3.1 Reynolds-averaged vertical profiles

In order to characterize the mean vertical structure of the CBL for different stability states, Reynolds-averaged mean vertical profiles from the $256^{3}$ simulations are displayed in figure 1. The mean horizontal velocity ( $U_{h}$ ) can be found in figure 1(a), the mean temperature in (b), the total (resolved $+$ SGS) heat flux in (c), the horizontal velocity variance in (d), the vertical velocity variance in (e) and the resolved vertical velocity skewness in (f). In (c–e), quantities are normalized with the convective velocity scale $w_{\star }$ and temperature scale $\unicode[STIX]{x1D703}_{\star }=Q_{0}/w_{\star }$ . One can see that the mean horizontal velocity ranges from approximately 1 to $12~\text{m}~\text{s}^{-1}$ throughout most of the depth of the CBL and the mean temperature from approximately 301 to 305 K, depending on the values of the forcings ( $U_{g}$ and  $Q_{0}$ ) imposed. The normalized surface heat flux (c) is similar throughout the depth of the CBL for all simulations considered, except that some differences are found near the inversion (where more weakly convective CBLs attain larger negative values in the heat flux). Our results are consistent with previous LES studies (Conzemius & Fedorovich Reference Conzemius and Fedorovich2006) that have demonstrated that the entrainment flux ratio $A_{R}=-\overline{w^{\prime }\unicode[STIX]{x1D703}^{\prime }}_{z_{i}}/\overline{w^{\prime }\unicode[STIX]{x1D703}^{\prime }}_{0}\approx 0.2$ in shear-free convective boundary layers, but is larger in sheared convective boundary layers where the additional TKE due to shear production is available to drive entrainment.

As found in previous studies (e.g. Salesky et al. Reference Salesky, Chamecki and Bou-Zeid2017), a monotonic decrease in horizontal velocity variance ( $\unicode[STIX]{x1D70E}_{h}^{2}/w_{\star }^{2}$ ) is observed with increasing $-z_{i}/L$ . However, the vertical velocity variance ( $\unicode[STIX]{x1D70E}_{w}^{2}/w_{\star }^{2}$ ) does not exhibit similar monotonic behaviour. One can see from figure 1(e) that a region of large $\unicode[STIX]{x1D70E}_{w}^{2}/w_{\star }^{2}$ occurs near the ground for the weakly convective cases (small $-z_{i}/L$ ), in addition to the maximum that occurs near $z/z_{i}=0.3$ –0.4. This low-level increase of $\unicode[STIX]{x1D70E}_{w}^{2}/w_{\star }^{2}$ near the ground has been observed in other studies of the sheared convective boundary layer (Sykes & Henn Reference Sykes and Henn1989; Dosio et al. Reference Dosio, Vilà-Guerau de Arellano, Holtslag and Builtjes2003; Salesky et al. Reference Salesky, Chamecki and Bou-Zeid2017) where it has been attributed to pressure redistribution of $\unicode[STIX]{x1D70E}_{u}^{2}$ (created by shear production near the ground) into $\unicode[STIX]{x1D70E}_{v}^{2}$ and  $\unicode[STIX]{x1D70E}_{w}^{2}$ . From the plot of vertical velocity skewness in figure 1(f), one can see that the vertical velocity in all of the CBLs considered has significant positive skewness, which corresponds physically to narrow, intense updrafts and wider but weaker downdrafts. The skewness of $w$ increases monotonically (for $z/z_{i}\leqslant 0.6$ ) with decreasing $-z_{i}/L$ . The implications of this interplay of buoyancy and shear for amplitude modulation as $-z_{i}/L$ increases will be discussed below.

3.2 Instantaneous quantities

In order to characterize how LSMs in the CBL modulate small-scale turbulence, it is instructive to first consider how structures in the instantaneous velocity field change with bulk stability. Although qualitative in nature, such visualizations provide rich insights on how the structural attributes of LSMs change with stratification. Figure 2 depicts instantaneous snapshots of the velocity field from a weakly ( $-z_{i}/L=3.1$ , (a,c,e)) and a highly ( $-z_{i}/L=1082$ , (b,d,f)) convective CBL simulation on the $256^{3}$ grid. Snapshots of the instantaneous streamwise velocity $\widetilde{u}^{\prime }/w_{\star }$ are displayed in the $x$ – $y$ plane in figure 2(a) and (b) and the instantaneous vertical velocity is displayed in figure 2(c) and (d), both plotted for $z/z_{i}=0.1$ . The elevation of the $x$ – $y$ slices plotted in figure 2(a–d) is denoted by a horizontal grey line in (e,f). In (e,f), the streamwise velocity is plotted in the $x^{\prime }$ – $z$ plane, where the $x^{\prime }$ – $y^{\prime }$ axis can be found overlaid on (a–d). The local $x^{\prime }$ – $y^{\prime }$ coordinate system is added to highlight an LSM, which is rotated counterclockwise relative to the $x$ -axis by virtue of the Coriolis force.

