3.1.1. Vertical profiles

Figure 2 shows the vertical profiles of some flow statistics plotted against the normalized height ($z/h$). Unless otherwise specified, the statistical quantities in figure 2 and in the rest of the article are ensemble averages over the selected 93 data blocks from near-neutral conditions, and the velocity scale $u_{*}$ is the friction velocity computed at the canopy top. To denote the spread, the error bars in figure 2 represent one standard deviation from the ensemble mean.

Figure 2.

Ensemble-averaged profiles of (a) mean velocity ($\bar {u}/u_{*}$), (b) velocity variances ($\sigma ^{2}_{\alpha }/u^{2}_{*}$, where $\alpha =$ $u$, $v$ or $w$), (c) momentum fluxes ($\overline {\alpha ^{\prime }w^{\prime }}/u^{2}_{*}$, where $\alpha =u$ or $v$), (d) correlation coefficients ($r_{\alpha w}$), (e) flux fractions ($\overline {u^{\prime }w^{\prime }}_{X}/u^{2}_{*}$) and ( f) time fractions ($T_{f\,X}$) corresponding to each of the four $u^{\prime }$–$w^{\prime }$ quadrants (here ${X}$ denotes any one of the four quadrants). The ordinate axis is the normalized height $z/h$, and $u_{*}$ is the friction velocity at the canopy top. The error bars in all the panels denote one standard deviation from the averaged values.

Overall, the statistics depicted in figure 2 are in agreement with field observations and large-eddy simulation experiments reported in previous studies (e.g. Finnigan Reference Finnigan2000; Dias-Junior, Marques Filho & Sa Reference Dias-Junior, Marques Filho and Sa2015; Patton et al. Reference Patton, Sullivan, Shaw, Finnigan and Weil2016; Banerjee, De Roo & Mauder Reference Banerjee, De Roo and Mauder2017; Brunet Reference Brunet2020). The normalized profile of the mean wind speed ($\bar {u}/u_{*}$) displays a sharp gradient in the upper part of this dense canopy, and attains a value of $\bar {u}/u_{*}=4$ at $z/h=1$, increasing up to 6 at $z/h=1.38$ in the RSL (figure 2a). The streamwise, cross-stream and vertical velocity variances ($\sigma _{u,v,w}^{2}/u^{2}_{*}$) also remain close to zero at heights deep within the canopy but approach the near-neutral values of $\sigma _{u,v}/u_{*} \approx 2$ and $\sigma _{w}/u_{*} \approx 1.25$ (Finnigan Reference Finnigan2000) at heights $z/h \geq$ 1 (figure 2b). Interestingly, the cross-stream velocity variances ($\sigma _{v}^{2}/u^{2}_{*}$) agree well with the near-neutral limit, while a small deviation is observed for the streamwise ones ($\sigma _{u}^{2}/u^{2}_{*}$).

The strong shear in $\bar {u}$ is reflected in the streamwise momentum flux ($\overline {u^{\prime }w^{\prime }}/u^{2}_{*}$) profile (figure 2c), where $\approx 90\,\%$ of momentum is absorbed in the upper 30 % of the canopy, with a constant value atop ($|\overline {u^{\prime }w^{\prime }}| \to u^{2}_{*}$ at $z/h \geq 1$). On the other hand, the cross-stream momentum fluxes ($\overline {v^{\prime }w^{\prime }}$) remain nearly zero throughout the canopy heights. This suggests, for our near-neutral canopy flow set-up, that the shear stress vector aligns well with the mean wind direction (Pan & Patton Reference Pan and Patton2017). If one considers the correlation coefficients ($r_{uw}$ and $r_{vw}$), the $r_{uw}$ values remain substantially larger than $r_{vw}$ while being slightly positive at the lowest three levels (figure 2d). The slight positive values of $r_{uw}$ at the lowest canopy heights agree well with the observations of Dupont & Patton (Reference Dupont and Patton2012). Moreover, at heights $z/h \geq 1$, $r_{uw}$ approximately approaches $- 0.4$, in agreement with Finnigan (Reference Finnigan2000) and Dupont & Patton (Reference Dupont and Patton2012). Hereafter, while referring to the momentum flux, we only imply $u^{\prime }w^{\prime }$.

Although in subsequent sections more information is provided about $u^{\prime }w^{\prime }$, the simplest way to start is by investigating the quadrant contributions to the time-averaged momentum flux values and how much time does the flow spend in each quadrant (see table 1 explaining the four different quadrants). To compute the amount of flux carried by each quadrant and the fraction of time spent in each, we use

(3.1)\begin{equation} \left. \begin{aligned} \frac{\overline{u^{\prime}w^{\prime}}_{X}}{u^{2}_{*}} & =\frac{{\sum}[(u^{\prime}w^{\prime})I_{X}]}{Nu^{2}_{*}},\\ T_{f\,X} & =\frac{{\sum}{I_{X}}}{N} \quad (X=E,S,OI,II), \end{aligned} \right\} \end{equation}

where

(3.2)\begin{equation} I_{X} = \begin{cases} 1, & \text{if }\{u^{\prime},w^{\prime}\} \in X,\\ 0, & \text{otherwise}, \end{cases} \end{equation}

and $N$ is the total number of points in a 30 min run (36 000 in our case). The results related to these aspects are presented in figure 2(e, f). By evaluating figure 2(e, f), it can be inferred that the differences in characteristics between the ejection and sweep motions remain most prominent for the mid-canopy heights, such as $z/h=0.63$, 0.70 and 0.90. However, as $z/h$ approaches 1 and above, this distinction becomes hardly noticeable. Specifically, at heights $z/h=0.63$ to 0.90, the flux contributions from the sweep quadrant remain larger than those from the ejection quadrant (figure 2e), despite the fact that the flow spends more time in the ejection quadrant (figure 2f). Therefore, at those heights, the short-lived sweeps appear to be a little more intense than the long-lived ejections.

3.1.2. P.d.f.s and quadrant analysis

To further unravel the statistical character of the velocity fluctuations ($u^{\prime }$ and $w^{\prime }$), we present the p.d.f.s of these quantities in figure 3(a,b). It is worth remarking that in these figures and the ones that follow, grey squares of various shades denote the heights within the canopy ($z/h<1$), whereas the heights at and above the canopy ($z/h \geq 1$) are represented by blue, red and pink circles. Before computing these p.d.f.s, we normalized $u^{\prime }$ and $w^{\prime }$ with their respective standard deviations, so that the results can be compared with a standard Gaussian distribution (see the cyan dash-dotted lines in figure 3a,b). In figure 3(a,b), the p.d.f.s $P(u^{\prime }/\sigma _{u})$ and $P(w^{\prime }/\sigma _{w})$ develop a prominent heavy tail (with respect to a Gaussian) towards the positive and negative side, respectively, at $z/h$ values between 0.63 and 0.90. Other than these heights, the p.d.f.s mostly remain close to Gaussian.

Figure 3.

(a,b) P.d.f.s for the normalized (a) streamwise velocity fluctuations $u^{\prime }/\sigma _{u}$ and (b) vertical velocity fluctuations $w^{\prime }/\sigma _{u}$. The standard Gaussian distribution is overlaid on (a) and (b) as cyan dash-dotted lines. (c,d) Vertical profiles of (c) skewness $\mathcal {S}_{u}$ and $\mathcal {S}_{w}$ and (d)  excess kurtosis $\mathcal {K}_{u}-3$ and $\mathcal {K}_{w}-3$ of the $u^{\prime }$ and $w^{\prime }$ signals. The three remaining panels demonstrate (e) p.d.f.s of instantaneous momentum fluxes $u^{\prime }w^{\prime }/u^{2}_{*}$, (f ) p.d.f.s of quadrant angles $\theta _{u^{\prime }w^{\prime }}$ and (g) quadrant distributions of $u^{\prime }w^{\prime }$. The descriptions of the different markers for the heights within (grey squares) and above (blue, red and pink circles) the canopies are depicted in the legend of panel (e). The dash-dotted green horizontal line in panel (f ) denotes the p.d.f. of a uniform distribution.

