Scale-dependent inclination angle of turbulent structures in stratified atmospheric surface layers

Abstract A large-scale spanwise and wall-normal array of sonic anemometers in the atmospheric surface layer is used to acquire all three components of instantaneous fluctuating velocity as well as temperature in a range of stability conditions. These data permit investigation of the three-dimensional statistical structure of turbulence structures. Based on a similar dataset, Krug et al. (Boundary-Layer Meteorol., vol. 172, 2019, pp. 199–214) reported a self-similar range of wall-attached turbulence structures under both unstable and near-neutral stability conditions. They considered only a wall-normal array and thus assessed statistical structure in the wall-normal direction, in relation to the streamwise wavelength. The present work extends the view of a self-similar range of turbulence structures, by including the statistical structure in the spanwise direction. Moreover, by analysing the phase shift between synchronized measurements in the spectral domain, it is inferred how a scale-dependent inclination angle in the streamwise/wall-normal plane varies with stability. Results suggest that the self-similar wall-attached structures have similar aspect ratios between streamwise/wall-normal scales and streamwise/spanwise scales such that $\lambda _x/\Delta z : \lambda _x/\Delta y \approx 1$ for both near-neutral and unstable conditions. Under the most unstable conditions, coherent structures with $\lambda _x/\delta = 1$ are inclined at angles as high as $65^{\circ }$ relative to the solid boundary, while larger scales $\lambda _x/\delta = 6$ exhibit inclination angles of approximately $35^{\circ }$. For near-neutral stability conditions, the angle tends towards $12^{\circ }$ for all scales. It is noted that in the near-neutral condition, the structure inclination angle and the aspect ratio – and thus the statistical modelling of coherent structures in the atmospheric surface layer – are highly sensitive to the value of the stability parameter.


Introduction
Townsend (1976) proposed a conceptual model for wall-bounded turbulence, the attached eddy hypothesis (AEH), which idealizes structures as a collection of inertia-driven self-similar eddies that are distributed randomly in the plane of the wall. Details of key assumptions and limitations associated with the AEH are covered in a recent review by . Based on the AEH, Perry & Chong (1982) proposed that coherent wall-attached eddies scale with the distance from the wall z, and their heights comprise a geometrical progression. Evidence in support of self-similarity and wall-scaling of wall-attached vortices has been reported in recent turbulent boundary layer (TBL) studies (e.g. Jiménez 2012; Hwang 2015; Baars, Hutchins & Marusic 2017). Here, we consider three hierarchy levels of randomly positioned regions of coherent velocity fluctuations, with each hierarchy shown in a different colour. For simplicity, we consider the volume of influence of eddies, in each level, to be characterized by L i , W i and H i in the x, y and z directions, respectively, with i = 1, 2, 3 denoting the ith hierarchy level. Figures 1(b) and 1(c) depict the aspect ratios in the streamwise/wall-normal plane A z ≡ L i /H i ∝ λ x / z and in the streamwise/spanwise plane A y ≡ L i /W i ∝ λ x / y, respectively. Baars et al. (2017) reported that in the neutral laboratory zero-pressure gradient TBL, the self-similarity is described by a streamwise/wall-normal aspect ratio λ x / z ≈ 14. A recent study by Baidya et al. (2019), in high-Reynolds-number pipe and boundary layer flows, indicated that the self-similar wall-attached structures follow the three-dimensional aspect ratio 14 : 1 : 1 in the streamwise, spanwise and wall-normal directions, respectively. More recently, Krug et al. (2019) explored the coherence for both velocity and temperature signals in the atmospheric surface layer (ASL). They found that the streamwise/wall-normal aspect ratio (A z ≡ λ x / z) decays with a logarithmic trend with increasing unstable thermal stratification; spanwise information was not explored in their study. Moreover, they found that in the case of stable thermal stratification, the turbulence structures do not adhere to a self-similar scaling, meaning that an aspect ratio is not defined.
Coherent structures reported in TBLs have been associated with characteristic inclination angles because of the mean shear. In the idealized (statistical) view of figure 1(b), inclination angle α in the x,z-plane reflects a phase shift τ in time series of velocity fluctuations at different z. Perry & Chong (1982) used a vortex skeleton approach and the Biot-Savart law to determine the inviscid velocity field of a representative eddy, termed the Λ-vortex. In neutral TBLs, hairpin vortices have been invoked commonly as the representative eddy, and Adrian, Meinhart & Tomkins (2000) suggested that these vortices are arranged together in groups called vortex packets. (For a comprehensive review of hairpin structures and their generating mechanisms, see Adrian 2007.) Christensen & Adrian (2001) found that the sequence of individual vortex heads forms an interface or shear layer that is, statistically, inclined away from the wall at angles between 12 • and 20 • . Laboratory results indicate the most probable inclination angle to be around 10 • -15 • (Adrian et al. 2000;Christensen & Adrian 2001;Baars, Hutchins & Marusic 2016). In the neutral surface layer, inclination angles ranging from 10 • to 20 • have been reported (Boppe, Neu & Shuai 1999;Carper & Porté-Agel 2004;Chauhan et al. 2013;Liu, Bo & Liang 2017). Also, Marusic & Heuer (2007) demonstrated the invariance of the inclination angle in wall-bounded flows with zero buoyancy (neutral conditions) over a wide range of Reynolds numbers through laboratory and field experiments. The discussion above pertains to the near-neutral case. However, in studies of the ASL, it has been observed that the inclination angle changes drastically under different stability conditions. The thermal stability of the ASL is generally characterized by the Monin-Obukhov stability parameter z s /L (Obukhov 1946;Monin & Obukhov 1954), where L = −u 3 τθ /(κgwθ) is the Obukhov length, κ = 0.41 is the von Kármán constant, g is the gravitational acceleration, and wθ is the surface heat flux, with w and θ the fluctuating wall-normal velocity and temperature components,θ the mean temperature, u τ the friction velocity, and z s the reference height for evaluating this parameter. Many studies have observed that the inclination angle of coherent structures becomes steeper with increasing unstable thermal stratification (Chauhan et al. 2013;Liu et al. 2017;Lotfy & Harun 2018). In addition, Chauhan et al. (2013) reported that for the case of stable stratification, the statistical inclination angle reduced below the typical values found for the near-neutral case. Recently, Salesky & Anderson (2020a) introduced an additional parameter to account for the loading and unloading of surface layer flux-gradient relations imposed by the passage of large-scale motions (LSMs). Meanwhile, Salesky & Anderson (2020b) developed a prognostic model for large-scale structures, where the inclination angle is the sum of the inclination angle observed in a neutrally stratified wall-bounded turbulent flow and the stability-dependent inclination angle of the wedge. Baars et al. (2016) indicates that in the neutral case, and for all scales λ x /δ > 0.5, the coherent scales obey a virtually constant inclination angle. In unstable conditions in the atmosphere, positive buoyancy lifts the structure away from the surface, leading to an increase in the statistical inclination angle (as averaged across all scales; see Chauhan et al. 2013;Liu et al. 2017). Now, in the unstable case, the dominance of buoyancy over shear is a function of wall-normal height, hence one expects the inclination angle to be scale-dependent.