Figure 2. Instantaneous velocity field from $256^{3}$ simulations. Results are plotted for a weakly convective CBL ( $-z_{i}/L=3.1$ ) in (a,c,e) and a highly convective CBL ( $-z_{i}/L=1082$ ) in (b,d,f). In (a,b), instantaneous streamwise velocity ( $\widetilde{u}^{\prime }/w_{\star }$ ) is displayed in the $x$ – $y$ plane, for $z/z_{i}=0.10$ . In (c,d), instantaneous vertical velocity ( $\widetilde{w}^{\prime }/w_{\star }$ ) is displayed in the $x$ – $y$ plane, for $z/z_{i}=0.10$ (where the $x^{\prime }$ – $y^{\prime }$ coordinate system is displayed in (a–d)). In (e,f) instantaneous streamwise velocity is displayed in the $x^{\prime }$ – $z$ plane with instantaneous $u$ – $w$ velocity vectors overlaid. The grey horizontal lines in (e,f) denote $z/z_{i}=0.10$ , and the solid black lines in (e) denote the structure inclination angle $\unicode[STIX]{x1D6FE}=17.7^{\circ }$ .

One can see that for weakly convective conditions, the streamwise velocity field in figure 2(a) is organized into high- and low-momentum streaks, which are typically aligned $10^{\circ }$ – $20^{\circ }$ to the left of the geostrophic wind in the Northern Hemisphere (LeMone Reference LeMone1973; Brown Reference Brown1980). Corresponding updrafts and downdrafts in the vertical velocity field, sometimes termed horizontal convective rolls (or HCRs) are visible in figure 2(c). In figure 2(e), we display the streamwise velocity (here the velocity component aligned with the roll axis) in the $x^{\prime }$ – $z$ plane, where the $x^{\prime }$ – $y^{\prime }$ axis is rotated $32^{\circ }$ counterclockwise from the $x$ – $y$ axis. Also displayed in (e) are inclined lines that indicate the characteristic inclination angle of surface-layer structures, $\unicode[STIX]{x1D6FE}=17.7^{\circ }$ , as calculated from the two-point correlation of streamwise velocity $R_{uu}(\unicode[STIX]{x0394}x,\unicode[STIX]{x0394}z)$ (discussed below in § 3.4). In (e), a low-momentum region in the streamwise velocity field is evident; ejections of low-momentum fluid occur in this region (as can be seen from the velocity vectors overlaid on the plot). Under weakly convective conditions the organization of the atmospheric surface layer is similar to what is found in the logarithmic region of a canonical wall-bounded turbulent shear flow, except that the low-momentum zones are rotated counterclockwise relative to the geostrophic wind and warm ( $\unicode[STIX]{x1D703}^{\prime }>0$ ) fluid concentrates in the low momentum zones (e.g. see figure 4 of Salesky et al. (Reference Salesky, Chamecki and Bou-Zeid2017)).

Under highly convective conditions ( $-z_{i}/L=1082$ , (b,d,f)), the CBL exhibits significantly different organization. Here the vertical velocity field organizes into open cells (similar to turbulent Rayleigh–Bénard convection), as can be seen in figure 2(d). The horizontal velocity field in figure 2(b) is comprised of ‘patches’ of high- and low-momentum fluid due to convergence and divergence into updrafts and downdrafts. In contrast to the slightly inclined surface-layer structures seen for weakly convective conditions in figure 2(e), here the structures are much more inclined (indeed, nearly vertical) since the vertical velocity field is dominated by updrafts and downdrafts. Thus as the CBL becomes increasingly convective, the vertical velocity field transitions from horizontal convective rolls to open cells (see Salesky et al. Reference Salesky, Chamecki and Bou-Zeid2017, for a further discussion). With this, the VLSMs in the streamwise velocity field vanish and are replaced with high- and low-momentum patches, and the inclination angle of surface-layer structures increases from the ${\sim}15^{\circ }$ found in neutrally stratified wall turbulence to nearly vertical (Hommema & Adrian Reference Hommema and Adrian2003; Carper & Porté-Agel Reference Carper and Porté-Agel2004; Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013). Nonetheless, the concept that the large-scale flow is composed of ‘building block’ structures persists across stability regimes. At the near-neutral and near-free convective limits, these ‘building blocks’ resemble LSMs, and convective cells, respectively. In this sense, one may expect that a large-scale amplitude modulation should be preserved across stability regimes, but that the identity of the modulator will transition from the large-scale streamwise velocity $u_{l}$ to the large-scale vertical velocity  $w_{l}$ . Spectrograms are convenient for describing the distribution of spectral density across wavelength and depth, and the following section demonstrates how the flow attains a vertically dominant state at the near-free convection limit.

