Abstract
Statistical descriptions of coherent flow motions in the atmospheric boundary layer have many applications in the wind engineering community. For instance, the dynamical characteristics of largescale motions in wall turbulence play an important role in predicting the dynamical loads on buildings, or the fluctuations in the power distribution across wind farms. Davenport (Quarterly Journal of the Royal Meteorological Society, 1961, Vol. 372, 194211) performed a seminal study on the subject and proposed a hypothesis that is still widely used to date. Here, we demonstrate how the empirical formulation of Davenport is consistent with the analysis of Baars et al. (Journal of Fluid Mechanics, 2017, Vol. 823, R2) in the spirit of Townsend’s attachededdy hypothesis in wall turbulence. We further study stratification effects based on twopoint measurements of atmospheric boundarylayer flow over the Utah salt flats. No selfsimilar scaling is observed in stable conditions, putting the application of Davenport’s framework, as well as the attachededdy hypothesis, in question for this case. Data obtained under unstable conditions exhibit clear selfsimilar scaling and our analysis reveals a strong sensitivity of the statistical aspect ratio of coherent structures (defined as the ratio of streamwise and wallnormal extent) to the magnitude of the stability parameter.
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1 Introduction and Context
Coherence quantities of atmospheric surfacelayer (ASL) turbulence are of great practical significance to the windengineering community as these are required for determining the dynamic action of atmospheric turbulence on windsensitive structures, such as tall buildings, long span bridges (Isyumov 2012), and wind turbines (Saranyasoontorn et al. 2004), or in predicting peakpower distributions across wind farms (Sørensen et al. 2007, 2008). Groundbreaking work on this subject was carried out by Alan G. Davenport and can be found in Davenport 1961a, b, 2002, Simiu and Scanlan (1996), Pasquill (1971) and Baker (2007). The particular aspect we focus on here is the degree and extent of the coherence of wind fluctuations in the vertical direction, which is quantified via the linear coherence spectrum
where, \(X\left( z;\lambda _x\right) = \mathcal {F}\left[ \psi (z)\right] \) is the Fourier transform of some fluctuating quantity \(\psi (z)\) and \(\lambda _x\) is a streamwise wavelength. The vertical position is z, and \(z_R\) denotes the reference position usually taken close to (or at) the surface. The asterisk * indicates the complex conjugate, \(\langle \rangle \) denotes ensemble averaging and \(\vert \vert \) is the modulus. It is noted that the numerator equals the square of the crossspectral magnitude, while the two energy spectra of \(\psi (z_R)\) and \(\psi (z)\) form the denominator. Since \(\gamma ^2\) only incorporates the magnitude of the crossspectrum, the value of \(\gamma ^2\) represents the maximum correlation for a specific scale \(\lambda _x\). Consequently, \(\gamma ^2\) equates to the fraction of common variance shared by \(\chi (z_R)\) and \(\chi (z)\) and we note that, by definition, \(0 \le \gamma ^2 \le 1\). As indicated in Eq. 1, \(\gamma ^2\) generally is a function of the positions z and \(z_R\) and the wavelength \(\lambda _x\) (Note that we restrict the discussion to streamwise scales here as, via Taylor’s hypothesis, this direction is the most accessible experimentally).
Davenport (1961a) hypothesised that the coherence is, (i) a function of the ratio \({\Updelta } z/\lambda _x\) only, where \(\Updelta z = zz_R\), and (ii) used an exponential decay to fit the dependence. Such a formulation is still widely used in the windengineering community to date (e.g., Baker 2007) and we will refer to it as Davenport’s hypothesis.
During the same era as Davenport, A. A. Townsend made his impact in the field of turbulent shear flows (Marusic and Nickels 2011), most notably with his attachededdy hypothesis (Townsend 1976; Perry and Chong 1982; Marusic and Monty 2019). A central tenet of the attachededdy hypothesis states that eddying motions in the logarithmic region of wallbounded flows are selfsimilar and that their size scales with their distance from the wall z. In the context of the ASL, reference to the attachededdy hypothesis has been made before, e.g., most recently by Li et al. (2018). Evidence in support of selfsimilarity and wallscaling has been reported throughout the boundarylayer community (see, for instance, Jiménez 2012; Hwang 2015; Marusic et al. 2017) and most recently by Baars et al. (2017) who investigated the vertical coherence of the longitudinal velocity fluctuations relative to a reference very close to the wall.
