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

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2. Prescription for turbulent viscosity

2.1. Reynolds viscosity as the general case

As 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 s (the radius of the orbit) and [FORMULA] (the azimuthal velocity), respectively.

As has been pointed out, for example, by Lynden-Bell & Pringle (1974) and Thompson et al. (1977), the high corresponding Reynolds number [FORMULA] 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 [FORMULA]. Typical values for [FORMULA] in laboratory flows are of the order of [FORMULA]. This limiting Reynolds viscosity can, in this case, be as high as

[EQUATION]

where [FORMULA] is a constant satisfying

[EQUATION]

In terms of previously introduced quantities we may write [FORMULA] and [FORMULA] so that [FORMULA].

In support of this choice of s as the natural length scale we note that it is the only length scale which is relevant for angular momentum transport and which contains information about the driving agent for the turbulence-namely the rotation field; likewise, the orbital velocity [FORMULA] is the only velocity scale containing such information.

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 [FORMULA] appropriate to the experimental situation (here R is the average cylinder radius and [FORMULA] and [FORMULA] are the relative angular velocity and gap size between the cylinders), RZ derive expressions for [FORMULA] as a function of relative gap size [FORMULA]. For small gap size they find [FORMULA] independent of gap size. For large relative gap size they find that [FORMULA] where their gradient Reynolds number [FORMULA] is essentially constant. Thus for small gaps the experimental data yield essentially the same value of [FORMULA] as we have adopted in Eq. (3). For large gap sizes, the constancy of [FORMULA] leads to essentially the same functional form as in Eq. (2) but with a significantly smaller value of [FORMULA]. RZ arrive at similar conclusions for the two regimes of gap sizes from an analysis of the torques exerted on the cylinders in cases where the flow is turbulent. We believe that the small gap limit is the more relevant to the accretion disk case, for which the speed of rotation is constrained at each radius by the gravitational field. The flux of angular momentum at each radius is thus determined by the imposed orbital velocity field. By contrast, the experimental configuration constrains the flow velocity only at the inner and outer radius with the flow in the gap region able to take up a velocity profile determined by viscosity and in which the angular momentum flux is independent of radius. In our opinion the experimental results provide strong support for a turbulent viscosity generated by hydrodynamically driven turbulence. While we recognize that more work is required on the question, we also believe that the small gap results provide significant support for the viscosity prescription given in Eqs. (2) and (3).

In the following we will refer to Eq. 2 as the [FORMULA]-ansatz and to the disk structure arising from this viscosity prescription as [FORMULA]-disks. We suggest that the [FORMULA]-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 [FORMULA]-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 [FORMULA].

2.2. [FORMULA]-viscosity as the limiting case for shock dissipation limited low mass accretion disks

If the accretion disk is such that the local sound speed is less than the turbulent velocities implied by the [FORMULA]-ansatz, i.e., [FORMULA], we may rewrite Eq. 2 as

[EQUATION]

where [FORMULA] and [FORMULA] are the maximum representative velocity and length scales allowed by local conditions. Furthermore, we may write

[EQUATION]

so that a restriction on [FORMULA] implies a constraint also on [FORMULA] and vice-versa.

If we consider turbulent elements in a smoothed out background gas with sound speed [FORMULA] we may impose the limit that the turbulent velocity will approach but may not exceed [FORMULA]. Thus Eq. 5 gives

[EQUATION]

or

[EQUATION]

with [FORMULA] a quantity smaller than but of order unity. This estimate of [FORMULA] 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 [FORMULA]-disk) hydrostatic equilibrium in the vertical direction implies

[EQUATION]

Using this in Eq. 7, we find for Shakura-Sunyaev disks

[EQUATION]

and hence that

[EQUATION]

with [FORMULA] 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 [FORMULA] will differ from that of Eq. 10. Note that the upper bound to [FORMULA] implies approximately isotropic turbulence. This is the standard [FORMULA]-ansatz but derived from considerations of rotationally generated turbulence.

It is worthwhile noting that this derivation of the Shakura-Sunyaev prescription not only yields its functional form, [FORMULA], but also the order of magnitude for the scaling parameter, [FORMULA], where [FORMULA] is close to but less than unity. This value is consistent with values derived by comparing [FORMULA]-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.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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