3.3 Spectrograms

Previous studies of amplitude modulation in turbulent boundary layers (e.g. Kim & Adrian Reference Kim and Adrian1999; Hutchins & Marusic Reference Hutchins and Marusic2007a ,Reference Hutchins and Marusic b ; Mathis et al. Reference Mathis, Hutchins and Marusic2009a ,Reference Mathis, Monty, Hutchins and Marusic b ; Anderson Reference Anderson2016) have found that spectrograms (premultiplied power spectra plotted as a function of dimensionless wavelength and wall-normal distance) provide evidence of very-large-scale motions if an outer peak exists at large wavelengths. In figure 3, we present spectrograms from selected $256^{3}$ simulations for weakly ( $-z_{i}/L=3.1$ (a,d,g)), moderately ( $-z_{i}/L=17.7$ (b,d,h)) and highly ( $-z_{i}/L=1082$ (c,f,i)) convective conditions, for the streamwise velocity ( $k_{x}\unicode[STIX]{x1D6F7}_{uu}/w_{\star }^{2}$ (a,b,c)), vertical velocity ( $k_{x}\unicode[STIX]{x1D6F7}_{ww}/w_{\star }^{2}$ (d,e,f)) and for the vertical heat flux ( $k_{x}\unicode[STIX]{x1D6F7}_{w\unicode[STIX]{x1D703}}/w_{\star }\unicode[STIX]{x1D703}_{\star }$ (g,h,i)). To these figures, we have added horizontal and vertical lines to denote the AM filter scale (§ 1.2) and Obukhov length, $-L/z_{i}$ , respectively.

Figure 3. Premultiplied spectrograms, plotted as a function of dimensionless wavelength and height from $256^{3}$ simulations. Panels (a,d,g) are plotted for $-z_{i}/L=3.1$ , (b,e,h) are plotted for $-z_{i}/L=17.7$ and (c,f,i) are plotted for $-z_{i}/L=1082$ . Spectrograms of streamwise velocity ( $k_{x}\unicode[STIX]{x1D6F7}_{uu}/w_{\star }^{2}$ ) are displayed in (a–c), spectrograms of vertical velocity ( $k_{x}\unicode[STIX]{x1D6F7}_{ww}/w_{\star }^{2}$ ) are displayed in (d–f) and spectrograms of vertical heat flux ( $k_{x}\unicode[STIX]{x1D6F7}_{w\unicode[STIX]{x1D703}}/w_{\star }\unicode[STIX]{x1D703}_{\star }$ ) are displayed in (g–i). In each panel, the horizontal dashed line denotes the cutoff wavelength used to separate the small scales from the large scales ( $\unicode[STIX]{x1D706}_{x}=z_{i}$ ) and the vertical dashed line denotes the normalized Obukhov length (i.e. $-L/z_{i}$ ).

The spectrogram of streamwise velocity under weakly convective conditions in figure 3(a) exhibits both an inner peak and an outer peak, similar to what has been found in high Reynolds number turbulent boundary layers in the absence of buoyancy. Here the outer peak occurs at $z/z_{i}\approx 0.1$ and $\unicode[STIX]{x1D706}_{x}/z_{i}\approx 6$ , and may be interpreted as the signature of VLSMs in the streamwise velocity field under weakly convective conditions. The outer peak in this case is smaller than the typically reported value of $\unicode[STIX]{x1D706}_{x}/z_{i}\approx 10$ in flows where buoyancy is absent (e.g. Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010b ); this occurs due to the combined effects of buoyancy and Coriolis. Recall that figure 2(a,c) showed that structures undergo a rotation due to Coriolis, but for the purpose of figure 3, the flow field is projected onto a spectrum of harmonic functions in  $x$ . This likely manifests in the spectrograms as an existential shortening of the structures at weakly convective cases. Nonetheless, we will show that the outer peak location shifts monotonically, and for increasingly unstable conditions the influence of this post-processing step diminishes since the flow attains a cell-like structure (figure 2 b,d). Other studies of neutrally stratified, Coriolis-free ABL turbulence (Fang & Porté-Agel Reference Fang and Porté-Agel2015; Jacob & Anderson Reference Jacob and Anderson2017) report VLSMs with longer streamwise extent ( ${\sim}20z_{i}$ ), and there is no need to repeat these cases in the present work.

As $-z_{i}/L$ increases, one can see (e.g. in figure 3 b) that the outer peak shifts to smaller wavelengths, until only a single peak is found under highly convective conditions in (c). These changes in the spectrograms correspond to the high- and low-momentum streaks in the $\tilde{u} ^{\prime }$ field transitioning to patches of high and low momentum fluid due to horizontal convergence and divergence from updrafts and downdrafts in the vertical velocity field (see figure 2 a,b).

Spectrograms of the vertical velocity $k_{x}\unicode[STIX]{x1D6F7}_{ww}/w_{\star }^{2}$ are plotted in figure 3(d–f). For weakly convective conditions in (d), the spectrogram of vertical velocity also exhibits an inner peak and an outer peak, with the outer peak found at $\unicode[STIX]{x1D706}_{x}/z_{i}\approx 6$ and $z/z_{i}\approx 0.5$ . Here the outer peak corresponds to horizontal convective rolls that occur in the $\widetilde{w}^{\prime }$ field (see figure 2 c). In contrast to the outer peak for streamwise velocity, which was found in the shear-dominated region of the CBL, $z<-L$ , the outer peak for vertical velocity is found in the buoyancy-dominated region, i.e.  $z>-L$ . As the CBL becomes increasingly convective, the outer peak for $\widetilde{w}^{\prime }$ shifts inward, until a single maximum is found near $z/z_{i}\approx 0.3$ and $\unicode[STIX]{x1D706}_{x}/z_{i}\approx 1$ . Note that the height at which the outer peak occurs is consistent with the maximum in $\unicode[STIX]{x1D70E}_{w}^{2}/w_{\star }^{2}$ at each stability, as can be seen in figure 1(e). And although the outer peaks for $k_{x}\unicode[STIX]{x1D6F7}_{uu}/w_{\star }^{2}$ and $k_{x}\unicode[STIX]{x1D6F7}_{ww}/w_{\star }^{2}$ occur at different heights, they do occur at approximately the same wavelength at each stability.

A similar set of plots for the heat flux $k_{x}\unicode[STIX]{x1D6F7}_{w\unicode[STIX]{x1D703}}/w_{\star }\unicode[STIX]{x1D703}_{\star }$ can be found in figure 3(g–i). Once again, an outer peak is found under weakly convective conditions (g) and then shifts inward to smaller wavelengths as $-z_{i}/L$ increases. Spectrograms for the intermediate cases (see tables 1 and 2) varied monotonically between the cases shown in figure 3 and are not displayed here for the sake of brevity. The implications of these changes in the spectral properties of the CBL with increasing $-z_{i}/L$ for flow modulation phenomena will be discussed further in § 3.5.

Table 1. Properties of large eddy simulations on $256^{3}$ grid, including geostrophic velocity ( $U_{g}$ ), kinematic surface heat flux ( $Q_{0}$ ), convective boundary-layer depth ( $z_{i}$ ), Obukhov length ( $L$ ), bulk stability parameter ( $-z_{i}/L$ ), friction velocity ( $u_{\unicode[STIX]{x1D70F}}$ ) and Deardorff convective velocity scale ( $w_{\star }$ ).

Table 2. Properties of large eddy simulations on $160^{3}$ grid, including geostrophic velocity ( $U_{g}$ ), kinematic surface heat flux ( $Q_{0}$ ), convective boundary-layer depth ( $z_{i}$ ), Obukhov length ( $L$ ), bulk stability parameter ( $-z_{i}/L$ ), friction velocity ( $u_{\unicode[STIX]{x1D70F}}$ ), Deardorff convective velocity scale ( $w_{\star }$ ) and number of ensemble members ( $N_{e}$ ).

3.4 Two-point correlations and inclination angles

In order to quantify changes in turbulence spatial structure with increasing instability, we here consider the correlation structure of the resolved streamwise velocity field and structure inclination angles. Two-point correlation maps of streamwise velocity are displayed in figure 4(a–c) for weakly, moderately, and highly convective conditions, respectively (for consistency, the selected stratification values are identical to those selected for figure 3). Here the two-point correlation of streamwise velocity is calculated via

(3.1) $$\begin{eqnarray}R_{uu}(\unicode[STIX]{x0394}x,\unicode[STIX]{x0394}z)=\frac{\langle \widetilde{u}^{\prime }(x,y,z)\widetilde{u}^{\prime }(x+\unicode[STIX]{x0394}x,y,z+\unicode[STIX]{x0394}z)\rangle }{\unicode[STIX]{x1D70E}_{\widetilde{u}(x,y,z)}\unicode[STIX]{x1D70E}_{\widetilde{u}(x+\unicode[STIX]{x0394}x,y,z+\unicode[STIX]{x0394}z)}},\end{eqnarray}$$

(e.g. Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013). Taylor’s hypothesis is used to convert a temporal lag to a spatial lag, e.g.  $\unicode[STIX]{x0394}x=U_{c}\,\unicode[STIX]{x0394}t$ , where the convection velocity $U_{c}$ is taken as the mean velocity at $z/z_{i}=0.2$ . Other values of the convection velocity were also considered in the computation of $R_{uu}$ ; we found that this had negligible impact on the resulting inclination angles. This same convection velocity will be used in the computation of two-point amplitude modulation coefficients below. Recall, too, from § 1.2 and (1.8)–(1.9), that we use a local coordinate system mapped to the mean flow direction at each elevation, and thus the reported correlation and inclination angles are true indicators of LSMs. That is, the reported statistics are not artificially shortened by virtue of a Cartesian coordinate system that is oblique to the actual streamwise flow direction.

Figure 4. Two-point correlation maps, $R_{uu}$ , plotted as a function of streamwise ( $\unicode[STIX]{x0394}x/z_{i}$ ) and vertical ( $\unicode[STIX]{x0394}z/z_{i}$ ) lags from $256^{3}$ simulations. In each panel, the black squares indicate the value of $\unicode[STIX]{x0394}x/z_{i}$ that corresponds to the maximum correlation for each value of $\unicode[STIX]{x0394}z/z_{i}$ . Solid lines are fit by linear regression to the maximum correlation for points in the range $\unicode[STIX]{x0394}z/z_{i}\in [0,0.2]$ and are used to calculate the inclination angle of the surface-layer structures. The horizontal dashed line in each plot indicates the location of the Obukhov length, i.e.  $-L/z_{i}$ . (a)  $-z_{i}/L=3.1$ , (b)  $-z_{i}/L=17.7$ , (c)  $-z_{i}/L=1082$ .

