4.1. Streamwise length scale

The streamwise length scales are obtained from the one-dimensional correlations of the streamwise velocity fluctuations $u$ and PM10 concentration fluctuations $c$ for the constant spanwise and wall-normal positions, i.e.

(4.1) \begin{eqnarray} R_{u u}(\triangle x) & = & \frac {\langle u(x, y, z) u(x+\triangle x, y, z)\rangle }{\sigma _{u(x, y, z)} \sigma _{u(x+\triangle x, y, z)}} \end{eqnarray}

and

(4.2) \begin{eqnarray} R_{c c}(\triangle x) & = & \frac {\langle c(x, y, z) c(x+\triangle x, y, z)\rangle }{\sigma _{c(x, y, z)} \sigma _{c(x+\triangle x, y, z)}} \end{eqnarray}

where $\triangle x = U_c \triangle \tau$ , $U_c$ is the convection velocity and $\triangle \tau$ is the time lead and lag to satisfy the correlation. To extract the spatial scale from two-point correlation, a nominal threshold is necessary. A small positive threshold value not only effectively avoids the influence of background noise (Ganapathisubramani et al. Reference Ganapathisubramani, Longmire and Marusic2006; Dennis & Nickels Reference Dennis and Nickels2011), but also can maintain sufficiently accurate correlations to capture the spatial features of large-scale structures (Krogstad & Antonia Reference Krogstad and Antonia1994). Unfortunately, there are no standardized criteria for threshold selection. Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999) suggested that the exact choice of threshold only affects the quantitative values but not the general vortex shape or the variation in the characteristic length. Therefore, a compromise threshold value of $R_{uu} = 0.2$ is selected in this study, which does not affect the results and conclusions.

Figure 7.

Spatial and temporal variations in the streamwise length scales of (a) the turbulence structures $L_{ux}$ and (b) dust concentration structures $L_{cx}$ during the sandstorm, where the red and green solid lines show the trends with time and wall-normal distance, respectively. (c) Evolution of $L_{ux}$ and $L_{cx}$ with time at a height $z/\delta=\, 0.03$ , where the blue and yellow solid lines indicate the results of the streamwise velocity and dust concentration, respectively. (d–f) Comparison of the variations in $L_{ux}$ and $L_{cx}$ with wall-normal distance, where the light red, light green and cyan background represent the rising stage, steady stage and declining stage; the blue and yellow solid symbols denote $L_{ux}$ and $L_{cx}$ , respectively. (e) Average $L_{ux}$ and $L_{cx}$ in the steady stage, where hollow symbols are the existing laboratory and ASL results of $L_{ux}$ (from Hutchins & Marusic Reference Hutchins and Marusic2007; Volino et al. Reference Volino, Schultz and Flack2007; Tutkun et al. Reference Tutkun, George, Delville, Stanislas, Johansson, Foucaut and Coudert2009; Lee & Sung Reference Lee and Sung2011; Hutchins et al. Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012; Liu et al. Reference Liu, Wang and Zheng2017b ).

Figure 8.

Same as figure 7 but for the fractional contribution of VLSMs to the total fluctuation intensity, where the black hollow star is the energy contribution of turbulence VLSMs given by Balakumar & Adrian (Reference Balakumar and Adrian2007), and the hollow cycle is that of sand-laden flow at a similar Reynolds number in Wang et al. (Reference Wang, Gu and Zheng2020).