Since the coherent structure in the ASL has a strong relationship with the stability parameter, this paper will address specifically the influence of stability on: (1) the streamwise/wall-normal aspect ratio A z ≡ λ x / z and the streamwise/spanwise aspect ratio A y ≡ λ x / y in § 3.1, and (2) the scale-dependent angle α in § 3.2, particularly under unstable conditions. Statistical relations for the aspect ratio and inclination angle for coherent turbulence fluctuations in the ASL are particularly relevant when analysing wind loading in the field of wind engineering (see Davenport 1961Davenport , 2002. 2. Turbulence dataset of the atmospheric surface layer 2.1. QLOA facility and available data The measurement data used throughout this article were acquired at the QLOA facility in western China, Gansu province, during three-month long measurement campaigns over two years (March to May in 2014 and2015). The QLOA consists of wall-normal and spanwise arrays of sonic anemometers, performing synchronous measurements of the three-dimensional turbulent flow field. Sonic anemometers (Gill Instruments R3-50 installed from s 2 to s 7 , and Campbell CSAT3B installed from h 1 to h 11 ; see figure 2) were employed to acquire time series data of the three components of velocity, as well as the air temperature, at a sampling frequency of 50 Hz. Continuous observations were conducted at the QLOA site for a duration of more than 3000 h, from which 89 h of data were selected to analyse the characteristics of the large-scale coherent structures under different stratification stability conditions. The wall-normal array consists of 11 sonic anemometers that were placed with a logarithmic spacing on a vertical radio-type tower. The spanwise array covered an overall distance of 30 m, with 7 anemometers that were placed at constant height (z = 5 m), with an equidistant spanwise spacing of 5 m. The spanwise and wall-normal coordinates for each of the 17 anemometers are provided in figure 2(b). It should be noted that the first sonic anemometer in the spanwise array (s 1 ) also functions as the fifth on the main tower (h 5 ), which means that we have 7 available anemometers in the spanwise array. The friction velocity u τ is inferred from u τ = (−uw) 1/2 at z = 5 m (calculated by the mean value from 7 sonic anemometers in the spanwise array). We assume an estimate for the surface layer thickness of δ = 60 m, following Hutchins et al. (2012). The 89 h of data remaining after preselection were subdivided into 89 segments of 1 h long time series. This database included 69 h of unstable data (z s /L < −0.01), 10 h of near-neutral data (−0.01 z s /L < 0.01), and 10 h of stable data (z s /L > 0.01). Recall that z s is the reference height used to define the stability parameter z s /L. For the benefit of comparison with previous works (Chauhan et al. 2013;Liu et al. 2017;Krug et al. 2019), the majority of our work uses z s = 2.5 m, unless otherwise specified. The demarcation of z s /L = −0.01 to distinguish between neutral and unstable thermal stratification is found 942 A38-4 commonly in the literature, but for all analysis in this paper, we present results as a function of z s /L; it will actually be shown later that this demarcation of z s /L = −0.01 is too permissive. The preselection criteria included wind direction (the wind direction had to be aligned with the x axis of the anemometer to within ±30 • ) and steadiness (statistically steady conditions based on the high-quality requirement by Foken et al. 2005). A de-trending operation is also added (to remove the large-scale synoptic trend). See Hutchins et al. (2012) and Wang & Zheng (2016) for full details of the preselection criteria.

Processing method
Two-point correlations can be computed on a per-scale basis in the Fourier domain using the linear coherence spectrum (LCS). For the fluctuating streamwise velocity signals u, the LCS is defined as The spatio-temporal transformation uses Taylor's hypothesis (Taylor 1938), where the local mean velocity is taken as the convection velocity. The asterisk * indicates the complex conjugate, denotes ensemble averaging, and | | designates the modulus. Scale-dependent phase information is embedded explicitly in the phase of the cross-spectrum φ uu . For (2.1), the LCS is defined based on u at two positions (z ref and z), separated in the wall-normal direction by z ≡ z − z ref . This coherence can also be computed across all other measured signals (v and θ) and also across spanwise separations y. The reference signal's height is denoted with the subscript 'ref' and is thus stated as z ref .
Since the LCS considers the magnitude of the complex-valued cross-spectrum, only the magnitude of coherence is considered (phase is covered later). Based on assumptions from the AEH (Baars et al. 2017;Krug et al. 2019), the coherence magnitude within a self-similar region in λ x ,z-space is expected to adhere to from which the statistical aspect ratio (in this case streamwise/wall-normal) then follows: Here, C 1 and C 2 are fitted parameters. Figure 3 shows an example of data obtained from the ASL to illustrate the process of the coherence spectrum. Figure 3(a) indicates the raw data for the streamwise velocity collected at h 5 = 5 m and h 6 = 7.15 m, under unstable conditions with z s /L = −0.52. A shorter time history of corresponding filtered signals is shown in figure 3(b), evidencing that signal h 6 leads h 5 . Thus a coherent velocity fluctuation is first sensed at the higher wall-normal location as a result of the structure inclination angle. The LCS for h 5 and h 6 is presented in figure 3(c) as a function of temporal frequency, as computed from the 1 h long time series data. Using Taylor's frozen turbulence hypothesis, the frequency axis can be converted to a streamwise wavelength: λ x ≡ U c /f . Here, U c is a convective speed, taken as the mean velocity at local height z. (The current dataset is fully submerged in the logarithmic region of the ASL. 10 1 10 0 10 -1 10 -2 f (Hz) 10 1 10 0 10 -1 10 -2 10 1 10 0 10 -1 10 -2 γ 2 L 0 0.5 1.0 10 2 10 1 10 0 10 -1 Here, the abscissa is converted to a spatial wavelength with the mean velocity at height z. (e) Scale-dependent phase of the cross-spectrum between h 5 and h 6 , which is converted to a physical inclination angle α as a function of a spatial wavelength in (f ). The blue dots and blue solid lines in (c-f ) indicate the frequency or wavelength corresponding to γ 2 L = 0.1. Filtered spectra (black line), overlaid on the grey raw spectra, utilize a bandwidth moving filter of 25 %.
Here, the convective speed of coherent structures agrees well with the mean velocity and is relatively scale-independent (Liu & Gayme 2020). Moreover, Baars et al. (2017) showed that when Taylor's hypothesis is used with the local mean velocity U(z), the coherence spectra agree with those computed from spatial DNS data.) The coherence spectrum of figure 3(c) can now be presented as a function of the streamwise wavelength, relative to the wall-normal separation distance z, as shown in figure 3(d). In addition to the coherence spectrum for h 5 and h 6 , the coherence spectra for all heights above h 6 (relative to h 5 again) are also shown to illustrate the wall-similarity that is to be investigated. Krug et al. (2019) noticed that only data at neutral and unstable thermal stratification conditions complied with (2.2), thus an aspect ratio was found only for those conditions. (A similar conclusion was reached for our QLOA site data, therefore we do not consider stable stratification with z s /L > 0.) In addition, Krug et al. (2019) found that the self-similar scaling applies also to fluctuations of the spanwise velocity v and the static temperature θ. Baidya et al. (2019) demonstrated that a scaling similar to (2.2) and (2.3) occurs in the spanwise direction, resulting in a streamwise/spanwise aspect ratio A y for the self-similar structure.
The phase spectrum aids in assessing the temporal shift between signals. Phase Φ( f ) in (2.4) is shown in figure 3(e) and can be used to extract a scale-by-scale inclination angle α (as shown in figure 3f ). That is, the temporal shift is τ = Φ( f )/(2πf ), where f is the mode frequency, and aids in computing the physical inclination angle through For the spectral analysis, the highest frequency resolved is set by the Nyquist frequency f s /2 = 25 Hz, where f s = 50 Hz is the sampling frequency. The lowest frequency is dictated by the interval length I used in the spectral analysis, and the longest interval used was I = 2 N samples with N = 15 (interval length ≈650 s). A composite approach with varying interval length (N = 8, . . . , 15) was used to generate the full spectra with as many ensembles as possible for the higher frequency portions of the spectrum (for N = 8 a total of around 1000 ensembles were used).

