Vertical Coherence of Turbulence in the Atmospheric Surface Layer: Connecting the Hypotheses of Townsend and Davenport
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.
Keywords
Atmospheric stability Atmospheric surface layer Eddy structure Spectral coherence1 Introduction and Context
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.
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 \(\gamma ^2\)).
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
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.
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
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).
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 Open image in new window and Open image in new window with respect to Open image in new window , 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 Open image in new window and Open image in new window tend to be slightly < 1 for \(z_s/L<1\), are statistically significant or simply owed to inaccuracies in determining Open image in new window . 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 Open image in new window and k
5 Concluding Remarks

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 Open image in new window 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 Open image in new window 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 Open image in new window 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 Open image in new window 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.
Notes
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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