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

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of a particular IC feature (anisotropy or lateral heterogeneity) across

a particular ray path, led to smaller values of Do, of order 0.1



1

(e.g., Creager, 1997; Poupinet et al., 2000; Vidale et al., 2000; Isse

and Nakanishi, 2002). A very different seismological approach

(Laske and Masters, 1999) examined the effects of such nonunifor-

mities on the spectra of nine free oscillations sampling the IC, and

yielded Do ¼ 0.01  0.21



1

. Because such vibrations are com-

paratively unaffected by errors in locating seismic events and by

local IC structure, and are independent of the earthquake source

mechanism, this result seems particularly robust and provides strong

evidence that Do is likely to be small, possibly not significantly dif-

ferent from zero.

The latter result is consistent with the inference that the IC is likely

to be gravitationally locked to the mantle, because the gravitational

torque exerted on the IC by mantle heterogeneities, when the IC and

mantle are not in their equilibrium orientation, greatly exceeds any

other (e.g., electromagnetic) torque that might act to maintain such a

misalignment (Buffett, 1996). A more extensive analysis of gravita-

tional coupling between IC and mantle by Xu et al. (2000) showed

that IC superrotation cannot be supported by such torques alone, and

that unless E < 3



the corresponding values of Do far exceed those

inferred from seismology. However, Buffett (1996) pointed out that

a nonzero Do could be sustained if the IC viscosity is low enough

(<5  10

Pa s) to allow its shape to deform as the IC moves out

of alignment with the mantle, but not so low as to quickly annul any

misalignment.

Any seismological indications of a value for Do, whether zero or

nonzero, constitute only a recent snapshot of its possible behavior on

the longer timescales of the geodynamo. Hollerbach (1998) concluded

that a value for Do will indicate the average angular velocity of the

OC just above the ICB, but is unlikely to suggest a reliable estimate

of magnetic field strength at the ICB unless seismology can detect

fluctuations in Do induced by torsional oscillations in the OC. At pre-

sent, seismological data is not inconsistent with, but can hardly be

regarded as constraining, recent dynamo models.

Periodic changes in orientation of i

relative to e

are a particu-

larly interesting subset of the Earth’s free oscillations, involving a

redistribution of angular momentum within the rotating planet. The

presence of the IC potentially adds two free modes to the Earth’s polar

motion spectrum: an ultralong period prograde inner core wobble

(ICW) and a nearly diurnal retrograde wobble accompanied by a much

larger, and prograde, long period free inner core nutation (FICN) The

first discussions of the ICW (Busse, 1970; Kakuta et al., 1975)

neglected gravitational coupling of the IC to the mantle. Models of

IC rotational dynamics, taking into account gravitation, fluid pressure

at the ICB, and elastic deformation using Earth properties from PREM,

predict periods for the FICN of 475 sd and for the ICW about 2400 sd

(Mathews et al., 1991). However, the first reported detection of the

FICN, in an analysis of data from very long baseline radio interfero-

metry (VLBI) by Mathews et al. (2002), indicated a period in the

range 930–1150 sd. Buffett et al. (2002) attributed the lengthening

over previous predicted periods to IC-OC electromagnetic coupling,

due to penetration of the IC by a magnetic field of 7 mT at the ICB.

So far the ICW remains below the level of detection in the data from

VLBI and gravimetry (Guo et al., 2005). For the case of a rigid IC

and mantle coupled only by gravitation and fluid pressure, Guo and

Ning (2002) found that nonzero Do has no effect on the FICN period

but slightly shortens the ICW period, and inclination E slightly length-

ens both periods. Dumberry and Bloxham (2002) showed that electro-

magnetic torques applied to the ICB by a radial magnetic field of a few

microteslas can drive the Markowitz wobble, a somewhat enigmatic

feature of the Earth’s polar motion in which the rotation pole traces

out (relative to the crust) a highly elliptical path with period about

30 years.

No longer treated with benign neglect, the rotational dynamics of

the IC now provides essential content in geodynamo models, an inter-

pretive challenge to seismologists probing the deepest interior of the

Earth, and intriguing new features for geodesists exploring the small

changes in the length of day and the motion of the pole.

Michael G. Rochester

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dynamo. Geomagnetizm i Aeronomiya, 4: 572–583.

Buffett, B.A., 1996. A mechanism for decade fluctuations in the length

of day. Geophysical Research Letters, 23: 3803–3806.

Buffett, B.A., and Glatzmaier, G.A., 2000. Gravitational braking of

inner-core rotation in geodynamo simulations. Geophysical

Research Letters, 27: 3125–3128.

Buffett, B.A., Mathews, P.M., and Herring, T.A., 2002. Modelling of

nutation and precession: effects of electromagnetic coupling. Jour-

nal of Geophysical Research, 107(B4): ETG 5.

Busse, F.H., 1970. The dynamical coupling between inner core and

mantle of the Earth and the 24-year libration of the pole. In

Mansinha, L., Smylie, D.E., and Beck, A.E. (eds.), Earthquake

Displacement Fields and the Rotation of the Earth. Dordrecht:

D. Reidel, pp. 88–98.

Creager, K.C., 1997. Inner core rotation rates from small-scale hetero-

geneity and time-varying travel times. Science, 278: 1284–1288.

Dumberry, M., and Bloxham, J., 2002. Inner core tilt and polar

motion. Geophysical Journal International , 151: 377–392.

Glatzmaier, G.A., and Roberts, P.H., 1995. A three-dimensional con-

vective dynamo solution with rotating and finitely conducting inner

core and mantle. Physics of the Earth and Planetary Interiors, 91:

63–75.

Gubbins, D., 1981. Rotation of the inner core. Journal of Geophysical

Research, 86: 11695–11699.

Guo, J.Y., Greiner-Mai, H., and Ballani, L., 2005. Spectral search for

the inner core wobble in Earth’s polar motion. Journal of

Geophysical Research, 110(B10): 10.1029/2004JB003377.

Guo, J.Y., and Ning, J.S., 2002. Influence of inner core rotation and

obliquity on the inner core wobble and the free inner core nutation.

Geophysical Research Letters, 29(8): 45.

Hollerbach, R., 1998. What can the observed rotation of the Earth’s

inner core reveal about the state of the outer core? Geophysical

Journal International, 135: 564–572.