The spatial and temporal variations in the streamwise length scale of the turbulence/dust concentration structures (denoted as $L_{ux}$ and $L_{cx}$ , respectively) during the sandstorm obtained by two-point correlation are shown in figure 7, where the qualitative evolution trends with time and height are given by red and green arrows, respectively. To compare the differences in the temporal evolution of $L_{ux}$ and $L_{cx}$ , the results at height $z/\delta=\, 0.03$ are plotted in figure 7(c). In the rising stage, $L_{ux}$ (blue line) are large and slightly increase (from 1.5 $\delta$ to 3.7 $\delta$ ) with initial increasing wind velocity, and then, with sandstorm development and the continuous increase in the wind velocity, $L_{ux}$ gradually decrease. After full development, $L_{ux}$ remain substantially invariant (approximately 1.6 $\delta$ ) in the steady stage, and the present results agree with the existing laboratory and ASL results within the experimental error range (as shown in figure 7 e). When the sandstorm attenuates during the declining stage, $L_{ux}$ decrease again. Namely, large-scale turbulence structures exist at the beginning of the sandstorm, but gradually decrease with sandstorm development in the rising stage, remain invariant during the steady stage and finally decrease with the attenuation of the sandstorm. In contrast, $L_{cx}$ (yellow line) evolve from zero at the beginning of the sandstorm to gradually increasing with sandstorm development, reaching a plateau (approximately 4.3 $\delta$ ) in the steady stage, and then decreasing with a decrease in wind velocity until it decreases to zero at the end of the sandstorm. The differences in temporal evolution between two-phase structures are mainly in the rising stage, where dust concentration structures emerge gradually with sandstorm development, while turbulent large-scale structures are present from the beginning. Hence, the variations in $L_{ux}$ and $L_{cx}$ with wall-normal distance at different stage are given in figure 7(d–f). Throughout the entire sandstorm, the outer-scaled streamwise length scale $L_{ux} /\delta$ is pronounced near the ground, and $L_{ux} /\delta$ appears to level off as the wake region is approached. Conversely, the variation trend of $L_{cx} /\delta$ is the mirror image of $L_{ux} /\delta$ and the difference with $L_{ux} /\delta$ decrease with height, as shown in figure 7(d–f). Moreover, $L_{cx} /\delta$ are less than $L_{ux}$ at the beginning of the sandstorm, but as the sandstorm develops, $L_{cx}$ gradually increase and exceed $L_{ux}$ near the ground, as shown in figure 7(d). When the sandstorm reaches the steady stage with relatively constant wind velocity and dust concentration, $L_{cx}$ at lower positions are greater than $L_{ux}$ , the difference between $L_{ux}$ and $L_{cx}$ decreases with height, and the two are close at higher positions, but $L_{ux}$ are slightly larger than $L_{cx}$ , as shown in figure 7(e). Note that later in the steady stage, $L_{ux}$ start to decrease before $L_{ux}$ . In the declining stage, $L_{cx}$ gradually decay to less than $L_{ux}$ at all heights. Qualitative differences between the variations in two-phase structure scales along height suggest there are differences in their formation mechanisms. In addition, at the beginning and end of the sandstorm, $L_{cx}$ equal zero at higher positions, indicating that the dust concentration structures originate from the ground and disappear at the positions far from the ground, as shown in figure 7(d,f). The streamwise time length scales obtained from the time autocorrelation are used to verify the streamwise spatial length scales obtained from the Taylor frozen hypothesis. The results show that the spatial and temporal evolution of the streamwise time length scales of turbulence/dust concentration structures is essentially the same as that of the streamwise spatial length scales, except that the streamwise time length scales of the velocity structures show an increasing tendency at the end of sandstorms that is opposite to the streamwise spatial length scales. The increase in the streamwise time length scales of velocity structures in the declining is a result of the fact the vortex structures take longer to pass through the observed position due to the lower wind velocity. Furthermore, to analyse the fractional contribution of the very large-scale structures to the total fluctuation intensity of the streamwise velocity and dust concentration fluctuations during the sandstorm, the spatial and temporal evolution of the fractional contribution are also shown in figure 8. The evolutions of fractional contribution are qualitatively consistent with that at the streamwise length scale.

Changes in turbulence driving factors behind the sandstorms can provide a reasonable explanation for the evolutions of two-phase structures. With the intrusion of cold fronts, cold air masses encounter warmer local air, which causes convection, implying the beginning of sandstorm (Dragani Reference Dragani1999; Helfer & Nuijens Reference Helfer and Nuijens2021). The strong quadratic interaction causes synoptic-scale fluctuations to break into large-scale turbulence structures, being larger than the structures in the steady stage and moving downwards (Liu et al. Reference Liu, Shi and Zheng2022), referred to as the top-down mechanism, resulting in a gradual increase in $L_{ux}$ with height, as shown in figure 7(d). Furthermore, at the initial rising stage, the wind velocity increases due to the energy carried by the cold air but the shear is small due to stronger atmospheric instability, resulting in the stretching of structures along the streamwise direction, namely, a short increase in $L_{ux}$ at the initial rising stage, as shown in figure 7(c). With sandstorm development, the wind velocity increases and shearing breaking is enhanced, causing large-scale turbulence structures to be broken into smaller ones. Additionally, with enhanced wall shear, the hairpin vortices created by wall bursts are aligned coherently along the streamwise direction, forming larger-scale coherent structures, referred to as the bottom-up mechanism (Adrian et al. Reference Adrian, Meinhart and Tomkins2000). This can also lead to increase in $L_{ux}$ and fractional contribution of VLSMs with height. When these two mechanisms are combined, $L_{ux}$ and fractional contribution of VLSMs are significant. The maximum contribution can reach up to 75 % at approximately the top of the logarithmic region in the rising stage, as shown in figure 8(d), which is much larger than results reported in previous document (the maximum at the same location is 60 %(Wang & Zheng Reference Wang and Zheng2016)).

When the cold air completely occupies the observation area, the flow is dominated by shear turbulence (Liu et al. Reference Liu, Shi and Zheng2022). The large-scale turbulence structures mainly originate from the self-organization process of the hairpin vortices (Balakumar & Adrian Reference Balakumar and Adrian2007; Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015). The streamwise length scale and the fractional contributions of the large-scale turbulence structures increase with height and present results in the steady stage are consistent with existing laboratory and ASL results (Balakumar & Adrian Reference Balakumar and Adrian2007; Hutchins & Marusic Reference Hutchins and Marusic2007; Volino et al. Reference Volino, Schultz and Flack2007; Tutkun et al. Reference Tutkun, George, Delville, Stanislas, Johansson, Foucaut and Coudert2009; Lee & Sung Reference Lee and Sung2011; Hutchins et al. Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012; Liu et al. Reference Liu, Wang and Zheng2017 b; Wang et al. Reference Wang, Gu and Zheng2020), as shown in figures 7(e) and 8(e), respectively. In the declining stage, with the end of the cold front transit event, the exhausted energy brought by the cold air mass leads to a decreasing wind velocity, making it challenging to sustain large-scale structures. Therefore, the large-scale turbulence structures gradually dissipate and energy gradually concentrates towards small-scale motions with the attenuation of the sandstorm.

The large-scale turbulence structures significantly contribute to the transport of dust particles and influence the distribution of dust particles in the flow field (Maxey Reference Maxey1987; Balachandar & Eaton Reference Balachandar and Eaton2010; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010). Consequently, the changes in the turbulence structures cause differences in the dust concentration structures at the different stages. At the beginning of the sandstorm, the wind velocity is low and the main driving factor is buoyancy-driven thermal turbulence (Liu et al. Reference Liu, Shi and Zheng2024), implying that dust particles cannot be released from the ground and carried into the air (Lim & Chun Reference Lim and Chun2006; Niu et al. Reference Niu, Qu, Zhang and Liu2012; Mockford et al. Reference Mockford, Bullard and Thorsteinsson2018). Additionally, the upstream dust particles have not yet reached the observation area. Therefore, there are no dust cluster structures, as shown in figure 7(b–d). With sandstorm development, the dust particles are released from sandy surfaces by enhanced shear turbulence and transported upward by ejections-Q2-type events (Marchioli & Soldati Reference Marchioli and Soldati2002; Zhang et al. Reference Zhang, Hu and Zheng2018). That is, dust concentration structures are organized from the ground surface, evolving towards higher atmospheric layer. The streamwise extent of high-concentration regions formed by dust clusters, i.e. $L_{cx}$ become larger with increasing wind velocity and dust concentration, as shown in figure 7(c). Moreover, as a number of large-scale dust concentration structures generate, their fractional contribution gradually increases, as shown in figure 8(c). When shear turbulence becomes dominant in the steady stage, sufficient dust particles form large-scale dust structures under the high-strain effect contributed by the turbulence structures, and these very large-scale dust concentration structures are dominant in dust concentration field and contribution to the total fluctuation intensity reaches 70 % near the ground, as shown in figures 7(e) and 8(e). This is different from the case of turbulence, where the energy contribution of VLSMs is strongest in the rising stage. In the declining stage, the atmosphere turns into a stable stratification, inhibiting the atmospheric turbulence and its diffusion (Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013; Liu et al. Reference Liu, Bo and Liang2017a ). The weakened shear is insufficient to lift enough dust particles from the ground and these dust particles cannot be transported to higher positions, thus their activity is limited to the near-ground region. As a result, the scales of dust structures gradually decrease with continuously decreasing wind velocity and large-scale dust structures cannot form away from the ground, as shown in figure 7(f).

Although dust particles exhibit a great capability to follow eddies, the observed decrease in $L_{cx}$ with height, in contrast to the behaviour of $L_{ux}$ , reveals a qualitative distinction between the two-phase structures. The interpretation could be that as the dust concentration decreases with height, the large-scale dust concentration structures cannot maintain coherence due to a lower concentration at a higher position and more small-scale structures are created, resulting in a decreasing scale with height. Additionally, larger dust particles near the ground have a larger time scale and filter out the effects of the smaller fluid scales (Marchioli & Soldati Reference Marchioli and Soldati2002), resulting in a lower correlation magnitude with velocity fluctuations than that of the dust concentration fluctuations (He & Liu Reference He and Liu2023). Moreover, the follow ability of particles increases with decreasing average particle size ( $d=\, 10$ $\unicode{x03BC} \mathrm{m}$ at $z/\delta=\, 0.006$ , while $d=\, 7$ $\unicode{x03BC}\mathrm{m}$ at $z/\delta=\,0.2$ ). Therefore, the difference between $L_{ux}$ and $L_{cx}$ decrease with height, as shown in figure 7(e). Additionally, the evolution of the fractional contribution of very large-scale fluctuations to the total fluctuation intensity in figure 8 provides further evidence supporting for the origin of two-phase coherent structures.

Figure 9.

Evolution of structure inclination angle of the turbulence structures ( $\gamma _u$ , blue line) and dust concentration structures ( $\gamma _c$ , yellow line). The light red, green and blue areas denote the ranges of turbulence structure inclination angle $\gamma _u$ in unstable, neutral and stable surface layers from Hommema & Adrian (Reference Hommema and Adrian2003); Morris et al. (Reference Morris, Stolpa, Slaboch and Klewicki2007); Marusic & Heuer (Reference Marusic and Heuer2007); Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012); Chauhan et al. (Reference Chauhan, Hutchins, Monty and Marusic2013); Liu et al. (Reference Liu, Bo and Liang2017a ) and He & Liu (Reference He and Liu2023). The light yellow area denotes the range of the dust concentration structure inclination angle $\gamma _c$ given by He & Liu (Reference He and Liu2023). The black solid line in the rising stage denotes the turbulence structure inclination angle $\gamma _u$ calculated by the empirical model $\gamma _{u}= \gamma _{0}+ 7.3\ln ( 1-70z/L )$ (where $\gamma _0$ is the inclination angle in the neutral regime) from Chauhan et al. (Reference Chauhan, Hutchins, Monty and Marusic2013). The yellow filled squares in the steady stage denote the dust concentration structure inclination angles $\gamma _c$ calculated by the relationship model ( $1/\tan \gamma _c = 1/\tan \gamma _u - 1$ ) given by He & Liu (Reference He and Liu2023).

4.2. Structure inclination angle

In the near-neutral flow, coherent structures have been associated with characteristic inclination angles because of the mean shear (Liu et al. Reference Liu, Bo and Liang2017 a). However, the buoyancy effect, addition of sand particles and flow acceleration inherent in the sandstorm process could influence the inclination angle. Consequently, the evolution of two-phase structure inclination angles (denoted as $\gamma _u$ and $\gamma _c$ , respectively) throughout the sandstorm process is shown in figure 9. In this study, the average structure inclination angle is obtained from the two-point correlations of the lowest signal with signals at higher wall-normal positions,

(4.3) \begin{eqnarray} \gamma =\arctan \left (\left \langle \triangle z / \triangle x^{*}\right \rangle \right )=\arctan \left (\left \langle \triangle z /\left (U_{c} \triangle t^{*}\right )\right \rangle \right ), \end{eqnarray}

where $\triangle x^{*}=U_c \triangle t^{*}$ and $\triangle t^{*}$ are the spatial and time delay of the peak locations of $R_{u u}$ or $R_{c c}$ obtained from (3.2) and (3.3).

It is seen that $\gamma _u$ gradually decrease with the development of the sandstorm. At the beginning of the sandstorm, $\gamma _u$ reaches 68 $^{\circ }$ . With sandstorm development, $\gamma _u$ gradually decrease and basically fall within the range given by the existing studies under unstable conditions (12 $^{\circ }$ –50 $^{\circ }$ , as shown by the light red area). Considering that the buoyancy effect could lead the head of coherent structures to be lifted up from the ground and become steeper (Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013; Salesky & Anderson Reference Salesky and Anderson2018), $\gamma _u$ calculated by an empirical model ( $\gamma _u=\gamma _0+7.3\ln (1-70z/L$ ), where $\gamma _0$ is the inclination angle in the neutral regime (Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013), are also included (marked by the black solid line in the rising stage) for comparison. The general trend of the present results agrees well with the empirical model, except in the initial stages. In the early stages of the sandstorm, the low wind velocity and the small velocity gradient under the unstable condition are not sufficient to allow turbulence structures to tilt substantially along the wall-normal direction. Moreover, strong buoyancy effect leads to a significant lifting on the structures. Thus, there is a large $\gamma _u$ . As sandstorm develops and wind velocity increases, on the one hand, the shear turbulence is enhanced and hairpin vortex packets are stretched by the velocity gradient of the streamwise wind velocity in the wall-normal direction, making the large-scale structures have a certain tilt angle with the ground (14 $^{\circ }$ according to the hairpin vortex packet model in Adrian et al. (Reference Adrian, Meinhart and Tomkins2000); on the other hand, the velocity gradient along the wall-normal direction increases with decreasing atmospheric instability, resulting in decreasing $\gamma _u$ because $\gamma _u$ is inversely proportional to the velocity gradient (Liu et al. Reference Liu, Bo and Liang2017a ; Lotfy & Harun Reference Lotfy and Harun2018). Additionally, the flow acceleration induces an increase in Reynolds shear stresses in the near-wall region (Talamelli et al. Reference Talamelli, Fornaciari, Westin and Alfredsson2002), which consequently contributes to a potential reduction in $\gamma _u$ due to its negative correlation with wall shear (Liu et al. Reference Liu, Bo and Liang2017 a). Nevertheless, the present results suggest that the effect of flow acceleration on $\gamma _u$ appears to be negligible when compared with the influence of thermal instability.

When the sandstorm develops into the steady stage, $\gamma _u$ remain essentially invariant at approximately 16.63 $^{\circ }$ . Although this value falls within the range of 11 $^{\circ }$ to 25 $^{\circ }$ given by near-neutral laboratory and ASL studies (as shown by the light green area), it is larger than the results of the near-neutral particle-free ASL (e.g. $\gamma _u = 13.7^{\circ }$ , Chauhan et al. (Reference Chauhan, Hutchins, Monty and Marusic2013) and $\gamma _u = 13.02^{\circ }$ , Liu et al. (Reference Liu, Bo and Liang2017 a)). This is consistent with the observations of Tay et al. (Reference Tay, Kuhn and Tachie2015) and He & Liu (Reference He and Liu2023) in the particle-laden flows, where the addition of sand particles causes an increase in $\gamma _u$ . In the declining stage, $\gamma _u$ are in agreement with the range (9 $^{\circ }$ –12 $^{\circ }$ , as shown by the light blue area) given by existing studies in stable condition but decrease with the attenuation of the sandstorm. When the atmosphere transforms into stable stratification, suppressing effect of the sinking cold and denser air results in a decrease in structure inclination angle (Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013; Liu et al. Reference Liu, Bo and Liang2017a ).

In stark contrast to the behaviour of $\gamma _u$ , $\gamma _c$ equals zero in the initial stage of the sandstorm and then $\gamma _c$ gradually increase and approach $\gamma _u$ with sandstorm development in the rising stage, which is similar to the evolutionary pattern of streamwise length scale $L_{cx}$ . This suggests that dust particles form flat and small structures early in the sandstorm. As shear turbulence enhances, more and more dust particles are captured by the turbulence structures and transported upwards, forming dust structures. Their inclination gradually exceed that of the turbulence structures. In the steady stage, $\gamma _c$ reach a plateau that is larger than $\gamma _u$ . This phenomenon is primarily attributed to the fact that the heavier particles near the ground tend to cross the path of the fluid flow and accumulate in the region of high strain and low vorticity (Maxey Reference Maxey1987; Eaton & Fessler Reference Eaton and Fessler1994; Wang et al. Reference Wang, Chen and Zheng2024), causing a particle–turbulence structure phase difference that varies with height (He & Liu Reference He and Liu2023), which culminates in $\gamma _c$ larger than $\gamma _c$ in the steady stage. The $\gamma _c$ obtained from the relationship model ( $1/\tan \gamma _{c} =1/\tan \gamma _{u}-1$ ) given by He & Liu (Reference He and Liu2023) are marked by yellow filled square symbols in figure 9. It is observed that, in the steady stage, $\gamma _c$ not only fall within the range given by He & Liu (Reference He and Liu2023), but also satisfy the relationship. In the stable declining stage, the suppressing effect also acts on the dust concentration structures, resulting in a decrease in $\gamma _c$ until it becomes zero at the end of sandstorm. Notably, in the middle and late stages of the sandstorm, $\gamma _c$ decreases earlier than $\gamma _u$ , echoing the $L_{cx}$ decreases earlier than $L_{ux}$ , which indicates that stable stratification has a stronger suppressing effect on dust transport than on turbulence.

