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

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COMPASS

A freely suspended, magnetized iron needle will attempt to align itself

with the ambient magnetic field vector (F); when made to float in a

bowl of liquid, or balanced horizontally on a pivot, it will orientate

itself along the local magnetic meridian (H, the horizontal part of the

vector). At what specific instant during the Middle Ages this attribute

was first discovered remains shrouded in the mists of time; China, the

Arab world, Greece, Italy, France, and Scandinavia have all claimed

primacy in this respect. The ubiquity of lodestone (or loadstone, var-

ious iron-rich, permanently magnetized minerals) at and near the

Earth’s surface must have enabled many cultures to become aware of

this mysterious, but highly useful, property.

Technically speaking, any contraption incorporating magnetic align-

ment to indicate direction can be considered a compass. The word

itself can be traced back to Mediterranean countries around the mid-

13th century, initially referring to sailing directions (both written

instructions and nautical charts). Two centuries later, its meaning had

expanded to include dividers applied in course plotting on a chart,

whereas in German-speaking regions the word additionally became

associated with pocket sundials, which contained a small dry-pivoted

magnetic needle, inset at the base, for meridional orientation. It would

take another century before the term would be predominantly asso-

ciated with the currently typical magnetized needles bearing a gradu-

ated card, often encased and in cardanic suspension. The following

discussion will outline some of the most notable aspects of this evolu-

tion, focussing in turn on its basic design, the observation compass,

technical improvements, and the problem of deviation.

Basic compas s design

The earliest unequivocal evidence of a magnetized piece of iron used

for orientation stems from China around the turn of the 12th century

A.D., in sources describing the floating magnetic “compass fish,”

and south-pointing needles magnetized by lodestone touch suspended

with a silk wire or balanced on a fingernail. The floating needle (in

a little water-filled saucer, marked with 12 or 24 wind directions)

was probably incidentally used at sea during overcast weather by both

Chinese, Korean, and Japanese navigators, until “dry” compasses were

introduced by European ships from the 16th century. However, South-

east Asian shipping was primarily coastal and monsoon-driven, limit-

ing reliance on the magnetic pathfinder. By contrast, contemporary

Arabian navigators traversing the Indian Ocean set their courses using a

32-point sidereal card without a needle, deriving directions instead from

the near-perpendicular rise and set of specific stars, such as Altair (East)

and Antares (Southeast). Their 13th-century adoption of the compass

on Mediterranean voyages was established through European contacts.

In western Europe, references to floating needles start to appear from

the 12th century, in accounts by A. Neckam (1187), P. de Maricourt (Pet-

rus Peregrinus, 1269), and in various ship’s inventories. Over the next

century, notions of magnetization and compass needles became so com-

monplace that they were increasingly used in a metaphorical sense in lit-

erature. During the 14th century, dry-pivoted instruments came

to replace their floating precursors and the needle was attached to

a starch-stiffened paper disk depicting the rose of the winds (6–8 in. in

diameter); its cardinal points were painted blue, the half cardinals red.

The direction of Jerusalem (east) was initially marked with a cross,

the others with the first letter of their associated Mediterranean

wind. The north-indicating letter “T” (for Tramontana) may have mutated

into the ornamental fleur-de-lys around 1500. Points division grew from

8 through 16–32 in the 16th century. From about 1650, standardized

designs engraved in copper plate replaced hand-drawn cards, while during

the 18th century, a more austere appearance with rim graduation became

the norm. At the time, maximum steering accuracy of square-rigged ves-

sels generally did not exceed one quarter point (2



With the advent of commercial cartography and triangulation, the

surveying compass rose to prominence on land. This instrument was

usually tripod-mounted and equipped with a plumb line to be aligned

with the sighted object. Its pivoted, straight needle rotated over a card

attached to the inside bottom of the case. In the 17th century, the

mounting of more elaborate sighting devices (double visors, cross-

hairs, small telescopes) allowed more precise readings to be taken.

Another compass-like instrument, used on land from the late 16th cen-

tury, was the versorium, an arrow-shaped iron needle, up to a foot

long, applied by natural philosophers such as W. Gilbert (q.v.) and

N. Cabeo to investigate electrical and magnetic phenomena, for exam-

ple, to detect the electrostatic attraction produced by rubbed amber.

At sea, the mariner’s needle became attached to the underside of the

card, providing instantaneous identification of all directions and a lar-

ger moment of inertia. By the late 16th century, the whole became

encased in a round or square bowl or box made of wood, copper, or

brass, suspended in brass or tin gimbals, inside a larger square box.

The box was covered with a glass or crystal lid, sealed with putty or

wax to keep out wind, dirt, and moisture, while its bottom was detach-

able for maintenance purposes, so the watertight top seal did not have

to be broken each time the needle required retouching.

The steering compass was housed on deck inside the binnacle,

which was horizontally divided into three compartments; the two outer

ones each held a compass, the middle one a lantern. When sailing craft

grew larger, the fore-and-aft line that indicated the ship’s heading

became less clearly distinguishable to the helmsman, so a correspond-

ing mark inside the compass bowl, the lubber’s line, was introduced

as a steering aid. Various production centers mushroomed, such as

Sicily, Genoa, and Venice (Mediterranean), Danzig (Baltic), and

Seville, Lisbon, La Rochelle, Bordeaux, Rouen, and Flanders (west

European coasts). From the 17th century, the capitals of the seagoing

nations likewise became the seat of compass making firms and indivi-

dual craftsmen, which soon came to dominate the manufacture of these

instruments.

To compensate for magnetic declination in steering, the needle

was sometimes affixed to the card at an angle to true north; another

physical solution was the rectifier, a second, rotatable card on top of

the first, with which changing declination values along a voyage could

be compensated by applying a variable (usually fixed-step) shift. To

compensate for magnetic inclination, which changes with latitude

along the voyage and dips the needle, thus increasing friction, a lump

of wax was attached to the card; from the 18th century, increasingly

sophisticated, adjustable counterweigh ts were used.

Regarding needle shape, at its crudest stage, a single piece of iron

wire was bent into rhombic form, the two ends forming an acute angle

at one apex, and the middle of the wire forming another at the opposite

end, allowing space in the middle for a cap to balance the needle on

the pivot. Alternatively, two separate curved pieces of iron could be

joined at the ends to form a lozenge or two empty parentheses. Single,

straight needles of well-tempered steel (enabling more intense, longer-

lasting magnetization), rose to prominence in the first half of the 18th

century. Metal caps then became replaced by glass and agate ones,

which suffered less wear from pivotal friction. Admiralties and com-

mercial shipping companies adopted standardized, often patented

designs (from 1766), for example, by G. Knight and R. Walker (18th

century), and by the firms of Stebbing, Weilbach, and Kleman

(19th century).

The observat ion compass

The main difference between a steering compass and an observa-

tion compass is a sighting device mounted on top of the box (see

Figure C13), to be aligned with a landmark (surveying and charting),

the ship’s wake (estimating leeway), or a celestial body (determining

magnetic declination). Such compasses could be set up on land or any-

where on deck using a wooden stool or special tripod. Despite great

66 COMPASS

variability of design, all types had cards graduated in degrees (or less)

to measure the horizontal arc between two imaginary great circles per-

pendicular to the horizon and meeting in the zenith. The arc distance

from true east was called amplitude; that from true north was called

azimuth. The most significant practical distinction between amplitude

compass and azimuth compass is that the former, older device only

allowed near-horizontal sightings directly over the card, whereas the

latter’s much taller visors, introduced in many countries in the second

half of the 18th century, permitted multiple observations to be taken in

short succession, and at a much greater altitude above the horizon. The

acceptance of azimuth as a superior measurement was, however, reli-

ant on adequate technology, the implementation of the new, longer cal-

culations, and awareness of the problems associated with amplitude

readings (limited opportunity, poor sighting conditions at the horizon

due to clouds and haze and atmospheric refraction).

The first description of an observation compass can be found in De

Maricourt’s “Epistle on the Magnet ” (1269). Apart from a silver or

brass transverse index, it carried a sighting rule with upright styles at

both ends to align with heavenly bodies. Similar 16th-century descrip-

tions are known from the Portuguese J. de Lisboa (1508), Italian L.

Sanuto (1588), Dutchman R. Pietersz van Twisk (1595), and Englishman

W. Barlowe (1597, with a rotatable verge ring carrying a brass ruler

with two upright, triangular sights). In France, two types of ampli-

tude compass existed concurrently, one with small pins fixed to the

rim of the box, the other with two small glass windows in opposite

sides, furnished with crosshairs. French navigators adopted azimuth

visors only from the 19th century onward. Some English 17th-century

amplitude compasses had a large, broad semicircle made of brass,

which extended beyond the compass rim and could be rotated around

it. Opposite this dial was a large foldable style, carrying a sight and a

hypotenusal thread from the top of the index down to the plate that

could cast a noonday shadow onto the card. This method was rarely

used at sea.

In terms of measurement accuracy, by taking the average or (more

often) the median of a series of magnetic declination readings obtained

in the course of a nautical day, a standard deviation of little under half

a degree was maintained by the majority of early-modern navigational

practitioners engaged in oceanic shipping. On land, with the aid of

more accurate instruments and fixed meridians engraved in stone, this

margin could sometimes be reduced to a few arc minutes. Around

1820, a more modern version of the azimuth compass was introduced

at sea, allowing celestial measurements near the zenith with a card

divided in half degrees.

Compass design improvements

Besides complaints about careless construction, low-quality materials,

skewed or warped cards, and the aforementioned wear on pin and cap,

the most serious defect of the instrument, mentioned time and again in

early-modern documents, was weak magnetization. Needles made of

iron wire (in use up to the early 18th century) required several retouch-

ings at sea on long voyages, which is why some vessels still carried

lodestones on board as well. Magnetization itself was deemed highly

dependent on the proper method; suggestions range from repeatedly

striking the needle with a hammer while placed on the lodestone,

through rotating its tips inside small holes made at the lodestone’s

poles, to (1– 12) stroking motions of the lodestone along (half or all

of ) the needle’s length. A separate debate focussed on the various

types and qualities of lodestone, using color, density, and origin as

decisive characteristics. With the 18th-century advent of steel needles

and artificial magnets (fully magnetized steel rods, much stronger than

natural magnets), the earlier dependence on lodestones, both by com-

pass makers and sailors, was greatly reduced.

A different way to improve directivity was to increase the number

of needles inside the instrument; by placing two, four, or even eight

straight needles in parallel under the card; their combined directive

property would greatly exceed that of any single one. The first con-

trolled experiment that compared the performance of single- and paral-

lel needle compasses dates from 1649, by C.J. Lastman and I. Blaeu,

two navigation officials employed by the Dutch East India Company.

The superior results of parallel needles were disseminated in leaflets

and navigation textbooks, and from 1655 until about 1710, Dutch

Eastindiamen routinely carried two or more such instruments.

After incidental descriptions in French and Spanish textbooks with-

out practical implementation, Danish naval officer and hydrographer

Lous rekindled interest from 1767, advocating a compass with four

flat, steel needles of 150 mm in length, held in a wooden or brass frame.

Similar designs were proposed by French mathematician C. de Borda

(1772) and astronomer P.-C. Le Monnier (1776), and by English captain

C.H. Lane (1788). However, it would take until 1840 before the new

British Admiralty’s standard compass would carry four straight steel

needles attached to a ring beneath the card, itself to be supplanted by

the improved card invented by W. Thomson (1878, adopted 1889),

which carried eight parallel needles suspended by wire. In subsequent

decades, this dry compass fought a losing battle against another design

improvement which would be universally adopted in the 20th century:

the liquid compass.

The liquid compass has the advantages of damping needle move-

ments and exterior disturbances and supporting the card to reduce fric-

tion, but at the cost of introducing such new complications as freezing,

leakage, and temperature-induced changes of volume or pressure.

After Royal Society experiments with lodestones acting under water

in the 1660s, E. Halley (q.v.) seems to have demonstrated a rudimen-

tary liquid compass there in 1690. The first comprehensive description

appeared in 1779, written by Dutch physician J. Ingen-Housz: a large,

sensitive needle inside a steel tube with a float, housed in a waterproof

casing filled with linseed oil. Several other designs were propounded

by instrument makers G. Wright (1781), Barton (1792), and C.C. Lous

(1795); liquid compass patents were, for example, issued to inventor

Francis Crow (1813), and the Danish compass production firm of

Weilbach (1830). The problem of variable fluid volume was finally

solved by W.R. Hammersley’s system of expansion chambers (1856).

A late-19th century liquid compass with two parallel artificial magnets

is depicted in Figure C14.

Compass deviation

The deflection of compass needles away from the geomagnetic vector

due to the attraction of magnetized iron nearby is known as compass

Figure C13 Cross section of an early-modern amplitude compass,

as used in oceanic navigation. 1, needle; 2, card; 3, cap;

4, pin; 5, balancing weight; 6, inclination counterweight; 7, glass

lid; 8, cardanic suspension; 9, compass bowl (inner box);

10, outer box; 11, rotatable verge ring; 12, fiducial line;

13, crosshair visor.

COMPASS 67

deviation (the term was coined by Arctic explorer J. Ross, ca. 1820).

The effect was first recorded by Portuguese navigator J. de Castro in

1538, who ascribed inconsistent readings of magnetic declination to

a cast iron gun on deck. From the 17th century onward, regulations

regarding compass use warned both mariners and surveyors against

wearing clothing attributes made of iron and prompted them to take

note of metal structures that might affect the readings. On the largely

wooden ships, the disturbance remained limited to incidental aberra-

tions of a few degrees at most, the cause of which was usually identi-

fied swiftly, and either removed or avoided thereafter.

The introduction of iron and steel in ship construction, however,

radically exacerbated the problem. Despite the advantages of stronger,

longer, cheaper, more spacious hulls, the increasing reliance on

structural iron parts (late 18th and first half of the 19th century) and

extensive armor plating (Crimea War, 1854–56), all introduced sub-

stantial sources of magnetic deviation, up to tens of degrees. The per-

iod from the first iron-hulled clippers (ca. 1800) up to all-steel ship

design (1877) was hallmarked by the scientific investigation of

deviational compass error, and the invention of various methods of

compensation.

Possibly the oldest method to investigate compass deviation was

“swinging the ship,” which involved turning the ship about a vertical

axis, and recording the incurred deviation on all headings. The proce-

dure was first suggested by American surveyor J. Churchman in 1790.

