2. Field Observations in the ASL

Meteorological and geographic observations in the ASL focus mainly on average statistics, and researchers in meteorology, oceanography and geography, which includes the physics of wind-blown sand, have observed the ASL flow on different underlying surfaces for many years. The earliest measurement of the ASL flow was performed in 1926 by British meteorologist F. J. Scrase (Reference ScraseScrase, 1930). Aswinging plate anemometer with a frequency of approximately 1.6 Hz was used to measure the streamwise and wall-normal wind velocities at a single point. Based on the observational data, the difference in the average Reynolds stresses at different heights was analysed; the results indicated that near the wall (1.5 m), the Reynolds stress is less than one-quarter of that farther from the wall (19 m). This observation is considered classic in the history of attempting to understand how momentum is transferred in the lower atmosphere (Reference BaradBarad, 1964). However, the ASL wind velocity measurement in 1926 was performed only at a single point and at a low sampling rate. Unfortunately, synchronous measurements at different locations have yet to be achieved. More recently, synchronous multipoint measurements of the wind velocity, temperature and humidity in the wall-normal direction were conducted in Kansas and Minnesota in the USA (Reference Kaimal and WyngaardKaimal & Wyngaard, 1989). The resulting wall-normal profiles of the velocity and temperature in the unstable stratified ASL can be used to verify and improve the Monin–Obukhov similarity theory. In general, the thermal stratifications of the ASL can be characterized by the Richardson number and the Monin–Obukhov stability parameter. However, the velocity measurement employed a wind cup at a sampling frequency of only 1 Hz, which makes it difficult to meet the high-resolution requirements for the analysis of turbulence signals. To increase the sampling rate of velocity measurements, Reference Zeng, Cheng, Hu and PengZeng, Cheng, Hu, and Peng (Reference Zeng, Cheng, Hu and Peng2010) employed sonic anemometers with a sampling frequency of 10 Hz to synchronously observe the wind velocity and the temperature in the wall-normal direction (at heights of 47, 120 and 280 m) within the urban boundary layer; a photo of the equipment is shown in Figure 1. The observations revealed the local meteorology and its impact on climate, and the authors proposed turbulence parametrization schemes for global weather forecasting models. Nevertheless, although the sampling frequency was increased, the underlying surface of the urban boundary layer and the flow environment were more complicated. Hence, their results are difficult to compare with the properties of the canonical flat TBL and thus cannot be used in the study of high-Reynolds-number wall turbulence.

Figure 1. Observation tower employed by (Reference Zeng, Cheng, Hu and PengZeng et al., 2010) to measure the vertical gradients in the urban boundary layer. This figure was taken from http://www.iap.ac.cn/xwzx/tpxw/201510/t20151010_4435945.html.

Geographical observations of wind-blown sand flows in the ASL began with Reference BagnoldBagnold (Reference Bagnold1941), who laid the foundations of wind-blown sand physics. Bagnold conducted field measurements on the wind velocity and sediment transport rate under wind-blown sand flows with a focus on measuring the latter. Bagnold's quantitative study on the physical mechanisms responsible for the formation and evolution of sand dunes created a precedent for the study of geomorphological dynamics and has since influenced researchers in various fields of wind-blown sand physics. However, measurement of the wind velocity in the ASL is performed to obtain the average velocity profile. Therefore, Pitot tubes distributed in the wall-normal direction (2.54, 10.16, 40.64 and 162.56 cm) were employed to measure the wind velocity at a sampling frequency of approximately 0.017–0.0056 Hz (the corresponding sampling interval is approximately 1–3 min). However, such a low sampling frequency makes it difficult to study high-Reynolds-number wall turbulence. Subsequently, many researchers observed the sediment flux and its influencing factors while focusing on the sediment transport rate, wind erosion, lift-off velocity of sand particles and threshold wind velocity of sand motion; details can be found in the paper by Reference ZhengZheng (Reference Zheng2009). In summary, both in meteorology and geography, ASL flow observations usually involve the average velocity; furthermore, the sampling frequency is relatively low and the underlying surface of the boundary layer is generally complex, which makes it difficult to analyse high-Reynolds-number wall turbulence. In addition, single-point or wall-normal-distributed measurements were employed, which cannot meet the requirements for analysing the spatial structure of such flow.

The ASL observations used to study high-Reynolds-number wall-bounded turbulent flow are highlighted by the SLTEST site, the details of which are provided by Reference MetzgerMetzger (Reference Metzger2002). Unlike observations in atmospheric sciences and geosciences, the observational data acquired at the SLTEST site can be used to investigate high-Reynolds-number wall-bounded turbulent flows because this site is located on the dry lake bed of the Great Salt Lake in Utah, which has a flat and open surface, and the sampling frequency of the wind velocity is relatively high (20 Hz). However, owing to the surface conditions in this area and the low wind velocity, the flow field comprises a single phase and is free of particles. The measurement array installed at the SLTEST site is shown in Figure 2, which shows that the SLTEST array can perform synchronous multipoint measurements in the spanwise and wall-normal directions, that is, measurements of the two-dimensional turbulent flow field. For example, using steady wind data obtained at the SLTEST site with $Re_{\tau } \sim 6\times 10^{5}\text {--}3\times 10^{6}$ for approximately 70 h under different stratification conditions (Reference Chauhan, Hutchins, Monty and MarusicChauhan, Hutchins, Monty, & Marusic, 2013; Reference Metzger and KlewickiMetzger & Klewicki, 2001), the turbulent statistics in the ASL were confirmed to exhibit scaling laws that are generally representative of the canonical zero-pressure-gradient flat TBL (Reference Kunkel and MarusicKunkel & Marusic, 2006). This indicates that the high-quality near-neutral ASL data acquired after applying a strict data processing procedure can be valid to study the canonical TBL, although the ASL flow may be affected by weather and environmental factors; the wind velocity and direction are uncontrollable and may be affected by significant thermal instability. In addition, based on the ASL data obtained at the SLTEST site, researchers found that VLSMs exist in the ASL (Reference Hutchins, Chauhan, Marusic, Monty and KlewickiHutchins, Chauhan, Marusic, Monty, & Klewicki, 2012; Reference Hutchins and MarusicHutchins & Marusic, 2007a; Reference Marusic and HutchinsMarusic & Hutchins, 2008), with an inclination angle independent of the Reynolds number but significantly dependent on the thermal stability (Reference Chauhan, Hutchins, Monty and MarusicChauhan et al., 2013; Reference Marusic and HeuerMarusic & Heuer, 2007). Researchers also found that the inner peak of turbulent kinetic energy increases with the Reynolds number (Reference Metzger and KlewickiMetzger & Klewicki, 2001), the turbulent kinetic energy exhibits log-linear behaviour with the wall-normal distance (Reference Marusic, Monty, Hultmark and SmitsMarusic et al., 2013) and LSMs/VLSMs modulate the amplitude of small-scale energy in the ASL (Reference Guala, Metzger and McKeonGuala et al., 2011; Reference Mathis, Hutchins and MarusicMathis, Hutchins, & Marusic, 2009). As is evident, research based on SLTEST measurements has greatly promoted the understanding of wall turbulence with a high Reynolds number. However, owing to the lack of measurements in the streamwise direction, the applicability of Taylor's hypothesis in the ASL has not been verified. Furthermore, because the SLTEST array can obtain only particle-free fluctuating velocity data, high-Reynolds-number two-phase wall-bounded turbulence has not been investigated. In addition, the short duration of the measurements at the SLTEST site and the relative scarcity of data make it difficult to obtain reliable turbulence statistics under high-Reynolds-number conditions.

Figure 2. Measurement array installed at the SLTEST site. This figure was adapted from Reference Hutchins, Chauhan, Marusic, Monty and KlewickiHutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012).

Unlike the SLTEST site, the QLOA is a unique field observation station that can achieve synchronous measurements of the electric field strength and the three-dimensional (streamwise, spanwise and wall-normal directions) wind velocity, sand concentration, temperature and humidity within the three-dimensional turbulent ASL flow, as detailed by Reference Wang and ZhengWang and Zheng (Reference Wang and Zheng2016). The QLOA site is flat and open, and skirts the path of strong northwest-moving monsoons and sandstorms. Therefore, the QLOA can not only observe high-Reynolds-number single-phase, particle-free flows but also conduct measurements on typical sand-laden two-phase wall-bounded turbulence, such as wind-blown sand flows/sandstorms. These sand-laden two-phase flow data make it possible to analyse the effects of particles on the statistical and structural characteristics of the flow field under high-Reynolds-number conditions and provide support for studies on the dust structure and transport in two-phase flows.

Figure 3 presents a photograph of the QLOA, which consists of a main tower with a height of 32 m, 11 and 12 auxiliary towers with a height of 5 m arranged parallel to the upwind direction of the incoming flow (that is, northwest–southeast) and the spanwise direction (which is orthogonal to the incoming flow), respectively, and another 11 auxiliary towers oriented east–west to capture another possible incoming flow direction. Compared with the SLTEST site, the QLOA incorporates two streamwise arrays arranged along different incoming flow directions. Therefore, the QLOA can not only acquire two-dimensional ASL observations in the wall-normal and spanwise directions but also realize synchronous measurements in the streamwise direction. Thus, the QLOA provides three-dimensional observations of the ASL flow in a spatial domain of 390 m in the streamwise direction, 60 m in the spanwise direction and 30 m in the wall-normal direction. The observation array in the streamwise direction facilitates the measurement of the streamwise length scale of the VLSMs in the ASL and accordingly provides the error and applicability of Taylor's hypothesis in estimating the scale of the VLSMs. Moreover, these three-dimensional ASL observations are helpful for inverting the spatial structures of the flow field (see the paper by (Reference Liu, Wang and ZhengLiu, Wang, and Zheng, 2019a), in which a plausible empirical model was proposed to characterize the general shape of the three-dimensional large log-layer structure), dust concentration and electric field.

Figure 3. Photograph of the QLOA.

To achieve the high-frequency synchronous measurement of multiple physical quantities, the QLOA is equipped with various measuring equipment, such as sonic anemometers (Gill Instruments R3-50 and Campbell CSAT3B) operating at a sampling frequency of 50 Hz, aerosol monitors (TSI, DUST TRAK II-8530-EP), temperature sensors and three-dimensional vibrating-reed electric field mills (VREFMs). These VREFMs were developed by the author's research group (Reference ZhengZheng, 2013), and the size of the probe is only one-tenth of that of a conventional atmospheric electric field meter. Therefore, the VREFMs can be more compactly installed to measure the three-dimensional electric field in a relatively narrow region without experiencing significant mutual disturbances among the sensors. Additionally, because the measuring principle of the electric field meter is based on a vibrating capacitor, disturbances in the electric field measurements arising from sand particles impacting the electrodes of the electric field meters can be effectively reduced. The various instruments collect data through multiple data acquisition instruments, which are synchronized in time by a global positioning system (GPS), thereby realizing the synchronous measurement of multiple physical quantities, such as the three-dimensional wind velocity, temperature and electric field intensity, in the three-dimensional ASL flow field.

Taking the entire duration of the sandstorm that occurred on April 17, 2017 as an example, the multiphysical data are presented in Figure 4. These synchronous measurements are helpful for comparatively analysing the relationships among the spatial distribution characteristics of different physical quantities, thereby revealing their coupling effects, such as the relationship among the volumetric dust structure, the electric structure and the turbulence structure, and thus their interaction mechanism. To date, continuous ASL observations have been conducted at the QLOA site for 5 years since its establishment, amounting to more than 7400 h of multiphysics synchronous observational data, which include nearly 600 h of high-quality steady wind data. Of these data, approximately 120 h pertain to fluctuating velocity data in the near-neutral, particle-free ASL, while the data in the stable and unstable ASL account for approximately 302 h; in addition, there are 83 h of sand-laden fluctuating velocity data in the near-neutral ASL and 90 h of sand-laden data in the non-neutral ASL. The Reynolds numbers in sand-free and sand-laden (including sandstorms) flows may reach $Re_{\tau } \sim 4.7\times 10^{6}$ and $Re_{\tau } \sim 5.4\times 10^{6}$, respectively, which are the highest known Reynolds numbers in these flows.

Figure 4. Continuous three-dimensional wind velocity (streamwise $u$, spanwise $v$ and vertical $w$), PM10 concentration ($C_{{PM10}}$), temperature ($T$) and relative humidity ($RH$), and three-dimensional electric field (streamwise $E_{x}$, spanwise $E_{y}$ and vertical $E_{z}$) data collected on April 17, 2017.

3. VLSMs in the Sand-Laden Two-Phase Flow

3.1 Existence of VLSMs in Sand-Laden Flows

Turbulent coherent structures represent a series of organized motions that are found in disorganized and complex turbulent flows, which mainly include hairpin vortices, quasi-streamwise vortices, inner streaks, LSMs and VLSMs, where LSMs/VLSMs are an important and perhaps dominant feature in the outer region of wall turbulence. The discovery of coherent structures in wall turbulence can be traced back to 1954, when S. Corrsin and A. L. Kistler found intermittent turbulence in the wake of a turbulent flow, which implies that turbulence is not a completely irregular and random motion (Reference Corrsin and KistlerCorrsin & Kistler, 1954). These coherent structures are responsible for the generation and dissipation of turbulence, important to the understanding of turbulence dynamics (Reference RobinsonRobinson, 1991), and contribute significantly to the transport of heat, mass and momentum (Reference Marusic, McKeon, Monkewitz, Nagib, Smits and SreenivasanMarusic, McKeon, et al., 2010). The LSMs in wall turbulence may have been found as early as the late 1950s when Grant observed long tails of time-delayed streamwise velocity fluctuations, which implied the potential presence of large-scale structures in shear turbulence (Reference GrantGrant, 1958). However, LSMs were not officially reported until Reference Kovasznay, Kibens and BlackwelderKovasznay, Kibens, and Blackwelder (Reference Kovasznay, Kibens and Blackwelder1970) clearly proposed the existence of LSMs on the streamwise scale reaching $2\text {--}3\delta$. Subsequently, Reference Kim and AdrianKim and Adrian (Reference Kim and Adrian1999) found VLSMs on the streamwise scale exceeding $3\delta$ or even $10\text {--}15\delta$ in wall turbulence, as shown in Figure 5. More recently, Reference Guala, Metzger and McKeonGuala et al. (Reference Guala, Metzger and McKeon2011) and Reference Hutchins, Chauhan, Marusic, Monty and KlewickiHutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) detected VLSMs in the particle-free flow of the ASL based on observational data ($Re_{\tau }=5\times 10^{5}\text {--}7.7\times 10^{5}$) obtained at the SLTEST site. Reference Wang, Zheng and TaoWang, Zheng, and Tao (Reference Wang, Zheng and Tao2017) further found VLSMs in sand-laden flow based on observational data ($Re_{\tau } \sim 5\times 10^{6}$) of a wind-blown sand-laden two-phase flow obtained at the QLOA. Therefore, LSMs and VLSMs (collectively known as superstructures (Reference Hutchins and MarusicHutchins & Marusic, 2007a)) are common in many different types of flows (Reference Blackwelder and KovasznayBlackwelder & Kovasznay, 1972; Reference Monty, Stewart, Williams and ChongMonty et al., 2007) and are associated with the higher and lower wavenumber peaks, respectively, in the premultiplied spectrum of the streamwise velocity fluctuation (Reference Kim and AdrianKim & Adrian, 1999; Reference Vallikivi, Ganapathisubramani and SmitsVallikivi, Ganapathisubramani, & Smits, 2015); therefore, these superstructures contribute significantly to the turbulent kinetic energy and Reynolds shear stress (Reference Balakumar and AdrianBalakumar & Adrian, 2007; Reference Guala, Hommema and AdrianGuala et al., 2006; Reference Wang and ZhengWang & Zheng, 2016). Moreover, LSMs/VLSMs also play an important role in mass transport (Reference Zheng, Zhang, Wang, Liu and ZhuZheng, Zhang, Wang, Liu, & Zhu, 2013), and the complex transport pathways of dust and other pollutants in the atmosphere may be associated with the dynamics of LSMs/VLSMs (Reference Jacob and AndersonJacob & Anderson, 2017). Therefore, the study of LSMs/VLSMs is not only of great significance to understanding wall turbulence but also helpful for revealing the physical mechanism responsible for mass transport in high-Reynolds-number wall turbulence.

Figure 5. Conceptual model of the VLSM generation process: hairpin vortices align coherently in the streamwise direction to form hairpin vortex packets, and packets align coherently to form VLSMs. Figure adapted from Reference Kim and AdrianKim and Adrian (Reference Kim and Adrian1999).

As described above, VLSMs are present within the sand-laden flow field; however, similar very-large-scale structures also exist in the sand concentration scalar field in wind-blown sand flows/sandstorms. Analyses of correlation contour maps and the premultiplied energy spectra of the streamwise velocity fluctuations based on observational data obtained at the QLOA suggest the presence of VLSMs on streamwise scales exceeding $3\delta$ in high-Reynolds-number ($Re_{\tau } \sim 5\times 10^{6}$) sandstorm flow fields. These VLSMs exhibit very long meandering features, as shown in Figure 6a, similar to the phenomenon observed by Reference Hutchins and MarusicHutchins and Marusic (Reference Hutchins and Marusic2007a) in the particle-free flow at the SLTEST site. In addition, the two-point correlation analysis of the fluctuating PM10 (particles with sizes less than 10 $\mathrm {\mu }$m) concentration measured during a sandstorm is shown in Figure 6b, which further reveals an inclined very-large-scale sand structure similar to a VLSM in the sandstorm flow field. This has been confirmed by LES results (Reference Zhang and ZhengZhang & Zheng, 2018). The streamwise distance shown in Figure 6 is converted from the temporal lead/lag using Taylor's hypothesis, where the convection velocity of the turbulence and sand structure is taken as the local mean velocity at the corresponding wall-normal position. This is consistent with the existing method used to study the characteristic length scale of VLSMs.

Figure 6. (a) Instantaneous streamwise velocity fluctuations across the spanwise array at $z \approx 0.033\delta$ based on observational data of a wind-blown sand-laden two-phase flow at the QLOA, with $Re_{\tau } \approx 5.4\times 10^{6}$ and the average PM10 concentration $C_{{PM10}} \approx 1.09$ mg m$^{-3}$. The $x$-axis is reconstructed using Taylor's hypothesis and a convection velocity based on the local mean. The length marked by the arrow shows only negative $u$ fluctuations ($u<-1$ m s$^{-1}$). (b) Two-dimensional correlation maps of the fluctuating PM10 concentration, where the dashed line represents a correlation coefficient of 0.05.

