

3H

O 422.39248

8 Co(NH

)

(SO

)

6H

O 395.22908

9 Fe(NH

)

(SO

)

6H

O 392.14288

10 K

294.18476

936 SUSCEPTIBILITY, MEASUREMENTS OF SOLIDS

Bibliography

Carlin, R.L., 1986. Magnetochemistry. Berlin: Springer-Verlag.

Drago, R.S., 1977. Physical Methods in Chemistry. Philadelphia:

W. B. Saunders Co.

Goodenough, J.B., 1963. Magnetism and the Chemical Bond. New

York: John Wiley & Sons.

Kittel, C., 1962. Elementary Solid State Physics: A Short Course. New

York: John Wiley & Sons.

Konig, E., and Landolt-Bornstein, 1966. In Hellwege. K.-H., and

Hellwege, A.M., (eds.), Magnetic properties of Coordination and

Organo-Metallic Transition Metal Compounds, Group II, Volume 2.

Berlin: Springer-Verlag.

Magnetic Susceptibility Balance Instructional Manual. Wayne, PA:

Johnson Matthey, 1991.

O’Connor, C.J., 1982. In Stiefel, E. (ed.), Magnetochemistry—

Advances in Theory and Experimentation. New York: John Wiley

and Sons; pp. 203–283.

Shoemaker, D.P., et al., 1996. Customized Experiments in Physical

Chemistry, 6th Edition. Lawrence, KS: Primus/McGraw Hill, p. 109.

Vonsovskii, S.V., 1974. Magnetism, Volume 1. Israel: Ketter Publish-

ing House Jerusalem Ltd (Translated by R. Hardin).

Woolcock, J., and Zafar, A., 1992. Microscale techniques for determi-

nation of magnetic susceptibility. Journal of Chemical Education,

69: A176–A179.

SUSCEPTIBILITY, PARAMETERS, ANISOTROPY

One of the main interests in the determination of the low field

anisotropy of magnetic susceptibility (AMS) is its value as a petrofab-

ric indicator. An important part of the interpretation of AMS measure-

ments in this context involves the orientation of the principal

susceptibilities. Several methods of calculating mean directions of

the principal susceptibilities, and associated regions of confidence

have been devised, but the tensor-averaging technique will provide a

better estimate in most circumstances (Ernst and Pearce, 1989). Also,

different methods have been proposed to estimate regions of confi-

dence around the mean principal susceptibilities. When the individual

measurements are not very different from the calculated mean (in

which case it is said that the uncertainty is not large), the regions of

confidence calculated with different methods are similar to each other

and exert a small influence in the interpretation of results. When the

uncertainty in the mean susceptibility is large, however, the regions

of confidence found with different methods can be very different

(Tauxe, 1998). Nevertheless, instead of entering in a pointless discus-

sion concerning which statistical method is better suited for every spe-

cific case, when a large scatter of directions of principal susceptibilities

is found it is more significant to check whether the individual data

used for the calculation of the average directions of susceptibility

might include systematic variations of principal susceptibilities

with a well-defined physical meaning. The identification of different

statistical populations made in this form not only provides a more

meaningful interpretation of AMS results than an interpretation based

in a very rigorous statistical treatment that does not acknowledge phy-

sical controls on the mineral fabric, but also leads to situations in

which the uncertainty associated with the average of each AMS popu-

lation is relatively small. Consequently, by adequately identifying sys-

tematic fluctuations in the AMS of a rock unit and associating them

with changes in the physical conditions controlling the magnetic fabric

(like for example at opposite sides of a tabular intrusive where shear

direction is expected to be different in orientation), it turns out that

the method used in the calculation of the associated regions of

confidence is relatively unimportant.

On the other hand, sometimes is necessary to concentrate our atten-

tion in the numeric components of the susceptibility tensor. This part

of the analysis may be somewhat problematic not only because of

the large number of AMS parameters that have been proposed over

time (Tables S4 and S5), but also because the parameters used in a par-

ticular analysis generally are selected based more on previous use than

on some objective criteria. Historically, five classes of AMS para-

meters have been defined: Bulk susceptibility, foliation, lineation,

shape, and degree of anisotropy. Among these, the bulk susceptibility

is the only one that has a well-defined physical meaning. Whether it is

calculated as the arithmetic or the geometric average of the three prin-

cipal susceptibilities, the bulk susceptibility is one of the three invar-

iants associated with any second rank tensor, and consequently it is a

measure of susceptibility independent of the coordinate system used

as a reference frame.

In contrast, the physical meaning of all the other AMS parameters is

less clear, and even it may be controversial. For instance, the lineation

parameters were proposed to supposedly yield a measure of the inten-

sity of alignment of elongated particles as detected through AMS

Table S4 Shape parameters

Parameters Name Range Group Source

1. ðk

 k

Þ=ðk

 k

Þ Prolateness 0 to ... (Khan, 1962)

2. ðk

 k

Þ=ðk

 k

Þ Oblateness 0 to ... (Khan, 1962)

3. a sinfðk

 k

Þ=ðk

 k

Þg

1=2

Angle V 0 to 90 (Graham, 1966)

4. ð2k

 k

Þ=ðk

 k

Þ Difference shape factor 1 to 1 (Jelinek, 1981)

5. ðk

 k

Þ=fðk

þ k

Þ=2  k

g q factor 0 to 2 S1 (Granar, 1957)

6. ðk

ðk

 k

ÞÞ=ðk

ðk

 k

ÞÞ 0to... (Janak and Hrouda, 1969)

7. ðk

ðk

 k

ÞÞ=ðk

ðk

 k

ÞÞ Shape indicator 0 to ... (Stacey et al., 1960)

8. ðk

 1Þ=ðk

 1Þ D factor 0 to ... (Hrouda, 1976)

9. ð2n

 n

Þ=ðn

 n

Þ Shape parameter T –1 to 1 (Jelinek, 1981)

10. k

Lineation 1 to ... S2a (Balsey and Buddington, 1960)

11. ðk

 k

Þ=k

Magnetic lineation 0 to 3 (Khan, 1962)

12. ðk

 k

Þ=k

Magnetic foliation 0–1.5 (Khan, 1962)

13. k

Foliation 1 to ... S2b (Stacey et al., 1960)

14. 1 ðk

Þ Foliation 0 to 1 (Porath, 1971)

15. k

0to... (Stacey et al., 1960)

16. k

E factor S2 (Hrouda et al., 1971)

17. k

 2k

þ 1 B parameter –1 to 1 (Cañón-Tapia, 1992)

¼ ln(k

), i ¼ 1,2,3, and k

¼ (k

þ k

)/3.