British naval surveyor M. Flinders, while traversing Tasmanian waters

in 1798, likewise noticed that the amount of compass error varied as a

function of course. On his next assignment in the region, he identified

(latitude-related) geomagnetic inclination as another factor to be con-

sidered. Subsequent captivity on the French isle of Mauritius (1803–

1810) allowed him to develop his theory of compass error further,

using his findings to correct and redraw much of his survey. Upon

his return to England, he suggested a counter attractor, the “Flinders

bar” (an iron rod of specific length placed inside the binnacle, opposite

the side of greatest deviation), which was eventually introduced on

board ship some 50 years later.

Research intensified from the 1810s, amongst others by explorer

W. Scoresby, astronomer F. Arago, Board of Longitude secretary Th.

Young, and Professor P. Barlow (q.v.), who developed a disk (“Bar-

low’s plate,” 1819) to quantify deviation aboard ship. Theoretical

aspects received attention in a series of treatises (1821–1838) by

French mathematician S.D. Poisson, relating geomagnetism and ship

magnetism using C.A. de Coulomb’s inverse-square law. Soon there-

after, Astronomer Royal G.B. Airy started a series of experiments

on iron ships, using combinations of bar magnets and iron chains as

counteracting physical correctors. In the second half of the century,

hydrographer Frederick Evans worked with mathematician A. Smith

on the problem, based on the Hydrographic Office’s first isogonic

chart (1858), resulting in the “Admiralty Manual for Deviations of

the Compass” in 1862.

Notwithstanding these efforts, deviation proved a highly complex

phenomenon that was never fully defeated. A different solution ensur-

ing accurate and safe navigation free from interference was eventually

found in the gyrocompass, based on the gyroscopic principle, which

forces the axis of any sufficiently fast spinning top to keep a fixed

orientation, for example, relative to the Earth’s axis (described in

1836 by E. Sang in Britain, and in 1852 by J.B.L. Foucault in France).

The first fully operational models were patented by H. Anschütz-

Kaempfe and E. A. Sperry in the early 20th century, and underwent

sea trials from 1907. Over the next few decades, the gyrocompass

proved a worthy successor to the magnetic compass, which became

relegated to the more modest role of auxiliary safety backup on large

ships. Nevertheless, the magnetic compass continues to serve the

majority of smaller, private vessels, as well as scouts, hikers, and other

travelers on land (see Figure C14). In addition, the historical accumu-

lation of magnetic compass readings recorded in ship ’s logbooks pre-

sently provides an invaluable source of data on geomagnetic secular

variation over the last four hundred years.

Art R.T. Jonkers

Bibliography

Bennett, J.A., 1987. The Divided Circle: A History of Instruments for

Astronomy, Navigation and Surveying. Oxford: Phaidon.

Blackman, M., 1983. The Lodestone: A Survey of the History and the

Physics, Contemporary Physics, 24 no. 4: 319–331.

Cotter, C.H., 1997. The Early History of Ship Magnetism: the

Airy-Scoresby Controversy, Annals of Science, 34: 589–599.

Day, A., 1967. The Admiralty Hydrographic Service 1795–1919.

London: Ministry of Defence.

Fanning, A.E., 1986. Steady as She Goes: A History of the Compass

Department of the Admiralty. London: Her Majesty’s Stationery

Office.

Freiesleben, H.C., 1978. Geschichte der Navigation. Wiesbaden: Steiner.

Hackmann, W.D., 1994. Jan van der Straet (Stradanus) and the Origins

of the Mariner’s Compass, In Hackmann, W.D., and Turner, A.J.

(eds.), Learning, Language and Invention: Essays Presented to

Francis Maddison. Aldershot: Variorum, 148–179.

Hewson, J.B., 1951. A History of the Practice of Navigation. Glasgow:

Brown, Son and Ferguson.

Hitchins, H.L., and May, W.E., 1955. From Lodestone to Gyro-

Compass. London: Hutchinson.

Körber, H.-G., 1965. Zur Geschichte der Konstruktion von Sonnenuh-

ren und Kompassen des 16. bis 18. Jahrhunderts, Veröffentlichun-

gen des Staatlichen Mathematisch-Physikalischen Salons 3. Berlin.

Maddison, F.R., 1969. Medieval Scientific Instruments and the Devel-

opment of Navigational Instruments in the Fifteenth and Sixteenth

Centuries. Agrupamento de Estudos de Cartografia Antiga 30.

Coimbra: Junta de Investigações do Ultramar.

McConnell, A., 1980. Geomagnetic Instruments before 1900: An Illu-

strated Account of their Construction and Use. London: Harriet

Wynter.

Merrifield, F.G., 1963. Ship Magnetism and the Magnetic Compass,

Navigation and Nautical Courses, Vol. 1, Oxford: Pergamon.

Schück, A., 1911–1915. Der Kompass. Hamburg: Schück.

Smith, J.A., 1992. Precursors to Peregrinus: The Early History of

Magnetism and the Mariner’s Compass in Europe, Journal of Med-

ieval History. 18:21–74.

Waters, D.W., 1958. The Art of Navigation in England in Elizabethan

and Early Stuart Times. London: Hollis and Carter.

Wolkenhauer, A., 1982. Der Schiffskompas in 16. Jahrhundert und die

Ausgleichung der Magnetischen Deklination, In Köberer, W. (ed),

Das Rechte Fundament der Seefahrt: Deutsche Beitrage zur

Geschichte der Navigation. Hamburg: Hoffmann and Campe.

Figure C14 An American 19th-century liquid steering compass

with two parallel needles (on left), and a Swedish late

20th-century handheld, dry-pivoted scouting compass (author’s

collection).

68 COMPASS

Cross- refere nces

Barlow, Peter (1776 – 1862)

Geomagnetism, History of

Gilbert, William (1544–1603)

Halley, Edmond (1656 –1742)

Voyages Making Geomagnetic Measurements

CONDUCTIVITY GEOTHERMOMETER

Intro ductio n

The concept of the electrical conductivity geothermometer comes from

the laboratory-demo nstrated dependence of the electrical conductivity

of rock material on temperature (Hinze, 1982). These laboratory

results are consistent with theoretical considerations.

Beneath various complications that arise in the upper crust of Earth,

rock material is generally considered to conduct electricity according

to the physical mechanism of electronic semiconduction. The partial

melting of rock material may also increase rock conductivity once con-

nectivity is established for the melt fraction (Shankland et al., 1981).

For electronic semiconduction, the relationship is the Arrhenius one

between electrical conductivity s and absolute temperature T, given by

(Parkinson and Hutton, 1989)

s ¼ s

exp ðE

= kT Þþs

expð E

= kT Þ (Eq. 1)

Here the first term in Eq. (1) describes extrinsic conduction, operating

through thermally activated crystal defects and impurities, and the sec-

ond term describes intrinsic conduction at higher tempe ratures. The

terms s

and s

are the high-temperature asymptotes for each type of

conduction, E

and E

are activation energies, and k is the Boltzmann ’s

constant.

The strategy is to exploit the phenomenon of electromagnetic induc-

tion at Earth ’s surface by natural source fields (Campbell, 1997) to

obtain a s profile. Then, in terms of parameters s

, s

, E

and E

deter-

mined experimentally from laboratory studies or on the basis of other

laboratory information, to obtain a T profile from the s profile. The

procedure is thus in two parts, which will now be discussed in turn.

Determ ining elect rical condu ctivity as a functi on

of dep th

The observed data are time-series of magnetic and electric fields mea-

sured at various places over the surface of Earth. Generally these

places are well-established magnetic observatories, sometimes supple-

mented by more temporary magnetotelluric stations. The primary sig-

nals have generally come to the solid Earth from outside it, and

because they change with time they induce secondary fields within

Earth, as the planet is an electrical conductor. At the surface, observa-

tions record both primary and secondary fields combined. A process of

data inversion is needed, based on the known physics of induction in

the Earth (Weaver, 1994), to give from these observations a profile

of electrical conductivity with depth.

