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Astron. Astrophys. 337, 105-112 (1998)

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3. Small-scale turbulent viscosity

We consider, for simplicity, both the typical size of the clouds, a, and the mean free path of the clouds, l, to obey an inequality:

[EQUATION]

The hydrodynamics of a cloudy accretion disk with respect to 2-D nonaxisymmetric shear perturbations is similar to that for a disk with the continuous distribution of the matter explored by Fridman (1989) who found the solution of this problem in a small-amplitude, short-wave limit

[EQUATION]

where [FORMULA] is the wave vector. In a local rotating coordinate system, the solution for the radial component of the perturbed velocity, [FORMULA], reads:

[EQUATION]

where

[EQUATION]

[EQUATION]

Here [FORMULA] is the angular velocity of the disk, t is the time elapsed since the perturbations were "turned on", and [FORMULA] is a (time-dependent) critical viscosity. When [FORMULA], viscosity does not play any role in the disk dynamics.

The shortwave perturbations under consideration behave as incompressible modes. The solution (3.3) describes how the vorticity decays with time in a viscous fluid. Some limiting cases of interest can be revealed from Eqs. (3.3)-(3.5). As [FORMULA], there appears the asymptotic solution [FORMULA], which has been known earlier (e.g.  Timofeev 1976, Zaslavskii et al.  1982). Another limiting case when viscosity is absent ([FORMULA]), deserves a more detailed consideration.

3.1. Non-viscous case ([FORMULA])

In the absence of viscosity, the solution (3.3) goes into

[EQUATION]

which was obtained by Lominadze et al.  (1988). In this case, solution (3.6) implies that

[EQUATION]

where

[EQUATION]

Eq. (3.7) is a 2-D analog of the Thomson theorem on the conservation of vorticity in an incompressible fluid flow (Fridman 1989).

The solution (3.6) is shown in Fig. 1 by the solid lines. Evidently, the velocity perturbation is growing (implying instability) if the denominator in (3.6) is decreasing. At the moment when the denominator has a minimum, i.e. [FORMULA], where [FORMULA] is given by

[EQUATION]

[FORMULA] reaches its maximum equal to

[EQUATION]

and then goes to zero at [FORMULA]. The growth of [FORMULA] at [FORMULA] is a result of a decrease of [FORMULA] while [FORMULA] keeps constant due to the Thomson theorem (3.7).

[FIGURE] Fig. 1. "Unstable" solutions, [FORMULA], represented by Eq. (3.3) for several values of [FORMULA]. The solid lines show a non-viscous case (Eq. 3.6). A general, non-zero viscosity solution (3.3) is shown by the dotted lines for [FORMULA] and by the dashed lines for [FORMULA], where [FORMULA] is defined by Eq. (3.5).

The minimum of the denominator in Eq. (3.6) at [FORMULA] implies that [FORMULA] as [FORMULA]. Since [FORMULA], the necessary condition for the perturbations to grow with time is given by

[EQUATION]

Therefore, in a differentially rotating disk, the growing short-scale spiral perturbations can only be leading.

To clarify possible misunderstandings, we note several points in the following:

In Appendix B, we show that in a differentially rotating disk there are no incompressible short-wave perturbations oscillating with the epicyclic frequency at all; in a solid-body rotating disk such perturbations are stationary, in agreement with our Eqs. (3.6)-(3.10).

In Appendix C, we demonstrate that Coriolis forces are irrelevant to the solution for velocity perturbations in a plane shear layer, which we found to be similar to short-wave, low-frequency perturbations in a rotating gaseous disk. We emphasize that Coriolis forces have nothing to do with the physical meaning of our solution. The latter is associated with vorticity conservation (in the limit of small viscosity).

We consider perturbations with wavelengths smaller than the disk's width. Consequently our case is close to a cylinder fluid layer. Meanwhile the short-scale perturbations under the condition (3.2) are incompressible (Fridman 1989). Therefore analytical results obtained in this paper as well as in our previous papers (Fridman 1989, Fridman and Ozernoy 1992) lend to support interpretation of well-known laboratory experiments for a liquid flow between two rotating cylinders. The latter demonstrate a nonlinear instability and developed turbulence (e.g., Lukashchuk & Predtechenskii 1984). Therefore, we have all the reasons to believe that, for short-wave perturbations, a nonlinear instability is likely to occur. Likewise, in Appendix C, we show that the dynamics of short-wave incompressible perturbations in a rotating disk is completely equivalent to that in a plane shear layer. This proves the equivalence of the nonlinear instability in a plane shear layer and in the situation investigated here.

It is instructive to compare the above instability with that considered by Goldreich & Lynden-Bell (1965) (referred to as GLB hereinafter) for a self-gravitating, differentially rotating disk. There are two basic differences between our and their situations. First, different branches of growing perturbations are considered: we deal with the vortex branch while GLB do with the sound-gravitational branch. Second, the conditions for the growth of perturbations are different, too. Under condition (3.10), the leading vortex perturbations grow, due to the conservation of vorticity, in an otherwise stable disk. GLB deal with the disk stable for axially-symmetrical perturbations, and the growth of their non-axially-symmetrical, long-wave perturbations proceeds due to a leading role of self-gravity. Toomre (1982) christened this growth as "swing amplification". It is worth noting that, in our case, a similar swing amplification occurs for the (vortex) short-wave perturbations and it does not require self-gravity.