Interestingly, it seems that, regarding the coherence, no crosswork exists between the two respective scientific communities to which Davenport (wind engineering) and Townsend (turbulent shear flow) belonged. To the authors’ knowledge, only Davenport himself noted the early work of Townsend, as Davenport (1961b, p. 209) states: “Some of the possible implications of this have been discussed by Townsend (1957).” In the present study we aim to connect the progress made in these communities regarding the understanding of the selfsimilar turbulent eddy structures in the ASL, as quantified by the coherence diagnostic. In doing so, we will show that the geometrical selfsimilarity implied in Davenport’s hypothesis is consistent with the attachededdy hypothesis. Further, we will demonstrate that also the functional form given in Eq. 2 agrees closely with a logarithmic dependence derived from the attachededdy model (Baars et al. 2017).
We start out by providing brief reviews of Townsend’s and Davenport’s hypotheses (Sect. 2) and demonstrate their conformity. Subsequently, we describe highfidelity velocity and temperature data taken along the vertical direction in the ASL over smooth terrain (Sect. 3). These data are used in Sect. 4 to infer the coherence statistics as a function of atmospheric stability. Throughout, we only employ the fluctuating components of the turbulence quantities; the streamwise (or longitudinal), spanwise and wallnormal velocity fluctuations are denoted by u, v and w, respectively, with associated coordinates x, y and z. Temperature fluctuations are denoted with \(\theta \), its mean by \(\Uptheta \).
2 Connecting the Hypotheses of Townsend and Davenport
2.1 Coherence Following Townsend’s AttachedEddy Hypothesis
Townsend (1976) envisioned that a wallbounded shear flow encompasses a range of selfsimilar ‘attached eddies’. The terminology ‘attached’ thereby implies that turbulence statistics scale with their distance from the wall, socalled zscaling. The exact types of characteristic eddies and whether they are truly attached is of secondary importance. The attachededdy description is applicable in the inertial region of the turbulent boundary layer, where the scales range from \(\mathcal {O}(100)\) viscous units \(\nu /U_\tau \) to the order of the boundarylayer thickness \(\delta \). In practice, the inertial or ‘logarithmic’ region of the ASL occupies the range from order of millimetres above the ground to \(\mathcal {O}(100\,\mathrm {m})\) and therefore is highly relevant to all windengineering applications. The ratio of the two length scales \(\nu /U_\tau \) and \(\delta \) forms the friction Reynolds number, \(Re_\tau \equiv \delta U_\tau /\nu \), where \(\nu \) is the kinematic viscosity and \(U_\tau = \sqrt{\tau _0/\rho }\) is the friction velocity, with \(\tau _0\) and \(\rho \) being the wallshear stress and fluid density, respectively. Note that throughout the superscript ‘+’ signifies normalization by ‘inner’ scales \(\nu /U_\tau \) and \(U_\tau \). A quantity analogous to the shear velocity, the wall conduction velocity, is given by \(\Uptheta _\tau = \varepsilon \partial _z \Uptheta \vert _{z=0}/U_\tau \), \(\varepsilon \) being the thermal diffusivity.
Here, we review key results of Baars et al. (2017), who examined twopoint measurements in the wallnormal direction for smooth terrain and wellcontrolled flow conditions. Baars et al. (2017) considered ASL measurements taken at the Utah SLTEST facility over salt flats at \(Re_\tau \approx 1.4 \times 10^6\) (Marusic and Heuer 2007); further details on these types of experimental campaigns to study highReynoldsnumber wallbounded turbulence can be found in the literature (Metzger et al. 2007; Hutchins et al. 2012; Wang and Zheng 2016; Yang and Bo 2017). A wallnormal array of five sonic anemometers was employed, situated above a wallshearstress sensor. This unique setup made it possible to investigate the coupling between the outerregion turbulence and the nearwall footprint in the fluctuating friction velocity. In most other cases only a nearwall velocity measurement is available, as will be considered later. The coupling of fluctuations at different heights and scales was examined in spectral space using the linear coherence spectrum defined in Eq. 1. One coherence spectrum is obtained per velocitypair \(u(z_R)\)–u(z), where the height z ranges from \(z^+ {\approx }3500\) up to \(z/\delta \approx 0.03Re_\tau \) (corresponding to physical dimensions of \(z = 2\) to 5 m, Marusic and Heuer 2007). Figure 1a shows the five \(\gamma ^2\) spectra as a function of \(\lambda _x/z\). Note that the streamwise wavelength \(\lambda _x\) has been computed following \(\lambda _x \equiv U(z)/f\), where U(z) is the mean streamwise wind speed and f is the temporal frequency.
A coherence spectrogram is formed by presenting the five individual coherence spectra as isocontours of \(\gamma ^2\) in the \((\lambda _x,z)\) plane (Fig. 1b). The isocontours increase in value with increasing wavelength, and the contours follow lines of constant \(\lambda _x/z\) (slope of 1), reflecting the collapse of the individual spectra in Fig. 1a. For reference, the energy spectrogram of the streamwise velocity fluctuations is shown with filled isocontours underneath the \(\gamma ^2\) spectrogram. Energy is presented in premultiplied form \(k_x\phi _{uu}/U^2_\tau \), where \(\phi _{uu}\) is the onesided power spectrum of u. Evidently, only a portion of the energy is statistically coherent with the nearwall measurement (the portion of the energy residing below nonzero contours of \(\gamma ^2\)).
Figure 1 demonstrates that only increasingly larger scales remain coherent with \(z_R\) when z moves upward. This trend is consistent with Townsend’s attachededdy hypothesis in the formulation of Perry and Chong (1982), where a hierarchy of selfsimilar eddies is assumed as sketched in Fig. 2a. Each consecutive hierarchy is subject to an arbitrary scaling factor \(\chi \). Figure 2b shows an idealized wallattached turbulent structure with wallnormal extent l and streamwise scale \(\Updelta x\). In spectral space this relates to the aspect ratio \( \propto \Updelta x/l\). When interpreting the coherence footprint of such a structure, it is important to recall that the coherence metric relates signal contributions at the same scale \(\lambda _x \propto \Updelta x\). In the present application, this implies a parallelogramlike eddy structure as shown in Fig. 2b, for which \(\Updelta x\) is the same at all z. The widely observed phase difference in coherence metrics along the vertical direction (e.g. Panofsky et al. 1974; Marusic and Heuer 2007; Chauhan et al. 2013; Liu et al. 2017; Salesky and Anderson 2018) is equivalent to the inclination angle \(\alpha \) of this structure as indicated in the figure. However, our definition of the aspect ratio only depends on \(\Updelta x\) and l, i.e. the maximum walldistance at which coherence at scale \(\Updelta x\) is observed. The aspect ratio is hence solely determined by the magnitude of the coherence and is insensitive to a phase shift (or inclination) between the turbulent fluctuations at z and \(z_R\).
The implications of the flow organization shown in Fig. 2a on the coherence spectrum with reference at the wall (\(z_R=0\)) is depicted in Fig. 2c. For each eddy hierarchy, there exists a minimum characteristic streamwise wavelength \(\lambda _{x,i} \approx \Updelta x_i\) at which the structure appears coherent. Since eddies within the same hierarchy appear randomly in space (or time) (Woodcock and Marusic 2015), a nonzero coherence (and also turbulent energy) exists for scales larger than \(\lambda _{x,i}\) within that hierarchy. Thus, for hierarchy i with a wallnormal extent of \(\chi ^{(i1)}l\), a nonzero contribution \(\gamma ^2_i\) to the coherence occurs in the region \(z < \chi ^{(i1)}l\) and . The magnitude of \(\gamma ^2_i\) may vary with \(\lambda _x\) and z, but for simplicity we assume a constant magnitude represented by a uniform grey scale per the \(\gamma ^2_i\) isocontour in Fig. 2c. Our conclusions, however, remain unaffected by eventual variations in \(\gamma ^2_i\) as long as these remain selfsimilar across hierarchies as implied by the selfsimilarity of the underlying eddy field. The full coherence spectrogram finally results from superposing the contributions of all hierarchies as reflected by increasing grey scales of the superposed transparent rectangles in Fig. 2c. Within a triangular region in \((\lambda _x,z)\) space, bounded by a minimum wallnormal height \(z = l\), a constant \(\lambda _x/z\) limit (at small wavelengths) and a constant \(\lambda _x\) limit (large wavelengths), the \(\gamma ^2\) isocontours align with lines of constant \(\lambda _x/z\). Within this region, the magnitude of \(\gamma ^2\) increases linearly with \(\ln (\lambda _x)\) (for constant z) and decreases with \(\ln (z)\) (for constant \(\lambda _x\)): a direct consequence of a geometrically selfsimilar structure. This implies that as a consequence of the attachededdy hypothesis assumptions, the coherence magnitude within the selfsimilar region adheres to
where \(C_1\), \(C_2\) are fit constants. The aspect ratio then follows from