Hollerbach, R., and Jones, C.A., 1993. A geodynamo model incorpor-

ating a finitely conducting inner core. Physics of the Earth and

Planetary Interiors, 75: 317–327.

Isse, T., and Nakanishi, I., 2002. Inner core anisotropy beneath Austra-

lia and differential rotation. Geophysical Journal International,

151: 255–263.

Kakuta, C., Okamoto, I., and Sasao, T., 1975. Is the nutation of the

solid inner core responsible for the 24-year libration of the pole?

Publications of the Astronomical Society of Japan, 27: 357–365.

Kuang, W., and Bloxham, J., 1997. An Earth-like numerical dynamo

model. Nature, 389: 371–374.

Laske, G., and Masters, G., 1999. Limits on differential rotation of the

inner core from an analysis of the Earth’s free oscillations. Nature,

402:66–69.

Mathews, P.M., Buffett, B.A., Herring, T.A., and Shapiro, I.I., 1991.

Forced nutations of the Earth: influence of inner core dynamics.

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Mathews, P.M., Herring, T.A., and Buffett, B.A., 2002. Modelling of

nutation and precession: new nutation series for nonrigid Earth

and insights into the Earth’s interior. Journal of Geophysical

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Poupinet, G., Souriau, A., and Coutant, O., 2000. The existence of

an inner core super-rotation questioned by teleseismic doublets.

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426 INNER CORE ROTATIONAL DYNAMICS

Su, W.J., Dziewonski, A.M., and Jeanloz, R., 1996. Planet within a pla-

net: rotation of the inner core of the Earth. Science, 274:1883–1887.

Vidale, J.E., Dodge, D.A., and Earle, P.S., 2000. Slow differential

rotation of the Earth’s inner core indicated by temporal changes

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Cross-references

Core Viscosity

Geodynamo, Numerical Simulations

Inner Core Anisotropy

Inner Core Composition

INNER CORE SEISMIC VELOCITIES

The inner core was discovered in 1936 by the Danish seismologist

Inge Lehmann (1888–1993) from the observation of short-period

P-wave arrivals in the shadow zone of the core, at a distance where

no P-wave would arrive if the core was totally fluid. Since her discov-

ery, major advances concerning the structure of the inner core have

been made thanks to seismological studies, combined with observa-

tional and computational results in mineral physics. A good knowl-

edge of inner core structure is important to specify the chemical and

mineralogical nature of the iron alloy which constitutes the center of

the Earth, and to constrain the Earth’s differentiation process. It also

has important outcomes for the thermodynamical and geodynamical

properties of the Earth.

Investigation tools: seismic waves and normal modes

The structure and properties of the inner core are mostly constrained

by two body waves: PKIKP, a P-wave transmitted through the inner

core, and PKiKP, a P-wave reflected at the inner core boundary

(ICB). They give information on the P-wave velocity and attenuation

inside the inner core, on the topography and properties of ICB, and

on the S-velocity and density contrasts at the ICB, as these parameters

influence the P-wave reflection and transmission coefficients. In the-

ory, the S-velocity is constrained by the phase PKJKP, which propa-

gates as an S-wave in the inner core, but its observation is still

controversial. The normal modes sensitive to inner core give informa-

tion on the mean P- and S-velocities and mean density in the inner

core. However, most of the modes sensitive to inner core structure

have stronger energy in the mantle and in the liquid core.

The phase PKIKP, also called PKPdf, is often compared to core phases

having their turning points inside the liquid core, namely PKPbc (which

turns at the base of the liquid core with a path in the mantle very close

to that of PKPdf), and PKPab (which turns in the middle of the liquid

core) (See Figure I8, and Kennett’s article on seismic phases). The dif-

ferential travel-times between PKPbc and PKPdf, or between PKPab

and PKPdf, are less sensitive than absolute travel-times to hypocenter

mislocations, origin time errors, and upper mantle heterogeneities.

PKPab, however, is strongly affected by the heterogeneities in the D

layer at the base of the mantle.

Mean radial model

The P-velocity profile inside the inner core is rather well con-

strained. The P-velocity increases from about 11.02  0.02 km s

1

the radius of ICB (1219  2 km) to 11.26 km s

1

at the center of the

Earth (Dziewonski and Anderson, 1981; Kennett et al., 1995). The P-

velocity jump at the ICB is about 0.6–0.7 km s

1

The S-velocity is mostly constrained from the periods of normal

modes, which first gave evidence of the rigidity of the inner core

(Dziewonski and Gilbert, 1971), and from the modeling of amplitudes

of core phases with synthetic seismograms (Müller, 1973). It increases

from 3.50 kms

1

at the ICB to 3.67 kms

1

at Earth’s center. This leads

to a high Poisson’s ratio s  0.44 inside the inner core (compared to

0.28–0.30 in the mantle), which is characteristic of solid metals.

The density is poorly constrained by the seismic data. The density

jump at the ICB is deduced from the P-wave reflection coefficient at

vertical incidence, and from the normal modes. A very large range of pos-

sible values have been obtained, the most recent ones being in the range

0.61.0 g cm

3

(Shearer and Masters, 1990; Cao and Romanowicz,

2004; Masters and Gubbins, 2004).

The velocity and density profiles are smooth, suggesting the

absence of chemical stratification inside the inner core. Assuming

the inner core results from the solidification of the liquid core, and tak-

ing into account the pressure at inner core condition (P ¼ 330 GPa at

the surface to 360 GPa at Earth’s center), the velocity and density

values favor an inner core composition depleted in light elements com-

pared to the liquid core. Estimation of the temperature in the inner

core, 5500–6500 K (see Core temperature), is strongly dependent on

the presence and nature of these light elements, as well as the power

released by compositional buoyancy, which is available to maintain

the geodynamo.

The attenuation g inside the inner core (or the quality factor Q ¼ 1/g)

is estimated from the broadening of the peaks of the core-sensitive

eigenmodes, and from the amplitude variations of PKPdf. Normal

modes generally lead to much higher Q-values than body waves. This

discrepancy is possibly due to the different contributions in bulk and

shear dissipation; it is however, not fully understood. For P-waves,

the comparison of the amplitudes of PKPdf and PKPbc leads to a

Q-value of the order of 200 in the uppermost 100 km of the inner core,

increasing to about 400 between 150 and 300 km below ICB. The pre-

sence of liquid inclusions is proposed, at least in the external part of

the inner core, to account for the low value of the quality factor (see

Romanowicz and Durek, 2000, for a review).

