3. The Origin of Planetary Magnetic Fields.
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Any model we care to construct in regard of the source of the Earth's magnetic field has a tough job meeting the broad array of constraints placed upon it. It must explain not only the geometry of the observed field, its tilt, its intensity, and its harmonic spectrum, but also the secular change in these elements (seesection 8) - the westward drift of isoporic focii, excursions, and reversals. It must be physically plausible; a model which postulates a giant bar magnet at the Earth's centre is, quite clearly, physically unrealistic. Our model of the source of the Earth's field has also to work within the framework of the planet's internal structure as defined by seismology. This consists of;
The interiors of the other terrestrial planets are broadly similar, with the following exceptions. The core of Mercury is very large in relation to the actual size of the planet and is expected to be almost entirely solid. The Martian core is likewise thought to be solid, although neither of these suggestions are by any means certain, and detailed seismological studies are required. The Lunar core, if there is indeed a core, is very small and probably solid. On the other hand, the core of Venus may be entirely liquid. Because Venus is marginally smaller than the Earth, lower internal pressures permit the core temperatures to be above the melting point all the way to the planet's centre.
The interiors of the Gas Giants, and the other satellites that have intrinsic fields, are quite different from the interiors of the terrestrial planets. The implications of their structures for the generation of magnetic fields will be taken up separately in the second half of this work, where the literature on their fields is reviewed (Sections 11-15).
Once we have a model that satisfies all of the above conditions it must then be a model that produces a field of the right shape and size from the available energy. Our model is no good if it guzzles more energy than is accessible from various internal sources (making suitable allowances for the efficiency of the generating mechanism in turning the energy obtained into magnetic energy).
The total heat loss through the surface of the Earth is ~4x10E13W, the majority of which is the result of radioactive decay in the crust. Energy in thethe Earth's core could come from a number of sources, chief among which is the energy released by the formation of the inner core. Taking the outer core fluid as a roughly binary mixture of iron and sulfur (but dominantly iron) we can see, from reference to the relevant phase diagram, that the first substance to crystallise on cooling will be pure iron. This iron precipitates out to form the solid inner core. The residual fluid, being lighter and more buoyant, rises through the outer core releasing ~2x10E12W of gravitational energy. The attendant cooling of ~100K over the age of the Earth releases latent heat of crystallisation - of the order of 10E12W. Also, as indicated earlier, one of the light elements in the core could be potassium. Were we to assume that 75% of the potassium the Earth accreted from the solar nebula was in the core, then the decay of the radiogenic component (40K) to 40Ar would also yield roughly 10E12W. Our field generation model therefore has somewhere in the region of a few times 10E12W available to it.
We have seen that magnetic fields can arise from permanent remanent fields in magnetized material below their Curie temperatures, and from the motion of charged particles in electric fields; but which of these two mechanisms can we call upon to explain the intrinsic magnetic field of the Earth and the other planets?
The planets could have acquired their own remanent fields as a result of external influence in one of two ways. Firstly, provided the solar nebula was sufficiently conductive, the trapping and concentration of interstellar field lines (the Galaxy has a magnetic field of ~10E-5 Gauss) inside condensing planets might create flux densities as great as a million gauss in body the size of Jupiter. The rate at which such a field would decay is then a function of the planet's size (L) and its conductivity (s, which is inversely proportional to the magnetic diffusivity). The decay timet
For a planet such as Jupiter this is a very long time - around 2 billion years - though it is rather shorter for a planet the size of the Earth - of the order of 1000 years. Clearly any such 'remanent' field acquired from the galactic field will have decayed long ago, and another explanation must be sought for the planet's intrinsic field.
We might, therefore, invoke the kind of remanent rock magnetization that was discussed in section 1. Although the ambient galactic field would be rather too weak to induce a significant remanence, the magnetic field of the Sun - in particular the young and highly energetic Sun - could conceivably magnetize the ferromagnetic parts of the planets. Unfortunately for this theory, the temperatures inside the Earth, and all of the other planets with intrinsic fields, are well above the Curie points of all magnetizeable materials. While the extremely weak fields of the Moon and Mars may be attributable to crustal remanent magnetism, this is most certainly not a satisfactory explanation for the origin of the Earth's field.
There are many different ways in which magnetic fields might plausibly be generated within the planets, and these are outlined and then discussed below. They are;
The first among these is the field that is generated by a rotating electric charge (the electron, for example, has a magnetic moment by virtue of its spin). Sadly, in order to generate fields of the size we observe the planets to possess requires that there be enormous electric fields (of the order of 10E6V/m for the Jovian field) in their atmospheres, and we have no evidence - especially not in our own atmosphere - for the existence of such fields.