SUSCEPTIBILITY, PARAMETERS, ANISOTROPY 937

measurements. This interpretation is only correct if the AMS is con-

trolled by elongated objects of similar (and previously known) aspect

ratios and for which the AMS bears a direct relation with the shape

of the object (normal shape-effect), but it becomes misleading under

other conditions. In particular, if the AMS is controlled by minerals

in which the physical dimensions of the grain are inversely related to

the intensity of susceptibility (inverse shape-effect Rochette et al.,

1991), the lineation becomes a measure of the grouping of particles

over one plane rather than reflecting the degree of clustering of the lar-

gest dimensions of the minerals. Alternatively, if the AMS of two

rocks is controlled by minerals with a normal shape-effect but different

average aspect ratios, the magnetic lineation of those two rocks will be

different even when the degree of clustering of the minerals is the

same. In an extreme case, a rock formed by very elongated minerals

that are not perfectly aligned might yield a larger lineation than a sec-

ond rock formed by a perfect alignment of not so elongated minerals,

therefore potentially leading to erroneous petrofabric interpretations.

These problems are not exclusive of the lineation parameter, as

ambiguities are found when the physical meaning of all the other types

of AMS parameters is examined in detail. Until now, there have been

two different approaches to alleviate this situation, both of which have

been unsuccessful in finding a general agreement within the scientific

community. The first attempt in suggesting criteria for the selection of

AMS parameters was made by Ellwood et al. (1988). They pointed out

that some methods of measurement favor the use of parameters that

include the differences of the principal susceptibilities while others call

for the use of their ratios (see also Hrouda and Jelinek, 1990).

Undoubtedly, this distinction has some merit within an instrumental

basis, but it overlooks the fact that most of the available parameters

use both ratios and differences of the principal susceptibilities in their

definitions. Furthermore, this approach has the problem that actual cal-

culation of different parameters often reduces in practice to a simple

arithmetical combination of the principal susceptibilities regardless

of the method used for the measurement of the susceptibility tensor

(Ellwood et al., 1988; Tarling and Hrouda, 1993). The second attempt

to devise useful criteria to guide the selection of AMS parameters was

made by Cañón-Tapia (1994). He used a mathematically oriented point

of view to compare the most important parameters proposed until that

time, and to estimate the influence that such parameters could have in

the final interpretation of results. Based on a theoretical comparison in

which individual parameters could be conveniently expressed as

families of lines, it was shown that many of the parameters proposed

in the literature are equivalent to each other and that the only differ-

ence between two apparently independent parameters can be reduced

to a difference in a numeric range. In particular, using this approach

it was shown that the foliation and lineation parameters are special

cases of the shape parameters, and therefore little is gained with their

use as separate categories. In contrast with the experimentally derived

criteria, the geometrically based comparison is applicable to any para-

meter defined as a combination of two or three of the principal suscept-

ibilities. Consequently, this set of criteria can be used in a general context

to guide the selection of AMS parameters in a more ordered form than

possible with other criteria available until now. Unfortunately, costume

derived from continual use of a particular type of AMS parameters seems

to be the dominant criteria in some cases. Consequently, foliation and

lineation parameters have continued to be used as independent types

and, in some instances, these two parameters are reported together with

a third shape parameter in the same work.

Although lacking an objective criteria behind their usage, in practice

it is found that the two most commonly used parameters are those

called P

and T (Tables S4 and S5, respectively). Another pair of para-

meters that has been used with increasing frequency, especially

because of its use in studies of lava flows, is formed by the A and B

parameters. These two pairs of AMS parameters yield different

numeric ranges in the analysis of AMS measurements (hence requiring

caution when comparing different works). Qualitatively, however,

these two pairs of AMS parameters yield equivalent results and there-

fore the main aspects in the interpretation will remain unaltered

irrespective of which pair of parameters was used (Cañón-Tapia, 1994).

In summary, despite the various methods available for the calcula-

tion of average orientations of principal susceptibilities and associated

regions of confidence, when attention is given to the physical factors

controlling the acquisition of the AMS of a rock, the differences

between alternative methods of calculation becomes negligible for

the interpretation of results. In contrast, little agreement has been

reached concerning the selection of AMS parameters. More details

concerning the physical value of individual AMS parameters will be

required before reaching a clearly satisfactory set of criteria that can

be used as a general guide for their selection.

Edgardo Cañón-Tapia

Bibliography

Balsey, J.R., and Buddington, A.F., 1960. Magnetic susceptibility ani-

sotropy and fabric of some Adirondack granites and orthogneisses.

American Journal of Science, 258A:6–20.

Cañón-Tapia, E., 1992. Applications to volcanology of paleomagnetic

and rock magnetic techniques. MSc thesis, University of Hawaii,

Honolulu, p. 146.

Cañón-Tapia, E., 1994. AMS parameters: guidelines for their rational

selection. Pure and Applied Geophysics, 142: 365–382.

Ellwood, B.B., 1975. Analysis of emplacement mode in basalt from

deep-sea drilling project holes 319A and 321 using anisotropy of

magnetic susceptibility. Journal of Geophysical Research, 80:

4805–4808.

Table S5 Parameters quantifying the degree of anisotropy

Parameters Name Range Group Source

1. k

Anisotropy degree P 1–... (Nagata, 1961)

2. 100ðk

 k

Þ=k

Percentage anisotropy 0–100 (Graham, 1966)

3. ðk

 k

Þ=k

Total anisotropy H 0–3 A1 (Owens, 1974)

4. ðk

 k

Þ=2k

Percentage anisotropy 0–1.5 (Khan, 1962)

5. ðk

 k

Þ=k

Absolute anisotropy 0–... (Rees, 1966)

6. (k

þ k

)/2k

Foliation 1–... A2 (Balsey and Buddington, 1960)

7. ððk

 k

Þ=2Þk

Magnetic excess 0–1(k

) (Granar, 1957)

8. exp{(2[(a

)

þ (a

)

þ (a

)

]}

1/2

Corrected anisotropy degree (P

)1–... A3 (Jelinek, 1981)

9. k

)

1/2

Emplacement factor 1–... (Ellwood, 1975)

10. 2k

=ðk

þ k

Þ Lineation degree 1–... A4 (Hrouda et al., 1971)

11. 100(1k

/2k

k

/2k

) A parameter 0–100 (Cañón-Tapia, 1992)

¼ ln(k

), i ¼ 1,2,3, and k

¼ (k

þ k

)/3, k

¼ (k

)

1/3

938 SUSCEPTIBILITY, PARAMETERS, ANISOTROPY

Ellwood, B.B., Hrouda, F., and Wagner, J.J., 1988. Symposia on mag-

netic fabrics: introductory comments. Physics of the Earth and Pla-

netary Interiors, 51: 249–252.

