  We present a comprehensive numerical study of the dynamics of relativistic
axisymmetric accretion tori with a power-law distribution of specific angular
momentum orbiting in the background spacetime of a Kerr black hole. By
combining general relativistic hydrodynamics simulations with a linear
perturbative approach we investigate the main dynamical properties of these
objects over a large parameter space. The astrophysical implications of our
results extend and improve two interesting results that have been recently
reported in the literature. Firstly, the induced quasi-periodic variation of
the mass quadrupole moment makes relativistic tori of nuclear matter densities,
as those formed during the last stages of binary neutron star mergers,
promising sources of gravitational radiation, potentially detectable by
interferometric instruments. Secondly, $p$-mode oscillations in relativistic
tori of low rest-mass densities could be used to explain high frequency
quasi-periodic oscillations observed in X-ray binaries containing a black hole
candidate under conditions more generic than those considered so far.
