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The Earth's magnetic fieldDavid Gubbins aa Bullard Laboratories, Department of Earth Sciences ,Madingley Rise, Madingley Road, Cambridge, CB3 0EZPublished online: 20 Aug 2006.
To cite this article: David Gubbins (1984) The Earth's magnetic field, Contemporary Physics,25:3, 269-290
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CONTEMP. PHYS., 1984, VOL. 25, NO. 3, 269-290
The Earth’s Magnetic Field
David Gubbins, Bullard Laboratories, Department of Earth Sciences, Madingley Rise, Madingley Road, Cambridge CB3 OEZ
ABSTRACT. There is a good observational record of the history of the Earth’s magnetic field throughout geological time. The physical processes occurring in the liquid core, where the field originates, are very complex however, and they are only just beginning to be understood. Recent progress has been made towards undelftanding the energetics of the core, which is a problem in thermodynamics; the dynamics, which is a problem in fluid mechanics in which magnetic forces are important; and also the slow variations of the magnetic field with time.
1. Introduction 1.1. Measurements of the Earth’s magnetic j e l d We know a remarkable amount about the history of the Earth’s magnetic field, which spans an enormous interval of time. As far as we can tell the field has existed in more or less its present form, that of a bar magnet aligned with the rotation axis, for virtually the whole of geological time, and yet it has undergone occasional rapid changes including complete reversals of polarity. This review is concerned with the physical principles underlying the magnetic field’s behaviour, but we begin with a very brief summary of the techniques used in measuring and inferring the past field, and their results.
Geomagnetism entered the space age in the 1960s with the launching of satellites to measure the total intensity of the field in high altitude orbits. These were followed, in 1979, by the much more sophisticated MAGSAT satellite, which measured all three components of the field at a lower altitude.
The satellites have given a very complete picture of the field in modern times, but very little information on the time changes. Good, permanent magnetic observatories have run for the last hundred years or so, and these are the main source of information on the slow time changes of the field, called the secular variation. Observatories give good data but poor geographical coverage, and have to be augmented by measure- ments at temporary (‘repeat’) stations and at sea. The main difficulty lies in maintaining a good baseline at a site for long periods of time.
Apart from some very early measurements in China, direct observations of the magnetic direction date from about 1550, when they became necessary for navigation. Absolute measurement of field strengths were not made until much later, in the early nineteenth century. This historical record, although incomplete and inaccurate, provides quite detailed evidence of the extent of secular variation.
Indirect inferences of the field make use of the magnetization of rocks and artefacts which may retain information about the field direction and strength at the time they were last magnetized. Archeornagnetism is now a well-established discipline (Aitken 1974) and magnetization can be used as an aid to date artefacts. Unless remains are found in the place where they were last magnetized, the directional information is lost; the present decay in the strength of the Earth‘s dipole moment has been going on for some time.
Sediments preserve the direction of the field when they are laid down, and a sedimentary column may be regarded as a tape recording of the changes in field direction. Lake sediments form rapidly and give a ‘high fidelity’ tape recording; marine
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sediments form more slowly. Lake sediments contain evidence of secular variation going back tens of thousands of years (Creer 1981) but the picture is very incomplete. Marine sediments contain a more complete record including polarity reversals, which appear to have occurred at random, at intervals of a million years or so, with constant polarity between each reversal.
Sediments do not make very good tape recorders, however, and it has not yet been possible to obtain reliable intensity information from them. Lavas become magnetized when they cool and they do sometimes preserve the intensity as well as the direction of the ancient field. Lavas only erupt spasmodically and the record is not complete; dating is very difficult, and it is often impossible to establish where a measurement should be placed in relation to the reversal time scale. Some lava sequences have preserved records of the field in transition during a reversal (e.g. Shaw 1975). The field appears to drop in strength over a period of ten thousand years or so, to flip polarity comparatively quickly, and grow slowly in strength again. There is a suggestion in recent work that the intermediate field retains symmetry about the rotation axis (Hoffman 1979).
As the reversal record becomes more complete, more ‘events’ and ‘excursions’ have been found. An ‘event’ is a reversal of very short duration; an excursion is best described as an ‘aborted reversal’, beginning as a reversal but reverting to the original polarity rather than changing sign altogether.
1.2. The Earth’s interior The magnetic field is known to originate inside the Earth. The properties of the deep interior have been worked out in some detail using a varied combination of geophysical and geochemical arguments. Seismology, the study of elastic waves set off by earthquakes and large explosions, has yielded accurate information for a few physical parameters: the density, pressure, gravitational acceleration, bulk and shear moduli and anelasticity (Q). Other parameters are very poorly known and rely on theoretical or indirect arguments. The pressure at the centre of the Earth reaches several megabars (3.6 x 1011 Pa), which is too high to be achieved in the laboratory, so that equations of state are a matter for some speculation.
