Non-linear evolution of a non-axisymmetric disc instability

A finite-difference hydrodynamics code is used to calculate the non-linear, time-dependent evolution of unstable non-axisymmetric modes in a narrow torus orbiting a black hole. The simulation is restricted to two dimensions in (r, ϕ) by assuming vertical hydrostatic equilibrium in the torus. The code is tested using axisymmetric radial breathing mode solutions, and by comparison with results obtained from linear perturbation analysis. The effects of differencing technique and grid resolution are considered. The evolution of three unstable modes demonstrates that the growth rate predicted by linear theory remains valid in the non-linear regime even for perturbations approaching $$\delta \varrho /\varrho = 1.$$ The end-point of the growth of an azimuthal mode of order m is the breakup of the disc into a series of m orbiting blobs, or ‘planets’.