Hydromagnetic dynamo theory

Post-publication activity

Prof. Axel Brandenburg, Nordita, AlbaNova University Center, Stockholm, Sweden, Stockholm, Sweden

Dynamo theory is a vast field with almost a hundred years of history, starting with early ideas by Joseph Larmor in 1919. Dynamos are particularly important in connection with understanding magnetic fields in astrophysics. Much of the work is at the level of analytic theories and numerical simulations. During the last decade also various liquid metal experiments have been performed.

Traditionally, dynamos are divided into

kinematic dynamos, where the flow can be considered given, and
nonlinear dynamos, where the flow is affected by the magnetic field through the Lorentz force.

The latter are sometimes also referred to as hydromagnetic dynamos, which emphasizes the importance of hydromagnetic interactions.

Kinematic dynamos

For kinematic dynamos the field strength is negligible, so the flow can be considered given. It must still be a solution to the Navier-Stokes equations, although this restriction is sometimes ignored. With given boundary conditions (e.g. vacuum outside the dynamo domain) this constitutes an eigenvalue problem, where the largest real part of the eigenvalue is the growth rate of the magnetic field. The dynamo is excited if the growth rate is positive.

Kinematic dynamos may be divided into laminar and turbulent dynamos. For laminar dynamos the velocity field is spatially smooth and often stationary. For these dynamos there is a further distinction between slow dynamos and fast dynamos, depending on whether in the limit of vanishing magnetic resistivity the growth rate of the magnetic field goes to zero (slow dynamo) or remains finite (fast dynamo). Examples of slow dynamos include the Roberts flow, which is a two-dimensional helical flow pattern, as it is realized approximately in the Karlsruhe dynamo experiment. An example of a fast dynamo is the ABC flow, although here the evidence is only numerical.

Turbulent dynamos can be divided into small-scale and large-scale dynamos. Small-scale dynamos work in principle with any random flow, but here we restrict ourselves to the physically relevant case of fully isotropic turbulence. Such flows constitute an idealization that allows analytic progress to be made. Such analytic theories can also be tested numerically, although the maximum mesh sizes restrict the largest achievable Reynolds numbers currently to around 1000. Real flows are always to some degree anisotropic due to the nature of the underlying forcing mechanism.

The onset of dynamo action is characterized by the magnetic Reynolds number, which is defined as R_m=u_rms/(ηk_f). Here, u_rms is the root-mean-square velocity in the dynamo-active domain, η is the magnetic diffusivity, and k_f is the wavenumber corresponding to the energy-carrying scale of the flow.

Small-scale dynamos

Small-scale dynamos can be studied analytically by solving evolution equations either for the correlation function <B_iB_j> or for the energy spectrum E(k); see, e.g., the review by Brandenburg & Subramanian (2005) with references to the original literature. In either approach the assumption of isotropy is usually invoked.

The critical magnetic Reynolds number for the onset of dynamo action is around R_m=35. The magnetic field has an approximate k^3/2 energy spectrum and is peaked at the resistive scale ~η/u_rms. The growth rate scales with R_m^1/2.

At low magnetic Prandtl numbers P_m=ν/η, the dynamo becomes harder to excite. This result does not, however, apply to dynamos with a mean flow (for example the Taylor-Green flow has a finite time average), or to flows with finite net helicity or anisotropy.

Large-scale dynamos

For large-scale dynamos the magnetic energy grows at scales large compared with the scale of the turbulence. This requires that there is scale separation, i.e. that the domain size is large compared with the size of the turbulent eddies.

The evolution equation for the large-scale field is obtained by averaging the Faraday equation (or induction equation; see also the article on magnetohydrodynamics), i.e. \[ { \partial{ \overline{\mathbf{B}} } \over \partial t } = \nabla \times \left( \overline{\mathbf{U}} \times \overline{\mathbf{B}} +\overline{ \mathbf{u} \times \mathbf{b}} - \eta\mu_0\overline{\mathbf{J}} \right), \] where the expression \(\overline{\mathbf{u}\times\mathbf{b}}\equiv {\overline{\mathbf{\mathcal{E}}}}\) is also referred to as the mean turbulent electromotive force. In the simplest isotropic case one has (e.g. Moffatt 1978) \[ \overline{\mathbf{\mathcal{E}}}=\alpha\overline{\mathbf{B}} -\eta_{\rm t}\mu_0\overline{\mathbf{J}}. \]

Here the first term is called the \(\alpha\ ;\) effect, while the second term corresponds to turbulent diffusion with a turbulent diffusivity, η_t. In the anisotropic case there can be many more terms (Rüdiger & Hollerbach 2004). One particularly important one that can also lead to dynamo action is the so-called \(\overline{\mathbf{W}}\times\overline{\mathbf{J}}\) effect (Rogachevskii & Kleeorin 2003).

