The first point to note is that the radio remnant is, roughly, in pressure equilibrium with its surroundings (for example Cygnus A, Arnaud et al. 1984). Sources will not, therefore, suffer catastrophic adiabatic losses (Gull & Northover 1973; Longair, Ryle & Scheuer 1973). Instead, it will diffuse and mingle with the surrounding thermal plasma. While this is a slow process, there is a long time available because most of the magnetic energy is input at early times. As a result, the magnetic field is expected to be well mixed with the thermal plasma at the current epoch.
The most likely driving force for the field dispersion is the growth of the cluster by hierarchical clustering, accreting companions and subclusters. As an extreme case, this is the same merger process that tangles the field and leads to radio halos (Tribble 1993). During and immediately after the merger event, the thermal plasma is likely to contain a large amount of turbulence, which can act in two ways. On large scales, bubbles of magnetized relativistic plasma are moved away from their initial location, distributing pockets of magnetic field throughout the cluster. On small scales, turbulence tangles the field and brings it into close contact with thermal plasma. Diffusion can then act on these small scales to complete the mixing of the thermal and relativistic plasma components.
The diffusion of the magnetic field into the thermal plasma necessarily involves dissipation, so that field is destroyed. However, for the same reason that we require small scale turbulence to drive the diffusion, because diffusion is inefficient on large scales, only the small scale field is destroyed. To see this, consider the evolution of the Fourier components of the field. Initially, the field is localized and ordered. Small scale turbulence adds power on small scales and enlarges the volume in which there is a magnetic field, by intermingling the thermal and relativistic plasmas. Then dissipation damps off the small scale components, leaving a large-scale field present in the thermal plasma.
The relic radio sources (Harris et al. 1993) may provide a link between the powerful radio sources and the fully mixed magnetized intracluster medium. Harris et al. suggest that the relic sources are remnants of old radio galaxies that have not yet dispersed. In this case the relic sources are remnants of powerful radio sources that have not yet been incorporated into the intracluster medium. Although not in rich clusters, the relics are in the neighbourhood of clusters. Currently in calm surroundings, they will be fully mixed into the intracluster medium when their local structure is accreted by a rich cluster. Bicknell, Cameron & Gingold (1990) have also considered the diffusion of magnetic field from a radio source into the surrounding thermal medium. Their detailed numerical model shows the essential features described above. Kelvin-Helmholtz instabilities give large-scale mixing of relativistic and thermal plasma, and a small amount of turbulence allows the field to diffuse into the thermal plasma.
There are two important extra features in the model presented in the current paper that are not present in the model of Bicknell et al. The first is that cluster mergers input turbulence into the intracluster medium, the large scale motions spreading the magnetic field throughout large regions of the cluster, and the small scale turbulence allowing efficient diffusion of the field into the thermal plasma. The second is that instead of considering a single source, all the sources present in a cluster throughout its history are allowed to contribute to the field in the cluster today.