## 1. Introduction

During the last 50 years, efforts have been made to study the spectral characteristics of atmospheric turbulence (Busch and Panofsky, 1968; Kaimal *et al.*, 1972; Højstrup, 1982; Mann, 1994) and the behaviour of the length-scale of turbulence with height and atmospheric stability (Kaimal *et al.*, 1972; Caughey and Palmer, 1979; Højstrup *et al.*, 1997; Lenderink and Holslag 2004; Peña *et al.*, 2010b). The investigation of the length-scale of the wind profile was initiated by Prandtl (1925), who introduced the concept of the mixing length when analyzing mean velocity profiles in laboratory flows. Prandtl's work has motivated the use of mixing-length theory for the entire boundary layer (Blackadar, 1962; Lettau, 1962; Delage, 1974; Gryning *et al.*, 2007). This has been used to derive wind profile parameterizations that fit wind speed observations performed beyond the surface layer both over land and water surfaces (Gryning *et al.*, 2007; Peña *et al.*, 2008; Peña *et al.*, 2010a) better than traditional surface-layer theory.

It is well-known (Stull, 1988; Kaimal and Finnigan, 1994; van de Wiel *et al.*, 2008) that both the mixing length and the peak of the turbulence spectra are measures of the size of the turbulent eddies –the first frequently related to their average size and the second related to the size of the eddies with the most energy. However, to our knowledge, there are no references apart from Peña *et al.* (2010b) in which a relation is established between the length-scales observed from the analysis of the spectral characteristics of atmospheric turbulence and those derived from the mean flow. A close study to ours on similarities between mixing and spectral length-scales is that of Cuxart *et al.* (2000), who compared a parameterized mixing-length model to spectral length-scales from large-eddy simulations (LESs) of the convective boundary layer (CBL).

The main idea in this paper is to derive and investigate the relations between the length-scale that controls the wind profile, the so-called mixing length, and the length-scale of the turbulence, namely the length-scale corresponding to the peak of the velocity spectrum. This is accomplished by spectral analysis of a number of 10 min sonic anemometer time series of wind velocity fluctuations at several heights over flat and homogenous terrain in Denmark. The analysis was performed over a wide range of atmospheric stability conditions and wind speeds, in order to observe the effects of both stability and friction Rossby number.

The length-scale formulations in this study are closely related to the explicit formulation of the length-scale used in first-order closure models of the atmospheric boundary layer, also called mixing-length models (Blackadar, 1962; Lacser and Arya, 1986). However, the simple explicit mixing-length approaches are today often replaced with length-scales determined from the more involved and physically realistic relationships that can be derived by combining the equation of turbulent kinetic energy with the equation of dissipation. Such a derivation is based on a more realistic physical representation of the energy dissipation as opposed to the mixing-length schemes. However, the length-scales are determined from two additional equations, so it becomes less transparent and not easily comparable to the length-scales in this study. The latter approach is often termed one-and-a-half-order closure (Mellor and Yamada, 1974; Detering and Etling, 1985; Apsley and Castro, 1997) and is known to provide a more realistic representation of the turbulence than first-order models (Holt and Raman, 1988).

In section 2, the behaviour with height and atmospheric stability of the mixing length within (section 2.1.1) and beyond (section 2.1.2) the surface layer is described, from which the dependency on friction Rossby number is shown. The expected relations between the mixing length and the length-scale of turbulence are introduced in sections 2.2.1 and 2.2.2. The site and measurements are described in section 3. Section 4.1 shows the results of the length-scale analysis for the diabatic conditions, and their application to the diabatic wind profile. In section 4.2 the length-scale analysis for a range of friction Rossby numbers, and its application to the adiabatic wind profile, is presented. Discussion and conclusions are given in the last two sections.