Grad-shafranov Approach to Axisymmetric Stationary Flows in Astrophysics

My lecture is devoted to the analytical results available for a large class of axisymmetric stationary flows in the vicinity of compact astrophysical objects. First, the most general case is formulated corresponding to the axisymmetric stationary MHD flow in the Kerr metric. Then, I discuss the hydrodynamical version of the Grad-Shafranov equation. Although not so well-known as the full MHD one, it allows us to clarify the nontrivial structure of the Grad-Shafranov approach as well as to discuss the simplest version of the 3+1-split language – the most convenient one for the description of ideal flows in the vicinity of rotating black holes. Finally, I consider several examples that demonstrate how this approach can be used to obtain the quantitative description of the real transonic flows in the vicinity of rotating and moving black holes.