Activity in this topic is now such that specialist workshops on Elec-

tromagnetic Induction in the Earth have been held in different parts of

the world every two years since the first one in Edinburgh, Scotland in

1972. Review and contributed papers are published, for example see

Weaver (1999). Data on electrical conductivity structure from electro-

magnetic observations are used to obtain information on geological

structure, composition, and history.

An important point is that the process of electromagnetic induction

in the Earth is frequency dependent. An indication of the depth of

penetration into a material by an induction process is given by the

electromagnetic “skin depth,” d ,

d ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

mso

(Eq. 2)

where o denotes angular frequency, s denotes the electrical conductiv-

ity of the material, and m denotes the permeability of the material

(usually taken as that of free space).

Thus lower frequency signals penetrate deeper into the Earth and

information on deep Earth conductivity depends on them. A natural

limit to the lowest frequencies which can be observed is imposed by

the spectrum of the source fields. Commonly the longest period stu-

died is 1 year, exploiting a periodicity which has its origin in the sea-

sonal cycle of Earth. A longer period of 11 years, which has its origin

in a periodicity seen in solar activity, is also sometimes used. A limit

may also be imposed by the length of observatory record needed to

obtain data for these lowest frequencies.

If the Earth had ideal spherical symmetry, and its properties (includ-

ing electrical conductivity) varied with depth only, the analysis of

observations to give structure would be more straightforward. This

situation, termed one-dimensional (1D), has mathematics which, while

far from simple, are not as complicated as the next stages of complex-

ity, which are termed two-dimensional (2D) and three-dimensional

(3D) respectively (Weaver, 1994). For a 2D situation, the conductivity

varies with two spatial parameters such as depth and one horizontal

direction; while in a 3D situation the conductivity may vary in all

directions.

Many case histories now demonstrate that departures from 1D are

common in the electrical conductivity structure of the crust and upper

mantle of Earth (Booker and Chave, 1989; Gough, 1989; Mareschal

et al., 1995). In addition to the possibility of “conductivity anomalies”

caused by partial melt, other notable causes of increased conductivity

are graphite and water (ELEKTB-Group, 1997).

However it is common to model the deeper regions of the Earth’s

mantle as 1D. This action is taken on the grounds that the material is

expected to be more homogeneous than the more easily inspected

crust.

An example of global conductivity profiles thus obtained by inverting

surface observations are those of Olsen (1999) shown in Figure C15.

Figure C15 Profiles of electrical conductivity of the Earth as a

function of depth, after Olsen (1999). The different profiles result

from different data inversion procedures.

CONDUCTIVITY GEOTHERMOMETER 69

In this case, as in others, the mantle-core boundary is taken to be a step

to effectively infinite conductivity, due to the molten metallic material

of the Earth ’s core.

Determ ining tem peratur e from electrical condu ctivity

When a conductivity profile with depth has been obtained, there is

then the question regarding what mineral data should be used to con-

vert the conductivity data to temperature. While there is general agree-

ment about the main mineralogical composition of Earth, some fine

points, upon which conductivity is highly dependent, remain uncer-

tain. Prime amongst these is the oxygen state, in Earth ’s mantle, of

the mineral olivine (Shankland and Duba, 1990; Constable and

Roberts, 1997).

Figure C16 from Shankland (1981) combines data on rock and

mineral conductivities. It shows firstly that electrical conductivity is

an Earth quantity which varies by orders of magnitude in common

rock materials. Then, within the range of conductivity values shown,

the two lines (on the log-log graph) for synthetic “hot-pressed olivine ”

give an indication of the range of values in Eq. (1) possible when

inverting conductivity data to give an “ electrogeotherm ” (and see Xu

and Shankland (1999) for more recent experimental results). Such

inversions are made on the basis (generally accepted by mineralogists)

that the mantle of Earth is comprised mainly of olivine.

A number of research workers have thus published temperature pro-

files for the Earth based on analyses of electromagnetic induction at

Earth ’s surface. The early result of Duba (1976), shown in Figure C17,

is instructive regarding the general magnitudes of the quantities

involved. Note that Constable (1993) explores the possibility that the

temperature at 410 km is less than as shown in Figure C17, for consis-

tency with experimental evidence that a phase transition at that depth

should occur at 1400



C. The electrogeotherm models of Const able

(1993), incorporating results from Constable et al. (1992), are shown

in Figure C18.

It is important to note also that at the high temperatures near the

base of the mantle, Eq. (1) becomes less temperature dependent, as

its asymptotic form for high temperatures is approached. Thus electri-

cal conductivity, even if known, becomes less useful in predicting

temperature at such high temperatures.

Conclusion

Increasing research into the various elements needed to construct

an electrical conductivity geothermometer has produced the result

(common in research) that the matter is more complicated than might

have at first been expected. While increased computer power enables

continued progress in the numerical modeling of complicated conduc-

tivity structures and of the inversion of observed data in terms of them

Figure C16 Conductivities of possible upper mantle constituents

(after Shankland (1981) and Rai and Manghani (1978)).

Figure C17 Olivine electrogeotherm for Earth’s upper mantle

compared with geotherms determined petrologically (after Duba,

1976).

Figure C18 Upper mantle electrogeotherms, after Constable

(1993). The different curves result from different data inversion

techniques, especially regarding a possible step increase in

electrical conductivity at depth 600 km.

70 CONDUCTIVITY GEOTHERMOMETER

(Fujii and Schultz, 2002), uncertainty about the details of the main

constituent of the mantle (olivine, and in particular its oxygen fuga-

city) remain and limit the application of the strategy.

In fact at mantle depths the relationship between electrical conduc-

tivity and temperature is increasingly being examined from another

view point. Rather than assume that the material in the mantle is known,

it is increasingly the case that the temperature is estimated from other

considerations. Then, such temperature estimates are used with labora-

tory measurements of different minerals to determine, from electrical

conductivity profiles, the mineralogy of the mantle (Shankland et al.,

1993; Constable, 1993).

Ted Lilley

Bibliography

Booker, J.R., and Chave, A.D., 1989. Introduction to special section

on the EMSLAB—Juan de Fuca experiment. Journal of Geophysi-

cal Research, 94: 14093–14098.

Campbell, W.H., 1997. Introduction to Geomagnetic Fields. Cam-

bridge: Cambridge University Press.

Constable, S., Shankland, T.J., and Duba, A., 1992. The electrical con-

ductivity of an isotropic olivine mantle. Journal of Geophysical

Research, 97: 3397–3404.

Constable, S.C., 1993. Constraints on mantle electrical conductivity

from field and laboratory measurements. Journal of Geomagnetism

and Geoelectricity, 45: 707–728.

Constable, S.C., and Roberts, J.J., 1997. Simultaneous modeling of

thermopower and electrical conduction in olivine. Physics and

Chemistry of Minerals, 24: 319–325.

Duba, A., 1976. Are laboratory electrical conductivity data relevant to

the Earth? Acta Geodaetica, Geophysica et Montanistica of the

Academy of Sciences of Hungary, 11: 485–495.

ELEKTB-Group, 1997. KTB and the electrical conductivity of the

crust. Journal of Geophysical Research, 102: 18289–18305.

