Since there is the excellent summary on stellar convection given by J.-C. Pecker at the Liège symposium on stellar evolution in 1959, where he gives a complete account of the existing literature, I shall in this summary not try to do justice to the history of stellar convection research nor shall I try to mention all the existing papers even if they are quite important. I shall rather attempt to give an objective picture of the present state of affairs in order to be able to discuss our problems with the aerodynamicists. I shall cite only those papers that have been explicitly used for the results presented here. Additional references to literature may be found in the article by J.-C. Pecker.

In the study of turbulent flows similarity arguments are used to explore the consequences of non-mechanistic assertions concerning the general behavior of the flow. For example, it is currently assumed that viscosity plays no role in the determination of the mean velocity profile of turbulent shearing flow far from a boundary. The consequences of this assumption are that the amplitude of the mean velocity will be determined by the momentum transported into such a region and that the velocity profile will be a solution to Euler's equations.

Speaking on localized velocity fields on the Sun, we mean the velocity fields in active regions on the disk. The experimental data used to picture these fields are based mainly on 1) the observations of Doppler shifts of spectral lines, 2) the form (asymmetry) of line profiles, 3) moving picture process in monochromatic, mainly Hα and H or K, light. However the big skin-time of solar plasma makes these data insufficient to get an adequate and complete idea about the motions in active regions, and we must also take into account the available observational data about magnetic fields in these regions. Somewhere the state of affairs permits one to disregard the possible influence of magnetic fields on the picture of velocity fields But these occasions are comparatively rare, because the main source of solar activity is closely, but in a not quite recognized way, connected with magnetic activity of the Sun. This is why we should in our talk consider as closely as possible both subjects—the motions in active regions and their magnetic fields. The best illustrations of the necessity of such a mode of consideration are the velocity fields in sunspots.

In many cases, the outer atmosphere or corona regions of stars are not dominated by the ordinary process of collisions between ions. Very roughly, collision-free phenomena become of interest when the mean free path for interparticle collisions is larger than the region of space being considered. For solar corona conditions, if we consider a density of 107 particles per cubic centimeter and a typical particle velocity of 500 kilometer per second, the mean free path is a few times the solar diameter. At the same velocity for a density of 100 particles per cubic centimeter, corresponding to the interplanetary density, the mean free path would be a thousand astronomical units. Since in both of these regions, we are interested in smaller dimensions, it is of interest to consider the properties of a plasma in which collisions are very rare.

Shock-waves represent one of the most important mechanisms for creating and heating a plasma. In classical non-dissipative gas dynamics, the formation of a shock is indicated by the progressive steepening of a finite-amplitude compressive wave front to the point where it becomes multivalued and consequently without physical meaning. This difficulty is avoided by the inclusion of dissipative effects, usually in the form of heat flow and viscosity. The dissipative mechanisms become more effective as the wave front steepens, and the result is a steady wave profile for which the non-linear and dissipative effects are counterbalanced. The scale length for the dissipative transition zone or wave profile is the mean-free-path; the actual thickness may range from one to several mean-free-paths, or even more for very weak shocks. Given the strength of the shock, the state on one side of the shock may be computed from the state on the other side directly from the laws of conservation of mass, momentum and energy (Hugoniot relations). Accordingly, the nature of the particular dissipative mechanism affects only the shape of the shock profile but not the end states.