3.2 Structure Inclination Angles

The LSMs/VLSMs are inclined structures that are oblique to the free stream, and the angle of inclination is usually referred to as the structure inclination angle. The structure inclination angle was originally studied by (Reference Kovasznay, Kibens and BlackwelderKovasznay et al., 1970), who examined the two-dimensional contour maps of two-point correlations based on the fluctuating streamwise velocity measured by hot-wire anemometers in the TBL at $Re_{\tau }=1240$ and found that LSMs have a moderate tilt in the streamwise direction. The observation of LSMs inclined at a shallow angle away from the wall was subsequently confirmed by Reference Blackwelder and KovasznayBlackwelder and Kovasznay (Reference Blackwelder and Kovasznay1972) and Reference FalcoFalco (Reference Falco1977) in the TBL. LSMs/VLSMs contribute significantly to the transport of turbulence, and thus, the complex transport pathways of heat, mass and momentum may be associated with the dynamics of inclined LSMs/VLSMs (Reference Jacob and AndersonJacob & Anderson, 2017; Reference Marusic, McKeon, Monkewitz, Nagib, Smits and SreenivasanMarusic, McKeon, et al., 2010). Moreover, the structure inclination angle is vital for accurately establishing the relationship between the velocity fluctuation and the wall shear stress (Reference Marusic, Mathis and HutchinsMarusic, Mathis, et al., 2010), and this relationship can be used to improve the wall model in an LES (Reference Marusic, Kunkel and Porte-AgelMarusic, Kunkel, & Porte-Agel, 2001). Therefore, the inclination angle has attracted the attention of many researchers (Reference Brown and ThomasBrown & Thomas, 1977; Reference Chauhan, Hutchins, Monty and MarusicChauhan et al., 2013; Reference Kovasznay, Kibens and BlackwelderKovasznay et al., 1970).

However, the structure inclination angles provided by different studies vary considerably. In the early days, Reference Brown and ThomasBrown and Thomas (Reference Brown and Thomas1977) inferred an $18^{\circ }$ angle in the TBL at $Re_{\tau }=3413$ from the location of the maximum correlation between the wall shear stress and the fluctuating streamwise velocities measured by a hot-film wall shear stress sensor and several hot wires. Subsequently, Reference Head and BandyopadhyayHead and Bandyopadhyay (Reference Head and Bandyopadhyay1981) suggested that these large-scale structures are composed of multiple hairpin vortices inclined at $15^{\circ }\text {--}20^{\circ }$ with respect to the wall in the TBL over the range of $500< Re_{\theta }<17\,500$ ($Re_{\theta }=U_{\propto }\theta /\nu$, where $U_{\propto }$ and $\theta$ denote the free stream velocity and momentum thickness, respectively). Advances in techniques have provided some improvements in this topic of research; in particular, the PIV technique has provided important new information about the structure inclination angle. For example, Reference TomkinsTomkins (Reference Tomkins1997) proposed a 12.3$^{\circ }$ angle in a laboratory TBL at $Re_{\tau }=2227$, while Reference Christensen and AdrianChristensen and Adrian (Reference Christensen and Adrian2001) suggested that the inclination angle may be $13^{\circ }$ and $14^{\circ }$ in a channel flow at $Re_{\tau }= 547$ and 1734, respectively, and Reference Adrian, Meinhart and TomkinsAdrian, Meinhart, and Tomkins (Reference Adrian, Meinhart and Tomkins2000) indicated that the inclination angle of a large ramp-like structure formed by hairpin vortex packets is approximately $3^{\circ }\text {--}35^{\circ }$, while individual hairpin vortices are inclined approximately $45^{\circ }$ with respect to the wall.

ASL observations have played an important role in obtaining more information on the VLSM inclination angle through experiments in high-Reynolds-number flows. In the neutral marine ASL, Reference Boppe and NeuBoppe and Neu (Reference Boppe and Neu1995) and Reference Boppe and NeuBoppe and Neu (Reference Boppe and Neu1999) observed a 15$^{\circ }$ inclination angle of large-scale structures. Reference Hommema and AdrianHommema and Adrian (Reference Hommema and Adrian2003) showed that the average inclination angle of VLSMs was $18.7^{\circ }$ in the neutral ASL at $Re_{\tau }=1.3\times 10^{6}$ through a smoke visualization experiment at the SLTEST site. (Reference Carper and Porté-AgelCarper and Porté-Agel, 2004) also examined data from the SLTEST site and found that the structure inclination angle is approximately $16^{\circ }$ in the neutral ASL by analysing the two-point correlations of the streamwise velocity fluctuations obtained by sonic anemometers. In addition, Reference Morris, Stolpa, Slaboch and KlewickiMorris, Stolpa, Slaboch, and Klewicki (Reference Morris, Stolpa, Slaboch and Klewicki2007) and Reference Guala, Metzger and McKeonGuala et al. (Reference Guala, Metzger and McKeon2011) suggested an $11^{\circ }$ angle at $Re_{\tau }=6\times 10^{5}$ and $5\times 10^{5}$, respectively, whereas Reference Hutchins, Chauhan, Marusic, Monty and KlewickiHutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) provided a $25^{\circ }$ angle at $Re_{\tau }=7.7\times 10^{5}$. (Reference Chauhan, Hutchins, Monty and MarusicChauhan et al., 2013) indicated that the inclination angle is $13.7^{\circ }$ in the neutral regime and varies significantly with the Monin–Obukhov stability parameter in the unstable ASL. In summary, in the neutral regime, the structure inclination angle ranges from $3^{\circ }$ to $35^{\circ }$ for $Re_{\tau }\sim O(10^{3})$, while it ranges from $11^{\circ }$ to $25^{\circ }$ for $Re_{\tau }\sim O(10^{6})$. Nevertheless, Reference Marusic and HeuerMarusic and Heuer (Reference Marusic and Heuer2007) postulated that the structure inclination angle does not depend on the Reynolds number given that no significant difference in the inclination angles ($13.8^{\circ }$ and $14.4^{\circ }$, respectively) was obtained from either a wind tunnel experiment at a low Reynolds number of $Re_{\tau }=1350$ or a neutral ASL experiment at a high Reynolds number of $Re_{\tau } \approx 1.8\times 10^{6}$. Thus, it is necessary to determine a suitable parameter to characterize the variation in the inclination angle and to ascertain the effects of particles on the inclination angle in wind-blown sand flows/sandstorms.

Studies based on long-term observations of sand-free and sand-laden flows at the QLOA have deepened the understanding of the VLSM inclination angle. A comparison between the structure inclination angles of sand-laden and sand-free flows is shown in Figure 7a, which demonstrates that the structure inclination angle in the sand-laden flow is much larger than that in the sand-free flow and decreases nearly linearly with the friction velocity (Reference Wang, Gu and ZhengWang et al., 2020). This linearly decreasing trend of the inclination angle is qualitatively consistent with the trend in a sand-free flow proposed by Reference Liu, Bo and LiangLiu, Bo, and Liang (Reference Liu, Bo and Liang2017). In addition, Reference Liu, Bo and LiangLiu, Bo, et al. (Reference Liu, Bo and Liang2017) provided a plausible explanation for why the friction velocity dominates the behaviour of the inclination angle: the structure inclination angle of an LSM/VLSM is actually the angle between the interface formed by two or more hairpin vortex heads (creating an LSM called a hairpin vortex packet) and the wall (Reference Adrian, Meinhart and TomkinsAdrian et al., 2000). In the same packet, the older hairpin vortices (being larger) are more likely to encounter a faster moving fluid, and thus, the older hairpin vortices move downstream faster than the younger, smaller vortices closer to the wall (Reference DennisDennis, 2015). As a consequence, the hairpin vortex packet is stretched by the velocity gradient, which forms an inclination angle. With an increase in the velocity gradient, this ‘stretching’ effect is enhanced, which results in a decrease in the inclination angle. The friction velocity reflects mainly the velocity gradient in the boundary layer. Therefore, the inclination angle decreases linearly with the friction velocity. Accordingly, the addition of a large number of sand particles into a sand-laden flow reduces the velocity gradient in the flow field, as shown in Figure 7b, which leads to an increase in the inclination angle. Note that experimental scatter is inevitably associated with these ASL measurements; however, it appears that these measurements can be used as a representation of the very-high-Reynolds-number behaviour. A detailed analysis of the effects of sand concentration on the mean shear requires more data for support. In addition to the particle concentration, the particle size has an effect on the mean shear and coherent structures. Reference Tsuji and MorikawaTsuji and Morikawa (Reference Tsuji and Morikawa1982) found that in cases of both large particle size (with a diameter of 3.4 mm) and small particle size (0.2 mm), the particles flatten the air-velocity profile (that is, they reduce the mean shear), and the flattening of the air-velocity profile is more remarkable in the case of small particle size. With smaller particle size (0.1–0.2 mm), Reference Righetti and RomanoRighetti and Romano (Reference Righetti and Romano2004) also observed reduced mean shear in the particle-laden turbulence. This may lead to the increased inclination angle of the large-scale coherent structure in the particle-laden turbulence, as suggested by Reference Tay, Kuhn and TachieTay et al. (Reference Tay, Kuhn and Tachie2015).

Figure 7. Variation in the structure inclination angle with the friction velocity in the neutral regime, taken from Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020), where the sand-free flow results are taken from Reference Liu, Bo and LiangLiu, Bo, et al. (Reference Liu, Bo and Liang2017).(b) Variation in the velocity gradient with the sand concentration in the sand-laden flow, taken from Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020).