Fujii, I., and Schultz, A., 2002. The 3D electromagnetic response of

the Earth to ring current and auroral oval excitation. Geophysical

Journal International, 151: 689–709.

Gough, D.I., 1989. Magnetometer array studies, earth structure, and

tectonic processes. Reviews of Geophysics, 27: 141–157.

Hinze, E., 1982. Laboratory electrical measurements on mantle rele-

vant minerals. Geophysical Surveys, 4: 337–352.

Mareschal, M., Kellett, R.L., Kurtz, R.D., Ludden, J.N., Ji, S., and

Bailey, R.C., 1995. Archaen cratonic roots, mantle shear zones

and deep electrical anisotropy. Nature, 375: 134–137.

Olsen, N., 1999. Long-period (30 days – 1 year) electromagnetic

sounding and the electrical conductivity of the lower mantle

beneath Europe. Geophysical Journal International, 138:179

–187.

Parkinson, W.D., and Hutton, V.R.S., 1989. The electrical conductivity

of the Earth. In Jacobs, J.A., (ed.), Geomagnetism. Vol. 3, Aca-

demic Press Inc., pp. 261–321.

Rai, C.S., and Manghani, M.H., 1978. Electrical conductivity of ultra-

mafic rocks to 1820 kelvin. Physics of the Earth and Planetary

Interiors, 17:6–13.

Shankland, T.J., 1981. Electrical conduction in mantle materials. In

O’Connell, R.J., and Fyfe, W.S., (eds.), Evolution of the Earth.Vol.5

of Geodynamics Series. Washington, D.C.: AGU, pp. 256–263.

Shankland, T.J., and Duba, A.G., 1990. Standard electrical conductiv-

ity of isotropic, homogeneous olivine in the temperature range

1200–1500



C. Geophysical Journal International, 103:25–31.

Shankland, T.J., O’Connell, R.J., and Waff, H.S., 1981. Geophysical

constraints on partial melt in the Upper Mantle. Reviews of Geo-

physics and Space Physics, 19: 394–406.

Shankland, T.J., Peyronneau, J., and Poirer, J.P., 1993. Electrical con-

ductivity of the Earth’s lower mantle. Nature, 366: 453–455.

Weaver, J.T., 1994. Mathematical Methods for Geo-electromagnetic

Induction. Baldock, UK: Research Studies Press.

Weaver, J.T., 1999. Collective review papers presented at the 14th

Workshop on Electromagnetic Induction in the Earth. Surveys in

Geophysics, 20: 197–200.

Xu, Y., and Shankland, T.J., 1999. Electrical conductivity of orthopyr-

oxene and its high pressure phases. Geophysical Research Letters,

26: 2645–2648.

Cross-references

Conductivity, Ocean Floor Measurements

Geomagnetic Deep Sounding

Magnetotellurics

Mantle, Electrical Conductivity, Mineralogy

CONDUCTIVITY, OCEAN FLOOR

MEASUREMENTS

The ocean floor presents a particularly harsh environment in which

to carry out electrical measurements, with pressures of up to 600 atm

(60 MPa), temperatures of around 3



C, and no possibility of radio

contact with instrumentation. Furthermore, seawater is a corrosive,

conductive fluid. Thus, progress in the field of electrical conductivity

studies has largely followed the availability of reliable underwater

technology and has not become truly routine until recently.

Most electromagnetic methods can be adapted for seafloor use, but

the high conductivity of seawater dominates both how data are col-

lected and how they are interpreted. Seawater conductivity depends

on salinity and temperature; in practice salinity variations are too small

to be significant and so to a good approximation seawater conductivity

is given by (3 þ T/10) S/m where T is temperature in degree celsius.

The bulk of the ocean thus has a resistivity of about 0.3 Ωm, with

warmer surface waters 0.2 Ωm.

DC resistivity is only practical for shallow investigations where the

seafloor conductivity is similar to or greater than the seawater. The

most common EM method applied to seafloor studies is the magneto-

telluric (MT) method (q.v.). First experiments date from the 1960s, and

Charles Cox, Jean Filloux, and Jimmy Larson (1971) report an MT

response from measurements made in the Pacific in 1965, using

1 km long cables on the seafloor and a new seafloor magnetometer

developed by Filloux. Filloux later also developed a system for mak-

ing electric field measurements using short (about 3 m) pipes acting

as salt bridges and a device to reverse the connection of electrodes

to the pipes, allowing any electrode self potential to be removed from

long period electric field variations (Filloux, 1987). Although this

technique is still important for studies of ocean currents using seafloor

E-field recorders, it has since proved unnecessary for MT studies, and

it is now common practice to simply mount electrodes at the ends of

four approximately 5 m plastic arms to give 10 m E-field dipoles. The

torsion fiber magnetometers originally used by Filloux to keep power

consumption low have been replaced in more recent instrumentation

by fluxgate sensors and induction coils.

There are advantages and disadvantages to the seafloor MT method.

On the advantage side, it is easy to make a low impedance, low noise

electrical contact with the environment. The sensor of choice for this is

silver-silver chloride nonpolarizable electrodes although a new carbon

fiber electrode has been developed for short period studies. The sea-

floor is also free of the cultural noise that can plague land MT surveys.

Access is good and permitting, if required, is usually valid for the

entire survey area. Arrays of seafloor MT recorders lend themselves

well to the new array processing techniques available.

On the other hand, the skin depth (exponential decay length for EM

fields) at 1 Hz in seawater is only 270 m and so the overlying ocean

removes the short period source fields, a problem exacerbated by the

red nature of the geomagnetic spectrum (see Geomagnetic temporal

CONDUCTIVITY, OCEAN FLOOR MEASUREMENTS 71

spectrum). Depending on water depth and the type of instrumentation,

data are limited to periods longer than 10 to 1000 seconds. Water

motion produces a Lorenz (E ¼ V  B) field that is prominent at tidal

periods and eventually dominates the induced fields (see Electromag-

netic ocean effects). Thus, seafloor MT is ultimately band limited.

The recent use of induction coil magnetometers (q.v.) on marine MT

instruments (Constable et al., 1998), coupled with high gain, AC-

coupled electric field amplifiers, optimizes the collection of higher

frequency data, but even with this system one is limited to about 10 s per-

iod data on the deep seafloor (in 4–6 km water).

The geomagnetic coast effect (q.v.) is very much more severe on the

ocean side. Electric charge accumulating on coastlines from the MT

fields leak into the conductive part of the mantle over a boundary layer

distance L given approximately by

ﬃﬃﬃﬃﬃﬃ

(Cox, 1980), where S is the

conductance of the oceans and T is the resistivity-thickness product

of the uppermost seafloor rocks. We know from CSEM studies that

T is about 10

Ωm

, so long period MT anomalies from the coast

effect are attenuated horizontally over an L of 3000 km. Undoubtedly,

there is electrical connection between the ocean and conductive mantle

at volcanic regions and possibly at subduction zones, but at passive

margins the coast effect is so strong that T can be estimated from land

MT data and global induction studies. For these reasons, it is important

in marine MT interpretations include coastlines and bathymetry during

modeling and interpretation.

