Toroidal LNRF-velocity profiles in thick accretion discs orbiting   rapidly rotating Kerr black holes
  We show that in the equatorial plane of marginally stable thick discs (with uniformly distributed specific angular momentum the orbital velocity relative to the locally non-rotating frames (LNRF) has a positive radial gradient in the vicinity of black holes with spin a>0.99979. The change of sign of the velocity gradient occurs just above the center of the thick toroidal discs, in the region where stable circular geodesics of the Kerr spacetime are allowed. The global character of the phenomenon is given in terms of topology changes of the von Zeipel surfaces (equivalent to the equivelocity surfaces in the tori with constant specific angular momentum distribution). Toroidal von Zeipel surfaces exist around the circle corresponding to the minimum of the equatorial LNRF velocity profile, indicating a possibility of development of some vertical instabilities in those parts of marginally stable tori with positive gradient of the LNRF velocity. Eventual oscillatory frequencies connected with the phenomenon are given in a coordinate-independent form. 
Zdenek Stuchlik, Petr Slany and Gabriel Torok