3.3 Characteristic Length Scale

The influence of the Reynolds number on the characteristic length scale of VLSMs is not clear. In the early days, Reference TrittonTritton (Reference Tritton1967) examined the two-point correlation of the streamwise velocity fluctuations in the streamwise direction in a wind tunnel TBL at $Re_{\tau } \approx 880$ and discovered that the streamwise length scale may exceed $5\delta$ and that the spanwise width scale is approximately $0.6\delta$ at the top of the logarithmic region. In an LES of a channel flow ($Re_{\tau } \approx 640$), Reference Moin and KimMoin and Kim (Reference Moin and Kim1982) studied the streamwise velocity correlation and inferred a streamwise length scale of approximately $6\delta$ with a corresponding spanwise width scale of approximately $0.5\delta$ at the wall-normal distance $z = 0.121\delta$. At the turn of the millennium, the proposal of the hairpin vortex packet model (Reference Adrian, Meinhart and TomkinsAdrian et al., 2000; Reference Kim and AdrianKim & Adrian, 1999) based on experiments in a pipe (at $Re_{\tau } = 1058\text {--}3175$) and the TBL (at $Re_{\tau } = 355\text {--}2000$) contributed to better understanding the mechanism responsible for the formation of LSMs/VLSMs. The basic idea of this model is that two or more hairpin vortices are aligned coherently in the streamwise direction, which creates an LSM (called a hairpin vortex packet) with a streamwise length scale of ${\sim }3\delta$; then, packets may also align with other packets to create a VLSM with a streamwise length scale of ${\sim }6\delta$.

With the development of experimental facilities and measurement techniques, the observable Reynolds number of a flow has greatly increased. Notably, Reference Hutchins and MarusicHutchins and Marusic (Reference Hutchins and Marusic2007a) compared the characteristic length scales at four different Reynolds numbers ($Re_{\tau } = 1120$, 7610, 14 380 and 19 960) by conducting experiments in the High Reynolds Number Boundary Layer Wind Tunnel (HRNBLWT) with a spanwise array of ten hot wires and discovered that the streamwise length scale for $Re_{\tau } = 1120$ seems to be slightly shorter than that for the other three cases. This finding may indicate that the VLSM scale is dependent on the Reynolds number. However, the results obtained by Reference Hutchins, Chauhan, Marusic, Monty and KlewickiHutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) from high-Reynolds-number ($Re_{\tau } \approx 7.7 \times 10^{5}$) ASL observations at the SLTEST site indicate that the outer-scaled streamwise velocity correlation agrees well with the documented laboratory TBL results provided by Reference Hutchins and MarusicHutchins and Marusic (Reference Hutchins and Marusic2007a), which provides possible evidence that these large-scale structures scale on the boundary layer thickness up to $Re_{\tau } \sim O(10^{6})$. Nevertheless, high-Reynolds-number experimental data are quite sparse and more work is needed.

Coherent structures exist throughout the boundary layer, and thus, it is very important to characterize the variation in the characteristic length scale with the wall-normal distance. Reference TrittonTritton (Reference Tritton1967) and Reference Moin and KimMoin and Kim (Reference Moin and Kim1982) found that the spanwise scale of the coherent structures increases with the wall-normal distance throughout the boundary layer, whereas the streamwise scale decreases with the wall-normal distance in the wake region in the TBL and channel flow at $Re_{\tau } = 880$ and 640, respectively. Subsequently, Reference Krogstad and AntoniaKrogstad and Antonia (Reference Krogstad and Antonia1994) published laboratory TBL results at $Re_{\tau }= 1850$ showing that the streamwise scale increases with the wall-normal distance according to the hyperbolic tangent function, i.e. $L_{x} \sim \mathrm {tanh}(12z/\delta )$. However, the variations in the characteristic length scale with the wall-normal distance given by different studies diverge. For example, Reference Tomkins and AdrianTomkins and Adrian (Reference Tomkins and Adrian2003) suggested that both the streamwise and spanwise length scales linearly increase with the wall-normal distance in the logarithmic region of the TBL at $Re_{\tau } = 2216$. This is consistent with Reference TownsendTownsend's (Reference Townsend1976) attached eddy hypothesis because the linear growth of the eddy size with distance from the wall is a necessary condition for self-similarity, where self-similarity is an assumption of the hierarchy of attached eddies Reference Perry and ChongPerry and Chong (Reference Perry and Chong1982); and recently summarized by Reference Marusic and MontyMarusic and Monty (Reference Marusic and Monty2019). The conclusions obtained from studies on the TBL at $Re_{\tau } = 690\text {--}2800$ indicate that the spanwise scale of the coherent structures increases linearly throughout the TBL and that the increase is more pronounced in the logarithmic region (Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and MarusicGanapathisubramani, Hutchins, Hambleton, Longmire, & Marusic, 2005; Reference Hutchins, Hambleton and MarusicHutchins, Hambleton, & Marusic, 2005). Based on experimental data in pipe ($Re_{\tau } = 933\text {--}4355$) and channel ($Re_{\tau } = 3178$) flows in conjunction with previously reported boundary layer data, Reference Monty, Stewart, Williams and ChongMonty et al. (Reference Esposito, Molinaro, Popa, Molfese, Cozzolino, Marty, Taj-Eddine, Di Achille, Franzese, Silvestro and Ori2007) found that the spanwise scale of the coherent structures linearly increases piecewise with the wall-normal distance (even in the wake region) for all available data, with $Re_{\tau } \sim O(10^{3}-10^{4})$ in different types of flows, as shown in Figure 8a. Recently, the DNS of a TBL at $Re_{\tau } \approx 920$ performed by Reference Lee and SungLee and Sung (Reference Lee and Sung2011) showed that the streamwise length scale increases as far as the top of the logarithmic region ($z/\delta = 0.2$) and then sharply decreases in the wake region, as shown in Figure 8b. This behaviour may be explained by the detached eddies in the attached eddy theory. Outside the log region of a boundary layer, coherent structures (e.g. hairpin packets proposed by Reference Kim and AdrianKim & Adrian (Reference Kim and Adrian1999)) that exist in the log region do not retain their coherence. The breakdown of these structures produces a population of smaller eddies, which may be detached eddies, vortex rings or some other outer-layer structure (Reference Katul and VidakovicKatul & Vidakovic, 1996; Reference Perry and MarusicPerry & Marusic, 1995). The length scale extracted from the two-point correlation represents the average scale of all turbulent coherent structures at a given distance from the wall. Therefore, the popular smaller scales may reduce the streamwise length scale.

Figure 8. (a) Variation in the spanwise width scale with the wall-normal distance. Figure adapted from Reference Monty, Stewart, Williams and ChongMonty et al. (Reference Monty, Stewart, Williams and Chong2007), where the solid curves are fitted to the open symbols for $z < 0.15\delta$ and the dashed curves are fitted to the open symbols for $z > 0.4\delta$. (b) Variation in the streamwise length scale with the wall-normal distance. This figure was adapted from Reference Lee and SungLee and Sung (Reference Lee and Sung2011).

Studies on flows at high Reynolds numbers and broad Reynolds number ranges based on observations of wind-blown sand flows/sandstorms at the QLOA have improved the understanding of the scale of coherent structures. The streamwise length scales calculated using the experimental data of Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020) in a wind-blown sand flow at the QLOA are shown as the yellow symbols in Figure 9. The results in this figure are in good agreement with those of a sand-free flow in the ASL (Reference Liu, Wang and ZhengLiu, Wang, & Zheng, 2017); that is, the streamwise length scale remains approximately invariant with the Reynolds number. However, combined with wind tunnel experimental data at low and moderate Reynolds numbers, Reference ZhengZheng (Reference Zheng2020) revealed that the streamwise length scale of the coherent structures tends to become gradually invariant with increasing Reynolds number spanning over four orders of magnitude $Re_{\tau } \sim O(10^{2}\text {--}10^{6})$, as shown by the pink symbols in Figure 9. The quantitative results show that when $Re_{\tau } < 1200$, the streamwise length scale increases with the Reynolds number, whereas when $Re_{\tau } > 1200$, the outer-scaled length scale no longer varies with the Reynolds number. In addition, Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020) revealed that the streamwise length scale and wall-normal length scale of the coherent structures in sand-laden flows increase with the wall-normal distance, which is qualitatively consistent with the relatively universal law governing the evolution of the spatial length scale revealed by Reference Liu, Wang and ZhengLiu, Wang, et al. (Reference Liu, Wang and Zheng2017) using data from the QLOA of high-Reynolds-number sand-free flows. Moreover, a quantitative comparison indicates that the addition of particles to a wind-blown sand flow reduces the wall-normal length scale by approximately 40 %.

Figure 9. Variation in the streamwise length scale of the coherent structure with the Reynolds number, where the results of sand-free flows in the ASL were taken from Reference Liu, Wang and ZhengLiu, Wang, et al. (Reference Liu, Wang and Zheng2017), the wind tunnel TBL results were taken from Reference ZhengZheng (Reference Zheng2020) and the sand-laden flow results were taken from Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020).

