3.1. Statistics

As both an in-depth statistical description of the canopy flow and a validation against the literature have already been presented by Patton et al. (Reference Patton, Sullivan, Shaw, Finnigan and Weil2016), only a brief overview of the main characteristics of the ABL flow developing over a vegetation canopy is given here.

The mean vertical structure of the simulated ABLs is characterized in figure 4 using vertical profiles of key statistical quantities such as the mean horizontal velocity, mean potential temperature, resolved heat flux, variance of the horizontal and vertical velocity components and the RSS $\langle u'w' \rangle$. Above the canopy, all the statistics of the present configuration are in very good agreement with those recently reported by Salesky & Anderson (Reference Salesky and Anderson2018) for similar values of $-z_i/L$. For brevity, only the characteristics of the weakly unstable (WU, $-z_i/L=4.3$) are discussed here. Statistical error of these statistics are presented in Appendix E.1. The horizontal wind speed shows a gradual increase with height while the mean potential temperature shows a decrease up to $z/z_i = 0.7$, in agreement with the imposed stability regime. However, inside the canopy, the vertical profile of the horizontal wind speed shows an inflection point near the top of canopy (Patton et al. Reference Patton, Sullivan, Shaw, Finnigan and Weil2016), in agreement with the now well-admitted presence of the shear-layer flow developing at canopy top. Consistent with the results of Salesky & Anderson (Reference Salesky and Anderson2018), both the variance of the horizontal wind speed and the resolved heat flux exhibit a monotonic decrease in the region up to $z/z_i=0.7$. However, given the finer spatial resolution of the present simulations and the presence of the canopy, these two statistics show a local maximum just above canopy top and a subsequent reduction with decreasing $z$. These two statistics show a decrease below canopy top with also an inflection point in their vertical profiles. As noted by Salesky & Anderson (Reference Salesky and Anderson2018), the vertical evolution of the variance of the wall-normal velocity component is more complex with first a local maximum close to canopy top at $z/z_i=0.04$ or $z/h=2$, a slight decrease followed by an increase with an absolute maximum at $z/z_i=0.4$. Above $z/z_i=0.4, \langle {w^\prime }^2\rangle /w_m^2$ shows a gradual decrease. The local extremum at the lowest location can be attributed to influence of the canopy and the associated strong shear and turbulence production while the gradual increase up to $z/z_i=0.4$ is an already observed feature of sheared convective boundary layers caused by TKE redistribution among velocity components (see Salesky & Anderson (Reference Salesky and Anderson2018) and references therein). The RSS shows negative values throughout the ABL with an extremum at canopy top and a gradual decrease in magnitude with increasing wall-normal distance. Combined with the wall-normal profile of the horizontal velocity which exhibits high shear at the same location, it indicates that the near-canopy region is a region of strong production of kinetic turbulent energy. Finally, in agreement with the literature, the skewness of the streamwise and wall-normal velocity components (not shown here) show negative and positive values, respectively, above the canopy, corresponding to the presence of strong ejection events. Within the canopy, streamwise and vertical velocity skewness shift to positive and negative values, respectively, consistent with the statistical predominance of strong sweeping events in this region (Patton et al. Reference Patton, Sullivan, Shaw, Finnigan and Weil2016).

Figure 4. Horizontal ensemble-averaged wall-normal profiles of $(a)$ horizontal wind speed $\langle u_h \rangle = (\langle u \rangle ^2 +\langle v\rangle ^2)^{1/2}$, $(b)$ temperature $\langle \theta \rangle$, $(c)$ resolved heat flux $\langle w^\prime \theta ^\prime \rangle /(w_m \theta _m)$, $(d)$ horizontal velocity variance $(\langle {u^\prime }^2 \rangle +\langle {v^\prime }^2\rangle )/w_m^2$, $(e)$ vertical velocity variance $\langle {w^\prime }^2 \rangle /w_m^2$ and $(f)$ RSS $\langle u^\prime w^\prime \rangle /w_m^2$ for all the cases (the WU case is shown in blue dashed lines).