Figure 5. Inclination angle ( $\unicode[STIX]{x1D6FE}$ ) of surface-layer structures calculated from $160^{3}$ and $256^{3}$ LES results, plotted as a function of Monin–Obukhov stability parameter $-\unicode[STIX]{x1D701}$ . The fit proposed by Chauhan et al. (Reference Chauhan, Hutchins, Monty and Marusic2013) is shown for comparison (solid line); dashed lines indicate $\pm 5^{\circ }$ from this empirical curve. Also displayed are values of $\unicode[STIX]{x1D6FE}$ calculated from unstable atmospheric surface-layer data from Carper & Porté-Agel (Reference Carper and Porté-Agel2004), Marusic & Heuer (Reference Marusic and Heuer2007) and Chauhan et al. (Reference Chauhan, Hutchins, Monty and Marusic2013).

In figure 4, contours of $R_{uu}$ are plotted as a function of the normalized horizontal and vertical lag; the black squares denote the streamwise lag that corresponds to the maximum correlation (i.e.  $\unicode[STIX]{x0394}x^{\ast }$ ) for each vertical lag. In order to determine the characteristic inclination of surface-layer structures at each stability, we perform a linear regression of the form $\unicode[STIX]{x0394}z=c_{1}\unicode[STIX]{x0394}x^{\ast }$ for $\unicode[STIX]{x0394}z/z_{i}\in [0,0.2]$ to determine the unknown coefficient  $c_{1}$ . The inclination angle is then given as $\unicode[STIX]{x1D6FE}=\tan ^{-1}(c_{1})$ . In figure 4, one can see that as $-z_{i}/L$ increases, the region of maximum correlation goes from being inclined in the $+\unicode[STIX]{x0394}x$ direction to nearly aligned with the $\unicode[STIX]{x0394}z$ axis, indicating that the steepness of surface layer structures increases with increasing $-z_{i}/L$ . This result is thoroughly consistent with the qualitative results displayed in figure 2 and the shift in the outer peak with increasing $-z_{i}/L$ that can be seen in figure 3: LSMs steepen and approach the classical convective cells as $-z_{i}/L\rightarrow \infty$ .

The change in surface-layer structures with increasing instability can be seen more clearly in figure 5, where the inclination angle $\unicode[STIX]{x1D6FE}$ is plotted as a function of the MO stability variable $-\unicode[STIX]{x1D701}=-z/L$ , at both grid resolutions. Also displayed in figure 5 is the empirical fit from the atmospheric surface-layer data of Chauhan et al. (Reference Chauhan, Hutchins, Monty and Marusic2013),

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}+7.3\ln (1-70\unicode[STIX]{x1D701}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}_{0}=12^{\circ }$ , dashed lines indicating $\unicode[STIX]{x1D6FE}\pm 5^{\circ }$ from this fit, and data collected under unstable conditions ( $\unicode[STIX]{x1D701}<0$ ) from several experimental studies are superimposed to broaden the context of the results (Carper & Porté-Agel Reference Carper and Porté-Agel2004; Marusic & Heuer Reference Marusic and Heuer2007; Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013). Here we observe that as $-\unicode[STIX]{x1D701}$ increases, the inclination angle increases from $10^{\circ }$ – $15^{\circ }$ to more than $50^{\circ }$ . While there is some scatter in the values of $\unicode[STIX]{x1D6FE}$ calculated from the LES results, the scatter we observe is comparable to what is found in the experimental data. This plot reinforces the conclusion that surface-layer structures become more steeply inclined with increasing instability, which can be interpreted as buoyancy modifying hairpin vortex packets to become lifted up from the ground (e.g. Hommema & Adrian Reference Hommema and Adrian2003). As discussed previously, spectrograms indicate that LSMs persist to highly convective conditions (figure 3 f). As such, we expect that inner-layer modulation by large-scale outer-layer structures should continue with increasing $-z_{i}/L$ , as long as there is a separation between the inner and outer peaks.

3.5 Modulation of small scales

Figure 6. Time series depicting modulation of small-scale streamwise velocity by large-scale streamwise velocity and large-scale vertical velocity for $-z_{i}/L=3.1$ simulation on $160^{3}$ grid at $z/z_{i}=0.15$ . (a) Time series of small-scale ( $u_{s}$ ) and large-scale ( $u_{l}$ ) streamwise velocity and large-scale envelope of small-scale streamwise velocity ( $E_{l}(u_{s})$ ), (b) time series of small-scale ( $w_{s}$ ) and large-scale ( $w_{l}$ ) vertical velocity, (c) time series of the filtered envelope ( $E_{l}(u_{s})$ ) of small-scale velocity with large-scale streamwise and vertical velocity also shown for comparison, (d) wavelet power spectra of streamwise velocity normalized by frequency and streamwise velocity variance. Here the dashed horizontal line denotes the cutoff frequency used to separate large from small scales. The amplitude modulation coefficients at this height are $R_{u_{l},u_{s}}=-0.60$ and $R_{w_{l},u_{s}}=0.62$ .