However, the streamwise length scale of coherent structures in the ASL obtained by spatiotemporally converting the measured data through Taylor's hypothesis is underestimated. Except for some of the flow images obtained by PIV in the laboratory (Reference Hutchins and MarusicHutchins & Marusic, 2007a; Reference Wang, Huang, Zhang, Wang and ZhaoWang, Huang, Zhang, Wang & Zhao, 2018), estimates of the streamwise length scale are still based mostly on Taylor's hypothesis. However, Taylor's hypothesis is applicable only on the premise that the convection velocity of the turbulence structure is consistent with the local mean velocity at the corresponding wall-normal position and that the structure does not deform during convection. Unfortunately, these assumptions are difficult to fully satisfy in actual flows, especially boundary layer flows. Therefore, the applicability of Taylor's hypothesis has become a hotspot in turbulence research (Reference Squire, Hutchins, Morrill-Winter, Schultz, Klewicki and MarusicSquire et al., 2017; see Reference He, Jin and YangHe, Jin, & Yang, 2017 for a review of relevant studies). Using the observational data from the streamwise array unique to the QLOA, Reference Han, Wang and ZhengHan, Wang, and Zheng (Reference Han, Wang and Zheng2019) reported that the directly measured streamwise length scale of the coherent structures is generally larger than that obtained by Taylor's hypothesis (see Table 4 in Reference Han, Wang and ZhengHan, Wang, et al. (Reference Han, Wang and Zheng2019), where the maximum relative error reaches 25 %). This is because the convection velocity of coherent structures in the logarithmic region of the ASL varies with the migration distance, as shown in Figure 10a, in which the convection velocity may reach 40 % higher than the local mean velocity when the migration distance exceeds approximately 300 m (where the migration distance represents the spatial displacement of coherent structure along the streamwise direction). Moreover, a coherent structure in the ASL experiences complete deformation after migrating a distance 1.6 times its own scale in the streamwise direction, as shown in Figure 10b, where

(1)\begin{equation} \gamma_{L}^{2}(\Delta x, f) = \frac{|\langle U_{A}(x, f) \overline{U_{B}(x + \Delta x, f)} \rangle\rvert^{2}}{\langle \rvert U_{A}(x, f) \rvert^{2} \rangle \langle \rvert U_{B}(x + \Delta x, f) \rvert^{2} \rangle} \end{equation}

is the linear coherence spectrum (Reference Baars, Hutchins and MarusicBaars, Hutchins, & Marusic, 2016) of the streamwise velocity fluctuations $u_{A} (x, t+\Delta t/U_{m})$ and $u_{B} (x+\Delta x, t)$, $U_{A}(x, f)$ and $U_{B}(x+\Delta x, f)$ are the Fourier transforms of $u_{A} (x, t+\Delta t/U_{m})$ and $u_{B} (x+\Delta x, t)$, $\langle \rangle$ denotes ensemble averaging, the overbar indicates the complex conjugate and $||$ designates the modulus.

Figure 10. (a) Variation in the convection velocity with the migration distance based on QLOA data and (b) coherent spectrum analysis. The measured convection velocity and the local mean velocity are represented by $U_{m}$ and $U$, respectively. Here, $\Delta x$ is the streamwise distance, $f$ is the frequency, and $\lambda =U_{m}/f$ is the streamwise wavelength.

3.4 Turbulent Kinetic Energy of VLSMs

The contributions of VLSMs to the turbulent kinetic energy, Reynolds shear stress and transport of material are significantly greater than those of other coherent structures. Turbulence exhibits multiscale motions and the energies of different-scale motions differ. Nevertheless, the dominant position of the corresponding motion in turbulence is reflected by the energy level. The LSMs/VLSMs initially received attention as a result of their large scale until Reference Kim and AdrianKim and Adrian (Reference Kim and Adrian1999) analysed the energy spectra in pipes over $1058 \leq Re_{\tau } \leq 3175$ and noticed that the high-wavenumber and low-wavenumber peaks in the energy spectra corresponded to LSMs and VLSMs in the outer layer, respectively. Moreover, the low-wavenumber peak becomes more obvious in higher-Reynolds-number flows (Reference Vallikivi, Hultmark and SmitsVallikivi, Hultmark, et al., 2015; Reference Wang and ZhengWang & Zheng, 2016), which means that the VLSMs in the outer layer have the most significant contribution to the turbulent kinetic energy in high-Reynolds-number wall turbulence, thereby becoming the dominant structural feature. Subsequently, researchers have successively investigated the energy fraction contributed by VLSMs to the total turbulent kinetic energy. For example, Reference Morrison, Mckeon, Jiang and SmitsMorrison et al. (Reference Morrison, Mckeon, Jiang and Smits2004) analysed pipe flows over $1500 \leq Re_{\tau } \leq 1.01 \times 10^{5}$ and revealed that the fraction of turbulent kinetic energy caused by VLSMs can reach 50 %. The pipe flow ($3815 \leq Re_{\tau } \leq 7959$) examined by Reference Guala, Hommema and AdrianGuala et al. (Reference Guala, Hommema and Adrian2006) indicated that the energy fraction of VLSMs can reach up to 65 %, with a 50 %–60 % contribution to the Reynolds shear stress. Reference Balakumar and AdrianBalakumar and Adrian (Reference Balakumar and Adrian2007) analysed experimental data in the TBL ($1476 \leq Re_{\tau } \leq 2395$) and channel flow ($531 \leq Re_{\tau } \leq 1584$), and proposed that the contributions of VLSMs to the turbulent kinetic energy and Reynolds shear stress were approximately 40 %–65 % and 30 %–50 %, respectively; they also stated that the variations in the energy contribution and shear stress carried by the VLSMs are functions of the Reynolds number for different flows, as shown in Figure 11. Numerical simulation results in recent years support this finding. For example, a DNS ($459 \leq Re_{\tau } \leq 732$) in the TBL (Reference Lee and SungLee & Sung, Reference Lee and Sung2011) demonstrated that the Reynolds shear stress contributed by VLSMs exceeds 45 %. Furthermore, the LES results of the TBL at $Re_{\tau } \sim 4000$ (Reference Deck, Renard, Laraufie and WeissDeck, Renard, Laraufie, & Weiss, 2014) show that VLSMs/LSMs significantly affect the wall shear stress and contribute more than 60 % of the turbulent friction coefficient (i.e. the contribution of turbulence to mean friction is obtained by the weighted integration of the Reynolds shear stress). The effect of VLSMs contributing such a high fraction of turbulent kinetic energy on the transport of material is also obviously critical. Especially in the ASL, VLSMs play a major role in the downward transport of momentum (Reference Horiguchi, Hayashi, Adachi and OnogiHoriguchi, Hayashi, Adachi, & Onogi, 2012), which will significantly affect the movement of both water vapour and CO$_{2}$ (Reference Cooper, Leclerc, Archuleta, Coulter, Eichinger, Kao and NappoCooper et al., 2006; Reference Serafimovich, Thomas and FokenSerafimovich, Thomas, & Foken, 2011).

Figure 11. Variations in the fractions of (a) kinetic energy and (b) shear stress carried by VLSMs as a function of the Reynolds number for channel (open symbols), zero-pressure-gradient boundary layer (ZPGBL, grey filled symbols) and pipe (black filled symbols) flows for the following fractions: open squares, $z/\delta \sim 0.1$; open upright triangles, $z/\delta \sim 0.2$; open inverted triangles, $z/\delta \sim 0.3$; open diamonds, $z/\delta \sim 0.5$; and open circles, $z/\delta \sim 0.7$ (taken from Reference Balakumar and AdrianBalakumar & Adrian (Reference Balakumar and Adrian2007)).

A new mechanism that explains how the VLSM energy changes with height based on the observation results of sand-laden two-phase flows in the ASL has been proposed. The variation in the energy fraction of VLSMs with height is very important because VLSMs commonly exist in the entire outer layer. However, this topic has been the subject of relatively little research. Recently, Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020) analysed the influences of sand particles on the turbulent kinetic energy and energy fraction of VLSMs based on the observational data of sand-laden two-phase flows from the QLOA, as shown in Figure 12 (yellow dots). The turbulent kinetic energy and energy fraction of VLSMs in high-Reynolds-number sand-laden flows increase with an approximately log-linear behaviour with height, which is qualitatively consistent with the particle-free flow results. However, a quantitative comparison shows that the normalized turbulent energy of VLSMs increases while the energy fraction of VLSMs decreases in sand-laden flows, which means that sand particles not only increase the energy of VLSMs but also significantly increase the total streamwise turbulent kinetic energy in sand-laden flows. The corresponding particle-free flow results can be found in the paper by Reference Wang and ZhengWang and Zheng (Reference Wang and Zheng2016), who revealed the mechanism by which the energy of VLSMs and the energy fraction increase with the wall-normal distance in an approximately log-linear manner, as shown in Figure 12 (black dots). The energy fraction of VLSMs can reach 60 % at the top of the logarithmic layer ($z = 0.2\delta$), which is larger than the energy fractions (approximately 40 %–50 % over $Re_{\tau } \sim O(10^{2}\text {--}O(10^{3}))$ obtained during the experiment conducted in the TBL by Reference Balakumar and AdrianBalakumar and Adrian (Reference Balakumar and Adrian2007) and Reference Lee and SungLee and Sung (Reference Lee and Sung2011), which also means that the contribution of VLSMs to the turbulent kinetic energy becomes more obvious at higher Reynolds numbers.

Figure 12. Variations in the contributions of VLSMs to the turbulent kinetic energy in a high-Reynolds-number flow in the ASL with height: (a) the energy fraction contributed by VLSMs to the total turbulent kinetic energy and (b) the normalized energy contributed by VLSMs in clean air (black) and sand-laden (yellow) flows, taken from Reference Wang, Gu and ZhengWang et al. (Reference Wang, Gu and Zheng2020).

Another important impact of research on the energy fraction contributed by VLSMs to the process of wind-blown sand flows/sandstorms is to reveal the nature of the meteorological phenomenon known as ‘gusty wind’. In 1983, the World Meteorological Organization (WMO) defined gusty wind as a fluctuation that occurs approximately once every several minutes (WMO, 1983). Moreover, Reference Zeng, Cheng, Hu and PengZeng et al. (Reference Zeng, Cheng, Hu and Peng2010) defined gusty wind as having a fluctuation period of 1–10 min under neutral stratified conditions. According to the analysis of observational data regarding the urban boundary layer, it has been proposed that gusty wind is coherent and plays an important role in the transport of both sand and dust. Based on sand-laden flow data from the QLOA, Reference Gu, Wang, Zhu and ZhengGu, Wang, Zhu, and Zheng (Reference Gu, Wang, Zhu and Zheng2019) compared the scales of VLSMs and LSMs with those of gusty wind in normalized energy spectra. They found that meteorological gusty wind contains a large number of synoptic waves but is missing the VLSMs and LSMs that have considerable contributions to the turbulent kinetic energy. In fact, most VLSMs and all LSMs are forfeited by gusty wind, especially under high-wind velocity condition, and the lost contributions of VLSMs and LSMs can reach 50 %.