3.2. Flow visualization

Instantaneous horizontal slices of the three velocity component fluctuations $u^\prime , v^\prime$ and $w^\prime$ normalized by $w_m$ at $z/h=$ 10, 5 and 1 ($z/z_i=$ 0.20, 0.10 and 0.02, respectively) are presented in figure 5($a$–$i$) with close up views of the lowest level in figure 5($j$,$k$,$l$) for the WU case. Well above the canopy (figure 5$a$–$f$), the fluctuations of the three velocity components are marked by the presence of large-scale structures elongated in the streamwise direction. Regions of narrow intense upward motion (figure 5$c$,$f$) are found to correspond to narrow streaks of low longitudinal velocity (figure 5$a$,$d$) and region of lateral convergence of the flow (figure 5$b$,$e$). Conversely, the regions of downdraft are broader, their locations match that of the high longitudinal velocity regions and correspond to region of lateral divergence of the flow. These structures are separated in the transverse direction by a distance on the order of $z_i$, which corresponds well to the presence of large convective rolls of streamwise axis (Lemone Reference Lemone1973). In the present study, the main axis of the convective rolls is oriented along the streamwise direction $x$ whereas observations reported that their axis is usually at an angle of about 20$^{\circ }$ from the ABL layer-average wind direction (Lemone Reference Lemone1973). It should be noted that smaller scales are embedded in these large-scale structures, and are likely to be under the influence of the larger scales as observed by Lemone (Reference Lemone1976) and recently confirmed by Patton et al. (Reference Patton, Sullivan, Shaw, Finnigan and Weil2016) and Salesky & Anderson (Reference Salesky and Anderson2018). At the top of the canopy ($z/h = 1$), $w'$ does not show the footprint of large structures scaling with $z_i$ but rather a finer-scale organization (figure 5$i$). Examination of the close up view (figure 5$l$) reveals the presence of both intense localized downdraft motions and weaker and broader updraft motions. At this wall-normal distance, the streamwise velocity $u'$ exhibits more anisotropic structures than $w'$, showing streaks of low- and high-speed elongated in the streamwise direction. Looking at the entire $x$–$y$ plane (figure 5$g$), narrow long low-speed streaks can be seen. Their locations match well that of the low streamwise velocity regions detected $z/h =$ 5 and 10 (figure 5$a$,$d$). Inspection of $v'$ reveals a strong imprint of the larger scales detected at higher levels, with embedded smaller scales resembling oblique streaks of negative and positive lateral velocity.

Figure 5. Instantaneous fluctuations normalized by $w_m$ of (left) $u$, (centre) $v$ and (right) $w$ in the $x$–$y$ plane at ($a$,$b$,$c$) $z/h=10$, ($d$,$e$,$f$) $z/h=5$ and ($g$,$h$,$i$) $z/h=1$ for the WU case. Panels ($j$,$k$,$l$) are close up views of the centre of the velocity field at $z/h=1$ corresponding to the black squares shown in ($g$,$h$,$i$).

From these qualitative observations, it can be concluded that the flow near the canopy top results from the superposition of large scales (of size of the order of $z_i$) onto smaller scales, with size of the order of $h$. This is in agreement with the observations of Raupach et al. (Reference Raupach, Finnigan and Brunet1996) and Patton et al. (Reference Patton, Sullivan, Shaw, Finnigan and Weil2016) that canopy eddies (consisting of compact strong intermittent sweeps and large weaker ejections) are under the influence of larger scales of the ABL, whose structure in the present case seems compatible with the presence of convective rolls. Longitudinal spacing of the structures present in the canopy-top region is also in qualitative agreement with the estimate $\varLambda _x/ L_s = 8.1$ of Raupach et al. (Reference Raupach, Finnigan and Brunet1996) which corresponds here to $5.2h$. It is worth mentioning that near canopy top, the presence of the larger scales of the ABL is mainly seen in both $u'$ and $v'$, the latter being likely to play a major role in generating canopy-eddies through the triggering of Kelvin–Helmholtz instabilities, despite leaving no trace in the statistics because of the homogeneous character of the flow in the lateral direction $y$.

3.3. Spectrograms

The qualitative conclusions based on the instantaneous slices presented in § 3.2 are now further investigated for the WU case through the analysis of the premultiplied one-dimensional spectra of the three velocity components, as a function of height $z$ and the spatial longitudinal or transverse wavelength, $\lambda _x$ or $\lambda _y$, respectively (figure 6). The wavelength corresponding to local maxima in the streamwise and transverse direction are denoted $\lambda _x^m$ and $\lambda _y^m$, respectively.

Figure 6. Pre-multiplied spectra of the three velocity components plotted as a function of the wall-normal distance $z$ and (left) the streamwise wavelength $\lambda _x$ and (right) the transverse wavelength $\lambda _y$ (normalized by $z_i$ or $h$) (WU case). Pre-multiplied spectra normalized by $w_m^2$ of $u, v$ and $w$ are shown in ($a$,$b$), ($c$,$d$) and ($e$,$f$), respectively. The vertical dashed line shows the cutoff wavelength chosen for scale separation. The red dashed line indicates the location of the local maximum of the spectra. Oblique dotted blacked lines correspond to trends $\lambda ^m \propto z^2$ in ($a$,$c$,$d$) and $\lambda ^m \propto z$ in ($b$,$e$,$f$) of the most energetic canopy scales in the region $1< z/h<5$. Spectra are scaled up by a factor of $10^4$ for clarity. For clarity, 20 contours are distributed logarithmically between 0.025 and the maximum of each spectrum. The horizontal dark-green long-dashed line corresponds to canopy top.