The major zones of the Earth that we need to consider for the present purposes are shown in figure 1. The inner core-outer core boundary is marked by a seismic discontinuity but the radius (121 5 km) is hard to tie down with precision. The inner core is hclievcd to be solid, the main evidence coming from free oscillations of the Earth (Masters and Gilbert 1981). The outer core IS liquid: it does not transmit shear waves. The liquid is mainly iron. Iron is the only element of sufficient abundance with the right density. High pressure experiments give a higher density for pure iron than holds in the outer core, and so the core is supposed to contain a few percent of lighter elements such as sulphur, oxygen or silicon. The density jump across the boundary with the inner core is in the range 0.2-1 gcm-3 from free oscillation data, which is too high to be due to a melting transition. The inner core probably contains a higher proportion of iron. Further details of the properties of the core are to be found in Jacobs (1975).
Temperatures in the core are controlled by the melting points of iron and the silicates of the overlying mantle. They are of the order of a few thousand degrees. The temperature in the crust and upper mantle can be deduced from mineral processes. The Curie point is exceeded at quite shallow depths, so that permanently magnetized material exists only in the crust.
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The Earth's magnetic j e l d 271
Figure 1. The magnetic field originates in the outer core, which is mainly liquid iron. The inner core is probably solid iron, and the mantle is composed of poorly conducting silicates. Most of the region beneath the crust is at too high a temperature to be magnetized.
We shall also need estimates of the transport properties of the liquid iron in the core. CJ, K and v (electrical and thermal conductivity and viscosity) are estimated from the fluid state theory, and are very uncertain. K is derived from c using the Wiedemann-
units. They are uncertain by a factor of 3 or more. The mantle can be treated as an electrical insulator for present purposes. The
conductivity of the upper mantle, down to 700 km, has been estimated from the electric currents induced there by external variations of the magnetic field, such as magnetic storms. At 700 km a sharp rise in conductivity occurs. The lower mantle may even be a semiconductor. The conductivity cannot be too high or the secular variation would be screened out, and in all probability the conductivity is two or three orders of magnitude down on that of the core.
Franzlaw,Valuesusuallyquotedare:rr=5 x lo5 Sm-',rc=25 Wm-' K - ' , \ I = = lo-6sI
1.3. Theory f o r the generation of the magnetic j e l d Any satisfactory theory for the magnetic field must answer two questions that arise from the behaviour of the magnetic field throughout geological time. Why does the field fluctuate so rapidly; and why has it not decayed away due to resistive losses in the core? Our best estimate for the electrical conductivity of the core suggests a decay time constant of about 10000 years, which is very short compared with the age of the Earth of 4 6 x lo9 years.
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The rapid time fluctuations are attributed to motions of the core fluid. The secular variation was known from very early times and Halley, in a remarkable paper that was written 300 years before the seismological discovery of the Earth’s liquid core, attributed the variations to a mobile interior of the Earth, although he thought in terms of moving, permanently magnetized rocks rather than induction by a conductor.
The dynamo theory, which asserts that induction will not only cause the field to fluctuate but will also act to sustain it against decay, is due to Larmor. Other suggestions have been made, but work done over the last hundred years has left only the dynamo theory as a viable proposition. A review and critique of theories that have fallen by the wayside has been given by Stevenson (1974).
The dynamo theory has now been worked out in more detail. The first issue to be settled was to establish that a homogeneous mass of fluid could act to put energy into the magnetic field. This so-called kinematic dynamo problem, which involves choosing some suitable fluid flow and determining its effect on a magnetic field without regard to the forces driving the flow, was finally solved theoretically for highly idealised models in 1958, and subsequent years have seen the construction of an increasing number of more realistic dynamo models.
The nature of the fluid flow in the core is a problem in fluid dynamics that has also received much recent attention; the Coriolis and magnetic forces are crucial in determining the flow. The dynamical dynamo problem involves the study of a complete system in which the field strength is regulated by the magnetic force reacting on the fluid motion. This formidable problem has been tackled by a few authors only very recently.
Any dynamo mechanism requires an energy supply, and the supply must be reasonable in the context of the Earth’s total heat budget. The energy balance can be studied using elementary thermodynamics. An early calculation was that of Bullard (1950) but there has been a revival of interest in the last decade. The separation of the theory into the kinematic dynamo, convection in the core, the dynamical dynamo and the energy source is a natural one. This paper does not follow the historical development of the subject, but deals with the justification of various parts of the theory in turn: first the energy supply must be adequate; secondly the fluid motion must be realistic (it must act as a dynamo and some account must be taken of the influence of the magnetic forces on the flow); finally, in section 4 we describe the etforts made to understand and interpret the time variations of the field.
1.4. The mathematical formulation The development of the dynamo theory for the Earth has amounted to the building of simple models. These models have to be theoretical because it is not possible to do meaningful laboratory experiments with magnetic fields and liquid conductors; the materials that are available simply do not conduct well enough for the experiment to be of a reasonable size. The lack of support from experimental observations has meant that the theories have been pursued with more mathematical rigour than would otherwise be the case.
The core is treated as a conductor that satisfies Ohm’s law for a moving medium
J=o(E+v x B) (1)
where J is the current density, B the magnetic field, and v the fluid velocity; it must also satisfy Maxwell’s equations although effects due to moving charges are neglected. This
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approximation is valid provided that v is much less than the velocity of light, as IS the case here. The equations are usually combined into the induction equation
dB/dt +(v.V)B=(B.V)v+(p,o)-’V2B (2)
Physically this equates the rate of change of B moving with the fluid, on the left hand side, and induction and diffusion on the right hand side. The kinematic dynamo problem involves solving (2) for B when v and ~7 are specified, and demonstrating that B does not decay with time.
The fluid dynamo of the core is usually modelled in the Boussinesq approximation, in which all effects of density variations except buoyancy are neglected. The equation of motion can be written as
p(dv/at + v VV + 2Q x V ) =(J x B)- V p + ~ v V ’ V + F (3)
where 2Q x v is the Coriolis force, J x B the magnetic force, v the kinematic viscosity and F is some force driving the fluid, such as the buoyancy of hot fluid in thermal convec0ion. In this case another equation for the temperature would be required. A study of convection involves solving ( 3 ) for v with B specified; the complete dynamical dynamo involves solving equations (2) and (3), with other necessary equations and boundary conditions, for v and B together.
For the Earth’s core it is most likely that the effects of viscosity and inertial terms are very small, leaving a force balance mainly between the Coriolis, magnetic, and applied forces. and the pressure. The relative importance of the various terms is measured by dimensionless numbers. For the core the Ekman number (the ratio of viscous to Coriolis forces) is and the Rossby number (inertial term: Coriolis force) is 10- ’. In the induction equation (2) the ratio of inductive to diffusive terms is measured by the magnetic Reynolds number. A fundamental requirement of the dynamo is that these last two terms be comparable, so that the decay of the field due to diffusive effects may be overcome by induction. A rough estimate gives a magnetic Reynolds number of 100 for the Earth’s core.
This very brief summary of the mathematical problem shows that it is a very difficult one. It contains all the complexity of the problem of finding a dynamical model of the atmosphere for weather prediction, but with the additional complication of a magnetic field. Progress has been made by studying simple systems rather than by an all-out attack on the full dynamical dynamo problem. For a detailed treatment of background to the relevant magnetohydrodynamic theory see Roberts (1 967).
2. The energy source 2.1. The thermodynamic approach Using thermodynamics we can study the energy balance for the dynamo without going into too much detail about the fluid flow or how the magnetic field is generated. For example, Bullard (1950) suggested that radioactive elements present in the core would provide enough heat to drive thermal convection, and that these convective motions would generate the magnetic field. The recent thermodynamic approach has been to regard the whole core as a heat engine, and to relate the magnetic field generated to the power input.
The first law of thermodynamics tells us that, in steady state, the energy supplied is equal to that flowing out, namely heat flowing into the mantle. The surprising fact is
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that neither the magnetic field nor the electrical heating enter into the global energy budget. The crucial point is that Ohmic losses remain inside the core as an additional heat source. Power estimates based on equating the supply with the electrical heating are fallacious.
One can argue on the basis of an exchange of entropy. Dissipative effects such as electrical heating always lead to an increase in entropy. Heat sources can balance this increase provided that heat is supplied to regions of high temperature. For the dynamo driven by thermal convection there is the simple inequality
(Backus 1975), where T is the temperature, Q is the total heat flux out of the core and 0 is the electrical heating given by:
The factor on the right hand side of (4) differs from the usual ‘Carnot’ efficiency factor, which has T,,, in the denominator and not Tmin, and it can exceed unity. This does not violate the first law because the heating, 0, remains inside the system.
Another important effect besides Ohmic heating, acting to increase the entropy and therefore impair the efficiency of the dynamo, is thermal conduction. The vigorous fluid motions required to generate magnetic fields act to stir the core into a state that is very close to adiabatic. The pressure gradient is known from seismology and an adiabatic temperature gradient can be estimated from the properties of iron. It amounts to 0-1 1 K per km, which leads to conduction of a large quantity of heat.
I t is a straightforward matter to derive equations relating diffusive contributions to the entropy as well as those from heat sources (Gubbins et al. 1979). Numerical estimates of all these quantities are difficult to make. CD is found from (5) but we have little idea how large J may be deep within the core. Gubbins et al. (1979) find @z 10’ ’ W with corresponding Q z 1013 W. This is 25% of all the heat issuing from the Earth’s surface, a very large proportion, and is probably too high to come from radioactive decay. Verhoogen (1 961) suggested that the heat may be supplied by the cooling and gradual freezing of the liquid outer core to form and enlarge the solid inner core, but 1013 W requires a high rate of cooling, and carries the implication that the inner core formed comparatively recently. There is a shortage of heat to drive the dynamo!
2.2. The gravitationally powered dynamo Braginsky (1963) extended Verhoogen’s ideas on core cooling by adding effects due to the outer core being a mixture of iron and some lighter constituent (then thought to be silicon). Freezing was supposed to separate a heavy fraction, containing a higher proportion of iron, explaining the greater density of the inner core, and leaving behind a light liquid fraction in the outer core that would be buoyant and lead to ‘composition- ally driven’ convection. This would drive the dynamo in similar fashion to thermal convection.
