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, because the temperatures of their interiors lie above the Curie temperature at which permanent magnetism disappears (Reif 1965). It goes without saying that stars and galaxies are not ferromagnetic. Magnetic fields cannot be dismissed as transient phenomena that 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 (Dunlop and Özdemir 2001; Ogg 2012). 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 timescale

$$\tau_{\rm ohm} = \mu_0\,\sigma\,L^2,$$ (8.100)

where $\sigma$ is the typical electrical conductivity, and $L$ is the typical lengthscale of the body, and this decay timescale is generally very small compared to the inferred lifetimes of astrophysical 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 $\sigma\sim 4\times 10^5\, {\rm S}\,{\rm m}^{-1}$ (Yoder 1995). 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 (Roberts and King 2013). 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 timescales (Mestel 2012).

The basic premise of dynamo theory is that all astrophysical bodies that contain anomalously long-lived magnetic fields also contain convecting, highly conducting, fluids (e.g., the Earth's molten core, the ionized gas that makes up the Sun), and it is the electric currents associated with the motions of these fluids that maintain the observed magnetic fields. At first sight, this proposal, first made by Larmor in 1919 (Larmor 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) ${\bf V}\cdot ({\bf j}\times{\bf B})$: in other words, 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 timescales far longer than the characteristic ohmic decay time.

Paleomagnetic data from marine sediment cores shows that the Earth's magnetic field is quite variable, and actually reversed polarity about $700,000$ years ago (Dunlop and Özdemir 2001; Valet, Meynadier, and Guyodo 2005). In fact, more extensive data shows that the Earth's magnetic field reverses polarity about once every ohmic decay timescale (i.e., a few times every million years) (Ogg 2012). The Sun's magnetic field exhibits similar behavior, reversing polarity about once every 11 years (Jones, Thompson, and Tobais 2010; Mestel 2012). An examination of this type of data reveals that dynamo magnetic fields (and velocity fields) are essentially chaotic in nature, exhibiting strong random variability superimposed on more regular quasi-periodic oscillations.

A thorough investigation of dynamo theory would be a far too difficult and time consuming task. Instead, we shall examine a far simpler version of this theory, known as kinematic dynamo theory, in which the velocity field, ${\bf V}$, is prescribed (Moffatt 1978; Krause and Rädler 1980). In order for this approach to be self-consistent, it must be assumed that the magnetic field is sufficiently weak that it does not affect the velocity field. Let us start from the MHD Ohm's law, modified by resistivity:

$${\bf E} + {\bf V} \times{\bf B} = \eta\,{\bf j}.$$ (8.101)

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

$$\frac{\partial {\bf B}}{\partial t} - \nabla\times({\bf V}\times{\bf B}) = \frac{\eta}{\mu_0}\nabla^2{\bf B}.$$ (8.102)

If the velocity field, ${\bf V}$, is prescribed, and unaffected by the presence of the magnetic field, then the previous 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 previous equation possess solutions in which the magnetic field grows exponentially in time? In trying to formulate an answer to this question, we hope to learn what type of motion of an MHD fluid is capable of self-generating a magnetic field.