Within the canopy, the wavelengths corresponding to the maximum of energy of the longitudinal and transverse spectra of both $u$ and $v$ are fairly constant with height, particularly in the lower half of the canopy ($0 < z/h < 0.5$). In this region of the canopy, the value of the wavelengths corresponding to the most energetic scales of the longitudinal component $u$ are found to be such that $\lambda ^m_x < \lambda ^m_y$. This might be interpreted as the footprint of the two-dimensional roller structures of spanwise axis generated by the Kelvin–Helmholtz instability at canopy top. However, the detailed investigation of the characteristics of the coherent structures within the canopy in beyond the scope of the present paper. Table 3 summarizes these wavelengths obtained in the upper half of the canopy. These length scales derived based upon the local spectral peak are larger than the integral length scales presented in Patton et al. (Reference Patton, Sullivan, Shaw, Finnigan and Weil2016), however, the length scale variation with wall-normal distance is consistent across both techniques. In the streamwise direction, the wavelength corresponding to the peak of the energy spectra of the vertical velocity component $w$ is constant throughout the canopy ($0< z/h<1$) and equal to $\lambda ^m_x/ h=4$, consistent with the integral length scales computed by Patton et al. (Reference Patton, Sullivan, Shaw, Finnigan and Weil2016). Conversely, in the transverse direction, the wavelengths of the most energetic structures increase with wall distance in the bottom half of the canopy and stays approximately constant in the upper half, with $\lambda ^m_y/ h\approx 4$ in this region. Finding similar values for $\lambda ^m_x$ and $\lambda ^m_y$ near canopy top confirms the isotropic character of the structures in $w$ noted in figure 5. In the streamwise direction, $\lambda _x / h=4$ corresponds to a streamwise spacing of canopy eddies $\varLambda _x / h=4$ (as defined by Raupach et al. Reference Raupach, Finnigan and Brunet1996) which falls into the range reported by Raupach et al. (Reference Raupach, Finnigan and Brunet1996) for comparable $L_s/h$ ($=0.64$ in the present case).

Table 3. Streamwise and spanwise wavelengths $\lambda _x^m$ and $\lambda _y^m$ corresponding to the most energetic scales of the three velocity components in the different regions of the flow, normalized by the canopy height $h$ or the boundary-layer depth $z_i$.

It must be noted that, for the present WU stability regime, inside the canopy, footprint of the larger scales from the ABL is only visible in the spectra in the transverse direction of $v$. Therefore, even if visible on instantaneous snapshots of the flow field, they are not energetic enough to stand out in this flow configuration. This contrasts with the spectral characteristics of the flow found in other stability regimes for which secondary peaks at ABL scales are present (Patton et al. Reference Patton, Sullivan, Shaw, Finnigan and Weil2016). However, further analysis is beyond the scope the present study. In the region $1< z/h<5$, all the spectra show a local maximum wavelength of the order of a few $h$ in agreement with the presence of canopy scales. Nevertheless, the primary spectral peak of the transverse velocity component corresponds to the larger scales from the ABL, which coexist with the canopy scale in this region. The wavelengths corresponding to the near-canopy structures increase with the wall distance either as $\lambda ^m \propto z^2$ for $u$ in the streamwise direction and $v$ in both directions or $\lambda ^m \propto z$ for $u$ in the transverse direction and $w$ in both directions (oblique dotted line in 6). In the region extending up to $z/h = 5$, the transverse wavelength corresponding to the local extrema of the pre-multiplied spectra of $w$ in the $x$ and $y$ directions shows a clear linear evolution with the wall-normal coordinate. This behaviour is also visible for the longitudinal component $u$, but only for the transverse direction. This linear increase with wall-normal distance of the most energetic scales is in agreement with the linear increase of the transverse spacing of low- and high-speed streaks and size of vortical structures reported in the surface layer of the ABL by previous studies (Lin et al. Reference Lin, McWilliams, Moeng and Sullivan1996a,Reference Lin, Moeng, Sullivan and McWilliamsb; Drobinski et al. Reference Drobinski, Carlotti, Newsom, Banta, Foster and Redelsperger2004).

Well above the canopy, the pre-multiplied spectra in the transverse direction reveal the presence of the footprint of large scales at $\lambda _y/ z_i = 1.0$ below $0.15 z_i$ for $u$ and $v$ and above $0.15 z_i$ for $w$. The presence of this peak in the spectra in the outer region of the boundary layer is consistent with the existence of streamwise rolls of large vertical and spanwise extent, and at the bottom of which the vertical component is minimum. Moreover, the lateral spacing corresponding to this wavelength of $1.0 z_i$ is in agreement with the observations from figure 5. According to Sykes & Henn (Reference Sykes and Henn1989), when $u_{*}/w_{*}$ is between 0.35 and 0.5, the aspect ratio of the rolls (defined as their width divided by their depth) is approximately 4. Therefore, in the present simulation, the ratio $u_{*}/w_{*}$ being equal to 0.45, one could use the value of the roll lateral spacing given by the large-scale lateral wavelength of $1.0 z_i$ to estimate the vertical extent of these structures to be $0.26 z_i$. In the longitudinal direction, there is a strong imprint of the larger scales showing at the wavelength $\lambda _x /z_i = 3.5$ for the three velocity components. Above $z/h = 5$, most of the energy is contained in the wavelength such that $\lambda _y / z_i \simeq 1.0$. Thus, in this region, the energy content in the spectral domain is consistent with the presence of large-scale structures corresponding to streamwise-oriented rolls whose vertical and spanwise extent scale with $z_i$. Pre-multiplied cospectra between the vertical velocity component $w$ and the longitudinal velocity component $u$ or the potential temperature $\theta$ as a function of the wall-normal distance and $\lambda _x$ or $\lambda _y$ are shown in figure 7. These plots clearly demonstrate the spectral separation between the near-canopy turbulence and the most energetic large scales from the ABL. The characteristic wavelengths identified in the pre-multiplied spectra of the velocity components are confirmed as well as the linear increase of the energetic scales with the wall-normal distance, entailing the presence of self-similar eddies. It is worth mentioning here that, when multiplied by the vertical gradient of the mean streamwise velocity, the RSS cospectrum provides direct information on the contribution of the turbulent scales to TKE production. It can already be anticipated that in the near-canopy region, TKE production mostly comes from the canopy scales themselves.

