Plasma instabilities and magnetic-field growth in clusters of galaxies

We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., beta>Re^{1/2}. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10^{-18} G to dynamically important strengths of ~10 microG on cosmologically trivial time scales (~10^8 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence.