A general characterization of free convection at a surface of characteristic length L is given by the Raleigh number:
At low Raleigh numbers the flow is laminar, although it may be unstable, until a transition at to a fully turbulent regime.
Let us consider a simple situation in which the air volume inside the dome enclosure sketched in fig. is affected by free convection generated by temperature differences between air and the floor of the dome. The floor-air temperature difference will generate a system of convection cells by which the heat is carried in the dome inner air volume. Since the flow conditions around a telescope are generally too complex to hope to come to rigorous modelisations of the areas interested by the seeing effect, it will be of interest to understand the similarity that rule the effect of changes of one or more parameters on the seeing, thereby deriving the criteria for interpreting the data obtained from mirrors and telescopes of different scales and conditions.
In an extreme simplification we will consider the height from the floor as the only geometrical parameter. Mean statistical values, may then be obtained through similarity theory (cf. section ). Following [Wyngaard], we take as scaling variables of the free convection field from a plane horizontal surface , q and z. There is no stability parameter and dimensional reasoning leads then to
where b is a constant. Rearranging gives:
which, within the constant factor, is equation () which had been derived for the unstable limit conditions of a turbulent boundary flow.
Equation () is a convenient expression to derive a relationship of the integrated seeing with the flow scaling variables. Noting that g and T, as well as air density and specific heat are not scalable, and assuming further that the height dependency will be constant through all scales, the scales and of seeing are related to the scale of surface flux (the normalized heat transfer rate) as
and, recalling equation (),
Noting that q is a likely function of temperature difference and upward flow speed, dimensional analysis gives:
Different expressions, depending on the particular conditions, are proposed to link the average value of flow speed in free convection. We will here take the simplest one:
Thus is approximately:
While this relationship is derived from dimensional reasoning, we note that the exponent of in equation () obtained from experiments of free convection on a horizontal surface actually varies from 1.25 to 1.6 depending on particular conditions. The exponents given by textbooks for free convection at a horizontal plate are respectively 1.33 (laminar flow) and 1.25 (turbulent flow). Equations () and () become in that case respectively:
[Giovannoni] gives also the experimental value of 1.6 for the case in which laminar free convection is influenced by an ambient stable stratification. This gives: