Dynamo models

(1/r^2) d(r^2 dB/dr)/dr + (1/) ((2/3)(B/r^2) d(r^2u)/dr + u dB/dr) = S(r), (8)

where u is the inflow velocity, S the source of energy input, and the (anomalous) resistivity. Unfortunately, a dynamo driven by galaxy wakes cannot generate the required fields (De Young 1992), and their choice of input model was a poor representation of a cooling flow cluster like Perseus. A real cooling flow typically has r (White & Sarazin 1988; Thomas, Fabian & Nulsen 1987), and a density profile ~ 1/r near the cluster centre steepening to ~ 1/r^2 at larger radii. This then implies that u is probably nearly constant.

I have modelled magnetic fields according to equation (8), for a variety of input parameters. The possible solutions are very varied. The results depend on the input parameters, and B is not necessarily everywhere positive. The simple flux freezing models give an equally good (if not better) fit to the data, with a much simpler model with fewer assumptions.

One problem is that equation (8) may not be a good description of the magnetic field in a cluster. An alternative is to consider a model in which a source S(r) generates magnetic energy that is then advected inwards by the flow and compressed. This situation is described by the equation

( d <B^2> / d t) = -u. <B^2> - (4/3) <B^2> .u + S(r). (9)

There are three differences between this model and that of equation (8). It is dimensionally correct, in that an energy source S(r) creates magnetic energy rather than flux. It is time dependent: the field does not achieve a steady state, as energy is continually being injected. Diffusion of the magnetic field has been neglected: note that the more centrally concentrated models of Burns et al. were those with the least diffusion.

Equation (9) can be recast in the form

B^2(r) = _r^{r_0} S(r') ( (r) / (r') )^4/3 d r' / -u(r') (10)

where the initial radius $r_0$ that gas now at r started from is determined by

_r^{r_0} d r' / -u(r') = t (11)

where t is the age of the system.

It is difficult to know what is a reasonable choice for the source term S(r). For turbulence generated by galaxy wakes, S(r) may be proportional to the galaxy density. For merger induced turbulence, little can be said about S(r). One possibility is to assume S(r) to be constant. The rationale for this is that the merger encounter starts at large radii, so there is no reason for the energy injection to be centrally concentrated.

Fig 2. Simulated cluster magnetic field as a function of radius for a dynamo model with inflow.

In Fig. 2 I present the results of one such model. In this model, S(r) is constant, the gas density is given by equation (7), and the inflow velocity is

u(r) = -2/[1+(r/0.25)^2], (12)

which gives a flow time of unity at r=0.25, with r_c set to unity. The overall result is not unexpected. Magnetic fields are generated throughout the cluster and are compressed by the inflow in the central regions. The flow has little or no effect beyond the central cooling region. This is because the flow only affects the magnetic field in regions where the flow time is comparable to or less than the age of the system. At large radii little flow has occurred and, so that the effect of compression and advection on the field is negligible.

On to:

• Images

• Section 4