Ernst, R.E., and Pearce, G.W., 1989. Averaging of anisotropy of mag-

netic susceptibility data. Geological Survey of Canada, Paper

89-9: 297–305.

Graham, J.W., 1966. Significance of magnetic anisotropy in Appala-

chian sedimentary rocks. In Steinhart, J.S., and Smith, T.J. (eds.),

The Earth Beneath the Continents. Washington, DC: American

Geophysical Union, pp. 627–648.

Granar, L., 1957. Magnetic measurements on Swedish varved sedi-

ments. Arkiv Geofysik, 3:1–40.

Hrouda, F., 1976. The origin of cleavage in the light of magnetic ani-

sotropy investigations. Physics of the Earth and Planetary Inter-

iors, 13: 132–142.

Hrouda, F., and Jelinek, V., 1990. Resolution of ferrimagnetic and

paramagnetic anisotropies in rocks, using combined low-field and

high-field measurements. Geophysical Journal International, 103:

75–84.

Hrouda, F., Janak, F., Rejl, L., and Weiss, J., 1971. The use of mag-

netic susceptibility anisotropy for estimating the ferromagnetic

mineral fabrics of metamorphic rocks. Geologische Rundschau,

60: 1124–1142.

Janak, F., F., and Hrouda, 1969. Research of magnetic susceptibility

and its anisotropy, Geofyzika, Brno.

Jelinek, V., 1981. Characterization of the magnetic fabric of rocks.

Tectonophysics, 79: T63–T67.

Khan, M.A., 1962. The anisotropy of magnetic susceptibility of some

igneous and metamorphic rocks. Journal of Geophysical Research,

67: 2873–2885.

Nagata, T., 1961. Rock magnetism. Tokyo: Maruzen.

Owens, W.H., 1974. Mathematical model studies on factors affecting

the magnetic anisotropy of deformed rocks. Tectonophysics, 24:

115–131.

Porath, H., 1971. Anisotropie der magnetischen Suszeptibilitat und

Sättigungsmagnetisierung als Hilfsmittel der Gefügekunde. Geolo-

gische Rundschau, 60: 1088–1062.

Rees, A.I., 1966. The effect of depositional slopes on the anisotropy of

magnetic susceptibility of laboratory deposited sands. Journal of

Geology,

74: 856–867.

Rochette, P., Jackson, M., and Aubourg, C., 1991. Rock magnetism

and the interpretation of anisotropy of magnetic susceptibility.

Reviews of Geophysics, 30: 209–226.

Stacey, F.D., Joplin, G., and Lindsay, J., 1960. Magnetic anisotropy

and fabric of some foliated rocks from SE Australia. Pure and

Applied Geophysics, 47:30–40.

Tarling, D., and Hrouda, F., 1993. The Magnetic Anisotropy of Rocks.

London: Chapman & Hall, p. 217.

Tauxe, L., 1998. Paleomagnetic Principles and Practice. Dordrecht:

Kluwer, p. 299.

SUSCEPTIBILITY, PARAMETERS, ANISOTROPY 939

TAYLOR’S CONDITION

Taylor’s condition

Taylor’s condition, first derived by J.B. Taylor in 1963, is a statement

about the electromagnetic torque within the Earth’s core, namely that

it must vanish when integrated over cylindrical shells parallel to the

axis of rotation. It is one of the most important results in geodynamo

theory.

Basic equatio ns

The two equations we will need are the induction equation:

¼rðU  BÞþr

B; (Eq. 1)

governing the evolution of the magnetic field B in the core, and the

Navier-Stokes equation (in its simplest, Boussinesq form):

]

þ U r



U þ 2

 U ¼rp þ E r

U þðrBÞ

 B þ F

; (Eq. 2)

governing the fluid flow U. In general there would also be an equation

for the thermal and/or compositional buoyancy that ultimately drives

the whole system, but for our discussion here we can take the buoy-

ancy force F

to be prescribed. The only feature that will matter is that

it is purely radial.

In these equations, length has been scaled by r

, time by r

=, U by

=r

, and B by

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Orm



, where r

is the outer core radius,  the mag-

netic diffusivity, O the Earth’s rotation rate, r the density, and m

the

magnetic permeability. The two nondimensional parameters then

appearing in (2) are the Rossby and Ekman numbers Ro ¼ =Or

and E ¼ n=Or

, where n is the viscosity. Inserting the numbers, we

find that Ro ¼ Oð10

9

Þ and E ¼ Oð10

15

Þ, indicating the extreme

dominance of rotation over inertia and viscosity.

One very basic question then is where does the Lorentz force fit into

this balance? Is it dominant, like the Coriolis force, or very small, like

inertia and viscosity? Formally it is certainly comparable to the

Coriolis force in (2), but that merely reflects our scalings for B and U.

So the question is, are these the right scalings, that is, do B and U come

out to be O(1) when measured in these

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Orm



and =r

units? As we

will see, Taylor’s condition touches upon precisely this point, and

suggests that B (and possibly U as well, see dynamo, model-Z) may

not necessarily come out to be O(1), but may scale as some power of

E instead.

Taylor’s analysis

If Ro and E are so small, we might try to simplify the Navier-

Stokes equation by dropping the inertial and viscous terms entirely,

yielding

 U ¼rp þðrB ÞB þ F

: (Eq. 3)

Taking the curl (and using also rU ¼ 0) yields

2

]

U ¼r ðrBÞB þ F

½; (Eq. 4)

where ðz; s; fÞ are standard cylindrical coordinates. One might sup-

pose then that the solution is simply U ¼ U

þ U

, with the so-called

magnetic and thermal winds given by

¼

r  ½ðr  BÞBdz; U

¼

rF

dz:

(Eq. 5)

Instead, it turns out that in general (3) has no solution at all. To see

why, consider its f-component:

2U 

¼

þ ½ðr  BÞB

(Eq. 6)

(it is at this stage that we use the fact that F

has no f-component).