In addition to the p.d.f.s, we present the skewness ($\mathcal {S}_{u,w}$) and kurtosis ($\mathcal {K}_{u,w}$) of $u^{\prime }$ and $w^{\prime }$ signals (figure 3c,d). Physically, skewness is associated with asymmetry in the p.d.f.s whereas kurtosis is related to intermittency. For a non-intermittent signal, kurtosis is 3, which corresponds to a Gaussian distribution. Therefore, to imply intermittency, instead of showing the kurtosis values directly, one can subtract 3 from them (figure 3d). Regarding skewness, the positive (negative) values denote whether the signal is more dominated by the large positive (negative) fluctuations, i.e. if the p.d.f. has a fat tail towards the positive (negative) side. From both p.d.f.s and skewness plots (figure 3a–c), it is abundantly clear that, at mid-canopy heights, the fluctuations in $u^{\prime }$ and $w^{\prime }$ signals are influenced by large positive and negative values, respectively. Because such extreme values are present, $\mathcal {K}_{u,w}$ values at $z/h=0.63$, 0.70 and 0.90 remain well above 3, thereby indicating strongly intermittent signals (figure 3d). Moreover, for both $u^{\prime }$ and $w^{\prime }$, the maximum $\mathcal {S}_{u,w}$ ($\approx \pm 1$) and $\mathcal {K}_{u,w}$ values ($\approx 7$) are located at a height $z/h=0.63$. This observation is in agreement with Dupont & Patton (Reference Dupont and Patton2012), which they interpreted as an indication of the penetration by the canopy-scale coherent structures resulting in strong sweeps and weak ejections at mid-canopy heights. Our results from figure 2(e, f) support this interpretation.

We next investigate how the height variations in $P(u^{\prime }/\sigma _{u})$ and $P(w^{\prime }/\sigma _{w})$ impact the distributions of the instantaneous momentum flux. Since we are interested in fluxes and not in correlation coefficients, $u^{2}_{*}$ is used as a scaling factor for the p.d.f.s of $u^{\prime }w^{\prime }$ rather than $\sigma _{u}$ and $\sigma _{w}$ (figure 3e). While observing figure 3(e), one can notice that $P[(u^{\prime }w^{\prime })/u^{2}_{*}]$ gradually develops large negative tails with increasing $z$, indicating a more frequent occurrence of the extreme values in $-u^{\prime }w^{\prime }$. This can be further confirmed through the vertical profiles of skewness in $u^{\prime }w^{\prime }$ (not shown). Such plots corroborate the fact that, except at the lowest three measurement heights, the skewness values are negative, meaning that large negative events (ejections and sweeps) dominate the $u^{\prime }w^{\prime }$ signal. Therefore, to better understand the role of ejections and sweeps in momentum transport, we introduce polar quadrant analysis. For a detailed description of this particular methodology, see Appendix A and figure 10 therein.

In figure 3( f) we show the ensemble-averaged phase angle p.d.f.s $P(\theta _{u^{\prime }w^{\prime }})$, corresponding to all nine measurement heights. The dotted green line in figure 3( f) represents a uniform distribution $P(\theta _{u^{\prime }w^{\prime }})=1/(2{\rm \pi} )$, whose significance will be discussed in figure 7(a–c). From figure 3( f) one can see that, for the lowest three heights, the behaviour of $P(\theta _{u^{\prime }w^{\prime }})$ is mainly dominated by the two peaks residing in the outward- and inward-interaction quadrants. But as the heights increase, the peaks in $P(\theta _{u^{\prime }w^{\prime }})$ shift towards the ejection and sweep quadrants. Moreover, at mid-canopy heights ($z/h$ between 0.63 and 0.90), the peaks of $P(\theta _{u^{\prime }w^{\prime }})$ remain larger for the ejection quadrant. This is consistent with figure 2( f), where one observed at those heights that the ejection motions occupied more time with respect to the other three quadrants. Apart from that, at heights $z/h \geq 1$, the sweep and ejection peaks in $P(\theta _{u^{\prime }w^{\prime }})$ are confined to the angles $-{\rm \pi} /4<\theta _{u^{\prime }w^{\prime }}<0$ and $3{\rm \pi} /4<\theta _{u^{\prime }w^{\prime }}<{\rm \pi}$, respectively. This outcome has an important consequence when one considers the flux distributions associated with the phase angles.

The flux distributions (see (A5) in Appendix A) are shown in figure 3(g), where they are normalized by $u^{2}_{*}$ and ensemble-averaged over all the near-neutral runs. By attentively observing the flux behaviours at heights $z/h= 0.63$ to 0.90, one can conclude that, while the p.d.f. peaks remain larger for the ejections (figure 3f), the flux contributions are higher for the sweeps (figure 3g). Note that this apparent distinction between the ejection and sweep quadrants gradually fades as one approaches the heights $z/h \geq 1$ (figure 3f ,g). All such findings are in accordance with figure 2(e, f). Furthermore, from figure 3( f ,g), it can be seen that the maximum flux contributions and p.d.f. peaks from the ejection and sweep quadrants are concentrated around those phase angles that lie on the left-hand side plane of the $-45^{\circ }$ slope line (see the dashed black line in figure 10). This indicates that the events from ejection and sweep quadrants, which contribute most to the momentum flux, are in fact associated with larger magnitudes of $u^{\prime }$ than $w^{\prime }$. Such features appear more prominent at heights $z/h \geq 1$, thereby promoting the increasing importance of $u^{\prime }$ fluctuations on the momentum transport as the heights approach the canopy top.

Hitherto, the statistics of velocity fluctuations ($u^{\prime }$ and $w^{\prime }$) and momentum ($u^{\prime }w^{\prime }$) are presented for heights within and above the GoAmazon canopy. The growing importance of ejection and sweep motions with increasing heights, and their differing contributions towards the time fractions and momentum fluxes, lead us to ask: Is there any particular time scale involved that differentiates the ejection and sweep characteristics observed so far? Answering such a question is not straightforward, since, during quadrant analysis, any time dependence is masked and therefore no information can be obtained about the time scales of the associated turbulence structures. Additionally, the ejection and sweep events are not distributed uniformly over time but rather appear intermittently (Narasimha et al. Reference Narasimha, Kumar, Prabhu and Kailas2007). To overcome these issues, persistence analysis is proposed, which can evaluate the time scales of the intermittently occurring momentum-transporting events.

3.2.1. Persistence p.d.f.s of velocity fluctuations

In figure 4(a,b), the persistence p.d.f.s of $u^{\prime }$ and $w^{\prime }$ signals are presented in log–log plots for all the nine measurement heights. Note that, before plotting the p.d.f.s, the persistence time $t_{p}$ was scaled with a global time scale $h/u_{*}$ and denoted as $\tau _{p}$. Since $h/u_{*}$ is height-invariant, such a scaling of $t_{p}$ would allow one to investigate how $P(\tau _{p})$ changes with height. This scaling also has a distinct advantage if one compares the results with those of Chamecki (Reference Chamecki2013), who used a similar strategy to normalize the $u^{\prime }$ and $w^{\prime }$ persistence p.d.f.s while studying non-Gaussian turbulence in canopy flows. To distinguish the heights within and above the canopy, an identical colour scheme to that in figure 3 is used in figure 4 as well.

Figure 4.

(a,b) Persistence p.d.f.s for the events in (a) $u^{\prime }$ and (b) $w^{\prime }$, corresponding to the heights within and above the canopy. The normalized persistence time scales $(t_{p}u_{*})/h$ are denoted as $\tau _{p}$ and their p.d.f.s are constructed over the ensemble of 93 half-hour blocks from near-neutral conditions. A power law of exponent $-1.5$ is fitted to the persistence p.d.f.s of $u^{\prime }$ events and shown as dash-dotted black lines. The green shaded regions in (a,b) denote the range of $\overline {\tau _{p}}$, over which it varies across the nine heights. (c,d) Vertical profiles of normalized integral time scales ($\varGamma _{x}$, where $x=u$ or $w$) for $u^{\prime }$ and $w^{\prime }$ signals, respectively. (e, f) Persistence p.d.f.s of $u^{\prime }$ and $w^{\prime }$, but where the time scales ($t_{p}$) are normalized by $\varGamma _{x}$. The grey shaded regions in (e, f) correspond to time scales $t_{p}/\varGamma _{x} \geq 1$. The power laws with exponents $-1.6$ and $-1.25$ (dash-dotted green lines) are fits redrawn from the SLTEST data (Chowdhuri et al. Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a).

By inspecting $P_{u^{\prime }}(\tau _{p})$ and $P_{w^{\prime }}(\tau _{p})$, one could immediately notice that these p.d.f.s follow a power law for nearly two decades up to a time scale $\tau _{p}<0.1$, and then there is an exponential drop. Although not explicitly mentioned, the plots of Chamecki (Reference Chamecki2013) suggest the same limit for a power-law distribution in the persistence p.d.f.s of $u^{\prime }$ and $w^{\prime }$. Moreover, Chamecki (Reference Chamecki2013) pointed out that in his dataset – at short time scales – the persistence p.d.f.s of either $u^{\prime }$ or $w^{\prime }$ did not display any substantially different behaviour with increasing heights. For the GoAmazon data, we also observe the same up to the time scales $\tau _{p}<0.1$.

Interestingly, the inter-event distribution times between earthquakes have also been reported to follow a power-law distribution and the range is found to be limited up to the mean values of such times (e.g. Kumar et al. Reference Kumar, Korkolis, Benzi, Denisov, Niemeijer, Schall, Toschi and Trampert2020). However, for canopy turbulence, the threshold time scale ($\tau _{p}=0.1$) up to which the power law exists in $P_{u^{\prime }}(\tau _{p})$ and $P_{w^{\prime }}(\tau _{p})$ is nearly 10 times larger than the mean values of $\tau _{p}$, denoted as $\overline {\tau _{p}}$. To illustrate this point, the green-shaded regions in figure 4(a,b) indicate the range of possible values over which $\overline {\tau _{p}}$ would vary when all the nine measurement heights are considered together. As one could notice from the green-shaded regions, for both velocity signals, $\overline {\tau _{p}}$ displays little to no variations with height ($\overline {\tau _{p}}=0.01$ to 0.03) with the values being significantly smaller than $\tau _{p}=0.1$. This apparent difference with earthquakes suggests that the characteristics of the persistence p.d.f.s are sensitive to the details of the physical system for which they are evaluated. Next, we provide further evidence that this sensitive dependence of persistence p.d.f.s on the configurations of the underlying system remains true even when one considers a turbulent flow setting.

For instance, the results of Chamecki (Reference Chamecki2013) were obtained for a near-neutral turbulent flow over a corn canopy having $h= 2.05$ m and an LAI of 3.3 m$^{2}$ m$^{-2}$. However, unlike Chamecki (Reference Chamecki2013), we do not observe any significant difference in the features of persistence p.d.f.s when positive and negative fluctuations in $u^{\prime }$ and $w^{\prime }$ are considered separately. From supplementary figure S1(a,b) (available at https://doi.org/10.1017/jfm.2022.414) one could infer that the p.d.f.s $P_{u^{\prime }>0}(\tau _{p})$ and $P_{u^{\prime }<0}(\tau _{p})$ behave almost similarly to $P_{u^{\prime }}(\tau _{p})$ in figure 4(a). An identical conclusion can be drawn if one considers $w^{\prime }$ signals (see figure S1c,d). Therefore, the statistical character of the persistence time scales of $u^{\prime }$ and $w^{\prime }$ signals appears to be different between the GoAmazon canopy and corn canopy of Chamecki (Reference Chamecki2013). Given this premise, one may additionally ask whether this difference also gets reflected in the power-law exponents of the persistence p.d.f.s, an aspect that deserves a thorough investigation.

In our case, since the power-law portions of persistence p.d.f.s collapse with height, a single exponent can be determined by performing a linear regression for the range $10^{-3} \leq \tau _{p} \leq 10^{-1}$ on the log–log plots of $P_{u^{\prime }}(\tau _{p})$. Although this is regarded as standard practice (e.g. Katul et al. Reference Katul, Albertson, Parlange, Chu and Stricker1994; Chowdhuri et al. Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a; Benavides et al. Reference Benavides, Deal, Rushlow, Venditti, Zhang, Kamrin and Perron2022), estimating the accuracy of the power-law exponent is not a trivial task and involves a lot of complicated steps (Clauset, Shalizi & Newman Reference Clauset, Shalizi and Newman2009). Nevertheless, to tackle this issue, we adopted a simple procedure. We first carried out the linear regressions for all the nine heights, thereby providing nine values of power-law exponents associated with each $z/h$. Note that, for all such fittings, the $R^{2}$ values were more than 0.98. In the next step, to estimate the error, we computed the spread (standard deviation) around the mean over nine values of power-law exponents, eventually yielding a value of $-1.5 \pm 0.03$. For illustration purposes, the black dash-dotted line in figure 4(a) represents the mean, which is $-1.5$. A similar process was repeated for $P_{w^{\prime }}(\tau _{p})$, and no discernible difference in the power-law exponent was observed when compared with $P_{u^{\prime }}(\tau _{p})$ (black dash-dotted lines in figure 4a,b).

Since Chamecki (Reference Chamecki2013) did not provide the values of power-law exponents, we compare our findings with other studies conducted on canopy turbulence (Cava & Katul Reference Cava and Katul2009; Lee Reference Lee2011; Cava et al. Reference Cava, Katul, Molini and Elefante2012) and on an aerodynamically smooth ASL flow. Specifically, for the ASL flow, we directly use the Surface Layer Turbulence and Environmental Science Test (SLTEST) experimental dataset, whose details are underlined in Chowdhuri et al. (Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a,Reference Chowdhuri, Kumar and Banerjeeb) and Chowdhuri, Prabhakaran & Banerjee (Reference Chowdhuri, Prabhakaran and Banerjee2020c). The SLTEST site was flat, horizontally homogeneous barring any obstacles, and having a smooth surface with an aerodynamic roughness length less than 5 mm (Chowdhuri et al. Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a,Reference Chowdhuri, Kumar and Banerjeeb,Reference Chowdhuri, Prabhakaran and Banerjeec). Therefore, the SLTEST dataset provides a perfect platform to compare the persistence properties with the GoAmazon site, which has roughness elements in the form of a tall canopy. As noted in previous studies, the near-neutral runs from the SLTEST experiment were selected based on the criterion $-z/L<0.2$ (Chowdhuri et al. Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a,Reference Chowdhuri, Kumar and Banerjeeb,Reference Chowdhuri, Prabhakaran and Banerjeec).

Without employing any scaling, the studies by Cava & Katul (Reference Cava and Katul2009) and Lee (Reference Lee2011) discovered that, in a near-neutral canopy flow set-up ($h=28$, 14 and 8.5 m and $\textrm {LAI}= 9.6$, 3 and 1 m$^{2}$ m$^{-2}$), the persistence p.d.f.s of velocity components showed a log-normal distribution. On the other hand, the studies by Cava et al. (Reference Cava, Katul, Molini and Elefante2012) and Chowdhuri et al. (Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a) used the Eulerian integral time scales ($\varGamma _{x}$, $x=u$ or $w$) to normalize $t_{p}$, due to the following physical reasons. First, the use of $\varGamma _{x}$ allows one to judge whether the $t_{p}$ values are smaller or larger than the inertial subrange, since the scales smaller than $\varGamma _{x}$ are generally associated with such behaviours (Kaimal & Finnigan Reference Kaimal and Finnigan1994; Manshour et al. Reference Manshour, Anvari, Reinke, Sahimi and Tabar2016). Second, the power-law exponents of the persistence p.d.f.s are typically explained in a framework of self-organized criticality and connected with the spectral slope in the inertial subrange, subjected to corrections accounting for small-scale intermittency (Cava et al. Reference Cava, Katul, Molini and Elefante2012; Chowdhuri et al. Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a; Huang, Katul & Hultmark Reference Huang, Katul and Hultmark2021). Therefore, it is prudent to use the values of $\varGamma _{u}$ and $\varGamma _{w}$ for renormalizing the persistence time scales, $t_{p}$.

We compute the integral time scales by following the standard procedure that involves integrating the autocorrelation coefficients of any turbulent signal up to the lags until the first zero-crossing occurs (Li, Katul & Bou-Zeid Reference Li, Katul and Bou-Zeid2012; Huang et al. Reference Huang, Katul and Hultmark2021). In Appendix B, $u$ and $w$ spectral information is presented, from which one can confirm that the time scales smaller or larger than the integral scales do indeed correspond to the inertial subrange or energy-production range (see figure 11d,e). Furthermore, to check whether our estimates of $\varGamma _{u}$ and $\varGamma _{w}$ match with Finnigan (Reference Finnigan2000), we convert those to normalized length scales using local mean wind speed ($\bar {u}$) and $h$. The vertical profiles of these quantities are shown in figure 4(c,d), where the error bars denote the spread around the ensemble mean. Typically, the profiles behave similarly to those in Finnigan (Reference Finnigan2000), with the integral scales of $u^{\prime }$ being substantially larger than those of $w^{\prime }$. Moreover, at $z/h=1$, the integral length scale of $u^{\prime }$ remains equal to the canopy height (figure 4c) whereas for $w^{\prime }$ a departure is observed from the stipulated value of $h/3$ (figure 4d). We next describe the features of persistence p.d.f.s when the time scales are scaled as $t_{p}/\varGamma _{x}$.

The overall character of $u^{\prime }$ and $w^{\prime }$ persistence p.d.f.s under the new scaling scheme (figure 4e, f) does not differ much from the old one (figure 4a,b). The power laws are followed up to the time scales $t_{p}<\varGamma _{x}$ and the exponents remain similar to before ($-1.5$). This is in sharp contrast to the momentum flux events, for which the effect of different scaling on its persistence time scales remains quite significant (discussed in § 3.2.4). If the power-law exponent of $-1.5$, as obtained for the GoAmazon dataset, is compared with that from Cava et al. (Reference Cava, Katul, Molini and Elefante2012), who considers many different canopy types, a large discrepancy could be noted. For instance, from Cava et al. (Reference Cava, Katul, Molini and Elefante2012) it becomes evident that, depending on the canopy type ($h=16$–28 m and $\textrm {LAI} = 7$–9.6 m$^{2}$ m$^{-2}$), the exponents of $P_{u^{\prime }}(t_{p}/\varGamma _{u})$ and $P_{w^{\prime }}(t_{p}/\varGamma _{w})$ can vary between $-1.6$ and $-1.9$, with often the two not being the same.

On the other hand, to assess the same in a near-neutral ASL flow over an aerodynamically smooth surface, we overlay $P_{u^{\prime }}(t_{p}/\varGamma _{u})$ and $P_{w^{\prime }}(t_{p}/\varGamma _{w})$ from the SLTEST experiment (Chowdhuri et al. Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a) on those already obtained from the GoAmazon canopy. This result is presented in supplementary figure S2(a,b), and also in figure 4(e, f), where we show only the power-law fits from the SLTEST dataset as green dash-dotted lines. For the near-neutral SLTEST dataset, Chowdhuri et al. (Reference Chowdhuri, Kalmár-Nagy and Banerjee2020a) estimated the power-law exponents of $P_{u^{\prime }}(t_{p}/\varGamma _{u})$ and $P_{w^{\prime }}(t_{p}/\varGamma _{w})$ to be equal to $-1.6$ and $-1.25$, respectively. However, when those are overlaid on the GoAmazon dataset, displaying an exponent of $-1.5$, there is hardly any difference for the $u^{\prime }$ signal (figure 4e and figure S2a). But for the $w^{\prime }$ signal, a conspicuous difference is noted between the GoAmazon and SLTEST datasets (figure 4f and figure S2b).

From the above discussion, it is apparent that the power-law exponents of the persistence p.d.f.s are far from universal owing to their dependence on the surface types over which a near-neutral atmospheric turbulent flow is studied. Other than the small time scales, the behaviour of the persistence p.d.f.s at large time scales (energy-production scales) is also of interest. For instance, at time scales $\tau _{p} \geq 0.1$ or $t_{p} \geq \varGamma _{w}$, a bulge is noticed in the persistence p.d.f.s of $w^{\prime }$, corresponding to the heights within the canopy (figure 4b, f). However, for $u^{\prime }$, such a distinctive feature remains largely absent (figure 4a,e). To examine the tails of the persistence p.d.f.s in greater detail, one can investigate the complementary cumulative distribution functions (c.c.d.f.s).

The c.c.d.f.s are denoted as $1-F_{x}(\mathcal {T})$, where $F_{x}(\mathcal {T})$ ($x=u^{\prime },w^{\prime }$) represents the conventional cumulative distribution function (c.d.f.) while $\mathcal {T}$ indicates the normalized time scales $\tau _{p}$ or $t_{p}/\varGamma _{x}$. The results related to the c.c.d.f.s of $\tau _{p}$ and $t_{p}/\varGamma _{x}$ for both the velocity signals are shown in supplementary figure S3. At large time scales ($\tau _{p} \geq 0.1$ or $t_{p} \geq \varGamma _{u}$), the c.c.d.f.s of $u^{\prime }$ are found to be weakly dependent on height (figure S3a,c). Conversely, for $w^{\prime }$, at $\tau _{p} \geq 0.1$, a clear separation is observed in their c.c.d.f.s between the heights within and above the canopy (figure S3b). It is interesting to note that this separation becomes a little less obvious when the $t_{p}$ values are scaled with $\varGamma _{w}$ (figure S3d).

So far, the deliberations rendered on the $u^{\prime }$ and $w^{\prime }$ persistence characteristics develop an intriguing premise regarding the nature of $t_{p}$, related to the four different quadrants of $u^{\prime }w^{\prime }$ signals. Certainly, one can ask the following:

1. (i) Owing to the interaction between the persistence patterns in the $u^{\prime }$ and $w^{\prime }$ signals, how would $P(t_{p})$ behave when the four different quadrants of the momentum-transporting events are considered separately?

2. (ii) How does the differing influence of heights on $P_{u^{\prime }}(t_{p})$ and $P_{w^{\prime }}(t_{p})$ affect the vertical evolution of the persistence patterns in $u^{\prime }w^{\prime }$ signals and is any quadrant-wise change observed in such patterns?

3. (iii) Does there exist any particular scaling that can collapse the persistence p.d.f.s of $u^{\prime }w^{\prime }$ signals, when all nine measurement heights are considered together?

3.2.2. Quadrant decomposition of the persistence time scale

To explore these research questions, in figure 5(a–d) we present the persistence p.d.f.s of the momentum-transporting events, separately for the four different $u^{\prime }$–$w^{\prime }$ quadrants (table 1). To compute these p.d.f.s, the individual residence times in the four quadrants are evaluated (see the graphical illustration in figure 1c), and thereafter have been normalized with a particular time scale. Similar to the $u^{\prime }$ and $w^{\prime }$ signals, we initially use the global time scale $h/u_{*}$ to normalize $t_{p}$ (denoted as $\tau _{p}$) so that the height variations can be clarified better. Note that, although the time scales of different quadrant events are investigated for the first time in canopy turbulence, there are a handful of studies in low-$Re$ engineering flows where such an analysis has been undertaken, albeit in a different form. For instance, Lozano-Durán, Flores & Jiménez (Reference Lozano-Durán, Flores and Jiménez2012) employed a concept similar to persistence to study the sizes of the positive and negative momentum-transporting events in a direct numerical simulation (DNS) of a turbulent channel flow. In their work, the authors defined the sizes of the events as connected regions in a three-dimensional space.

Figure 5.

(a–d) Persistence p.d.f.s of $\tau _{p}$ for the four different $u^{\prime }$–$w^{\prime }$ quadrant events, namely (a) ejection, (b) sweep, (c) outward interaction and (d) inward interaction, corresponding to the heights within and above the canopy. The dash-dotted green lines with triangle markers denote the results from a randomly shuffled sequence of the $u^{\prime }$ and $w^{\prime }$ signals. (e–h) Momentum flux distributions plotted against $\tau _{p}$, depicted separately for the four different quadrants (in corresponding order). Absolute values are used for the momentum flux distributions to clearly demonstrate the differences in their characteristics as the heights change.

From figure 5(a) and (b), which depict $P_{u^{\prime }w^{\prime }}(\tau _{p})$ from ejection and sweep quadrants, respectively, one could notice that these p.d.f.s remarkably collapse onto each other for all the nine heights within and above the canopy. The goodness of the collapse can even be verified from the c.c.d.f.s of the same, shown in supplementary figure S4(a,b). This finding is non-trivial due to the reasons underlined next. First and foremost, irrespective of the fact that at heights deep within the canopy almost no streamwise momentum is transported ($\overline {u^{\prime }w^{\prime }} \to 0$, figure 2c,e), the time scales of ejection and sweep events display a height invariance. To confirm whether this outcome is a consequence of coherent structures in canopy flows, we generate randomly shuffled surrogates of $u^{\prime }$ and $w^{\prime }$ signals and recompute $P_{u^{\prime }w^{\prime }}(\tau _{p})$. To briefly explain the shuffling procedure, one can select the time series of any signal and then operate a random permutation to disrupt the underlying temporal arrangement, thereby creating a surrogate dataset that does not possess any relationship among the signal data points (Chowdhuri, Iacobello & Banerjee Reference Chowdhuri, Iacobello and Banerjee2021). In this process, the signal's p.d.f. remains precisely conserved although the appearance of the data points becomes random.

We find that the persistence p.d.f.s generated from randomly shuffled sequences of $u^{\prime }$ and $w^{\prime }$ (green dash-dotted lines with triangle markers) markedly disagree with the observed ones (except for the smallest $\tau _{p}$ values, which most likely represent noise) at all the nine heights. Since any temporal coherence is destroyed through random shuffling, the disagreement of $P_{u^{\prime }w^{\prime }}(\tau _{p})$ with its randomly shuffled surrogate physically suggests that the coherent structures, which constitute the ejection and sweep motions, extend their footprints deep within the canopy. Despite the presence of such coherent motions at heights deep inside the canopy, they do not transport any momentum, as almost all are absorbed by the foliage (Dwyer, Patton & Shaw Reference Dwyer, Patton and Shaw1997). In the words of Townsend (Reference Townsend1961) and Bradshaw (Reference Bradshaw1967), these motions could be referred to as inactive ones since they have a negligible impact on the flux transport, i.e. they contribute insignificantly to the total flux.

