3.1. Intermittency via network assortativity

In order to characterise the statistical features of large-scale intermittency, we begin by investigating the p.d.f.-based higher-order moments of the analysed signals. Figure 2(a,b) shows the skewness ($\mathcal {S}$) and kurtosis ($\mathcal {K}$) of the $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$ signals against the stability ratio $-\zeta$. For the $T^{\prime }$ signals, $\mathcal {S}$ and $\mathcal {K}$ significantly exceed their Gaussian values (0 and 3, respectively) for the highly convective conditions ($-\zeta >1$). Whereas, in the near-neutral regime ($-\zeta <0.1$) the $T^{\prime }$ signals remain almost Gaussian. On the other hand, for the $u^{\prime }$ signals, the $\mathcal {S}$ and $\mathcal {K}$ values are very close to their Gaussian counterparts irrespective of stability. However, for the $w^{\prime }$ signals, a slight deviation from Gaussianity is observed as compared with the $u^{\prime }$ signals.

Figure 2.

(a) Skewness, $\mathcal {S}$, and (b) kurtosis, $\mathcal {K}$, of $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$ against the stability parameter, $-\zeta$. In (c) the p.d.f., $p(\hat {T})$, of the normalised temperature fluctuations, $\hat {T}=T^{\prime }/\sigma _{T}$ (where $\sigma _{T}$ is the standard deviation of $T^{\prime }$), are shown for the three stability classes (SC): SC1, $0<-\zeta <0.1$; SC3, $0.3<-\zeta <0.5$; SC6, $-\zeta >2$. The solid black line corresponds to the standard Gaussian distribution. (d) Assortativity coefficient, $r$, from the $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$ signals against $-\zeta$. The green line indicates the $r$ values for a Gaussian white noise (GWN). In (e,f) the ratios between $r$ from the original $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$ signals and their mirrored ($r_{M}$) and shuffled ($r_{S}$) counterparts, respectively, are shown.

The near-Gaussian behaviour of the $u^{\prime }$ and $w^{\prime }$ signals suggest that the velocity fluctuations are characterised by a low degree of large-scale intermittency, which agrees with previous studies (e.g. Chu et al. Reference Chu, Parlange, Katul and Albertson1996). But, in the buoyancy-dominated regime ($-\zeta >1$), the large skewness and kurtosis of the temperature signals inform that there are intermittent bursts of large positive values amidst the frequent occurrences of small negative fluctuations. To confirm this further, in figure 2(c) we show the temperature p.d.f.s for the three representative stability classes. We notice that at $-\zeta >2$, the $T^{\prime }$ p.d.f. significantly differs from a Gaussian distribution with anomalous occurrences of large positive $T^{\prime }$ values.

Despite the fact that p.d.f.-based analyses of large-scale intermittency have been widely used, they cannot fully explain the role of flow organisation on the signal's intermittent characteristics (Tsinober Reference Tsinober2014). Contrarily, the assortativity coefficient, $r$, quantifies large-scale intermittency by retaining information on the temporal structure of the signals. In figure 2(d) the behaviour of $r$ is shown for the $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$ signals against $-\zeta$. For $T^{\prime }$, the $r$ values undergo a gradual transition between two distinct plateaus of constant values for the near-neutral ($-\zeta <0.1$) and highly convective ($-\zeta >1$) conditions. Particularly, in the near-neutral regime the $r$ values of $T^{\prime }$ remain close to a Gaussian white noise. Since a decrease in $r$ is sensitive to an increasing large-scale intermittency of the signal, the temperature fluctuations in a buoyancy-dominated ASL flow ($-\zeta >1$) appear more intermittent as compared with a shear-dominated ASL flow ($-\zeta <0.1$). We also observe that $r$ remains almost invariant with $-\zeta$ for $w^{\prime }$, while a slight increase in $r$ is observed for $u^{\prime }$.