Figure 7. Pre-multiplied cospectra plotted as a function of the wall-normal distance $z$ and the (left) streamwise wavelength $\lambda _x$ and (right) the transverse wavelength $\lambda _y$ (normalized by $z_i$ or $h$) (WU case). The negative of the pre-multiplied cospectra of $w$ and $u$ (e.g. vertical momentum flux) normalized by $w_m^2$ are shown in ($a$,$b$) and pre-multiplied cospectra of $w$ and $\theta$ (e.g. vertical heat flux) normalized by $w_m\theta _m$ are shown in ($c$,$d$). The vertical dashed line shows the cutoff wavelength retained for scale separation. The red dashed line indicates the location of the local maximum of the spectra. Cospectra are scaled up by a factor of $10^4$ for clarity. For clarity, 20 contours are distributed logarithmically between 0.025 and the absolute maximum of each cospectra. The horizontal dark-green long-dashed line corresponds to canopy top.

3.4. Evidence of AM of small scales

In the previous section, we have shown that the most energetic structures in the canopy region and in the ABL are well separated in spectral space. Here, we demonstrate in the WU case the existence of AM of the canopy turbulence by the larger scales of the ABL using the same method as that proposed by Bandyopadhyay & Hussain (Reference Bandyopadhyay and Hussain1984). To do so, we use instantaneous $u '$ at the centre of the spatial domain ($y/h = 0$), at $z/h = 1$. A two-dimensional low-pass filter with cutoff wavelengths of $\lambda _x /z_i = 0.57$ and $\lambda _y /z_i = 0.29$ is employed to decompose $u^\prime$ into its large- and small-scale components $u^L$ and $u^S$, respectively, such that $u^\prime = u^L + u^S$. The large-scale envelope $E^L(u^S)$ of the small-scale signal is then computed by low-pass filtering $(u^S)^2$ with the same cutoff wavelength (Basley et al. Reference Basley, Perret and Mathis2018). Example signals obtained via this scale decomposition are shown in figure 8. The similarity between $u^L$ and $E^L(u^S)$ is evident and their resemblance is confirmed by the significant level of correlation between the two signals; which for the present case, the correlation coefficient between these two signals is 0.7, which clearly indicates AM of canopy turbulence by large ABL scales, similar to that found by Mathis et al. (Reference Mathis, Hutchins and Marusic2009) in the near-wall region of a FPBL. In order to quantify the AM of the small scales by the large scales of the flow, the single point AM coefficient, defined as the correlation coefficient between the low-pass filtered envelope of a small-scale quantity $X^S$ and the large-scale velocity component $u_i^L$, measured at the same point, is computed as (Mathis et al. Reference Mathis, Hutchins and Marusic2009)

(3.1)\begin{equation} AM(u_i^L,X^S)= \frac{\langle u_i^L E^L(X^S) \rangle} {\sigma_{u_i^L} \sigma_{E^L(X^S)}}. \end{equation}

In order to fully characterize the AM mechanism, $AM(u_i^L,X^S)$ is computed for all the velocity components but also for the small-scale kinetic energy defined as $(\xi ^S)^2=[(u^S)^2+(v^S)^2+(w^S)^2]/2 = \delta$KE$^{(1)}$. Consistent with previous studies of various types of wall-bounded flows (e.g. Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014; Blackman & Perret Reference Blackman and Perret2016; Basley et al. Reference Basley, Perret and Mathis2018; Salesky & Anderson Reference Salesky and Anderson2018), all small-scale velocity components (and consequently the small-scale kinetic energy) are amplitude modulated by the large-scale velocity components (figure 9). The wall-normal distribution of the AM coefficients depends on the large-scale component considered, while for one given large-scale component, all the small-scale quantities are influenced in the same manner, as demonstrated by AM correlation coefficients with similar wall-normal variation and magnitude. When considering $AM(u^L,X^S)$ (figure 9$a$), we find positive values with a peak at canopy top, meaning that canopy-scale turbulence is amplified by high-intensity large-scale structures of the streamwise velocity component $u^L$. $AM(u^L,X^S)$ changes sign at $z/h=2$ (or $z/z_i=0.04$), a behaviour similar to that found by Salesky & Anderson (Reference Salesky and Anderson2018). Non-zero values of $AM(v^L,X^S)$ (figure 9$b$) result from having not performed any local rotation has been performed to align the horizontal velocity components with the mean wind direction at each height (as was done in Salesky & Anderson (Reference Salesky and Anderson2018), for instance). Despite that difference, in the region above the canopy and when taking into account the vertical displacement induced by the canopy, the present levels of $AM(u^L,X^S)$ compares well to Salesky & Anderson (Reference Salesky and Anderson2018) (not shown here). A similar trend is found for the large-scale wall-normal velocity component $AM(w^L,X^S)$ as for $AM(u^L,X^S)$, but with opposite sign (figure 9$c$). Contrary to what Salesky & Anderson (Reference Salesky and Anderson2018) observed in their ABL flow without canopy, significant negative values of $AM(w^L,X^S)$ are found in the canopy region for all three small-scale velocity components and kinetic energy. Canopy-scale turbulence is therefore suppressed when upward (positive $w^L$) LSM exists within the canopy, while enhancing turbulence in the region above the canopy.

