1 Introduction

In the past decade, thunderstorms have been subjects of widespread concern in the field of wind engineering. These phenomena are composed of sets of cells, whose development goes through three main stages: cumulus, maturity and dissipation. When a thunderstorm occurs, the downdraft lashing against the ground will produce strong non-stationary radial outflows and ring vortices. This air movement is called a "downburst" (Fujita 1981), which can be divided into "macroburst" and "microburst" according to its horizontal size. The composition of the outflow field and the vortex flow field of thunderstorms could cause extremely high wind speeds that often produce huge loads to impinge on the surface buildings, posing a major threat to structural safety. Relevant research conducted in the past decades showed that downbursts not only brought damage to the built environment but also have surprisingly destructive effects on aircraft and other flying objects (Solari et al. 2015; Elawady et al. 2017).

As awareness has increased of the potential disasters of downbursts, researchers have carried out extensive investigations on thunderstorm wind characteristics through field measurements (Chaudhuri et al. 2020; Jiang et al. 2022), wind tunnel tests (Asano et al. 2019; Fang et al. 2023) and numerical simulations (Selvam and Holmes 1992; Iida et al. 2015). Specifically, the collaborative team of Hangan and Romanic have conducted extensive research on the climatic environment of downbursts and their interaction with boundary layer winds by comparing downburst models with full-size downbursts (Romanic et al. 2020, 2022; Canepa et al. 2022). For example, Romanic et al. (2020) proposed a new downburst scaling method to discuss the similarity of the transient characteristics between the downburst model simulated in the WindEEE Dome simulator at Western University and a full-scale downburst event in the Mediterranean Sea, based on the criteria of the scaling method. Two full-scale downburst events recorded in Genoa and Livorno, Italy, were partially reconstructed using this scaling method, and the heights of the corresponding measured wind speed points were accurately predicted. However, the method ignores the effect of atmospheric boundary layer winds and can only be scaled based on a subset of the measurement points rather than the entire wind field. Canepa et al. (2022) have studied the three-dimensional interaction between downbursts generated by large-scale impinging jets and atmospheric boundary layer winds. The ABL winds were simulated by a series of boundary layer plates to create a uniform flow in the experimental section, and the near-surface flow dynamics were studied using particle image velocimetry (PIV) measurements. The results show that the ABL-like winds affect the flow dynamics of the impinging jet, leading to changes in velocity distribution, turbulence intensity and recirculation zone length. Solari’s team has fully explored the wind profile and turbulence characteristics of thunderstorms through long-term observations of a large number of thunderstorm events (Solari et al. 2015; Zhang et al. 2019; Canepa et al. 2020). For example, Canepa et al. (2020) analyzed the vertical profile characteristics of thunderstorm downbursts by measuring the vertical wind profiles during 10 thunderstorm events in 2015–2018 using a LiDAR wind profiler in the northern Mediterranean. The results show that the vertical profiles of wind speed, wind direction and turbulence intensity of thunderstorm downbursts are significantly different from those of non-thunderstorm winds. It is found to be more turbulent and to have a stronger horizontal wind component near the ground. Sterling’s team has carried out a good amount of meaningful work on thunderstorm models (Jesson and Sterling 2018; Jesson et al. 2019). For instance, Jesson and Sterling (2018) developed a simple vortex model to simulate thunderstorm downbursts and evaluated the model parameters using available experimental data. The model was found to be able to reproduce the main features of the flow field in physically simulated downburst flows with a reasonable degree of accuracy. The model is able to accurately capture key features of downburst flow dynamics such as vortex ring formation and subsequent outflow diffusion. In addition, Choi (2004) conducted field measurements and experimental research on the wind speed profile during thunderstorms. These researchers collected data from two different thunderstorms using cup anemometers and sonic anemometers for measurements at different levels. The results showed that the wind speed profile during a thunderstorm was distinct from the typical logarithmic profile associated with neutral atmospheric conditions. Shu et al. (2015, 2017) compared the vertical wind profile and gust factor distribution characteristics of typhoons, monsoons and thunderstorms based on the data collected from a near-surface anemometer and a Doppler radar profiler. The observed vertical mean wind profile of the typhoons and monsoons showed logarithmic growth below 500 m, while that of thunderstorms represented obvious jet characteristics. In contrast, the gust factors of tropical cyclones, thunderstorms and monsoons ranged from high to low. Devi et al. (2020) presented the wind profile characteristics obtained from the stratospheric–tropospheric radar at Gauhati City as well as the data validation procedure utilizing wind parameters collected from radiosondes. The structural constant C2n (a parameter controlled by fundamental meteorological variabilities such as temperature, pressure and specific humidity) was determined using radar Bragg-scattered signals to assess the dynamics of the atmospheric system, and the resulting changes caused by the thunderstorm on this parameter were then shown. In order to potentially use these effects as input to a thunderstorm prediction model, the storm-induced effects were detected in the wind field-derived variabilities. Furthermore, based on wind field analysis of monsoon or extreme weather events, researchers have also actively and extensively explored other related fields, such as wind energy utilization and meteorological environment, etc. Using a mesoscale weather research and forecasting model (ARW dynamics solver, version 4.3.1) in conjunction with a 3D Price-Weller-Pinkel ocean model and a wind farm parameterization model (driving the mesoscale inter-farm interaction), Wang et al. (2022, 2023) evaluated the wind farm wake and power output under diverse wind and terrain. The findings indicate a strong correlation between wind and topography and the farm wake. The projected development mode of positive turbine power with cautious farm power exhibits remarkably minimal wake interference across farm clusters and is suggested as a priority path for the offshore wind sector towards carbon–neutral and cleaner energy. Besides, based on data available between 1960 and 2009, Shi and Cui (2012) explored the features of high-impact weather events, including the frequency and extreme value of rainstorms, typhoons, thunderstorms, strong winds, tornadoes, fog, haze, and hot days in Shanghai, China. The frequency, geographic distribution, and effects of meteorological catastrophes on people and property between 1984 and 2009 were also covered. Shanghai has become more susceptible to high-impact weather and meteorological disasters due to global climate change, urbanization, and rapid economic development. In particular, precipitation extremes, summer high temperatures, haze, and typhoons made more strategies for natural disaster mitigation and/or adaptation quite useful and necessary for local government and the general public in the future.

From a wind engineering perspective, relevant studies have used two main types of methods to identify thunderstorms. The first category relies on records of thunderstorms or related meteorological information such as thunder, lightning, rainfall and sudden temperature drops. This method is very straightforward, but the information on thunder and lightning is often quite random. The second category identifies thunderstorm winds by mean and maximum wind speeds and derived information such as gust factors, which requires relatively little raw thunderstorm information. Based on numerical simulations, wind tunnel experiments, and field measurements, existing studies have provided a large number of valuable investigations on the occurrence mechanism, meteorological characteristics, and wind field characteristics of thunderstorms. There have been many attempts to analyze thunderstorms and obtain their major features to evaluate related effects. However, the characteristics such as small temporal and spatial scales and randomness of occurrence make it difficult to monitor downbursts. Due to the complexity of atmospheric phenomena and the limited dataset of measurements, many uncertainties remain to be clarified in the understanding and modeling of the downburst wind field. In addition, severe thunderstorms may become more frequent as a result of Earth activity and Sun–Earth interactions, such as climate change (Brooks 2013; Liu et al. 2016) and changes in sunspot numbers (Schlegel et al. 2001). Analysis of the characteristics of the downburst wind field is conducive to predicting thunderstorms and improving the wind-resistant design of civil structures, which is urgently needed in areas affected by extreme weather events such as thunderstorms.

Among various research methods for downburst wind, field measurement is considered the most direct and reliable way to investigate thunderstorm characteristics. Furthermore, the characteristic analysis of the downburst wind field is the premise for studying thunderstorm-induced load distributions and related wind-resistant design methods. Therefore, this study uses data recorded during thunderstorms by a radar wind profiler at the Hong Kong Observatory and a near-ground anemometer installed at the Hong Kong International Airport (HKIA) to conduct a detailed feature analysis of the mean wind speed vertical profile and turbulence characteristics within the atmospheric boundary layer.

2 Observation station and dataset

In this study, the thunderstorm winds in Hong Kong are jointly observed using two pieces of equipment, namely, the Doppler radar wind profiler and the near-ground anemometer, as shown in Fig. 1. First, several meteorological indicators during thunderstorms are investigated based on field observations and daily logs. Then, multiple thunderstorm events in Hong Kong are selected according to the relevant indicators. Subsequently, the mean wind profile characteristics within the altitude range of kilometers are analyzed using the detection data from the Doppler profiler and the turbulent wind characteristics are investigated using the observations from the near-ground anemometer.