Similarly, a rotating ferromagnet becomes magnetized along its rotation axis by the gyromagnetic effect. This effect is desperately weak, though; a ferromagnet rotating one hundred time a second produces a field of just 10E-5 Gauss, and certainly no more than 10E-10 of the Earth's field can be the result of the gyromagnetic effect.
Differential rotation in a rapidly rotating metallic liquid is able to generate a magnetic field; the field strength being proportional to the amount by which the angular velocity changes over the mean free path of a single electron. This is effect might be important as the Earth is known to have a core of liquid iron and the outer planets are believed to have layers of liquid metallic hydrogen in their interiors. As with gyromagnetism, however, the effect is very weak indeed and field produced by differential rotation are negligible in comparison to the observed field of the Earth. Such fields are weaker by twelve orders of magnitude than the terrestrial field.
Yet another mechanism based on rotation was proposed by Blackett, who theorised that all massive rotating bodies should possess intrinsic fields as a result of some new, and unknown, physical law. Blackett observed that, on a log-log plot of angular momentum against magnetic moment, one could draw a straight line upon which, broadly speaking, most planets fell. Mars and Venus plot well off this line, and the Earth only plots on the line when the angular momentum of both the Earth and the Moon are considered together, highlighting the apparent lack of a physical basis for the hypothesis (which is widely touted a 'magnetic Bode's law'). The hypothesis may be compared with another, more physically sound relationship between angular momentum and magnetic moment which is derived from dynamo theory. A log-log plot ofr1/2WRc^4 (where r is core density, Rc is core radius, and W is rotation rate) against magnetic moment (normalized to the terrestrial moment) again produces a straight line, but one which matches the planets' observed moments far better. Blackett disproved his own hypothesis in 1952 when he failed to observe a field being generated by a spinning gold sphere.
Other theories for the generation of planetary magnetic fields invoke electric currents driven by thermal gradients. Walter Elsasser, in 1939, suggested that the thermoelectric effects in the Earth's liquid outer core might generate a magnetic field as a result of the temperature difference between warm upwelling material, and cold downwelling material. Calculations made in the 1950's, however, indicate that convection would need to be so vigorous to yield the desired effect that the amount of heat transported would be greater than the heat available. David Stevenson performed the same calculations for Jupiter in 1974 and found that, in principle, thermoelectric currents inside Jupiter were possible, and could be contributing to the planet's sizeable magnetic moment.
When a radial thermal gradient occurs in the presence of a pre-existing axial magnetic field then electric currents are set up which generate a magnetic field that augments the existing field. This is actually a combination of two effects, the galvanomagnetic effect and the thermomagnetic effect, neither of which, alone or in concert, can match the Earth's observed field.
In an analogous vein, thermoelectric currents in the outer core (induced by a pre-existing field) could be inducing Hall currents in the lower mantle. The magnetic field that these Hall currents generate would then supplement the original field. This avenue has not been explored as a means of generating the terrestrial magnetic field as the conductivity and the Hall coefficient of the lower mantle are not well constrained. It is considered unlikely that the envelopes of molecular hydrogen at both Jupiter and Saturn are sufficiently conductive to support Hall currents.
That leaves only one remaining possibility, that of an electromagnetic induction mechanism acting in the fluid outer core - not altogether dissimilar to the common or garden bicycle dynamo. Sir Joseph Larmor was the first to put forward such a suggestion as early as 1919, and important contributions were made by investigators such as Walter Elsasser and Edward Bullard throughout the 1930's, 40's, and 50's; foremost being those of Eugene N. Parker in his 1955 treatise on astrophysical dynamos.
The most simple and widely used analogy for the geodynamo (and the one used in the 1940's prior to the application of the principles of magnetohydrodynamics) is the homopolar disc dynamo illustrated below. The figure on the left (overleaf) shows a metal disc rotating through the field lines of a bar magnet. Lorentz forces (B X v) acting on electrons in the conductive disc cause a charge displacement from the centre of the disc to the edge which, provided the circuit is completed as shown, will allow a current to flow. If we now remove the bar magnet and replace it with an electromagnet in series with our circuit (as in the illustration on the right) then, provided there is an initial 'seed' field to start the current flowing, the field induced in the electromagnet by the current will reinforce the flow and become self-exciting. It would be easy to mistake this for some kind of perpetual motion machine until it is made explicit that it is the mechanical energy of the rotating disc which is being transformed into magnetic energy. Another important point to note is that this self exciting dynamo only works because it is asymmetric; the mechanism would cease to work, for example, if another electromagnetic coil were introduced into the circuit on the other side of the disc. The implication that this has for the geodynamo is that it cannot be symmetrical, an idea that was introduced in 1934 by Thomas G. Cowling. (Although Cowling's theorem is rather more complicated than that, it can be reduced to the statement that axisymmetric dynamos are not self-sustaining.)
