Self-similarity means the main body is similar to its fragments. It predicts a power law relation between physical quantity and scales. For wind speed, the physical quantity is the energy spectrum, which has a power law relation with scale ( Kolmogorov, 1941; Kader and Yaglom, 1991; Kaimal and Finnigan, 1994). It reveals the self-similarity of various scales of eddies in the inertial range ( Frisch, 1995). The inertial range is usually below 4 min for wind speed ( Van der Hoven, 1957; Stull, 1987). This is a small-scale range. When the inertial range is exceeded and for larger scales, the self-similarity is broken.

However, the large-scale fluctuations of 4-30 min are important for the transport process in the atmospheric boundary layer ( Sakai et al., 2001). This paper tries to recover the self-similarity of large-scale fluctuation, from the perspective of the shape of the time series, rather than the perspective of the energy spectrum. In the time series of wind speed, there are various scales of fluctuations. Almost all these fluctuations have a similar shape-a ramp structure. It suggests the self-similar characteristic for these ramp structures, and a possible power law relation between a certain physical quantity of the ramp structure and the time scale. Antonia et al. (1982) used to study the ramp structure of temperature, and pointed out that it reflects a kind of coherent structure. Coherent structures are often used to study flux transport ( Chen et al., 1997; Katul and Hsieh,1997) and gust ( Cheng et al., 2007; Zeng et al., 2010), but not the self-similarity character. From the analysis results of several ramp structures, we suspect that this physical quantity is the linear slope of the ramp structure. To confirm this hypothesis, sufficient data are analyzed and the results are presented in this paper.

The detail about the linear slope of the ramp structures is introduced in section 2.1. Before calculating its linear slope, an integrated ramp structure must be precisely extracted from the data. However, the time series is very disordered, which makes it extremely hard to distinguish an integrated ramp structure. Therefore, we must use some reliable methods to find the ramp structure. The discrete wavelet transform can manifest ramp structures clearly in data; and this is introduced in section 2.2. Data are collected from three different underlying surfaces: grassland (from Xilin Gol grassland, height 70 m), city (from Beijing 325 m Meteorological Tower, height 47 m), and lake (from Baiyangdian Lake, height 12 m). More details about these data are introduced in section 2.3. In section 3, we show the computed results of the linear slopes of the ramp structures, and verify our hypothesis.

Ramp structure of wind speed contains two parts: an increase part and a decrease part. Both parts show a linear characteristic and have a linear slope, k. Using the discrete wavelet transform, ramp structures can manifested clearly in the time series of wind speed ( Hu, 1998; Li et al., 2001; Cheng and Hu, 2005). Their linear slope, k, can then be calculated. Time scale is expressed by δ _t in this paper.

2.1 Linear characteristic of ramp structure and its linear slope, k

The time series of wind speed are composed of large and small ramp structures, as shown in Fig. 1. In this figure, three ramp structures have been extracted, and these are shown in black frames. We can see that the large ramp structure contains small ramp structures; the small ramp structure contains smaller ones, and so on.

The ramp structure contains two parts: an increase part and a decrease part. The body of the increase/decrease part always shows a linear characteristic. Its linear slope, k, can be computed by linear regression. We compute the k of the three ramp structures extracted in Fig. 1, and these are shown in Fig. 2. kappears to have an inverse power law relation with δ _t, during both the increase and decrease parts. Next, we extract many ramp structures from the data, to verify this power law relation.

Figure 1

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	Figure 1 Ramp structures of wind speed. Data collected from Xilin Gol grassland (height: 30 m). V is the wind speed (units: m s-1) of the prevailing wind direction; t is time (units: s).

Figure 2

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	Figure 2 Three ramp structures extracted from Fig. 1, and the linear slope, k(units: m s-2), and scale, δ t (units: s), of their increase/decrease parts.

However, the time series of wind speed sometimes seem very complicated and disordered, which makes it extremely hard to distinguish ramp structures; so, we need a reliable method to manifest the ramp structures. The discrete wavelet transform, as we shall see next, is a reliable method.