The energy supply in this case comes from the gravitational energy of the Earth itself, which is reduced by rearrangement of mass. There is a fundamental difference between the thermodynamics of this gravitationally-powered dynamo and the heat- driven dynamo. While convection can transport heat through the system with great
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The Earth’s muynetic jield 275
efficiency, but without necessarily generating any magnetic field at all, the potential energy liberated from compositional convection can only escape the system by being converted first to kinetic energy and then to heat via some dissipative process. Since electrical heating is a major dissipative process in the core, viscosity being so small, a great part of the gravitational energy released will go into electrical heating. The gravitationally powered dynamo is therefore much more efficient than the heat-driven dynamo in that the same magnetic field can be sustained with less energy.
There is still the need to maintain an adiabatic temperature gradient in the core, and thermal conduction provides a sizeable entropy gain that impairs the dynamo efficiency. It is not possible to separate the discussion of the gravitational energy from that of the heat sources in the core, which will also contribute to maintaining the adiabatic gradient. There is also a new effect, due to molecular diffusion of the lighter elements through the liquid of the outer core, that will act to increase entropy. The magnitude of this effect is unknown, but it may be large, and it also acts to impair the efficiency of the dynamo because diffused iron will not drive convection or generate a magnetic field.
Gubbins et al. (1979) estimate that a modest magnetic field could be sustained by gravitational energy with a net Q of 5 x 10” W. This would not entail rapid freezing of the core, unlike the thermal dynamo. The calculations depend critically on the exact size of the density jump across the inner core-outer core boundary, which determines the contrast in composition between the two cores. Other calculations for the gravitational dynamo and other effects such as the loss of the Earth’s rotational energy are discussed in the review of Gubbins and Masters (1979).
2.3. The thermal state of the core The importance of having a liquid mixture in the outer core has prompted speculation on effects that may arise in a mixture. Three temperature gradients are important: the actual temperature gradient, the adiabat and the melting point gradient.
If the temperature gradient is less than the adiabatic then convection of any sort is impossible, and only horizontal motions and oscillations occur. Loper (1978 a) has pointed out that compositional convection will maintain the adiabat by stirring the core, and in the absence of separate heat sources, heat will be carried downwards by the compositional convection, and back up by thermal conduction down the adiabatic gradient. A number of interesting effects may occur in this doubly-diffusive system. Again, Fearn and Loper (1981) have suggested that the light elements may distribute themselves in such a way as to produce a stable density gradient in certain parts of the core, where convection could not occur.
It is quite possible for convection to carry fluid into a zone where the temperature is below the melting point. In this case two-phase convection results, rather like the process occurring during a thunderstorm, and convection takes place in a slurry at the melting point (see, for example, Busse 1972). One region where this might well occur is just above the inner core boundary, and a slurry may occur in the boundary layer (Loper and Roberts 1981).
The cooling rate at the inner core boundary may be favourable for the formation of dendrites (Fearn et al. 1981). This would suggest a ‘soft’ inner core, at least near its surface, with many fluid inclusions. Seismic evidence for Q in the inner core is conflicting (Doornbos 1974 and Masters and Gilbert 1981).
Unfortunately there is very little direct evidence for any of these interesting effects. For the time being, at least, theoretical speculation has outstripped observation.
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Figure 2. Some features that may exist in the core:
1. Solid iron inner core. 2. ‘Soft’ outer shell of inner core with fluid inclusions due to dendritic growth.
3,5. Slurry, liquid and solid phases mixed at the solidus temperature. 4. Liquid iron and some lighter elements such as oxygen, sulphur. 6 Region of liquid core where convection may not occur, either because the temperature
gradient is below the adiabat or there is an accumulation of light material at the top of the core.
2.4 The thermal history of the core It will have become apparent to the reader by now that modern theories for the power supply to the dynamo are intimately linked with the evolution of the core. In fact the magnetic field provides useful constraints on the thermal history of the Earth as a whole. There must always have been a sufficiently large throughput of heat to maintain a field, and yet the cooling rate cannot have been so rapid as to lead to freezing of the entire core.
Cooling of the core is controlled by the temperature in the mantle which is itself believed to cool by convection, although the convective velocities are very much slower than those in the core. The mantle has a high heat capacity and its cooling is probably largely independent of the core!
Recent work on the core has treated the two as independent: Loper (1978b) assumed a constant heat flux across the core boundary while Gubbins et al. (1979) assumed a known cooling rate there. If mantle cooling is too slow the temperature difference across the whole core may become less than that required to maintain the
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adiabat. Either convection will be driven compositionally, or part of the core ceases to convect. Gubbins et al. (1982) calculate that the upper few hundred kilometres of the core may become stable if the mantle were to stay at a constant temperature. Such stability would affect the periods of free oscillations and so could, in principle, be detected seismologically.