3.2. General case (non-zero viscosity)

The growth of perturbations with accounting for viscosity, which is described by Eq. (3.3), is shown in Fig. 1 by the dotted and dashed lines. The "instability" described by the exact solution (3.3) is of very specific kind as [FORMULA] tends to zero at [FORMULA]. Nevertheless, during a finite time [FORMULA] given by Eq. (3.8) the perturbations are in the growing regime and their amplitude increases by a factor of [FORMULA], which can be [FORMULA] if [FORMULA].

Let us suppose that by the moment [FORMULA] when the amplitude of any perturbation arrives at its maximum, the viscosity does not play any essential role, i.e.  [FORMULA]. Though this growth in the amplitude proceeds during the time interval [FORMULA] only, it could onset the local turbulence. In this case, turbulence can be established everywhere in the disk as a superposition of the spiral perturbations originated in different points of the disk and on different moments of time; all of them have experienced a growth during [FORMULA].

The turbulence appeared as a result of the growth of the perturbations can be characterized by some turbulent viscosity, which generally is much larger than the molecular one. It is possible that a steady-state regime will emerge in which

[EQUATION]

In this regime, the decay due to viscosity is strong enough to provide a steady-state level of the turbulence so that the amplitude of the perturbations is kept more or less constant in time, i.e. 

[EQUATION]

Substituting Eqs. (3.11) and (3.12) into Eq. (3.3) with taking into account Eq. (3.5) and assuming that, by order of magnitude, [FORMULA], we arrive at the following transcendental equation:

[EQUATION]

Taking the logarithm of this equation one finds:

[EQUATION]

It is straightforward to see that the basic contribution into the turbulent viscosity is given by the perturbations of the smallest [FORMULA]'s; therefore [FORMULA]. From Eqs. (3.2) and (3.12) one has [FORMULA], whence [FORMULA], i.e. 

[EQUATION]

By substituting [FORMULA] into Eq. (3.15) the latter can be rewritten in the form:

[EQUATION]

It is instructive to compare the expression for the Bohm diffusion coefficient [FORMULA] with our Eqs. (3.14) and (3.15) for [FORMULA] A well-known estimation of the Bohm diffusion coefficient for a strong turbulence plasma is given by (see e.g., Kadomtsev, 1964, or a review by Horton, 1984):

[EQUATION]

where [FORMULA] is the maximum linear growth rate of the drift instability, and [FORMULA] is the minimum wave number, also from a linear theory. They are much the same: in our case [FORMULA] and instead of [FORMULA] we substitute [FORMULA] Kadomtsev obtained [FORMULA] by using the relationship [FORMULA] where [FORMULA] is the Larmor radius. At [FORMULA], [FORMULA] has a maximum. But the Larmor radius in plasma corresponds to the epicyclic radius in graviphysics. The latter, in fact, is the thickness of disk, [FORMULA], and this, actually, is being used here.

This viscosity whose value is given by Eq. (3.15) or (3.16) was called by Fridman & Ozernoy (1992) anomalous in the same sense as one introduces anomalous resistivity and anomalous diffusion in plasma physics: The origin of this viscosity is in fully developed turbulence which is established in the saturation regime described above.

Eq. (3.15) or (3.16) could be derived from dimensional arguments (of course, without the numerical coefficient) as an estimation of viscosity in a rotating disk with turbulent motions. However, we should emphasize that without an analysis such as one given above it would be impossible to reveal an underlying physical mechanism for the origin of such turbulence.

It is important to test whether the necessary condition for developing of small-scale turbulence

[EQUATION]

is met. To this end, let us find the ratio [FORMULA] as a function of [FORMULA], i.e.  of the basic parameter that characterizes the number of interactions per one revolution:

[EQUATION]

Asymptotically,

[EQUATION]

i.e.  in both limits, [FORMULA] and [FORMULA], the viscosity coefficient is much less than the critical value given by Eq. (3.15) or (3.16). The function (3.18) reaches its maximum at [FORMULA], and this maximum is given by

[EQUATION]

Even in this, the least favorable case [when [FORMULA] as one can see from Eq. (2.4)] [FORMULA] is less than [FORMULA] by a factor of 2 or so, which is enough for small-scale turbulence to appear. Therefore the range of physical conditions under which the viscosity due to fully developed turbulence should dominate is indeed very broad.

Unlike the classical example of gravitational instability, our mechanism for the growth of shortwave perturbations has a much higher level of viscous stabilization, as it follows from the value of critical viscosity calculated above in comparison with that for a self-gravitating disk. Indeed, the growth of shortwave perturbations leads to the increase of the amplitude by a factor of

[EQUATION]

This strong inequality follows from the fact that [FORMULA] where [FORMULA] is the characteristic time of the growth of perturbations and T is the period of the disk revolution. According to the perturbation theory implemented to examine instability, the condition [FORMULA] holds (otherwise the zero-approximation of perturbation theory is not fulfilled: the equilibrium condition is broken for the time less then that of one revolution of the disk). As a result of the above inequality, a strong growth of perturbation takes place, which leads to the development of short-scale turbulence and the appearance of turbulent viscosity. A very large factor of the growth given above explains why the value of the critical turbulent viscosity, which stops the growth of perturbations, turns out to be much larger than that for the instability of a self-gravitating disk (e.g., Fridman & Polyachenko 1984, p. 41).

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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