A conducting liquid is governed by the Navier-Stokes equation with a Lorentz force term, Ohm's law, and Maxwell's equations. (Backus 218)
| Equation | What it describes |
|---|---|
| Navier-Stokes | The motion of a fluid |
| Lorentz force | The force of magnetism pushing the fluid |
| Ohm's law | The generation of electrical currents due to the magnetic and electric field |
| Maxwell's equations | The interacion of the electric and magnetic fields with themselves and their generation by electrical currents |
The Navier-Stokes equation alone is very difficult, and sometimes impossible, to solve directly. The addition of the magnetic interaction makes the system considerably more complicated. Any progress requires either making some simplifying assumptions or plugging it into a computer. It is difficult to know what assumptions are valid because there are many uncertainties regarding the conditions of the Earth's core.
With a bit of algebra it is possible to eliminate the electrical current term to make an equation that directly relates the magnetic field to the velocity. The result is the magnetic induction equation: (Merrill 310)
The right hand side of the equation consists of a dissipative term and an interaction term. The relative strengths of these two terms determine the general behavior of the system. If the fluid is poorly conducting or is not moving then equation is governed only by the dissipative term. In this case the magnetic field will decay freely and eventually disappear. If there were no currents in the core the Earth's magnetic field would lose half of its strength every 10000 years. (Backus 227)
The other extreme is a fluid that is perfectly conducting or is moving very rapidly. In this case the dissipative term will vanish and the fluid will have very strong magnetic self-interaction. With a perfect conductor the fluid movement is perfectly correlated with the magnetic field - a condition known as frozen flux. If an imaginary tube is drawn around some magnetic field lines and this tube is allowed to flow along with the fluid, it will always enclose the same number of field lines. (Backus 231-239)
In some ways the perfect conductor model is a good way to understand what goes on inside the Earth and in some ways it is not. (Demorest 9) Overall though it is a good framework for determining what sorts of fluid motions can support dynamos.
A lot of effort has been put forth to come up with an example of an idealized solution to these equations that would describe a self-supporting dynamo. The first line of attack is usually to try and find the simplest possible solution and in this case it would seem reasonable to look for a solution that is symmetric about the Earth's axis of rotation. This proved fruitless and eventually Cowling proved that such a solution is not even possible. Others then were able to prove the impossiblility of other classes of solutions. (Backus 261) For a long time things were not looking good for the dynamo theory and many were wondering if there was a solution at all. Eventually some contrived examples were worked up that proved at least the possibility of a solution. (Backus 261) More recently some satisfactory dynamical models have been worked up and both laboratory and computational experiments are starting to show some promising results. (VKS, Glatzmaier)