Similarity Hypotheses for the Atmospheric Surface Layer Expressed by Non-Dimensional Characteristic Invariants – A Review

Abstract

Abstract:In this paper, similarity hypotheses for the atmospheric surface layer (ASL) are reviewed using nondimensional characteristic invariants, referred to as -numbers. The basic idea of this dimensional -invariants analysis (sometimes also called Buckingham's -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) MoninObukhov 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 -numbers occur that result in corresponding similarity functions. In contrast to that, Prandtl-Obukhov-Priestley scaling will lead to only one 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 dimensionalinvariants analysis, elementary laws of -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.

On the limiting effect of the Earth's rotation on the depth of a stably stratified boundary layer

“…A large number of illustrative examples from different areas of study, ranging from fractal curves to the theory of elasticity and including geophysical fluid dynamics, are considered by Barenblatt (1982Barenblatt ( , 1996. The atmospheric surface-layer flux-profile relationships as viewed from the self-similarity standpoint are discussed in Kramm and Herbert (2009). A close analogy between the self-similarity of the second kind and the renormalization group theory that found some use in turbulence studies is discussed in Goldenfeld et al (1989).…”

Hans A. Panofsky’s Integral Similarity Function—At Fifty

“…where it is assumed that complete similarity is established (e.g., [12,13]). For which is valid for neutral stratification, Formula (1.2) immediately provides One of the notable advantages of HAP's profile formula is obvious.…”

“…   (e.g., [13]). When the transfer of water vapor across the ASL plays a notable role like over water surfaces and wet soil or vegetation, it is indispensable to use the following expression for the Obukhov stability length [29]: , is equal to that of water vapor.…”

“…It reads [20], Ellison [21], Yamamoto [22], Panofsky [23], and Sellers [24]). As Obukhov already suggested it in 1946, the KEYPS formula was eventually renamed in O'KEYPS formula (e.g., [13,25,26]). 0     (e.g., [28,33,35,36,40], and Figure 1).…”

“…0     (e.g., [28,33,35,36,40], and Figure 1). From a physical point of view the O'KEYPS formula seems to be more preferable than the Businger-Dyer-Pandolfo relationship because the O' KEYPS formula can be related to the local balance equation of the turbulent kinetic energy if the conditions of stationary state and horizontal homogeneity are fulfilled (e.g., [13]). Since the O'KEYPS formula seems to be bulky, Carl et al [45] and Gavrilov and Petrov [46] eventually proposed the expression…”