The similarity between $r$ and p.d.f.-based statistics as a function of $-\zeta$ validates our metric, which is able to capture the main features of the underlying dynamics. On this note, the NVGs can be further used to assess whether the large-scale intermittency in the signal is related to the positive or negative fluctuations via studying the assortativity coefficients of the original ($r$) and mirrored ($r_{M}$) signals. Figure 2(e) shows the ratio $r/r_{M}$ that characterises the asymmetry in large-scale intermittency between positive and negative fluctuations from a network perspective. For $T^{\prime }$ signals, $r/r_{M}$ decreases from the shear-dominated regime towards the buoyancy-dominated regime. Physically this indicates that, for highly convective regimes ($-\zeta >1$), the mirrored signal is less intermittent, given that $r_{M}>r$ (see figure S1 in the supplementary material available at https://doi.org/10.1017/jfm.2021.720). Moreover, since the changes in $r_{M}$ with $-\zeta$ remain significantly less as compared with $r$ (see figure S1), the behaviour of $r/r_{M}$ tends to mimic the behaviour of $r$. Therefore, in highly convective conditions the large-scale intermittency in $T^{\prime }$ is dictated by positive fluctuations (i.e. peaks, $T^{\prime }>0$) rather than the negative ones (i.e. pits, $T^{\prime }<0$). This finding is consistent with the fact that the p.d.f.s of $T^{\prime }$ in the highly convective conditions show a heavy tail towards the positive fluctuations, as evident from figure 2(c). Moreover, the ratio $r/r_{M}$ for the two velocity components remains almost constant and approximately equal to 1, suggesting a peak-pit symmetry of the signals for any $-\zeta$ in agreement with results shown in figure 2(a,b). It is worth remarking that, despite the aforementioned overall agreement, the NVGs are able to isolate the intermittent behaviour of positive and negative fluctuations, while higher-order moments do not distinguish extreme events as peaks and pits. For instance, the kurtosis (or any other even-order moment) does not distinguish between the positive and negative extreme events, while the skewness does not account for only extreme events but both small and large values in the signal.

To further explore the role of the temporal structure of the signals on large-scale intermittency, we computed the ratios of $r$ values between the original and randomly shuffled signals, $r/r_{S}$. To compute $r_{S}$, one shuffled surrogate series was generated for each time series of $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$; however, a larger number of surrogates do not lead to significant changes in $r_{S}$. Since random shuffling destroys the temporal dependencies while preserving the signal p.d.f. (and, therefore, all the p.d.f. moments), the deviation from 1 in $r/r_{S}$ is an indication of how strong is the impact of the temporal structure on large-scale intermittency. Figure 2(f) illustrates that $r/r_{S}$ significantly decreases for $T^{\prime }$ and increases for $u^{\prime }$ towards $-\zeta >1$, while no discernible change is observed for $w^{\prime }$. It is thus apparent that, in the buoyancy-dominated regime, the flow organisation plays an important role to determine the temporal arrangement of $u^{\prime }$ and $T^{\prime }$ values in the signals. However, since shuffling destroys any dependencies in a time series, a more detailed analysis is required to shed light on the relative contribution of linear and nonlinear effects undergoing in the flow, which impact the signal's intermittent features.

3.2. Role of nonlinearity on convective dynamics

To investigate the role of nonlinearity, first we investigate the behaviour of the degree distribution (i.e. the probability to find a node with a given degree value, $\kappa$), since it is strongly related to the temporal structure of the mapped signals (Lacasa et al. Reference Lacasa, Luque, Ballesteros, Luque and Nuno2008; Liu et al. Reference Liu, Zhou and Yuan2010). In fact, the degree distribution from the NVG is sensitive to the signal's p.d.f. and on the linear and nonlinear dependencies in the signals (Manshour Reference Manshour2015; Iacobello et al. Reference Iacobello, Scarsoglio and Ridolfi2018). To account for the linear and nonlinear effects, we evaluated the degree distributions of $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$ from two different types of surrogate signals: (i) random shuffling, and (ii) iteratively adjusted amplitude Fourier transform (IAAFT). In the IAAFT method, surrogate data are generated that do not contain nonlinear effects but preserve the linear effects described by the auto-correlation in the time series (Lancaster et al. Reference Lancaster, Iatsenko, Pidde, Ticcinelli and Stefanovska2018). To achieve such an objective, the Fourier amplitudes of the time series are kept intact (thereby preserving the auto-correlation), but the associated Fourier phases are replaced by a random uniform distribution between 0 to 2${\rm \pi}$. The randomness in the Fourier phases destroys any nonlinear structure of the time series. However, due to the randomisation of the Fourier phases the p.d.f. of the time series is altered to be Gaussian. Hence, to preserve both the p.d.f. and amplitude spectrum, the Fourier amplitudes and the signal's p.d.f.s are adjusted iteratively at each stage of phase randomisation until the resultant signal has the same power spectrum and p.d.f. as the original one. Lancaster et al. (Reference Lancaster, Iatsenko, Pidde, Ticcinelli and Stefanovska2018) provides a step-by-step implementation of the IAAFT algorithm, which has also been applied to turbulence studies (Poggi et al. Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004; Basu et al. Reference Basu, Foufoula-Georgiou, Lashermes and Arnéodo2007; Cabrit, Mathis & Marusic Reference Cabrit, Mathis and Marusic2013; Keylock Reference Keylock2017).