In the late 1970s, Charles Cox (1981) proposed replacing the high-

frequency energy lost to the magnetotelluric signal with a deep towed

human-made transmitter, and carried out the first experiments in 1979

(Spiess et al., 1980). This controlled source electromagnetic (CSEM)

sounding method complements the MT method, because not only is

it sensitive to shallower structure, but also it is preferentially sensitive

to resistive seafloor, whereas the MT signal is induced in conductive

rocks. In the most common application of this technique, a horizontal

electric transmitting dipole is towed close to the seafloor, and the

resulting electric fields are measured as a function of range and fre-

quency by a number of seafloor electric field recorders. At a typical

transmission frequency of 1 Hz, the skin depth in seawater is so small

that receivers placed more than a few kilometers from the transmitter

record only signals that have propagated through the more resistive

seafloor. For two decades the marine CSEM method remained an exo-

tic technique used to study deep seafloor and mid-ocean ridges, but

during the first years of the 21st century it was enthusiastically

embraced by the petroleum industry. Hydrocarbon reservoirs are rela-

tively resistive (10–100 Ωm) compared to the marine sediments in

which they are often found (around 1 Ωm), and this resistivity contrast

combined with CSEM sounding’s ability to detect thin resistive layers

makes the method useful for mapping oil and gas reservoirs (Constable

and Weiss, 2005).

Nigel Edwards (1985) and colleagues developed a marine applica-

tion of a land technique called magnetometric resistivity (MMR)

sounding, in which a bipolar vertical electric field transmitter extends

from the ship to near the seafloor. A low-frequency current is trans-

mitted and the resulting magnetic fields are measured by seafloor

recorders some distance away. Since the essentially DC current flow

from the transmitter is distorted by the resistive seafloor rocks, the

horizontal magnetic fields become a measure of seafloor conductivity.

Figure C19 illustrates the three types of measurements discussed

here. The recorders are essentially the same in every case, except that

for long-period MT measurements a fluxgate magnetometer would

supplement or replace the induction coil sensors. Figure C20 illustrates

the current understanding of oceanic conductivity structure. Seawater,

as noted, is around 0.3 Ωm except above the thermocline near the

surface. The upper 400–600 m of oceanic crust is composed of extrusive

volcanics, lava flows, and pillow lavas. Corresponding to seismic layer 2,

these rocks are highly porous and have resistivities between 1 and 10 Ωm.

They are more conductive at mid-ocean ridges (MORs) because pore-

water is heated by volcanic action. Any sediments overlaying the basalts

will have a resistivity between that of seawater and about 1 Ωm.

At a depth of about 500 m, porosity, permeability, and electrical con-

ductivity all drop as the lithology changes to intrusive basaltic volcanics

(dikes and sills). Away from ridges, resistivity will be a few tens of Ωm

in this layer. At around 2 km below the seafloor, we find the gabbros

that make up most of the crust. Away from the MOR axes, resistivity

increases to several tens of thousands of Ωm by the time mantle depths

are reached; whether this increase occurs at the top of the gabbroic layer,

or deeper in the crust, is hard to determine because resistivity is mono-

tonically increasing below the seafloor and the resolving kernels of

EM methods are intrinsically smooth. At fast MOR spreading centers,

and at currently active places on slowly spreading ridges, this layer con-

tains the magma chamber that fuels crustal accretion. Resistivity within

the magma chamber will be between that of basaltic magma (0.3 Ωm)

and about 30 Ωm, depending on melt content.

Off-axis, crustal conductivity is simply a reflection of porosity, and

any conductivity contrast at the moho, which would typically be below

600



C, will be small, and the lithospheric mantle, like the lower crust,

is many thousands of Ωm. It is possible that the oceanic upper mantle,

depleted in volatiles during crustal formation at mid-ocean ridges, is

about an order of magnitude more resistive than continental mantle.

At the base of the thermal lithosphere (from 6 to 100 km deep,

depending on plate age), conductivity increases to between 100 and

500 Ωm consistent with the electrical conductivity of dry subsolidus

Figure C19 Experimental layout for seafloor-controlled source EM (CSEM) sounding (left), magnetometric resistivity (MMR) sounding

(center), and magnetotelluric (MT) sounding (right). In all cases the receiver instruments are very much the same; for MT sounding a

CSEM or MMR transmitter is not required.

72 CONDUCTIVITY, OCEAN FLOOR MEASUREMENTS

olivine near its melting point, or perhaps a little more conductive (see

Conductivity geothermometer ). Many early marine MT studies

reported a highly conductive layer thought to correspond to the asthe-

nosphere between about 100 and 200 km deep. Although there may

well be a conductivity anomaly associated with the base of the litho-

sphere, the very high conductivities originally modeled are probably

an artifact of using one-dimensional interpretation on data that are

affected by the coast effect (q.v.). The controversy that once surrounded

this issue seems to be abating, however, and more recent multisite, mul-

tidimensional studies often include the coastlines explicitly and tend not

to require an asthenospheric conducting layer.

Both laboratory studies of high pressure mantle minerals (see

Mantle, Electrical conductivity, Mineralogy) and induction studies

using long-period magnetic observatory records (see Geomagnetic

deep sounding) show that the lower mantle (believed to be composed

of silicate perovskite and mangnesio-wüstite) is very much more

conductive than the upper mantle and is about 0.2–0.5 Ωm. Whether

this conductivity increase occurs entirely at 670 km or starts shallower

in the transition zone is unclear. Recent laboratory measurements on

the high-pressure phases of olivine suggest that conductivity should

increase at 440 km and some induction studies support this. However,

the jump in conductivity at 670 km is so large that smooth inversions

for mantle conductivity inevitably exhibit an increase starting at

shallower depths.

Steven Constable

Bibliography

Constable, S., Orange, A., Hoversten, G.M., and Morrison, H.F., 1998.

Marine magnetotellurics for petroleum exploration 1. A seafloor

instrument system. Geophysics, 63: 816–825.

Constable, S., and Weiss, C.J., 2005. Mapping thin resistors (and

hydrocarbons) with marine EM methods: Insights from 1D model-

ing. Geophysics, 71: 643–651.

Cox, C.S., 1980. Electromagnetic induction in the oceans and infer-

ences on the constitution of the Earth. Geophysical Survey, 4:

137–156.

Cox, C.S., 1981. On the electrical conductivity of the oceanic litho-

sphere. Physics of the Earth and Planetary Interiors, 25: 196–201.

Cox, C.S., Filloux, J.H., and Larsen, J.C., 1971. Electromagne-

tic studies of ocean currents and electrical conductivity below

the ocean-floor. In A.E. Maxwell (ed.), The Sea. New York: Wiley,

pp. 637–693.

Edwards, R.N., Law, L.K., Wolfgram, P.A., Nobes, D.C., Bone, M.N.,

Trigg, D.F., and DeLaurier, J.M., 1985. First results of the MOSES

experiment: Sea sediment conductivity and thickness determina-

tion, Bute Inlet, British Columbia, by magnetometric off-shore

electrical sounding. Geophysics, 50: 153–160.

Filloux, J.H., 1987. Instrumentation and experimental methods for

oceanic studies. In J.A. Jacobs (ed.), Geomagnetism, London:

Academic Press, pp. 143–248.

Spiess, F.N., Macdonald, K.C., Atwater, T., Ballard, R., Carranza, A.,

Cordoba, D., Cox, C., Diaz Garcia, V.M., Francheteau, J., Guerrero, J.,

Hawkings, J., Haymon, R., Hessler, R., Juteau, T., Kastner, M., Larson,

R., Luyendyk, B., Macdougall, J.D., Miller, S., Normark, W.,

Orcutt, J., and Rangin, C., 1980. East Pacific Rise: hot springs and geo-

physical experiments. Science, 207:1421–1433.

Cross-references

Coast Effect of Induced Currents

Conductivity Geothermometer

EM, Industrial Uses

EM, Marine Controlled Source

Geomagnetic Deep Sounding

Geomagnetic Spectrum, Temporal

Magnetotellurics

Mantle, Electrical Conductivity, Mineralogy

Ocean, Electromagnetic Effects

CONVECTION, CHEMICAL

Convection is the differential motion which occurs due to the action of the

gravitational body force on density inhomogeneities within a fluid body.

The density of a fluid body is a function of its pressure, temperature,

and chemical composition. Pressure variations cannot cause convection;

they play a secondary role, at most. “Chemical convection” refers to

motions which arise due to spatial variations in density arising from var-

iations in the chemical content of the fluid; an alternate and equivalent

name is “compositional convection.” Similarly, motions which arise due

to spatial variations in temperature are called “thermal convection.”

The study of convection is facilitated by assuming that the density,

r, is independent of pressure and is a linear function of both tempera-

ture, T, and chemical content (i.e., mass or volume fraction of one of

two chemical constituents), C; for example,

r ¼ r

 r

a T  T

½r

b C  C

½;

where a is the coefficient of thermal expansion, b is the compositional

expansion coefficient, and a subscript 0 denotes a constant (reference)

value. For normal fluids a > 0 (hot fluid is lighter than cold), while

the sign of b depends on whether C measures the fraction of denser

or less dense constituent. Commonly, C is taken to quantify the less

dense constituent and then b > 0. In the core, C quantifies the nonme-

tallic constituent and b is of unit order.

Figure C20 Approximate seafloor conductivity profiles below a

mid-ocean ridge (broken line) and older, normal oceanic

lithosphere (solid line).

CONVECTION, CHEMICAL 73

If the surfaces of constant density are not perpendicular to the local

acceleration of gravity, convection invariably occurs. On the other

hand, if these surfaces are everywhere perpendicular to gravity, then

a quiescent state is always possible. An important question is whether

convection can and will occur in such a fluid body. The fluid is stable

if all perturbations of arbitrary form and amplitude decay with time.

On the other hand, if any small perturbation of the quiescent state

leads to motions which increase in amplitude with time, the fluid is

said to be (convectively) unstable.

The tendency for a fluid body to convect is measured by the

Rayleigh number:

Ra ¼

DrðÞgh

where g is the acceleration of gravity, h is the vertical extent of the

body, Dr is a measure of the density perturbations (excluding those

due to pressure), n is the kinematic viscosity of the fluid, and k is

the diffusivity of the factor causing density variations (either tempera-

ture or composition). Often the symbol D is used in place of k when

variations are due to composition. Commonly Dr is defined such that

a positive value denotes lighter fluid beneath denser and a negative

value denotes denser fluid beneath lighter.

There is a dynamic similarity between the situation in which the

density of the fluid depends only on the relative proportions of two

chemical constituents of the fluid (with Dr/r

¼bDC) and that in which

density depends only on the temperature (with Dr/r

¼ aDT ). In either

case the fluid is convectively unstable when the Rayleigh number

exceeds a critical value, the magnitude of which depends on the shape

and nature of the boundaries of the fluid body (see Chandrasekhar,

1961). If the density is a function of both temperature and compo-

sition, then the stability of the fluid body depends on two Rayleigh

numbers, measuring the density changes due to these two variables,

and is significantly more complicated (see Turner, 1974).

The existence of Earth’s magnetic field is very strong evidence that

convection occurs in the outer core, as there is no plausible explana-

tion for the origin of this field other than dynamo action driven by con-

vective motions (see Geodynamo ). The outer core convects because it

is cooled and the principal sources of buoyancy are the latent heat and

light material released as the inner core grows by solidification of the

denser component of the liquid outer core (see Geodynamo, energy

sources and Core convection). That is, convection is likely due to both

thermal and chemical (or compositional) differences; both the thermal

and compositional Rayleigh numbers are likely to be positive, particu-

larly near the inner-core boundary.

Chemical convection of another sort can occur within the uppermost

portion of the inner core. It is likely that the inner core is not comple-

tely solid, but rather a mixture of solid and liquid, sometimes called a

mush. The solid and liquid phases are in phase equilibrium, placing a

constraint on the temperature and composition of the liquid phase.

Ignoring pressure effects, this may be characterized by a simple linear

liquidus relation:

T ¼ T

 G C  C

½;

where G is a constant quantifying the rate at which the temperature of

pure iron is decreased by addition of (nonmetallic) impurities in the

core. Using this constraint to eliminate composition from the equation

for density,

r ¼ r

 r

a 



T  T

½:

For almost all materials, the change of density with composition is suffi-

ciently large that b > aG. In this case, the normal density-temperature

relation is reversed; in a mush, cold fluid is less dense than warm.

In the core, pressure effects are important; and in the above formu-

las, one must replace the temperature by the potential temperature,

or equivalently, normalize the temperature with the adiabatic tempera-

ture (see Core, adiabatic gradient). While the actual temperature in the

core increases with depth, the normalized temperature decreases with

depth. (This is why the outer core solidifies at the bottom, rather than

the top.) The point to be emphasized is that this situation prevails in

the mush at the top of the inner core and the liquid phase is prone to

convective instability. Within the mush, thermal and chemical effects

are no longer independent and it is not proper to consider thermal

and chemical convections separately. However, given the dominance

of density changes due to chemical differences, convection is properly

characterized as chemical.

Chemical convection within mushy zones is commonly observed in

the casting of metallic alloys (e.g., see Worster, 1997). The situation in

Earth’s core is simulated when an alloy is cooled from below and the

denser phase freezes first. In this situation, convection within the mush

has a curious structure. Upward motion induces melting of the crystals

of the solid phase, thereby reducing the resistance to flow. This pro-

vides a positive feedback leading to the formation of discrete chimneys

in the mush. The cold, chemically buoyant fluid in the mush convects

by moving laterally to and rising up these chimneys; mass is conserved

by a downward return flow of warm, chemically denser fluid into the

mush away from the chimneys. It is uncertain whether chimney con-

vection occurs in the core, where both Coriolis and Lorentz forces

act on the moving fluid.

David Loper

Bibliography

Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability,

London: Oxford University Press.

Turner, J.S., 1974. Double-diffusive phenomena, Annual Review of

Fluid Mechanics, 6:37–56.

Worster, M.G., 1997. Convection in mushy layers, Annual Review of

Fluid Mechanics, 29:91–122.

Cross-references

Core Convection

Core, Adiabatic Gradient

and F layers of the Earth

Geodynamo

Geodynamo, Energy Sources

CONVECTION, NONMAGNETIC ROTATING

Various physical processes are responsible for maintaining motion,

possibly turbulent, in the Earth’s fluid core (see Core motions), which

is ultimately responsible for the geodynamo (q.v.). Here, however,

we focus attention on thermal convection (see Core convection) driven

by an unstable temperature gradient. Compressibility will be ignored

except where density variations lead to buoyancy forces, the so-called

Boussinesq approximation (q.v.). The basic model (Chandrasekhar, 1961)

generally adopted in theoretical studies concerns a self-gravitating

sphere, gravitational acceleration gr, in which fluid is confined to a

spherical shell, inner radius r

, and outer radius r

. Relative to cylindrical

polar coordinates ðs; f; zÞ, the system rotates rapidly about the z-axis

with constant angular V ¼ O

z ð0; 0; OÞ. The Boussinesq fluid, density

r, kinematic viscosity n, has coefficient of thermal expansion a and

thermal diffusivity k. The fluid is assumed to contain a uniform distribu-

tion of heat sources with thermal boundary conditions, which leads to a

spherically symmetric temperature exhibiting an adverse radial tempera-

ture gradient br.