Based on laboratory data at \(Re_\tau \approx 14{,}000\), Baars et al. (2017) obtained \(C_1 \approx 0.302\) and \(C_2 \approx \,0.796\), which results in . These values were seen to be consistent with numerical data at \(Re_\tau \approx 2000\) (Sillero et al. 2013) and the ASL data at \(Re_\tau \approx 1.4 \times 10^6\) (Marusic and Heuer 2007) (see the corresponding trend line in Fig. 1). All cases represent the turbulent boundary layer under neutral stability conditions and were performed over smooth walls/terrain. Since (2) holds for three orders of magnitude in \(Re_\tau \), it can be concluded that a wallattached, selfsimilar structure is indeed ingrained in the u velocity field. Similar conclusions for more isolated ranges in \(Re_\tau \) were made by Morrison and Kronauer (1969) and Bullock et al. (1978) for pipe flow and more recently by del Álamo et al. (2004) for channel flow. Generally, all these results explicitly show the (wall)coherent nature of the turbulence in the logarithmic region (at ). In particular, for all ASLtype applications (\(Re_\tau \sim 10^5\) to \(10^6\)), where the relative range of scales that are coherent will grow, these results reflect largescale turbulent structures comprising significant lifetimes in the streamwise direction (Cantwell 1981; Robinson 1991; Hutchins and Marusic 2007).
Thus far, the coherence trend has been discussed only relative to the wall such that \(z_R\rightarrow 0\). However, in typical tower micrometeorological studies, the reference measurement is taken at \(z_R \sim 1\) m (which is well within the logarithmic region for typical atmospheric conditions). To illustrate how an offwall position at height \(z_R\) affects the idealized coherence trend in the attachededdy picture envisioned by Townsend, we increased the number of discrete hierarchies to 10 in Fig. 2d. For any given \(z_R\), only the wallattached turbulent structures that extend beyond \(z_R\) are coherent with \(z_R\) (their corresponding coherence contours are blueshaded). The coherence trend above \(z_R\) remains unaffected if only wallattached structures are considered.
As a final remark, we point out that the above considerations made for the streamwise velocity component should also apply to the spanwise velocity field and temperature (Perry and Chong 1982; Krug et al. 2018).
2.2 Comparing Davenport’s Hypothesis to Townsend’s
Davenport (1961a) presented a trend in the wallnormal coherence of streamwise velocity fluctuations u based on observations from typical tower micrometeorological data. It was evident from the data that both a decreasing wavelength \(\lambda _x \equiv U/f\) and an increase in vertical separation \(\Updelta z = z  z_R\) made the turbulent quantities less coherent. He hypothesized that for a given stability, the coherence should only depend on the ratio of \(\Updelta z\) and \(\lambda _x\). The implied geometrical selfsimilarity is obviously equivalent to the attached eddy framework discussed above if \(\Updelta z \approx z \), which is approached either for \(z_R \rightarrow 0\) or for \(z \gg z_R\) in experiments. The Davenport formulation, however, also entails selfsimilarity for any reference point, not just the wall, and therefore additionally encompasses also selfsimilarity of ‘detached’ structures. Noting that the dropoff in coherence with increasing \(\Updelta z/\lambda _x\) resembles an exponential decay, Davenport (1961a) gave the following empirical expression
where a is a decay parameter. Here, a factor of two is added in the exponent compared to the original formulation, as Davenport proposed the relation for \(\gamma \) (rootcoherence) and we use \(k \equiv 2a\) for brevity. Just as in other later works, we prefer \(\gamma ^2\), since the squared coherence is proportional to the fraction of energy that is coherent over \(\Updelta z\). With respect to the fitting constant in (4), Davenport initially quoted \(a = 7.7\) for the ‘vertical coherence’ (\(\Updelta z\) separations) of the u component in neutral conditions. Slightly updated values and extensions of the formulation to other velocity components and the temperature field have been reported in the ensuing literature (Pielke and Panofsky 1971; Davison 1976; Berman and Stearns 1977). It is generally accepted that a varies with surfacelayer stability in the sense that it is small in strong convection and large in neutral or stable air, and we address this aspect in the following. A representation of (4) with \(k = 23\) (Panofsky 1973; Naito and Kondo 1974) is included in Fig. 1a and is seen to match closely with (2) and the data for neutral conditions.
Since the introduction of (4) in 1961, many researchers have tested their data against Davenport’s hypothesis. Studies range from research focusing on the vertical coherence of various velocity components in tower micrometeorological data (Davenport 1961a; Panofsky and Singer 1965; Pielke and Panofsky 1971; Naito and Kondo 1974; Panofsky et al. 1974; Brook 1975; Seginer and Mulhearn 1978; Kanda and Royles 1978; Soucy et al. 1982; Bowen et al. 1983; Saranyasoontorn et al. 2004), investigations including the lateral/spanwise coherence (Kristensen and Jensen 1979; Ropelewski et al. 1973; Panofsky and Mizuno 1975; Perry et al. 1978; Kristensen 1979; Kristensen et al. 1981; Schlez and Infield 1998), the coherence of temperature fluctuations (Davison 1976) and even mesoscale applications (typically in the horizontal directions) (Hanna and Chang 1992; Woods et al. 2011; Vincent et al. 2013; Larsén et al. 2013; Mehrens et al. 2016). Together, these measurements cover a great variety of terrain and topography. Here we wish to restrict the discussion to the effect of stability and limit the analysis to the base case over smooth terrain, where high fidelity data are available from experiments at the Utah salt flats.
Before we introduce this dataset (see Sect. 3), we briefly consider stability effects in more detail. For this purpose, we have replotted the parametrizations according to (2) and (4) already included in Fig. 1a in Fig. 3 along with the Davenport parametrizations at varying k. As mentioned before, increasing stability corresponds to increasing k and the linear plot of \(\gamma \) vs. \(z/\lambda _x\) in Fig. 3a clearly illustratres how this leads to a faster decay of coherence. From the plot of \(\gamma ^2\) vs. \(\lambda _x/z\) in Fig. 3b it becomes apparent that a change in k approximately leads to a horizontal shift of the curves. In the framework of Baars et al. (2017), such a shift implies a change in aspect ratio of the wallattached structures and we investigate this aspect in more detail below.
3 Dataset
The experimental dataset employed herein has been recorded by Marusic and Heuer (2007), Marusic and Hutchins (2008) and was previously employed in a study of the neutral ASL by Hutchins et al. (2012) and investigation of the stability dependence of the structure inclination angle by Chauhan et al. (2013). We refer to these papers for details beyond the short overview provided here.
The complete dataset consists of a continuous recording over several days (26 May to 4 June 2005) at the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility in the salt flats of western Utah. A measurement tower held nine logarithmicallyspaced sonic anemometers (Campbell Scientific CSAT3) at distances ranging from 1.42 m to 25.69 m above ground, which recorded all three velocity components along with temperature. All measurements were synchronized and recorded at a sampling rate of 20 Hz. In addition, there was a spanwise array at \(z_s = 2.14\) m above ground with nine anemometers of the same type evenly spaced over 30 m from the tower. Data from this array are only employed here to characterize the stability of the ASL.
Prior to further analysis, the data are corrected for wind direction and a detrending procedure is applied (see Hutchins et al. 2012, for details). While the detrending is necessary to remove slow temporal trends in the data, it inherently also compromises the coherence at very long scales, which needs to be kept in mind when interpreting the data. After data selection a total of 63 1h long segments, corresponding to the dataset used in Chauhan et al. (2013), remains. The stability of each segment is characterized using the Monin–Obukhov stability parameter \(z_s/L\) with the Obukhov length scale
where the von Kármán constant \(\kappa = 0.41\), g is the acceleration due to gravity, and \(U_\tau = (\overline{uw})^{1/2}\) is the friction velocity. All quantities are evaluated from an average of the total of 10 sonic anemometers at \(z_s = 2.14\) m. Most of our data lie in the unstable regime \(z_s/L<0\) but a few data points also have \(z_s/L>0\).