In figure 3(a–c) we show the complementary cumulative distribution functions (CCDFs) of the degree, $1-P(\kappa )$ (where $P(\kappa )$ is the degree CDF), from the original, shuffled and IAAFT surrogates of $u^{\prime }$, $w^{\prime }$ and $T^{\prime }$, for three representative stability classes (see Appendix for CCDFs of $\kappa$ from all the six stability classes). Complementary cumulative distribution functions are typically used to overcome the intrinsically fat-tailed behaviour of degree distributions and to highlight the contribution from high $\kappa$ values (Newman Reference Newman2018). Irrespective of the stability class, the CCDFs of original signals (orange lines) and their randomly shuffled surrogates (blue lines) remain far apart from each other, revealing the strong effect of the signal's temporal structure on the network topology (Iacobello et al. Reference Iacobello, Scarsoglio and Ridolfi2018). On the other hand, the difference between the CCDFs from original and IAAFT signals (green lines in figure 3a–c) are less evident. Only for temperature, $T^{\prime }$, a remarkable difference in CCDFs is observed in the highly convective regime (figure 3c), which systematically disappears as the near-neutral regime is approached (figure 3a).

Figure 3.

The CCDFs, $1-P(\kappa )$, of the degree, $\kappa$, from the original, shuffled and IAAFT signals corresponding to three representative stability classes (SC): (a) $0<-\zeta <0.1$, (b) $0.3<-\zeta <0.5$ and (c) $-\zeta >2$. For clarity purposes, the distributions are shifted vertically by five and ten decades for $w^{\prime }$ and $u^{\prime }$, respectively. (d) The L1-norm, $\mathcal {L}_{log}$, between the difference of original and IAAFT signals for the six SC. (e) Ratio, $r/r_{I}$, against $-\zeta$ between $r$ from the original signals and $r_{I}$ from the IAAFT signals. For comparison, the ratio $r/r_{S}$ of the $T^{\prime }$ signal (figure 2f) is included in (e) as a green line.

The comparison between the original degree distributions and their IAAFT surrogates is useful to assess the net role of nonlinearity on the signal's temporal structure, since these surrogates retain only the signal's p.d.f. and its linear correlations. Therefore, the results of figure 3(a–c) suggests that the effects of nonlinearity remain quite different on the temporal structure of velocity and temperature fluctuations, despite being part of the same nonlinear system (i.e. atmospheric turbulence at a large Reynolds number). In order to quantify the net effect of nonlinearity in $P(\kappa )$, we calculated the L1-norm of the difference between the logarithms of $P(\kappa )$, i.e. $\mathcal {L}_{\log }=\|\log [P(\kappa )_{{I}}]-\log [P(\kappa )_{{O}}] \|_1$, where $I$ and $O$ subscripts refer to IAAFT and original signals, respectively. The L1-norm is a simple yet useful metric to compare the difference between two probability distributions (Hasson et al. Reference Hasson, Iacovacci, Davis, Flanagan, Tagliazucchi, Laufs and Lacasa2018). The logarithm is taken since $P(\kappa )$ have an exponential tail so that in computing the L1-norm we balance the contribution to $P(\kappa )$ coming from low and high $\kappa$ values.

In figure 3(d) we present the $\mathcal {L}_{\log }$ values between the original and IAAFT signals as a bar plot for the same six stability classes. In the near-neutral regime ($0<-\zeta <0.1$) the impact of nonlinearity is rather low and similar for all the signals. For the $T^{\prime }$ signals, the $\mathcal {L}_{\log }$ values gradually increase as the highly convective regime ($-\zeta >2$) is approached. This implies that the temporal structure of the temperature fluctuations becomes highly sensitive to the nonlinear dependencies in such conditions. On the other hand, for the velocity signals, we observe that the L1-norm for the $w^{\prime }$ signals is slightly larger than the $u^{\prime }$ signals. This outcome suggests that the $w^{\prime }$ signals appear more affected than the $u^{\prime }$ signals by nonlinear dependencies for large $-\zeta$ values.

So far, we have reported through the degree distribution that the NVGs not only are sensitive to the full temporal structure (figure 3a–c), but are also able to highlight the impact of nonlinear dependencies in the signals (figure 3d). Therefore, since the NVG changes with nonlinearity, the network assortativity coefficient, $r$, – which accounts for signal intermittency – is also expected to change. To evaluate such a variation in $r$, figure 3(e) illustrates the ratio, $r/r_{I}$, computed between $r$ for the original signals, and $r_I$ for its IAAFT surrogates. Note that, in accordance with Poggi et al. (Reference Poggi, Porporato, Ridolfi, Albertson and Katul2004), we used 10 IAAFT surrogates for each signal to compute the $r_I$ values. Values of $r/r_{I}$ close to 1 indicate that there is a negligible impact of nonlinearity on the signal intermittency, since $r$ accounts for both linear and nonlinear effects, while $r_{I}$ only for linear effects. For the $T^{\prime }$ signals, $r/r_{I}$ decreases from 1 in the shear-dominated regime to around 0.6 in the buoyancy-dominated regime. Interestingly, $r/r_I$ for the $T^{\prime }$ signals is almost coincident with the corresponding $r/r_S$ (green line in figure 3e), which accounts for both linear and nonlinear effects on the signal's temporal structure. Hence, it indicates that the temperature intermittency – especially in the highly convective regime – is mainly characterised by the nonlinear dependencies in the signal, while the linear effects are negligible.

Contrarily, for the velocity signals, the effect of nonlinearity on $r$ is different. For $u^{\prime }$ signals, the ratio $r/r_{I}$ remains almost constant with $-\zeta$ (figure 3e), although its $r/r_{S}$ values increase with $-\zeta$ (figure 2f). This suggests that the changes in the temporal structure of $u^{\prime }$ are primarily governed by linear dependencies – described by the amplitude spectrum – as the highly convective regime is approached. For the ASL flows, it is known that there is an increase in large-scale spectral energy with respect to the small scales, as the turbulence production due to buoyancy is increased (Wyngaard Reference Wyngaard2010). This relative increase in large-scale spectral energy results in smoother $u^{\prime }$ signals, given the small scales are less energetic. In general, smoother signals correspond to more regular NVGs and higher assortativity values (Iacobello et al. Reference Iacobello, Scarsoglio and Ridolfi2018, Reference Iacobello, Marro, Ridolfi, Salizzoni and Scarsoglio2019, Reference Iacobello, Ridolfi and Scarsoglio2021). Thus, the increasing trend shown in figure 2(f) for $u^{\prime }$ is mainly related to linear effects and is a result of a change in the spectral energy at small and large scales. For $w^{\prime }$, on the other hand, we see that the ratio $r/r_I$ decreases in the highly convective regime. Similar to $u^{\prime }$, as the convective regime is approached, the large-scale spectral energy increases with respect to the small scales (Wyngaard Reference Wyngaard2010), thus making the $w^{\prime }$ signal smoother. Linear effects, therefore, would make the assortativity increase while (as shown in figure 3e) nonlinear effects make it decrease. The net effect is an apparent independence of the assortativity on the full temporal structure with $-\zeta$, as clearly observed in figure 2(f). In other words, there is a balance between the linear and nonlinear effects which govern the temporal structure of the $w^{\prime }$ signals in the buoyancy-dominated regime.