We calculate the structure and short-term evolution of a gamma-ray burst (GRB) central engine in the form of a turbulent torus accreting onto a stellar mass black hole. Our models apply to the short GRB events, in which a remnant torus forms after the neutron star-black hole or a double neutron star merger and is subsequently accreted. We study the two-dimensional, relativistic models and concentrate on the effects of the black hole and flow parameters as well as the neutrino cooling. We compare the resulting structure and neutrino emission to the results of our previous one-dimensional simulations. We find that the neutrino cooled torus launches a powerful mass outflow, which contributes to the total neutrino luminosity and mass loss from the system. The neutrino luminosity may exceed the Blandford-Znajek luminosity of the polar jets and the subsequent annihilation of neutrino-antineutrino pairs will provide an additional source of power to the GRB emission.
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