Next, we will employ the SLTEST dataset to investigate how stability affects the selfsimilarity of coherent structures and how this changes the aspect ratio of the structures in the flow. We use the lowest measurement point as reference throughout, i.e. \(z_R = 1.41\) m from now on.
4 Results
4.1 Stability Dependence of the SelfSimilar Scaling
We start out by considering a representative case with stable stratification \(z_s/L>0\) in Fig. 4a. Evidently, the stable stratification has a significant effect on the spectral energy distribution of the streamwise velocity fluctuations. Even more importantly, however, these changes are seen to also propagate to the coherence spectrogram. While the coherence levels are generally lower, the isolines are also seen to deviate notably from a slope of 1, which would be expected for selfsimilarity as discussed above (recall Fig. 1). This equally holds for plotting \(\gamma ^2\) vs. z (as implied by the attachededdy hypothesis), as well as vs. \(\Updelta z\) following Davenport’s hypothesis. The same observations could be made for the other stable data points (in total we have 11), other velocity components and temperature (not shown here). Remarkably, this is already the case for relatively moderate values of \(z_s/L\). In fact, the case at \(z_s/L= 0.10\) shown in Fig. 4 corresponds to the most stable conditions in the dataset. Based on these findings, the application of both (2) and (4) does not appear justified for stable stratification and will not be pursued further.
The situation is different for unstable configurations, where the heat flux is directed upward, as can be seen from Fig. 4b. Also, here, the turbulence field is significantly affected by buoyancy as evidenced by considerably higher fluctuation levels. In contrast to the stable case, however, the coherence isocontours in this case are observed to follow a slope of 1 for large enough \(z^+\) consistent with the presence of selfsimilar wallattached structures. The fact that such a scaling is only obtained for sufficiently large \(z^+\) is to be expected since, for small separations, also structures that are not attached to the wall will be coherent (trivially \(\gamma ^2 = 1\) at all scales \(z = z_R\)) and pure wallscaling is only recovered once their influence has decayed. For large \(\Updelta z\), there is no difference between plotting \(\gamma ^2\) as a function of z and \(\Updelta z\), consistent with the expectation at \(\Updelta z\gg z_R\). A more interesting observation can be made at smaller \(\Updelta z\), for which \(\Updelta z\gg z_R\) does not hold. Even for this range, the slope of the \(\gamma \) isocontours is now approximately one, indicative of selfsimilar scaling, all the way to the smallest vertical separation distance (noting that \(\Updelta z = 0\) is not shown due to the logarithmic axis). This means that selfsimilarity is now also observed where it was compromised by the contribution of nonattached structures when plotting with reference to the wall. The extended scaling region is therefore, in fact, indicative of selfsimilarity of nonwallattached structures as suggested by the formulation of Davenport. Judging by the fact that the red lines for the rampup in coherence are relatively straight, detached and wallattached structures have the same aspect ratios. As a side note to the discussion here, we point out that the eventual decay of coherence for \(\lambda _x>10^6\) is likely an artefact of the detrending procedure since a similar effect was not observed in laboratory data (see Baars et al. 2017).
Our dataset allows us to investigate selfsimilar scaling also in the spanwise velocity component and the temperature fields. Such data are not commonly available in laboratory experiments and have therefore not been are used previously in this context. We plot the results corresponding to the unstable case in Fig. 4b but now for v and \(\theta \) in Figs. 5a, b, respectively. With maybe some minor limitations in the scaling vs. \(z^+\) in the vcomponent, the situation is very similar to that observed in the plot for the streamwise component in Fig. 4b. This provides evidence of selfsimilar scaling also in the spanwise velocity component and the temperature field. Especially the fact that the temperature field adheres to a similar geometrical organization appears remarkable in view of how different the energy spectrogram looks in this case. Compared to the velocity counterparts, the relative contributions at high \(\lambda _x\), as well as further away from the wall, are considerably lower for the scalar spectrogram.
