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coefficients over the next 5 years. The 1995 epoch IGRF has Gauss coefficients truncated at degree 10 (corresponding to 120 coefficients) and degree 8 for the secular variation; this is regarded as a practical compromise to produce a well-determined main field model. The IGRF 1995 epoch model coefficients up to degree 4 are listed in Table 1.1; for degrees greater than 4 the magnitude of the coefficients falls off quite rapidly with increasing degree. Harmonics of order zero are referred to as zonal harmonics, with coefficients g]°, gj, g? > etc which are the coefficients for the geocentric axial dipole, geocentric axial quadrupole, geocentric axial octupole, and so on, respectively. All the other terms are the nonzonal harmonics. For convenience, the coefficients are typically referred to as if they were the harmonic; thus, g"' is used to refer to the harmonic of degree I and order m. The main field is dominated by the geocentric axial dipole term ( gf ), then the equatorial dipole ( gj and h{ ). The latter causes the main dipole to be inclined to the axis of rotation by about IOV20. As a simplistic separation, the Gauss coefficients less than degree 14 are generally attributed to sources in the Earth's liquid core and those greater than degree 14 to sources in the Earth's crust. See Merrill et al. (1996) for more details on the Gauss coefficients and their analysis.

The dynamo theory of the Earth's magnetic field originates from a suggestion of Larmor (1919) that the magnetic field of the Sun might be maintained by a mechanism analogous to that of a self-exciting dynamo. Elsasser (1946) and Bullard (1949) followed up this suggestion proposing that the electrically conducting iron core of the Earth acts like a self-exciting dynamo and produces electric currents necessary to maintain the geomagnetic field. The action of such a dynamo is simplistically illustrated by the disc dynamo in Fig. 1.6. If a conducting disc is rotated in a small axial magnetic field, a radial electromotive

Fig. 1.6. The disc dynamo. A torque is applied to rotate a conducting disc at angular speed to in a magnetic field aligned along the axis of the disc. An electric current, induced in the rotating disc, flows outward to the edge of the disc where it is tapped by a brush attached to a wire. The wire is wound back around the axis of the disc in such a way as to reinforce the initial field.

force is generated between the axis and the edge of the disc. A coil in the external circuit is placed coaxial with the disc so as to produce positive feedback so that the magnetic field it produces reinforces the initial axial field. This causes a larger current to flow because of the increased emf and the axial field is increased further, being limited ultimately by Lenz's law, the electrical resistance of the circuit, and the available mechanical power. The main point is that starting from a very small field, perhaps a stray one, it is possible to generate a much larger field.

In the simple disc dynamo of Fig. 1.6, the geometry (and therefore the current path) is highly constrained and all the parts are solid. That makes solution of the relevant equations, and understanding of the process, relatively simple. In the

Fig. 1.7. Production of a toroidal magnetic field in the core, (a) An initial poloidal magnetic field passing through the Earth's core is shown on the left, and an initial cylindrical shear motion of the fluid (i.e., with no radial component) is shown on the right, (b) The interaction between the fluid motion and the magnetic field in (a) is shown at three successive times moving from left to right. The fluid motion is only shown on the left by dotted lines. After one complete circuit two new toroidal magnetic field loops of opposite sign have been produced. After Parker (1955).

Magnetic Field

Velocity Field

Magnetic Field

Velocity Field

Fig. 1.7. Production of a toroidal magnetic field in the core, (a) An initial poloidal magnetic field passing through the Earth's core is shown on the left, and an initial cylindrical shear motion of the fluid (i.e., with no radial component) is shown on the right, (b) The interaction between the fluid motion and the magnetic field in (a) is shown at three successive times moving from left to right. The fluid motion is only shown on the left by dotted lines. After one complete circuit two new toroidal magnetic field loops of opposite sign have been produced. After Parker (1955).

Earth there is a homogeneous, highly electrically conductive, rapidly rotating, convecting fluid that forms the dynamo. This highly unconstrained situation, together with the need to include equations such as the equation of state of the fluid and the Navier-Stokes equation, means that the geodynamo problem is exceptionally difficult to solve. Despite this, major advances have been made in recent years. Although the details are necessarily complex, several of the major concepts are reasonably accessible.

