Extracting of Bursting Fluctuations in the Time Series of Atmospheric Turbulence by the Statistical Analysis

It has been found that the probability density functions (PDFs) of atmospheric turbulence are generally to be non-Gaussian and their tails are much longer than a Gaussian distribution (Liu et al., 2010, Liu et al., 2011). Longer tails mean that large fluctuations will appear more frequently than Gaussian predictions and the observed time series seem to be bursting. 

Many interesting phenomena are related to bursting fluctuations. One is the cumulative effect. It means that although bursting fluctuations have smaller probabilities than background signals, they have larger magnitudes and thus could cause noticeable effects (Mahrt, 1998). Another interesting phenomenon is the local effect. It means that influential effects are focused on a small fraction of time interval. In some cases, the local effect will be important for our lives. In the nocturnal boundary layer, the bursting vertical mixing can bring the ozone aloft to the surface and this would lead to a local pollutant event (Salmond and McKendry, 2005). The mechanism of bursting fluctuations is still unknown. In CBL, it may be related to thermals and in stable NBL, it has found that the bursting fluctuations (also called intermittent turbulence) could be caused by some dynamical instability, such as strong wind shear near surface (Blackadar, 1979), low lever jet (Derbyshire, 1990) and gravity waves (Nappo, 1991).

Bursting fluctuations in the time series of atmospheric turbulence are always mingled with noise and background turbulence. If we want to study their characteristics from observations, we should extract bursting fluctuations from original observations. The most commonly used method for extracting bursting fluctuations is conditional sampling analysis, where fluctuations above a threshold are considered to be bursting signals (Doran, 2004). However, many works set the threshold artificially, and different thresholds may cause apparently conflicting results (Schumann and Moeng, 1991). This becomes one major criticism of conditional sampling analysis (Mason et al., 2002).

In this paper, we propose a new method to extract the bursting fluctuations in the time series of atmospheric turbulence (Liu et al., 2014). Unlike the commonly used conditional sampling analysis, this method defines a threshold by using a systematical method and tries to reduce the artificiality in this process. It defines threshold by the position where the types of probability density functions (PDFs) of turbulent fluctuations begin to change character from stable distributions to truncated stable distributions. Absolute values of fluctuations greater than the threshold are considered to be bursting fluctuations. We have used this method to extract the bursting fluctuations in the time series of vertical velocity in CBL (see Figure 1) and intermittent turbulent heat fluxes in stable NBL (see Figure 2). Results show that our method works well.

It should be mentioned here that Katul et al. (1994) have used a similar method to extract bursting fluctuations. Different from our method, they use a Gaussian distribution as a reference distribution and data deviate from Gaussion are considered to be bursting fluctuations. The reasons that we use a stable distribution as a reference distribution are as follows. First, Gaussion distribution is a symmetric distribution where the background turbulence and noise are apparently asymmetric and can't be fitted with Guassian. Second, stable distributions can be asymmetric with some parameters and can fit the background turbulence and noise well (Liu et al., 2010; Liu et al., 2011). Third, mathematically, the Gaussion distribution is a limit prediction of the central limit theorem. The stable distribution is a limit prediction of the generalized central limit theorem and the Gaussian distribution is in fact a special situation of the stable distribution. This makes our choice is more reasonable than Gaussian.

 

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