4. Amplitude Modulation and Its Application

In addition to the ubiquity and importance of LSMs/VLSMs, researchers pay attention to these outer-region structures because they have a significant amplitude modulation effect on small-scale near-wall motions (Reference Brown and ThomasBrown & Thomas, 1977; Reference Rajagopalan and AntoniaRajagopalan & Antonia, 1980). The discovery that LSMs/VLSMs modulate the amplitudes of small-scale motions not only contributed to a better understanding of the mechanism responsible for the production of near-wall turbulence (i.e. this process is not completely ‘autonomous’ when referring to the near-wall cycle (Reference Jiménez and PinelliJiménez & Pinelli 1999; Reference Schoppa and HussainSchoppa & Hussain, 2002)) but also provided an effective way to predict the behaviour of near-wall turbulent motions (Reference Marusic and HutchinsMarusic & Hutchins, 2011; Reference Marusic, Mathis and HutchinsMarusic, Mathis, et al., 2010; Reference Marusic, Monty, Hultmark and SmitsMarusic et al., 2013). That makes sense because the scales of the two kinds of structures are different enough for the near-wall elementary structures to spend most of their ‘lives’ in an environment defined by a single large-scale modulation (Reference JiménezJiménez, 2013). In addition, the modulation effect of LSMs/VLSMs leads to a weakened control effectiveness on near-wall coherent structures (Reference Deng, Huang and XuDeng, Huang, & Xu, 2016), and thus, manipulating the LSMs/VLSMs may provide an effective approach for controlling the turbulence (Reference Hwang and SungHwang & Sung, 2017). Therefore, amplitude modulation has recently become a research hotspot and has attracted the attention of many researchers (Reference Jacobi and MckeonJacobi & Mckeon, 2013; Reference Luhar, Sharma and McKeonLuhar, Sharma, & McKeon, 2014; Reference Talluru, Baidya, Hutchins and MarusicTalluru, Baidya, Hutchins, & Marusic, 2014). In summary, studies on amplitude modulation may be divided into three stages: the early discovery of this phenomenon, quantitative research, and recent applications and promotion.

The phenomenon of amplitude modulation is common in different types of flows. Initially, Reference Brown and ThomasBrown and Thomas (Reference Brown and Thomas1977) compared low-frequency fluctuations with rectified high-frequency smoothed signals in the TBL at $Re_{\tau } = 3413$ and found a high-frequency large-amplitude fluctuation occurring near the maximum in the slowly varying part of the signal, which indicates that the low-frequency fluctuation amplitude is associated with high-frequency fluctuation. In numerous different types of shear flows (including boundary layers, mixing layers, wakes and jets), Reference Bandyopadhyay and HussainBandyopadhyay and Hussain (Reference Bandyopadhyay and Hussain1984) also found significant coupling between scales in all flows by decomposing the streamwise velocity fluctuation and examining the correlations between the low-frequency and the high-frequency components. Recently, amplitude modulation was highlighted by Reference Hutchins and MarusicHutchins and Marusic (Reference Hutchins and Marusic2007b), who provided a detailed description of the amplitude modulation process; that is, the amplitudes at small scales were larger within large-scale positive fluctuations, whereas the amplitudes at small scales became relatively quiescent within large-scale negative fluctuations.

The amplitude modulation coefficient, as a mathematical tool, was originally proposed by Reference Mathis, Hutchins and MarusicMathis et al. (Reference Mathis, Hutchins and Marusic2009). The amplitude modulation coefficient can quantify the degree of the amplitude modulation effect of LSMs/VLSMs on small-scale motions. The corresponding flow chart of the complete decoupling procedure is illustrated in Figure 13. Reference Mathis, Hutchins and MarusicMathis et al. (Reference Mathis, Hutchins and Marusic2009) compared the results of one- and two-point analyses and suggested that the single-point amplitude modulation coefficient provides a reasonable estimate of the degree of modulation. Based on this quantitative method, Reference Mathis, Hutchins and MarusicMathis et al. (Reference Mathis, Hutchins and Marusic2009) and Reference Mathis, Monty, Hutchins and MarusicMathis, Monty, Hutchins, and Marusic (Reference Mathis, Monty, Hutchins and Marusic2009) examined HRNBLWT ($Re_{\tau } = 2800\text {--}19\,000$), pipe ($Re_{\tau } = 3015$), channel ($Re_{\tau } = 3005$) and ASL ($Re_{\tau } = 6.5 \times 10^{5}$) experimental data and found that the amplitude modulation coefficient exhibits almost no difference between internal and external wall-bounded flows but exhibits strong dependence on the Reynolds number and wall-normal distance. Furthermore, the amplitude modulation coefficient increased in a log-linear fashion with the Reynolds number in the buffer layer ($20< z^{+}<100$), but a more significant influencing factor (i.e. the wall-normal distance) has not yet been considered.

Figure 13. Flow chart for the calculation of the amplitude modulation coefficient, taken from (Reference Mathis, Hutchins and MarusicMathis et al., 2009).

The application of amplitude modulation in the prediction model of near-wall turbulent motion further highlights the importance of the amplitude modulation effect. Based on studies of the amplitude modulation coefficient, Reference Marusic, Mathis and HutchinsMarusic, Mathis, et al. (Reference Marusic, Mathis and Hutchins2010) and Reference Mathis, Hutchins and MarusicMathis, Hutchins, and Marusic (Reference Mathis, Hutchins and Marusic2011) proposed a time series predictive model for the near-wall fluctuating streamwise velocity; the equation is shown in Figure 14. Subsequently, Reference Marusic and HutchinsMarusic and Hutchins (Reference Marusic and Hutchins2011), Reference Inoue, Mathis, Marusic and PullinInoue, Mathis, Marusic, and Pullin (Reference Inoue, Mathis, Marusic and Pullin2012) and Reference Mathis, Marusic, Chernyshenko and HutchinsMathis, Marusic, Chernyshenko, and Hutchins (Reference Mathis, Marusic, Chernyshenko and Hutchins2013) presented a fluctuating wall-shear stress predictive model. With these predictive models, the near-wall streamwise velocity and the wall-shear stress fluctuations can be predicted using only the large-scale velocity information measured in the logarithmic region. Recently, (Reference Howland and YangHowland and Yang, 2018) made theoretical predictions on the model parameters within the wall-shear stress model and then applied them to wall-modelled LES by relating the amplitude modulation process to the Townsend attached eddy hypothesis. These models are important given that it is difficult to accurately measure and simulate turbulent motion close to a solid boundary. The proposal of these near-wall models further highlights the importance of amplitude modulation; therefore, recent research on amplitude modulation still attracts the attention of many researchers.

Figure 14. Mathematical formulation of the predictive model for the near-wall fluctuating streamwise velocity, taken from Reference Mathis, Hutchins and MarusicMathis et al. (Reference Mathis, Hutchins and Marusic2011).

Reference Talluru, Baidya, Hutchins and MarusicTalluru et al. (Reference Talluru, Baidya, Hutchins and Marusic2014) calculated the amplitude modulation coefficient of large-scale streamwise velocity fluctuations on all three components of the small-scale velocity with measurements from cross-wire probes in the TBL at $Re_{\tau } = 15\,000$ and found that the modulation of the small-scale energy by large-scale structures is relatively uniform across all three velocity components. Reference Luhar, Sharma and McKeonLuhar et al. (Reference Luhar, Sharma and McKeon2014) and Reference Tsuji, Marusic and JohanssonTsuji, Marusic, and Johansson (Reference Tsuji, Marusic and Johansson2016) investigated the amplitude modulation of pressure fluctuations and found a relatively small modulation effect between large- and small-scale pressure fluctuations. Reference Squire, Baars, Hutchins and MarusicSquire, Baars, Hutchins, and Marusic (Reference Squire, Baars, Hutchins and Marusic2016) and Reference Pathikonda and ChristensenPathikonda and Christensen (Reference Pathikonda and Christensen2017) explored the inner–outer interactions in rough-wall TBL flows and suggested that a rough wall increases the amplitude modulation coefficient Reference Nadeem, Lee, Lee and SungNadeem, Lee, Lee, and Sung (Reference Nadeem, Lee, Lee and Sung2015) and Reference AndersonAnderson (Reference Anderson2016) performed numerical simulations with similar results). Recently, Reference Yao, Huang and XuYao, Huang, and Xu (Reference Yao, Huang and Xu2018) investigated the amplitude modulation effect by the DNS of turbulent channel flows and indicated that extraordinarily high fluctuation events are provoked by the modulation effect. Reference Salesky and AndersonSalesky and Anderson (Reference Salesky and Anderson2018) presented evidence of amplitude modulation phenomena in the unstably stratified (i.e. convective) ASL by conducting an LES; they suggested that the modulation effect of the large-scale streamwise velocity decreases monotonically, while the modulating influence of the large-scale vertical velocity remains significant because the spatial attributes of the flow structures change from streamwise to vertically dominated. Reference Wang, Pan and WangWang, Pan, and Wang (Reference Wang, Pan and Wang2020) examined the characteristics of the streamwise component of the fluctuating wall-shear stress in a smooth-wall TBL and inferred that the LSMs/VLSMs in the outer region mainly apply an amplitude modulation effect instead of a footprint effect on the wall-shear stress. However, none of the abovementioned studies on amplitude modulation involved particle-laden two-phase flows, and the decoupling procedure employed a nominal cut-off wavelength (usually taken as $\delta$) to divide the fluctuating velocity into large- and small-scale components. Consequently, very little is known about the impacts of particles on the amplitude modulation effect and which scales of motion dominate the amplitude modulation.