If a magnetic field exists in a perfectly conducting medium, then when the medium moves, it carries the magnetic field lines along with it according to the frozen-in-field theorem of Alfven (1942, 1950). Although the core fluid is not a perfect conductor, there is still a strong tendency (certainly over short time scales) for the fluid to drag magnetic field lines along with it. This is central to dynamo theory because differential motions of the fluid stretch the magnetic field lines and thereby add energy to the magnetic field. Because the fluid is not a perfect conductor the magnetic field will diffuse away with time, and so it is necessary for there to be dynamo action to add energy back into the magnetic field to overcome this diffusion. Another central concept is that of poloidal and toroidal fields. Toroidal fields have no radial component and so it is not possible to observe at the Earth's surface a toroidal field in the Earth's core. Conversely, a poloidal field does have a radial component and the geomagnetic field at the Earth's surface is poloidal. The magnetic field can be written as the sum of a poloidal field and a toroidal field, and many of the concepts of dynamo theory revolve around the question of how to generate a toroidal field from a poloidal field and, conversely, how to generate a poloidal field from a toroidal field.

Figure 1.7 illustrates how a toroidal magnetic field can be generated from an initial poloidal magnetic field using a process referred to as the (D-effect. If the core fluid motion has a toroidal component (relative to the overall rotation of the Earth), then the highly conducting fluid drags the magnetic field lines along with it in its toroidal motion as shown in Fig. 1.7b. This stretches the magnetic field lines, thereby adding energy to the magnetic field, and draws the poloidal field lines out into toroidal loops. However, the co-effect cannot generate a poloidal field from an initial toroidal field. Another process, known for historical reasons as an a-effect, is required for this.

The simplest picture of how the a-effect can occur is provided by convection in the core together with Alfven's frozen flux theorem and helicity, as is illustrated in Fig. 1.8. The toroidal field will be affected by an upwelling of fluid. As the field line moves with the fluid the upwelling will produce a bulge, which stretches the field line. The field line is in tension so, just like an elastic band, energy is required to stretch the field line. By this process energy is added to the magnetic field. The Coriolis force will act to produce a rotation (known as helicity) in the fluid as it rises, counterclockwise in the northern hemisphere. The field line will be twisted with this rotation and a poloidal magnetic loop will be

Fig. 1.8. Production of poloidal magnetic field in the northern hemisphere. A region of fluid upwelling, illustrated by dotted lines on the left interacts with toroidal magnetic field (solid line). Because of the Coriolis effect the fluid exhibits helicity, rotating as it moves upward (thin lines center). The magnetic field line is carried with the conducting liquid and is twisted to produce a poloidal loop as on the right. After Parker (1955).

produced after 90° of rotation. Because the field gradients are large at the base of the loop, it can detach from the original field line to produce a closed flux loop. The process is inherently statistical, but eventually poloidal loops of this sort merge to produce a large poloidal loop. The above turbulent process provides a simple visualization of the generation of poloidal field from toroidal field. This particular turbulent process may not be the only contributor to the a-effect in the Earth's core (e.g., Roberts, 1992).

The combined action of the processes illustrated in Figs. 1.7 and 1.8 is referred to as an aoo-dynamo. It is worth noting that the a-effect can also generate poloidal field from an initial toroidal field. Thus it is possible to have a2- and a2co-dynamos. Readers are referred to Merrill et al. (1996) for more details.

Roberts (1971) and Roberts and Stix (1972) pointed out that if the large-scale velocity shear that causes the co-effect is symmetric with respect to the equator and if the a-effect is antisymmetric with respect to the equator (as might be expected since the Coriolis force changes sign across the equator), then the dynamo can be separated into two noninteracting systems made up of specific families of spherical harmonics. Gubbins and Zhang (1993) refer to these as the antisymmetric and symmetric families. Spherical harmonics whose degree and order sum to an odd number belong to the antisymmetric family and those whose degree and order sum to an even number belong to the symmetric family. The situation shown in Fig. 1.7 is the simplest one in which the initial poloidal field is antisymmetric with respect to the equator.

### 1.1.4 Variations of the Dipole Field with Time

The intensity of the dipole field has decreased at the rate of about 5% per century since the time of Gauss' first spherical harmonic analysis (Leaton and Malin, 1967; McDonald and Gunst, 1968; Langel, 1987; Fraser-Smith, 1987) (Fig. 1.9a). Indeed, Leaton and Malin (1967) and McDonald and Gunst (1968)