Figure 8. Instantaneous velocity signals showing AM at $z/h=1$. $(a)$ instantaneous fluctuations $u^\prime$ of the streamwise velocity component, $(b)$ large-scale component of $u^\prime$, $(c)$ fluctuations of the low-pass-filtered envelope $E^L({u_i^\prime }^S)$ of the small-scale velocity components (solid line) ${u^\prime }^S$, (dashed line) ${v^\prime }^S$ and (dotted line) ${w^\prime }^S$, $(d)$ (solid black line) small-scale TKE $(\xi ^S)^2$ and (dotted red line) low-pass filtered envelope of $(\xi ^S)^2$. See text for the definition of the employed low-pass filter. (WU case).

Figure 9. The AM coefficients of the three small-scale velocity components ($\blacksquare$, red) $u^S$, ($\bullet$, blue) $v^S$, ($\blacktriangle$, green) $w^S$ and ($\blacktriangledown$, black) the corresponding TKE $(\xi ^S)^2$ by $(a)$ the large-scale streamwise velocity component $u^L$, $(b)$ the large-scale spanwise velocity component $v^L$ and $(c)$ the large-scale wall-normal velocity component $w^L$ (WU case). The beige shaded area corresponds to the canopy region.

3.5. Small-scale TKE and RSS budget

The transport equations for TKE and RSS obtained using the multi-level formalism presented in § 2.4 with a four-level decomposition are used to further analyse the interaction between large and small scales. Only the WU case is analysed here, the influence of atmospheric stability being addressed in § 3.6. We focus on terms representing interscale spatial transport and transfer of TKE and RSS between components of the velocity field (table 2), namely $T_t^S$ the (spatial) turbulent transport of small-scale kinetic energy by the small scales themselves, $T_t^L$ the (spatial) turbulent transport of small-scale kinetic energy by the large scales ${\boldsymbol{u}^L}, P^M$ the production of small-scale kinetic energy by the mean flow ${\boldsymbol{u}^M}, T_r^L$ the interscale transfer of energy between the large scales ${\boldsymbol{u}^L}$ and the small scales ${\boldsymbol{u}^S}$ (with $T_r^L>0$ if ${\boldsymbol{u}^S}$ receives energy from ${\boldsymbol{u}^L}$), and $D_r$ the contribution of drag to small-scale kinetic energy, which involves interaction between ${\boldsymbol{u}^S}, {\boldsymbol{u}^L}$ and ${\boldsymbol{u}^M}$. The small-scale TKE production term due to buoyancy $B^S$ is also shown for completeness. In addition to the analysis of the transport term of the small-scale kinetic energy, a similar analysis is conducted for the small-scale RSS $-\langle u^Sw^S \rangle$. Since the RSS is typically negative, we include a minus sign in our definition so that a term in the balance equation with a positive value corresponds to an increase in the small-scale shear stress magnitude. For each of these terms, vertical profiles of both their mean value and standard deviation have been computed and are shown in figure 10 to analyse their relative importance on average but also from an instantaneous point of view. In particular, comparing the standard deviation to the mean value of a given term highlights the possible occurrence of instantaneous forward and backscatter of kinetic energy or RSS.

Figure 10. Wall-normal evolution of ($a$,$c$) the mean and $(b,\!d)$ standard deviation of the interscale transfer terms ($\blacksquare$, red) $T_t^S$, ($\bullet$, blue) $T_t^L$, ($\triangle$, pink) $P^M$, ($\blacktriangledown$, orange) $T_r^L$, ($\diamond$, green) $D_r$ and (solid black line) $B^S$ (as in table 2) for ($a$,$b$) the small-scale TKE $(\xi ^S)^2$ and ($c$,$d$) RSS $-u^Sw^S$ (WU case). Positive values indicate a gain for the small scales.

