Humpy LNRF-velocity profiles in accretion discs orbiting nearly extreme Kerr black holes. A possible relation to QPOs

Change of sign of the LNRF-velocity gradient has been found for accretion discs orbiting rapidly rotating Kerr black holes with spin a > 0.9953 for Keplerian discs and a > 0.99979 for marginally stable thick discs. Aschenbach (2004) has identified the maximal rate of change of the orbital velocity within the "humpy" profile with a locally defined critical frequency of disc oscillations, but it has been done in a coordinate-dependent form. We define the critical "humpy" frequency H in general relativistic, coordinate independent form, and relate the frequency defined in the LNRF to distant observers. At radius of its definition, so-called "humpy" radius r_h, the "humpy" frequency H is compared to the radial (R) and vertical (V) epicyclic frequencies and the orbital frequency of the disc. For Keplerian thin discs, we show that the epicyclic resonance radii r_31 and r_41 (with V:R = 3:1 or 4:1) are located in vicinity of r_h where efficient triggering of oscillations with frequencies ~ H could be expected. Asymptotically (for 1-a < 10^(-4)) the ratio of the epicyclic and Keplerian frequencies and the humpy frequency is nearly constant, i.e., almost independent of spin, being for the radial epicyclic frequency R:H ~ 3:2. For thick discs the situation is more complex due to dependence on distribution of the specific angular momentum l determining the disc properties. For l = const tori and 1-a < 10^(-6) the frequency ratios of the humpy frequency and the orbital and epicyclic frequencies are again nearly constant and independent of both a and l, being for the radial epicyclic frequency R:H close to 4. In the limiting case of very slender tori (l ~ l_ms) the epicyclic resonance radius r_41 ~ r_h for spin 1-a < 2x10^(-4).