The homopolar disc dynamo model is relatively easily transposed to the Earth's outer core. The conducting disc is replaced by the liquid iron in the outer core (conductivity ~10E6 mho/m) and the motion is provided by the convection of this fluid. Thus the ultimate source of energy is the heat which is driving this convection. We can assume that a 'seed' field was induced either by a concentration of interstellar flux lines, or by the Sun's field, and that, as convecting conductive material cut these field lines, currents were generated which created their own magnetic field and so the system became self-exciting. One of the problems with this simplistic model is that the fluid motions in the northern and southern hemispheres are symmetrical and this prevents the self-exciting dynamo from working. However, in the real core, the fluid is rotating along with the rest of the Earth and is subject to the coriolis force. The coriolis force acts in opposite directions in each hemisphere, lowering the symmetry of the system and allowing the dynamo to work.
Homogenous dynamo theory, as it is know, fails to consider a phenomenon, discovered by Hannes Alfvén, whereby the flux lines are carried along with a conducting fluid. A magnetic field will diffuse out of a given material at a rate that is inversely proportional to the conductivity. The magnetic diffusivity(lm) is given by;
lm = (m0s0)^-1
where s0 is the conductivity.
Therefore, for a material that is a very good conductor, the magnetic diffusivity is very small and field lines diffuse out so slowly that they behave as if they are frozen in place. When the conductor is a fluid then the field lines will flows wherever the fluid flows. One consequence of this is that field lines in a conducting fluid are capable of behaving like elastic strings, and can support magnetic waves - known as Alfvén waves. In order for large scale motions in the core to dominate over the magnetic (also known as ohmic) diffusion -i.e. for the field to be stable against dissipation - then a dimensionless number known as the magnetic Reynolds number (Rm) must be greater than about 10.
Rm = Lv/lm
wherev is the velocity of convective motion
andL is the characteristic length scale of the convection cells.
Balancing this is the need for diffusion to be fast enough to keep the field line topology relatively simple without inextricably tangling them up, and reducing the efficiency of the dynamo.
Alfvén's frozen-in field theorem leads to two very important, and relatively simple, effects in the core that are the foundations of kinematic dynamo theory. First it is necessary to define the two types of field which will be discussed; poloidal fields and toroidal fields.
What Alfvén's theorem shows us is that differential rotation of the liquid outer core will drag existing poloidal field lines along with the conducting fluid to form, after one complete rotation, a toroidal field. The mechanism by which a toroidal field is created from a poloidal field is known as thew-effect, for the simple reason that it relies on the angular velocity (w) of the fluid.
A second important effect is that resulting from the interaction of helical fluid flow with an existing field line. The effect is illustrated in figure 10 overleaf, where the helical velocity along the z axis is split into its two components v1 and v2. When a magnetic field (B0) is applied along the x axis then the cutting of field lines by motion along v1 produces the current loop J1. This current produces its own magnetic field (B1) which is directed along the y axis. The interaction between B1 and v2 then generates an electric field (E') directed along the x axis. The upshot of all this is that a helical flow in a magnetic field creates a mean electric field in the same direction as the original magnetic field, and also generates an orthogonal magnetic field. The mean electric field is often written as;
It is from the above expression that the a-effect takes its name. The a-effect is extremely important because not only does it allow us to convert turbulence in the core into a useful dynamo mechanism, but it permits both the generation of poloidal fields from toroidal fields and the generation of toroidal fields from poloidal fields.
Here, at last, we have a way of producing a self-sustaining magnetic field, simply by considering the motions of a fluid conductor. In point of fact we have more than one way;
This is not the place to critique the relative merits of the three types of dynamo; each has both advantages and disadvantages in terms of their ability to simulate reversals, excursions, and the secular variation, though it is generally thought that the a^2w-dynamo is the most physically plausible. As our understanding of, and ability to model, magnetohydrodynamic fluids improves, then so our models of the terrestrial dynamo improve. The latest (1995) dynamo model of Glatzmaier and Roberts has the toroidal field confined to the inside of a cylinder that is tangential with the inner core (i.e. north and south of the inner core), with tangential columns of helical convection rolls in the annulus outside this column. The model has the advantage that it is able to match some of what we know (or believe) about reversals, such as the drop in field intensity during the reversal and the time lag between the reversal of the toroidal and poloidal components of the field.