Large-scale dynamos are amenable to a mean field treatment where one considers only the averaged equations using, for example, an azimuthal average relevant for axisymmetric fields. In analytic studies one often uses instead ensemble averages.

Nonlinear dynamos

The nonlinear regime is reached when the Lorentz force begins to affect the fluid motions. When a kinematic dynamo has achieved appreciable amplitudes at the end of its exponential growth, the Lorentz force will usually begin to quench the dynamo and lead to some equilibration. This is the normal situation. There are, however, two rare exceptions: there can be so-called self-driving dynamos (where a suitable flow only exists because of the Lorentz force) and so-called self-killing or suicidal dynamos (where the Lorentz force destabilizes the flow that led to the exponential growth in the kinematic regime). An important example of a self-driving dynamo is the dynamo that works as a result of the magnetorotational instability (described below). Another example are the numerical models of the geodynamo (Glatzmaier & Roberts 1995).

Saturation of small-scale dynamos

Figure 1: Magnetic and kinetic energy spectra from a nonhelical turbulence simulation with P_m=1. The kinetic energy is indicated as a dashed line (except for the first time displayed where it is shown as a thin solid line). At early times the magnetic energy spectrum follows the k^3/2 Kazantsev law, while the kinetic energy shows a short k^-5/3 range. Adapted from Brandenburg & Subramanian (2005).

In the kinematic regime the magnetic energy spectrum develops a k^3/2 power law at large scales, so the spectral magnetic energy peaks at small scales. As the dynamo saturates, the magnetic energy spectrum approaches the k^-5/3 power law of the turbulent flow and saturates. Thus, also the magnetic energy spectrum gradually develops a k^-5/3 power law that is familiar from Kolmogorov turbulence. Simulations can at present only show the beginnings of this development; see Figure 1.

Saturation of large-scale dynamos

What we said about the saturation of small-scale dynamos does in principle also apply to large-scale dynamos. However, in certain cases large-scale dynamos can saturate slowly or at substantially lower field strengths due to the effects of magnetic helicity conservation. This applies in particular to the case of closed or periodic boundary conditions that are often used in numerical simulations. Writing the induction equation ∂B/∂t=-∇×E in terms of the magnetic vector potential A, where B=∇×A, we have ∂A/∂t=-E-∇φ, where φ is the electrostatic potential. This way we obtain an evolution equation for the magnetic helicity <A·B> of the form \[ {\partial\over\partial t}\langle\mathbf{A}\cdot\mathbf{B}\rangle= -2\langle\mathbf{E}\cdot\mathbf{B}\rangle. \] Using the fact that E=-U×B+J/σ, where σ=1/(ημ₀) is the electric conductivity, we see that \[ {\partial\over\partial t}\langle\mathbf{A}\cdot\mathbf{B}\rangle= -2\eta\mu_0\langle\mathbf{J}\cdot\mathbf{B}\rangle, \]

so the magnetic helicity can only evolve resistively. This is what slows down the saturation of those large-scale dynamos where magnetic helicity plays a role and what can thus lead to catastrophically low saturation levels.

The evolution equation for the magnetic helicity of the large scale field, \(\overline{\mathbf{B}}\ ,\) is similar to that for the total field, except that the mean electromotive force produces additional magnetic helicity proportional to \[ \langle\overline{\mathbf{\mathcal{E}}}\cdot\overline{\mathbf{B}}\rangle =\alpha\langle\overline{\mathbf{B}}^2\rangle -\eta_{\rm t}\mu_0\langle\overline{\mathbf{J}}\cdot\overline{\mathbf{B}}\rangle \ .\]

The evolution equation for the magnetic helicity in the small scale field, <a·b>, must then be amended by the same term, but with the opposite sign so that the sum of both terms adds up to the original helicity equation. This leads to the production of excess small scale magnetic helicity which, in turn, modifies the total (magnetic and kinetic) small scale helicity and thus quenches the α effect (Pouquet, Frisch, Leorat 1976; Blackman and Field 2002)

Astrophysical bodies escape this type of quenching by developing magnetic helicity fluxes inside the dynamo domain and out through the boundaries. In the case of the sun (e.g. Blackman and Brandenburg 2003) the shedding of magnetic helicity may occur mostly via coronal mass ejections, but this is still an open research area. P. J. E. Peebles and Bharat Ratra (2003). The cosmological constant and dark energy. Reviews of Modern Physics 75: 559–606. Archived from the original on 2003. http://www.arxiv.org/abs/astro-ph/0207347. .