4.3. Self-similarity

To inspect the self-similarity and assess the wall-attached aspect, the LCS is adopted as a statistical tool for effectively detecting self-similar eddies (Baars et al. Reference Baars, Hutchins and Marusic2017; Baidya et al. 2019>; Krug et al. Reference Krug, Baars, Hutchins and Marusic2019; He & Liu Reference He and Liu2023). The LCS defined as

(4.4) \begin{eqnarray} \gamma _{\mathrm {LCS}}^{2}\left (z, z_{r e f}, \lambda _{x}\right ) = \frac {\left |\left \langle \tilde {u}\left (z ; \lambda _{x}\right ) \overline {\tilde {u}^{*}\left (z_{r e f} ; \lambda _{x}\right )}\right \rangle \right |^{2}}{\left \langle \left |\tilde {u}\left (z ; \lambda _{x}\right )\right |^{2}\right \rangle \left \langle \left .\tilde {u}\left (z_{r e f} ; \lambda _{x}\right )\right |^{2}\right \rangle } = \frac {\left |\Phi _{u u}\left (z, z_{r e f} ; \lambda _{x}\right )\right |^{2}}{\Phi _{u u}\left (z ; \lambda _{x}\right ) \Phi \left (z_{r e f} ; \lambda _{x}\right )}, \end{eqnarray}

where $z_{r e f}$ and $z$ are the spatial measurement heights for the reference and traversing position; $\tilde {u}^{*}(z; \lambda _x)$ is the Fourier transform of velocity fluctuation $u(z)$ , $\lambda _x$ is the streamwise wavelength, the asterisk $\ast$ denotes the complex conjugate, $ \mid \ \mid$ designates the modulus and $ \lt \gt$ denotes ensemble averaging. Moreover, $\Phi _{u u} (z, z_{r e f}; \lambda _{x} )$ and $\Phi _{u u} (z ; \lambda _{x} )$ are the cross-spectrum and power spectral spectrum, respectively. The LCS is the square of a scale-specific correlation coefficient and can assess the linear coherence between the velocity fluctuations $u$ of each scale (Baars et al. Reference Baars, Hutchins and Marusic2017). According to the definition $0 \le \gamma _{LCS}^2 \le 1$ , $\gamma _{LCS}^2 = 0$ indicates that the scale is not correlated and $\gamma _{LCS}^2 = 1$ denotes perfect. This coherence can also be estimated across all other measured signals, such as spanwise velocity, temperature (Krug et al. Reference Krug, Baars, Hutchins and Marusic2019) and dust concentration fluctuations (He et al. Reference He, Liu and Zheng2024).

The $\gamma _{LCS}^2$ spectra of streamwise velocity fluctuations $u$ and dust concentration fluctuation $C_{PM10}$ are shown as a coherence spectrogram in figures 10(a–c) and 10(d–f) , respectively. As seen, the isocontour map of $\gamma _{LCS}^2 (z, z_{ref}; \lambda _{x})$ is a function of $\lambda _{x}$ and $\triangle z$ ( $\triangle z = z - z_{ref}$ is the distance between the reference and traversing positions). The reference height $z_{ref} =0.006\delta$ is adopted in this study. It should be noted that a lower reference position allows for better capture of the wall-attached hierarchical organization of eddies, but it is still difficult to measure very close to the ground in field observations. For a large value of $\triangle z$ ( $z \gg z_{ref},$ ), there is no obvious difference between plotting $\gamma _{LCS}^2$ as a function of $z$ and $\triangle z$ (Krug et al. Reference Krug, Baars, Hutchins and Marusic2019; He et al. Reference He, Liu and Zheng2024). Therefore, in this study, normalization with $\triangle z$ is employed, as this provides a more extensive scaling region. Following the conceptual model of $\gamma _{LCS}^2$ isocontours conducted by Baars et al. (Reference Baars, Hutchins and Marusic2017), when they superpose each hierarchy of self-similar scales, the coherence contours are additive, implying that there exists a triangular region in the ( $\lambda _x, \triangle z$ )-space where the isocontours of $\gamma _{LCS}^2$ align with lines of constant $\lambda _x /\triangle z$ (as shown by the red dashed line). These regions can be obviously observed in figures 10(b), 10(c) and 10(e), indicating that the velocity and dust concentration fluctuations at the steady stage and the velocity fluctuations at the declining stages may contain self-similar wall-attached structures. The geometrical self-similar wall-attached structures indicate that the value of $\gamma _{LCS}^2$ linearly increases with $\log (\lambda _x)$ at a constant height $z$ and decreases with $\log (z)$ for a constant streamwise wavelength $\lambda _x$ in the self-similar region. The region adheres to