Stability dependence of aspect ratio
The linear coherence spectra for u, v and θ as functions of λ x / z and λ x / y for the unstable case with z s /L = −0.52 are given in figures 4(a) and 4(c), respectively. As reported by Krug et al. (2019), the LCS collapses on one common curve over a range of λ x / z and λ x / y. By fitting (2.2) to these regions to obtain C 1 and C 2 , the aspect ratio A can be assessed. For this particular unstable case, we might expect the positive buoyancy to cause the self-similar structures in the hierarchy to lift more aggressively from the wall, extending the wall-normal coherence for a given λ x scale, hence reducing A z u . Indeed, for the unstable case considered in figure 4(a), this yields an A z u that is significantly lower (A z u = 3.6) than the value A z u ≈ 14 reported for laboratory neutral conditions (Baars et al. 2017) and in close agreement with the aspect ratio reported in Krug et al. (2019) for similar values of the stability parameter. The resulting aspect ratios for u, v and θ from all 79 h datasets (covering a range of stabilities from 0.007 −z s /L 1.04) are plotted as functions of the stability parameter in figure 4(b) for streamwise/wall-normal aspect ratios A z , and in figure 4(d) for streamwise/spanwise aspect ratios A y . In all cases, a clear trend emerges between aspect ratio and stability parameter, and a log-linear trend is fitted to the extracted data (black solid curves). These fitted trends are consistent with those of Krug et al. (2019) (black dashed curves in figure 4b), indicating that the self-similar scaling under near-neutral and unstable conditions is a universal phenomenon. The relatively small differences in the slopes of the solid (current study) and dashed (Krug et al. 2019) trend lines, visible in particular for A z u and A z θ , are ascribed to experimental uncertainty in the field measurements: e.g. differences between test sites and the fact that the fitting procedure was performed on a different number of data points, residing at different values of −z s /L. As an extension to the results of Krug et al. (2019), the streamwise/spanwise aspect ratios of figure 4(d) seem to also exhibit log-linear trends, although in these cases the scatter in results is greater. Baidya et al. (2019) indicates that the aspect ratio is A z u : A it is found that A z u : A y u = 0.90 : 1 for the near-neutral case z s /L = −0.01, and that this ratio changes to A z u : A y u = 1 : 1 for a strong unstable stability condition z s /L = −1. By recalling that A z u ≡ λ x / z, the data here indicate that these self-similar eddies for u follow an aspect ratio λ x : y : z ≈ 6.4 : 0.90 : 1 in the near-neutral condition (z s /L = −0.01). It is worth highlighting that again the aspect ratio is sensitive to even very weakly unstable conditions. Therefore, the value for u measured in the ASL is less than the result λ x / z = 14 from Baars et al. (2017) and Baidya et al. (2019) in neutral laboratory conditions, as was also noticed by Krug et al. (2019), whose prediction implies that λ x / z = 14 will be attained only for |z s /L| ≈ 0.0003 (but for the current QLOA dataset, we have data only for |z s /L| 0.007). Finally, the aspect ratio for u shows that λ x : y : z ≈ 4.25 : 1 : 1 under a more unstable condition (z s /L = −1), demonstrating that positive buoyancy has a lifting effect, increasing the size of coherent structures in the wall-normal and spanwise directions relative to its streamwise extent.

Stability dependence of structure inclination angle
Per the phase spectrum in figure 3(f ), the scale-dependent phase is now analysed for scales within the range 1 < λ x /δ < 10 and for a similar z as before, z = h 5 − h 1 = 4.1 m. Larger wavelength information in the phase spectra is prone to noise issues due to the limited ensembles available for constructing the spectra, while smaller wavelengths generally have a lower coherence. Figure 5(a) indicates the phase spectra in terms of inclination angle as functions of wavelength λ x /δ, for all different stability parameters −z s /L. Note that all measured angles in figure 5(a) are positive, corresponding to forward-leaning structures. Though there are some clear outliers in these plots, certain trends are visible, as evidenced by the coloured symbols that visualize data at constant scales λ x /δ = 1, 2 and 6. Clearly, these different wavelengths exhibit different dependencies of α with the stability parameter, with λ x /δ = 1 exhibiting much steeper angles α in the most unstable cases, and markedly shallower angles as near-neutrality is approached. In general, it is also noted that longer structures (larger wavelength) will exhibit smaller inclination angles, especially noticeable at stronger convective conditions.
To quantify the trends in the data, a functional form is adopted from Chauhan et al. (2013), who used a log-linear trend of convective data in a boundary layer to model the variation of α with stability z s /L. Note that their angle was inferred from two-point cross-correlation maps and hence is scale-invariant. As such, the only difference is the introduction of a scale-dependent coefficient, here denoted as C 0 = C 0 (λ x /δ) in (3.1).