Moreover, up to $\tau _{p}<0.1$, the same power law with an exponent $-1.5$, as observed in $P_{u^{\prime }}(\tau _{p})$ and $P_{w^{\prime }}(\tau _{p})$, also appears in the persistence p.d.f.s of $u^{\prime }w^{\prime }$ events from ejection and sweep quadrants (figure 5a,b). This power-law exponent does not match with the observations of Katul et al. (Reference Katul, Albertson, Parlange, Chu and Stricker1994) and Chowdhuri et al. (Reference Chowdhuri, Prabhakaran and Banerjee2020c) from a near-neutral ASL flow. Both of these studies found that, at short time scales, the persistence p.d.f.s of momentum- and heat-transporting events follow a power law with an exponent $-1.4$. For a visual demonstration, one can consult supplementary figure S2(c), keeping in mind that the scaling of $t_{p}$ is with $\varGamma _{w}$. In that plot, the persistence p.d.f.s of $u^{\prime }w^{\prime }$ events (considering all four quadrants together) from the SLTEST dataset are overlaid on the GoAmazon one, and a clear difference between the two could be noted in the power-law section.

Apart from these high-$Re$ atmospheric flows ($Re \approx 10^{7}$), Kailasnath & Sreenivasan (Reference Kailasnath and Sreenivasan1993) found that, in the inertial layer of a moderate-$Re$ (between 2800 and 13 000) flat-plate boundary layer, the persistence p.d.f.s of the $u^{\prime }w^{\prime }$ signal closely followed the same p.d.f.s of $w^{\prime }$, with both displaying a nearly identical single-exponential function. Therefore, no indication of a power law was evident in their studies. Conversely, from the DNS results of a turbulent channel flow ($Re \approx 5000$), Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) found that the p.d.f.s of streamwise lengths of negative momentum events (comprising ejections and sweeps) followed a power-law distribution with an exponent between $-4/3$ and $-5/3$. These results together with those from ASL and canopy flows extend our previous conclusion on $u^{\prime }$ and $w^{\prime }$ persistence p.d.f.s to the $u^{\prime }w^{\prime }$ signals. In other words, they reinforce the fact that the statistical character of these p.d.f.s is indeed dependent on the flow type that is being studied. Physically, the reason behind such a dependence may lie in the structural discrepancies between the RSL of canopy flows and the inertial sublayer (ISL) of a rough- or smooth-wall boundary layer. In the ISL of wall-bounded turbulent flows, the ejection and sweep motions are typically associated with hairpin structures (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Ganapathisubramani, Longmire & Marusic Reference Ganapathisubramani, Longmire and Marusic2003; Hommema & Adrian Reference Hommema and Adrian2003; Adrian Reference Adrian2007; Fiscaletti, De Kat & Ganapathisubramani Reference Fiscaletti, De Kat and Ganapathisubramani2018) and, by employing persistence analysis, one expects to provide insights into their time scales (Kailasnath & Sreenivasan Reference Kailasnath and Sreenivasan1993). However, the presence of a canopy in the RSL alters the structures of these hairpins (Finnigan, Shaw & Patton Reference Finnigan, Shaw and Patton2009). Notwithstanding different opinions on how exactly such modification occurs (Bailey & Stoll Reference Bailey and Stoll2016), it appears that these modified hairpin structures cause a difference in the persistence results when one compares the RSL and ISL flows.

Continuing with figure 5, one can see that, contrary to the ejection and sweep quadrants, $P_{u^{\prime }w^{\prime }}(\tau _{p})$ from the counter-gradient quadrants (outward and inward interaction) show a considerable amount of height dependence (figure 5c,d). Such a sensitive dependence on $z$ is more clearly visible in the corresponding c.c.d.f.s (supplementary figure S4c,d). By further comparing the features of persistence p.d.f.s among the four different quadrants, we find that, at heights deep within the canopy, the occurrence of large-duration events from counter-gradient quadrants is statistically as likely as those from the ejection and sweep quadrants. On the other hand, at $z/h \geq 1$, $P_{u^{\prime }w^{\prime }}(\tau _{p})$ of outward- and inward-interaction quadrants drop off quite rapidly at large $\tau _{p}$ values as compared to the co-gradient quadrants. This observation suggests that, at heights above the canopy, the long-duration events are primarily dominated by the ejection and sweep motions. Note that such separation in persistence p.d.f.s between the co- and counter-gradient quadrants mainly happens at large $\tau _{p}$ values ($\tau _{p}>0.1$), while at small $\tau _{p}$ ($\tau _{p}<0.1$) the quadrant-wise distinction remains hardly noticeable. Unlike the exponents, this finding is in qualitative agreement with the results obtained from near-neutral ASL flows (Chowdhuri et al. Reference Chowdhuri, Kumar and Banerjee2020b,Reference Chowdhuri, Prabhakaran and Banerjeec) and with the DNS of a turbulent channel flow (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012).

Overall, the quadrant-decomposed features of persistence help one to comprehend the interaction between the structures in the $u^{\prime }$ and $w^{\prime }$ signals, generating the dynamics of the momentum-transporting events. While plotting the persistence p.d.f.s corresponding to the ejection and sweep quadrants, we note that these p.d.f.s are height-invariant. However, the same is not true for the outward- and inward-interaction quadrants, for which a strong sensitivity to the heights is noticed. Concerning the velocity signals, the p.d.f.s of persistence time scales associated with $u^{\prime }$ show little to no height dependence (figure 4a). On the other hand, a clear bulge is observed in the persistence p.d.f.s of $w^{\prime }$ at time scales $\tau _{p} \geq 1$ (figure 4b). Therefore, at those time scales, the persistence p.d.f.s of $w^{\prime }$ differ between the in- and above-canopy measurement locations, meaning that they depend on $z$, unlike the $u^{\prime }$ one. One may ask why such a bulge observed in the persistence p.d.f.s of the $w^{\prime }$ signal does not get reflected in the time-scale distributions of co-gradient motions? Given that height invariance is the common factor, we hypothesize that the distributions of the persistence time scales of ejection and sweep motions are most likely dominated by the positive and negative patterns in the $u^{\prime }$ signal. In contrast, the patterns in the $w^{\prime }$ signal exert their influences to decide the nature of the persistence p.d.f.s of counter-gradient quadrants, ultimately resulting in the observed differences with height.

Notwithstanding these valuable insights, one needs to remember that the persistence analysis provides information on the time-scale distribution of the flux events but not on the flux values themselves. This limitation arises because the amplitude variations are not accounted for in the persistence p.d.f.s. Hence, to establish the role of quadrant-wise persistence patterns towards momentum transport, additional analysis is required (see § 2.2.2).

3.2.3. Flux contributions from persistence patterns

Figure 5(e–h) individually depict the quadrant contributions to the momentum flux against the normalized persistence time scale $\tau _{p}$, corresponding to all the nine measurement heights. Similar to persistence p.d.f.s, the momentum flux distributions are far from a random configuration, thus accounting for the dynamical flow structures. Moreover, upon considering amplitudes, one can see that, notwithstanding the excellent collapse observed in the p.d.f.s of ejection and sweep time scales (figure 5a,b), momentum flux distributions against $\tau _{p}$ show a clear dependence on $z$ (figure 5e, f). To emphasize such dependences, the absolute values of $\langle u^{\prime }w^{\prime } {\mid} \tau _{p} \rangle /u^{2}_{*}$ are plotted in figure 5(e–h) on a logarithmic scale.

At time scales $\tau _{p}<0.1$, flux distributions corresponding to ejections and sweeps display a distinct power-law behaviour (figure 5a,b). However, in these plots, the power-law exponents appear to be different from $1.5$ (shown in dash-dotted black lines). There are a couple of consequences of this result – the first one is from a modelling standpoint while the other one is speculative. Regarding the first, persistence p.d.f.s and associated flux distributions display exponents whose absolute values do not match. As discussed in § 2.2.2, this indicates that the information about persistence is not sufficient and amplitude variability of the instantaneous momentum fluxes should be taken into account while computing the flux contributions from individual events of different time scales. We shed further light on this issue in § 3.3.

Considering the second point, which is related to Brownian motions (Castellanos et al. Reference Castellanos, López, Senties and Anabitarte2013), one may argue that the presence of amplitudes causes a departure from a local gradient-diffusion parametrization (Banerjee et al. Reference Banerjee, De Roo and Mauder2017; Li, Liu & Huai Reference Li, Liu and Huai2022) and thus precludes the phenomenological use of a turbulent diffusivity (analogous to Brownian motion) to describe momentum transport. However, more evidence is needed to support this argument and providing such evidence remainsbeyond the scope of this study. Interestingly, unlike ejections, for the sweep motions, the slopes of $\langle u^{\prime }w^{\prime } {\mid} \tau _{p} \rangle /u^{2}_{*}$ gradually become steeper as $z/h$ approaches 1. This behaviour of the sweep events documents a change in the small-scale dynamics from heights within the canopy to above it.

At larger time scales, the power-law behaviour disappears, with $\langle u^{\prime }w^{\prime } {\mid} \tau _{p} \rangle /u^{2}_{*}$ attaining peaks at $\tau _{p} \approx 0.2$ and $\tau _{p} \approx 0.1$ for the ejection and sweep quadrants, respectively (figure 5e, f). On the other hand, no clear peaks are noticed for the counter-gradient quadrants and the flux distributions remain mostly confined to the scales $\tau _{p}<0.1$ at $z/h \geq 1$ (figure 5g,h). This is consistent with figure 5(c,d) where it was shown that, at heights above the canopy (contrary to those within), the large-time-scale ($\tau _{p}>0.1$) events become highly infrequent for outward- and inward-interaction quadrants. As a whole, the features of figure 5 make one wonder whether the apparent distinction that exists between the persistence properties of co-gradient and counter-gradient quadrants could be resolved if a different scaling were to be used other than $h/u_{*}$. Put differently, it remains unclear if the persistence time scales of co-gradient motions scale differently than the counter-gradient ones.

3.2.4. An alternative scaling with integral scales

Since the persistence time scales of $u^{\prime }w^{\prime }$ events are a result of the interaction between the positive and negative patterns in the component signals ($u^{\prime }, w^{\prime }$), there could be two alternative possibilities to scale the $t_{p}$ of the four different quadrants. From our foregoing discussion in § 3.2.2, it appears that the time scales associated with the co- or counter-gradient motions may scale separately with $\varGamma _{u}$ or $\varGamma _{w}$.

To confirm this hypothesis, one first attempts to scale $t_{p}$ with $\varGamma _{u}$. However, under such scaling, both the persistence p.d.f.s and c.c.d.f.s (supplementary figure S5) exhibit a poorer collapse with height than what was being reported in figure 5(a–d). For instance, regarding the counter-gradient quadrants, one notices that the persistence time scales inside the canopy well exceed $\varGamma _{u}$, while the opposite remains true for heights $z/h \geq 1$ (figure S5). On the other hand, the distributions of persistence time scales collapse the best for all four quadrant events when normalized by the integral time scale of the vertical velocity ($\varGamma _{w}$). For demonstration purposes, the persistence p.d.f.s of $t_{p}/\varGamma _{w}$ are presented in figure 6(a–d) while the corresponding c.c.d.f.s can be found in supplementary figure S6. To denote the apparent mismatch with the SLTEST dataset, the dash-dotted lines (black and green) in figure 6(a) indicate both the power-law exponents $-1.5$ and $-1.4$.

Figure 6.

(a–d) Persistence p.d.f.s for the four different $u^{\prime }$–$w^{\prime }$ quadrant events, where the time scales are scaled as $t_{p}/\varGamma _{w}$. In panel (a), a power law with exponent $-1.4$ (dash-dotted green lines) is shown, which is a fitting result to the SLTEST data as obtained from Chowdhuri et al. (Reference Chowdhuri, Prabhakaran and Banerjee2020c). (e, f) Cumulative contributions to the time fractions and momentum fluxes shown against the time scales ($t_{p}/\varGamma _{w}$) of the $u^{\prime }w^{\prime }$ events. The two green dash-dotted vertical lines denote the time scales $t_{p}/\varGamma _{w}=0.5$ and 2, respectively. The grey shaded regions in all panels correspond to time scales $t_{p}/\varGamma _{w} \geq 1$.

The collapse of quadrant-wise persistence p.d.f.s under the $\varGamma _{w}$ scaling physically suggests that eddy structures whose time scales are comparable to $\varGamma _{w}$ participate in the momentum transport (either actively or inactively) both within and above the GoAmazon canopy. In fact, one can argue that this result from persistence analysis adds significant value to the information obtained from the $u$–$w$ co-spectra (see the discussion in § 3.3.1). Although previous researchers have commented that the integral scale of vertical velocity is comparable to the canopy height $h$ (Finnigan et al. Reference Finnigan, Shaw and Patton2009), our results show that if $h$ is used in place of $\varGamma _{w}$ the persistence p.d.f.s of counter-gradient events do not collapse with height. Contrarily, for ejections and sweeps, this difference is insignificant. While converting $\varGamma _{w}$ to a length scale, Taylor's hypothesis is needed, which often is questionable in canopy turbulence (Finnigan et al. Reference Finnigan, Shaw and Patton2009; Everard et al. Reference Everard, Katul, Lawrence, Christen and Parlange2021). It could be possible that the discrepancy between the two scalings for the counter-gradient quadrants is a result of the failure of Taylor's hypothesis. Interestingly, the co-gradient quadrant results remain unaffected by this. Moreover, when normalized by $\varGamma _{w}$, the distinction between the power-law and exponential drop in $P_{u^{\prime }w^{\prime }}(t_{p}/\varGamma _{w})$ becomes quite evident at $t_{p}/\varGamma _{w}=1$, irrespective of the quadrant type.

From these newly scaled persistence p.d.f.s, we can conclude that the $u^{\prime }w^{\prime }$ events remain statistically self-similar up to the time scales $t_{p}/\varGamma _{w}<1$. However, such self-similarity is destroyed at scales $t_{p}/\varGamma _{w} \geq 1$ due to the presence of large-scale coherent structures. Therefore, it can be stipulated that $t_{p}/\varGamma _{w}=1$ represents a threshold to separate two distinct event classes, one whose statistical characteristics show a power-law behaviour and one for which a deviation is noted from the power law. Furthermore, to assess whether the statistics of momentum flux events remain sensitive to the choice of the threshold while separating the two event classes, one may consult figure 6(e, f).

In figure 6(e, f), the cumulative contributions to the time fractions ($F[T_{f} {\mid} (t_{p}/\varGamma _{w})]$) and momentum fluxes ($F[u^{\prime }w^{\prime } {\mid} (t_{p}/\varGamma _{w})]$) are shown, if one considers the persistence time scales of all four quadrants together. In order to explain the features better, as an example, we focus on the $x$-axis value $t_{p}/\varGamma _{w}=1$. Corresponding to this time scale, the $y$-axis value in figure 6(e) indicates how much fraction of the time would be occupied if only those events were taken into account whose time scales lie in the range $t_{p}/\varGamma _{w} \leq 1$. A similar inference could be drawn by focusing on figure 6( f) but for the momentum flux contributions. Because the cumulative effects are presented, the $y$-axis values in figure 6(e, f) converge towards 1 and the time-averaged momentum flux ($\overline {u^{\prime }w^{\prime }}/u^{2}_{*}$, figure 2c), respectively.

Regarding the sensitivity to the threshold, the two vertical green dash-dotted lines in figure 6(e, f) denote the time scales $t_{p}/\varGamma _{w}=0.5$ and $t_{p}/\varGamma _{w}=2$. By reading the values on the vertical axes associated with these scales, it can be identified how much the time fractions and flux contributions would have changed if, instead of fixing the threshold at $t_{p}/\varGamma _{w}=1$, one had fixed it at 0.5 or 2. Upon completion of such a task, nearly a ${\pm }20\,\%$ change can be noted for both these quantities. Keeping this in mind, we next present results to address the connection between the persistence time scales and bulk statistical features of momentum transport reported in figures 2 and 3.

3.3.1. Short- and long-lived events

In figure 7, we demonstrate how the two event classes, distinguished by their normalized time scales $t_{p}/\varGamma _{w}<1$ (short-lived events) and $t_{p}/\varGamma _{w} \geq 1$ (long-lived events), affect the bulk properties of momentum transport (shown in figures 2e, f and 3f ,g). To construct these plots, we conditionally sample the $u^{\prime }$ and $w^{\prime }$ values, pertaining to the events that persist for times smaller or larger than the threshold, $t_{p}/\varGamma _{w}=1$. Henceforth, we denote the signals sampled from the short- and long-lived events as $\{u^{\prime }_{S}, w^{\prime }_{S}\}$ and $\{u^{\prime }_{L}, w^{\prime }_{L}\}$, respectively. In this regard, one may note that $u^{\prime }=u^{\prime }_{S}+u^{\prime }_{L}$, $w^{\prime }=w^{\prime }_{S}+w^{\prime }_{L}$ and $u^{\prime }w^{\prime }=u^{\prime }_{S}w^{\prime }_{S}+u^{\prime }_{L}w^{\prime }_{L}$.

Figure 7.

(a,b) P.d.f.s of quadrant angles for events whose duration remains in the range (a) $t_{p}/\varGamma _{w}<1$ (short, $u^{\prime }_{S}w^{\prime }_{S}$) and (b) $t_{p}/\varGamma _{w} \geq 1$ (long, $u^{\prime }_{L}w^{\prime }_{L}$). (c) Normalized Shannon entropies of the quadrant angles for the two event classes. (d,e) Quadrant distributions of the momentum flux for the two event classes. ( f)  The contributions to the time fractions and momentum fluxes from the short- ($S$) and long-lived ($L$) events.

To investigate the impact of these two event classes on quadrant statistics, figure 7(a,b) represent the phase angle p.d.f.s separately for the short- and long-lived events. These plots are designed in such a way that, when summed over $P(\theta _{u^{\prime }_{S}w^{\prime }_{S}})$ and $P(\theta _{u^{\prime }_{L}w^{\prime }_{L}})$, one would recover the phase angle p.d.f.s of figure 3( f). Similar to figure 3( f), we use 60 bins of $\theta _{u^{\prime }w^{\prime }}$ to compute $P(\theta _{u^{\prime }_{S}w^{\prime }_{S}})$ and $P(\theta _{u^{\prime }_{L}w^{\prime }_{L}})$. Irrespective of the event type, when 93 half-hour blocks are considered together, the samples corresponding to each quadrant angle p.d.f. turn out to be close to $10^{5}$, a significantly large number to ensure statistical robustness.

As opposed to $P(\theta _{u^{\prime }_{L}w^{\prime }_{L}})$, one cannot recognize any apparent asymmetry among the four different quadrants in $P(\theta _{u^{\prime }_{S}w^{\prime }_{S}})$. Particularly, for the large-scale events, the $P(\theta _{u^{\prime }_{L}w^{\prime }_{L}})$ values at all but the lowest three heights remain clearly skewed towards the ejection quadrants with respect to sweeps (figure 7b). These p.d.f. features can be further connected to the turbulence organization. For instance, an equipartition of the phase angle p.d.f.s among the four different quadrants indicates that the organizational structure of turbulence associated with such events is close to random. In order to quantify the organizational state of these two event classes, we compute the normalized Shannon entropy of $\theta _{u^{\prime }_{S}w^{\prime }_{S}}$ and $\theta _{u^{\prime }_{L}w^{\prime }_{L}}$. Using information theory (Shannon Reference Shannon1948; Banerjee et al. Reference Banerjee, Vercauteren, Muste and Yang2018; Ghannam et al. Reference Ghannam, Poggi, Bou-Zeid and Katul2020), the normalized Shannon entropy ($H_{N}$) of $\theta _{u^{\prime }w^{\prime }}$ can be defined as

(3.3)\begin{equation} H_{N}(\theta_{u^{\prime}w^{\prime}})={-}\frac{1}{\ln(N_{b})}\sum_{i=1}^{N_{b}} P_{i}(\theta_{u^{\prime}w^{\prime}})\ln[P_{i}(\theta_{u^{\prime}w^{\prime}})], \end{equation}

where $N_{b}$ is the number of bins over which the $\theta _{u^{\prime }w^{\prime }}$ values are divided (60 in our case), and $P_{i}(\theta _{u^{\prime }w^{\prime }})$ is the probability of occurrence of a particular binned value $\theta _{u^{\prime }w^{\prime }}$. For a uniform distribution, $H_{N}(\theta _{u^{\prime }w^{\prime }})$ is equal to 1, and therefore the departure from unity could be regarded as a metric quantifying discrepancy with an equipartition configuration of $\theta _{u^{\prime }w^{\prime }}$.

In figure 7(c), we plot the vertical profiles of $H_{N}$ values corresponding to the short- (blue lines with squares) and long-lived (red lines with circles) events. These profiles confirm that, relative to the short-lived events, the long-lived ones gradually become more organized as $z/h$ approaches 1. We also compare these values with $H_{N}(\theta _{u^{\prime }w^{\prime }})$ (black lines with triangles), obtained from the full signal without any scale separation. Since $H_{N}(\theta _{u^{\prime }w^{\prime }}) \approx 1$, one can conclude that, without separating the events based on their persistence time scales, it is impossible to extract information about the organization pattern of the coherent structures. After establishing this important point, to further unravel the event characteristics, we scrutinize the plots involving quadrant flux distributions in the phase-angle space (figure 7d,e).

Similar to figure 7(a,b), the summed flux distributions of short- and long-lived events in figure 7(d,e) yield the total, as shown in figure 3(g). Indeed, the flux distributions in the phase-angle space portray a sharp contrast between the short- and long-lived events. Excluding the lowest three heights, the sweep quadrants control the momentum transport at scales $t_{p}/\varGamma _{w}<1$ (figure 7d). In fact, for the mid-canopy heights ($z/h$ between 0.63 and 0.9), the flux contributions from the short-lived sweep events greatly exceed the ejections. Accordingly, this finding suggests that the influence of sweep motions over the flux fractions at mid-canopy heights (figure 2e) is caused by those events which persist for times $t_{p}/\varGamma _{w}<1$. On the contrary, the dominance of ejections in the time fractions (figure 2f) can be attributed to those events whose persistence time scales are $t_{p}/\varGamma _{w} \geq 1$.

However, for $z/h$ values between 0.9 and 1.38, flux contributions from ejections and sweeps at scales $t_{p}/\varGamma _{w}<1$ remain significantly small when compared to the long-lived counterparts. Conversely, for the counter-gradient quadrants, in agreement with figure 5(g,h), they only exert their influence on momentum transport for the short-lived events while appearing almost absent at scales $t_{p}/\varGamma _{w} \geq 1$. To quantify these features, we investigate the contributions from these two event classes (considering all the four quadrants together) to the time fractions and momentum fluxes with increasing $z/h$ (figure 7f).

In figure 7( f), the quantity $T_{f \vert x}$ denotes the contributions to the time fractions from the events whose time scales are selected based on the condition $x$, i.e. whether $t_{p}<\varGamma _{w}$ (short-lived) or $t_{p}\geq \varGamma _{w}$ (long-lived). Similarly, $(\overline {u^{\prime }w^{\prime } {\mid} x})/u^{2}_{*}$ in figure 7( f) indicates normalized contributions to the time-averaged momentum flux from the short- or long-lived events. We use different coloured lines to differentiate between the vertical profiles of time fractions and flux contributions in figure 7( f) (see the legend). Since time fractions are positive definite quantities while the momentum fluxes are predominantly negative, the vertical zero line in figure 7( f) separates the two. Furthermore, if the flux contributions are summed from these two event classes, one would obtain the total time-averaged momentum flux. To illustrate this point, we show black triangles in figure 7( f), which represent the vertical profile of $(\overline {u^{\prime }w^{\prime }})/u^{2}_{*}$ in figure 2(c) without any error bars.

From figure 7( f), one can see that the long-lived events occupy around 40 % of the time, while the rest is dominated by the short-lived ones. This allows one to infer that not much change exists in the time fractions between these two event classes, as both remain close to 0.5. However, flux contributions from these two events show remarkable differences. For instance, at heights with $z/h$ values between 0.9 and 1.38, almost all the momentum fluxes are carried within the long-lived events ($\approx 80\,\%$), while the short-lived ones only contribute marginally ($\approx 20\,\%$). The appearance of long-lived events as the primary carrier of momentum flux agrees well with the results obtained from near-neutral ASL flows (Chowdhuri et al. Reference Chowdhuri, Prabhakaran and Banerjee2020c) and low-$Re$ DNS of turbulent channel flows (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). In spite of not being conceptually the same, the $u$–$w$ co-spectra in figure 11( f) (in Appendix B) indicate that nearly all the momentum transport is associated with periods larger than $\varGamma _{w}$. An analogous connection between the event-based and Fourier analyses was observed by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) too for a turbulent channel flow. They mentioned that the length scales obtained from their event-based analysis could indeed be related to the spectrum of wall shear stress, thereby implying that such scales were the carriers of the Reynolds-stress fluctuations. Moreover, in the context of atmospheric turbulence, Huang et al. (Reference Huang, Katul and Hultmark2021) presented an analogy between the turbulence spectra and the shape of persistence p.d.f.s. Therefore, from this perspective, the fact that persistence time scales smaller than $\varGamma _{w}$ hardly contribute to the momentum is in qualitative agreement with the assumption of quasi-isotropic conditions in inertial-subrange turbulence (Wyngaard & Coté Reference Wyngaard and Coté1972).