(4.5) \begin{eqnarray} \gamma _{L C S}^{2} & = & C_{1} \ln \left (\lambda _{x} / \triangle z\right )+C_{2}, \end{eqnarray}

where $C_1$ and $C_2$ are constants. The statistical aspect ratio (streamwise/wall-normal) then follows

(4.6) \begin{eqnarray} AR = \frac {\lambda _x}{\triangle z} \mid _ {\gamma _{L C S}^{2}} = \exp \left ( -\frac {C_2}{C_1} \right ). \end{eqnarray}

The $\gamma _{LCS}^2$ spectra of figure 10 can be presented as a function of $\lambda _x / \triangle z$ , as shown in figure 11. As expected, within the self-similar region, all LCSs collapse. By inspection, in the flow field, the completely separated LCS lines at the different heights in the rising stage (as shown in figure 11 a) imply that the turbulence structures do not satisfy self-similarity. However, all LCS lines collapse well in a certain wavelength range in the steady stage (as shown in figure 11 b), indicating that these turbulence structures exhibit self-similarity as described by the attached eddy model. As shown in figure 11(c), in the declining stage, the coherence at the higher positions attenuates to a very low level, but there is still a region with collapsed LCS lines near the ground, indicating that the self-similarity is satisfied in the near-ground under weakly stable stratification. For the dust concentration fluctuations, collapsed LCS lines can be observed in figure 11(e) but are absent in figure 11(d,f), indicating that wall-attached self-similar dust cluster structures only occur in the steady stage. Identifying the scaling region by the range of this collapse (marked in red in the figure) allows us to fit (4.5) and (4.6) to the data and to extract the statistical aspect ratio, as shown by the grey solid line in figure 11(b,c,e).

Figure 10.

Coherence spectrogram of streamwise velocity fluctuations $u$ (a–c) and dust concentration fluctuations $C_{PM10}$ (d–f) in the different stages of the sandstorm, where (a) and (d) is the rising stage, (b) and (e) is the steady stage and (c) and (f) is the declining stage.

Figure 11.

Obtained LCSs are plotted as a function of $\lambda _{x} /\triangle z$ . The layout is the same as in figure 10.

The observed self-similarity indicates that the LCS lines at different heights follow the same increasing pattern with increasing wavelength; that is, the aspect ratio remains invariant across different the heights. The evolution of the aspect ratio of the streamwise velocity fluctuations ( $AR_u$ ) and the dust concentration fluctuation ( $AR_c$ ) at the different heights during the sandstorm are plotted in figure 12. Notably, the self-similarity region gradually decreases with height because only larger hierarchies can be captured when the traversing points become higher, which leads to a significant error in the aspect ratios obtained by fitting, thus the results for the two positions farthest away from the reference positions are not shown in figure 12. In particular, the relative contributions at high $\lambda _x$ , as well as farther away from the ground, are considerably lower for the $C_{PM10}$ spectrogram compared with the streamwise velocity, which is consistent with the temperature fluctuation (Krug et al. Reference Krug, Baars, Hutchins and Marusic2019). Additionally, $AR_c$ at the beginning and end of the sandstorm are not considered in figure 12(b) because the dust concentration levels at this point are low enough to form significant dust clusters structures.