Next, integrate (6) over the so-called geostrophic surfaces C(s) shown

in Figure T1, consisting of concentric cylindrical shells parallel to the

z-axis. The pressure-gradient term then vanishes, since

df ¼ 0: (Eq. 7)

The Coriolis term also vanishes, since

U 

dS is the net flux across

C(s), which must be zero if the fluid is incompressible, as in the

Boussinesq approximation here. We are thus left with Taylor’s celebrated

result, stating that B must satisfy

CðsÞ

½ðr  BÞB

dS ¼ 0: (Eq. 8)

That is, when considered pointwise, all four forces contribute to (3),

but when the corresponding torques are integrated over these particular

surfaces C(s), the buoyancy, pressure-gradient and Coriolis terms all

drop out, so the only remaining term must then also be zero. We see

therefore that unless B satisfies (8), and on each such surface C(s),

(3) simply has no solution at all.

Now suppose that B does satisfy (8), so a solution exists. There is

then the additional complication that this solution is not unique, since

one can add to it an arbitrary geostrophic flow U

ðsÞ

, which after all

satisfies ]U=]z ¼ 0; rU ¼ 0, and the no normal flow boundary con-

ditions. Equation (3) thus has the awkward feature that it has either no

solution at all if (8) is not satisfied, or else an infinite number of solu-

tions if it is.

This arbitrariness in the geostrophic flow can fortunately be removed,

by requiring that B should not only satisfy (8) at some particular

instant, but that it should evolve in time in such a way that (8) continues

to be satisfied. That is, we want not only (8), but also d=dt(8), or

CðsÞ

rBðÞ

þr



 B





dS ¼ 0: (Eq. 9)

To see how this determines U

, we recall first (1), giving ]B=]t in

terms of B and U. We remember further that except for U

, the rest

of U is known, from (5). If we therefore write

U ¼ U

ðsÞ

þ U

; (Eq. 10)

then inserting (10) into (1), and then (1) into (9), we end up with an

(horribly complicated) equation in which the only unknown quantity

is U

. In this way (3) has a unique solution after all, and in the process

(8) continues to be satisfied into the future. Taylor therefore envisioned

the geodynamo evolving through such a sequence of states.

Alternatives to Taylor’s analysis

Unfortunately, no one has ever succeeded in evolving a numerical geo-

dynamo model according to Taylor’s prescription. Instead, most work-

ers have proceeded more cautiously, not setting Ro and E identically to

zero. If one restores viscosity, at least in the Ekman boundary layers,

then an asymptotic analysis (Tough and Roberts 1968) of these layers

yields:

CðsÞ

½ðr  BÞB

dS ¼ E

1=2

4ps

ð1  s

1=4

ðsÞ; (Eq. 11)

indicating that the integrated electromagnetic torque does not have to

vanish identically, but can instead be balanced by this small but non-

zero viscous torque. (One might suppose that U

and U

would have

associated viscous torques as well. However, the constants of integra-

tion in (5) can be chosen such that their contributions vanish.)

Equation (11) can then be solved for U

in terms of B. According to

this prescription therefore, the field evolves according to (1), and at each

instant in time the flow is still given by (10), but with U

now given by

(11) rather than Taylor’s more complicated formula. The advantage of

this approach is that it is not only easier to implement numerically

(although still nontrivial), it is also more general than Taylor’s approach.

In particular, by postulating an expansion of the form

B ¼ B

þ E

1=2

þ E B

þ; (Eq. 12)

and assuming that B

satisfies (8), we can recover Taylor’s solution

(and in the process also obtain the higher-order viscous corrections

to it). This is not the only possibility though. We can also postulate

an expansion of the form

B ¼ E

1=4

þ E

3=4

þ E

5=4

þ; (Eq. 13)

where now B

does not satisfy (8). In doing so, we obtain the so-called

Ekman states, which do not exist at all in Taylor’s analysis, because

there E  0. Malkus and Proctor (1975) then conjectured that in gen-

eral the field would start out in the Ekman regime, but would undergo

a transition to the Taylor regime for sufficiently strong forcing. That

such a transition can indeed occur was demonstrated numerically by

Hollerbach and Ierley (1991).

There are also additional possibilities beyond these two states; for

example, Hollerbach (1997) showed that an intermediate state can per-

haps exist in which the leading order field scales as E

1/8

. If one allows

the existence of boundary layers other than the Ekman layers impli-

citly contained in (11), one can also obtain the so-called model-Z states

(see dynamo, model-Z).

The geophysical significance of this plethora of possible states is

that it is believed that the geodynamo is not necessarily in a single

state throughout its entire evolution. Most of the time it is probably

in a Taylor state (or possibly a model-Z state), since those are the only

two states in which B ¼ Oð1Þ, as we would like it to be. During rever-

sals or excursions though, it could conceivably be in one of the weaker

states. Indeed, what triggers these events may well be a breakdown of

the previously existing Taylor state.

Finally, we note that fully three-dimensional numerical simulations

are unfortunately not yet in a parameter regime where one could test

these various ideas; numerically accessible Ekman numbers are so

large that one cannot clearly distinguish between these different states.

It is only in mean-field models such as that of Hollerbach and Ierley

(1991) that one can reduce E sufficiently to obtain clear distinctions

between different states. Because Taylor’s condition is inherently axi-

symmetric though, these results should carry over to the fully three-

dimensional models, if only we could reduce E further there as well.

Rainer Hollerbach

Bibliography

Hollerbach, R., 1997. The dynamical balance in semi-Taylor states.

Geophysical and Astrophysical Fluid Dynamics, 84:85–98.

Figure T1 The geostrophic surfaces C (s) over which (8) is to be

integrated. Note also how these surfaces change abruptly across

the so-called tangent cylinder, the particular surface (indicated by

the dotted lines) just touching the inner core.

TAYLOR’S CONDITION 941

Hollerbach, R., and Ierley, G.R., 1991. A modal a

-dynamo in the

limit of asymptotically small viscosity. Geophysical and Astrophy-

sical Fluid Dynamics, 56: 133–158.

Malkus, W.V.R., and Proctor, M.R.E., 1975. The macrodynamics of

a-effect dynamos in rotating fluids. Journal of Fluid Mechanics,

67: 417–443.

Taylor, J.B., 1963. The magnetohydrodynamics of a rotating fluid and

the Earth’s dynamo problem. Proceedings of the Royal Society of

London, Series A, 274: 274–283.

Tough, J.G., and Roberts, P.H., 1968. Nearly symmetric hydromag-

netic dynamos. Physics of the Earth and Planetary Interiors, 1:

288–296.