A more quantitative analysis is best achieved by replotting the data in the style of Fig. 1a. This is done in Fig. 6 for the two velocity components and the temperature in the unstable case with \(z_s/L = 2.80\). Note that here we use the normalization with \(\Updelta z\) as this provides a more extensive scaling region. Using z instead provides similar results—albeit with larger uncertainties. Plotting the results as presented in Fig. 6 scrutinises the aspect of selfsimilarity, since for this case a collapse of curves at different \(\Updelta z\) is expected. As can be seen from the figure, such a nominal collapse is indeed observed for all quantities plotted. Identifying the scaling region by the range of this collapse (marked in red in the figure) allows us to fit the expression (2) to the data and to extract the statistical aspect ratio. These results are also included in Fig. 6 and it is obvious that the resulting to is significantly lower than the value of obtained under neutrally stable conditions. It therefore seems that unstable conditions drastically reduce the aspect ratio of coherent structures in the flow.
We systematically investigate this trend by applying the same fitting procedure to all unstable datasets available. The results obtained from doing so are presented for u, v, and \(\theta \) in Fig. 7a–c. In all three cases, there is a clear trend of decreasing with increasing magnitude of the stability parameter up to \(z_s/L \approx 1\). For even higher values of \(z_s/L\), the aspect ratio consistently attains approximately constant values in all three quantities in the range 2 to 3. It is worth highlighting how sensitive the aspect ratio is to even only weakly unstable conditions. Indeed, the data appear well represented by an entirely empirical, logarithmic fit for \(z_s/L <1\). The greyshaded region corresponds to \(z_s/L<0.1\), which is a commonly used limit for approximately neutral conditions (e.g. Högström et al. 2002; Metzger et al. 2007; Hutchins et al. 2012). As our results show, varies by up to 50% in this region, lending support to more stringent criteria such as \(z/L<0.06\) applied in Wang and Zheng (2016) at least for certain statistical quantities. We do note, however, that the precise value of the stability parameter depends on the choice of the reference height z (we use \(z = 2.14\) m) as L is determined by wall quantities only. This choice is somewhat arbitrary for the multipoint statistics presented here and was motivated by data availability and consistency with previous studies in the present case. The dependence is linear however, and the values can therefore easily be transformed to other reference heights.
Finally, we briefly comment on the differences in the aspect ratios obtained from the three different quantities. We already established from Fig. 7a–c that the general trends are consistent across u, v and \(\theta \). In Fig. 7d, we plot and with respect to , and in general all ratios are close to 1. It will require further research to determine whether the slight deviations from 1, e.g., both and tend to be slightly < 1 for \(z_s/L<1\), are statistically significant or simply owed to inaccuracies in determining . Differences between the coherence decay in different velocity components have nevertheless also been reported in the windengineering community where Berman and Stearns (1977) and similarly Pielke and Panofsky (1971) report somewhat smaller decay rates (lower k) for the vcomponent as compared to the ucomponent.
4.2 Relationship Between and k
The relationship between the statistical aspect ratio derived based on (2) and the parameter k in (4) is addressed in the following. An analytical relationship is readily derived by setting (2) and (4) equal at a reference point \(\xi \equiv {\lambda _x}/{z}\large _{\text {ref}}\) and assuming \(z = \Updelta z\). In this case, we obtain
and from Fig. 6, \(\xi = 20\) is a reasonable choice for the present data as it corresponds to \(\gamma ^2 \approx 0.5\). A representation of (6) with this parameter is included in Fig. 8a along with results for \(k_i\) obtained from fitting (4) to the same data points used to determine (here and in the following \(i = u,v,\theta \)). Overall, the agreement of (6) with the data is very good and there is little scatter even between different components and the scalar. For the region of interest, the somewhat unwieldy relation (6) is very well approximated by a linear function and the corresponding linear dependence is included in Fig. 8a. Figure 8b and c compare our results for \(k_u\) and \(k_v\) to values reported in the windengineering literature and collected by Panofsky (1973). In these figures the region corresponding to stable conditions is shaded as our detailed analysis revealed a lack of selfsimilar scaling as implied by both (2) and (4). The scatter in the literature data is considerable between different observation sites likely reflecting differences in topography. In this sense, our data from the SLTEST can be considered as a reference case with minimal roughness and topographical effects. Consequently, the scatter in the present data is significantly lower. For both, u and v, the new data points are largely consistent with the dashed lines labelled ‘Trend others’, which is a trend line (by eye) for all measurements but those at the Kennedy Space Center drawn by Panofsky (1973). The data from the Kennedy Space Center (and the corresponding trend lines) lie at consistently and significantly higher values of \(k_i\) than our results. Nonetheless, the fact that \(k_i\) levels off at an approximately constant value beyond \(z_s/L=1\) is a feature that appears to be shared by all available datasets. Once again, we emphasize that the exact values of the stability parameter might change slightly depending on which reference height is chosen; unfortunately, information on the reference heights could not be retrieved for all literature datasets introducing an element of uncertainty.