As expected, $AR_u$ remain essentially invariant with height in the steady stage, elucidating the self-similarity of turbulence coherent structures. This is readily explicable because the turbulence structures in the steady stage mainly originate from the self-organization process of the hairpin vortices (Baars et al. Reference Baars, Hutchins and Marusic2017; Marusic & Monty Reference Marusic and Monty2019). The present result $AR_u=\, 12.19$ agrees with the particle–laden flow result ( $AR_u = 12.8$ , He et al. Reference He, Liu and Zheng2024) but is smaller than the particle-free flow results ( $AR_u = 14$ , Baars et al. (Reference Baars, Hutchins and Marusic2017); $AR_u = 13.9$ , Duan et al. (Reference Duan, Zhang, Zhong, Zhu and Li2020); $AR_u = 14.1$ , He et al. (Reference He, Liu and Zheng2024)), as shown in figure 12(a). In contrast to the steady stage, $AR_u$ gradually increases with the sandstorm development at the rising stage, but decreases with height, indicating that self-similarity is not satisfied. The general trend in present $AR_u$ in the rising stage follows the empirical equation ( $AR_u=-3.1\log _{10}(-z/L)+3.0$ , as marked by the red square (Krug et al. Reference Krug, Baars, Hutchins and Marusic2019)), i.e. $AR_u$ increase with the weakening stratification instability. However, the $AR_u$ at different heights cannot be qualified by this equation. Note that the rising stage is characterized by sharp flow acceleration in addition to the changes in thermal stability. The flow acceleration significantly enhances the near-wall Reynolds shear stresses (Fernholz & Warnack Reference Fernholz and Warnack1998; Piomelli et al. Reference Piomelli, Balaras and Pascarelli2000), making the large-scale coherent structures more elongated near the ground (Talamelli et al. Reference Talamelli, Fornaciari, Westin and Alfredsson2002). Consequently, the turbulence structures near the ground in the rising stage exhibit a larger aspect ratio; that is, $AR_u$ decreases with height. In the declining stage, $AR_u$ are relatively invariant near the ground, but decrease at higher positions. This trend can be attributed to the attenuated shear turbulence, especially suppressed by gravity when stably stratified, results in a scale reduction and self-similar wall-attached structures cannot extend to the highest measured position (approximately 0.2 $\delta$ ) (Chauhan et al. Reference Chauhan, Hutchins, Marusic and Monty2010,Reference Chauhan, Hutchins, Monty and Marusic2013; Liu & Zheng Reference Liu and Zheng2024). Namely, the turbulence structures adhere to self-similarity near the ground but are not satisfied at higher positions under the weak stable stratification because there is no longer a coherent region at higher positions.

Figure 12.

Evolution of aspect ratios at different heights during the sandstorm: (a) streamwise velocity fluctuation $u$ (denotes $AR_u$ ); (b) dust concentration fluctuation $C_{PM10}$ (denotes $AR_c$ ). The black hollow symbols denote the existing laboratory and ASL data (from Baars et al. Reference Baars, Hutchins and Marusic2017; Krug et al. Reference Krug, Baars, Hutchins and Marusic2019; Duan et al. Reference Duan, Zhang, Zhong, Zhu and Li2020; He et al. Reference He, Liu and Zheng2024) and the red square denotes the result of the empirical formula from Krug et al. (Reference Krug, Baars, Hutchins and Marusic2019). The blue dashed line denotes the present aspect ratio of the streamwise fluctuation $u$ .

Figure 12(b) suggests that the dust structures exhibit self-similarity in the steady stage, as evidenced by the independence of $AR_c$ on the height, which is similar to the behaviour of turbulence structures. The present aspect ratio $AR_c=\, 11.39$ is close to $AR_c=\, 12.0$ reported by He et al. (Reference He, Liu and Zheng2024). Dust particles with great capability to follow eddies in the flow are entrapped by wall-attached turbulence structures with low momentum, and cluster with each other in the low-speed regions, evolving with them in a self-similar way (Zhu et al. Reference Zhu, Pan, Wang, Liang, Ji and Wang2021). This indicates that the self-similarity of dust structures arises from the self-similarity of multiscale turbulence structures (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010; Baker et al. Reference Baker, Frankel, Mani and Coletti2017; Zhu et al. Reference Zhu, Pan, Wang, Liang, Ji and Wang2021). However, statistical $AR_c=\, 11.39$ is smaller than $AR_u=\, 12.19$ in the steady stage. Comparing the definitions of inclination and aspect ratio, the decrease of the aspect ratio can be anticipated by the increase of inclination. Dust structures are not self-similar in the rising and declining stages, which is expected because the self-similarity of turbulent structures is absent in the entire measured region. Moreover, akin to $AR_u$ , $AR_c$ decreases with height. Nevertheless, there is a distinctive difference between $AR_c$ and $AR_u$ near the ground, where $AR_c$ is not only significantly larger than $AR_u$ , but also evolves with time in the opposite way to $AR_u$ .