A full study of the thermal evolution of the core requires that core and mantle be considered together. Mollett (1983) has considered the effect of a two-layer convecting mantle and the core model of Gubbins et al. (1 979). His calculations suggest that it may be possible to drive the dynamo by heat alone if the inner core is a recent feature. Initial cooling rates are very high because the high temperatures lead to vigorous mantle convection. It is therefore possible to supply enough heat to power a thermal dynamo before formation of the solid inner core allows compositional convection to take over
3. The dynamo problem 3.1. Convection in the core The dynamics of convection in the core will probably be the same whether buoyancy is of thermal or compositional origin, and almost all detailed work has concentrated on thermal convection. The important terms in equation (3) are the Coriolis, magnetic and pressure forces, as well as the buoyancy force F driving the motion.
There are two important theorems governing the fluid mechanics of a rapidly rotating system with a magnetic field. The first is the Proudman-Taylor theorem, which applies in the absence of magnetic fields, and states that fluid motion cannot vary with the coordinate along the rotation axis. G. I. Taylor devised a very elegant experiment to demonstrate this. He showed that when a short cylinder is towed along the bottom of a rotating tank of water the whole column of water above the cylinder is dragged along with it.
Magnetic forces can exert torques on the fluid. It is impossible for torques to be opposed by a pressure gradient, nor can torques aligned with the rotation axis be opposed by Coriolis forces. The magnetic field will only be consistent with a force balance between Coriolis, magnetic and pressure forces if these torques vanish. This condition is usually called J. B. Taylor's condition (Taylor 1963) and is written as
j, (J x B),dS = 0
where S is any cylindrical surface whose axis is along JL. If (6) is not satisfied there will be rapid acceleration of these cylinders of fluid until some field configuration is achieved which does satisfy (6).
The importance of buoyancy is measured by the dimensionless Rayleigh number, which gives the ratio of buoyancy to diffusive effects and which must exceed a certain critical value before convection sets in. Work on convection begins with finding the critical Rayleigh number and the pattern of convection at onset. Coriolis and magnetic forces, acting separately, inhibit convection, that is they increase the critical Rayleigh number, but the two forces applied together are destabilizing and produce a lower critical Rayleigh number (Chandrasekhar 1961). This is because the two forces oppose each other to some extent, rather than opposing the buoyancy forces.
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The effect of rapid rotation on convection in a sphere was studied by Roberts (1 967) and Busse (1970) by analytical means, and by Busse and Carrigan (1976) experiment- ally. Convection sets in as rolls which are aligned with the rotation axis due to the ProudmawTaylor theorem (figures 3, 4). The motion cannot be two dimensional everywhere because of the effect of the boundaries at each end of a roll. The result is that the rolls become small and drift in a prograde direction (in the same sense as the rotation), the enhanced viscous and inertial forces breaking the Proudman-Taylor constraint. The drift rate depends on the inclination of the boundary, which becomes steeper away from the axis, and so the rolls are not round but take on the curved shape shown in figure 3. For a further discussion of this nonlinear theory see Soward (1977). Numerical calculations have given more detail and confirmed many of the general results (Gilman 1977).
The Earth’s core is believed to contain a strong toroidal magnetic field. This field is not directly observable in the insulator outside because it is associated with radial electric currents. A very large toroidal field has been detected on the surface of the Sun. I f the toroidal field is many times larger than the dipole field, then magnetic forces will be strong enough to seriously influence the fluid flow. The fluid flow tends to align itself with very strong fields, leading to a more axisymmetric structure than that of figure 3.
i
4 .
Figure 3. Convection rolls in a rapidly rotating sphere are aligned with the rotation axis and have a characteristic curved cross section. The rolls drift in the same direction as the rotation (shown by arrow) due to the influence of the boundaries.
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(4 Figure 4. Convection rolls in a perspex sphere of water that is rotating at several hundred rpm.
The water contains particles that align with the flow and reflect light, making the flow pattern visible View (a) is along the rotation axis, (b) from the side.
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The problem of convection in a rotating sphere with an imposed toroidal field has been studied recently and the work is reviewed by Busse (1983). Again, the magnetic force can act to oppose the Coriolis force and allow convection at a lower Rayleigh number. It is not known which regime holds in the core.
3.2. Kinematic dynamo theory Having discussed the sort of fluid motion that might be expected in the core, we can investigate the flows for dynamo action. Dynamo theory experienced a breakthrough in the 1960s, and now realistic working dynamos are the rule rather than the exception. This work has already received extensive review and further details are to be found in the books by Moffatt (1978), Krause and Radler (1980) and Parker (1979).
It is well known that early work on the dynamo yielded negative results in the form of anti-dynamo theorems: it is impossible to have a dynamo with an axisymmetric, steady magnetic field (Cowling 1934) or a purely toroidal velocity (Bullard and Gellman 1954). These results are central to dynamo theory but are no longer regarded as a block to progress; the convection is expected to have both radial motion and to be non-axisymmetric. There has been some recent interest in Cowling’s theorem and Hide and Palmer (1982) have succeeded in generalizing the proof to cover time dependent magnetic fields and compressible flow with variable electrical conductivity.