The primary findings regarding the transport of small-scale turbulent kinetic energy in the WU configuration can be summarized as follows (figure 10$a$). As expected, the production term $P^M$ is always positive, confirming that the mean flow is a source term of TKE. The spatial transport term $T_t^S$ acts as a source term within the canopy (with a peak at the same height as the production term $P^M$), and a sink term just above the canopy. The small-scale component $({\boldsymbol{u}^S})$ transports its energy from above into the canopy. As ${\boldsymbol{u}^S}$ represents in this region the most energetic scales (figure 6), which is consistent with previous studies of TKE budgets within and above canopies which did not decompose each term by scale (Brunet et al. Reference Brunet, Finnigan and Raupach1994; Dwyer, Patton & Shaw Reference Dwyer, Patton and Shaw1997). The turbulent transport term $T_t^L$ of $(\xi ^S)^2$ by ${\boldsymbol{u}^L}$ is one order of magnitude smaller than the other terms, which agrees with the findings of Kawata & Alfredsson (Reference Kawata and Alfredsson2018) in the case of a plane Couette flow. $T_t^L$ is positive within the canopy, meaning that the large scales ${\boldsymbol{u}^L}$ transport small-scale energy into the canopy layers of the flow, and is negative above the canopy demonstrating that large-scale turbulence transports small-scale kinetic energy away from the roughness sublayer. $T_t^L$ becomes positive for $z/h>8$ with very low magnitude. The interscale energy transfer term $T_r^L$, which indicates nonlinear scale interaction (e.g. triadic interaction), is positive for all $z$ and peaks at canopy top. Intense mean wind speed shear in this region promotes small-scale TKE generation via the work of the small-scale Reynolds stresses against the spatial gradients of the large-scale velocity components. In the mean, TKE is transferred from large to small scales (classic forward scatter of energy), consistent with the concept of the energy cascade process. It can be noted that the sum of $T_r^L$ and $T_t^L$, which represents the global interaction between large and small scales, peaks (with a positive value) at canopy top and is dominated by the contribution of $T_r^L$ in this region. Within the canopy, the drag term $D_r$ acts as a sink of turbulent kinetic energy throughout the entire canopy (and is zero above) as expected. Finally, it is worth mentioning that the AM coefficient $AM(u^L,\xi ^S)$ involves the same quantity $u_i^Su_i^Su_j^L$ as the spatial transport term $T_t^L$, which suggests that $AM(u^L,\xi ^S)$ preferentially points to the influence of spatial transport rather than pure nonlinear interscale energy transfer. However, this last point only holds when the AM coefficient is based on the term $u_i^Su_i^Su_j^L$ (as in Basley et al. (Reference Basley, Perret and Mathis2018) or Blackman & Perret (Reference Blackman and Perret2016)) and not when defined using the low-pass-filtered envelope of the small scales calculated as the Hilbert transform of the small-scale signals, as in Mathis et al. (Reference Mathis, Hutchins and Marusic2009) or Salesky & Anderson (Reference Salesky and Anderson2018).

The standard deviations of these interaction terms controlling the transport of small-scale TKE are of a similar order of magnitude as their respective mean values or even greater by a factor of four to five for $T_t^S$ and $T_t^L$ (figure 10$b$). In decreasing order of standard deviation, the five different transport terms rank as follows within the canopy: $T_t^S > T_t^L > P^M > D_r > T_r^L > B^S$. From an instantaneous or local point of view, intense events of both forward- and backscatter of energy exist, and are dominated by spatial transport, and then by interscale transfer $T_r^L$. The standard deviation of $T_t^L$ being larger than that of $T_r^L$ shows that instantaneous spatial transport dominates in the redistribution of energy.

Turning focus now to the small-scale RSS (figure 10$c$), the main results are the following. Similar conclusions as for the small-scale TKE terms can be drawn concerning $T_t^S, P^M$ and $D_r$, which do not involve large-scale – small-scale interactions. The spatial transport term $T_t^L$ is positive just below $z/h=1$, negative for $1 < z/h < 6-7$ and positive (close to zero) above. Small-scale RSS is therefore transported away from the near-canopy region by ${\boldsymbol{u}^L}$. Positive values of the interscale transfer term $T_r^L$ for $z<3h$ (e.g. in the roughness sublayer) indicate forward interscale transfer (e.g. from ${\boldsymbol{u}^L}$ to ${\boldsymbol{u}^S}$), while above $z/h\simeq 3, T_r^L$ becomes negative indicating a backscatter mechanism with RSS transfer from ${\boldsymbol{u}^S}$ to ${\boldsymbol{u}^L}$, consistent with Kawata & Alfredsson (Reference Kawata and Alfredsson2018). Above $z/h\simeq 4$, the sum $T_t^L + T_r^L$ remains negative (not shown here), suggesting a net loss of small-scale Reynolds shear stress to the large scales. Finally, conclusions drawn on the standard deviation of all the terms from the analysis of the small-scale TKE remain valid for those of the RSS (figure 10$d$).

