Annu. Rev. Astron. Astrophys. 1996. 34: 155-206
Copyright © 1996 by . All rights reserved

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4.4. Basic Galactic Dynamo Models

The simplest form of the mean field (alpha2 Omega) dynamo equation (2) that retains the basic physics (e.g. Parker 1979, Roberts & Soward 1992) is, in dimensionless form,

\frac{\partial
 \overline{\mathrm{\bf B}}}{\partial t} = \nabla \times 
 (R_{\omega} \overline{\mathrm{\bf u}} \times \overline{\mathrm{\bf B}} +
 R_{\alpha} F \alpha \overline{\mathrm{\bf B}} - G \eta_t , \nabla \times
 \overline{\mathrm{\bf B}} ) , (5)

where F(bar{B}, Omega) = (1 + bar{B}2 / Beq2)-1 is the simplest form of "alpha-quenching" and G(bar{B}, Omega) = 1. Distances and times are measured in units of h* and h*2 / etat*, respectively, where bar{u} = Omega × r and alpha and etat are normalized by appropriately chosen characteristic values denoted by asterisks. Dimensionless numbers

R_{\omega} = h_{\ast}^2
 \Omega_{\ast} / \eta_{t \ast} , \quad R_{\alpha} 
 = h_{\ast} \alpha_{\ast} / \eta_{t \ast} (6)

characterize the amplification of magnetic field by shearing of the mean velocity field and the alpha-effect, respectively. Using Equation (3), alpha and etat can be expressed through observable parameters of the galaxies such as the rotation curve, rms velocity and scale, and the thickness of the ionized disk (a function of r). The quenching effects also require that the gas density is specified as a function of position. Equation (5) must be supplemented by boundary conditions. In models that treat the disk alone, these are usually vacuum boundary conditions in which one assumes that the turbulent magnetic diffusivity outside the disk is infinite. This proves to be a reasonable approximation to reality (Moss & Brandenburg 1992), as etat varies by perhaps a factor of about 50 between the disk and the halo (see Brandenburg et al 1993, Poezd et al 1993). However, more advanced treatments employ the embedded disk model (Stepinski & Levy 1988). This includes a spherical galactic halo and appropriate boundary conditions are imposed at the surface of the halo, whereas the disk is modeled by appropriate distributions of bar{u}, alpha, and etat. This concept has proved sufficiently adaptable to accommodate developing requirements, such as the inclusion of a flared disk, an alpha-effect extending into the halo (Section 7.1), and/or a galactic wind (Section 7.2).

Initial conditions for (5) are often chosen to correspond to a weak seed field. Exponentially growing solutions then arise, bar{B} propto exp(Gamma t), provided the dynamo number D = Ralpha Romega exceeds a certain value Dcrit approx 10. Using Equation (3) one can show that D appeq 9(h* Omega* / v)2. For h* appeq 500 pc, Omega* appeq 20 km s-1 kpc-1, and v appeq 10 km s-1 we obtain D appeq 10, so that the dynamo is expected to operate under typical galactic conditions. For D >> Dcrit, the growth rate is estimated as gamma appeq CD1/2 etat / h*2 appeq C(alpha* Omega* h*)1/2, with C a quantity of order unity depending on the galaxy model. A typical model gives gamma-1 appeq 5 × 108 yr; this is a lower estimate for the dynamo timescale. [We note, however, that the timescale for the magnetic shear instability is the inverse Oort a-value (Balbus & Hawley 1992), which is somewhat shorter (108 yr). This mechanism leads to dynamo action (Brandenburg et al 1995b) that would lower the effective value of gamma-1.]

All classical dynamo models predict that the large-scale field in the outer parts of the disks in spiral galaxies has quadrupole (S0) symmetry, that is, both bar{B}r and bar{B}phi are even in z, whereas bar{B}z is odd (Parker 1971, Vainshtein & Ruzmaikin 1971). This mode is dominant in a disk (but not in a sphere). A dipole (A0) mode, with both bar{B}r and bar{B}phi odd in z and bar{B}z even, can be dominant near the axis of the disk. The large-scale field is amplified until alpha becomes significantly quenched, which occurs when bar{B} is of order Beq, typically a few µG.

Field evolution is qualitatively different if the initial field is a random field with strength close to Beq. There is then no kinematic stage, because alpha-quenching is immediately important. The dynamo acts then to change the scale and spatial distribution of the field. An example of typical evolution of the magnetic field in a spiral galaxy as envisaged by the standard dynamo model is illustrated in Figure 7.

Figure 7

Figure 7. Face-on views showing the evolution of the magnetic field in a model of M83 (from KJ Donner & A Brandenburg, in preparation). The lower panel gives an edge-on view for t = 8.1 Gyr.

Over the past 5 to 10 years a large number of galactic dynamo models have been developed. The minimum ingredient of such models is a flat geometry. Such models were first computed in the 1970s, but computers can only now reach the regime applicable to the theory of asymptotically thin disks (Walker & Barenghi 1994 and references therein). Galactic models share the somewhat frustrating property that nonaxisymmetric solutions are always harder to excite than axisymmetric ones (Ruzmaikin et al 1988a, Brandenburg et al 1990, Moss & Brandenburg 1992). Not even the inclusion of anisotropies seems to change this conclusion (Meinel et al 1990). Stable nonaxisymmetric solutions have only been found if alpha and etat vary azimuthally (Moss et al 1991, 1993a;, Panesar & Nelson 1992). The inclusion of nonlinear effects demonstrated that mixed parity states can persist over rather long times, even comparable with galactic lifetimes (Moss & Tuominen 1990, Moss et al 1993a). When eta-quenching is included (G neq 1), linear calculations show that A0 and S1 modes may be more readily excited (Elstner et al 1996).

In most of these models alphaij and etaijk were adopted using qualitative forms of (3) and (4), calibrated by observations. Significant conceptual progress has been made recently by deriving all these functions consistently from the same turbulence model, which includes stratification of density and turbulent velocity, derived from a condition of hydrostatic equilibrium (Schultz et al 1994, Elstner et al 1996). One should not forget, however, that such models still rely on important approximations and simplifications (e.g. FOSA and the lack of a reliable turbulence model).

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