Many astrophysical bodies possess magnetic fields. The Earth has one, the geomagnetic field, and it is apparently as old as the Earth. Since the temperature throughout most of the Earth is above the Curie points of all known materials, the Earth is not a permanent magnet. The geomagnetic field must be produced by electric currents flowing mainly within its core, which is a good electrical conductor. Lacking sources, such currents would die out in about 10,000 years. It is now believed that the currents are maintained by electromagnetic induction, as in a commercial dynamo, the moving parts of the geodynamo being motions in the outer core, which is fluid.
The first question in dynamo theory is whether self-excitation is possible in a body having such a low degree of symmetry as the Earth’s core. This well-studied “kinematic dynamo problem” defines a nontrivial non-self adjoint eigenvalue problem, which will be discussed after the basic electrodynamic theory has been summarized. The second question is whether motions of sufficient vigor, and which have the “helicity” required for field generation, would arise naturally in the Earth’s outer core. This nonlinear “magnetohydrodynamic dynamo problem” raises some interesting questions in rotating magnetoconvection. The third question is whether the required energy source is available to power the dynamo. For bodies such as the solar convection zone, energy is available in abundance, but the geodynamo’s needs are less easily met.
Other topics will be touched on, including fast dynamos, auto-reversing dynamos, and chaotic dynamos.