Anisotropy

Since the first observation of large-scale travel-time anomalies of

PKPdf (Poupinet et al., 1983), there have been many studies revealing

the existence of a strong anisotropy in the inner core. Paths parallel to

Figure I8 Main seismological phases sampling the inner core, and

reference phases sampling the liquid outer core which are used

for comparison (see Seismic phases).

INNER CORE SEISMIC VELOCITIES 427

the Earth’s rotation axis (“polar paths”) are about 3.5% faster than

those parallel to the equatorial plane. This anisotropy is well estab-

lished from absolute travel-times of PKPdf (e.g., Morelli et al.,

1986; Shearer, 1994; Su and Dziewonski, 1995), and from differential

travel-times of core phases (e.g., Creager, 1992) (Figure I9). It takes

about 1210 s for a P-wave to cross the Earth in the equatorial plane,

7 s less to cross it along the pole axis, once the influence of Earth

ellipticity has been corrected. The anomalous splitting of inner

core-sensitive modes is another argument in favor of P-anisotropy

(Woodhouse et al., 1986; Tromp, 1993; Romanowicz et al., 1996; Ishii

et al., 2002); it may also give insight into S-anisotropy (Beghein and

Trampert, 2003).

The question has arisen whether the anomalous propagation times

and mode splittings could originate outside the inner core. A structure

in the liquid core, in particular heterogeneity or anisotropy inside

the inner core tangent cylinder or beneath the polar caps, which is

compatible with normal modes (Romanowicz and Bréger, 2000), is

ruled out by the propagation times of core phases (Souriau et al.,

2003), and seems in any case hardly acceptable from a geodynamical

point of view. On the other hand, an anomalous ellipticity of the inner

core is ruled out by the PKiKP travel-times (Souriau and Souriau, 1989).

The favored explanation for anisotropy is the preferred orientation

of anisotropic iron crystals. The hexagonal closed-packed form, Fe-E,

is often proposed (Stixrude and Cohen, 1995), but other structures

of iron are also possible. The orientation of elliptical fluid inclusions

provides another possible explanation. The mechanism able to gener-

ate a preferred orientation is however debated; processes such as con-

vection in the inner core, solidification in the presence of a magnetic

field, solidification texturing at the ICB, or growth in a stress field,

have been proposed (see reviews and discussions in Buffett, 2000;

Bergman, 2003).

The depth variation and lateral variation of anisotropy are important

for understanding its physical nature and origin. Early studies pro-

posed an anisotropy with cylindrical symmetry and possibly a slightly

tilted symmetry axis (Creager, 1992; Su and Dziewonski, 1995). This

tilt is however not strongly required by the data (Souriau et al., 1997).

The variation with depth of anisotropy reveals that the uppermost part

of the inner core is isotropic (Song and Helmberger, 1995). The thick-

ness of the isotropic layer seems to vary with longitude, from 100 km

beneath the western hemisphere to about 400 km beneath the eastern

hemisphere (Creager, 1999; Garcia and Souriau, 2000) (Figure I10).

This difference explains the apparent hemispherical pattern of the ani-

sotropy (Tanaka and Hamaguchi, 1997). However, an artifact due to

strong heterogeneities in the lowermost mantle along some particular

polar paths (in particular the very anomalous path from South Sand-

wich Islands to Alaska) cannot completely be ruled out.

At greater depth, there are some presumptions that the anisotropy

may be different in the central 300–400 km of the inner core. Ishii

and Dziewonski (2002) observe an orientation of the slow axis at



from the pole axis for waves penetrating deep in the inner core.

From a normal mode inversion, Beghein and Trampert (2003) find a

fast axis perpendicular to the Earth’s rotation axis in the lowermost

400 km, which could suggest the presence of a different phase of iron.

Figure I9 Evidence for anisotropy in the inner core. Top:

Travel-time anomalies (with respect to a mean model) of the inner

core seismic phase PKPdf, compared to the liquid core phase

PKPbc, as a function of the angle of the ray with respect to the

Earth’s rotation axis. Data sample the uppermost 500 km of

the inner core. A faster propagation (low residuals) is observed for

polar paths. The curve corresponds to a uniform structure with

hexagonally symmetric anisotropy. Bottom: Same for absolute

PKPdf travel-times extracted from bulletin data, for rays turning

at different depths below inner core boundary (ICB). Note the

absence of anisotropy at the top of the inner core. Data have been

binned to avoid the predominance of some oversampled regions.

Figure I10 A possible model of inner core anisotropy explaining

body wave data. Meridian cross-section. Structure with 3%

anisotropy and fast axis parallel to Earth rotation axis surrounded

by an isotropic layer of variable thickness. A different form of

anisotropy may be present in the central 300–400 km of the

inner core.

428 INNER CORE SEISMIC VELOCITIES

Heterogeneities

Even though most of the propagation time anomalies are well

explained by anisotropy, it is important to estimate whether heteroge-

neities could also contribute to these anomalies. Voigt’s isotropic aver-

age of P-wave velocity seems almost invariant between the western

and eastern hemispheres (Creager, 1999), indicating the same chemical

composition of the two hemispheres. However, in the uppermost

50 km, a hemispherical P-heterogeneity has been detected by model-

ing the PKP waveforms sampling the ICB region (e.g., Niu and

Wen, 2001). The absence of heterogeneity in the range of wavelengths

200–1000 km is evidenced from stochastic methods: the heterogeneity

level is less than 0.3% at any depth (Garcia and Souriau, 2000). In the

range 50–200 km, many studies have reported the possible existence

of heterogeneities. However, it is not possible to prove that the hetero-

geneities effectively take place in the inner core, and not in the mantle.

On the other hand, the splitting functions of inner core-sensitive

normal modes exhibit large-scale variations, a result which seems in

contradiction with the previous ones. Note, however, that modes are

mostly sensitive to S-heterogeneities, and that single modes give

access only to the even part of the structure of the inner core (Laske

and Masters, 2003).

