Numerical study of the non-linear stage of thermal instability in cooling flows - II. The non-linear perturbation in the case of a spherically symmetric background flow

We have performed two-dimensional numerical hydrodynamical calculations of thermal instability in cooling flows in a spherical coordinate system, in order to investigate the evolution of the non-linear axisymmetric perturbation. If the relative density contrast is smaller than the critical value the perturbation decays, due to the formation of the vortex ring before the perturbation cools. We carry out one-dimensional spherically symmetric calculations, and show that even if the density contrast is smaller than the critical value, in the case of the spherically symmetric perturbation it can cool. Our numerical results show that the non-radial motion is very important for the evolution of the non-linear perturbation. We give the criterion for the amplitude and size of perturbations which are thermally unstable. We show that only if the density contrast of the perturbation is very large, can the thermal instability develop in the spherically symmetric background cooling flow. These results indicate that the apparently pervasive and extensive mass deposition occurring in real cooling flows remains unexplained and what makes thermal instability occur - for example a magnetic field may be needed.