74 CONVECTION, NONMAGNETIC ROTATING

The basic static equilibrium state just described may be buoyantly

unstable. The ensuing fluid motion, velocity u , causes temperature per-

turbations T of the basic state (see Core temperature ), which satisfy the

heat conduction equation

T ¼ b r  u þ kr

T ; (Eq. 1a)

where D

 ] = ] t þ u r is the material derivative: here b is a constant

for the case of heating by a uniform distribution of heat sources in a

full sphere but may be a function of the radius r for more general situa-

tions. In turn, the ensuing density variations ar T drive the motion

itself, which is governed by

u þ 2 V  u ¼rð p =r Þþg a T r þ nr

with r u ¼ 0 ; (Eq. 1b)

where p is the fluid pressure, and both r and g are constants. The vor-

ticity equation obtained by taking its curl is

z ð2 V þ z Þru ¼g a r rT þ nr

z ; (Eq. 1c)

where z ru is the relative vorticity (2 V þ z is the absolute vor-

ticity). Generally the inner and outer spherical boundaries are regarded

as isothermal ðT ¼ 0 Þ and impermeable ðr  u ¼ 0 Þ, whereas both

the cases of stress-free and no-slip boundaries are often considered

separately.

The physical system is characterized by four dimensionless num-

bers, namely the aspect ratio  , the Ekman number, the Prandtl

number, and the Rayleigh number:

¼ r

= r

; E ¼ n =2 O r

; P ¼ n =k ; R ¼ g ab r

= kn (Eq. 2)

These parameters arise naturally for the linear convection that occurs

at the onset of instability. For a given system such as the Earth ’ s core,

we must regard  , E, and P as given. Interest then focuses on what

happens with increas ing R. The initial bifurcation occurs at the so-

called critical Rayleigh number R

. The strength of the finite amplitude

that ensues is measured by some convenient parameter such as the

Rossby number U

O based on a typical velocity U

The ons et of instabili ty

The linear problem for the onset of instability has a long history. The

early studies (Chandrasekhar, 1961) focused on axisymmetric con-

vection. Later it was realized (Roberts, 1968) that, in the geophysi-

cally interesting limit of small Ekman number ð E  1 Þ, the onset is

characterized by nonaxisymmetric wave-like modes proportional to

exp ið mf  ot Þ with m ¼OðE

1= 3

Þ and ð r

=n Þo ¼OðE

 2= 3

Þ; they

occur when R ¼OðE

 4=3

Þ .

Since the rotation (as measured by the smallness of E ) is so rapid, the

flow is almost geostrophic; 2 V  u rðp= r Þ (see Eq. (1b)). True

geostrophic flow satisfies the Proudman-Taylor theorem (q.v. ) and is

independent of z , i.e., 2 V ru ¼ 0 (see Eq. (1c) ). In a spherical shell,

geostrophic flow is purely azimuthal and not convective. The system

overcomes this difficulty by occurring on the small Oð E

1= 3

Þ azimuthal

length scale upon which it is quasigeostrophic (i.e., independent of z on

the Oð E

1 =3

Þ length scale of the convection but dependent on z on the

Oð r

Þ length scale of the core radius):

p 2r OC; u rC

z þ W

z ; z ðr

C Þ

z þrW

(Eq. 3)

where r

r

 ]

= ] z

.Notethatru ¼ 0 does not imply that

] W = ] z ¼ 0. Instead it simply means that there are small extra components

of u,orthogonalto

z ,oforder E

1/ 3

W ignored in Eq. (3). Similar remarks

apply to the representation of z . Within the framework of these approxima-

tions, the linearized form of the z- c om po ne nt s o f t h e v o r t ic it y Eq. (1c) and

t h e e qu at io n o f m ot io n ( 1 b ) a re

ð]

 nr

Þr

C þ 2O ]

W ¼ g a]

T (Eq. 4a)

ð]

 nr

ÞW  2 O ]

C ¼ g a zT ; (Eq. 4b)

respectively, where ]

 ]= ] t etc. The corresponding form of the heat

conduction Eq. (1a) is

ð]

 k r

Þ T ¼ bð ]

C þ zW Þ: (Eq. 4c)

Correct to leading order it is sufficient to solve the system (4) subject

to the impermeable boundary condition

r  u  ]

C þ zW ¼ 0on r ¼ r

and r

; (Eq. 5)

namely the inner and outer boundaries.

In the early solutions of Eq. (4) for a full sphere, it was assumed that

motion is localized in the vicinity of some cylinder s ¼ s

on a short

radial scale, yet long compared to the azimuthal length scale

Oð s

1 =3

Þ. Accordingly, the approximation r

¼ s

2

]

was also

made (Roberts, 1968; Busse, 1970). The localized convection has a

column ar structure, for which the evolution of its axial vorticity

r

C is governed by Eq. (4a) . The buoyancy torques g a ]

T that

drive motion are most effective near the equator, where the radial

component of gravity is strongest. Nevertheless, the large tilt of the

boundaries at the ends of the column necessarily break the constraints

of the Proudman-Taylor theorem and inhibit motion. This effect is

quantified by the term 2O]

W , which describes the stretching of abso-

lute vorticity and is minimized at the poles where the boundary tilt

vanishes. In consequence, the preferred location of the convection is

at mid latitudes, where s

is roughly a half.

A heuristic appreciation of the solution is obtained by making the

geostrophic ansatz that C and T are independent of z (i.e., assume

Eq. (4b) is simply 2O]

C ¼ 0) and that W is linear in z, which accord-

ing to the boundary condition (5) implies W ¼ðz=h

Þ]

C, where

h ¼ðr

 s

1=2

is the column half height. Linked to these heuristic

assumptions is the neglect of W in Eq. (4c), which then leaves the

model equations (Busse, 1970)

ð]

 nr

Þr

C  2ðO=h

Þ]

C ¼ ga]

T; (Eq. 6a)

ð]

 kr

ÞT ¼ b]

C (Eq. 6b)

In the absence of viscosity n ¼ 0 and buoyancy forces gab ¼ 0,

Eq. (6a) possesses eastward propagating Rossby wave solutions,

which under the long radial length assumption r

¼ s

2

]

have

frequency o ¼ 2Os

m. When dissipation and buoyancy are rein-

stated, the dispersion relation becomes

ðP

o þ ik

Þ½ð

o þ ik

ÞE

1

sð1 

1

=k þ

R ¼ 0; (Eq. 7)

where we have introduced the dimensionless parameters

o ¼ r

o=n,

k ¼ r

m=s and

s ¼ s=r

. The onset of convection occurs at the

minimum value of R over

s and k subject to the constraint that

o is

real. This recovers our earlier critical value estimates

 0:5,

¼OðE

1=3

Þ,

¼OðE

2=3

Þ, and R

¼OðE

4=3

Þ (Busse, 1970).

The marginal waves, like the Rossby waves travel eastward, and

indeed have the Rossby wave character (not to be confused with an

alternative class of solutions, namely the equatorially trapped inertial

waves discussed later) in the small Prandtl number limit. For that rea-

son, the waves that occur at finite P are referred to as thermal Rossby

waves.

CONVECTION, NONMAGNETIC ROTATING 75