Fig. 1
figure 1

Location of observation sites and surrounding terrain features

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2.1 Doppler radar wind profiler

The Doppler radar observation station is built on Cheung Chau Island (CCH), which is located southwest of Hong Kong with an altitude of 72 m. To the south of CCH is the South China Sea, dotted with some small islands in the open sea area. Its northern semi-plane is a typical mountain landform, with several mountain peaks of more than 800 m, which has a significant shielding effect on the northerly wind. The profiler model is installed at an altitude of 99 m (27 m above the ground). During observation, the profiler system emits three beams of light every 30 s, and each beam can be measured at 50 vertical levels distributed in the range of 213–5371 m. The Doppler system automatically calculates the mean horizontal wind speed and direction over a 10-min period. Using a sliding window technique, the 10-min mean wind records are updated every 1 to 2 min. In practice, this boundary-layer wind profiler can operate at up to 1299 MHz and is capable of collecting gusts in seconds and mean winds in minutes.

2.2 Near-ground anemometer

The near-ground wind measurements are collected from an anemometer located at Hong Kong International Airport (HKIA). Previous studies have shown that low-level wind shear may pose a threat to departing/arriving aircraft (Liu et al. 2010). In order to further understand the characteristics of wind shear around the airport, field observations of such wind phenomena are required. As shown in Fig. 1, the HKIA is located at the Pearl River Estuary of China. The airport is surrounded to the north and west by seawater, while the south is a complex mountain landscape represented by Lantau Island. The HKIA has been equipped with various meteorological instruments at different locations. The anemometer adopted herein (R1W) is installed on a mast (10 m above the ground) at the end of an airport runway. It is cup-shaped (Thies 4.3303 model), with a sampling frequency of 1 Hz, and the detection accuracies of the wind speed and direction measurements are 0.3 m/s and 1° respectively.

3 Occurrence of thunderstorms

In previous studies related to thunderstorms, researchers have observed that the occurrence of downbursts is often characterized by changes in some meteorological factors, like wind speed, wind direction and temperature and humidity. Based on the long-term synchronous observation data of the Doppler radar wind profiler on CCH and the near-surface anemometer at HKIA at the beginning of this century (2008–2010), combined with the weather forecast and daily logs of events, this study selected meteorological records of four thunderstorm days in the summer and autumn of 2008 (i.e., June 2, July 8, August 6 and September 2) to analyze the characteristics of relevant climatic factors in detail. Representative radar reflectivity images reflecting the evolution of thunderstorms during the 4 days are shown in Fig. 2, which shows that under thunderstorm conditions, the observation sites are surrounded by distinct clusters of precipitation clouds; these clusters may be banded, spiral or irregularly shaped. Adjacent areas of the site are sometimes accompanied by local lightning, indicating that thunderstorms are occurring or are about to occur at the observation site.

Fig. 2
figure 2

Representative radar reflectivity maps showing the evolution of thunderstorms during 4 days of observation. (Red star indicates the observation site)

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Figure 3 shows the time histories of instantaneous wind speed and mean wind direction during the four selected days. Considering the downburst wind features a violent change and short duration, the instantaneous wind speed adopted herein refers to the maximum 1-s wind speed within a 10-min duration. At 04:20, 15:10, 13:30 and 13:00 respectively, there was a sharp increase in wind speed during these 4 days (Fig. 3a). This observation is consistent with that reported by Choi (2004). Meanwhile, note that the mean wind direction at the corresponding times also exhibits local mutations, although the variation of T3 is relatively less obvious (Fig. 3b). This may be because the wind direction changes too quickly during downbursts. On the contrary, no significant variation in the mean wind direction is detected in the mechanical anemometer observations.

Fig. 3
figure 3

Variation in meteorological elements of four observed thunderstorms

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In addition to horizontal wind speed and direction, this section further analyzes the change trend of vertical wind speed and signal-to-noise ratio (SNR) during thunderstorms. The vertical wind speed recorded by the radar profiler can provide information about the intensity of the downward airflow. When the horizontal wind speed reaches the peak, the enhancement of the downward vertical wind speed indicates that the storm is passing through the observation point. As shown in Fig. 4a and b, with the downburst occurrence, there is a significant downward vertical wind speed (shaded area) at both the near-ground level (200 m) and at a high altitude (1200 m). Simultaneously, the SNR also undergoes a sudden change during the downburst (Fig. 4c and d). Along with the rainfall, the SNR increases gradually (shown in the shaded area), which indicates the enhancement of the remote sensing signal of the profiler through raindrop reflection.

Fig. 4
figure 4

Variation of vertical wind speed and SNR

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As illustrated in Figs. 3 and 4, during thunderstorms, multiple meteorological factors, including but not limited to wind speed and direction, vertical wind speed and SNR, will change to some extent. Therefore, when determining the occurrence of a thunderstorm, these factors can be combined for a comprehensive judgment. By monitoring the development trend of multiple factors, the likelihood of thunderstorm occurrence could be predicted.

Thunderstorms usually occur in the summertime, and their duration varies from several minutes to hours (Grinn-Gofroń and Strzelczak 2013). To further explore the evolution characteristics of thunderstorms, based on the thunderstorm data synchronously recorded by the Doppler profiler of the Hong Kong Observatory and the anemometer at the HKIA, this study will compare the characteristics of mean winds from a spatial dimension and the fluctuating wind characteristics from a temporal dimension. Moreover, to analyze the thunderstorm over a continuous period of time and to avoid the interference associated with non-thunderstorm winds, this study chose four thunderstorm durations of 2 h (encompassing the major phase of a thunderstorm) from each of the four aforementioned thunderstorm days to explore the characteristics of downbursts. The specific segments of thunderstorm winds are listed in Table 1.

Table 1 Selected segments of four thunderstorms
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4 Mean wind characteristics based on the Doppler radar dataset

Based on the wind dataset recorded by the wind profiler at the Hong Kong Observatory, the mean wind speed distribution along the measured height in the selected 4 days during which thunderstorms occurred is shown in Fig. 5. To better distinguish between thunderstorm and non-thunderstorm periods, dashed boxes are added to the figure to represent the concentration periods of thunderstorms, except for T2 where most data are missing at higher altitudes (Fig. 5b). The blank spots in the figure indicate suspicious or false data. Note that the wind speeds during thunderstorms are significantly higher than those during the period without a thunderstorm. In addition, as illustrated in the figure, during a thunderstorm, the wind speed gradually increases with increasing altitude, but after reaching a certain height (say, 2 km), the wind speed would decrease, evidently observed in T1 and T4. The maximum mean wind speed during a thunderstorm can be as large as nearly 40 m/s (T3). In contrast, during T4, the maximum mean wind speed is relatively small, approximately 10 m/s.

Fig. 5
figure 5

Filled color contours of mean wind speeds

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Correspondingly, the mean wind direction distribution along altitude in the four thunderstorm days is shown in Fig. 6. During periods without a thunderstorm, the wind direction changes significantly along the height. However, in comparison, the change of wind direction during thunderstorms along the vertical direction is not obvious (dashed boxes in Fig. 6a–c). This variation in the effect of wind veer may be related to the unique structure of thunderstorms.

Fig. 6
figure 6

Filled color contours of mean wind directions

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Figures 5 and 6 compare the distribution characteristics of wind speed and direction with altitude during a thunderstorm and non-thunderstorm periods. To explore the development trend of the wind gradient during a thunderstorm, Fig. 7 illustrates the mean wind speed profiles of the four selected thunderstorms on the basis of 10-min intervals. Note that due to data loss at high altitudes, only the profiles with relatively complete structures within the entire measured height range are displayed. Overall, during the four thunderstorms, the extreme wind speeds occurred in the lower atmosphere, with a local peak at a relatively low height in the wind profile. Despite the fact that the four thunderstorms occurred on separate days, the heights of the extreme winds are basically the same within 1–1.8 km (shaded area in the figure), this observation in this study is somewhat higher than those reported in some previous literature in which the heights were around 400 m (Shu et al. 2017). While the wind speed rises with altitude below the designated heights, the evolution of the wind profiles during the various thunderstorms exhibits some variances above the heights. Specifically, within the height range above 2 km, the wind profiles of T1 (Fig. 7a) show a gradual decreasing trend while those of T2 and T3 fluctuate continuously within a certain range (Fig. 7b and c), and the profiles of T4 first decrease and then increase gradually (Fig. 7d), with maxima even exceeding the corresponding local peak value at the lower atmosphere.

Fig. 7.
figure 7

10-min mean wind speed profiles

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As shown in Fig. 7, the evolution trends of wind profiles during these thunderstorms are different. This distinction might be related to the interference from non-thunderstorm winds. Previous studies have shown that a thunderstorm cycle may only last thirty seconds to a few minutes (Canepa et al. 2020). Therefore, to focus on the structure of the thunderstorm itself, this section specifically focuses on the central moment of thunderstorm occurrence, and takes a 2-min interval to characterize the core area of a thunderstorm. Subsequently, the corresponding wind profiles are presented in Fig. 8. Compared with the 10-min mean wind profiles (shown in Fig. 7), the profiles in the core area of thunderstorms are more concentrated and their shapes are more similar. However, there is no obvious difference in the evolution characteristics between these two kinds of wind profiles. From another perspective, the similarity of these two types of profiles highlights the variability of thunderstorm wind profiles. In most thunderstorm wind profiles, the height of local peaks is relatively steady. However, once this height is exceeded, the development pattern of the profiles is not constant. It might gradually decline, vary consistently, or climb once again.

Fig. 8
figure 8

Mean wind speed profiles within the cores of thunderstorms

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5 Wind gust characteristics based on near-ground measurements

5.1 Time history of fluctuating wind speeds

To explore the characteristics of fluctuating wind fields during a thunderstorm, the wind measurement time series are first decomposed into mean and fluctuating (longitudinal-u and lateral-v) wind time series based on wind speed and direction information. For the decomposition of wind speeds, the conventional method considers winds as a stationary process to obtain a constant mean component (Luo et al. 2020), and another method considers winds to be a nonstationary process to derive a slowly varying mean component (Canepa et al. 2020). This study will use both methods to decompose the measured wind speed and then compare the turbulence characteristics of downbursts. These two methods are expressed in Eqs. (1-1 to 1-4) and (2-1 to 2-5) respectively, and the sketch of the decomposition process is shown in Fig. 9.

$$U = \overline{{\left[ {\mathop u\nolimits_{x}^{2} (t) + \mathop u\nolimits_{y}^{2} (t)} \right]^{1/2} }}$$
(1-1)
$$\psi = \arcsin \frac{{\overline{{\mathop u\nolimits_{x} \left( t \right)}} }}{U} + step( - \overline{{\mathop u\nolimits_{y} \left( t \right)}} ) \times 180^{{\text{o}}}$$
(1-2)
$$u(t) = u_{x} (t)\cos \psi + u_{y} (t)\sin \psi - U$$
(1-3)
$$v(t) = - u_{x} (t)\sin \psi + u_{y} (t)\cos \psi$$
(1-4)

where ux(t) and uy(t) are the measured wind signals along east–west and north–south directions, respectively; U and ψ are the horizontal mean wind speed and direction respectively; the overbar represents the mean value and “step” indicates the unit step function; u(t) and v(t) are the derived fluctuating components.

$$u_{x} (t) = \overline{V}_{x} + V_{x}^{\prime }$$
(2-1)
$$u_{y} (t) = \overline{V}_{y} + V_{y}^{\prime }$$
(2-2)
$$\overline{u}(t) = \sqrt {\overline{V}^{2} x(t) + \overline{V}^{2} y(t)}$$
(2-3)
$$u^{\prime } (t) = - V_{x}^{\prime } (t)\sin \overline{\beta }(t) - V_{y}^{\prime } (t)\cos \overline{\beta }(t)$$
(2-4)
$$v^{\prime } (t) = V_{x}^{\prime } (t)\cos \overline{\beta }(t) - V_{y}^{\prime } (t)\sin \overline{\beta }(t)$$
(2-5)

where \(\overline{V}_{x}\) and \(\overline{V}_{y}\) are the slowly-varying mean wind speeds in the east–west and north–south directions respectively; \(V_{x}^{\prime }\) and \(V_{y}^{\prime }\) are the corresponding residual fluctuations; \(\overline{u}(t)\) is the horizontal mean wind velocity vector; \(\overline{\beta }(t)\) is the measured wind direction; \(v^{\prime } (t)\) and \(u^{\prime } (t)\) are the longitudinal and lateral turbulent fluctuations respectively.