A38-9
This function will be illustrated later, but can capture the faster increase of α with increasing −z s /L, for smaller scales. (In the work of Chauhan et al. (2013) this coefficient was constant and equaled C 0 = 7.3.) So we have The form of (3.1) guarantees that α approaches the invariant value of α 0 when −z s /L → 0 (here, α 0 is taken as 12 • , and recall that z s = 2.5 m). A curve-fitted plane to those spectra is shown in figure 5(b), in which this fit has the form in (3.1); iso-contours of α are depicted in figure 5(c), and show that under the most unstable conditions, coherent structures of λ x /δ = 1 are inclined at angles as high as 65 • . It should be noted from figure 5(a) that for small near-neutral values of the stability parameter, it is not always possible to compute an inclination angle α for the smaller wavelength (λ x /δ = 1) from the phase spectra since the coherence across z drops below γ 2 L = 0.1. Figure 6 shows the variation of α as a function of the stability parameter −z s /L for three different length scales, λ x /δ = 1, 2, 6. Inclination angles computed from the phase spectra are shown by the coloured circles, and the blue lines show the surface fit to the data given by (3.1). In addition, the black open circles in figure 6, which show the inclination computed from the two-point correlations for all 11 measurement heights, indicate that the current data are consistent with the similarly computed results from Chauhan et al. (2013), shown by the black open squares. The black solid line in figure 6 shows the fit proposed by Chauhan et al. (2013) based on the two-point correlation contours. As such, this original fit includes all scales, and is skewed disproportionately towards larger-scale features at higher z. It is clear from figure 6 that the increasing buoyancy lifts all scales to larger inclination angles, although the smaller-scale structures considered (λ x /δ = 1) exhibit steeper angles at all values of the stability parameter as compared to the larger features (λ x /δ = 6). This means that the scale-dependent coefficient C 0 = C 0 (λ x /δ) in (3.1) increases systematically as we focus on smaller scales. The coloured curves in figure 6 show the curve fits to the data based on the fit proposed in (3.1), with the constant α 0 and value of the scale-dependent coefficient C 0 given in the figure legend. It should be noted here that for the limited scale range discussed here, 1 < λ x /δ < 10, all of these scales are large and associated with the upper end of attached motions and superstructures (Hutchins & Marusic 2007). Although with increasing stability all scales are lifted compared with the neutral condition, the low end of this range (λ x /δ = 1) exhibits the steepest angles, reaching α ≈ 70 • at z s /L = −1.0. Baars et al. (2016) indicated that the inclination angle of the large-scale structures in the neutral laboratory boundary layer is scale-independent with value α = 14.7 • . Though the data in figure 6 do suggest that α becomes scale-independent in the limit of small |z s /L|, the angle seems to be closer to α ≈ 12 • for the current data.
The results summarized in figure 6 are computed for a linear coherence spectrum between the reference location z ref = 0.9 m and the location z = 5 m ( z = 4.1 m). In addition, the stability parameter presented on the abscissa of figure 6 is computed based on conditions at z s = 2.5 m. Before considering further the form of the fit described by (3.1), we must consider the sensitivity to z s , and z. Generally, the stability parameter z s /L depends linearly on height, which would suggest that the scale-dependent structure inclination angles α will also increase with wall-normal height z (since as we move away from the wall, buoyancy effects will increase in dominance relative to shear). To account for this, and to form a stability parameter that better reflects the altitude at which the phase spectrum is evaluated, we propose a fractional stability parameter, dubbed z F /L, where  Chauhan et al. (2013); asterisks from R uτ u of Marusic & Heuer (2007); diamonds of Carper & Porté-Agel (2004); open pentagram from R uu of Marusic & Heuer (2007).

Wall-normal direction
Maintaining z F at the value given by (3.2), when changing z or z ref , ensures that we assess stability at the same fractional location 0.39 z when varying the height of inspection z. We can now substitute z F for z s in (3.1): all heights above, ranging from 1.71 m to 30 m, with thinner grey lines indicating lower values of z. We can extract α from the phase spectra only in situations where the coherence γ 2 L is greater than 0.1; this condition gives rise to the sharp drops/rises in figure 8(a) as at different λ x /δ, varying subsets of data are available for the calculation of α. In general, it is noted that the fractional location for the stability height described above does a reasonable job of collapsing the C 0 curves for all z; C 0 decreases with λ x /δ, indicating that the smaller-scale structures tend to have larger inclination angles in the convective conditions. Based on the approximate collapse observed in figure 8(a), we can approximate crudely the λ x /δ dependence of C 0 with which is shown by the blue solid line in figure 8(a). Finally, by combining (3.4) with (3.3), we obtain the scale-dependent structural inclination angle as (3.5) Figure 8(b) shows the influence of wall-normal offset z (with z ref fixed at 0.90 m) on the computed inclination angle α as a function of stability for the wavelengths λ x /δ = 1, 2, 6 for u. A larger z leads to higher α, but this can be accounted for by considering the fractional stability parameter. The curves, showing (3.5), describe the variation of α with z s /L, λ x /δ and z reasonably well.
