- The injection range is the typical range of scales at which kinetic energy is injected into turbulent motions. In turbulent convection, injection of kinetic energy is due to the work of the buoyancy force. The scale most representative of the injection range under these conditions is called the Bolgiano scale (Bolgiano, 1959; L’vov, 1991; Chillá et al., 1993; Rincon, 2006). It can be shown, based on purely dimensional arguments and scaling considerations for heat transport in turbulent convection, to be almost always of the same order (up to some order one prefactor) as the local typical scale height (Rincon, 2007). In an incompressible thermal convection experiment, this corresponds to the distance between the hot and cold plates, but in the strongly stratified SCZ, a more sensible estimate is the local pressure scale height. Close to the surface, the Bolgiano scale is therefore comparable to or slightly larger than the granulation scale (Section 3.1 below). As one goes to deeper layers, the pressure scale height gets larger and larger as a consequence of the strong stratification, and so does the Bolgiano scale. So, in the SCZ, the injection scale basically increases with depth and ranges from 1 Mm close to the surface to 100 Mm close to the bottom of the SCZ.

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, 1995). 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, 1998; Clyne et al., 2007). Figure 1 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.

Let us now discuss the ordering of dissipation scales in the SCZ. The most important one is obviously the viscous dissipation scale 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 and the thermal dissipation scale . Note that all these scales are local and change with depth in the inhomogeneous SCZ.

- A rough estimate for the viscous dissipation scale can be obtained from the Kolmogorov
phenomenology of turbulence (Frisch, 1995) via the expression , where
stands for the injection scale and , being the typical velocity at the injection
scale
^{3}. In the SCZ, where (Rieutord, 2008), we find at the surface, assuming and (the typical granulation scale and velocity). At the bottom of the SCZ, where the injection scale is much larger, one can estimate similarly that . Hence, is everywhere extremely small and not available to observations. - In MHD, the relative value of the magnetic dissipation scale with respect to the viscous
cut-off scale depends on the ordering of dissipative processes (see, e.g., Schekochihin
et al., 2007) in the fluid. When the magnetic diffusivity is much larger than the kinematic
viscosity , as is the case in the Sun, one may use (Moffatt, 1961), where
is called the magnetic Prandtl number
^{4}and we have assumed a Kolmogorov scaling for the velocity field. The magnetic diffusivity in the subsurface layers of the Sun is (Spruit, 1974; Rieutord, 2008), so . Consequently, close to the surface (see also Pietarila Graham et al., 2009). This is also very small in comparison to the resolution of current observations, but is much larger than . Close to the bottom of the SCZ, , so . - The thermal dissipation scale is very important in the solar context, as it is the largest of
all dissipation scales in the problem. In the SCZ, the thermal diffusivity is everywhere much
larger than the kinematic viscosity , so the thermal Prandtl number is very
small. Under these conditions, we may estimate from the expression
^{5}, once again assuming a Kolmogorov scaling for the velocity field. Thermal diffusion in the Sun is insured by photons, and thus depends strongly on the opacity of the fluid. In the deep SCZ, , so . In the very surface layers, is comparable to the scale of granulation , at which heat advection and radiation are comparable (see Section 3.1 below). This is not small anymore in terms of solar observations, but remains nevertheless smaller than the typical scale of supergranulation .

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

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