Two approaches to the dynamo problem have been successful: both involve averaging the induction equation in some way so as to produce an equation governing the averaged field. The Braginsky dynamo involves an average over longitude, 4, and relies on small departures from axial symmetry to sustain the field. The averaged field is axisymmetric. The turbulent dynamo, and the closely related work on the dynamo action of motions that are periodic in space, involves an average over a volume in space-the lattice cell of the motion for periodic flows. The average field is one that is smooth in space, but it is not subject to Cowling’s theorem and can be axisymmetric.
Let a bar denote an average. The averaged induction equation (2) may be written as
aBpt = R,(V x E + v x (V x B) + v2B (7) E is the mean emf generated by the non-axisymmetric or turbulent flow. In simple cases E may be written as
E=aB
Thus there is a mean emf induced parallel to the mean field, which is of course contrary to ordinary induction in which the induced emf is perpendicular to both the magnetic field and the direction of motion of the conductor.
The presence of the additional emf E means that V and B are no longer subject to the anti-dynamo theorems. In particular, B can be axisymmetric and V toroidal. An analytical solution to (7) is available when CI = constant, V = 0.
In the case of turbulence CI is related to the helicity of the flow h.
h measures the degree to which streamlines are twisted into a helical form, hence the name helicity. Helical motions are now recognized as being efficient for field generation, but they are not essential: dynamo action has been proved for some flows that have no helicity.
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The Earth’s magnetic field 28 1
Figure 5. Differential rotation (solid lines, 5 (a)) can act to twist a dipole field into a toroidal field. Reconnection (5b-c) occurs with diffusion. The toroidal field has opposite sense in the two hemispheres. Although the field is intensified in this way, it will ultimately decay unless the dipole is regenerated by some other inductive process, such as the a-effect.
Much effort has gone into solving equation (7) for various choices of M and V. Analytical solutions are very rare, and most of the work involves numerical calculation. Helicity in the Earth’s core is due to rotation and z is expected to have opposite signs in the northern and southern hemispheres. tl =ao cos 0, where M~ is a constant and 8 the co-latitude, is a common choice. V includes the large scale motions, the most important of which is differential rotation of the core liquid, which can produce a large toroidal field from the dipole (figure 5). The mean emf crB can also produce toroidal field, but it alone can produce dipole field from toroidal field, thus completing a cycle by which the field can be regenerated.
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282 D. Gubbins
Figure 6. Streamlines of an axially symmetric flow that generates a magnetic field by dynamo action, after Gubbins (1973). Streamlines take the form of helices wrapped around the axis of symmetry, giving the flow a large helicity.
Dynamos that rely on differential rotation are described as am-dynamos; those that rely on c i alone are of a2-type. a2-dynamos tend to give steady fields whereas am- dynamos give oscillatory fields. A t first the 2’ model was proposed for planetary fields and the X(’J model for the Sun, but this is certainly an oversimplification. Adding an auisymmetric meridional circulation, like a large convective cell, can produce steady liclih from x , ) dynamos. Meridian circulation can also determine whether the dynamo produccs ;I predominantly dipolar field or a quadrupolar field (Proctor 1977). Other studies have examined more complicated forms for the mean emf E (Busse and Miin 1979 and Krause and Radler 1980).
There are now some numerical solutions of the full induction equation that do not rely on either the Braginsky or the turbulent dynamo formalism. Fluid motion with axial symmetry can act as a dynamo provided that the magnetic field itself is not axisymmetric. If v is independent of # then equation ( 2 ) has solutions for B proportional t o exp(irmh). Other fields can grow with time (Gubbins 1972). The fluid motion is helical, the streamlines curving around the axis of symmetry as in figure 6. The motion is similar to that of the turbulent dynamo. Kumar and Roberts (1975) have achieved the
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The Earth’s magnetic Jield 283
goal set by Bullard and Gellman (1954) by showing that differential rotation and simple convective cells can maintain a mainly dipolar field. Their model is similar to the Braginsky dynamo.
3.3. Dynamical dynamos In the kinematic formulation the magnetic field can continue to grow indefinitely because no account is taken of the back-reaction of the magnetic field on the fluid flow. A major goal of work on the dynamical dynamo is to achieve a state of equilibrium, if one exists. Equilibriation can come about because the fluid velocity is slowed down by the magnetic forces to a point where the magnetic Reynolds number is maintained at the critical value without much change in form; or it may lead to a more drastic change in flow pattern to one that is less efficient at generating magnetic field. This distinction can be used to classify the dynamical models studied so far.
Busse (1975) has given a complete dynamo model based on thermal convection. The convection rolls (figure 3) contain flow along them, due to the sloping boundaries. The overall flow is helical and gives a periodic dynamo with a dipole field and a modest toroidal field. The magnetic forces are included in the equation of motion under the assumption that the field strength is small, and an equilibrium solution is obtained by asymptotic methods. Busse (1976) went on to propose equilibrium field strengths for the planets. Gubbins (1975) calculated solutions ofa dynamo driven by a constant body force by numerical means.
The numerical calculations are very time-consuming and have some limitations. The extremes of high rotation, Rayleigh number and magnetic fields all lead to solutions that are very difficult to represent accurately in space and time, and in practice the ranges of parameters that can be studied is quite limited.
