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Atmospheric Disturbance Characteristics in the Lower-middle Stratosphere Inferred from Observations by the Round-Trip Intelligent Sounding System (RTISS) in China


doi: 10.1007/s00376-021-1110-2

  • Through multi-order structure function analysis and singularity measurement, the Hurst index and intermittent parameter are obtained to quantitatively describe the characteristics of atmospheric disturbance based on the round-trip intelligent sounding system (RTISS) in the lower-middle stratosphere. According to the third-order structure function, small-scale gravity waves are classified into three states: stable, unstable, and accompanied by turbulence. The evolution of gravity waves is reflected by the variation of the third-order structure function over time, and the generation of turbulence is also observed. The atmospheric disturbance intensity parameter RT is defined in this paper and contains both wave disturbance ($ {H}_{1} $) and random intermittency ($ {C}_{1} $). RT is considered to reflect the characteristics of atmospheric disturbance more reasonably than either of the above two alone. In addition, by obtaining the horizontal wavenumber spectrum from the flat-floating stage and the vertical wavenumber spectrum from the ascending and descending stages at the height range of 18–24 km, we found that when the gravity wave activity is significantly enhanced in the horizontal direction, the amplitude of the vertical wavenumber spectrum below is significantly larger, which shows a significant impact of gravity wave activity on the atmospheric environment below.
    摘要: 基于国内的往返式智能探空系统(RTISS),通过多阶结构函数分析和奇异测度,获得了Hurst指数和间歇性参数,以定量描述中下平流层的大气扰动特征。根据三阶结构函数,小尺度重力波被分为稳定、不稳定和破碎三种状态。重力波的演化表现为三阶结构函数随时间的变化,也观察到湍流的产生。本文定义了大气扰动强度参数RT,包含波扰动(H1)和随机间歇性(C1),RT被认为比单独使用上述两者能更合理地反映大气扰动的特征。此外,通过获得18-24 km高度范围内的平漂阶段的水平波数谱和上升和下降阶段的垂直波数谱,我们发现当重力波活动在水平方向显著增强时,下方18-24 km高度范围内的垂直波数谱的幅振幅明显增大,说明重力波活动对下方大气环境影响显着。
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  • Figure 1.  Schematic diagram of the RTISS. L is the distance from the station to the radiosonde, Ω and θ are the azimuth and elevation, respectively, and the black curve is the detection trajectory of WH on 30 October 2018.

    Figure 2.  (a) The trajectory of the flat-floating phase and (b) the floating height variation with time.

    Figure 3.  The third-order structure function of the flat-floating stage from (a) WH2, (b) YC2, and (c) CS2. The drawn third-order structure functions are absolute values, where the red dots represent negative values and the blue dots represent positive values.

    Figure 4.  The multi-order structure function (q = 1, 2, 3, 4, 5) with the separation scale r calculated from the horizontal velocity component $ {u_L} $ from (a) WH2, (b) YC2, and (c) CS2 in the flat-floating stage.

    Figure 5.  The variation of horizontal velocity component $ {u_L} $ along the meridional (zonal) distance $ L $ from (a) WH2, (b) YC2, and (c) CS2, where $ {{{r}}_1} $, $ {r_2} $, $ {r_3} $, and $ {r_4} $ correspond to the intervals with significantly large fluctuations.

    Figure 6.  Velocity increments calculated from beginning to end in the WH2 data series where the separation distances are (a) 64 m, (b) 512 m, (c) 4096 m, and (d) 32 768 m, respectively.

    Figure 7.  Third-order structure function on different segments for WH2.

    Figure 8.  (a) The variation of multi-order singularity measure with spatial scale for WH2 data. (b) The variation of slope with order $ q $ for WH2 data. (c)–(d) The same as WH2 but for YC2. (e)–(f) The same as WH2 but for CS2. The red dotted line corresponds to $ K\left( q \right) = 0 $, and the solid black line (and dashed line) represents the tangent slope of the turbulent (gravity wave) area at $ q $ = 1.

    Figure 9.  The third-order structure function of the flat-floating stage from other data series. The purple dashed line and the green dashed line represent the result of linear fitting, and the slope value is marked beside each line. When there is an obvious change in slope, the dashed lines of different colors are used to distinguish it.

    Figure 10.  (a) Hurst parameter (blue curve) and intermittency (red curve). (b) The spectral amplitude from the flat-floating stage (blue curve) and the vertical range of 18–24 km (red curve). (c) The intensity of atmospheric disturbance RT (blue curve) and the average height in the flat-floating stage (red curve) in 12 sets of data. The numbers in the figure correspond to the serial numbers in Table 1.

    Table 1.  Data information after decomposition and re-interpolation in the flat-floating phase.