Regarding the modulation effect of VLSMs, the author's team performed a detailed analysis of the sand-free and sand-laden flows of the ASL and obtained some novel results. Using experimental data with different sand concentrations in the near-neutral ASL obtained from long-term observations at the QLOA, Reference Liu and ZhengLiu and Zheng (Reference Liu and Zheng2020) found that VLSMs also exhibit an amplitude modulation effect on the small-scale motions in sand-laden flows, but the amplitude modulation coefficient in a sand-laden flow is significantly reduced compared with that in a sand-free flow, as shown in Figure 15. This may be because the addition of sand particles into a two-phase flow does not change the scales of the most energetic motions but does change the distribution of energy between turbulent motions at different scales. In addition, Reference Liu, Wang and ZhengLiu, Wang, and Zheng (Reference Liu, Wang and Zheng2019b) proposed that not all of the length scales of turbulent motions have an amplitude modulation effect among them; rather, several specific motions have an effect: the most energetic motions with scales larger than the wavelength of the energy spectral peak in the lower-wavenumber region contribute significantly to the amplitude modulation effect, while the motions with small scales (shorter than the wavelength of the higher-wavenumber peak) are strongly modulated.

Figure 15. Colour contour maps showing the variations in the amplitude modulation coefficient with the length scales of the large-scale ($\lambda _{x}>a\delta$) and small-scale ($\lambda _{x}< b\delta$) components compared with the premultiplied energy spectra of the streamwise velocity fluctuations in a (a) sand-laden flow and (b) sand-free flow, taken from Reference Liu, Wang and ZhengLiu et al. (Reference Liu, Wang and Zheng2019b).

Reference Marusic, Mathis and HutchinsMarusic, Mathis, et al. (Reference Marusic, Mathis and Hutchins2010) proposed the Marusic–Mathis–Hutchins (MMH) near-wall wind velocity prediction model, the basic idea of which is to superpose and modulate a large-scale structure in the outer region onto a small-scale structure near the wall to predict the fluctuating wind velocity signal near the wall, which is typically difficult to measure. Therefore, this method can predict the streamwise velocity time series only below the input signal height (usually taking the centre height of the logarithmic region $z^{+} = 3.9Re_{\tau }^{1/2}$), which is in the outer region. In the wind tunnel and ASL with boundary layer thicknesses of 0.3 and 150 m, respectively, these heights correspond to 0.02 and 0.3 m, respectively. However, in practical applications, it is often necessary to know the wind velocity fluctuation and the sand/dust transport at 2 m or even tens of metres above the input signal height. Moreover, the determination of both the MMH model parameters and the universal signal depends on high-precision calibration experiments, and the calibration process requires synchronous measurements at the measuring point in the centre of the logarithmic region and at different heights below. A hot-wire probe with high spatial and temporal resolutions is used for the wind speed measurement to ensure that the minimum viscous-scale vortices can be resolved and are close to the wall surface to ensure that the near-wall measurement signals can be obtained. However, owing to the complexity of the ASL surface and the limitations of experimental means, it is very difficult to carry out high-precision calibration experiments in the field. In addition, owing to the constant adjustment of modulation-related parameters in the process of determining the universal signal, the modulation coefficient of the large-scale fluctuation in the outer region to the near-wall standard signal is zero, so this process cannot be solved directly and the iterative adjustment process is tedious.

To describe the fluctuation characteristics of the ASL velocity field, Reference Han, Liu, Bo and ZhengHan, Liu, Bo, and Zheng (Reference Han, Liu, Bo and Zheng2019) extended the MMH model to the ASL clear-air field to obtain a modified prediction model based on the measured time series of the streamwise velocity field at a certain height $z_{0}$ (called the reference height); the purpose of this model was to predict the fluctuating streamwise velocity at any height. The modified model considers the effect of the high-Reynolds-number ASL on LSMs/VLSMs and its further effect on small-scale fluctuations. The model provided in Reference Han, Liu, Bo and ZhengHan, Liu, et al. (Reference Han, Liu, Bo and Zheng2019) is written as

(2)\begin{equation} u^{ + }(z) = \overline{u}^{ + }(z) + u_{s * }^{ + }(z)(1 + R(z) k(z) u_{ML}^{ + }(z_{M},\gamma_{L})) + k(z) u_{ML}^{ + }(z_{M},\gamma_{L}),\end{equation}

where $u^{+}(z)$ is the predicted streamwise velocity, $\overline {u}^{+}(z)$ is the mean velocity given by the logarithmic law, $u_{s*}^{+}(z)$ is the small-scale turbulent signal generated by the harmony superposition method, $R(z)$ is the amplitude modulation coefficient, $k(z)$ is related to the energy ratio between the large- and small-scale turbulent signals, $u_{ML}^{+}(z_{M},\gamma _{L})$ is the large-scale turbulent signal shifted forward in the streamwise direction according to the large-scale structure inclination angle $\gamma _{L}$, and $z_{M}$ represents the centre of the logarithmic region. The predicted results of the model are in good agreement with the results of field observations, as shown in Figure 16. Because the model does not need complex calibration experiments and the model parameters are universal, the fluctuating velocity prediction model is easy to apply.

Figure 16. Comparison between the model prediction results and measured results at (a) 0.9 m; (b) 1.71 m; (c) 7.15 m; and (d) 30 m, taken from Reference Han, Liu, Bo and ZhengHan, Liu, et al. (Reference Han, Liu, Bo and Zheng2019).

5. Electric Field Measurements in Dust Storms

Particle charging is a ubiquitous phenomenon in industry (e.g. fluidized beds (Reference Sippola, Kolehmainen, Ozel, Liu, Saarenrinne and SundaresanSippola et al. 2018)), nature (e.g. volcanic eruptions and dust storms (Reference Mather and HarrisonMather & Harrison, 2006; Reference Zhang and ZhouZhang & Zhou, 2020b)), and even interplanetary and interstellar space (e.g. protoplanetary disks and dusty plasmas (Reference Blum and WurmBlum & Wurm Reference Blum and Wurm2008)). Dust storms are natural laboratories for the study of particle charging in particle-laden turbulent flows, with friction Reynolds numbers reaching as high as $Re_{\tau } \sim O(10^{6})$, and dust storms containing polydisperse sand and dust particles ranging in size from $O(10^{-6})$ to $O(10^{-3})$ m. The ambient electric field density in dust storms reaches ${\sim }10\text {--}100$ kV m$^{-1}$, which is 2–3 orders of magnitude higher than that of the fair-weather atmospheric electric field (e.g. Reference HarrisHarris, 1967; Reference KamraKamra, 1972; Reference RudgeRudge, 1913; Reference StowStow, 1969; Reference Zhang and ZhouZhang & Zhou, 2020b). Numerous studies have shown that the electrostatic forces to which particles are subjected are comparable to their gravitational force, which has considerable influences on the particle lifting process, preferential particle concentration, and particle transport (e.g. Reference Kok and RennoKok & Renno, 2006; Reference Renzo and UrzayRenzo & Urzay, 2018; Reference Schmidt, Schmidt and DentSchmidt, Schmidt, & Dent, 1998; Reference Zheng, Huang and ZhouZheng, Huang, & Zhou, 2003). For example, Reference Kok and RennoKok and Renno (Reference Kok and Renno2006) showed that electrostatic forces between particles can significantly reduce the critical wind speed for particle lifting and even directly lift particles away from the wall. Reference Renzo and UrzayRenzo and Urzay (Reference Renzo and Urzay2018) conducted numerical simulations and discovered that electrostatic forces can remarkably affect the preferential concentration of particles in turbulent flows. Reference Zheng, Huang and ZhouZheng et al. (Reference Zheng, Huang and Zhou2003) showed that in wind-blown sand flows, the electrostatic force increases the dust mass flux when the dust particles are negatively charged, while the electrostatic force decreases the dust mass flux when the dust particles are positively charged. In addition, the theoretical model of Reference Zhou, He and ZhengZhou, He, and Zheng (Reference Zhou, He and Zheng2005) shows that the transmission of electromagnetic waves in dust storms is significantly affected by the amount of electrical charge on the dust particles.

