Cooling flows are turbulent (Heckman et al. 1989; Loewenstein & Fabian 1990). In part, this turbulence is due to the multiphase nature of the gas, with denser gas parcels falling in and stirring the rest of the gas. The origin of the density fluctuations is difficult to explain without magnetic fields, as otherwise conduction or gas motions erase the inhomogeneities (Balbus 1991). Conduction in a tangled magnetic field automatically generates a multiphase medium (Tribble 1989, 1992b). The turbulence in the inner regions will stir the gas, stretching and possibly amplifying the magnetic field.
This turbulent stirring will also isotropize the magnetic field. Simple inflow gives a field that is predominantly radial (Soker & Sarazin 1990). The turbulent velocities are much greater than the mean inflow velocity, which is of order 10 km/s, so that a characterization of the flow as a smooth steady inflow is incorrect. Rather, an individual element of gas in the flow is in constant motion. Its shape and orientation are constantly changing, but its mean position falls slowly inwards on a 10^10 yr timescale. The overall effect is isotropic compression of the field. As the gas cools further, the magnetic field becomes stronger and eventually becomes both dynamically and energetically important (Tribble 1991a).
Note that, outside of the central cooling flow region where the field is compressed and amplified by the cooling flow, there is no evidence for a decline in magnetic field strength with radius. The RM profile is consistent with constant magnetic properties, although a decline in field strength could be matched by a rise in scale length (Kim et al. 1991). The overall profile of the Coma radio halo (Kim et al. 1990) is not sharply peaked, so that either the magnetic field strength does not fall much with increasing radius, or the density of relativistic electrons must increase with radius (which seems unlikely).
The origin of the relativistic electrons must also be considered, for even if the electrons have recently diffused into the central region of enhanced field, they can be at most 5x10^8 yr old. This assumes that the break in the electron energy spectrum is given by inverse Compton losses, but that they are now radiating in a 10 field at =90 cm. Either a currently active source (such as the active nucleus in NGC 1275) is required, or the electrons have been reaccelerated.
Reacceleration in the turbulent cooling flow is possible. Second order Fermi acceleration (Fermi 1949) may be sufficient, as the level of turbulence is high. If the clouds have velocities v then the average energy gain per collision is
E / E 2 (v/c)^2 5x10^-6 (v/500 km/s)^2. (3)
This can maintain the relativistic particles in a steady state as long as the collisions are sufficiently frequent. The collision time must be shorter than
t_coll = 5x10^-6 (v/500 km/s)^2 t_age. (4)
For parameters appropriate to the Perseus cluster, second order Fermi acceleration can balance losses as long as the time between collisions is less than 1000 yr.
To be effective, second order Fermi acceleration therefore requires the typical fluctuation scale in the cooling flow to be a few hundred pc or less. Infalling dense regions are broken up by hydrodynamical stresses (Nulsen 1986), until they reach sizes of ~200 pc when further breakup is prevented by magnetic fields (Tribble 1991a). Resolved Faraday rotation observations detect magnetic structures on scales of a few kpc (Dreher, Carilli & Perley 1987; Taylor et al. 1990; Perley & Taylor 1991), although considerable finer scale structure could be present and remain undetected. Therefore, second order Fermi acceleration is just feasible in cooling flows. At the very least, reacceleration will slow the fading of the radio emission and modify the radio spectrum.
Given that second order Fermi acceleration can almost sustain the relativistic electron population, one then naturally wonders whether any more efficient processes are possible. First order acceleration will occur due to straightforward compression in the large scale cooling flow, but this appears inadequate as this will accelerate the electrons on the cooling flow timescale which is very long ~10^10 yr, although Becker (1992) has considered this possibility and concludes that first order acceleration can lead to a sustained minihalo. His success is primarily due to his choice of a 350 km/s inflow velocity at his input radius of 10 kpc, which is rather high for a multiphase cooling flow. Also possible is acceleration within small blobs cooling out of the inhomogeneous flow. As the gas blobs cool they contract and any relativistic electrons that are trapped in the blob will be compressed and accelerated. This may well be important as the cooling and hence contraction and reacceleration timescale for an individual blob of gas can fall below 10^8 yr, so that cooling blobs could accelerate electrons which then diffuse out to the rest of the cooling flow. Other processes that may be important are magnetic field reconnection and shocks in the cooling blobs. I conclude that there are a variety of processes in cooling flows (especially a strong cooling flow like Perseus) that may be able to keep relativistic electrons radiating at observable frequencies.