Fig. 9
figure 9

Illustration of the decomposition process of winds

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The decomposed fluctuating wind time histories are shown in Figs. 10 and 11, corresponding to the constant and slowly-varying mean components respectively. The overall development trend of the two types of fluctuating components is basically the same, but the turbulent fluctuation that considers nonstationarity is more compact and changes more rapidly than does the stationary fluctuating process. Note that the fluctuating wind speeds of thunderstorms T1, T2 and T4 are relatively close with a maximum value of approximately 6 m/s, and the fluctuating wind speed of T3 is slightly larger, close to 14 m/s. This relative relation of values is consistent with the original wind measurements shown in Fig. 3. In addition, the magnitudes of the longitudinal and lateral fluctuating wind speeds during the thunderstorm are similar, and the magnitude relation between the two fluctuating directions is also observed by He et al. (2020) during a typhoon.

Fig. 10
figure 10

Fluctuating components based on constant mean wind speeds

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Fig. 11
figure 11

Fluctuating components based on slowly-varying mean wind speeds

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5.2 Data quality control

The fluctuating characteristics of wind speed are closely related to the atmospheric thermal stability condition. Therefore, before analyzing the characteristics of the fluctuating wind field, it is generally necessary to judge the thermally neutral stability of air flows. The Obukhov length (Businger et al. 1971) and Richardson number (Golder 1972) are two commonly used judgment methods. In the wind engineering field, based on the statistical analysis of a large number of field observations, researchers have found that the atmosphere usually contains significant non-stationary components when the wind speed is low. However, when the wind speed increases to a certain value, the neutral condition is considered a reasonable assumption. The critical wind speed suggested by different researchers is different, for example, the value of 10 m/s is recommended in the Engineering Science Data Unit (ESDU) (1985) and Masters et al. (2010) adopted a smaller value of 5 m/s as the threshold value. Generally, parts of the wind speeds that are under these critical values will be removed in analyses of fluctuating wind characteristics. Considering that the recorded wind speeds of thunderstorms in this study are not very large, and the duration of the thunderstorm itself is also relatively short, the fluctuation features corresponding to the lower wind speeds are retained in the following discussions, and the influence of atmospheric thermal instability on the fluctuating characteristics of the thunderstorms is considered.

5.3 Gust factor and turbulence intensity

The gust factors of the thunderstorms are shown in Fig. 12 (u1 and v1 denote the stationary fluctuation state and u2 and v2 represent the nonstationary state, hereafter). For the nonstationary fluctuating process, the distributions of gust factors in the longitudinal and lateral directions are concentrated, and there is clear stratification in the two fluctuating components, while those of stationary fluctuations (especially for T1 and T4, Fig. 12a and d) are relatively scattered and interweave between the two directions. As mentioned in the previous section, when the wind speed is slow, it is often related to the thermal instability of the atmosphere. Therefore, the gust factors of the stationary fluctuating process in T1 and T4 are volatile. For instance, when the wind speed is less than 5 m/s (T4), the gust factor has a large dispersion range, while when the wind speed exceeds 5 m/s, the gust factor tends to be stable. For both the fluctuating processes, the longitudinal gust factors of T2 and T3 are within 1.3–1.7 and 1.4–1.9 respectively, and the lateral ones are 0.3–0.5 and 0.4–0.8 respectively. By contrast, the gust factors in the two directions of T1 and T4 are scattered between 1–2 and 0–1 respectively. In addition, judging from the variation in gust factors with wind speed, there is no specific correlation between them.

Fig. 12
figure 12

Variation in gust factors for the four thunderstorms

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The turbulence intensity during the four thunderstorms is shown in Fig. 13 (denoted as Ir, the subscript r refers to u or v). Similar to the distribution of gust factors, the difference in turbulence intensity between these two fluctuating processes is also evident. Compared with the gust factor, the values of turbulence intensity in two directions (longitudinal and lateral) are closer, especially for the nonstationary fluctuating process, the values are almost the same. Specifically, for the nonstationary fluctuation, the variation range of turbulence intensity values is within 0.1–0.3, while for the stationary process, the range of T1 and T4 is widely distributed between 0.1–0.6, and an individual value corresponding to the low wind speed even approaches 1 (Fig. 13a). In addition, although the distribution ranges of the turbulence intensity for the two fluctuating processes are different, with increasing wind speed, the turbulence intensities seem to show a gradual decreasing trend. This differs from the development trend of the gust factor with wind speed.