We now consider the extent to which amplitude modulation occurs in the convective atmospheric boundary layer. Before exploring the dependence of AM on height ( $z/z_{i}$ ) and on stability ( $-z_{i}/L$ ), we first provide an example of the amplitude modulation of small-scale streamwise velocity ( $u_{s}$ ) by both large-scale streamwise ( $u_{l}$ ) and vertical velocity ( $w_{l}$ ). This is done to illustrate the temporal variability of the quantities that enter into the calculation of the AM coefficient in (1.3)–(1.7) (e.g. $E_{l}^{\prime }(u_{s})$ , and $u_{l}$ or $w_{l}$ ). In figure 6, we plot time series of streamwise and vertical velocity and wavelet spectra for the $-z_{i}/L=3.1$ simulation on the $160^{3}$ grid, at $z/z_{i}=0.15$ . In figure 6(a), we display time series of the large-scale ( $u_{l}$ ) and small-scale ( $u_{s}$ ) streamwise velocity components as well as the large-scale envelope of the small-scale streamwise velocity $E_{l}(u_{s})$ . The corresponding large- and small-scale components of vertical velocity are shown in (b). In (c), $u_{l}$ , $w_{l}$ and $E_{l}^{\prime }(u_{s})$ are included on the same plot, so one can see visual evidence of the correlation between the large-scale velocity components and the large-scale envelope of $u_{s}$ . Finally in (d), we display wavelet power spectra of the streamwise velocity, plotted as a function of large eddy turnover time. The wavelet power spectrum is computed by first calculating the wavelet transform of the streamwise velocity  $u$ , defined via convolution with a spectrum of wavelet functions:

(3.3) $$\begin{eqnarray}\widehat{u}(z;t_{s},t)=\int u(z,t^{\prime })\,\unicode[STIX]{x1D713}^{\ast }\left(\frac{t^{\prime }-t}{t_{s}}\right)\text{d}t^{\prime },\end{eqnarray}$$

where $\unicode[STIX]{x1D713}(t_{s},t)$ is the wavelet function with characteristic time scale, $t_{s}$ , and $(\cdot )^{\ast }$ denotes the complex conjugate. We employ the Morlet wavelet

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D713}(t/t_{s})=\text{e}^{\text{i}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}t/t_{s}}\text{e}^{-|t/t_{s}|^{2}/2},\end{eqnarray}$$

for its high-frequency resolution where the complex frequency $|\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}|=6$ (Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015). Given the complex wavelet coefficients, one can then calculate the wavelet power spectrum:

(3.5) $$\begin{eqnarray}E_{u}(z;t_{s},t)=\frac{|\widehat{u}(z;t_{s},t)|^{2}}{t_{s}}.\end{eqnarray}$$

In figure 6(c) there is visual evidence of a negative correlation between large-scale streamwise velocity $u_{l}$ and the filtered envelope of small-scale $u$ ( $E_{l}^{\prime }(u_{s})$ ). Here the envelope of $u_{s}$ increases due to modulation by a low-momentum LSM (i.e.  $u_{l}<0$ ) and decreases due to modulation by a high-momentum LSM ( $u_{l}>0$ ). This is consistent with previous studies of amplitude modulation in turbulent boundary layers in the absence of buoyancy (e.g. Mathis et al. Reference Mathis, Hutchins and Marusic2009a ,Reference Mathis, Monty, Hutchins and Marusic b ; Marusic et al. Reference Marusic, Mathis and Hutchins2010a ). At this height, the single-point amplitude modulation coefficient is $R_{u_{l},u_{s}}=-0.60$ . From figure 6(b,c), one can see that AM due to large-scale vertical velocity also occurs, but here amplitude modulation is positive ( $R_{w_{l},u_{s}}=0.62$ ). We observe that small-scale streamwise velocity fluctuations are excited in large-scale updraft regions ( $w_{l}>0$ ) and suppressed in large-scale downdraft regions ( $w_{l}<0$ ), due to the positive correlation between $w_{l}$ and the envelope of the small-scale velocity fluctuations ( $E_{l}(u_{s})$ ). It is also interesting to consider how the energy content in $u$ varies as a function of frequency and time relative to $u_{l}$ and  $w_{l}$ . One can see from figure 6(c,d) that regions of $w_{l}>0$ and $u_{l}<0$ correspond to a shift of energy in $E_{u}$ to higher frequencies (see, e.g. $tw_{\star }/z_{i}=4$ , 13–14, 16), reinforcing the picture that these large-scale structures modulate small-scale turbulence. As noted in § 1.2, the horizontal velocity components have been mapped to the local mean flow direction at each elevation, and thus the physics associated with the passage of LSMs are accurately captured (for high Rossby number cases, the local mean flow would be precisely aligned with the $x$  axis).