    NumberDirectionStep (m)Distance (km)
    1-WH1zonal1324.5
    2-WH2zonal1667.7
    3-WH3meridional1390.3
    4-CS1zonal1376.0
    5-AQ1meridional1129.0
    6-AQ2zonal1653.6
    7-YC1meridional12105.5
    8-YC2meridional861.9
    9-GZ1meridional1587.6
    10-GZ2meridional937.7
    11-GZ3meridional954.9
    12-CS2meridional1150.4
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Manuscript received: 17 March 2021
Manuscript revised: 29 June 2021
Manuscript accepted: 24 August 2021
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Atmospheric Disturbance Characteristics in the Lower-middle Stratosphere Inferred from Observations by the Round-Trip Intelligent Sounding System (RTISS) in China

    Corresponding author: Zheng SHENG, 19994035@sina.com
  • 1. College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China
  • 2. Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing 210094, China
  • 3. Unit No. 95806 of Chinese People’s Liberation Army, Beijing 100076, China

Abstract: Through multi-order structure function analysis and singularity measurement, the Hurst index and intermittent parameter are obtained to quantitatively describe the characteristics of atmospheric disturbance based on the round-trip intelligent sounding system (RTISS) in the lower-middle stratosphere. According to the third-order structure function, small-scale gravity waves are classified into three states: stable, unstable, and accompanied by turbulence. The evolution of gravity waves is reflected by the variation of the third-order structure function over time, and the generation of turbulence is also observed. The atmospheric disturbance intensity parameter RT is defined in this paper and contains both wave disturbance ($ {H}_{1} $) and random intermittency ($ {C}_{1} $). RT is considered to reflect the characteristics of atmospheric disturbance more reasonably than either of the above two alone. In addition, by obtaining the horizontal wavenumber spectrum from the flat-floating stage and the vertical wavenumber spectrum from the ascending and descending stages at the height range of 18–24 km, we found that when the gravity wave activity is significantly enhanced in the horizontal direction, the amplitude of the vertical wavenumber spectrum below is significantly larger, which shows a significant impact of gravity wave activity on the atmospheric environment below.

摘要: 基于国内的往返式智能探空系统(RTISS),通过多阶结构函数分析和奇异测度,获得了Hurst指数和间歇性参数,以定量描述中下平流层的大气扰动特征。根据三阶结构函数,小尺度重力波被分为稳定、不稳定和破碎三种状态。重力波的演化表现为三阶结构函数随时间的变化,也观察到湍流的产生。本文定义了大气扰动强度参数RT,包含波扰动(H1)和随机间歇性(C1),RT被认为比单独使用上述两者能更合理地反映大气扰动的特征。此外,通过获得18-24 km高度范围内的平漂阶段的水平波数谱和上升和下降阶段的垂直波数谱,我们发现当重力波活动在水平方向显著增强时,下方18-24 km高度范围内的垂直波数谱的幅振幅明显增大,说明重力波活动对下方大气环境影响显着。

    • A disturbance in the atmosphere can be regarded as the superposition of fluctuations of various scales, ranging from turbulent scale to planetary wave scale. As a vital causing of atmospheric disturbance, gravity wave (GW) activity has been extensively studied. The propagation and development of GWs have an important impact on the atmospheric circulation at higher latitudes (Lindzen, 1981; Fritts et al., 2014) and play an important role in the material transport and energy transmission between the stratosphere and the troposphere (Zhao et al., 2019; Chang et al., 2020; Sheng et al., 2020; He et al., 2021), leading to interactions and energy cascade processes among different scales. At present, the detection means of GWs are relatively abundant. Remote sensing measurements from satellite platforms can cover a worldwide range (Wu et al., 2006; Preusse et al., 2008; Alexander et al., 2010; Wright et al., 2016), but the spatial resolution is limited. Observations of temperature and wind from radiosondes have extensive data accumulation and high resolution and can be used to analyze GWs (Wang and Geller, 2003; Murphy et al., 2014; Yoo et al., 2018) and turbulence (Luce et al., 2014; Zhang et al., 2019). Although rocket detection can reach a higher altitude range (up to 60 km), the release cost is much higher than for radiosondes, and it is not suitable for long-term observation (Lübken et al., 1993; Lübken, 1997). Radar and lidar can achieve continuous observations at a wide range of altitudes (Dhaka et al., 2003; Stober et al., 2012; Sato et al., 2014; Love and Murphy, 2016; Kantha et al., 2017), however, they are restricted to a few specific sites. Aircraft detection can be used to study the characteristics of GWs in the horizontal direction and is usually performed only for specially designed campaigns (Wroblewski et al., 2007; Zhang et al., 2015).

      In the upper troposphere and lower stratosphere, the influence of clear air turbulence on economic aircraft is crucial, and this fact has directly prompted many related studies in this height regime (Sharman et al., 2012). Wroblewski et al. (2007) studied turbulence and Kelvin–Helmholtz billows in stably stratified shear flow in the upper troposphere. Lu and Koch (2008) analyzed horizontal velocity fields using spectrum and structure function analysis and found clear scale interactions between turbulence and small-scale GWs. Of course, the area within the upper troposphere and lower stratosphere contains dynamic processes of multiple scales ranging from several thousand kilometers to several millimeters, covering from planetary waves to small-scale turbulence. Therefore, there are abundant studies on the dynamic structure within this height range (Nastrom and Gage, 1985; Bacmeister et al., 1996; Hooper and Thomas, 1998; Cho and Lindborg, 2001; Zhang et al., 2015; Söder et al., 2021).

