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tokens: outward flux is not conserved; its losses are a source for the photon angular momentum flux (Fig. 4 ). But in the model solution, the disc can only supply the flux from the relatively small amount of material within the ISCO, because just outside the ISCO the disc velocity is outward: it is not accreting! This cannot go in indefinitely – unless the outward flux is somehow being fed externally. The fact that the integrated total luminosity of our solutions formally diverges as t → ∞ is a clue: something must be feeding this. This could in principle be a magnetic couple to the black hole itself (or a vertical accretion process), but how would a turbulent/viscous model resolve this issue? Our solutions must not be the true t → ∞ disc solutions; they represent instead an extended phase of disc evolution in which the accretion is stalled. While this may be very useful for understanding observations, it raises an important point of physics: how does a real disc ‘unstall’ itself? Figure 4. The normalized angular momentum flux divergences, stemming from the three terms in equation ( 47 ), at a particular time (τ = 0.67) in the disc evolution. Throughout the bulk of the disc, the angular momentum loss balance is predominantly between the radial flow (the first term) and the turbulent stresses (second term). In the region r ≲ 8 r g , however, the photon losses (third term) dominate over the radial flow stress. This balance can be maintained as an extended phase, but not indefinitely. 4.3 Modelling the angular momentum gradient in the inner disc The presence of a non-zero radial velocity means that the azimuthal motion is only approximately Kerr metric circular. We have argued that the correction is second order in the fluctuation amplitude, but , the derivative of the angular momentum, appears as a singular denominator in the evolution equation. Since strict adherence to the idealized model leads to long-term unsustainable behaviour, it is of interest to consider a generic change to the angular momentum gradient of the form where now U φ indicates the mean fluid angular momentum of equation ( 10 ), and is the idealized angular momentum of a Kerr circular orbit. The function ε( r ) is left somewhat arbitrary, but it must satisfy two simple properties: (i) it reaches a maximum at the ISCO, and (ii) it vanishes sharply at large radius. The idea is that the assumption of purely circular orbits is a good approximation at large radii, but become progressively worse as the ISCO is approached. The development of this theory is the topic of the companion paper. We note here, however, that a modification of the angular momentum gradient in the inner disc regions would also affect the steady-state modelling of relativistic thin discs. The steady-state solution of equation ( 10 ) leads to a disc temperature profile where F 1 and F 2 are constants related to the mass and angular momentum flux throughout the disc, respectively. A substantial modification of in the