Cross-references

Anelastic and Boussinesq Approximations

Core, Boundary Layers

Dynamo, Model-Z

Geodynamo, Numerical Simulations

Inner Core Tangent Cylinder

Thermal Wind

THELLIER, E

MILE (1904–1987)

Introduction

Émile Thellier (Figure T2), a pioneer in rock magnetism and archeomag-

netism, was born February 11, 1904 at Mont-en-Ternois (Pas-de-Calais),

France and died May 11, 1987 in Paris. Following studies at the École

Normale Supérieure de Saint-Cloud (1924–1926), he taught at the

École Primaire Supérieure de Bourges (1927–1930). In spite of a heavy

teaching load, 21h per week, he successfully completed the Licence ès-

sciences physiques of the Faculté des Sciences de Paris, where he would

spend the remainder of his career. His outstanding performance led his

professors to urge him to embark on research. At the Sorbonne in the

laboratory of Prof. Charles Maurain, he completed a Diplôme d’Études

Supérieures (1931), studying the magnetism of baked clay, and the

Agrégation de Sciences physiques (1932).

It had been known since at least the time of Brunhes (1906) that the

thermoremanent magnetization (TRM) acquired by baked clays and

volcanic rocks paralleled the direction of the geomagnetic field acting

when they cooled and survived later reversals of the field. Maurain

proposed to Thellier the idea of using the TRM of ancient pottery to

deduce not only the direction but also the strength of the Earth’s field

at the time of firing. This program Thellier later carried out brilliantly

during a long career, but his first desire was to reproduce TRM in the

laboratory and understand its properties. His success in this undertak-

ing makes him the true “discoverer of magnetic memory,” in the words

of his great confrere, Louis Néel. Working with clays from the famed

Sèvres pottery factories, and later with volcanic rocks, Thellier min-

utely and exhaustively determined how TRM forms, producing a

remarkable Docteur ès-Sciences thesis (Thellier, 1938).

This landmark paper, which has lost none of its immediacy today,

launched his academic career at the Institut de Physique du Globe de

Paris, where he became successively Physicien-adjoint (1943), Maître

de Conférences (1945), Professeur (1948) and Director (1954–1966).

From 1967, he moved his research to the Observatoire du Parc

Saint-Maur, in the suburbs of Paris, where he established a CNRS Labor-

atoire de Géomagnétisme, which he molded into one of the world’s

leading rock magnetic and paleomagnetic research centers. Thellier

received many honors in his lifetime. He was named an Officer of the

Legion of Honour (1957), elected to the Académie des Sciences (1967),

and awarded four prizes by the academy and one by the Université de

Paris. His first love, to the end of his life, remained his research and the

laboratory he had created.

Thellier’s laws of thermoremanent magnetization

Thellier worked exclusively with samples containing very fine parti-

cles of minerals like magnetite (Fe

) and hematite (a-Fe

) which

we now know contain only one magnetic domain (single-domain par-

ticles) or at most a very few domains. For such particles he established

a series of universal properties (Thellier, 1938, 1941; Thellier and

Thellier, 1941). A master experimentalist, he perfected astatic, transla-

tion, and rotating-sample magnetometers of unprecedented sensitivity.

Without such instruments, it would have been impossible to satisfy

himself or others of the universality of laws such as the additivity of

partial TRMs because the clays he used in his earliest work and the

hematites used by his student Roquet (Roquet and Thellier, 1946;

Roquet, 1954) were weakly magnetic compared to the volcanic rocks

favored by others (Koenigsberger, 1938; Nagata, 1943).

The Thellier “laws” are most clearly stated in Thellier (1946), a

masterpiece of mature reflection about results obtained in the preced-

ing one and a half decades and their implications for theories of mag-

netism. [For alternative statements, see Roquet (1954, p. 21–23) and

Aubouin and Coulomb (1987).] Quoting Thellier (translated from the

French original), we have the following general properties.

1. TRM (intensity) is proportional to field (strength) for weak fields.

2. All partial TRMs are parallel to the field H in which they were

acquired.

3. A partial TRM acquired by cooling in H through the tempera-

ture interval (T

, T

) and in zero field through all other temperature

intervals is unaffected by reheating in zero field to a tempera-

ture T

and completely disappears after reheating to T

4. A partial TRM (T

, T

, H ) is independent of other partial TRMs

acquired in temperature intervals outside (T

, T

), which may be

due to fields differing from H in strength and direction.

5. All these partial TRMs add geometrically but each of them retains

true autonomy and an exact memory of the temperatures and field

in which they were acquired.

Figure T2 Thellier, E

mile (1904–1987).

942 THELLIER, E

MILE (1904–1987)

The first two properties were known or assumed before Thellier’s time

(e.g., Folgerhaiter, 1899). The latter three are entirely novel and are

usually referred to as Thellier’s laws. Thellier’s statements recognize

the vectorial nature of partial TRM additivity and independence, and

form the basis for using thermal demagnetization to separate natural

remanent magnetization (NRM) vectors of different ages in paleomag-

netism. These statements go well beyond the usual scalar laws given

in textbooks (e.g., Dunlop and Özdemir, 1997). Collectively, the five

laws form a firm fundamental justification for determining the direc-

tion and strength of the paleomagnetic field from the NRM of rocks

and baked materials, provided the NRM fraction utilized was produced

originally as one or more partial TRMs carried by single-domain or

nearly single-domain size grains. Number 3 is the basis for magnetic

paleothermometry (e.g., Pullaiah et al., 1975).

Thellier and the theory of thermoremanent

magnetization

Louis Néel was intrigued by the apparent generality of the Thellier

laws, clearly related to the fine size of the magnetic carriers and not

to their chemistry. In 1949, he published his celebrated theory of ther-

mal relaxation of single-domain grains, which very elegantly and com-

pletely explained Thellier’s observations. Partial TRM was seen to

be acquired sharply at a blocking temperature T

during cooling, as

a grain passed from a superparamagnetic to a thermally blocked state,

and demagnetized (“unblocked”) at the identical temperature T

during

reheating. From this single property flow the three Thellier laws of

reciprocity (blocking and unblocking as reciprocal processes), inde-

pendence and additivity. This powerful and far-reaching theoretical

picture remains the foundation of magnetic theory today, in both geo-

physics and magnetic recording.

