5. Geodynamo Modeling

Dynamo models that solve the induction equation alone are called `kinematic', and several such geodynamos models have been integrated. The main challenge today is to solve the fully dynamic dynamo problem, sometimes also called `the MHD dynamo problem' or `the fully self--consistent dynamo problem', in which the induction equation is solved and the equation of motion for the fluid. This nonlinear problem raises formidable difficulties. Because of Cowling's theorem, a true MHD dynamo model should be 3D, but a supercomputer is then required to integrate it numerically. Axisymmetric (2D) models can be solved on workstations, but the magnetic field will decay to zero unless the emf generated by the omitted asymmetric fields and flows is reintroduced in some way, through an effect.

Axisymmetric effect models are of two extreme types, and dynamos, together with a range of models between them. In an model, zonal field creates meridional field and vice versa; in an --model, the --effect creates meridional field from zonal field, but the zonal field is created by an effect, i.e. by the inductive effects of zonal shearing motions. Sometimes models are called ``strong field dynamos'' since the zonal field, which is locked inside the conductor, is large compared with the observed meridional field, in contrast to the ``weak field'' dynamos of type where the strength, , of the meridional field is characteristic of the strength of the entire field. A strong field dynamo functions only if the product of and , as measured by the so-called ``dynamo number,'' D, exceeds in magnitude a certain marginal value, ; for an model to function the effect magnetic Reynolds number, , a dimensionless measure of , must be large enough. While dynamos are usually steady, dynamos tend to be oscillatory, but they too may become steady, and more efficient (as judged by a smaller value of ), when a sufficiently strong meridional flow is present. Meridional flow is produced by Lorentz forces or by core--mantle coupling; see also Bergman and Madden [1993].

Zonal shearing motion is comparatively easily excited in rotating fluids, for example by pole--equator temperature differences; the effect and the zonal field may therefore be strong. It used to be said that an appeal to an invisible zonal field is a return to armchair science, but galaxies are transparent to observation. Their fields are predominantly toroidal and seem to be produced by an mechanism; see of Krause et al. [1993]. It is in principle possible to detect a zonal field in Earth's core through the electric fields it creates outside the core, in particular the potential difference between the two ends of a trans-oceanic cable [ Lanzerotti et al., 1992, 1993, 1994]. In practice, the obscuring, long period, inductive effects of ocean currents have so far prevented a convincing demonstration [ Runcorn and Winch, 1991).

The axisymmetric force balance in an MHD dynamo is not easily understood. Many studies of 2D ``intermediate'' dynamo models have been launched to elucidate it. These are so named because, while they do not address the full MHD problem, they take a step beyond kinematic models. An source is invoked to maintain the axisymmetric field. Whether an , or dynamo results depends on the dynamical balance. In an dynamo the primary balance is geostrophic, i.e. between Coriolis and pressure forces; the magnetic field strength, B, is determined by a secondary balance, e.g. between the Lorentz and viscous forces, which gives , where is core viscosity. In a strong field dynamo, the magnetic field plays a role in the primary balance and therefore mT, a relation confirmed by Benton [1992]. Benton argued further that a typical zonal flow velocity would be m s, which is similar to the speed with which some features of the geomagnetic field drift westward.

St. Pierre [1993b] demonstrated that, when a strong field branch exists, weak field solutions are likely to be nonlinearly unstable. Although his plane layer model is geometrically too simple to represent the geodynamo, it is a convective MHD dynamo of similar physical type. It clearly demonstrates the existence of a strong field branch, one that also operates subcritically, i.e. at smaller thermal forcing than that at which kinematic dynamo action is first possible. The model of St. Pierre is fully 3D, as is the spherical 3D model of Glatzmaier and Roberts [1994] described below.

The axisymmetric force balance in intermediate models is so dominated by magnetic and Coriolis forces that inertial forces are often omitted. One of two extreme scenarios arise, or perhaps some intermediate scenario. At one extreme is the model-Z state [ Braginsky, 1975, 1991, 1994] which relies on the coupling of core to mantle and in which the geostrophic motions in the core are large. At the other extreme is the Taylor state [ Taylor, 1963] in which core--mantle coupling is insignificant, but in which a certain integral demand (the Taylor constraint) must be satisfied. Sometimes models of either type can exist under the same conditions of excitation. Model-Z is energetically the more expensive to run, because of core--mantle friction, and the external fields it produces therefore tend to be smaller than in the corresponding Taylor--type model. There are therefore two contenders for the geodynamo, a strong field (model-Z) mechanism and a very strong field (Taylor--type) mechanism. In trying to decide between these, it is usually supposed for simplicity that core-mantle coupling is viscous --- it is the existence of this coupling rather than its precise nature that is significant. St. Pierre [1993a] has examined the stability of Taylor states.

