  We present general relativistic hydrodynamics simulations of constant
specific angular momentum tori orbiting a Schwarzschild black hole. These tori
are expected to form as a result of stellar gravitational collapse, binary
neutron star merger or disruption, can reach very high rest-mass densities and
behave effectively as neutron stars but with a toroidal topology (i.e.
``toroidal neutron stars''). Our attention is here focussed on the dynamical
response of these objects to axisymmetric perturbations. We show that, upon the
introduction of perturbations, these systems either become unstable to the
runaway instability or exhibit a regular oscillatory behaviour resulting in a
quasi-periodic variation of the accretion rate as well as of the mass
quadrupole. The latter, in particular, is responsible for the emission of
intense gravitational radiation whose signal-to-noise ratio at the detector is
comparable or larger than the typical one expected in stellar-core collapse,
making these new sources of gravitational waves potentially detectable. We
discuss a systematic investigation of the parameter space both in the linear
and nonlinear regimes, providing estimates of how the gravitational radiation
emitted depends on the mass of the torus and on the strength of the
perturbation.