By way of a summary, figure 9 shows an illustration of the scale-dependent structure inclination angle and aspect ratio for both neutral (subscript n) and unstable (subscript u) thermal stratification conditions. The illustrations show the streamwise extent of the structure (in physical space following the simplification that its length L scales with half the wavelength, thus L ∼ λ x /2) and its wall-normal extent with height H; the streamwise/wall-normal aspect ratio of the structure in the x,z-plane adheres to A z u of figure 4(b). When concentrating on the neutral stability condition first (bottom part of figure 9), both scales drawn exhibit the same inclination angle α n . That is, the statistical inclination angles of large and small scale comprise the same forward leaning behaviour.
When considering unstable thermal stratification -in figure 9, just one value of z s /L = −0.50 is considered, evaluated at z s = 2.5 m -the inclination angle is scale-dependent and also varies with z (the procedure for plotting this structure is summarized in the Appendix). First, note that the coherent wall-attached structures are still self-similar per the definition used in this paper, based on the structure's streamwise wavelength relative to its wall-normal extent. But the aspect ratio reduces compared to the neutral case, and this depends on the degree of thermal stratification per the relations shown in figure 4(b), thus (L ns /H ns = L nl /H nl ) > (L us /H us = L ul /H ul ). Here, subscripts s and l refer to the small-and large-scale structures visualized, respectively. Concentrating on the inclination angle (e.g. the phase shift between a reference height z ref and a height of inspection z), it was found that for unstable stratification, it increases with increasing z. Hence α ul2 > α ul1 , and the coherent shape of the structure is characterized by a leading/trailing edge front that is curved and becomes steeper with height. Moreover, smaller-scale structures exhibit steeper angles for the case of equal z (e.g. α us > α ul1 ). The trends for this one stability condition are visualized with the 'α-surface' in the centre of figure 9. There, the fractional stability z F is a surrogate for the z trend.

Conclusion
Wall-normal and spanwise arrays of sonic anemometers deployed in the atmospheric surface layer enable examination of the linear coherence spectrum, γ 2 L , as a function of the streamwise wavelength (λ x ), spanwise offset ( y) and wall-normal offset ( z). This in turn offers the opportunity to explore the three-dimensional form of the wall-attached self-similar structure for the streamwise velocity u, which illustrates that the self-similar wall-attached structures follow an aspect ratio λ x / z : λ x / y ≈ 1 under near-neutral and unstable conditions. It is found that the aspect ratio λ x / z is greater for the near-neutral case, and becomes progressively smaller as instability increases. Hence similar length (λ x ) structures in unstable conditions will be taller and wider than their near-neutral counterparts. The phase of the cross-spectrum provides a scale-by-scale structure inclination angle. We find that this inclination angle is invariant with scale for the near-neutral case, but with increasing positive buoyancy becomes increasingly scale-dependent. For unstable conditions (e.g. z s /L = −0.5), all scales are inclined at steeper angles compared to the near-neutral case, with the smaller scales with λ x /δ = 1 exhibiting inclination angles that are approximately twice those of larger scales, λ x /δ = 6. A structure outline is then generated through considering a sequence of local heights z. For every height, the angle relative to the fixed reference height is determined from (3.5) and (3.2); this is illustrated by the series of ten grey lines at the left edge of the structure in figure 10. Note that the structure is defined only up to a height that is dictated by the aspect ratio condition, following the trend line in figure 4(b). Thus the wall-normal extent up to which the coherent structure reaches is found by computing z from A z u ≡ λ x / z = −1.9 log 10 (−z s /L) + 3.5.