Heterogeneities at very short wavelength have been detected in

the uppermost 300 km of the inner core from the energy present in the

coda of the PKiKP waves (Vidale and Earle, 2000). The observations

are explained by scatterers of size 2 km, with velocity contrast of

1.2%. The modeling of the inner core attenuation also suggests the

presence of scatterers: small heterogeneities of length scale 10 km

with velocity perturbations of about 8%, due to the boundaries

between single or ordered groups of crystals with various orientations,

could explain the mean level of attenuation, as well as the apparent

anisotropy in velocity and attenuation (Cormier and Li, 2002).

Conclusion

Our knowledge of inner core structure has been considerably improved

during the last two decades, thanks to an increasing number of digital

records with worldwide distribution, and to the availability of about

40 years of bulletin data. Major observational difficulties come from

the perturbing effect of the D” layer, at the base of the mantle and from

the uneven sampling of the Earth, in particular the scarcity of polar

paths, that are crucial for investigating the anisotropy. The deployment

of permanent ocean bottom observatories will help to overcome this

difficulty in a near future.

There is general agreement about a 3% to 3.5% anisotropic internal

structure with fast axis parallel to the Earth’s rotation axis, surrounded

by an isotropic layer. Other features, such as the hemispherical varia-

tion of the thickness of the isotropic layer, or the possibility of a differ-

ent anisotropy in the deeper inner core, are still debated. Numerous

questions remain unanswered, concerning in particular the S-velocity

model, the P-anisotropy near the center of the Earth, and the physical

cause of the anisotropic P-wave velocity.

An accurate knowledge of the inner core heterogeneities appears

also necessary for being able to quantify a possible superrotation of

inner core with respect to the mantle. Such analyses rely on the possi-

bility to track inner core heterogeneities over several years or several

decades. Although the results are still controversial, with most recent

rotation rates in the range 0.2 to þ0.4



1

(Souriau and Poupinet,

2003), they will certainly improve in the future with the availability

of longer and denser series of data.

Annie Souriau

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Cross- refere nces

Core Composition

Core Density

Core Temperature

, Seismic Properties

Inner Core Anisotropy

Inner Core, PKJKP

Inner Core Rotation

Inner Core Tangent Cylinder

Seismic Phases

INNER CORE TANGENT CYLINDER

The Earth ’ s core consists of a solid inner core of radius r

¼ 1 220 km,

and a fluid outer core of radius r

¼ 3 480 km. The obvious geometry

to study it would therefore appear to be spherical (or perhaps spheroi-

dal, if one insists on taking into account the slight flattening induced

by the rotation). However, one could also view it in terms of cylindri-

cal geometry, in which case a new feature emerges, namely the inner

core tangent cylinder, the cylinder just touching the inner core and par-

allel to the axis of rotation. Figure I11 shows this tangent cylinder C,

as well as how the outer core is then divided into regions inside and

outside C, with the region inside C further subdivided into two parts

above and below the inner core. The purpose of this article is to dis-

cuss why it might be sensible to consider the Earth ’s core in terms

of cylindrical geometry, and what the significance of this tangent

cylinder therefore is.

Cyli ndrica l co ordinates

To see why cylindrical coordinates in fact arise quite naturally, we

begin by considering the Navier-Stokes equation governing the fluid

flow in the outer core. In its simplest, Boussinesq form, this takes

the form

]

þ U r



U þ 2

 U ¼rp þ E r

U þ F

þ F

;

(Eq. 1)

where F

and F

are the magnetic and buoyancy forces, respectively,

the Rossby number Ro ¼=Or

measures the importance of inertia,

and the Ekman number E ¼ n= O r

the importance of viscosity, both

relative to the Coriolis force 2

 U : Here V is the Earth ’s rotation

rate ð 7 :27  10

 5

rad s

 1

Þ; 2m

 1

and n  10

 6

 1

the

magnetic diffusivity and viscosity, respectively. Inserting the numbers,

we therefore find that Ro ¼ Oð 10

 9

Þ and E ¼ Oð10

15

Þ, indicating the

extreme dominance of rotation over inertia and viscosity.

If the dominant balance in the Navier-Stokes equation is thus

 U rp þ F

þ F

; (Eq. 2)

then taking the curl (and using rU ¼ 0) one obtains

 2

]

U rðF

þ F

Þ: (Eq. 3)

So we see that even though the geometry of the core is most obviously

defined in terms of spherical coordinates ðr; y; fÞ, the dynamics of the

outer core naturally, indeed almost inevitably, lead to cylindrical coor-

dinates ðz; s; fÞ. And as we saw above, once one acknowledges the

relevance of cylindrical coordinates, one invariably must consider the

tangent cylinder as well.

Figure I11 The circles represent the inner and outer core

boundaries; the dashed lines then show the location of the

inner core tangent cylinder. The dotted lines are the paths over

which (Eq. 4) is to be integrated, with the dots at the ends

indicating where the no normal flow boundary conditions are to

be imposed. Note how these boundary conditions change

abruptly across the tangent cylinder. See also Hollerbach and

Proctor (1993) for the details of how this constant of integration

is determined (or not).

430 INNER CORE TANGENT CYLINDER

Shear layer s on C?

One might suppose that the solution of (Eq. 3) is then simply

U ¼

rðF

þ F

Þdz

þ U

ðs; fÞ; (Eq. 4)

where the constant of integration U

ðs; fÞ is to be determined by the

appropriate boundary conditions (no normal flow at r ¼ r

and r

There are in fact severe difficulties with this approach (see Taylor’s

condition). However, here we simply wish to consider what happens

as one crosses C. In particular, the boundary conditions to be imposed

change discontinuously as one crosses C, as indicated in Figure I11.If

the boundary conditions change abruptly though, one must expect that

the constant of integration U

ðs; fÞ will also change, so according to

(Eq. 4) the flow itself will be discontinuous everywhere along the tan-

gent cylinder.

Any finite viscosity will of course smooth these discontinuities out

into shear layers of small but finite thickness. Figure I12 shows two

examples of such layers, at E ¼ 10

5

. The panel on the left shows

contours of the angular velocity in the so-called Stewartson layer pro-

blem, in which one imposes a slight differential rotation on the inner

core (such as may indeed exist in the Earth—see inner core rotation).

