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tokens: field, and the interaction between them in a conservative way using Godunov, finite-difference methods (Rezzolla Zanotti 2013 ). The radiation is evolved adopting the M1 closure scheme (Levermore 1984 ). The simulations assume axisymmetry and run in 2.5 dimensions, that is, allowing for non-zero azimuthal vector components. We assume a black hole mass of 10 M ⊙ and evolve the models for roughly 1.2 s, corresponding to ∼165 orbital periods at the pressure maximum of the initial model. The numerical domain extends from r min = 5 to r max = 14 along the radial direction, and from θ = π/2 − 0.3 to θ = π/2 + 0.3 along the θ direction, and it is covered by a uniform (in radius and polar angle) grid composed of 448 × 448 grid-points. We note that the radial resolution is therefore Δ r = 0.02. Standard outflow/no-inflow boundary conditions are adopted at all the boundaries. 2 For simplicity, in this work we adopt only the scattering opacity κ s = 0.34 cm 2 g −1 , and we account for the Comptonization using the ‘blackbody’ method (see Sadowski Narayan 2015 ). Since we do not assume any external heating source, the tori are expected to cool down by emitting the radiation that supports them. 3 RESULTS 3.1 Torus dynamics All the models have been evolved up to t = 25 000, corresponding to ∼165 orbital periods computed at the maximum density radius r c = 8.35. Two purely hydrodynamic models, namely Hydro43 and Hydro53, have been also studied, in order to provide a comparison with respect to the radiative models, which is especially useful in the spectral analysis of Section 3.2 . Because of the radial perturbation introduced, the tori start oscillating in the potential well. However, since the initial models are significantly below the marginally stable configuration, no appreciable mass accretion is induced by the oscillations and the periodic behaviour can last for a long time. In Fig. 1 , a full oscillation sequence at t ∼ 0.1 s is monitored through six snapshots for the model Rad30. The left- and the right-hand panels report the distribution of the rest-mass density and of the radiation energy density, respectively. The arrows in the radiative panels reflect the free-streaming radiation leaving the torus surface. As expected, during the evolution the energy density of the radiation remains concentrated towards the centre of the torus, where the optical depth is the largest. Radiation diffuses out following the gradient of radiative energy density and ultimately leaves the surface of the torus. The distribution of the out coming radiation is determined by the shape of the photosphere. In the case of the tori studied here, most of the radiation is emitted roughly in the direction perpendicular to the equatorial plane. Figure 1. Illustration of a typical oscillation sequence for the model Rad30 through six snapshots (from top to bottom). Left-hand panels: distribution of the rest-mass density with the velocity represented by orange arrows. Right-hand