In this paper, similarity hypotheses for the atmospheric surface layer (ASL) are reviewed using nondimensional
characteristic invariants, referred to as π-numbers. The basic idea of this dimensional π-invariants analysis
(sometimes also called Buckingham’s π-theorem) is described in a mathematically generalized formalism. To illustrate
the task of this powerful method and how it can be applied to deduce a variety of reasonable solutions by the formalized
procedure of non-dimensionalization, various instances are represented that are relevant to the turbulence transfer across
the ASL and prevailing structure of ASL turbulence. Within the framework of our review we consider both (a) Monin-
Obukhov scaling for forced-convective conditions, and (b) Prandtl-Obukhov-Priestley scaling for free-convective conditions.
It is shown that in the various instances of Monin-Obukhov scaling generally two π-numbers occur that result in
corresponding similarity functions. In contrast to that, Prandtl-Obukhov-Priestley scaling will lead to only one π number
in each case usually considered as a non-dimensional universal constant.
Since an explicit mathematical relationship for the similarity functions cannot be obtained from a dimensional π-
invariants analysis, elementary laws of π-invariants have to be pointed out using empirical or/and theoretical findings. To
evaluate empirical similarity functions usually considered within the framework flux-profile relationships, so-called integral
similarity functions for momentum and sensible heat are presented and assessed on the basis of the friction velocity
and the vertical component of the eddy flux densities of sensible and latent heat directly measured during the GREIV I
1974 field campaign.