      Ground-based and space-based techniques are usually unable to measure the wave intrinsic frequency (in the Lagrangian frame of reference), which can reflect important wave characteristics. The use of superpressure balloons (SPB) does not suffer this limitation and provides a new research method (Hertzog et al., 2002; Boccara et al., 2008). However, the “old” SPB had a frequency limit (Vincent et al., 2007), and only recent campaigns have recorded at higher frequency (Schoeberl et al., 2017). The RTISS discussed in this article uses a zero-pressure balloon, which has a lower cost and higher sampling frequency, and for the first time achieved three-stage detection of rising, flat-floating, and falling. On the basis of previous research (He et al., 2020), the evolution of GWs and turbulence is thoroughly studied and discussed in this manuscript, and a new way for diagnosing the atmospheric disturbance state is proposed. In previous studies, the structure function has been used to connect the scale directly with the actual measurement results, since the structure function at a specific distance can reflect the direct measurement of the corresponding wind velocity. Considering that Kolmogorov's theory predicts that the law of –5/3 slope will be followed in the turbulent inertia region (Kolmogorov, 1991), it is reasonable to expect the wind variance of a GW to be between –1 and –3 (Cho and Lindborg, 2001). Compared with spectral analysis, the most obvious advantage of the third-order structure function is that it is possible to calculate energy flux by only using one-dimensional data series (Lindborg, 1999). In order to further quantify the disturbance characteristics of turbulence and small-scale GWs, the Hurst parameter is obtained by a multi-order structure function, and the intermittent parameter is obtained by a singular measurement (Marshak et al., 1997).

    2.   Observations and Methods
    • The data from the intelligent round-trip sounding system experimental project carried out in China is used to conduct the corresponding research. The detection process is shown in Fig. 1, including the three stages of rising, flat-floating, and falling, and is able to complete the observation in the vertical direction and the continuous high-frequency observation at a stable height in the horizontal direction. In the ascending stage, the outer ball is used to provide upward buoyancy while carrying a radiosonde for real-time observation and collection of data; the outer ball explodes at a predetermined height, and then the inner ball provides buoyancy in the flat-floating stage to balance gravity. The inner ball is a zero-pressure balloon and reaches a balance between its own gravity and buoyancy at the height of the designed ascent limit. After reaching the predetermined position, the system separates the radiosonde and the inner ball through the fuse device. The radiosonde falls, guided by a parachute, and recording measurements on its way down. In the floating phase, the balloon performs a quasi-Lagrangian measurement with a frequency of 2 s−1. The radiosonde is equipped with a sensor module consisting of sensors for temperature, humidity, and air pressure measurements. The Beidou navigation satellite system is used for longitude, latitude, and altitude measurement, and the horizontal wind speed and direction are calculated from this.

      Figure 1.  Schematic diagram of the RTISS. L is the distance from the station to the radiosonde, Ω and θ are the azimuth and elevation, respectively, and the black curve is the detection trajectory of WH on 30 October 2018.

      The detection system has different working principles in the three stages of rising, flat-floating, and falling (Cao et al., 2019). In the ascending phase, the balloon is subjected to buoyancy, gravity, and air resistance in the vertical direction. In the horizontal floating stage, the adaptive flat-floating process is realized by controlling an appropriate net lift force from the ground. Under this condition, the vertical force is dynamically balanced, and the balloon trajectory can be regarded as an approximate horizontal motion. In the descending stage, the radiosonde descends under the parachute, from the low-density atmosphere into the high-density atmosphere.

      We obtained 12 sets of data retrieved in Wuhan, Anqing, Yichang, Ganzhou, and Changsha in 2018, which are: WH1, WH2, and WH3 (22, 23, and 30 October); AQ1 and AQ2 (19 October and 16 November); YC1 and YC2 (18 October and 22 October); GZ1, GZ2, and GZ3 (31 October, 6 November and 9 November); CS1 and CS2 (19 October 19 and 16 November), respectively. The data in the ascending and descending stages are evenly interpolated to a vertical step of 12 m. The balloon trajectory in the flat-floating phase and the height variation with time are shown in Fig. 2. The balloon trajectories in the flat-floating phase are interpolated to x, y coordinates relative to the start of the flat-floating phase in a local geophysical coordinate system (it can also be said to be the projection of the meridional and zonal). In the CS1 data, there is a significant drop in the height after the start of the flat-floating phase, and this segment is deleted in the subsequent calculations. Originally sampled equidistant in time, the sampling distance of the interpolated data is adjusted to the average spatial sampling during the respective flight. Specific information for the flat-floating phase is shown in Table 1.

      Figure 2.  (a) The trajectory of the flat-floating phase and (b) the floating height variation with time.

      NumberDirectionStep (m)Distance (km)
      1-WH1zonal1324.5
      2-WH2zonal1667.7
      3-WH3meridional1390.3
      4-CS1zonal1376.0
      5-AQ1meridional1129.0
      6-AQ2zonal1653.6
      7-YC1meridional12105.5
      8-YC2meridional861.9
      9-GZ1meridional1587.6
      10-GZ2meridional937.7
      11-GZ3meridional954.9
      12-CS2meridional1150.4

      Table 1.  Data information after decomposition and re-interpolation in the flat-floating phase.

    • The perturbation profile of atmospheric elements contains abundant fluctuation information. The energy corresponding to different scales can be obtained intuitively and clearly using Fourier spectral transformation, and the spectral amplitude and spectral slope can quantitatively describe the spectral structure, reflecting the disturbance characteristics of the atmosphere. Here we use the potential temperature perturbation profile to perform spectral transformation. For non-stationary random processes with stationary increments, the statistical characteristics of increments can be defined as (Marshak et al., 1997):

