科研之友
微信
新浪微博
Facebook
分享链接
摘要
We study linear transient dynamics in a thin Keplerian disc, employing a method based on a variational formulation of the optimization problem. It is shown that in a shearing sheet approximation due to a prominent excitation of density waves by vortices, the most rapidly growing shearing harmonic has an azimuthal wavelength lambda(y), of the order of the disc thickness H, and its initial shape is always nearly identical to a vortex having the same potential vorticity. Also, in the limit lambda(y) >> H, the optimal growth G alpha (Omega/kappa)(4), where Omega and kappa denote local rotational and epicyclic frequencies, respectively. This suggests that the transient growth of large-scale vortices can be much stronger in areas with non-Keplerian rotation (e. g. in the inner parts of relativistic discs around black holes). We estimate that if the disc is already in a turbulent state with effective viscosity given by the Shakura parameter alpha < 1, the considered large-scale vortices with wavelengths H/alpha > lambda(y) > H have the most favourable conditions to be transiently amplified before they are damped. At the same time, turbulence is a natural source of the potential vorticity for this transient activity. We extend our study to a global spatial scale, showing that global perturbations with azimuthal wavelengths more than an order of magnitude greater than the disc thickness are still able to attain the growth of dozens of times in a few Keplerian periods at the inner boundary of the disc.
