12.3. Role of Viscosity

In the above dynamical mechanism for the formation of rings, gas plays the essential role, because of its dissipative character. Even considering the case of a non-self-gravitating disk, spiral structure is hardly driven in the stellar component by a bar potential. Stellar particles will first follow a very transient spiral structure by crowding of orbits in the relaxation phase, and then will settle around the periodic orbits, aligned either parallel or perpendicular to the bar, but along its symmetry axes. There will then be no gravity torques exerted on the stars, and no angular momentum transfer. In a self-gravitating disk, gravitational instabilities contribute to heat the disk, so that a pure stellar disk soon becomes dynamically hot and cannot sustain a long lasting spiral structure. The dissipative gas component is therefore required to cool down the disk, and maintain the spiral structure. This induces a phase-shift between the density and the potential, and gravity torques act to transfer angular momentum, which is at the basis of ring formation.

But if dissipation is present, the action of viscous torques themselves could play a role in the radial flow of gas. The idea was already mentioned by Randers (1940), and has been widely developed in the context of accretion disks (e.g. Frank et al. 1985). Lynden-Bell & Pringle (1974) showed that the evolution of any gaseous disk can be characterized by an expansion of the outer parts, which carry most of the angular momentum, and consequently a collection of an ever increasing mass towards the center. At the limit, a negligible mass is ejected and takes away all the angular momentum. This comes from the fact that for a given total angular momentum, the least energy of a rotating disk is obtained for a uniform rotation.

Most galaxies are in differential rotation, and the angular velocity decreases with radius. In viscous exchanges (collisions) particles in the outer parts will gain angular momentum and energy, in order for the galaxy to tend to the uniform rotation state of least energy. But the specific angular momentum (per unit mass) is a monotonically increasing function of radius. Therefore, angular momentum is distributed to the regions of the disk that are already richer in angular momentum, and this is not a stable process. To remain in equilibrium, the matter that receives angular momentum has to increase its radius. This implies also that its speed decreases, and the system is going away from the desired uniform rotation state. The process is divergent, and no equilibrium will be reached until all the matter has fallen towards the center.

How does the viscosity act to make the system evolve this way? We have only advanced until now energetic principles. Viscosity (or equivalently here cloud collisions) produce shearing forces on each gaseous ring, that compose to produce a torque. Shearing forces tend to cancel the relative velocity between two adjacent gaseous rings. Since is decreasing with radius, the torque will transfer angular momentum to the outer parts. The ideal state, where there would be no viscous forces, is again the uniformly rotating disk. The analysis above demonstrates however that uniform rotation is unstable in the presence of viscosity.

Viscous torques then provide a mechanism to form gas rings in galaxies: Icke (1979) noticed that if the rotation is sufficiently uniform inside the turnover radius of the rotation curve (R_max), then the shear is zero inside, and viscous torques external to the turnover radius will accumulate gas in a ring at R_max. ⁽⁴⁾ The ring is short-lived, however, since all the gas is bound to go towards the center. But the time-scale of this process is longer as the rotation is more uniform in the center. Simulations of the effect of viscous torques in galactic disks have succeeded in forming a gaseous ring at turnover (Icke 1979; Silchenko & Lipunov 1987; Lesch et al. 1990). The ring is smooth, with no sharp boundary, and is reminiscent of the molecular ring in the Milky Way, between 4 and 8 kpc. Its radius is shrinking with time. If it is possible that viscous torques can explain some of the rings observed, they cannot account for the presence of several sharp rings in the same galactic disk, and especially not the outer rings.

In addition, the efficiency of viscous torques has only been assumed in those simulations, but not firmly established. The physical nature of viscosity in gaseous galactic disks is not well known. To transfer angular momentum over large scales, large relative velocities are required. The molecular viscosity is of course negligible. But there are large random velocities between interstellar clouds, which are supported by star formation energy. These motions give rise to the turbulent viscosity. The estimation of the efficiency of viscous torques in galactic disks was attempted by Lynden-Bell & Pringle (1974), who concluded that viscosity had little importance over a Hubble time. The estimated time-scales were 10⁹ yrs at 1 kpc radius, and 10¹¹ yrs at 10 kpc.

An interesting idea is that the viscosity is gravitational in origin: large-scale instabilities heat the gas component, and fuel small-scale random motions, like the local effective viscosity of Lin & Pringle (1987). The gas has the tendency to reduce its velocity dispersion by dissipation. When this dispersion V falls below the critical value c_r necessary to ensure stability against axisymmetric perturbations (i.e., the parameter Q = V / c_r falls below 1), then instabilities occur in the disk, and stir up the random motions of the gas. This acts as a feedback mechanism to maintain large velocity dispersions.

Attempts to estimate the efficiency of viscous torques in gas simulations in galaxies have always led to the conclusion that gravity torques dominate over viscous torques in the transfer of angular momentum (Combes et al. 1990a; Barnes 1991). The corresponding time-scale is one or two orders of magnitude shorter (von Linden et al. 1994). The best proof of the predominance of gravity torques is the formation of outer rings, by accumulation of gas at the OLR. The effect of the viscous torques is only to bring the gas towards the center, while in the simulations the gas is driven outwards, from CR to OLR.