MHD Dynamo Theory

Many stars, planets, and galaxies possess magnetic fields whose origins are not easily explained. Even the ``solid'' planets could not possibly be sufficiently ferromagnetic to account for their magnetism, since the bulk of their interiors are above the Curie temperature at which permanent magnetism disappears. It goes without saying that stars and galaxies cannot be ferromagnetic at all. Magnetic fields cannot be dismissed as transient phenomena which just happen to be present today. For instance, paleomagnetism, the study of magnetic fields ``fossilized'' in rocks at the time of their formation in the remote geological past, shows that the Earth's magnetic field has existed at much its present strength for at least the past $3\times 10^9$ years. The problem is that, in the absence of an internal source of electric currents, magnetic fields contained in a conducting body decay ohmically on a time-scale

$$\tau_{\rm ohm} = \mu_0\,\sigma\,L^2,$$ (780)

where $\sigma$ is the typical electrical conductivity, and $L$ is the typical length-scale of the body, and this decay time-scale is generally very small compared to the inferred lifetimes of astronomical magnetic fields. For instance, the Earth contains a highly conducting region, namely, its molten core, of radius $L\sim 3.5\times 10^6$m, and conductivity $\sigma\sim 4\times 10^5\, {\rm S}\,{\rm m}^{-1}$. This yields an ohmic decay time for the terrestrial magnetic field of only $\tau_{\rm ohm}\sim 2\times 10^5$ years, which is obviously far shorter than the inferred lifetime of this field. Clearly, some process inside the Earth must be actively maintaining the terrestrial magnetic field. Such a process is conventionally termed a dynamo. Similar considerations lead us to postulate the existence of dynamos acting inside stars and galaxies, in order to account for the persistence of stellar and galactic magnetic fields over cosmological time-scales.

The basic premise of dynamo theory is that all astrophysical bodies which contain anomalously long-lived magnetic fields also contain highly conducting fluids (e.g., the Earth's molten core, the ionized gas which makes up the Sun), and it is the electric currents associated with the motions of these fluids which maintain the observed magnetic fields. At first sight, this proposal, first made by Larmor in 1919, sounds suspiciously like pulling yourself up by your own shoelaces. However, there is really no conflict with the demands of energy conservation. The magnetic energy irreversibly lost via ohmic heating is replenished at the rate (per unit volume) ${\bf V}\cdot ({\bf j}\times{\bf B})$: i.e., by the rate of work done against the Lorentz force. The flow field, ${\bf V}$, is assumed to be driven via thermal convention. If the flow is sufficiently vigorous then it is, at least, plausible that the energy input to the magnetic field can overcome the losses due to ohmic heating, thus permitting the field to persist over time-scales far longer than the characteristic ohmic decay time.

Dynamo theory involves two vector fields, ${\bf V}$ and ${\bf B}$, coupled by a rather complicated force: i.e., the Lorentz force. It is not surprising, therefore, that dynamo theory tends to be extremely complicated, and is, at present, far from completely understood. Fig. 21 shows paleomagnetic data illustrating the variation of the polarity of the Earth's magnetic field over the last few million years, as deduced from marine sediment cores. It can be seen that the Earth's magnetic field is quite variable, and actually reversed polarity about $700,000$ years ago. In fact, more extensive data shows that the Earth's magnetic field reverses polarity about once every ohmic decay time-scale (i.e., a few times every million years). The Sun's magnetic field exhibits similar behaviour, reversing polarity about once every 11 years. It is clear from examining this type of data that dynamo magnetic fields (and velocity fields) are essentially chaotic in nature, exhibiting strong random variability superimposed on more regular quasi-periodic oscillations.

Figure 21: Polarity of the Earth's magnetic field as a function of time, as deduced from marine sediment cores.

Obviously, we are not going to attempt to tackle full-blown dynamo theory in this course: that would be far too difficult. Instead, we shall examine a far simpler theory, known as kinematic dynamo theory, in which the velocity field, ${\bf V}$, is prescribed. In order for this approach to be self-consistent, the magnetic field must be assumed to be sufficiently small that it does not affect the velocity field. Let us start from the MHD Ohm's law, modified by resistivity:

$${\bf E} + {\bf V} \times{\bf B} = \eta\,{\bf j}.$$ (781)

Here, the resistivity $\eta$ is assumed to be a constant, for the sake of simplicity. Taking the curl of the above equation, and making use of Maxwell's equations, we obtain

$$\frac{\partial {\bf B}}{\partial t} - \nabla\times({\bf V}\times{\bf B}) = \frac{\eta}{\mu_0}\nabla^2{\bf B}.$$ (782)

If the velocity field, ${\bf V}$, is prescribed, and unaffected by the presence of the magnetic field, then the above equation is essentially a linear eigenvalue equation for the magnetic field, ${\bf B}$. The question we wish to address is as follows: for what sort of velocity fields, if any, does the above equation possess solutions where the magnetic field grows exponentially? In trying to answer this question, we hope to learn what type of motion of an MHD fluid is capable of self-generating a magnetic field.