      Since the turbulence theory focuses on the increment of velocity, the velocity increment between $ {x} $ and $ x + r $ is defined as $ {\text{d}}{u_L}\left( {x,r} \right) $ and $ {\text{d}}{u_T}\left( {x,r} \right) $, where $ {u_L} $ and $ {u_T} $ represent the velocity components parallel and perpendicular to the separation distance r (consistent with decomposition direction), respectively. The second-order velocity structure function can be defined as:

      where $ \left\langle . \right\rangle $ is the ensemble average. Considering that the Fourier power spectrum alone cannot distinguish different theories, based on Kolmogorov theory, the third-order structure function is used here to quantify and compare the disturbance information contained in the atmospheric wind field (Lindborg, 1999; Cho and Lindborg, 2001). The third-order structure function not only can eliminate the arbitrariness of the universal constant in the power law expressed by the second-order structure function in physical space, but it also can reflect the direction of the energy cascade through the sign of the value, where negative values represent downscale energy cascades and positive values represent upscale energy cascades. Considering that transition in the stratosphere occurs within a thin layer (Benavides and Alexakis, 2017), the third-order structure function can be written as (diagonal part):

      where $ \varepsilon $ is the energy dissipation rate. In order to systematically describe the random atmospheric process, referring to the multi-order structure function and singularity measurement method (Davis et al., 1994; Marshak et al., 1997), the q-order structure function can be written as:

      Through multi-order structure function analysis and singularity measurement, the Hurst index and intermittent parameter are obtained. The specific calculation method is described in Lu and Koch (2008).

    3.   Results and discussion
    • Using the method introduced in section 2, we calculated the third-order structure function for all 12 sets of data. Firstly, the third-order structure function $ {S_3}\left( r \right) $ is calculated over the separation distance $ r = l \times {2^n} $, $ n = 0,1...,N $, where $ l $ is the average step. $ N $ is determined by the maximum data length. The experimental results of Lu and Koch (2008), by computing and analyzing the third-order structure functions for the aircraft observational data, verify the consistency of Kolmogorov theory well in the application of structure functions within the inertial subrange (Cho and Lindborg, 2001). In the turbulent inertial region, $ {S_3}\left( r \right) $ follows the r slope. In the GW subrange, $ {S_3}\left( r \right) $ follows the r2 slope when turbulence occurs, while it follows the r3 slope without turbulence. We firstly select three sets of data for third-order structure function analysis. The results are shown in Fig. 3. The data sequences are from WH2, YC2, and CS2, respectively. The absolute value of the third-order structure function is plotted in the figure, where red represents negative values and blue represents positive values. The black lines represent the expected r3, r2, and r1 power spectra, respectively. There is obvious turbulent activity in Fig. 3a. In the large scale, it is an upscale energy cascade, and in the small scale, it is a downscale energy cascade. There is an obvious r slope in the spatial scales within 128 m. In the spatial scales larger than 128 m, the whole curve follows the r2 slope. In Fig. 3b, the curve in the main separation distance range obviously follows the r3 slope and the upscale energy cascade dominates in the entire scale range, indicating that there is stable GW activity in the data series. In Fig. 3c, the whole curve is inclined to the r2 slope, with the first half (11–400 m) more inclined to the r3 slope and the second half (>400 m) more inclined to the r2 slope. There is an upscale energy cascade on small scales and a downscale energy cascade on large scales. According to Lu’s results, the r2 slope should indicate the coexistence of turbulence and GWs, but there is no r slope on a small scale. Here, we think there may be two reasons. One is that the horizontal scale of turbulence is too small, and it is not reflected in the resolution of the observation data. The second is that it is at the transitional stage from GW to turbulence, which has not caused the breaking of GWs and the generation of turbulence. In this paper, GW activity can be divided into three different states: stable GWs, where the $ {S_3}\left( r \right) $ follows the r3 slope; unstable GWs, where the $ {S_3}\left( r \right) $ follows the r2 slope; and GWs that coexist with turbulence, where $ {S_3}\left( r \right) $ follows the r slope in the turbulent subrange and r2 slope in the GW subrange.

      Figure 3.  The third-order structure function of the flat-floating stage from (a) WH2, (b) YC2, and (c) CS2. The drawn third-order structure functions are absolute values, where the red dots represent negative values and the blue dots represent positive values.

    • For different GW activity characteristics, does the displayed multi-order structure function satisfy the linear relationship? Are there multiple scale ranges corresponding to different linear relationships? How can these phenomena be explained? Taking WH2, YC2, and CS2 as examples, the results of the multi-order structure function obtained are shown in Fig. 4. When calculating the Hurst parameter, choosing the appropriate fitting interval is a critical step. In other words, linear regression should be performed between inner scale $ \eta $ and outer scale $ R $ in log-log coordinates, where $ \eta = 4l $. We focus on the curve of $ q $ = 5 to select this fitting interval, which emphasizes the most intense events. On different data sets, we can observe different numbers of scale breaking points which show a significant change in the linear relationship for $ {S_q}\left( r \right) $ and $ r $. For WH2, there are some obvious small fluctuations in the fifth-order structure function with several scale breaking points (significant changes in the fitting slope), which is thought to be caused by different energy cascade directions in multiple scales. For YC2, there is an obviously smoother curve, and an approximately linear relationship can be seen within 3 km, which is consistent with the single energy cascade direction. For CS2, it has a linear relationship weaker than YC2 and stronger than WH2, with only one scale breaking point, which is within 20–30 km. In order to make the parameters obtained from different data sets comparable, the outer scales are all selected in the range of several kilometers since there may be multiple points that satisfy the linear relationship.

      Figure 4.  The multi-order structure function (q = 1, 2, 3, 4, 5) with the separation scale r calculated from the horizontal velocity component $ {u_L} $ from (a) WH2, (b) YC2, and (c) CS2 in the flat-floating stage.