What is not generally recognized is that Thellier himself anticipated

Néel’s ideas, albeit in qualitative rather than quantitative form, three

years earlier. Again quoting from Thellier (1946): “These facts, which

do not as yet seem to have attracted the attention of theoreticians, seem

to me to demand a mechanism of immobilization of elementary mag-

netic moments below a temperature y, this temperature having a strong

variation from point to point within a body. One can imagine the body

to be constituted of elementary domains, perhaps very tiny, each with

spontaneous magnetization. Above y, their moments will be free (rota-

tion, wall displacement or reversal) and the magnetic state of the body

will establish itself by the combined action of the field and of thermal

agitation (a sort of paramagnetism). Below y, the elementary moments

will be bound to equilibrium positions from which they undergo only

reversible displacements in weak fields. The temperature y will vary at

each point in the body, perhaps with the dimensions and the shape of

the crystalline grains, and will be broadly distributed between the Curie

point and room temperature. One can thus explain thermoremanence

by the progressive fixing, in the course of cooling, of moments, which

find themselves held fast when they pass through their individual

temperature y. One can also account for the shape of thermomag-

netic curves, in particular the strong increase in susceptibility below

the Curie point.”

In this remarkable paragraph are all the essential ingredients of

Néel’s picture: blocking temperature y; unblocked or superparamag-

netic state above y in which thermal agitation establishes an equili-

brium; freezing in of the equilibrium in cooling with only reversible

changes possible below y; and distribution of y values due to variable

sizes and shapes of grains, explaining the spectrum of partial TRMs

comprising the total TRM of the body. Thellier even anticipates the

explanation of the Hopkinson peak in susceptibility at high tempera-

ture. Although Thellier’s ideas were not crystallized into a quantitative

theory, his intuition was correct in every essential.

Néel did not directly acknowledge these early theoretical ideas but

his appreciation of Thellier’s enormous experimental contributions is

clear in his commentary on a paper summarizing all of Thellier’s

results (Thellier, 1951, p. 217; translated from French): “Among the

phenomena demonstrated in the magnetization of baked clays and

lavas, we should distinguish carefully (total) TRM on the one hand

and the independent magnetizations acquired in nonoverlapping tem-

perature intervals (i.e., partial TRMs) on the other... The second is a

much more remarkable phenomenon and it is to Thellier that we

owe our experimental knowledge of it.”

Anhysteretic magnetization and AF demagnetization

Thellier’s earliest scientific achievements were made in collaboration

with three remarkable women, whose contributions tend to be over-

looked. First was his wife Odette Thellier, who participated in experi-

ments showing that thermal demagnetization of TRM to a temperature

resulted in precisely the same intensity as suppressing the field at T

during initial cooling from the Curie point T

(i.e., pTRM (T

,H )

(Thellier and Thellier, 1941). From these experiments evolved the

pTRM reciprocity law. Odette Thellier was also her husband’s full

partner in archeomagnetic research on both paleodirections and

paleointensities throughout the 1940s and 1950s, culminating in their

famous paper, Thellier and Thellier (1959) (see following section).

Thellier’s second collaborator, Juliette Roquet, extended investiga-

tions to synthetic minerals of both fine and coarse grain sizes and nat-

ural materials containing different size fractions. Her thesis (Roquet,

1954) is the first systematic rock magnetic study of grain-size trends

in isothermal and thermal properties of partial TRM and other rema-

nences. It set a standard that has yet to be surpassed, thanks to her ded-

ication and the exacting standards set by her mentor.

Third was Francine Rimbert, who constructed one of the first AF

demagnetizers and used it to demonstrate that anhysteretic remanent

magnetization (ARM) is almost as strong and resistant to alternating

fields as TRM (Thellier and Rimbert, 1954). They pointed out the need

to scrupulously eliminate any extraneous fields during AF demagneti-

zation, including the ambient geomagnetic field, to avoid contaminat-

ing NRM by unwanted ARMs. This was the pioneering study in AF

demagnetization, which was then developing as a standard paleomag-

netic cleaning method.

Rimbert’s thesis (Rimbert, 1959), another monumental piece of

work testifying to her ability and her supervisor’s stringent expecta-

tions, is a complete experimental and theoretical study of ARM, partial

ARMs, and AF demagnetization. It is the ultimate justification of AF

bias methods for encoding magnetic information, for example in ana-

log audio and video recording, and of AF cleaning of successive gen-

erations of primary and secondary NRMs in rocks. Laws analogous to

those of Thellier for partial TRM have since been verified for fine-

grained magnetic oxides (Dunlop and West, 1969) and ferromagnetic

thin films (Papusoi and Apostol, 1979), ARM playing the role of

TRM and AF substituting for temperature.

Archeomagnetism and the Thellier-Thellier

paleointensity method

Archeomagnetism is the study of the direction and intensity of the geo-

magnetic field during historical times. Directional work is on the face

of it more straightforward than determining paleointensity because loss

of part of the primary NRM does not affect the direction of the remain-

der. Loss of NRM intensity, on the other hand, translates directly into a

diminished estimate of paleointensity; a problem the Thellier-Thellier

method was designed to overcome.

In spite of the comparative ease of directional measurements and the

early beginning made by Émile and Odette Thellier, they published

during the 1940s and 1950s only sporadic determinations of inclina-

tion I (for pots and bricks for which only paleohorizontal at the time

of firing was known) or both I and declination D (for fixed objects

such as the kilns in which firing took place). This was not for want

of experimental work but because of the Thelliers desire to have a

complete and trustworthy picture of the variations of D and I through

time before publishing a master curve that would certainly be widely

THELLIER, E

MILE (1904–1987) 943

used by archeologists as a dating tool. Thus it was only after investiga-

tions were complete for about 100 sites in France, Turkey, Cambodia,

and North Africa that Émile Thellier revealed in a presentation at the

1971 IUGG assembly in Moscow the fruits of all their labors, curves

of secular variation of the field throughout the past 2000 years.

The Thelliers moved more quickly to establish the historical record

of field intensity. For paleodirectional work, Émile Thellier had per-

fected a sensitive spinner magnetometer on a grand scale, still in use

today, capable of measuring the NRMs of intact pots with dimensions

as large as 50cm without the need to subsample these priceless arche-

ological treasures. Paleointensity work required a methodological

invention of equal ingenuity, the Thellier-Thellier protocol, still the

standard method used today.