In the axisymmetric Stewartson layer problem, the only component

of the flow that would be discontinuous in the E ! 0 limit is the angu-

lar velocity U

=s, that is, a component of the flow tangential to C. If

one considers the more general nonaxisymmetric problem, one finds

that even the normal component U

may be discontinuous. The right

panel in Figure I12 shows such an example, and sure enough, we note

that the layer that resolves the discontinuity at finite E is more severe

than it was in the Stewartson layer problem.

Both the axisymmetric and nonaxisymmetric layers shown in

Figure I12 are purely nonmagnetic. If one includes a magnetic field,

it seems likely that the tension in the field lines would tend to suppress

such strong shear layers. At least in the nonaxisymmetric case, how-

ever, there is also a price to pay: Hollerbach and Proctor (1993)

showed that a shear layer can only be avoided if the forcing (that is,

þ F

) satisfies a certain integral constraint on C, and Hollerbach

(1994b) showed further that the magnetic field does indeed evolve

so as to satisfy this constraint. It would thus appear that there probably

are no shear layers on C (the axisymmetric Stewartson layer is simi-

larly suppressed by a magnetic field), but that instead one has certain

integral constraints on C. The dynamical significance—and indeed

the physical interpretation—of these constraints is not fully understood

though.

Figure I12 The panel on the left is from Hollerbach (1994a),

and shows contours of the angular velocity U

=s in the

axisymmetric Stewartson layer problem. The panel on the

right is from Hollerbach and Proctor (1993), and shows

contours of U

in a nonaxisymmetric shear layer problem.

Both layers are at E ¼ 10

5

Figure I13/Plate 10b Contours of the radial magnetic field at the surface of the Earth’s core in 1980, northern hemisphere, in equal

area projection. The circle denotes the end of the tangent cylinder. Blue indicates a region where field lines enter the core, yellow to

red where they exit, and the zero contour is a “magnetic equator” where the field is horizontal. Note the concentration of flux into two

lobes in the northern hemisphere just outside the tangent cylinder, and the lack of flux inside the tangent cylinder.

INNER CORE TANGENT CYLINDER 431

Inside versus outside C

There are also a number of issues where it is not so much C itself, but

rather whether one is inside or outside that matters. For example, the

fact that the region inside C is further subdivided into parts above

and below the inner core means that inside C Taylor’s condition consists

of two distinct integral constraints (at any given cylindrical radius s),

whereas outside C there is only one integral over the entire extent of

the cylindrical shell. What effect this has on the geodynamo is not

known for certain, but given the general importance of Taylor’s condi-

tion, it could be significant.

Another difference between inside versus outside C is how the

height of fluid columns parallel to the rotation axis varies with s.

Inside C, dH/ds is positive, whereas outside it is negative. The reason

this may be important is that so-called Rossby waves (e.g., Salmon,

1998) propagate in the direction rH  V, that is, westward inside

C, eastward outside. However, as with the Stewartson layer, classical

Rossby waves are a nonmagnetic phenomenon more usually asso-

ciated with oceanography or meteorology, so it is not entirely clear

how relevant they are to the Earth’s core. Similar effects have also been

observed though in convection-driven hydromagnetic waves (Zhang and

Gubbins, 2002), so it seems that this distinction between inside and

outside C does indeed persist into the magnetic regime as well.

Speaking of convection, numerous simulations have shown that

the overall pattern of convection appears to be quite different inside

versus outside C. (See Core convection or Kono and Roberts, 2002

for a summary of some of these results.) The reasons for this are not

always apparent, but one effect that is likely to play a role is simply

that inside C gravity and the rotation vector are largely parallel,

whereas outside C they are predominantly perpendicular. Given the

powerful influence that both of these effects have on convection (with

gravity driving the whole process, of course), it seems likely that their

relative orientation will also be important.

The inner core may also impart some stability to the geodynamo,

because the magnetic field can only change inside it by diffusion,

whereas in the outer core it can change by the much more rapid pro-

cess of advection (Hollerbach and Jones, 1993). This longer timescale

of the inner core has been used to explain the preponderance of geo-

magnetic excursions (q.v.), or aborted reversals, over full reversals of

the geomagnetic field (Gubbins, 1999). The tangent cylinder may

enhance this stabilizing effect by providing a larger, cylindrical

volume of “dead” space where the flow does not affect the magnetic

field to any great extent. In particular, we note that the inner core by

itself constitutes only 4% of the whole core volume, whereas the total

region inside the tangent cylinder constitutes 18%. If the fluid inside C

is relatively quiescent, it would therefore considerably boost this stabi-

lizing influence.

Observed effects on the Earth’s field

Surprisingly, the effect of the inner core tangent cylinder is spectacu-

larly visible in the Earth’s magnetic field when projected down to

the core surface. Figure I13/Plate 10b shows the vertical component

Figure I14/Plate 10a Contours of the same magnetic field as in

Figure I13. The inner core is shown in red and the tangent

cylinder is indicated, along with the circle where it intersects the

surface in the northern hemisphere.

Figure I15/Plate 10c The same magnetic field at the core surface in Aitoff projection. Note the four main lobes, symmetrical about the

equator and distributed outside the tangent cylinder.

432 INNER CORE TANGENT CYLINDER

of the magnetic field at the surface of the core in the northern

hemisphere. The dark circle is the top of the tangent cylinder, with

radius 20



. Figure I14/Plate 10a shows the same field with a cutaway

liquid core to illustrate the tangent cylinder. The vertical field is stron-

gest not at the pole, which would be the case for an axial dipole, but in

two lobes just outside the tangent cylinder. Similar lobes are present in

the southern hemisphere: these are the four main lobes of the geomag-

netic field responsible for the predominance of its dipole character;

they are less mobile than the features at lower latitude associated with

the westward drift (q.v.).

The pattern of lobes is symmetrical about the equator, Figure I15/

Plate 10c , suggestive of straight convection rolls touching the inner

core (see Convection, nonmagnetic rotating ). Numerical simulations

and kinematic dynamo calculations show magnetic flux being con-

centrated at the surface by shear around the tangent cylinder and by

downwelling associated with the convection rolls (see Geodynamo,

numerical simulations and Gubbins et al. , 1999). It appears that the

geomagnetic field is not generated strongly inside the tangent cylinder,

and that most of the generation occurs just outside, as in some of the

models. Note that the radial field is almost zero within the tangent

cylinder in the northern hemisphere, and in Figure I13/Plate 10b there

is a similar null patch in the southern hemisphere.