The average magnetic field generated by turbulence, or by the Braginsky dynamo, will exert a force and modify the mean flow. This mean flow can inhibit the dynamo process. Malkus and Proctor (1975) discuss a simple cr-dynamo in which c1 is held constant and unaffected by the magnetic forces. Their work suggested that equilibrium would be achieved with a mean field that satisfies the Taylor constraint, equation (6), so that viscous and inertial forces are unimportant. However, subsequent models, particularly ‘model z’ of Braginsky (1976, 1978), have all involved viscous or inertial effects, in some way, in the equilibrium state.
3.4. Oscillations Long term oscillations in the magnetic field, such as the present change in dipole moment, probably reflect the behaviour of the whole dynamo. A dynamical model of a dynamo may exhibit oscillations about its equilibrium state (e.g. Gubbins 1975). Busse (1982) has studied oscillations of a plane layer dynamo. These nonlinear oscillations are likely to be a permanent feature of the Earth’s dynamo and can occur spontaneously in simple systems. They may be related to geomagnetic reversals.
Theoretical work on reversals has been restricted to very simple models. It is now known that Bullard’s disk dynamo exhibits reversals, some of which bear a remarkable resemblance to true geomagnetic behaviour. Work so far has concentrated on studies of very simple reversing systems rather than on accurate modelling of the Earth. For references see Krause and Roberts (1 98 1).
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4. Secular variation 4.1. Theories for secular variation The secular variation has a time constant of a thousand years or less, and is supposed to occur separately from the basic dynamo processes. Theories of the secular variation have, for the most part, involved dynamo theory only in that they require a field and fluid flow as a basic state.
The dominant feature of the secular variation has, for the past few centuries, been a westward drift. Bullard et al. (1950) suggested that this was caused by rotation of the mantle relative to the core, and attributed the relative rotation to the action ofmagnetic forces on the weakly conducting lower mantle.
The secular variation is more complex than a simple rotation of the field. Recent work has concentrated on wave motions and instabilities that may take place in the core. There is a review of this work by Acheson and Hide (1973). If we suppose the core to contain a strong toroidal field of about 100 gauss, neglect diffusion and viscosity, and assume plane wave solutions of the form exp i(k * x - tot) , then equations (2) and (3) yield the dispersion relation
,
In the core we have
and one of the roots of equation (10) becomes close to (2R. k)/k, representing inertial waves slightly modified by magnetic field. The other root is
w -(B * k)2k/2(R. k)pop
It represents ‘slow’ waves with periods of up to a thousand years. These waves, which are similar to Braginsky’s MAC waves (Magnetic, Archimedean, Coriolis), are associated with the secular variation. These disturbances show a preference to propagate westwards rather than eastwards, which could account for the westward drift (Acheson and Hide 1973). It is also possible for both this mechanism and the Bullard theory to be operating together. The coupling of the core to the mantle, which is mostly due to magnetic forces, is another feature that may influence the secular variation. The topography of the core mantle boundary will have an influence on the flow, in the same way that mountains on the earth’s surface can influence the weather. Hide and Malin (1970) have found a correlation between the gravity field, which may reflect undulations on the core-mantle interface, and the magnetic field.
Neither the theories of the origin of secular variation, nor the observations, are complete enough to allow a proper comparison at this stage.
4.2. Kinematics of the secular variation Some progress may be made in interpreting observations by treating the secular variation kinematically and ignoring the dynamics. Roberts and Scott (1965) addressed the problem of finding the actual fluid motions at the core surface. If electrical diffusion may be neglected, then the field lines are ‘frozen’ to the conductor and must move with it, thereby acting as a tracer for the flow. Diffusion will be negligible for sufficiently
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Figu
re 7.
C
onto
urs w
here
the
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al c
ompo
nent
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by
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ove
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. Thi
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Def
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65, d
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286 D. Gubbins
short periods of time (much shorter than the diffusion time scale of about 10 000 years), and for sufficiently long wavelengths. The approximation seems reasonable at first sight.
If the core is treated as a perfect conductor, current sheets may form making tangential components of B discontinuous across the boundary. Attention has therefore been confined to the radial component, B,, and equation (2) becomes
dB,/dt + V h . (vB,) = 0 (12)
where Vh denotes the horizontal gradient, diffusion has been neglected, and equation (12) applies at the core mantle boundary where v,=O. The problem posed by Roberts and Scott (1965) was to find v from equation (12) given measurements of B,.
Equation (12) can have many solutions for v (Backus 1968). There are also restrictions on the field for equation (12) to have any solution at all. These represent restrictions due to the assumption of perfect conductivity. First, curves on which B, vanishes are permanent features: they cannot merge, disappear, or form, although they can move about. There appear to be about five of these ‘null flux’ curves on the core boundary at present (figure 7). Furthermore, the total flux through the areas bounded by null flux curves cannot change. For an area Si this condition can be written as
B,dS=O Is. (13)
A whole class of velocities that are solutions to equation (14) can be found provided (1 3) is satisfied.
