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The earliest results concerning the turbulence structure in a turbulent boundary layer with very unstable thermal stratification are due to Prandtl (1932). These results were developed further and made more precise by Obukhov (1946, 1960), Monin & Obukhov (1954) and Priestley (1954, 1955, 1956, 1960). All of these authors dealt with a surface layer of the Earth's atmosphere on hot summer days. Such a layer is the most easily accessible example of an unstably stratified boundary layer and it will be the main concern in this paper too. The theoretical predictions by the above-mentioned authors seemed at first to be confirmed by the available experimental data but in the late 1960s it became clear that at least some of the predictions disagreed strongly with the experimental information.A more elaborate theory was proposed by Betchov & Yaglom (1971) who used a suggestion of Zilitinkevich (1971). According to this theory, within an unstably stratified boundary layer there are three special sublayers where turbulence structure is self-preserving and obeys rather simple power laws. The new theory explained the disagreement between some of the deductions from the old theory and the data. However, the data available in 1971 were insufficient for the confirmation of the new theory and it was even supposed by Betchov & Yaglom (1971) that their theory could not be applied to atmospheric surface layers on hot summer days.Much new experimental data concerning unstably stratified boundary layers has been obtained in recent years; in particular, extensive experimental information was collected during the summers of 1981–1987 at the Tsimlyansk Field Station of the Moscow Institute of Atmospheric Physics. This paper is a survey of the deductions from the theory by Betchov & Yaglom which concern the mean fields and the one-point fluctuation moments in unstably stratified boundary layers, and a comparison of these deductions with the data available in 1989. It is shown that the data agree more or less satisfactorily with the theoretical predictions and permit one to obtain estimates for a number of coefficients that enter the theoretical equations.

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General dimensional and similarity arguments are applied to derive a heat and mass transfer law for fully turbulent flow along a rough wall. The derivation is quite analogous to Millikan's (1939) derivation of a skin-friction law for smooth-and rough-wall flows and to the derivation of the heat and mass transfer law for smooth-wall flows by Fortier (1968a, b) and Kader & Yaglom (1970, 1972).The equations derived for the heat or mass transfer coefficient (Stanton number) Ch and Nusselt number Nu include the constant term β of the logarithmic equation for the mean temperature or concentration of a diffusing substance. This term is a function of the Prandtl number, the dimensionless height of wall protrusions and of the parameters describing the shapes and spatial distribution of the protrusions. The general form of the function β is roughly estimated by a simplified analysis of the eddy-diffusivity behaviour in the proximity of the wall (in the gaps between the wall protrusions). Approximate values of the numerical coefficients of the equation for β are found from measurements of the mean velocity and temperature (or concentration) above rough walls. The equation agrees satisfactorily with all the available experimental data. It is noted that the results obtained indicate that roughness affects heat and mass transfer in two ways: it produces the additional disturbances augmenting the heat and mass transfer and simultaneously retards the fluid flow in the proximity of the wall. This second effect leads in some cases to deterioration of heat and mass transfer from a rough wall as compared with the case of a smooth wall at the same values of the Reynolds and Prandtl numbers.

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