Using the above method, we obtain 809 ramp structures from the Xilin Gol grassland data, 782 ramp structures from the Beijing 325 m Meteorological Tower data, and 804 ramp structures from the Baiyangdian Lake data. We compute their linear slope, k, (for both the increase and decrease parts) and scale, δ _t. These are shown in Fig. 4. The decrease parts are on the left of the figure, and their linear slope, k, are negative. The increase parts are on the right, and their k are positive. In this figure, there seems to be a power law relation between k and δ _t for both the decrease and increase parts: or (the latter matches Fig. 4 as the x-axis is k). The angle brackets represent the average and θ is the power exponent. The smaller figures within Fig. 4 are logarithmic charts. They show that within a certain range of dispersion, log(| k|) has a linear relation with log( δ _t), showing that a power law relation does indeed exist. Although three sets of data are collected from different underlying surfaces, their power law relations are all obvious, except for slight differences in their power exponents. This is a universal relation which is free from an environmental effect.

Figure 3

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	Figure 3 A sketch of how to manifest ramp structures from large scale to small scale.

Figure 4

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	Figure 4 Analysis result of data from (a) Xilin Gol grassland (height: 70 m; power exponents, θ, are 0.75 (decrease part) and -0.76 (increase part)); (b) Beijing 325m Meteorological Tower (power exponents, θ, are 0.96 (decrease part) and -0.90 (increase part)); and (c) Baiyangdian Lake (height: 12 m; power exponents, θ, are 0.79 (decrease part) and -0.76 (increase part)).

We then compute the power exponents, θ, by linear regression. θ has opposite sign to k. The θof the decrease parts are 0.75 (Xilin Gol grassland), 0.96 (Beijing 325 m Meteorological Tower), and 0.79 (Baiyangdian Lake). The θof the increase parts are -0.76 (Xilin Gol grassland), -0.90 (Beijing 325 m Meteorological Tower), and -0.76 (Baiyangdian Lake). For both the decrease and increase parts, their linear slope, k, all have an inverse power law relation with scale, δ _t.

From the perspective of self-similarity, this power law relation means that the physical quantity has self-similar characteristic. In our research, this physical quantity was the linear slope, k, of the ramp structure. The lower limit of the self-similar scale range was 2 s. The upper-limit scale is higher than expected. It reaches 27 min in the Xilin Gol grassland data, while the upper-limit scale of the inertial range is only 4 min. The large-scale fluctuations that contribute to the transport process are contained in this range. The self-similarity of the large scale is recovered in our work.

This paper investigates the self-similar characteristic for the ramp structure of wind speed. There are two important quantities of ramp structures: linear slope, k, and scale, δ _t. Data analysis of three different underlying surfaces all show a power law relation between k and δ _t. This suggests that the k of the ramp structure has a self-similar characteristic. The data give a self-similar scale range from 2 s to 27 min.

This work extends the self-similar range to a larger scale, while it is often considered that the self-similarity is broken in large scales. It is a little like the work of extended self-similarity ( Benzi et al., 1993), which also extends the inertial range to a larger scale by using a simple algebraic transformation. This reveals the hidden self-similarity in large scales. Our work also reveals this hidden self-similarity from the perspective of the linear slope of the ramp structure. We suppose that this self-similarity may also exist in other fields such as economics, biology, and so on. We intend to verifying this in the future.

However, in the extraction process we do not obtain all the ramp structures that constituted the time series. On the one hand, some ramp structures are too disordered to be distinguished; so, we have to abandon them. On the other hand, the wavelet method we use has its own limitation. It depends on manual intervention to judge the fitting effect. This process takes far too much time if we choose all the ramp structures; therefore, we only choose some of them. However, it is not purposive behavior and the choice is random. The wavelet method needs further development to get rid of the manual intervention. In addition, as a limitation of sample frequency, we cannot obtain smaller ramp structures. To avoid the sampling distortion, we only use ramp structures that are above 2 s. To verify this self-similarity of smaller scales, this research needs higher sample frequency data.

Acknowledgements. This work is supported by the National Natural Science Foundation of China (Grant No. 91215302), "One-Three-Five" Strategic Planning (wind power prediction) of the Institute of Atmospheric Physics, Chinese Academy of Sciences (CAS) (Grant No. Y267014601), and the Strategic Project of Science and Technology of CAS (Grant No. XDA05040301).