This lack of field near the poles probably explains why virtual

paleomagnetic poles (VGPs ), those determined from a single site mea-

surement of the magnetic field direction assuming a geocentric axial

dipole (q.v.), usually lie some distance (typically 10



) from the geo-

graphic pole. The same is true of the present-day field. Only when a

time average is done to produce a paleomagnetic pole, by averaging

many samples from a rock unit that spans many thousands of

years, is a reliable estimate obtained for the geographic pole (see Pole,

paleomagnetic ), and even then the match is not perfect. The departure

of the VGP from the geographic pole is caused by the nonaxisym-

metric nature of the field near the poles; this largely averages out with

time as the main lobes drift. The departure of the resulting paleomag-

netic pole is usually attributed to a nondipolar time average. For

further discussion of the time average see Gubbins (1998).

Rainer Hollerbach and David Gubbins

Bibliogr aph y

Gubbins, D., 1998. Interpreting the paleomagnetic field. In Gurnis, M.,

Buffett, B., Wysession, M., and Knittle, E. (eds.), The Core-

Mantle Boundary Region. AGU Geophysical Monograph, Geo-

dynamics Series. Washington, DC: American Geophysical Union,

pp. 167– 182.

Gubbins, D., 1999. The distinction between geomagnetic excursions

and reversals. Geophysical Journ al International , 137,F1–F3.

Gubbins, D., Barber, C.N., Gibbons, S., and Love, J.J., 1999. Kine-

matic dynamo action in a sphere. II. Symmetry selection. Proceed-

ings of the Royal Society of London, Series A, 456: 1669 – 1683.

Hollerbach, R., 1994a. Magnetohydrodynam ic Ekman and Stewartson

layers in a rotating spherical shell. Proceedings of the Royal

Society of London, Series A, 444: 333– 346.

Hollerbach, R., 1994b. Imposing a magnetic field across a nonaxisym-

metric shear layer in a rotating spherical shell. Physics of Fluids , 6 :

2540 –2544.

Hollerbach, R., and Jones, C.A., 1993. Influence of the Earth ’s inner

core on geomagnetic fluctuations and reversals. Nature, 365:

541– 543.

Hollerbach, R., and Proctor, M.R.E., 1993. Nonaxisymmetric shear

layers in a rotating spherical shell. In Proctor, M.R.E., Matthews,

P.C., and Rucklidge, A.M. (eds.), Solar and Planetary Dynamos.

Cambridge: Cambridge University Press, pp. 145– 152.

Kono, M., and Roberts, P.H., 2002. Recent geodynamo simulations

and observations of the geomagnetic field. Reviews of Geophysics ,

40(4): 1013, doi:10.1029/2000RG000102 .

Salmon, R., 1998. Lectures on Geophysical Fluid Dynamics. Oxford:

Oxford University Press.

Zhang, K., and Gubbins, D., 2002. Convection-driven hydromagnetic

waves in planetary fluid cores. Mathematical and Computer Mod-

elling , 36: 389– 401.

Cr oss-referenc es

Anelastic and Boussinesq Approximations

Convection, Nonmagnetic Rotating

Core Convection

Dynamos, Kinematic

Earth Structure, Major Divisions

Geocentric Axial Dipole Hypothesis

Geodynamo, Numerical Simulations

Geomagnetic Excursion

Inner Core Rotation

Pole, Paleomagnetic

Proudman-Taylor Theorem

Taylor ’s Condition

Westward Drift

INNER CORE, PKJKP

Gener al background

The inner core was detected by Lehmann in 1936 (Lehmann, 1936)

based on observations of P-waves through the inner core (later called

PKIKP phases) that were unexplained by the Earth models of that

time. In her noteworthy paper Lehmann concluded: “ The question of

the existence of the inner core cannot however be regarded solely from

a seismological point of view, but must be considered also in its other

geophysical aspects. ” Indeed, in 1940 Birch hypothesized that the

inner core may be the result of a liquid-solid phase change in iron

(Birch, 1940). A solid inner core implies that shear waves can be

locally present, and the phase PKJKP with a shear wave leg in the

inner core should exist (Figure I16 ). The letter J in this phase notation

denotes the shear wave path in the inner core, whereas I represents

an inner core P-wave path. Similarly, the letter K denotes a P-wave

passage through the outer core, and P and S stand for mantle P- and

S-wave legs, respectively.

Thus, to confirm the inner core’s solidity, the search for observa-

tions of PKJKP started in the 1940s. In his article on the history of

inner core research, Bolt (1987) recalls Bullen’s attempts to find

PKJKP (see also Bullen, 1951), and numerous other seismologists

have tried to identify PKJKP as well. The detection of PKJKP, how-

ever, has proven to be extremely difficult. The solidity of the inner

core was eventually established in 1971 based on normal mode obser-

vations (Dziewonski and Gilbert, 1971), but identification of PKJKP

remained a challenge to many seismologists.

PKJKP identifications

PKJKP is difficult to detect because of its small amplitude. This is a

consequence of its large spreading (see Figure I16), inefficient P-to-S

and S-to-P conversions at the inner core boundary, and significant

inner core attenuation (Doornbos, 1974). Furthermore, interference

with other larger amplitude phases severely hampers a proper interpre-

tation. PKJKP cannot be identified on indiv idual seismograms, and

therefore stacking of large number of seismograms is required. If dif-

ferent stations are used in the stack, slowness information can be

obtained, which is essential for an unambiguous identification.

Julian et al. (1972) were the first to report observations of PKJKP.

They stacked data of five earthquakes with magnitudes ranging from

6.0–6.3 recorded by the Large Aperture Seismic Array (LASA) in

INNER CORE, PKJKP 433

Montana (USA). This first PKJKP claim was, however, controversial

because the inferred inner core shear velocity is at odds with the shear

velocity determined from the normal mode data (3.5–3.6 km s

1

). It

has been speculated that the observed signal was in fact the SKJKP

phase instead of PKJKP, but the data were not reanalyzed.