On null flux curves equation (12) reduces to
We are able to find the component of v perpendicular to the null curve (that is, along V , B,) from equation (1 4).
4.3. Finding the secular variation at the core mantle boundary Finding core motions requires a knowledge of the field and its rate of change at the core mantle boundary, which involves extrapolating the observations down through the mantle. The mantle is usually assumed to be an insulator for this purpose (although there has been some recent interest in the effect of mantle conduction on the very short period secular variation see Backus, 1983). I n an insulator B is a potential field, so that if it is measured everywhere on the earth’s surface i t can be found anywhere in the insulator: a result of potential theory. The sources of the field are inside the core, and short wavelengths fall off rapidly with distance from the core. Unfortunately, this makes it very difficult to reconstruct the short wavelengths from surface data: the downward continuation process is unstable. Downward continuation is usually achieved by finding the coefficients of the spherical harmonic expansion and multiplying each coefficient by the appropriate power of (u 0 where ( I is the Earth’s radiub, the core radius. The instability appears in the high degree coefficients, that are multiplied by larger factors depending on the degree.
The instability can be removed by seeking only fields that are in some sense ‘smooth’ on the core boundary (Shure et al. 1982). For example the observations may
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be fitted directly to a core held that sacrifices the condition fB,2dS < s, where the integral is over the whole of the core mantle boundary and s is chosen so that the fit to the data is satisfactory. This process leads to a stable field but it does not help in identifying the short wavelengths at the core boundary. Gubbins (1983 a) has developed a method that yields similar results by placing a priori restrictions on the unknown short wavelength fields. In this way it is possible to assign error estimates to all of the spherical harmonic coefficients. The a priori information is not strong enough to allow calculation of an error estimate for point values of the core field, but it is enough to place meaningful error bounds on averages, like the integrals in equation (13). The errors on these integrals, calculated from data for the last 20 years, suggest that the assumption of perfect conductivity is valid. Gubbins (1983 b) has extended the analysis to produce core fields that fit the data and satisfy all the conditions of equation (13) exactly. This is a necessary preliminary step before calculating core motions.
4.4. Results for core motions Around 1969 there occcurred a very rapid change in the secular acceleration (Ducruix et al. 1980). The effect seems to have been worldwide but was most convincing in Europe where the density of magnetic observations is greatest. Le Mouel et al. (1981) have proposed that the effect is correlated with a change in the length of day. This would indicate a change in relative rotation between the core and mantle, and Le Mouel and Courtillot (1981) have proposed a theory for it. Gubbins (1983 b) has analysed the behaviour in terms of fields frozen to a perfectly conducting core and finds the pattern too complicated to be due to simple rotation. The dipole moment underwent a sharp change in acceleration in 1969, which cannot be attributed to relative rotation. This rapid change of field may yield new information on the conductivity of the mantle.
Upwelling is an interesting property of the core flow that is detectable despite the non-uniqueness associated with finding complete core motions. V h . v measures convergence or divergence of flow at a point, and for an incompressible flow this indicates upwelling or downwelling of fluid beneath the surface. Rewriting equation (12) in the form
aB,/at + v . v,B,v,. v = o (15)
we see that at extrema or saddle points of B, the second term vanishes and the secular variation is determined solely by upwelling. Whaler (1980) plotted the zero contour of secular variation on the core surface and found that it passed close to all the extrema, suggesting that upwelling was small compared with other properties of the flow (figure 8).
If there is no upwelling then integrals of B, over all contours of B,, not just the null flux curves as in equation (15), must vanish. Furthermore these conditions are sufficient to allow calculation of velocities consistent with the secular variation. This work uses only data from a few magnetic observatories. Further studies with more complete data sets may yet reveal upwelling.
5. Summary The various branches of geomagnetism contain some formidable problems and there is still much work to be done. Much progress has been made in understanding the dynamics of the earth’s core, however, and there is now some hope that theories may soon be complete enough to allow more direct comparisons with observations.
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T h e Earth's magnetic jield 289
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KRAUSE, F., and ROBERTS, P. H., 1981, Adu. Space Res., 1,231-240. KUMAR, S., and ROBERTS, P. H., 1975, Proc. R . SOC. Lond. A, 314,235-258. LE MOUEL, J.-L., and COURTILLOT, V., 1981, Phys. Earth Planet. Interiors, 24, 236-241. LE MOUEL, J.-L,, MADDEN, T. R., DUCRUIX, J., and COURTILLOT, V., 1981, Nature, 290,763-765. LOPER, D. E., 1978a, J. Geophys. Res., 83, 5961-5970. LOPER, D. E., 1978 b, Geophys. J . R. Astr. SOC., 54, 389404. LOPER, D. E., and ROBERTS, P. H., 1981, Phys. Earth. Planet. Interiors, 24, 302-307. MALKUS, W. V. R., and PROCTOR, M. R. E., 1975, J . Fluid Mech., 67, 417444. MASTERS, T. G., and GILBERT, F., 1981, Geophys. Res. Lett., 8, 569-571. MOFFATT, H. K., 1978, Magnetic fteld generation in electrically conducting juids. Cambridge
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