The Thellier-Thellier method compares NRM, produced thermally

in an unknown ancient field H

, with a TRM produced in a known

laboratory field H

. This basic idea had been put forward earlier by

Folgerhaiter (1899) and Koenigsberger (1938) but no trustworthy

results emerged. Folgerhaiter, quoted by Thellier (1938, 287), says

(translated from French) “One could also arrive at some conclusion

about the intensity of the terrestrial field by reheating ancient vases

and comparing the ancient and presently acquired intensities of mag-

netization; but measurements made on vases heated and reheated in

repeated experiments have shown me that this method leads to too

uncertain results.” The reason for the uncertainty is alteration of the

chemistry and physical state of the magnetic minerals resulting from

heating, demonstrated very clearly by Koenigsberger’s progressive

heating experiments on igneous rocks.

The novelty of the Thelliers’ procedure is in the interweaving of

pairs of heating-cooling steps to successively higher temperatures,

instead of a single heating-cooling to T

. In their original version

(Thellier and Thellier, 1959), both heatings to a particular temperature

were carried out in the presence of a laboratory field (the ambient

Earth’s field in their experiments) but the sample was rotated 180



between heatings. In the currently most used version (Coe, 1967),

the first heating-cooling is in zero field and the second in H

. In the

Coe version, the first heating serves to demagnetize that part of the

NRM with T

 T

, while the second heating replaces this loss with

a partial TRM (T

). The NRM and partial TRM are not gener-

ally in the same direction, so that they must be obtained by vector sub-

traction of the results of the two heatings. In the original version, equal

partial TRMs in opposite directions are acquired in the two heatings,

but the vector subtractions are still straightforward. Although the mod-

ified version gives a neater segregation between NRM loss and partial

TRM gain, the original version has some bonuses: the two heatings

have perfect symmetry and no null field is needed.

The Thellier-Thellier method is firmly rooted in Thellier’s three

laws of partial TRM. This protocol has three tremendous advantages,

not matched by other techniques:

1. There is a built-in test of the TRM origin of the NRM, namely con-

stancy of the ratio of NRM lost/partial TRM gained over successive

heating steps.

2. Portions of the NRM that are unreliable can be recognized and dis-

carded. A common contaminant at low T

is viscous remanent mag-

netization (VRM) produced by the present Earth’s field. Alteration

of mineral microstructure or chemistry tends to occur at high T

3. Linear least-squares fitting to the set of acceptable NRM and partial

TRM data can be used to obtain the mean paleointensity ratio

and its associated error. This procedure, the Arai plot, was

introduced later, by Nagata et al. (1963) but it is a natural conse-

quence of the linear replications in the Thellier-Thellier method.

The name of Koenigsberger has been associated by some with this

method and given precedence over the Thelliers themselves (so-called

KTT method). This misrepresents the facts. In four of Koenigsberger’s

papers published in German journals between 1930 and 1936 and

in his summary work in English (Koenigsberger, 1938), there is no

indication that he was aware of partial TRMs or their properties. The

only paper by Thellier he cites is on determining directions. He did

carry out stepwise heatings with H

parallel or antiparallel to NRM

but these were separate, not interwoven, experiments, and there was

no attempt to use the results to estimate field strength. Most of his

samples altered so much in the first set of heatings that there was no

symmetry between þH

and –H

curves.

Koenigsberger espoused the idea that NRM spontaneously decayed

with time (what we would nowadays call viscous decay) so that the ratio

¼ NRM/kH would be systematically lower than Q

¼ TRM/k

for rocks of increasing age. Here k, k

are susceptibilities measured

before and after TRM acquisition and H is the local present Earth’s field.

He viewed this as an age determination, not a paleointensity, method.

Thellier (1938, p. 293) correctly ascribed the differences between Q

and Q

to increased susceptibility resulting from alteration during heat-

ing and not to spontaneous decay of NRM. Thellier went on to suggest

that the ratio Q

¼ (NRM/TRM) (k

/k) might serve as a rough estimate

of the paleointensity ratio H

. This is a slight improvement on

Folgerhaiter’s prescription, H

¼ NRM/TRM, in that it takes some

account of alteration of a sample through the ratio k

/k, but it is far from

being an earlier incarnation of the powerful and sophisticated Thelliers’

method.

Viewing the considerable alteration evidenced by the differences

between k

and k in Koenigsberger’s igneous samples, Thellier soon

came to the reasonable conclusion that the suggested correction proce-

dure was unjustified, particularly since the quantitative relation

between remanence and susceptibility depends strongly on mineralogy

and grain size. He says (Thellier, 1938, p. 293) “The susceptibility has

changed markedly as a result of heating for many of these rocks; they

should be rejected for the purpose of studying the intensity of the ter-

restrial field.” Since then, many intricate and ingenious schemes have

been proposed for “undoing” the effects of alteration during heating

but none in the final analysis gives results that the most paleomagne-

tists would trust. There is really no substitute for the Thellier-Thellier

method (under which we include microwave heating methods that heat

the magnetic minerals but not the rock matrix), nor for the uncompro-

mising standards set by Émile Thellier.

David J. Dunlop

Bibliography

Aubouin, J., and Coulomb, J., 1987. La vie et l’oeuvre d’Émile

Thellier. La Vie des Sciences. Comptes rendus de l’Académie des

sciences (Paris), Série génerale, 4(6): 607–610.

Brunhes, B., 1906. Recherches sur la direction d’aimantation des

roches volcaniques. Journal of Physique, Série 4, 5: 705–724.

Coe, R.S., 1967. Paleointensities of the Earth’s magnetic field deter-

mined from Tertiary and Quaternary rocks. Journal of Geophysical

Research, 72: 3247–3262.

Dunlop, D.J., and Özdemir, Ö., 1997. Rock Magnetism: Fundamentals

and Frontiers. Cambridge and New York: Cambridge University

Press.

Dunlop, D.J., and West, G.F., 1969. An experimental evaluation of

single-domain theories. Reviews of Geophysics, 7: 709–757.

Folgerhaiter, G., 1899. Sur les variations séculaires de l’inclinai-

son magnétique dans l’antiquité. Journal de Physique, Série 3, 8:

5–16.

Koenigsberger, J.G., 1938. Natural residual magnetism of eruptive

rocks. Terrestrial Magnetism and Atmospheric Electricity, 43:

119–130, 299–320.

Nagata, T., 1943. The natural remanent magnetism of volcanic rocks

and its relation to geomagnetic phenomena. Bulletin of Earthquake

Research Institute, University of Tokyo, 21:1–196.

Nagata, T., Arai, Y., and Momose, K., 1963. Secular variation of the

geomagnetic total force during the last 5000 years. Journal of Geo-

physical Research, 68: 5277–5281.