5 Concluding Remarks
The main findings of the present study are summarized as follows:

We have demonstrated that implications of Townsend’s attachededdy hypothesis for the coherence trend are consistent with Davenport’s hypothesis. This applies to the geometrical selfsimilarity of wallattached structures as well as to the fact that the empirically derived exponential decay in Davenport’s formulation matches closely with a logarithmic expression that follows directly from the aspect of selfsimilarity.

The selfsimilarity implied by Davenport is even more comprehensive and also encompasses structures that are not attached to the wall. Evidence of such a selfsimilar behaviour could indeed be observed for the high \(Re_\tau \) SLTEST data employed herein.

The selfsimilarity assumptions/hypotheses do not seem to hold for stable data. Neither z nor \(\Updelta z\)scaling is observed in this case, which implies that there is no selfsimilarity for stable data. This is clearly observed in our results since we compute the coherence spectrum (a continuous function of scale with a finely frequencydiscretized fast Fourier transform approach). In the literature, the coherence is often computed at coarsely spaced frequency discretizations with fits based on a few data points only. Our results provide clear evidence that the stable ASL has no selfsimilar coherence and hypotheses of Townsend and Davenport should not be applied in this case. We point out that it is predominantly the selfsimilarity aspect that fails in the stable regime while there is still nonzero coherence. The departure from selfsimilarity occurs far from extreme (‘zless’) conditions at relatively weak stratification, for which Monin–Obukhov similarity theory holds. While we do not observe selfsimilarity for any of our stable data, it appears likely that for very weak stable stratification selfsimilarity may be recovered. Unfortunately, we cannot determine such a threshold from the present dataset.

Consistent with expectations based on the attachededdy hypothesis framework, selfsimilar scaling was not only observed for the ucomponent, but also for spanwise velocity fluctuations v and temperature fluctuations \(\theta \).

Even relatively weak unstable stratifications drastically reduce the statistical aspect ratio for all quantities investigated here. Based on our results, we were able to parametrize this trend in terms of a logarithmic decay for \(z_s/L<1\) and constant values for \(z_s/L>1\).

Generally, the trend of decreasing with decreasing stability is intuitively consistent with the fact that buoyancy supports the upward motion of structures from the wall. The question remains as to whether the nature of the structures themselves changes under very unstable conditions (e.g. towards convection celltype motion) as may be suspected based on the change in trend for around \(z_s/L =1\). A conclusive answer in this regard cannot be provided from the present analysis. It is, however, remarkable that the slope \(C_1\) is largely insensitive to \(z_s/L\) in our data. Physically, the parameter \(C_1\) can be interpreted as a measure for the relative contribution of attached structures to the overall turbulence intensity. The fact that this quantity remains unaltered seems to indicate that, at least in the parameter range accessed here, the fundamental flow organization does not change significantly. This notion is substantiated by the observation that also selfsimilarity still holds in the unstable regime.

We established a simple linear relation between the aspect ratio and k in the Davenport formulation, such that the fit parameter k can be interpreted as an aspect ratio. Comparison of our results for the stability dependence of k with the literature reveals significantly lower scatter for the highfidelity ASL data over smoothterrain presented here. As such, the present dataset serves as a basecase for the vertical coherence over any other type of terrain.
As a final remark, we point out that even though the present study is limited to vertical coherence, connecting Davenport’s to Townsend’s framework for this case also places predictions and applications to horizontal coherence on a stronger footing.
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Acknowledgements
The authors acknowledge financial support by the Australian Research Council and by the University of Melbourne through the McKenzie fellowship program. We further thank Dr. Kapil Chauhan for making the detrended data available to us.
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Krug, D., Baars, W.J., Hutchins, N. et al. Vertical Coherence of Turbulence in the Atmospheric Surface Layer: Connecting the Hypotheses of Townsend and Davenport. BoundaryLayer Meteorol 172, 199–214 (2019). https://doi.org/10.1007/s10546019004454
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DOI: https://doi.org/10.1007/s10546019004454