This then is the background against which much of the recent work on intermediate geodynamos may be viewed. Hollerbach and Ierley [1991] analyzed an intermediate dynamo of type and showed that, as exceeds its marginal value, , the solution is at first viscously controlled. As is further increased, a second critical value, , is reached at which Taylor states appear. When Hollerbach et al. [1992] carried out a parallel study for an model, they uncovered a more complex situation. Despite very simple choices of and , they found that, as the dynamo number D increases beyond , the solution is at first viscously controlled but that, as D increases through a second critical value , oscillations arise in which the Taylor balance is struck during part of each cycle but in which viscous coupling is essential during the remainder.

Braginsky and Roberts [1994b] continued earlier investigations of one particular model. They observed a transition from Taylor-type behavior to model-Z-type behavior as D increases. An dynamo model integrated by Glatzmaier and Roberts [1993] developed an interesting bifurcation as D was increased. Against the background of an approximately steady dipole component, an oscillatory quadrupole field causes the meridional field lines to bunch up alternately in one hemisphere and the other. The role of the dipole and quadrupole families of solutions of the geodynamo equation in geomagnetic field reversals is an oft recurring theme, and was raised again in a novel way by Hoffman [1991]. Questions of parity coupling in models also arose in the work of Hollerbach [1991].

In integrating a kinematic geodynamo model, Braginsky [1964] found that the fields induced outside the TC differed substantially from those generated inside it. Dynamic (intermediate) models have recently been studied by Hollerbach and Jones [1993a]. They find that most dynamo action takes place outside the TC, a conclusion that may depend on their choices of and since it was not confirmed by the recent 3D integrations of Glatzmaier and Roberts [1994]. The model of Hollerbach and Jones was used to benchmark that of Glatzmaier and Roberts [1993]; the agreement was nearly perfect. Hollerbach and Jones [1993b, 1994] argued that the SIC plays a potent role in the reversal mechanism; its electromagnetic inertia diminishes chaos in the FOC. Glatzmaier and Roberts [1994] agreed. The effects of conducting boundaries were investigated by Hirsching and Busse [1993].

Although the emphasis of the subject has moved towards MHD models, kinematic geodynamos are still being profitably studied. In particular, Hagee and Olson [1991] have suggested an interesting connection between the observed secular variation and certain types of anisotropic models; see also Kono and Roberts [1994]. A general method for solving weak field MHD models when conditions for kinematic dynamo action are only marginally exceeded, has been explored by Kono and Roberts [1991, 1992].

Lateral variations in the temperature of the CMB bring about concomitant changes in the electrical conductivity of the lower mantle so that new current paths are allowed and old ones forbidden. The axisymmetry assumed in most geodynamo modeling is destroyed and with it the applicability of Cowling's theorem [ Busse, 1992]]. A zonal shear can readily create zonal magnetic field from meridional field through the effect, but it is incapable, in an axisymmetric system, of creating meridional field from zonal field. This, however, is no longer true when longitudinal inhomogeneities destroy the axial symmetry. A zonal shear can then produce zonal field from meridional field and vice versa. This fact enabled Busse and Wicht [1991] and Wicht and Busse [1993] to construct new, simple models of dynamo action that make use of the broken symmetry and which work through zonal shear alone.

An unusual approach to the geodynamo problem was initiated by Ruzmaikin et al. [1993]. They divide the fluid domain into fixed cells, each of which randomly amplifies or destroys field by dynamo action; nonlinearity, diffusion and correlations between cells are then added. An initially smooth field becomes intermittent, the field concentrating mainly in a few cells, the location of which changes with time, a phenomenon they liken to the motion of geomagnetic field anomalies. Roberts [1992] introduced a ``mapping method'' that has been successfully tested against axisymmetric [ Nakajima and Roberts, 1994a] and asymmetric [ Nakajima and Roberts, [1994b] dynamo models; see also Nakajima et al. [1993a].

The resources of the NSF Pittsburg Supercomputing Center were enlisted to generate the first 3D time--dependent, fully self--consistent numerical solution of the MHD equations that describes thermal convection and magnetic field generation in a low--viscosity rapidly--rotating spherical shell with a solid conducting inner core. The resulting solution, reported by Glatzmaier and Roberts [1994], serves as a crude simulation of the geodynamo, crude because because the truncation was too severe and because geophysicially realistic values for some parameters were not numerically accessible (e.g. was several orders of magnitude too large, though still apparently not very influential). The heat flux from the core was taken to be W and the integration was continued over approximately three magnetic diffusion times, during which the field showed no signs of disappearing. Field generation takes place mainly within and near the TC. The pattern and amplitude of the radial magnetic field at the CMB is qualitatively similar to that of the Earth. The toroidal field energy is rather larger than the poloidal field energy but the maximum amplitudes attained by the two fields are comparable (T); the maximum fluid velocity is of order m s. An irregular exchange of field between hemispheres takes place, similar to that found in the 2D model of Glatzmaier and Roberts [1993]. Its timescale is about 10% of the magnetic diffusion time of the FOC. Excitingly, the dynamo sometimes reverses its polarity spontaneously. Preliminary to doing so, the poloidal field in the SIC has to reverse; if it does not do so, the reversal is aborted (as in geomagnetic excursions). It is hard not to be excited by such similarities between the computed model and the real Earth. Perhaps the answer to the challenging sentence that opened this review is at last within sight?