Fig. 13
figure 13

Variations in turbulence intensity for the four thunderstorms

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In the discussion on the characteristics of the typhoon wind fields, researchers have found that there was a degree of correlation between gust factor and turbulence intensity, and have proposed some mathematical models, one of which introduced the logarithmic ratio of the mean and gust duration and has been widely used (Ishizaki 1983; Cao et al. 2009; He et al. 2022):

$$GF_{i} (\tau ,T) = 1 + k_{1} \times I^{{k_{2} }} \times \ln (T/\tau ), \, i = u$$
(3-1)
$$GF_{i} (\tau ,T) = k_{1} \times I^{{k_{2} }} \times \ln (T/\tau ), \, i = \nu , \, \omega$$
(3-2)

where k1 and k2 are fitting coefficients.

Generally, when the value of k2 is set as 1, the relationship between the gust factor and turbulence intensity is converted into a linear form. Given this, the correlation between these two shown in Fig. 14 is linearly fitted. The gust factor and turbulence intensity of the two fluctuating processes basically conform to a linear relationship when the two scattered points in T1 are ignored (in the Fig. 14a), although the data points of T1 and T4 are relatively scattered. This also means that for the fluctuating characteristics of a downburst, given that one of the gust factor and turbulence intensity is known, the distribution of the other can be estimated by using a linear relationship.

Fig. 14
figure 14

Correlations between the gust factor and turbulence intensity in longitudinal and lateral directions

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5.4 Turbulence integral length scale

The turbulence integral length scale is a spatial dimension used to define the turbulence structure of a wind flow, representing its average eddy size. There are various methods to calculate the turbulence integral length scale (L), among which an integration algorithm based on the autocorrelation function is commonly used (Flay and Stevenson 1988), the calculation equation is as follows:

$$L_{i} = \frac{U}{{\sigma_{i} }}\int_{0}^{\infty } {R(\tau )} d\tau ,{ (}i{ = }\mu {, }\upsilon {, }\omega {)}$$
(4)

where U is the mean wind speed and σ is the corresponding standard deviation; R(τ) indicates the autocorrelation function of fluctuating winds. Herein, the τ is taken as the upper limit of the integral, with the R(τ) gradually decreasing until it reaches 0.05.

The turbulence integral length scales of the four thunderstorms are shown in Fig. 15 (denoted as Lr, the subscript r refers to u or v). In general, the turbulence integral scales of the two types of fluctuating winds are significantly different. Although the gust factor and turbulence intensity show a certain deviation due to the influence of atmospheric stratification instability associated with the low wind speed (as shown in Figs. 13 and 14), this effect does not appear to be reflected in the turbulence integral scale, because the distribution range of the scales of the four thunderstorms is almost the same. The atmospheric thermal instability, as well as the unsteady nature of the thunderstorms themselves, have a significant effect on the turbulence characteristics of the thunderstorm winds. Also, different turbulence parameters also cause their trends to differ. The gust factor and turbulence intensity are two dimensionless parameters, so differences between storms can be observed after normalization (as shown in Figs. 12 and 13), but the turbulence integration scale is an absolute parameter, and its large value and inherent scatter might mask the relatively small differences between storms. Additionally, the integral scale of the stationary turbulent fluctuation is scattered between 0 and 400 m, which is similar to the observation results of He et al. (2020) during a typhoon, while the integral scale of the non-stationary fluctuation is concentrated in the range of 0 to 100 m, which is in good agreement with the derived results of Zhang et al. (2019) for thunderstorms.

Fig. 15
figure 15

Distributions of longitudinal and lateral turbulence integral length scales

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5.5 Energy distribution

Spectral analysis is widely used to determine the energy distribution of gust winds in the frequency domain. In recent decades, various spectrum models have been proposed to describe the distribution of gust characteristics. Among them, the von Karman spectrum (Von Karman 1948) is generally regarded as the best analytic model of isotropic turbulence (ESDU 1985), and its equation is defined as follows:

$$\frac{{f^{\prime}S_{uu} }}{{\sigma_{u}^{2} }} = \frac{{4n_{u} }}{{\left( {1 + 70.8n_{u}^{2} } \right)^{{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0pt} 6}}} }}, \, n_{u} = L_{u}^{x} f^{\prime}/V_{z}$$
(5-1)
$$f^{\prime}\frac{{S_{ii} }}{{\sigma_{i}^{2} }} = \frac{{4n_{i} \left( {1 + 755.2n_{i}^{2} } \right)}}{{\left( {1 + 283.2n_{i}^{2} } \right)^{{{{11} \mathord{\left/ {\vphantom {{11} 6}} \right. \kern-0pt} 6}}} }}, \, n_{i} = L_{i}^{x} f^{\prime}/V_{z} , \, i = v \, or \, \omega$$
(5-2)

where \(f^{\prime}\), σ and S denote frequency, power spectral density and standard deviation, respectively; Vz stands for the mean wind speed at a height of z and L is the calculated turbulence integral length scale.