      To specifically understand this phenomenon, the horizontal velocity component $ {u_L} $ is shown in Fig. 5 for the three sets of data. $ {{{r}}_1} $, $ {r_2} $, $ {r_3} $, and $ {r_4} $ correspond to the intervals with significant inclinations accompanied by a relatively large increase or decrease in the speed increment on these intervals, which leads to the ‘saturation point’ of the fifth-order structure function in Fig. 4. With the increase of $ r $, the fifth-order structure function obviously increases first, which is due to increments in the obviously larger inclination (corresponding to the width interval of $ {{{r}}_1} $, $ {r_2} $, $ {r_3} $, and $ {r_4} $, respectively). When increasing to a certain value, the fifth-order structure functions of the three sets of data will tend to saturate, contributing a fixed number of disturbance events to the spatial average. For YC2, the fifth-order structure function will increase with the continuous increase of $ r $. The distance between the maximum and minimum of $ {u_L} $ corresponds to $ {r_2} $ and $ {r_3} $, respectively. This is consistent with the saturation points on the curve in Fig. 4b at the separation distances of 16 km and 32 km, respectively. As for CS2, while the $ \left| {\delta u\left( r \right)} \right| $ in the second half increases with the increasing r, the $ \left| {\delta u\left( r \right)} \right| $ in the first half will decrease more, which caused a downward trend in the fifth-order structure function when exceeding the saturation points at the separation distance of 22 km.

      Figure 5.  The variation of horizontal velocity component $ {u_L} $ along the meridional (zonal) distance $ L $ from (a) WH2, (b) YC2, and (c) CS2, where $ {{{r}}_1} $, $ {r_2} $, $ {r_3} $, and $ {r_4} $ correspond to the intervals with significantly large fluctuations.

      Here, we choose $ {H_1} = H(1) = \zeta \left( 1 \right) $ as the Hurst parameter. A linear regression is performed between the inner scale $ \eta $ and outer scale $ R $ in log-log coordinates to obtain $ \zeta \left( q \right) $ from the first-order structure function. $ {H_1} $ can represent the roughness (nonstationarity) of random processes in the atmosphere. The value is between 0 and 1. Larger values reflect smoother data sequences. The Hurst parameters of WH2, YC2, and CS2 are 0.50, 0.71, and 0.76, respectively.

      To give the results more credibility, we use the WH2 data as an example to specify that averaging across different scales is statistically sufficient to support the results when calculating structural functions. At each fixed separation distance r, the calculated speed increments $ \delta {u_L} $ are sequentially moved backward from the starting point, so even at the maximum separation distance, there is enough data set of $ {\text{d}}{u_L} $. Figures 6ad correspond to all velocity increments calculated from beginning to end where the separation distances are 64 m, 512 m, 4096 m, and 32 768 m, respectively, shown as the 3rd, 6th, 9th, and 12th sequence points in Fig. 4a. With the increase of separation distance, the number of corresponding velocity increments decreases gradually. However, all remain at a number sufficient to meet the statistical characteristics. It can be seen from the figure that as the separation distance r increases, the overall fluctuation of $ {\text{d}}{u_L} $ increases, which is caused by the obvious inclined segment in the speed curve, leading to the gradual increase of $ {S_q}\left( r \right) $. Therefore, only small-scale GWs are discussed here. The slope of the third-order structure function on a larger scale (tens of kilometers or even larger) may be affected by the fluctuation of flat-floating height, so we will not discuss it here.

      Figure 6.  Velocity increments calculated from beginning to end in the WH2 data series where the separation distances are (a) 64 m, (b) 512 m, (c) 4096 m, and (d) 32 768 m, respectively.

    • Since the data is continuously measured, we think it is feasible to divide the entire data sequence into several segments. The detection range in the horizontal direction is within tens of kilometers, and the detection time is generally a few hours. Within this temporal and spatial resolution, the divided segments can represent different flat-floating stages, which can also reflect the wave evolution characteristics over time. Since the generation of turbulence has been observed in the WH2 data, the distance of flat-floating is too long. We hope to refine the phase of the observed turbulence, so the target data is segmented and then the third-order structure function is calculated separately. The total number of data points for WH2 is 4234. Figures 7ai correspond to the segments of different flat-floating stages. The number of data points on each segment is the same, and the horizontal distance of the corresponding segment increases gradually with time.

      Figure 7.  Third-order structure function on different segments for WH2.

      Considering that the occurrence of turbulence is related to the appearance of the r slope in a small-scale range, it is found through observation that there is an r slope in Figs. 7ce, and the downscaling energy cascade gradually expands with time, from large scale to small scale. The inconsistency of energy transmission directions on different scales also causes the occurrence of instability of GWs. However, turbulence can only occur if the inconsistency of the energy cascade occurs within the corresponding inertial scale (For the third-order structure function on different segments, the inconsistency of energy cascade occurs ranging from tens of meters to hundreds of meters when the turbulence occurs, while that occurs on a larger scale when there is no turbulence). For example, Fig. 7a shows an upscale energy cascade on all scales. Then, from a large-scale wave source, the downscale energy cascade gradually transfers to a small scale (Figs. 7b and c), which eventually leads to the increase of fluctuating instability and the generation of turbulence. After the turbulence is weakened, the GW gradually returns to be stable, and the energy cascade direction at all scales returns to be the same as shown in Fig. 7i. Also, the result shows that the data form different segments do have differences in slope and energy cascade, but the results calculated using the entire segment of data (unsegmented) can reflect turbulence information (if it exists). Using the third-order structure function of the entire data segment (Fig. 3a) can reflect the characteristics of the overall flat-floating data and will not miss the short-term turbulent activity on individual segments.