Next, we turn our attention to how both single- and two-point amplitude modulation coefficients depend on height ( $z/z_{i}$ ) and on atmospheric stability ( $-z_{i}/L$ ). Profiles of the single- and two-point AM coefficients of $u_{s}$ (a,b), $w_{s}$ (c,d) and $\unicode[STIX]{x1D703}_{s}$ (e,f), and surface fluxes $(u\,w)_{s}$ (g,h), and $(w\,\unicode[STIX]{x1D703})_{s}$ (i,j), can be found in figure 7, due to modulation by large-scale $u$ (a,c,e,g,i) $w$ (b,d,f,h,j). In each panel solid lines are used to indicate single-point AM coefficients, and dashed lines are used to indicate two-point AM coefficients, where the reference height used to calculate the two-point coefficients, $z_{ref}/z_{i}=0.2$ , is indicated by the vertical grey line.

Figure 7. Single-point (solid lines) and two-point (dashed lines) amplitude modulation coefficients from $160^{3}$ simulations. The vertical grey line in each panel denotes the reference height ( $z_{ref}/z_{i}=0.2$ ) used when calculating the two-point coefficients. Panels (a,c,e,g,i) denote amplitude modulation by large-scale streamwise velocity ( $u_{l}$ ) and panels (b,d,f,h,j) denote modulation by large-scale vertical velocity ( $w_{l}$ ). (a,b) Modulation of small-scale streamwise velocity ( $u_{s}$ ), (c,d) modulation of small-scale vertical velocity ( $w_{s}$ ), (e,f) modulation of small-scale temperature ( $\unicode[STIX]{x1D703}_{s}$ ), (g,h) modulation of small-scale momentum flux ( $uw_{s}$ ), (i,j) modulation of small-scale heat flux ( $w\unicode[STIX]{x1D703}_{s}$ ). In (a), black diamonds denote $R_{u_{l},u_{s}}$ from smooth-wall channel flow data (taken from Mathis et al. Reference Mathis, Monty, Hutchins and Marusic2009b , their figure 3a).

In figure 7(a), modulation of small-scale $u$ by large-scale $u$ , i.e.  $R_{u_{l},u_{s}}$ , is displayed for the suite of simulations on the $160^{3}$ grid. The smooth-wall channel flow data of Mathis et al. (Reference Mathis, Monty, Hutchins and Marusic2009b ) are also included in this panel for comparison. One can see from this panel that the single-point AM coefficient attains positive values close to the ground, and changes sign between $z/z_{i}=0.05$ –0.1, depending on the stability. As $-z_{i}/L$ increases, the magnitude of the single-point AM coefficient decreases, until the modulation is negligible. Note that the most highly convective case ( $-z_{i}/L=1041$ ) has been omitted from these plots because the AM profiles are extremely noisy; this appears to occur because there is no separation between the outer and inner peak and the concept of amplitude modulation is no longer meaningful in this case.

The behaviour of the two-point AM coefficients, indicated by the dashed lines, is similar, except that (as expected) the two-point coefficients attain smaller values farther from the reference height. These results indicate that as the outer peak in the spectrogram of $u$ shifts to smaller wavelengths, the scale separation between the outer peak and inner peak decreases, such that the amplitude modulation of small-scale turbulence is significantly reduced. Thus one can conclude that amplitude modulation due to $u_{l}$ is a significant dynamical feature of weakly convective CBLs, where VLSMs exist in the streamwise velocity field. However, as $-z_{i}/L$ increases, surface-layer structures become shorter and more inclined, and amplitude modulation due to $u_{l}$ becomes negligible.

Figure 8. Depth-integrated single- (a,b) and two-point (c,d) AM coefficients by large-scale streamwise (a,c) and vertical (b,d) velocity. Integrals are shown on the ordinate and correlations are denoted on the abscissa, where the panel (a) inset denotes the colour scheme (for clarity, the colour scheme is identical to that of figure 7). Panel (a) includes the depth-integrated integral of the Mathis et al. (Reference Mathis, Monty, Hutchins and Marusic2009b ) data points.

Profiles of AM coefficients where modulation by large-scale vertical velocity is considered are given in figure 7(b,d,f,h,j). In contrast to modulation by large-scale  $u$ , amplitude modulation due to $w_{l}$ is positive throughout much of the depth of the CBL, the magnitude of $R$ is larger for most quantities, and $R$ remains large as $-z_{i}/L$ increases. Physically, this means that amplitude modulation by large-scale $w$ occurs both under weakly convective conditions, where the updrafts are organized into long, linear features (HCRs) and under highly convective conditions, where buoyant plumes occur and the $w$ field organizes into open cells on large scales. As discussed above, the physical interpretation is that small-scale turbulence is excited in updraft regions, and suppressed in downdraft regions. This is conceptually analogous to the intermittent periods of small-scale excitation and quiescence between large-scale low- and high-momentum regions, respectively (Mathis et al. Reference Mathis, Hutchins and Marusic2011).