3.6. Influence of stability

Atmospheric stability's influence on the wall-normal distribution of the AM coefficients between the large-scale velocity components $u_i^L$ and the small-scale TKE $(\xi ^S)^2$ is shown in figure 11. The same trends as for the WU case are found for $AM(u^L,\xi ^S)$ and $AM(w^L,\xi ^S)$, for all the stability cases, namely positive (negative) values of $AM(u^L,\xi ^S)$ ($AM(w^L,\xi ^S)$) within the canopy and negative (positive) above. Atmospheric stability seems to influence more the wall-normal evolution of $AM(v^L,\xi ^S)$ with a sign change in $AM(v^L,\xi ^S)$ between the NN case and all other stability cases below $z/h=2$. This difference is attributed to the influence of the Coriolis forces on the spanwise component $v$ which decreases with increasing $-z_i/L$ and the fact that no coordinate rotation was performed to align the horizontal velocity component with the local mean wind. As found by Salesky & Anderson (Reference Salesky and Anderson2018), at any wall-normal location, the magnitude of $AM(u^L,\xi ^S)$ (and $AM(v^L,\xi ^S)$) decreases with increasing $-z_i/L$, even dropping to zero for $z/h > 8$ (or $z/z_i>0.16$) for the most unstable case (figure 11$a$). Regarding the influence of the large-scale wall-normal velocity component $w^L$ (figure 11$c$), the general trend is a reduction of the magnitude of $AM(w^L,\xi ^S)$ with increasing $-z_i/L$ but with non-zero values, indicating a persistent influence of the large-scale updraft on the canopy-scale turbulence. Nevertheless, our results demonstrate clear differences from Salesky & Anderson (Reference Salesky and Anderson2018) who found little influence of stability on $AM(w^L, \xi ^S)$ when interrogating turbulent flows interacting with parameterized roughness, which suggests a key influence of explicitly resolving processes produced by the inflection-point instability at canopy top. The mean transport terms of the small-scale TKE budget are analysed as a function of $-z_i/L$ (figure 12). As expected given the decrease of the imposed geostrophic wind with increasing $-z_i/L$, the production of small-scale TKE $P^M$ drops to zero with increasing $-z_i/L$. In the most unstable cases, the flow is no longer shear driven but buoyancy driven through the influence of the term $B^S$. This term, whose magnitude is almost independent of the atmospheric stability (with a peak value around $0.012h/w_m^3$ except for the FC case which peaks at $0.015h/w_m^3$), becomes the predominant source term of small-scale TKE with increasing $-z_i/L$. Spatial transport of small-scale energy by the large scales $T_t^L$ drops to zero with increasing $-z_i/L$ while spatial transport of small-scale energy by the small scales $T_t^S$ also decreases with $-z_i/L$ but stays non-zero in the roughness sublayer. The height of the (positive) peak of $T_t^S$ remains the same for all $-z_i/L$. The region of negative $T_t^S$ above the canopy extends to higher heights with increasing $-z_i/L$. It is therefore consistent with the existence of a mixing-layer region in which small-scale energy is transported by small scales from above the canopy into the canopy, with a wall-normal extent of a few times $h$. The interscale energy transfer term $T_r^L$ remains positive, consistent with the classic energy cascade process. Through drag, the canopy remains a sink term of kinetic energy (negative values of $D_r$) as expected. Decomposing the global drag term $D_r$ by scale (as in (2.19)) shows that the major contribution comes from the interaction between the small scales and the mean velocity field, followed by the interaction among small scales (not shown). Energy exchange via interaction between the large and small scales is two orders of magnitude smaller than the leading term, indicating that the influence of the large scales on the small scales through drag is negligible compared with that of mean shear. Finally, the conclusions associated with interrogating the standard deviation of the interaction terms from the WU case (§ 3.5) remain valid with increasing instability (not shown); a result indicating that no matter the stability regime, large and small scales interact instantaneously and predominantly through spatial transport. Regarding the transport of small-scale RSS (figure 13), all transport terms drop progressively to zero with increasing $-z_i/L$ while their standard deviations remain high and almost independent of the stability regime. In the FC case, the source term $B^S$ also becomes zero as the covariance $\langle \theta _v^S u^S \rangle$ (as well as $\langle \theta _v^S v^S\rangle$) drops to zero due to the isotropic character in the horizontal plane of the convection cells. To highlight the importance of the mean interscale transfer of TKE or RSS, the term $T_r^L$ is shown in figure 14 normalized by the production terms $P^M + B^S$. As mentioned above, all flow configurations show forward scatter of TKE from the large to the small scales with increasing relative magnitude with increasing $-z_i/L$ (figure 14$a$). It must be noted that as the influence of the mean shear decreases, a peak of $T_r^L/(P^M + B^S)$ appears at canopy top, reaching almost 60 % for the FC case. This indicates that as the mean shear production term vanishes, the large scales become a non-negligible source of TKE for the near-canopy structures. Except for the NN case, which shows an almost constant with height forward scatter of RSS (close to 2 %), backscatter of RSS exists for all other atmospheric stability cases above $z/h\simeq 2$ with an increasing relative importance with increasing $-z_i/L$ (figure 14$b$).

Figure 11. The AM coefficients $AM(u_i^L,\xi ^S)$ between the small-scale TKE $(\xi ^S)^2$ and the large-scale $(a)$ streamwise velocity component $u^L$, $(b)$ spanwise velocity component $v^L$ and $(c)$ wall-normal velocity component $w^L$, for each atmospheric stability cases listed in table 1.

