The aim of this paper is to investigate the possible correlations between the integral scale of the turbulent stratified flows in the atmospheric boundary layer and parameters characterizing topological features of the wind velocity field, such as fractal dimension and its stability properties, studied through the bulk Richardson number. We are aware that there is a lack of investigations between the integral scale and the fractal dimension. The size of the integral scale of the horizontal and vertical components and fractal dimension of wind velocity near the earth's surface in the boundary layer are determined. Also, these magnitudes are compared between them and vs. other parameters such as the bulk Richardson number. It is assumed that the turbulence is the primary agent that causes changes in the boundary layer. In turbulent flows it is observed that time series of meteorological variables such as wind velocity, temperature, pressure and other atmospheric mechanical magnitudes fluctuate in a disordered way, with peaks extremely sharp and irregular in space and time variations. The complicated nature of these series indicates that the motion of the air is turbulent. If we take a good look at the variety of fluctuations of different periods and amplitudes observed in them, we could explain the complicated structure of turbulence. The irregularity of the time series obeys the existence of different sizes and timescales and also the nonlinear transfer of energy that exists between them in the turbulent flows (Monin and Yaglom, 1971).

The irregular behavior of these flows is also due to waves and turbulence that are often superimposed onto a mean wind (Stull, 1988). If we filter the mean wind and waves in the appropriate range, we will only have turbulence. Some previous works present results of this procedure (Tijera et al., 2008). In this work, the series of wind velocities in the three directions x, y and z recorded by the anemometer are divided into series of non-overlapping 5 min length. Each of these series applies the necessary rotations to get the x axis in the mean wind direction (mean v is zero) and zero mean vertical velocity (w vertical component) (Kaimal and Finnigan, 1994). We filter horizontal and vertical mean wind velocity, obtaining the time series of fluctuations of the velocity in both directions (u′=u-u‾, w′=w-w‾).

When we observe these time series such as wind velocity, they vary in an irregular shape, and in spite of their complexity present a self-similarity structure (Frisch, 1995). This is a common property of the fractals, so that wind velocity could be considered a fractal magnitude. The modern physical notion of fractals is largely known due to Mandelbrot (1977, 1985), but the mathematical notion of curves lines or sets having noninteger dimensions is much older (Hausdorff, 1918; Besicovitch, 1929). An analysis that compares the Haussdorff dimension and Kolmogorov capacities of self-similar structures with noninteger fractal dimensions (Kolmogorov capacity or box counting dimension) was presented by Vassilicos (Vassilicos and Hunt, 1991). The wind velocity vs. time are irregular curves of this type, with noninteger dimensions. These values correspond to the fractal dimension. A way of measuring the complexity of these series is by means of fractal dimensions. The fractal dimension of wind components is a characteristic of turbulent flow, as has been shown in previous works where its relation to stability was highlighted (Tijera et al., 2012).

In this paper the integral scales of u′ and w′ components are compared. The scales are calculated using sonic anemometer data from three elevations 5.8 (∼ 6), 13 and 32 m above the ground at the main tower site of the SABLES 98 field campaign. Turbulent motion of the atmospheric flows occurs through a broad range of scales, from the smallest ones that are usually defined as the scales at which the motion dissipates into heat due to the viscosity of the fluid, to the larger scales corresponding to the integral scale. The integral scale can be defined in several ways: the larger scale of the flow, or the scale above which the Fourier transform has a slope inferior to a -5/3 slope, such as where the turbulent kinetic energy (TKE) is maximum. Micrometeorological studies have found integral scales varying in a huge range, from around 100 to 1000 m (Teunissen, 1980; Kaimal and Finnigan, 1994).

We study the anisotropy of the turbulent atmospheric flows in these scales by comparing the integral scale of fluctuations of the velocity component along the mean wind direction and the vertical component at three different levels above the ground (5.8, 13.5, 32 m).

The irregular behavior of the atmospheric turbulent fluxes in the boundary layer at large Reynolds numbers leads us to be interested in calculating their fractal dimension. Fractal dimension could help us to classify the irregularity of these flows. The more irregular the flow, the greater its fractal dimension. Turbulent flows are characterized by the formation of many eddies of different length scales. Theses irregularities are due to the superimposition of eddies of different sizes, and this is related to a broad range of scales that exist in turbulence. These scales vary from the smallest scales as dissipative scales to larger scales as integral scales. This paper is concerned with the analysis of the relationship between the integral scale and the fractal dimension, as well as with the relationship between the integral scale with the bulk Richardson number, which provides a measure of the degree of stability in the flow, and how this turbulent flow is prone to developing instabilities. It is also used as a criterion for the existence or non-existence of turbulence in a stably stratified environment (a large positive value over a critical threshold is indicative of a decaying turbulence or a complete non-turbulence) (Arya, 2001).

In this section we describe the methodology applied to calculate the fractal dimension and the integral scale. The estimation of the fractal dimension of time series has been the most commonly used criterion to measure their chaotic structure; there exist different works in that direction (Grassberger and Procaccia, 1983; Shirer et al., 1997). One of the methods most commonly used to estimate the fractal dimension of atmospheric flows has been the mean slope method through the box-counting dimension using mean slopes of the graph of ln N(L) vs. ln (L) for small ranges of L, where N(L) is the number of boxes of side L necessary to cover the different points that have been registered in the physical space (velocity–time) (Falconer, 2000; Peitgen et al., 2004). As L→0, then N(L) increases, and N meets the following relation: NL≅kL-d. The value d is the box-counting dimension that is an approximation of the Hausdorff dimension and is calculated approximately by means of least-square fitting of the representation of log N(L) vs. log L, obtaining the straight line regression given by the following equation: log⁡NL=log⁡k-dlog⁡L. The fractal dimension d will be given by the slope of this equation as shown in Fig. 1.

In this paper we focus on calculating the integral scales for horizontal and vertical component fluctuations u′ and w′, and we studied their variations with respect to the fractal dimension and the bulk Richardson number, a turbulent parameter of stability.

These integral scales have been estimated using the normalized autocorrelation function and a Gaussian fit. The velocity autocorrelation function as a function of τ (lag number) for the u′ component is R(τ)=u′(t)u′(t+τ)‾u′2(t)‾. The integral timescale is TL=∫0∞R(τ)dτ≈∫0τ′R(τ)dτ. The integral timescale provides a measure of the scales of eddies in the x direction of a flow field. In Eq. (2) we observed that τ′ denotes the last lag in the data series. In boundary layer observations this timescale can be related to a length by multiplying the mean wind velocity by the timescale. This requires the assumption of frozen turbulence known as Taylor's hypothesis (Panofsky and Dutton, 1984). The integral length scale can be defined as λ=v‾TL. The used method is based on a Gaussian fit of the normalized autocorrelation function R(τ) and we calculated the value of τ that verifies the following equation: τ-∫0τRτdτ=∫ττ′R(τ)dτ. Fig. 2 shows the Gaussian fit for an example of a data series of wind velocities with τ that verifies Eq. (6). This value allows us to calculate the integral timescale by multiplying it by the time interval between each lag.

The data set was recorded at the Research Centre for the lower Atmosphere (CIBA in the Spanish acronym), located in Valladolid province (Spain), and was measured in the SABLES 98 experimental campaign. This research center was set up primarily to study the atmospheric boundary layer. The campaign took place from 10 to 27 September 1998 (Cuxart et al., 2000). This experimental site is a quite flat and homogeneous one that forms a high plain of nearly 200 km2, surrounded by crop fields and some small bushes strewn over the ground. The Duero River flows along the southeastern border of the high plain. The synoptic conditions during the period of study of 8 consecutive days (from 14 to 21 September) were controlled by a high-pressure terrain system that produces thermal convection during the diurnal hours and moderate to strong stable stratification during the nights.

In this work, data from sonic anemometers measured at a sampling rate of 20 Hz installed at 5.8 (∼ 6), 13 and 32 m are analyzed; 5 min non-overlapping series are used to evaluate the different parameters. At a rate of 20 data points per second, sonic anemometers can resolve integral scales between about 10 and 2000 m of the u′ horizontal component and between 1 and 1000 m of the w′ vertical component, depending on the height at which the sonic anemometer is positioned and at the wind speeds typically measured in the SABLES 98 experiment. We detect vertical scales over a broad range of scales from 1 to 1000 m. The integral scales here are calculated based on the autocorrelation function, the mean wind velocity and integral timescale, and each of them can be expected to vary significantly. As the integral scales are the larger scales of turbulent flows, it is possible to detect vertical scales larger than heights at which the sonic anemometer is located.

We have calculated the fractal dimension and the integral scale of the horizontal and vertical components using wind velocity data from sonic anemometers at three different heights: 5.8, 13 and 32 m. The numerical results show slight significant differences in the diurnal and night cycles when the variation of the integral scale is analyzed vs. the fractal dimension. Atmospheric stratification is analyzed for the three heights through the bulk Richardson number, finding the three classical types of stratification along the diurnal cycle. It would be interesting for future works to study the growth of the integral scale in stratified flows and whether this could be due to the inverse cascade on both the diurnal and nighttime cycles. The main conclusions of this study are as follows.

Although all data appear quite scattered in this work, the average values of these magnitudes show interesting results. During the diurnal hours the average values of the integral scale of the horizontal component increase with the increase in fractal dimension to around 1.25 at 5.8 and 13 m height. At these heights we have found linear fits between these magnitudes with high coefficients of correlation. While at 32 m the linear fit is not so evident, the maximum scales are in the 1.15–1.20 range of the fractal dimension. One of the possible explanations for this behavior is that during the diurnal hours the average values of the integral scale increase due to the unstable stratification. During the night hours the average values of the integral scale decrease with the increase in the fractal dimension. These values also fit a straight regression line at the three analyzed heights. During the nights the existence of the stable stratification decreases the integral scale, with an increase in the fractal dimension to an approximate value of 1.2.

For the vertical component of the integral scale the results are similar, even though with slight differences. The average values have fit a quadratic function. During the diurnal hours the average values of the integral scale reach a maximum around the value of the fractal dimension of 1.15 at the three heights. From this value the integral scale decreases with the increase in the fractal dimension to a few meters. The different degree of stratification along diurnal hours will be reflected in that different behavior from the value of 1.15. At nights when stability is normally high, the integral scale decreases with increasing fractal dimension in a parabolic way.

In the unstable and neutral stratification the integral scales are higher than the integral scales under the influence of the stable stratification.

To characterize the anisotropy of turbulent flows we have used the comparison of integral scales of horizontal and vertical components, showing that the scale of the u′ component is almost always larger than the scale of the w′ component at the same height.

Data are not available form a public data repository (they are raw data with 20 Hz sampling rate and then huge files). However, all the information about them can be found in the reference Cuxart et al. (2000). On the other hand, original data used in this work can be provided upon request to author.