The electrostatic force exerted on a dust particle is equal to the product of the particles’ charge-to-mass ratio and the electric field strength at its spatial location; hence, the particles’ charge-to-mass ratio and the spatial structure of the electric field are two important physical quantities for quantifying the particle charging phenomena in dust storms. Currently, two methods, namely, the trajectory deflection method (e.g. Reference Miura, Koyaguchi and TanakaMiura, Koyaguchi, & Tanaka, 2002) and the Faraday cage method (e.g. Reference Schmidt, Schmidt and DentSchmidt et al., 1998), are employed to measure the charge-to-mass ratio of particles. The former method indirectly determines the particles’ charge-to-mass ratio by measuring the horizontal deflection of the trajectory of the charged particles under an applied horizontal electric field, and the magnitude of this horizontal deflection is proportional to the particles’ charge-to-mass ratio; however, as the accuracy of this method depends on the precise measurement of particle trajectories (usually using PIV, e.g. Reference Waitukaitis, Lee, Pierson, Forman and JaegerWaitukaitis, Lee, Pierson, Forman, & Jaeger, 2014), it cannot be used for dust storm measurements. The latter method determines the average charge-to-mass ratio of particles during a period by measuring the total charge and mass of the particles collected in a Faraday cage in real time. This method has been widely used to measure the charge-to-mass ratios of sand and dust particles (e.g. Reference Bo, Zhang and ZhengBo, Zhang, & Zheng, 2014; Reference Schmidt, Schmidt and DentSchmidt et al., 1998) within the saltation layer (i.e. within a height of $\sim$10 cm above the surface). However, it is difficult to apply this method at higher altitudes, where the particle concentration is small and the wind shear stress is strong enough that particles can hardly be collected in a Faraday cage (e.g. Reference Harrison, Barth, Esposito, Merrison, Montmessin, Aplin and FarrellHarrison et al., 2016). Therefore, the electrical properties of suspended dust particles in dust storms are currently unclear, and the particles’ charge-to-mass ratio is usually assumed to be a constant of ${\sim }\pm 60\ \mathrm {\mu }$C kg$^{-1}$ in many numerical models (e.g. Reference Schmidt, Schmidt and DentSchmidt et al., 1998; Reference Zhang and ZhouZhang & Zhou, 2020a; Reference Zheng, Huang and ZhouZheng et al., 2003); this magnitude is consistent with the measurements of (Reference Schmidt, Schmidt and DentSchmidt et al., 1998). Although Reference Williams, Nathou, Hicks, Pontikis and BartholomewWilliams, Nathou, Hicks, Pontikis, and Bartholomew (Reference Williams, Nathou, Hicks, Pontikis and Bartholomew2009) inferred that the charge structure of dust storms may be unipolar or bipolar (if particle charging occurs primarily at the ground or in midair, respectively), they neglected the effects of turbulence, which plays an important role in particle charging (e.g. Reference Grosshans and PapalexandrisGrosshans & Papalexandris, 2017; Reference Renzo and UrzayRenzo & Urzay, 2018). Consequently, they did not obtain detailed information about the electrical properties of the suspended particles.

On the other hand, although Reference RudgeRudge (Reference Rudge1913) conducted observations of atmospheric electric field changes during dust storms as early as 1913, the spatial structure of the aeolian electric field during a dust storm remains puzzling owing to its complexity (Reference Harrison, Barth, Esposito, Merrison, Montmessin, Aplin and FarrellHarrison et al., 2016; Reference ZhengZheng, 2013). As shown in Figure 17, the existing observations of the aeolian electric field are concerned mainly with the vertical component, and these measurements were performed at a single point ${\sim }1\text {--}2$ m above the ground (Reference EspositoEsposito, 2016; Reference HarrisHarris, 1967; Reference KamraKamra, 1972; Reference RudgeRudge, 1913; Reference StowStow, 1969; Reference Yair, Katz, Yaniv, Ziv and PriceYair, Katz, Yaniv, Ziv, & Price, 2016); thus, the spatial structure of the aeolian electric field was not retrieved. In particular, the direction of the vertical component of the aeolian electric field is still under debate. That is, measurements have uncovered upward-oriented (e.g. Reference HarrisHarris, 1967; Reference RudgeRudge, 1913; Reference StowStow, 1969), downward-oriented (e.g. Reference EspositoEsposito, 2016), and even alternatively oriented vertical electric fields (Reference KamraKamra, 1972; Reference Williams, Nathou, Hicks, Pontikis and BartholomewWilliams et al., 2009).

Figure 17. Field observations of the aeolian electric field in dust storms by Reference Esposito, Molinaro, Popa, Molfese, Cozzolino, Marty, Taj-Eddine, Di Achille, Franzese, Silvestro and OriEsposito et al. (Reference Esposito, Molinaro, Popa, Molfese, Cozzolino, Marty, Taj-Eddine, Di Achille, Franzese, Silvestro and Ori2016). (a)Instrument stations deployed in the field observations, which were conducted in the Merzouga Desert, Morocco in 2014. The inset shows the installation of the instruments in 2013. (b) Time series of the vertical electric field at a height of 2 m measured on 16 July 2013. The positive direction of the vertical electric field is oriented upward. This figure was adapted from Reference Esposito, Molinaro, Popa, Molfese, Cozzolino, Marty, Taj-Eddine, Di Achille, Franzese, Silvestro and OriEsposito et al. (Reference Esposito, Molinaro, Popa, Molfese, Cozzolino, Marty, Taj-Eddine, Di Achille, Franzese, Silvestro and Ori2016).

To obtain the electrical properties of the suspended particles at high altitudes and image the fine structure of the electric field in dust storms, a set of real-time, simultaneous, multipoint observations of the wind velocity, PM10 dust concentration and electric field were performed at the QLOA (Reference Zhang, Bo and ZhengZhang, Bo, & Zheng, 2017; Reference Zhang and ZhouZhang & Zhou, 2020b). First, as shown in Figure 18, the space charge density and dust mass concentration 5 m above the ground are weakly correlated at short time scales (i.e. $<$1 min), while they show a highly linear correlation ($\text {wavelet coherence} \geq 0.8$) at long time scales (exceeding 10 min) at different stages of the dust storm. Because the ratio of the space charge density to the dust mass concentration characterizes the average charge-to-mass ratio of the suspended particles, a strong correlation at a time scale exceeding 10 min indicates that the average charge-to-mass ratio does not change with the dust concentration over this scale. Such a strong correlation is usually termed the large-scale electrification equilibrium phenomenon of dust particles in dust storms (e.g. Reference Harper and DufekHarper & Dufek, 2016; Reference Zhang and ZhengZhang & Zheng, 2018). Reference Zhang and ZhengZhang and Zheng (Reference Zhang and Zheng2018) showed that for a given ambient temperature and relative humidity, there is a significant fluctuation in the average charge-to-mass ratio below the 10 min scale, while the average charge-to-mass ratio is considered to remain unchanged above the 10 min scale.

Figure 18. Wavelet coherence between the space charge density (estimated from the divergence of the three-dimensional electric field measurements) and the PM10 dust mass concentration at a measurement point 5 m above the ground. Panels (a–c) correspond to dust storms observed on 17 April, 20 April and 22 April 2017, respectively. The horizontal dashed line corresponds to a time scale of 10 min and the white solid line is the cone of influence for wavelet analysis. This figure was adapted from Reference Zhang and ZhengZhang and Zheng (Reference Zhang and Zheng2018).

Furthermore, Reference Zhang and ZhouZhang and Zhou (Reference Zhang and Zhou2020b) employed Tikhonov regularization and proposed a method to reconstruct the spatial structures of the space charge density and electric field in dust storms based on locally measured electric field data in the ASL, thereby providing a method for studying the electrical structure of dust storms. The reconstructed results in Figure 19a indicate that the space charge density in dust storms exhibits a universal mosaic pattern (i.e. a mosaic of positively and negatively charged regions), which implies that the charge-to-mass ratio of dust particles varies spatially. Surprisingly, this mosaic charge structure is quite different from the unipolar or bipolar charge structures inferred from previous work (Reference Williams, Nathou, Hicks, Pontikis and BartholomewWilliams et al., 2009). This is because particle charging is generally size-dependent; i.e. large particles tend to be positively charged, whereas small particles tend to be negatively charged (Reference Lacks and SankaranLacks & Sankaran, 2011). In dust storms, negatively charged small particles with an $S_{t}$ (Stokes) number less than ${\sim }O(1)$ accumulate in the low-speed region owing to turbulence, while particles with an $S_{t}$ number much larger than ${\sim }O(1)$ are distributed more uniformly owing to their large inertia (Reference Lacks and SankaranLacks & Sankaran, 2011). It is likely that these different responses of charged particles to turbulence are the main physical mechanism that generates the mosaic charge structure of dust storms (e.g. Reference Cimarelli, Alatorre-lbarguengoitia, Kueppers, Scheu and DingwellCimarelli, Alatorre-lbarguengoitia, Kueppers, Scheu, & Dingwell, 2014; Reference Renzo and UrzayRenzo & Urzay, 2018). Moreover, as shown in Figure 19b, the electric field of the dust storms is three-dimensional at different locations. Therefore, field measurements should focus on the three-dimensional electric field and not only on the vertical component. Note that the measurement points in the streamwise direction play an important role in the reconstruction model. As shown in Figure 19c, the relative error between the subsampling and complete sample reconstructions reaches 40 % when the streamwise measurement points (i.e. streamwise towers 5 m above the ground) have been removed. This means that such subsampling reconstructions are unreliable. In fact, because the space charge density in a dust storm is not uniformly distributed in the streamwise direction, a reconstruction with insufficient streamwise data cannot fully reproduce the real structures.

Figure 19. Reconstruction of the electrical structure of a dust storm, where the electric field data are from observations collected on 16 April 2017 at 13:20:00 (UTC+8). (a) Reconstruction of the three-dimensional structure of the space charge density $\rho _{inv}$ on 16 April 2017 at 13:20:00 (UTC+8), where the isosurfaces are shown at a space charge density magnitude of 0.02 $\mu$C $m^{-3}$. The positive and negative surfaces are coloured red and blue, respectively. Contour slices at $x = 5$ m are coloured based on the space charge densities. (b) Reconstruction of the three-dimensional structure of the electric field on 16 April 2017 at 13:20:00 (UTC+8), where the lines are electric field lines and the contour clouds are stained with the logarithm of the electric field size. Slices at $x = 0$, $y = 0$ and $z = 4$ m are coloured based on the log magnitude of the three-dimensional electric field. (c) Relative error between the $\rho _{inv}$ obtained from the reconstructions with subsampling and complete sampling, where the subsampling size $m$ ranges from 5 to 18 for the three different data sets of dust storms $1\text {--}3$. Dust storms $1$, $2$ and $3$ occurred on 16, 17, and 20 April 2017 (UTC+8), respectively. Error bars represent the standard deviation of the relative error over 10 random subsampling reconstructions. This figure was adapted from Reference Zhang and ZhouZhang and Zhou (Reference Zhang and Zhou2020b).