944 THELLIER, E

MILE (1904–1987)

Néel, L., 1949. Théorie du traînage magnétique des ferromagnétiques

en grains fins avec applications aux terres cuites. Annales de Géo-

physique, 5:99–136.

Papusoi, C., and Apostol, P., 1979. Propriétés d’indépendance des

rémanences magnétiques hystérétiques et anhystérétiques dans les

couches électrolytiques de Co-Ni-P. Physica Status Solidi (a), 54:

477–486.

Pullaiah, G., Irving, E., Buchan, K.L., and Dunlop, D.J., 1975. Magne-

tization changes caused by burial and uplift. Earth and Planetary

Science Letters, 28: 133–143.

Rimbert, F., 1959. Contribution à l’étude de l’action de champs alterna-

tifs sur les aimantation rémanentes des roches. Revue de

l’Institut Francais du Petrole et Annales des Liquides Combustibles,

14:17–54, 123–155.

Roquet, J., 1954. Sur les rémanences magnétiques des oxydes de fer et

leur intérêt en géomagnétisme. Annales de Géophysique, 10: 226–

247, 282–325.

Roquet, J., and Thellier, É., 1946. Sur des lois numériques simples,

relatives à l’aimantation thermorémanente du sesquioxyde de fer

rhomboédrique. Comptes rendus de l’ Académie des sciences

(Paris), 222: 1288–1290.

Thellier, É., 1938. Sur l’aimantation des terres cuites et ses applica-

tions géophysiques. Annales de. l’Institut de Physique du Globe,

Université de Paris, 16: 157–302.

Thellier, É., 1941. Sur les propriétés de l’aimantation thermorémanente

des terres cuites. Comptes rendus de l’Académie des sciences

(Paris), 213: 1019–1022.

Thellier, É., 1946. Sur la thermorémanence et la théorie du métamag-

nétisme. Comptes rendus de l’Académie des sciences (Paris),

223:

319–321.

Thellier, É., 1951. Propriétés magnétiques des terres cuites et des

roches. Journal de Physique et le Radium, 12: 205–218.

Thellier, É., 1967. Notice sur les titres et travaux scientifiques de

M. Émile Thellier, Professeur à la Faculté des Sciences de

Paris. 28 pp.

Thellier, É., and Rimbert, F., 1954. Sur l’analyse d’aimantations fos-

siles par action de champs magnétiques alternatifs. Comptes rendus

de l’Académie des sciences (Paris) , 239: 1399–1401.

Thellier, É., and Thellier, O., 1941. Sur les variations thermiques de

l’aimantation thermorémanente des terres cuites. Comptes rendus

de l’Académie des sciences (Paris) , 213:59–61.

Thellier, É., and Thellier, O., 1959. Sur l’ intensité du champ magné-

tique terrestre dans le passé historique et géologique, Annales Géo-

physique, 15: 285–376.

Cross-references

Archeomagnetism

Demagnetization

Magnetic Susceptibility

Magnetization, Anhysteretic Remanent (ARM)

Magnetization, Natural Remanent (NRM)

Magnetization, Thermoremanent (TRM)

Magnetization, Viscous Remanent (VRM)

Nagata, Takesi (1913–1991)

Neél, Louis (1904–2000)

THERMAL WIND

The term thermal wind refers to steady or slowly time-varying shear

flow, driven by buoyancy forces due to lateral gradients in density

and modified by planetary rotation through the Coriolis acceleration.

Thermal winds in the Earth’s fluid outer core are important in the geo-

dynamo. These flows generate toroidal magnetic field in the outer

core, drive anomalous rotation of the solid inner core, and contribute

to the observed secular variation of the geomagnetic field, particularly

the westward drift. Thermal winds may also play a key role in geo-

magnetic polarity reversals. This article reviews the basic properties

of thermal winds in the core and describes how these flows affect

the geomagnetic field.

Examples of thermal wind flows

Examples of thermal wind flows abound in Nature. The general circula-

tion of Earth’s atmosphere is a thermal wind to a first approximation, and

is driven by lateral variations in density associated with the pole-to-

equator zonal temperature gradient. Some major ocean current systems

such as the Gulf Stream are driven partly by lateral density gradients

and qualify as thermal wind flows, the density gradients associated with

these currents arising from both temperature and salinity variations.

Lateral density variations are certainly present in the outer core, due to

the combined effects of thermal and compositional variations. Because

rotation influences fluid motion in the core as much as it does in the

ocean and atmosphere, thermal winds are likely to be important parts

of core flow. However, since the outer core is a much deeper fluid and

is affected by convection and the forces associated with Earth’s magnetic

field, thermal winds in the core are expected to differ from their

atmospheric and oceanic counterparts in certain respects.

The thermal wind equation

A thermal wind equation appropriate for Earth’s outer core can be

obtained from the vorticity conservation equation in the limit where

the fluid inertia and the viscous force are negligible in comparison

with effects of the Coriolis acceleration, density gradients, and the

Lorentz force due to Earth’s magnetic field. Here the focus is on the

zonal (the longitude-averaged) component of motion in the longitude

(east-west) direction, which is usually the largest part of the thermal

wind. Assuming a Boussinesq fluid, the equation for this component

of motion near the surface of the core is (Davidson, 2001)

¼

r

J 

;

where u

is the zonal velocity in the f-direction (measured eastward),

r is fluid density, y is co-latitude, O is the angular velocity of rotation

in the z-direction,

J and

B are electric current density and magnetic

field intensity, respectively, and g is the acceleration of gravity at the

core’s outer radius R. The thermal wind equation is written here in

terms of density, but as its name suggests, it is often applied to fluids

where the density is a function of temperature only. In these cases

¼ar

;

where T is temperature and a is the thermal expansion coefficient of

the fluid.

Before describing thermal winds in the core, it is useful to consider

a rotating fluid in which lateral density gradients and electric currents

are negligible. In this case the thermal wind equation reduces to the

Proudman-Taylor constraint:

¼ 0:

Fluid motions satisfying this constraint are sometimes called columnar,

because they have zero shear in the axial direction, parallel to the

rotation axis. In a rotating fluid with strong lateral density gradients

or strong Lorentz forces, the Proudman-Taylor constraint does not

hold. The above equation dictates that the flow contains axial shear

in proportion to these effects. Other terminology is sometimes used

for these flows. In Meteorology and Oceanography, the term thermal

wind usually refers specifically to the shear, not the flow itself

THERMAL WIND 945