It took more than 60 years after the detection of the inner core

before two new reports of PKJKP were made. Both papers rely on

advanced signal detection techniques combined with data from large

magnitude (7.9– 8.2) deep earthquakes. Okal and Cansi (1998) applied

a multichannel correlation method to data of the 1996 Flores Sea event

for eight stations of the French short-period seismic network. They

identified PKJKP and the surface-reflected pPKJKP phase. Deuss

et al. (2000) used 47 globally recorded broadband seismograms for

the same Flores Sea event and employed a nonlinear stacking techni-

que based on phase coherency. They compared the data stack with the-

oretical stacks computed from normal mode seismograms: (1) for a

model with a solid inner core, and (2) for a model with a fluid inner

core. Evidently, the difference between these two synthetic stacks is

due to the presence of J-phases. Deuss et al. (2000) found that, due

to interference, PKJKP cannot be identified unambiguously in the data

of the Flores Sea event, but the combination of pPKJKP and SKJKP

can be identified in the data.

Although recent claims have now been made, it remains difficult to

observe the J-phases. For instance, Deuss et al. (2000) showed that the

global broadband data do not allow identification of the J-phases for

the magnitude 8.2 Bolivia earthquake of 1994 due to interference

effects of other signals. For the same reason they dispute the PKJKP

identification of Okal and Cansi (1998) in their data set.

Implications of PKJKP observations

PKJKP and other J-phases may now be detected under very favorable

conditions: a very big earthquake (M  8), a station distribution with a

large number of stations in the appropriate distance range, a favorable

source depth and radiation pattern, and use of a sophisticated stacking

technique. The detection itself, however, is not crucial anymore as the

shear velocity is determined from normal mode data. Yet, observations

of J-phases can still be important because they contain information

about attenuation and anisotropy, constraining the physical state of

the inner core. The issue of inner core attenuation is briefly addressed

by Deuss et al. (2000). They find that the shear-wave attenuation as

given by reference model PREM (Dziewonski and Anderson, 1981)

is in agreement with their data. It corresponds to a relatively high level

of attenuation possibly due to partial melting (Loper and Fearn, 1983).

Inner core shear wave anisotropy may be investigated by directional

dependence of the PKJKP observations (e.g., parallel and perpendicu-

lar to the Earth’s rotation axis). After the detection, this could be the

next challenge in PKJKP research. However, normal mode measure-

ments proved to be more efficient to determine these properties

(Beghein and Trampert, 2003).

All in all, PKJKP is a phase that does not easily reveal the informa-

tion it contains.

Hanneke Paulssen

Bibliography

Beghein, C., and Trampert, J., 2003. Robust normal mode constraints

on inner core anisotropy from model space search. Science, 299:

552–555.

Birch, F., 1940. The alpha-gamma transformation of iron at high

pressures, and the problem of the Earth’s magnetism. American

Journal of Science, 238: 192–211.

Bolt, B.A., 1987. 50 years of studies on the inner core. EOS, Transac-

tions of the American Geophysical Union, 68:73–81.

Bullen, K.E., 1951. Theoretical amplitudes of the seismic phase

PKJKP. Monthly Notices of the Royal Astronomical Society, Geo-

physics Supplement, 6: 163–167.

Deuss, A., Woodhouse, J.H., Paulssen, H., and Trampert, J., 2000.

The observation of inner core shear waves. Geophysical Journal

International, 142:67–73.

Doornbos, D.J., 1974. The anelasticity of the inner core. Geophysical

Journal of the Royal Astronomical Society, 38: 397–415.

Dziewonski, A.M., and Anderson, D.L., 1981. Preliminary reference Earth

model. Physics of the Earth and Planetary Interiors, 25:297–356.

Dziewonski, A.M., and Gilbert, F., 1971. Solidity of the inner core of

the Earth inferred from normal mode observations. Nature, 234:

465–466.

Julian, B.R., Davies, D., and Sheppard, R.M., 1972. PKJKP. Nature,

235: 317–318.

Lehmann, I., 1936. P’, Bur. Centr. Seismol. Int. A., Trav. Sci., 14:87–115.

Loper, D.E., and Fearn, D.R., 1983. A seismic model of a partially mol-

ten inner core. Journal of Geophysical Research, 88:1235–1242.

Okal, E.A., and Cansi, Y., 1998. Detection of PKJKP at intermediate

periods by progressive multichannel correlation. Earth and

Planetary Science Letters, 164

:23–30.

Cross-references

Core Composition

Inner Core Anisotropy

Inner Core Composition

Inner Core Seismic Velocities

Lehmann, Inge (1888–1993)

INSTRUMENTATION, HISTORY OF

Geomagnetic instruments before 1800: instrum ents

in early navigation and science

Scientists and various practitioners have used a bewildering array of

magnetic instruments over the last thousand years or so: the mariner’s

compass (many types), circumferentors, declinometers, variation

Figure I16 Schematic ray paths of PKJKP through mantle (P),

outer core (K), and inner core (J).

434 INSTRUMENTATION, HISTORY OF

compasses, dip circles or inclinometers, theodolite magnetometers,

variometers, bifilar magnetometers, Lloyd’s and Schmidt’s balances,

earth inductors, and more recently, nuclear magnetic resonance magneto-

meters, saturable-core magnetometers, induction magnetometers, and

more. This rich history, however, is characterized by short bursts of

inventive activity followed by longer periods of diffusion of instrumental

designs. Because this volume includes articles on the more common con-

temporary magnetic instruments (see Archaeology, magnetic methods;

Compass; Magnetometers, laboratory; Magnetic surveys, marine;and

Observatories, instrumentation), this article emphasizes the earlierhistory

of magnetic instruments and their use. It sets some current instruments

against this backdrop (see also History of geomagnetism).

Although magnetism as a property of the lodestone was known since

antiquity in both China and Greece, the first magnetic instruments

unequivocally appear in the historical record in the 12th century CE

(see Compass). Strong evidence now shows that the magnetic compass

was in use in Europe around 1150 and in China around 1100 (Smith,

1992, pp. 25 and 33–38), with evidence the Chinese knew something

of the magnetic needle’s directionality for centuries. The first accounts

that can be dated are the Meng Chhi Pi Than, a chronicle written by

Shen Kua in 1088, and Alexander Neckam’s De Nominibus Utensilium,

written between 1175 and 1183. Shen Kua not only mentioned a

“wet” compass (a needle floating on water) and balancing a needle on

a horizontal edge, but also magnetic declination. Neckam’s was the first

European reference to a compass needle on a pivot (a “dry” compass),

but it is possible he had never used one and was only reporting an

increasingly common practice on ships at sea. The best known magnetic

investigator of the 13th century, Petrus Peregrinus (Pierre de Mari-

court), wrote his famous Epistola ... de Magnete in 1269. Peregrinus

was a consummate experimenter and careful thinker. Familiar with the

compasses used and discussed for about a century, he developed

improved wet and dry compasses, with a scale of 360



attached (Smith,

1992, pp. 70–72). His significance lies in the clarity and completeness

of his summation of magnetic knowledge.