Figure 12. Wall-normal profiles of the mean interscale transfer terms ($\blacksquare$, red) $T_t^S$, ($\bullet$, blue) $T_t^L$, ($\triangle$, pink) $P^M$, ($\blacktriangledown$, orange) $T_r^L$, ($\diamond$, green) $D_r$ and (black solid line) $B^S$ for the small-scale TKE $(\xi ^S)^2$, for all the atmospheric stability cases listed in table 1. Positive values indicate a gain for the small scales. Please note that the $x$-axis range changes between panels.

Figure 13. Wall-normal profiles of the mean interscale transfer terms ($\blacksquare$, red) $T_t^S$, ($\bullet$, blue) $T_t^L$, ($\triangle$, pink) $P^M$, ($\blacktriangledown$, orange) $T_r^L$ and ($\diamond$, green) $D_r$ and (black solid line) $B^S$ for the small-scale RSS $-u^Sw^S$, for all the atmospheric stability cases listed in table 1. Positive values indicate a gain for the small scales. Please note that the $x$-axis range changes between panels.

Figure 14. Mean interscale transfer term $T_r^L$ normalized by the sum of the shear and buoyancy production terms $P^M+B^S$ for $(a)$ small-scale TKE $(\xi ^S)^2$, and $(b)$ the small-scale RSS $-u^S w^S$, as a function of the wall-normal distance, for the five stability regimes. See corresponding text for the definition of each term.

This section's discussion clearly demonstrates atmospheric stability's influence on the vertical flow structure and subsequently the mechanisms controlling energy transport. Increasing instability ($-z_i/L \rightarrow +\infty$) results in reduced mechanical turbulence production by reducing mean wind speed shear magnitudes in the lower regions of the flow. However, it has been shown (e.g. Patton et al. Reference Patton, Sullivan, Shaw, Finnigan and Weil2016; Salesky & Anderson Reference Salesky and Anderson2018) that increasing $-z_i/L$ and changing the stability regime from neutral to free convection drastically changes the instantaneous structure of the flow. The elongated roll structures visible in weakly unstable and near-neutral regime (figure 5) indeed transition to cellular convective structures with increasing $-z_i/L$ (not shown). Complementary to our analysis of the interscale transfer terms (see discussion surrounding figures 12 and 14), we now study the influence the evolution ABL-scale structure with increasing atmospheric instability (i.e. the evolution from rolls to cells) has on interscale energy transfer $T_r^L$ by interrogating stability's influence on contributions from subcomponent terms comprising $T_r^L$ that involve horizontal gradients of the large-scale velocity field. In the following, $T_{r,xy}^{u_i L}$ denotes the component $(G_3-I){\mathcal {L}}_{ix}^{L}({\delta u_i^L}/{\partial x}) + (G_3-I){\mathcal {L}}_{iy}^{L}({\delta u_i^L}/{\partial y})$ of $T_r^L$ (with no implicit summation over repeated index $i$ here). Here, $T_{r,xy}^L$ represents the contribution of the horizontal gradients of the three large-scale velocity components to $T_r^L$. The contribution of the vertical gradient of the LSM is represented by the difference $T_r^L - T_{r,xy}^{L}$. Wall-normal profiles of $T_r^L, T_{r,xy}^{u_i L}$ (with $i=1$, 2 or 3) and $T_{r,xy}^{L}$ are presented in figure 15.

Figure 15. Contribution of the terms involving horizontal gradients of the large-scale velocity field to the TKE interscale transfer term $T_r^L$ as a function of the wall-normal distance, for the five stability regimes. See corresponding text for the definition of each term.

First, for any stability regime, the difference $T_r^L - T_{r,xy}^{L}$ indicates that the terms involving the vertical gradient of the LSM are responsible for the forward scatter peak at canopy top (difference between the orange line, $T_r^L$ and the dark-green diamond ($\diamond$, green) $T_{r,xy}^L$) which is consistent with the classic turbulence kinetic energy production process, namely the work of small-scale shear stress against vertical gradient of large-scale velocity components. Above $z/h=2$, the horizontal gradients of the large-scale motion are major contributors to interscale energy transfer, with contribution from the different velocity components varying with $-z_i/L$. In particular, there is an increasing contribution of the vertical large-scale velocity component $w^L$ to interscale transfer with increasing $-z_i/L$. The contribution of the horizontal velocity components ($u^L$ and $v^L$) peaks at $z/h \simeq 2$ or a bit lower for the free-convective case while the influence of $w^L$ starts higher if present. For all stability cases, the global trend is the following: there is a major contribution of the horizontal gradients of ($\triangle$, pink) $u^L$, then ($\blacktriangledown$, red) $v^L$ and then ($\bullet$, blue) $w^L$, the latter being close to zero for weak convective regimes. The influence of $w^L$ indeed becomes significant and predominant only for the two most unstable cases and impacts the region $z/h>2$, consistent with the analysis of the AM coefficient $AM(w^L,\xi ^S)$ as a function of $-z_i/L$.