In general, long-lived events are the signatures of the coherent structures and, particularly for the ISL of a wall-bounded flow, they have been linked to the presence of attached eddies or hairpin structures (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). Notwithstanding the fact that in the RSL of a canopy flow the hairpin structures get modified (Finnigan et al. Reference Finnigan, Shaw and Patton2009), we do find an agreement between ISL and RSL flows when the contributions of long events to the momentum fluxes are considered. Since $u^{\prime }w^{\prime }$ persistence p.d.f.s differ between the two flows but the flux contributions qualitatively match, this points towards a complicated role of amplitude variations in momentum transport. In fact, the amplitudes of the long-lived events should be substantially stronger than those of the short-lived ones, so that, despite their similar contributions to $T_{f}$, they nearly carry almost all the flux (figure 7f). Therefore, from both figures 5 and 7, the importance of amplitudes is abundantly evident. This outcome also raises a couple of questions whose answers are not trivial, such as the following:

1. (i) Does there exist a range of persistence time scales for which the effect of $u^{\prime }w^{\prime }$ amplitudes is felt most strongly while evaluating the event contributions to the momentum transport?

2. (ii) Do the scale-wise amplitude variations in $u^{\prime }w^{\prime }$ depend on the quadrant type of the flux events? For example, does the amplitude variability associated with the time scales of ejection events appear stronger than that of the sweeps?

To tackle these issues, we separate the effect of amplitude variations from persistence by developing a method that involves surrogate datasets.

3.3.2. Amplitude variations

To explain our method, let us consider a 60 s time series of the instantaneous momentum flux (shown as black lines in figure 8) at a particular height, $z/h=0.7$. In figure 8, the different quadrant states are colour-coded using the same scheme as in figure 1(c). Now, to generate the surrogate data, all the points of $u^{\prime }w^{\prime }$ are replaced by the quantity

(3.4)\begin{equation} u^{\prime}w^{\prime}_{s}=\frac{\sum{u^{\prime}w^{\prime}}}{N_{P}-N_{N}} \, \text{sgn}{(u^{\prime}w^{\prime})}, \end{equation}

where $u^{\prime }w^{\prime }_{s}$ are the constant surrogate values, $N_{P}$ and $N_{N}$ are the number of positive and negative samples, and $\textrm {sgn}$ represents a function that is $+1$ (for $u^{\prime }w^{\prime }>0$) or $-1$ (for $u^{\prime }w^{\prime }<0$) depending on the argument. Note that, if the values of $u^{\prime }w^{\prime }_{s}$ are summed and then divided by the total number of samples, one would simply obtain the time-averaged momentum flux for that 30 min run. Furthermore, if the persistence p.d.f.s of the four quadrant states are evaluated from $u^{\prime }w^{\prime }_{s}$, they remain exactly the same as the original ones. Therefore, by employing such surrogate data, momentum transport at any time scale can only be explained through persistence, without focusing on the amplitude variability.

Figure 8.

A 60 s section time series of absolute values of $u^{\prime }w^{\prime }$ is shown to illustrate the concept of ignoring the amplitude variations in the flux values. The four different quadrant states are colour-coded, as indicated in the legend.

Consequently, if one compares the flux distributions plotted against $t_{p}/\varGamma _{w}$ between the original and surrogate data of $u^{\prime }w^{\prime }$, then any existing difference could be directly related to amplitude variability. Since cumulative distributions are preferred when any two statistical quantities are compared (Chamecki Reference Chamecki2013), we compute the accumulated values of the fluxes against the persistence time scales, separately for the four quadrants. Thereafter, for each quadrant, we take the absolute difference between the cumulative flux distributions as obtained from the original and surrogate datasets. This is a standard procedure while comparing any two statistical distributions (Chowdhuri et al. Reference Chowdhuri, Iacobello and Banerjee2021). Mathematically, such an operation can be expressed as

(3.5)\begin{equation} {\mathcal{L}}_{1} {\mid} (t_{p}/\varGamma_{w}) = \big| F[u^{\prime}w^{\prime} {\mid} (t_{p}/\varGamma_{w})] - F[u^{\prime}w^{\prime}_{s} {\mid} (t_{p}/\varGamma_{w})] \big|, \end{equation}

where the $F$ functions indicate the cumulative flux distributions at a particular scale $t_{p}/\varGamma _{w}$ and ${\mathcal {L}}_{1} {\mid} (t_{p}/\varGamma _{w})$ is the absolute difference between the original ($u^{\prime }w^{\prime }$) and surrogate ($u^{\prime }w^{\prime }_{s}$) datasets at that scale.

In figure 9(a–d), we show how ${\mathcal {L}}_{1} {\mid} (t_{p}/\varGamma _{w})$ varies quadrant-wise when plotted against the persistence time scales $t_{p}/\varGamma _{w}$ and normalized heights $z/h$. For a better visualization, we use the logarithm of ${\mathcal {L}}_{1} {\mid} (t_{p}/\varGamma _{w})$ while plotting the contours (figure 9a–d). To highlight the threshold scale $t_{p}/\varGamma _{w}=1$ and $z/h=1$, two black dotted lines (horizontal and vertical) are drawn in figure 9(a–d). The red-shaded contours in figure 9(a–d) denote the regions where the largest difference exists between the two cumulative distributions, i.e. where the amplitudes play a stronger role.

Figure 9.

(a–d) Quadrant decomposition of the absolute differences between $u^{\prime }w^{\prime }$ distributions (${\mathcal {L}}_{1} {\mid} (t_{p}/\varGamma _{w})$) shown against the persistence time scales $t_{p}/\varGamma _{w}$, when evaluated for the original and surrogate time series of $u^{\prime }w^{\prime }$. Note that the original time series preserve both the persistence and amplitude effects, whereas the surrogate ones only contain the persistence effects. For better visualization, the logarithms of ${\mathcal {L}}_{1} {\mid} (t_{p}/\varGamma _{w})$ are used. (e) Vertical profiles of the summed $\mathcal {L}_{1}$ values shown separately for each quadrant.

By inspecting the contour plots (figure 9a–d), it is recognized that for the lowest three levels local amplitude effects are negligible across any time scales, as no presence of red regions can be detected. This indicates that there is almost no temporal variability in the $u^{\prime }w^{\prime }$ time series at heights deep within the canopy. Moreover, since the time-averaged momentum flux is itself low at those heights (figure 2c), the instantaneous flux values remain low too due to the consequence of how the surrogate data are defined in (3.4). Therefore, despite the fact that the organizational structure of the flux events remains the same as at the other heights (evident from the vertical collapse of persistence p.d.f.s in figure 6), no contribution is made to the momentum transport at any time scale. Such a phenomenon shares a resemblance with the concept of inactive turbulence as proposed by Townsend (Reference Townsend1961) and Bradshaw (Reference Bradshaw1967).

However, as the heights increase, the range of time scales involved with the red-shaded regions also tends to increase. In fact, at heights $z/h \geq 1$, a broad range of time scales between $0.1 < t_{p}/\varGamma _{w} < 10$ engulf the red regions, indicating the strong presence of amplitudes at such scales. Interestingly, this apparent increase in the range of time scales with height is quite clear for the ejection quadrants as opposed to the rest. For the other three quadrants, a slight break in the red regions is observed between the mid-canopy and above-canopy heights, thereby implying a subtle change in the amplitude effects.

To further characterize the height variations, in figure 9(e), we show the vertical profiles of $\sum {{\mathcal {L}}_{1} {\mid} (t_{p}/\varGamma _{w})}$ (summed over all $t_{p}/\varGamma _{w}$ values), separately for the four quadrants (shown as different coloured lines). Although the amplitude effects become increasingly important with $z$, unlike figure 9(a–d), we do not observe any particular dominance from whichever of the four quadrants. Nevertheless, upon closer inspection, the ${\mathcal {L}}_{1}$ values of ejection motions are found to be somewhat greater than the sweeps. Thus, the amplitude effects appear to be a little stronger for the ejections as compared to the sweeps.

In conclusion, our results from surrogate analysis provide a novel viewpoint by pointing out the range of time scales that are associated with amplitude variability in $u^{\prime }w^{\prime }$ time series. Moreover, we observe that this range is sensitive to the quadrant type and measurement heights. This outcome is compelling enough to make one ponder what type of eddy processes might give rise to such a phenomenon. In wall-bounded turbulence, local amplitude variability in either velocity fluctuations or instantaneous momentum flux is usually associated with a modulation effect by the large-scale structures from the outer layer (Hutchins & Marusic Reference Hutchins and Marusic2007; Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015; Gomit, De Kat & Ganapathisubramani Reference Gomit, De Kat and Ganapathisubramani2018; Laskari et al. Reference Laskari, De Kat, Hearst and Ganapathisubramani2018). On the other hand, for canopy turbulence, Perret & Patton (Reference Perret and Patton2021) have identified the similar presence of an amplitude modulation effect but with a change in its mechanism for heights within and above the canopy. To explain this change, they propose a theory that involves forward and backward scattering of turbulent kinetic energy. Quite possibly, our observation, showing that the range of time scales involved with amplitude variability of $u^{\prime }w^{\prime }$ increases with height, could be related to such mechanisms. However, pursuing that further is beyond the scope of this study.