2.2 Scales in turbulent convection

Solar convection is in a highly turbulent state. Global Reynolds numbers Re = LV ∕ν, based on the (large) vertical extent of the convective layer and typical convective velocities, range from 1010 to 1013 in the SCZ. To fix ideas, laboratory experiments on turbulent convection are currently limited to rms 7 Re < 10 (e.g., Niemela et al., 2000). As the general topic of this review is the origin and structure of the supergranulation pattern, what we are mostly interested in here are the remarkable length scales of such a turbulent flow. For this purpose, we assume that solar convection is simply thermal convection of an electrically conducting fluid and that the flow is incompressible (which it is not, as a consequence of the important stratification of the SCZ). Hydrodynamic, incompressible turbulent flows in the laboratory are characterised by two length scales: the injection scale and the viscous dissipation scale.

A perhaps more intuitive physical picture of the previous argument is given by cold downflows diving from the Sun’s surface. Such flows can cross a significant fraction of the convective zone because of the nearly isentropic state of the fluid (Rieutord and Zahn, 1995Jump To The Next Citation Point). This underlines the fact that the driving of turbulence spans a wide range of scales in a continuous way as one looks deeper in the SCZ. Note that the expanding thermal plumes undergo secondary instabilities along their descending trajectories, producing a intricate mixture of vorticity filaments (see, e.g., Rast, 1998Clyne et al., 2007Jump To The Next Citation Point). Figure 1View Image provides a simple sketch of this process. As we shall see in Section 6.3, numerical simulations of solar convection have provided a very neat confirmation of this phenomenology.

View Image

Figure 1: Left: the entropy profile as a function of depth, as estimated by numerical simulations or crude mean-field models like the mixing length theory. Right: section of a cool plume diving from the surface. As it penetrates into the isentropic background, the plume increases both its mass and momentum flux by turbulent mass entrainment (represented by curly arrows). Its horizontal scale grows proportionally to depth, the aperture angle of the cone being around 0.1. At a given depth, the typical size of energetic eddies is like the width of the plume while a mean flow at the scale of the depth is also generated. From this model, we see that the length scale characterising the buoyant flow at a given depth increases monotonically with depth (image by Mark Rast, see External Linkhttp://www.vapor.ucar.edu/images/gallery/RastPlume.png and Clyne et al., 2007).

Let us now discuss the ordering of dissipation scales in the SCZ. The most important one is obviously the viscous dissipation scale ℓν but, as we are considering thermal convection in an electrically and thermally conducting fluid, we also need to consider two other dissipative scales: the magnetic dissipation scale ℓ η and the thermal dissipation scale ℓ κ. Note that all these scales are local and change with depth in the inhomogeneous SCZ.

To summarise, the ordering of scales close to the solar surface is as follows:

ℓν ≪ ℓη ≪ ℓκ ∼ LB ∼ LG ≪ LSG .

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