    • In addition to the Hurst parameter, intermittency (Marshak et al., 1997) has also been calculated to reflect the statistical characteristics of atmospheric random processes. The larger the intermittence parameter, the more singular the data series. Taking WH2, YC2, and CS2 as examples, the results are shown in Fig. 8. Figure 8a shows the relationship between the multi-level singularity measurement $ \left\langle {\varepsilon {{\left( r \right)}^q}} \right\rangle $ and the spatial scale $ r $ in log-log coordinates. The slope $ K\left( q \right) $ is calculated and shown in Fig. 8b. Similarly, according to the fifth-level singularity measurement $ \left\langle {\varepsilon {{\left( r \right)}^5}} \right\rangle $, the turbulence fitting interval and the GW fitting interval are distinguished (i.e., $ \left[ {\eta ,{R_{\text{t}}}} \right] $ and $ \left[ {{R_{\text{t}}},{R_{\text{w}}}} \right] $, the subscript t and w represent the outer scale of turbulence and GWs, respectively), where the linear relationship is satisfied, and the calculated $ K\left( q \right) $ follows the relationship $ K\left( 0 \right) = K\left( 1 \right) = 0 $. The turbulence scale reaches 512 m, and the scale range of the fitted GW is 512 m to 4 km, indicating that the intermittency calculated here is limited to small-scale GWs. The corresponding intermittencies $ {C_1} $ representing turbulence and GW events are 0.18 and 0.12, respectively. Figures 8c and d and Figs. 8e and f are the same as Figs. 8a and b but for YC2 and CS2. It can be seen from the comparison that the stable GW has a slightly smaller $ {C_1} $ (0.1) than the unstable GW (0.16). As for the broken GW, the intermittency is calculated separately on each scale, and $ {C_1} $ is very small for the GW scale but increases significantly for the turbulence scale.

      Figure 8.  (a) The variation of multi-order singularity measure with spatial scale for WH2 data. (b) The variation of slope with order $ q $ for WH2 data. (c)–(d) The same as WH2 but for YC2. (e)–(f) The same as WH2 but for CS2. The red dotted line corresponds to $ K\left( q \right) = 0 $, and the solid black line (and dashed line) represents the tangent slope of the turbulent (gravity wave) area at $ q $ = 1.

    4.   Integrated analysis of all data
    • Lu and Koch (2008) suggests that the presence of such an upscale energy cascade indicates the presence of Kelvin–Helmholtz instability. They provides supporting explanations that may also help to explain the results of this study. Since $ {S_3}\left( r \right) = \left\langle {\delta {u_L}\left[ {{{\left( {\delta {u_L}} \right)}^2} + {{\left( {\delta {u_T}} \right)}^2}} \right]} \right\rangle $, negative values represent $ \delta {u_L} $<0, indicating that the velocity field is converging (or decelerating), while positive values represent $ \delta {u_L} $>0, indicating that the velocity field is diverging (or accelerating) on the spatial scale of r. Therefore, we believe that the inconsistency of convergence and divergence at different scales leads to the instability of the GW (i.e., the bidirectional energy cascade). It should be noted that the curve of the third-order structure function will show obvious fluctuations at large separation distances, and the linear relationship is not obvious. Therefore, the description of the slope here is limited to the scale range of tens of meters to several kilometers. We analyzed the other nine sets of data in the same way, and the results are shown in Fig. 9.

      Figure 9.  The third-order structure function of the flat-floating stage from other data series. The purple dashed line and the green dashed line represent the result of linear fitting, and the slope value is marked beside each line. When there is an obvious change in slope, the dashed lines of different colors are used to distinguish it.

      The relationship between the third-order structure function and the separation distance from log-log coordinates is shown in Fig. 9. In the small-scale range (within 1 km), they have a more obvious linear relationship. In the larger-scale range, there are sawtooth fluctuations on some profiles (such as CS1, YC1, YC2, and GZ3), indicating that with the increase of separation distance, the existence of some inclination angles will cause dramatic changes in the local velocity difference, resulting in the apparent deviation of the linear relationship. A similar relationship can be seen in Fig 5, and a similar discussion is given in section 3.2. The root cause of this phenomenon is that the actual height of the balloon during the flat-floating stage is changing in real time, so it may cause the sounding system on a small segment of the flat-floating trajectory to swing slightly up and down. We should always keep this in mind when using the RTISS to analyze atmospheric disturbance characteristics.

      For the part with jagged fluctuations, linear regression is used to fit a straight line to determine the slope of the corresponding part. The slope value is marked beside the fitted dashed line, and although the actual fitting slope may not completely coincide with the reference slope values of 1, 2, and 3, the different states of GWs can also be distinguished from the specific value. When performing slope matching to classify the state of GWs, it is best to take the part with an obvious linear relationship as the main reference (small-scale range). For example, profile YC2 in Fig. 3 follows the r3 slope on a small scale (<2 km) and the r2 slope on a large scale (>2 km). We consider this example to be a stable GW, following the r3 slope. Although there is an energy cascade in the opposite direction at r = 16.4 km when compared to the surrounding points, it can only be speculated that there may be an unstable disturbance on a large-scale range, and the energy of the wave source will be gradually transferred to the small-scale GW through the wave–wave interaction. However, this study focuses on small-scale GWs within several kilometers, and the r2 slope on a larger scale is not considered in the classification criteria.

