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Astron. Astrophys. 357, 1123-1132 (2000)

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1. Introduction

One of the major shortcomings of the current theoretical descriptions of accretion disks is lack of detailed knowledge about the underlying physics of viscosity in the disk. This problem is significant because almost all detailed modelling of the structure and evolution of accretion disks depends on the value of the viscosity and its dependence on the physical parameters. There is general agreement that molecular viscosity [FORMULA] is totally inadequate and that some kind of turbulent viscosity is required. Moreover, the Reynolds number in the disk flow is extremely high in any astrophysical context and this in itself is likely to lead to strong turbulence regardless of the details of the actual instability involved.

However, there is far less certainty about how to prescribe such a turbulent viscosity in the absence of a proper physical theory of turbulence. Most investigators adopt the so-called [FORMULA]-ansatz introduced by Shakura (1972) and Shakura & Sunyaev (1973) that gives the viscosity ([FORMULA]) as the product of the pressure scale height in the disk (h), the velocity of sound ([FORMULA]), and a parameter [FORMULA] that contains all the unknown physics:


One interprets this as some kind of isotropic turbulent viscosity [FORMULA] where [FORMULA] is an (a priori unknown) length scale and [FORMULA] an (a priori unknown) characteristic velocity of the turbulence. One may then write [FORMULA]. On general physical grounds neither term in parentheses can exceed unity so that [FORMULA]. If initially [FORMULA], shock waves would result in strong damping and hence a return to a subsonic turbulent velocity. The condition [FORMULA] would require anisotropic turbulence since the vertical length scales are limited by the disk's thickness, which is comparable to h.

While it is always, in a trivial way, possible to calculate a value [FORMULA], a parameterization of this sort for [FORMULA] is only useful if the proportionality parameter, [FORMULA], is (approximately) constant. One can expect this to happen only if the scaling quantities are chosen in a physically appropriate manner. Models for the structure and evolution of accretion disks in close binary systems (e.g., dwarf novae and symbiotic stars) show that Shakura & Sunyaev's parameterization with a constant [FORMULA] leads to results that reproduce the overall observed behaviour of the disks quite well. Time dependent model calculations of the outbursts of dwarf novae (e.g., Meyer & Meyer-Hofmeister 1984) and X-ray transients (e.g., Cannizzo 1996) demonstrate that, over a wide range of physical states of a disk in different phases of its evolution, the derived values of the viscosity parameter [FORMULA] do not vary by more than approximately an order of magnitude and are not too different from unity. As a result of this success, the [FORMULA]-ansatz is now used in essentially all accretion disk models.

It is noteworthy that, despite this success, the [FORMULA]-ansatz retains no information about the mechanism generating the turbulence but only about physical limits to its efficiency in a disk. In fact we would expect any high Reynolds number astrophysical shear flow to exhibit some kind of turbulent viscosity regardless of whether or not it happens to be in a disk. We therefore conclude that a more general prescription underlies the [FORMULA]-ansatz for accretion disks.

In recent years, Balbus & Hawley (1991) and their collaborators (e.g., Hawley et al. 1995) have shown that for non-selfgravitating magnetic accretion disks, an instability exists that can give rise to turbulence with the required formal dependence and-if only marginally--the required amount. We also note that in substantial regions of proto-stellar and proto-planetary disks, the charge density is unlikely to be high enough to sustain a significant magnetically mediated viscosity, although this phenomenon may be relevant elsewhere.

Whether a purely hydrodynamic turbulence can sustain the viscosity in the angular momentum profile of an accretion disk and can result in an angular momentum transport towards regions with larger specific angular momentum is still a matter of debate. Balbus et al. (1996), for instance, argue against it, based on numerical experiments, albeit for a rather low effective Reynolds number. Dubrulle (1992) and Kato & Yoshizawa (1997), among others, argue in favor of it, mainly based on analytical considerations. Experiments dating back to the 1930s on the Couette-Taylor flow between co-axial rotating fluids Wendt (1933), Taylor (1936a, b)show clearly the existence of a purely hydrodynamic instability. While the flow is essentially incompressible, turbulence is generated above a critical Reynolds number, independent of the radial profile of angular momentum 1. A modern review has been given by DiPrima & Swinney (1985). Most recently Richard & Zahn (1999) (hereafter RZ) have undertaken a reanalysis of Taylor's experimental results and, for high Reynolds number flow, have interpreted them in terms of a turbulent viscosity (see also Sect. 2).

In this contribution, we adopt the view that hydrodynamically driven turbulence can sustain the viscosity in accretion disks. We suggest, in Sect. 2, a viscosity prescription, the [FORMULA]-ansatz, that represents the maximum attributable to hydrodynamic turbulence. We show that in the limit of low mass, thin disks, hydrodynamic turbulence will result in the Shakura-Sunyaev prescription. We then discuss the implications of the proposed formulation for the structure and evolution of selfgravitating disks, noting that even for these disks, the viscosity prescription differs from the [FORMULA]-ansatz and hence removes a difficulty first noted by Paczynski (1978). Finally, we discuss protostellar, galactic and galactic center disks as examples where the [FORMULA]-ansatz may be relevant.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000