## 2. Prescription for turbulent viscosity## 2.1. Reynolds viscosity as the general caseAs noted in Sect. 1 the need for some kind of turbulent
viscosity in accretion disks is generally recognized, as is the very
high Reynolds number of the flow in the absence of such a viscosity.
Here, we wish to investigate in particular the case of accretion disks
where the magnetic fields do not play an important role. In these
circumstances, it seems reasonable to assume that the turbulence is
driven by the velocity field in the disk, which itself has
characteristic length and velocity scales As has been pointed out, for example, by Lynden-Bell & Pringle (1974) and Thompson et al. (1977), the high corresponding Reynolds number should lead to the generation of turbulence and hence to a steady enhancement in the effective viscosity. This will continue until the Reynolds number has been reduced to approximately its critical value . Typical values for in laboratory flows are of the order of . This limiting Reynolds viscosity can, in this case, be as high as where is a constant satisfying In terms of previously introduced quantities we may write and so that . In support of this choice of This approach receives further support from the reanalysis by RZ of the Wendt (1933) and Taylor (1936a, b) experiments on turbulent viscosity generated in the flow between coaxial rotating cylinders. We note, however, that it is difficult to make precise comparisons between accretion disks and rotating cylinders in view of quite different constraints on the fluid flow. Using a definition of appropriate
to the experimental situation (here In the following we will refer to Eq. 2 as the -ansatz and to the disk structure arising from this viscosity prescription as -disks. We suggest that the -ansatz is the most appropriate initial formulation for accretion disks since it is directly connected to the driving mechanism. It establishes the maximum value of the viscosity that can arise from hydrodynamically driven turbulence. The actual viscosity may, however, be limited to lower values by such phenomena as shock dissipation of turbulent energy if the implied turbulent velocities exceed the local sound speed. As we show in Sect. 2.2, this yields the -ansatz in conditions relevant to these non-selfgravitating accretion disks. However, it leads to a different prescription in shock limited selfgravitating disks. In the Couette-Taylor case, all velocities were subsonic (the flow was essentially incompressible) so that no additional constraints applied. This would also be the case in astrophysical disks in which . ## 2.2. -viscosity as the limiting case for shock dissipation limited low mass accretion disksIf the accretion disk is such that the local sound speed is less than the turbulent velocities implied by the -ansatz, i.e., , we may rewrite Eq. 2 as where and are the maximum representative velocity and length scales allowed by local conditions. Furthermore, we may write so that a restriction on implies a constraint also on and vice-versa. If we consider turbulent elements in a with a quantity smaller than but of order unity. This estimate of may be interpreted as the distance a hydrodynamically driven turbulent element can travel before losing its identity due to shock dissipation. In a standard geometrically thin, non-selfgravitating accretion disk (i.e., a Shakura-Sunyaev, or -disk) hydrostatic equilibrium in the vertical direction implies Using this in Eq. 7, we find for Shakura-Sunyaev disks with again not too much smaller
than unity. This derivation of the Shakura-Sunyaev scaling, starting
from the assumption of a Reynolds driven turbulence, depends on the
disk mass being negligible, i.e., a vertical hydrostatic equilibrium
of the form of Eq. 8 has to apply. For selfgravitating disks,
Eq. 8 no longer applies, and thus the functional form of
will differ from that of Eq. 10.
Note that the upper bound to
It is worthwhile noting that this derivation of the Shakura-Sunyaev prescription not only yields its functional form, , but also the order of magnitude for the scaling parameter, , where is close to but less than unity. This value is consistent with values derived by comparing -disk models with observations of disks, for instance in dwarf novae Cannizzo, Shafter & Wheeler (1988). From the above, it is clear that the viscosity in accretion disks depends not only on the generation of hydrodynamic turbulence but also on the limitation arising from the requirement that the turbulence be subsonic. It also depends on whether or not the disk is selfgravitating. In the following section, we use he same logic to investigate the viscosity prescription in selfgravitating accretion disks, in which turbulence is limited by shock dissipation. © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |