**Publication date:** Mar 2007

**Abstract:**

Context: 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. Such a "humpy" LNRF-velocity profiles occur just above the

marginally stable circular geodesic of the black hole spacetimes. Aims: 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 that should be corrected. Methods: We

define the critical "humpy" frequency νh in general

relativistic, coordinate independent form, and relate the frequency

defined in the LNRF to the distant observers. At radius of its

definition, the resulting "humpy" frequency νh is compared

to the radial νr and vertical νv epicyclic

frequencies and the orbital frequency of the discs. We focus our

attention to Keplerian thin discs and perfect-fluid slender tori where

the approximation of oscillations with epicyclic frequencies is

acceptable. Results: In the case of Keplerian discs, we show that

the epicyclic resonance radii r3{:1} and r4{:1}

(with ν_v{:}ν_r=3{:}1, 4{:}1) are located in vicinity of the

"humpy" radius rh where efficient triggering of oscillations

with frequencies νh 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 a, being for the radial epicyclic frequency

ν_r{:}νh ˜ 3{:}2. In the case of 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 ν_r{:}νh close to 4. In the

limiting case of very slender tori (ℓ˜ℓms) the

epicyclic resonance radius r4{:1}˜ rh for

all the relevant interval of 1-a<2× 10-4. Conclusions: .The hypothetical "humpy" oscillations could be related

to the QPO resonant phenomena between the epicyclic oscillations in both

the thin discs and marginally stable tori giving interesting predictions

that have to be compared with QPO observations in nearly extreme Kerr

black hole candidate systems. Generally, more than two observable

oscillations are predicted.

**Authors:**

Stuchlík, Z.; Slaný, P.; Török, G.;