The power spectrum analysis results (power spectral density, PSD) of the two fluctuating processes are shown in Figs. 16 and 17 respectively. In general, the distribution characteristics of the four thunderstorms in the frequency domain feature relatively high energy in the low-frequency part while low energy in the high-frequency part. For the stationary fluctuation winds (Fig. 16a), the longitudinal power spectra of T2, T3 and T4 show a trend of increasing first and then decreasing, while that of T1 is almost a continuing decline. The lateral spectra of T2 and T3 also increase at first and then decrease, while those of T1 and T4 continue to decline (shown in Fig. 16b). In contrast, the power spectra of non-stationary fluctuating winds (Fig. 17) present a uniform trend that is upward first and downward later. In addition, to explore the similarity between the power spectrum of a thunderstorm and the classical spectrum model, the spectral results of T3 are selected and illustrated in the Figs. 16c, d, 17c and d for comparison with the von Karman spectrum. To facilitate comparison, the power spectral density of T3 is normalized using the turbulence integral scale obtained in the previous Sect. 5.4. As illustrated in these figures, the spectral curves derived from the three spectral analysis methods (Welch, Yule-Walker and MTM) agree well with each other. Additionally, for the stationary fluctuating process, the energy distribution in the longitudinal direction is in good agreement with that of the von Karman spectrum (Fig. 16c), while there is a certain difference in the lateral direction (Fig. 16d). This is because the crosswinds are often affected by atmospheric instability and topography, resulting in a different shape of lateral wind speed spectrum to the von Karmen spectrum. The energy in the high-frequency part of the lateral component of T3 is slightly higher than that of the von Karman spectrum. For the non-stationary fluctuating process (in Fig. 17c and d), the power spectra do not coincide with the von Karman spectrum, which indicates that the von Karman spectrum is more suitable for fitting the characteristics of stationary fluctuating winds.

Fig. 16
figure 16

Normalized PSDs of stationary fluctuating components

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Fig. 17
figure 17

Normalized PSDs of nonstationary fluctuating components

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6 Concluding remarks

This study involves a comprehensive investigation of the wind field characteristics during thunderstorms. The mean wind profile characteristics within an altitude range of several kilometers are analyzed using the detection data from a Doppler profiler, and the turbulent wind characteristics are investigated using the observations from a near-ground anemometer. The main conclusions of this study are summarized as follows:

  1. (1)

    According to the long-term field measurements during thunderstorm events, multiple meteorological factors, including horizontal wind speed and direction, temperature, humidity and air pressure that have been frequently mentioned in previous studies, as well as vertical wind speed and SNR observed in this study, would change to some extent. These factors should be combined for a comprehensive judgment on the occurrence of thunderstorms.

  2. (2)

    Despite the fact that the thunderstorms occurred on separate days, the heights of the extreme winds are basically the same within 1–1.8 km, this observation is somewhat higher than that reported in some previous literature. This difference needs to be verified and compared with more field data of thunderstorms in both the lower and higher layers. While the wind speed rises with altitude below the designated heights, the evolution of the wind profiles during the various thunderstorms exhibits some variances above the heights.

  3. (3)

    The influence of atmospheric stratification instability brought on by the low wind speed causes some variations in the gust factor and turbulence intensity, but this effect does not appear to be reflected in the turbulence integral scale, as the distribution range of the scales during multiple downbursts is nearly identical. Furthermore, the integral scale of the stationary turbulent fluctuation is scattered, similar to observations during a typhoon, while that of the non-stationary fluctuation is concentrated, in good agreement with derived results for thunderstorms.

  4. (4)

    For the stationary fluctuating winds, there may be differences in the evolution of the power spectra between the longitudinal and lateral directions, due to perturbations caused by atmospheric instability and topography. On the other hand, the spectra of the non-stationary fluctuating winds show a uniform trend, initially upward and later downward. Additionally, the von Karman spectrum is more suitable for fitting the characteristics of the stationary fluctuating component.