5.3 Large-scale instabilities and collective interactions

Besides thermal convection scenarios, a few other theoretical arguments have been put forward to explain the origin of the solar supergranulation. These ideas are all based on the possible collective effects of small-scale structures such as granules, which might lead to a large-scale instability injecting energy into the supergranulation range of scales.

5.3.1 Rip currents and large-scale instabilities

The first work along this line of thought was published by Cloutman (1979). He proposed to explain the origin of supergranulation using the physical picture of rip currents on the beaches of oceans: the repeated breaking of waves on beaches induces currents (rip currents) flowing parallel to the coast line. On the Sun, he identified breakers with the rising flows of granules breaking into the stably stratified upper photosphere.

The rip current model provides an illustration of the suggestion of Rieutord et al. (2000Jump To The Next Citation Point) that the collective interaction of solar granules may give rise to a large-scale instability driving supergranulation flows. The idea finds its root in theoretical work on energy localisation processes in nonlinear lattices (Dauxois and Peyrard, 1993) and large-scale instabilities (“negative eddy viscosity instabilities”) of periodic flows, such as the Kolmogorov flow (Meshalkin and Sinai, 1961Sivashinsky and Yakhot, 1985) or the decorated hexagonal flow (Gama et al., 1994). Asymptotic theory on simple prescribed vortical flows can be performed under the assumption of scale separation (Dubrulle and Frisch, 1991) between the basic periodic flow and the large-scale instability mode. In such theories, the sign and amplitudes of the turbulent viscosities is found to be a function of the Reynolds number. For instance, an asymptotic theory based on a large aspect ratio expansion was developed by Newell et al. (1990) for thermal convection. In this problem, large-scale instabilities take on the form of a slow, long-wavelength modulation of convection roll patterns. Their evolution is governed by a phase diffusion equation with tensorial viscosity. In the case of negative effective parallel diffusion (with respect to the rolls orientation), the Eckhaus instability sets in, while the zigzag instability is preferred in the case of negative effective perpendicular diffusion.

5.3.2 Plume interactions

Another way of explaining the origin of supergranulation assumes that the pattern results from the collective interaction of plumes. The word plume usually refers to buoyantly driven rising or sinking flows. Plumes can be either laminar or turbulent, however the turbulent ones have by far received most of the attention because of their numerous applications (see Turner, 1986).

The first numerical simulations of compressible convection at high enough Reynolds numbers (e.g., Stein and Nordlund, 1989aRast and Toomre, 1993) quite clearly showed the importance of vigorous sinking plumes. These results prompted Rieutord and Zahn (1995Jump To The Next Citation Point) to study in some details the fate of these downdrafts. Unlike the downflows computed in early simulations, solar plumes are turbulent structures, which entrain the surrounding fluid (see Figure 1View Image). As Rieutord and Zahn (1995) pointed it out, the mutual entrainment and merging of these plumes naturally leads to an increase of the horizontal scale as one proceeds deeper.

In this context, toy models have been elaborated to investigate the properties of “n-body” dynamical advection-interactions between plumes. For instance, Rast (2003bJump To The Next Citation Point) developed a model in which a two-dimensional flow described by a collection of individual divergent horizontal flows (“fountains”) mimicking granules is evolved under a simple set of rules governing the merging of individual elements into larger fountains and their repulsion8. For some parameters typical of the solar granulation (individual velocities and radius of the fountains notably), he argued that the clustering scales of the flow after a long evolution of the system resembled that of mesogranulation and supergranulation. A similar model incorporating simplified magnetic field dynamics was designed by Crouch et al. (2007Jump To The Next Citation Point). They observed some magnetic field organisation and polarity enhancement at scales similar to that of supergranulation in the course of the evolution of the model. The main caveat of these toy models is of course that they do not rely on the exact dynamical physical equations.

The previous concepts and models are appealing but theoretical and numerical support has been lacking so far to weigh their relevance to the supergranulation problem. In particular, analytical developments for fluid problems have been restricted to low Reynolds numbers and very simple analytical flow models. It is unclear if and how some quantitative progress can be made along these lines. Future high-resolution numerical simulations may offer some insight into the nonlinear dynamics at work at supergranulation scales but, as will be shown in Section 6, they have not yet given us any indication that large-scale instabilities or nonlinear granulation dynamics can generate a supergranulation flow. In particular, all hydrodynamic simulations so far tend to show that plume merging is a self-similar process that does not naturally produce a flow at well-defined scale that could be identified with supergranulation.

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