      Combined with the criteria mentioned above, all 12 sets of data can be divided into three stages: 1) stable GWs: WH1, CS1, AQ1, AQ2, YC1, and YC2; 2) unstable GWs: GZ1, GZ2, GZ3, and CS2; 3) GWs that coexist with turbulence: WH2 and WH3. Among them, GZ1 and GZ2 follow the r2 slope but do not show a bidirectional cascade. We believe that there are two possible reasons for this: 1) The energy cascade has spread to larger or smaller scales. Within the range of the selected r scale, the velocity field has tended to be consistent, while there may be a bidirectional energy cascade process outside the r scale range; 2) As r increases, the distance between it and an adjacent r increases exponentially, and there may be a bidirectional cascade on other scales that are not shown (since we found bidirectional energy transmission in the segmented part) , but this phenomenon was masked over the entire data sequence.

    • According to the methods used in the analyses of the example cases, the Hurst parameter and intermittency are calculated for all the remaining datasets. The results are shown in Fig. 10a. Combining the potential temperatures (not shown here) of the flat-floating stage from the 12 sets of data, we carefully observed the potential temperature disturbance under different parameter combinations $ \left( {{H_1},{C_1}} \right) $. It is found that within the same horizontal length range, the smaller the $ {H_1} $, the more wave packets (disturbance events) exist, and the larger the $ {C_1} $, the stronger the artifacts that are superimposed on the waves. This can also be reflected by the ratio of the maximum value of the disturbance to the average value (Davis et al., 1994; Marshak et al., 1997). It can be seen from Fig. 10a that $ {H_1} $ in unstable GWs and broken GWs is generally lower than in stable GWs. As for $ {C_1} $, it can be either higher or lower for GWs at different stages. In the range of small-scale GWs (within a few kilometers), smaller $ {H_1} $ values correspond to more wave packets (considered as a high-frequency wave), and vice versa.

      Figure 10.  (a) Hurst parameter (blue curve) and intermittency (red curve). (b) The spectral amplitude from the flat-floating stage (blue curve) and the vertical range of 18–24 km (red curve). (c) The intensity of atmospheric disturbance RT (blue curve) and the average height in the flat-floating stage (red curve) in 12 sets of data. The numbers in the figure correspond to the serial numbers in Table 1.

      In order to roughly understand whether the wave activity in the horizontal direction has an impact on the atmospheric environment below, we use the data from the ascending and descending stages of the RTISS to compare with the data from the flat-floating stage. In order to make the comparison reasonable, for all sets of experimental data, the horizontal wavenumber spectrum is obtained in the flat-floating stage, and all data series from the ascending and descending stages at the height range of 18–24 km are selected to calculate the vertical wavenumber spectrum. Calculated spectral slope and spectral amplitude are shown in Fig. 10b, where the spectral amplitude in the vertical direction is the average of the rising and falling segments. A significant phenomenon is that the variation trend of the spectral amplitude in the flat-floating stage is basically consistent with the atmosphere below. Since for the rising and falling data, the selected height range is the same, there should be a link between the vertical wavenumber spectrum of this height interval and horizontal wavenumber spectrum from a higher height. The result shows that the stronger the spectrum amplitude in the flat-floating stage, the stronger the spectrum amplitude is in the height range of 18–24 km. We believe that the cause of this phenomenon may be the consistency of energy transmission by GWs in the vertical direction within this height range.

      Considering that more disturbance events with greater disturbance intensity on the horizontal profile will lead to greater energy contained in the GW, it is reasonable to use the ratio of the two parameters to represent the intensity of the wave. Therefore, the parameter that quantitatively describes the intensity of atmospheric disturbance is defined as: ${{{\text{RT}} = {H_1}} /{\sqrt {{C_1}} }}$ (the root value of C1 is taken here to make the variation trend more obvious), which is shown in Fig. 10c, where the blue curve represents the ratio and the red curve represents the average height. The average height has a variation trend that is opposite to the spectral amplitude, indicating that in the lower-middle stratosphere, as the height increases, the GW activity in the horizontal direction decreases. $ {\text{RT}} $ follows the same trend (i.e., the smaller the value, the larger the spectral amplitude, containing a stronger energy of atmospheric disturbance). However, it should be noted that the relationship between the spectral amplitude and $ {\text{RT}} $ in individual cases is not matching. Considering that the spectral amplitude is obtained using the center wavenumber, which may make the calculated spectral amplitude not reflect the energy of the wave correctly if the dominant wavelength is not in the fitting interval, this result is acceptable. For example, WH2 and CS1 are at similar flat-floating heights. No obvious difference can be seen from the spectral amplitudes, but turbulence occurs on WH2, and atmospheric disturbance is more intense. Therefore, the disturbance intensity of WH2 is significantly larger than that of CS1. Similarly, AQ2 has a lower flat-floating height and a larger spectral amplitude than AQ1. However, the RT value of AQ1 is smaller than AQ2. Though in terms of the third-order structure function, the slope and smoothness of AQ1 and AQ2 are similar, there is an inconsistency of energy transmission at the tail of AQ1, and this may be the reason for the stronger disturbance in AQ1. In conclusion, it seems more reasonable to use RT to represent the atmospheric disturbances, since it has reliable physical meaning.