From 1200 until around 1600, use of the compass and knowl-

edge of its phenomena were restricted mainly to navigators and

cartographers, although poets and philosophers discussed it frequently

from the mid-13th century. Early 15th century German makers of sun-

dials had a new use of the compass, the orientation of portable

sundials. The mid- to late 16th century was a period of instrumental

innovation and investigation of the nature of environmental magnet-

ism, indeed, the beginning of geomagnetism. The Portuguese Pedro

Nunes and João de Castro and the Englishmen Robert Norman,

William Borough, and William Gilbert (q.v.) stirred interest in Earth’s

magnetism partly because of phenomena revealed by instrumental

improvements.

These magnetic workers represent a mixing of the practical goals

of navigators and more esoteric goals of improving the instruments

and studying the phenomena revealed by them. Nunes and de Castro

developed an improved “variation compass” and began paying closer

attention to magnetic declination in the 1530s (Fara, 1998, p. 136).

Norman focused on the dipping needle, conceived the dip circle (first

described in print in 1581), and concluded that the seat of magnetism

is within the Earth. Borough concentrated on improving instruments

for measuring magnetic declination and Gilbert assembled the most

complete résumé of magnetic experiments, a summa of magnetic

knowledge to the time. No fundamental changes in instrument design

occurred until near 1800. Nevertheless, two fundamental discoveries

were made using improved instruments: the discovery by Henry

Gellibrand (q.v.) in 1634 that declination varies over time and by

George Graham in 1722 that declination varies daily according to a

set pattern. During the 18th century artisans modified the variation

compass with lighter needles, a box to enclose the needle, and the

addition of reading microscopes and vernier scales. Another 18th

century innovation was the tall azimuth visor, which allowed mariners

to sight celestial objects high above the horizon.

Navigators and surveyors continued into the 18th and 19th centuries

to be among the primary users of compasses. Their compasses “com-

pensated” for declination by rotating the compass card so the needle

pointed east or west of north (and by other means) and for inclination

by weighting one end of the needle. In the 18th century, John England

(of London) introduced steel needles in 1705, as did the Dutch in 1731

and P. Le Maire and d’Après de Mannevillette in 1745. Gowin Knight

and John Smeaton, among others, continued to experiment with new

designs (Fara, 1998, p. 136). More improvements were aimed at ease

of use and dependability. Innovations were often encouraged by the

British Board of Longitude, the French Academy of Sciences, and

other institutions.

Ever since Robert Norman first discussed it in the late 16th century,

the dip circle had been the least reliable magnetic instrument. The ver-

tical needle was supported by a metal axis with pointed ends, inserted

in glass or crystal cups. Friction often stopped the needle short of the

direction of dip. Instrument makers tried many methods to minimize

this friction, from lining the axle points and cups with gold or bell

metal to adding “friction” wheels (McConnell, 1980, pp. 16–17;

Good, 1998a, pp. 175–177).

Natural philosophers meanwhile, fascinated by the detail and com-

plexity of the phenomena, worked with artisans to modify instruments.

In 1777, Charles Coulomb proposed using a thread to suspend the needle

and he derived the mathematical theory of torsional suspensions (see

Oscillations, torsional ). Jean Jacques Cassini soon used this type

instrument at the Paris Observatory; Tiberius Cavallo, Christopher

Hansteen (q.v.), and ultimately Carl Friedrich Gauss (q.v.) in the

1830s, elaborated on and perfected fiber-suspension instruments

(Multhauf and Good, 1987, pp. 5–20; Good, 1998b). The instrument

makers responsible for these devices included Étienne Lenoir, Thomas

Jones, and George Dolland.

Instrument makers produced a variety of instruments for both navi-

gation (and surveying) and for scientific investigation. The former

instruments stressed durability and ease of use, the latter precision.

Scientific use, however, included both fixed observatories—where size

of the instrument was no concern—and expeditions by land or sea, in

which size mattered. From the 1780s on, expeditions included geo-

magnetic research more often, though not until the late 19th century

were dedicated geomagnetic surveys more commonplace.

The first completely new line of instruments to appear after

the compass and dip circle started out simply as a different way to

use these two instruments. George Graham observed the oscillating

needle of a dip circle in 1723 but did not draw any conclusion regard-

ing the intensity of magnetic force (McConnell, 1980, p. 26). Frederick

Mallet used an oscillating compass needle in 1769 to measure the hor-

izontal magnetic force. And in 1776 Jean Charles Borda discussed

oscillations of a dipping needle as a measure of total magnetic force.

An oscillating needle indicated magnetic intensity in the same way that

a pendulum measured gravity: the shorter the period, the stronger the

force. Until the 1830s, however, the results of different instruments

could not be compared and were called “relative” measurements.

Soon many investigators were improving intensity instruments,

developing methods to counteract problems like friction and the asym-

metry of needles. Soon, too, scientists carried out magnetic intensity

measurements on several expeditions: Jean-Françoise La Pérouse

made measurements lost at sea in the 1780s. E.P.E. De Rossel, with

the D’Entrecasteaux expedition, counted dip circle oscillations in

1792 and 1793 during a circumnavigation. Alexander von Humboldt

(q.v.) continued and popularized the study of magnetic intensity during

his expedition to South America, and Humboldt and J.L. Gay-Lussac

continued this around Europe (Parkinson, 1998, p. 448). Humboldt

used both instruments but began a trend among researchers to prefer

to measure horizontal intensity with an oscillating, suspended needle.

This led to a famous and popular instrument made by Christopher

Hansteen in 1824, a needle suspended by a fiber. But the type was

already widely established and used by others, too, including Coulomb

INSTRUMENTATION, HISTORY OF 435