The simplest explanation for reversals in kinematic dynamo theory is that convection ceases in the outer core for a sufficiently long time period that the field can decay, almost to zero. In the absence of convection, the temperature gradient in the outer core would gradually increase until such time as the fluid once again became unstable to convection. At this time a new poloidal field would form whose polarity would depend on the polarity of any residual field in the spot where convection restarts. One of the problems with this is that the free-decay time is rather longer than the observed duration of reversals. In addition, the broad looping motions of the magnetic poles prior to, and following, reversal are not explained by this model.
More widely favoured are dynamic-reversing models. Such models have solutions which emerge from the mathematics that are either oscillatory (the field reverses polarity frequently and spontaneously) or steady (the field maintains a constant polarity for long time periods). Two extensive periods in the Earth's history - discussed in section 8 - during which the geomagnetic polarity remained constant for several tens of millions of years, the Cretaceous normal superchron and the Kiaman reversed superchron, are viewed as steady periods in an otherwise oscillatory system. The question of whether the terrestrial dynamo is a genuinely chaotic system, or simply an oscillatory system, is one which remains for the future to answer.
Powering the Geodynamo.
The amount of energy needed to sustain the Earth's magnetic field depends to a large extent on the magnitude of the toroidal field in the core, a field which we have no way of directly observing (though in theory we can detect eddy currents induced in the mantle by such fields). Various models put the strength of the toroidal field anywhere between 1 Gauss and 1000 Gauss. At present the consensus is that the toroidal field is roughly ten times larger than the poloidal field in the core, yielding a field strength of approximately 100 Gauss. The most basic approach to the problem of determining the energy consumption of the dynamo, given by John A. Jacobs in 1972, is to divide the total amount of energy available in the magnetic field by the time it would take to dissipate if there were to be a cessation of field regeneration. The rate of energy consumption (dE/dt) is then given by;
dE/dt = H^2V/t
whereH is roughly the magnitude of the toroidal field in the core
V is the volume of the dynamo
andt is the dissipation time of the core field.
Taking H = 100 Gauss ( = 10E-2 Tesla = 0.01/4px10E-7 = 7958A/m), V = 1.69x10E20m^3 (the volume of the outer core minus the volume of the inner core), and t = 10E11 seconds (~3000 years) then dE/dt = 10E17 ergs/s or 10E10W.
We believe, however, that the dynamo is only about 0.1% efficient (and certainly no more than 4% efficient) in its conversion of thermal energy to magnetic energy. This means that the dynamo actually needs an input of around 10E12W. As was discussed at the start of this section, there are several sources in the core that put out around 10E12W, and the total is believed to more in the region of 10E13W (1/4 of the heat flow through the surface). Thus there would seem to be enough energy available to drive even an inefficient geodynamo.
For the cases of the other terrestrial planets then several of the parameters are subject to change. Both Mercury and Mars should have quite small liquid regions in their cores and this is expected to make a kinematic dynamo unlikely. It is no surprise then that Mars shows no sign of possessing an intrinsic field. The fact that Mercury has a relatively large magnetic field of its own is, however, something of a puzzle. The core of Venus, while large enough, probably doesn't have a significant energy source (in the form of a crystallising inner core), and the core of the Moon is believed to be entirely solid.
For Jupiter, taking the toroidal field strength as ~1000 Gauss (an internal field >10E4 Gauss will exert a sizeable outward pressure) so as to maintain the ratio of the poloidal to toroidal fields of 1:10, yields and energy consumption (dE/dt) = 10E11W. Given Jupiter's enormous internal heat flux (~8x10E17W) then the magnetic field inside the planet may be more like that of star. That we don't see features analogous to sunspots can be attributed to the insulating layer of molecular hydrogen.
When we come to consider the magnetic fields of Uranus and Neptune we find that the length scales (for convection) are so large, in order to maintain the field against ohmic diffusion, that the fields must be generated, not in their cores, but in their superheated liquid mantles. This is hypothesised to be the explanation for the large dipole tilts of their fields and the large apparent offsets of their dynamo source regions as determined by spherical harmonic analysis.
To further appreciate the power of dynamo theory we are going to have to learn more, not only about the magnetic history of the Earth, but also about the interiors of the terrestrial planets and their palaeomagnetic records. No doubt, in fifty years from now, we will be in a far better position to comment on the dynamos of the solar system.
© A.D. Fortes. 1997.