    • Indeed, since the sounding system is still in the experimental stage, there is no guarantee that each release process will go well or that it can obtain good results. In contrast, the overpressure balloons used in other countries are part of a more mature system. Therefore, we chose to conduct a few case studies first, before collecting extensive data, to provide preliminary results in support of future research. Before performing the analysis, data screening was carried out to exclude individual data segments with significant changes in height. and the fluctuation range of flat-floating height is basically within a few hundred meters (individual data series can reach more than 1 km). Since the structure function is a statistical feature and we only focus on small-scale GWs (within a few kilometers) and turbulence, the deviation of the structure function which may be caused by the obvious change of the floating height on a larger scale (the influence of floating height on the statistical characteristics of velocity increment gradually increases, as shown in Fig. 6) is not within the scope of our discussion. Therefore, the results are reasonable and reliable.

      Also, it should be noted that in some subsequent studies, the data from overpressure balloons were averaged, because their trajectories approximately coincide (in a stable zonal circulation background field). But our detection system is a short-term regional stratospheric detection. Stable zonal circulation, like that found in the polar regions, is rarely seen over China; the trajectories over China are very irregular and need to be processed. However, the twelve sets of case study data used in this study are released at different locations, heights, and trajectories. Our point of view is that no matter what kind of processing method is used, error is unavoidable, and according to the characteristics of the flat-float segment data, the decomposition of the latitude and longitude can enable full use of the data.

      We have mainly considered two reasons for choosing to use the potential temperature spectrum: first, considering that the potential temperature is used to describe the Kelvin-Helmholtz billows and is related to the evolution of GWs and turbulence (Wroblewski et al., 2007; Balsley et al., 2018; Fritts et al., 2018), compared to wind speed data, it may produce better results (and it does according to our results); second, we chose to use potential temperature instead of wind to minimize the influence of balloon bobbing and to avoid the derivatives from GPS locations. Due to the obvious rise and fall of height in individual flat-drifting sections, the influence of the inherent trend item of wind speed changing with height will affect the de-trend and fitting results and may cause inaccuracy of the wind speed spectrum.

    5.   Summary
    • In this study, the data collected from the experimental RTISS project were used to observe and analyze the characteristics of atmospheric disturbance in the lower-middle stratosphere. Using the analysis of energy cascade and the slope of the third-order structure function, the disturbance results obtained from the twelve sets of data are divided into three stages according to the evolution of GWs: stable GWs, where the $ {S_3}\left( r \right) $ follows the r3 slope; unstable GWs, where the $ {S_3}\left( r \right) $ follows the r2 slope; and GWs that coexist with turbulence, where $ {S_3}\left( r \right) $ follows the r slope in the turbulent subrange and the r2 slope in the GW subrange. For spectral analysis, the disturbance characteristic reflected by the spectral amplitude is closer to $ {H_1} $. Considering that the disturbance includes not only the roughness of the data sequence but also some small-scale disturbances that deviate from the average fluctuation, which can be reflected by intermittent parameters, it is more accurate and reasonable to use $ \left( {{H_1},{C_1}} \right) $ to represent the disturbance characteristics in the atmosphere. The segmented third-order structure function applied to the WH2 data can clearly reflect the evolution process of small-scale GWs, which is accompanied by the generation of turbulence.

      Using multi-order structure functions and singular measures, $ {H_1} $ and $ {C_1} $ are calculated and used together to quantitatively describe the characteristics of atmospheric disturbance, where $ {H_1} $ represents stability and smoothness and $ {C_1} $ represents intermittency and singularity. This is the first time that a set of the Hurst parameter and the intermittency describing atmospheric disturbance has been obtained in the lower-middle stratosphere. Broken GWs and unstable GWs generally correspond to lower $ {H_1} $ value than stable GWs, while the value of $ {C_1} $ is not closely related to the stage of the GW. In addition, the atmospheric disturbance intensity parameter $ {\text{RT}} $ is defined in this paper and contains both wave disturbance ($ {H_1} $) and random intermittency ($ {C_1} $). $ {\text{RT}} $ is considered to reflect the characteristics of atmospheric disturbance more reasonably. The smaller the value, the greater the intensity of atmospheric disturbance. Combining the data of the flat-floating stage with the data of the rising and falling stages, we found that the data with more unstable disturbance information from the flat-floating stage corresponds to significantly enhanced energy in the atmospheric environment below, which is presumed to reflect the consistency of energy transmission by GWs in the vertical direction within this height range. Of course, more work focusing on this can be carried out in the future.

      An interesting phenomenon is that when performing singular measurements, the scales of calculated intermittent parameters are all within a few kilometers, which is the same as small-scale GWs, indicating that the intermittency characteristics obtained here are used to describe these waves. Our results fill the gap of consistent and high-frequency observations of atmospheric disturbance at this altitude. Also, we attempt to classify the GW states from the perspective of structural function for the first time and can provide valuable observation references for related research. In the future, related research and more round-trip sounding system data can be used to conduct a more comprehensive analysis of stratospheric atmospheric disturbance characteristics.

      Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant No. 41875045), the National Key Research and Development Project (2018YFC1506200), and the Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210056). Thanks for the support provided by "Western Light" Cross-Team Project of Chinese Academy of Sciences, Key Laboratory Cooperative Research Project. Additionally, we would also like to thank the editors and the anonymous reviewers for their insightful and helpful comments. Data can be download in 4TU.ResearchData (http://doi.org/10.4121/uuid:868c025a-1b30-42a8-b722-e6cb66b0ae22).

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