In order to summarize the figure 7 result, and to assess the extent to which AM occurs in the CBL in a global sense, we have also plotted depth-averaged values of the AM coefficients, which are shown in figure 8. That is, we plot histograms of the quantity $\overline{R}=(1/z_{i})\int _{0}^{z_{i}}R_{b_{l},a_{s}}\,\text{d}z$ , where $R_{b_{l},a_{s}}$ is the AM coefficient between the large-scale quantity $b_{l}$ and the small-scale quantity  $a_{s}$ . Note that by definition, $\overline{R}\in [-1,1]$ since $R_{b_{l},a_{s}}\in [-1,1]$ . This figure helps to better characterize global changes in AM with $-z_{i}/L$ , since discerning differences in the figure 7 was difficult in some cases. Figure 8(a,c) and (b,d) shows modulation of small-scale amplitude by $u_{l}$ and  $w_{l}$ , respectively; where values of $\overline{R}$ calculated from the single-point AM coefficient are displayed in (a,b) and values calculated from the two-point AM coefficient are displayed in (c,d). Perhaps the most striking result can be seen in the single-point correlations of $u_{l}$ with small-scale quantities in (a), where $|\overline{R}|$ is the largest for weakly convective (e.g.  $-z_{i}/L=3.1$ ) conditions and AM becomes negligible ( $\overline{R}\approx 0$ ) as $-z_{i}/L$ increases. This can be seen very clearly for $R_{u_{l},u_{s}}$ where the neutral channel flow data (Mathis et al. Reference Mathis, Monty, Hutchins and Marusic2009b ) are displayed together with the CBL LES results in (a).

In contrast, the average AM coefficients between $w_{l}$ and small-scale quantities exhibit very little dependence on $-z_{i}/L$ for all quantities considered. This is true for both single (b) and two-point (d) correlations, respectively. This demonstrates that $w_{l}$ is a ‘buoyancy proof’ modulator and AM due to $w_{l}$ occurs in an global sense as long as sufficient scale separation exists between the inner and outer peaks. It is not surprising that, for near-neutral cases, both $u_{l}$ and $w_{l}$ modulate small-scale quantities, since $u_{l}$ and $w_{l}$ are both so inherently tied to the identity of LSMs (where low- and high-momentum regions exhibit $\{u_{l}<0,w_{l}>0\}$ and $\{u_{l}>0,w_{l}<0\}$ , respectively). As surface heating increases, however, and the flow structure transitions to buoyant plumes (and $u^{\prime }$ becomes decorrelated from  $w^{\prime }$ , e.g. figure 2 b,d), it is entirely reasonable for $w_{l}$ to be the only plausible modulator. The correlations demonstrate this, and the results are consistent with all preceding results shown here (figures 2 to 5).

The effects of unstable stratification on LSMs and amplitude modulation are summarized in the conceptual diagram presented in figure 9. A canonical neutrally stratified high- $\mathit{Re}$ boundary layer is depicted in figure 9(a), a moderately convective ABL in (b) and a free convective ABL in (c). In (a,b), the surface layer is denoted by the dashed horizontal line. (Note that the diagram is not drawn to scale in order to emphasize AM within the surface layer.) For neutral stratification (a), the inclination angle of the hairpin vortex packet is $\unicode[STIX]{x1D6FE}\approx 15^{\circ }$ . In the surface layer ( $z/z_{i}\lesssim 0.1$ ), positive AM of a small-scale signal occurs in regions where $u_{l}^{\prime }>0$ and $w_{l}^{\prime }<0$ and negative AM occurs where $u_{l}^{\prime }<0$ and $w_{l}^{\prime }>0$ . As the ABL becomes unstably stratified (b), the inclination angle of surface-layer structures increases and LSMs become shorter than what is found in the neutral case. Once again AM in the surface layer is positive in regions where $u_{l}^{\prime }>0$ and $w_{l}^{\prime }<0$ and negative in regions where $u_{l}^{\prime }<0$ and $w_{l}^{\prime }>0$ . Finally, in the free-convective case (c), the LSMs shorten further and the inclination angle of surface-layer structures is $\unicode[STIX]{x1D6FE}\approx 90^{\circ }$ . In this case, $u_{l}$ is no longer an effective modulator. Near the ground, positive AM does occur in regions where $w_{l}^{\prime }<0$ and negative AM occurs in regions where $w_{l}^{\prime }>0$ . Here it also should be noted that $w_{l}^{\prime }>0$ regions are narrower than the $w_{l}^{\prime }<0$ regions, due to the positive skewness of $w$ (e.g. figure 1 f).

Figure 9. Conceptual diagram depicting the effects of buoyancy on LSMs and amplitude modulation in boundary layers with unstable thermal stratification. (a) Neutral case (canonical high- $\mathit{Re}$ boundary layer). (b) Shear–buoyancy transition where $U_{g}\neq 0$ and $Q_{0}>0$ . (c) Free convection where $U_{g}=0$ and $Q_{0}$ is finite. The figure is not drawn to scale in order to emphasize amplitude modulation within the surface layer (denoted by the horizontal dashed line). Note that $x^{\prime }$ denotes the direction aligned with the axis of the LSMs.

Finally, we emphasize that the strongest correlation is found between $w_{l}$ and small-scale momentum fluxes, $(uw)_{s}$ . Since $(uw)_{s}=u_{\unicode[STIX]{x1D70F}}^{2}$ at the wall, this result may have implications for development of surface flux models for use in wall-modelled LES (Pope Reference Pope2000; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010b ).