Applications

Sun

Figure 2: Longitudinally averaged radial component of the observed solar magnetic field as a function of cos(colatitude) and time. Dark (blue) shades denote negative values and light (yellow) shades denote positive values. Note the sign changes both in time and across the equator. Adapted from Brandenburg & Subramanian (2005), using Kitt Peak Magnetogram data provided by R. Knaack (Zürich).

The sun has a magnetic field that manifests itself in sunspots through Zeeman splitting of spectral lines. It has long been known that the sunspot number varies cyclically with a period between 7 and 17 years. The longitudinally averaged component of the radial magnetic field of the sun shows a markedly regular spatio-temporal pattern where the radial magnetic field alternates in time over the 11 year cycle and also changes sign across the equator. One can also see indications of a migration of the field from mid latitudes toward the equator and the poles. This migration is well seen in a sunspot diagram, which is also called a butterfly diagram, because the pattern formed by the positions of sunspots in time and latitude looks like a sequence of butterflies lined up along the equator (see Figure 2).

Distributed versus overshoot dynamos

There is at present no clear consensus as to whether the solar dynamo operates in the entire convection zone of the sun (distributed dynamo) or whether it works preferentially at the bottom of the convection zone in the so-called tachocline, where the differential rotation changes sharply into a rigidly rotating profile.

Advection dominated dynamos

In recent years the idea of advection-dominated or flux transport dynamos has been developed. Here the cycle period and the propagation direction of the magnetic activity pattern is determined by a large scale meridional circulation. There is now observational evidence that such a circulation of suitable magnitude and direction does indeed exist.

Galaxies

Observations of radio emission of spiral galaxies show well developed large scale patterns. The flows responsible for dynamo action in galaxies include both the differential rotation and supernova-driven turbulence. The e-folding time for dynamo action may be a significant fraction of the age of the universe, so the initial seed magnetic fields must not be too weak. As possible sources of seed magnetic fields outflows from active galaxies and starburst galaxies, primordial magnetic fields, and battery effects are being discussed. Primordial magnetic fields may have been generated during one of the phase transitions during the early Universe. For reviews see Beck et al. (1996) and Widrow (2002).

Accretion discs

The rotation law in accretion discs is Keplerian, resulting from a balance between centrifugal and gravitational forces (v_φ²/r=GM/r²), where v_φ is the azimuthal velocity, r is the radius from the central object, G is Newton's gravitational constant, and M is the mass of the central object.) Such a rotation law implies that the specific angular momentum increases with radius, so such discs are hydrodynamically stable, but in the presence of a magnetic field, points that are separated in space may be coupled nonlocally. Under such conditions two points in a Keplerian orbit that are being pulled together will actually move further apart from each other. This is the essence of the magnetorotational instability (MRI).

In practice, because of large Reynolds numbers, the MRI leads to turbulence. This turbulence, in turn, can lead to dynamo action. Simulations in the presence of stratification have shown that there can be an α effect, although the sign is opposite to the one naively to be expected.

Early Universe

There are several mechanisms that could potentially generate magnetic fields during one of the early phase transitions (e.g. the electroweak phase transition). In the absence of any additional driving, the flows would be primarily a consequence of the driving from the Lorentz force. Since that time, and before gravitational clumping takes over, the magnetic field would only be decaying. The field might however be helical, in which case a magnetic cascade would be possible that would transfer magnetic energy to larger scales.

Laboratory Plasma Dynamos

Dynamo effects are being discussed in connection with various plasma experiments. These are usually relatively short-lived events (nanoseconds to microseconds) that are initiated by an electric discharge. Prime examples are the Reversed Field Pinch (e.g. Ji and Prager 2002) and Spheromak (e.g. Bellan 2001) configurations, where velocity and magnetic field fluctuations generated from a current-driven instability (current parallel to the magnetic field) correlate to produce a turbulent electromotive force. This generates a poloidal field via an analogous α effect to that discussed above. Here, unlike in the case of kinematic velocity driven dynamos, the α effect is driven by an externally imposed magnetic fieldFigure 2

Laboratory Liquid Metal Dynamos

Over the past few years it has been possible to demonstrate sustained self-excited dynamo action in the laboratory using liquid sodium (Gailitis et al. 2001, Stieglitz & Müller 2001, Monchaux et al. 2007). Although the magnetic Reynolds numbers are currently lower than what is possible with simulations, the fluid Reynolds numbers are much higher than what can be achieved in simulations because of the very small magnetic Prandtl number. In this respect we may expect a lot of new results to come from laboratory experiments.