As mentioned earlier, the canopy convector effect was introduced by while studying the interaction of vegetation cover with the surface radiation balance for the Yatir forest. The annual average incoming solar radiation in the Yatir forest is about 238 W m-2, comparable to that in the Sahara, but the net radiation (Rn) is about 35 % higher than the Sahara (RY10) due to the lower albedo of the forest. However, both remote sensing and local measurements indicated that the surface temperature of the forest canopy in Yatir is lower than the surface temperature of the nearby non-forested area – on annual average by about 5 K. This is striking, as firstly, the lower albedo (by 0.1) of the forest than that of the surrounding shrubland translates into an approximately 24 W m-2 increase in radiation load on the forest canopy. Secondly, the cooler canopy surface suppresses the upwelling longwave radiation, resulting in an additional increase in radiation load by about 25 W m-2. The combined annual increase in radiation load by about 50 W m-2 associated with the Yatir afforestation in the Negev is quite high and is comparable to the net annual radiation difference between the Sahara and Denmark, for example (RY10). Thirdly, the latent heat flux of evapotranspiration (LE), the obvious cooling and energy dissipation mechanism in temperate forests, is not an option since water is virtually unavailable for about 7 months a year. Thus sensible heat flux (H) is the only major heat dissipation route, translating into a Bowen ratio (H/LE) as high as 20 or more – unlike temperate forests with a Bowen ratio ≈ 1. In the Yatir forest, the entire net solar radiation flux (up to 800 W m-2) is equilibrated by a massive sensible heat flux (H) of similar magnitude. Note that this high H cannot be explained by the difference between surface and air temperature (ΔT=Ts-Ta) as the canopy surface is cooler than the surrounding desert surface in this case, but the air temperatures above desert and forest canopy are similar. To expound this apparent contradiction of larger sensible heat flux for smaller ΔT, it is important to recall that, when adopting the simplified big-leaf representation of the forest as a single surface, H=-ρCpTa-TsrH, where ρ and Cp are the density and specific heat capacity of air, respectively, Ta is air temperature, Ts is canopy surface temperature, and rH is the apparent canopy aerodynamic resistance to heat transfer (the word “apparent” is used to indicate that this property is a construct of the formulation and not a direct physical property). Hence the large H is not explained by the temperature difference (ΔT) but by a decreased rH. Thus the semi-arid forest with its low tree density and large surface area becomes an efficient low aerodynamic resistance “convector” that is well coupled to the atmosphere above . This “canopy convector effect” (CCE) is adequate enough to support the massive sensible heat flux larger than the surrounding Negev desert, still maintaining a relatively cooler (than the desert) surface temperature (of the canopy top). It is worth noting here that Eq. () offers a very simplistic description of the complex mixing process in the surface layer; however, it should be interpreted as a zeroth-order representation of the corresponding processes. RY10 identified the difference of roughness between the desert and forest as the underlying mechanism of the CCE by arguing that rH∝1/PAI where PAI denotes the plant area index. However, in this work, we attempt to identify a more fundamental mechanism behind the CCE which is more strongly connected to the feature of canopy turbulence. Therefore we hypothesize that even with the same physical roughness, variation of the aerodynamic roughness is a sufficient condition for displaying the CCE. This difference of aerodynamic roughness for the same physical roughness (of the same vegetation canopy) can be generated by changing the intensity of atmospheric stratification . Thus observing the variation of the canopy aerodynamic resistance to heat transfer (rH) with atmospheric instability is a sufficient condition to demonstrate the generality of the CCE. To be more precise, if rH is found to decrease with increasing unstable stratification, that would exhibit the fact that canopy turbulence effectively reduces the aerodynamic resistance to cope with heat stressed environments; i.e., the canopy convector effect would manifest itself.

Therefore, to summarize the canopy convector effect in simpler terms, it can be mentioned that the darker and colder canopy surface reduces albedo, which leaves more of the incoming energy on the canopy surface. However, the organization of these dark leaf surfaces is such that they are spread over a relatively thick canopy depth (relative to grassland or shrubland, where all leaves are condensed in a much thinner layer). Because canopy in dry forests is sparse, wind can easily penetrate it and can easily exchange heat with the leaf surfaces. Therefore, forests would have intrinsically lower aerodynamic resistance to heat transfer than shorter biomes because of the higher roughness. Moreover, the same forest (with the same physical roughness) could have higher aerodynamic roughness and consequently lower aerodynamic resistance to heat transfer for more heat stressed conditions. Given Eq. (), that would mean higher heat flux. Thus while the CCE would always be present in a forest compared to a grass or shrubland because of the obvious roughness difference, we establish that the CCE can also be present within the same forest for different conditions of heat stress – which is a more subtle point and will be further discussed in the following sections by using large-eddy simulation (LES).

LES provides a useful and meanwhile standard tool for studying canopy turbulence under different conditions of atmospheric stratification. A recent publication by studied the influence of different atmospheric instability classes on coupled boundary layer–canopy turbulence. In this work, those same instability classes are simulated to put our hypothesis to the test.

Apart from the LES outcomes, it is also important to study whether the existing parameterizations of rH can exhibit the CCE. Parameterizations of rH in the literature use Monin–Obukhov similarity theory (MOST) extensively. MOST can provide corrections for the vertical profile of the mean longitudinal velocity u and potential temperature (Ta-Ts) under thermal stratification, which deviates from the traditional log-law under neutral conditions. Thus under MOST, with the assumption that the vegetation is low, dense, and horizontally homogeneous, u=u*κln⁡z-dz0m-ψmζ,ζ0m, and Ta-Ts=Pr0T*κln⁡z-dz0h-ψhζ,ζ0h, where u* is the friction velocity, κ is the von Kármán constant, z is the height from the ground, d is the zero-plane displacement height, often approximated as (2/3)hc as per the literature , and ζ=(z-d)/L is called the stability parameter. L is the Obukhov length, computed as L=-u*3Taκgw′T′‾, where g=9.81 m s-2, the gravitational acceleration. w′T′‾ is the sensible heat flux – assumed to be constant in the surface layer . Negative ζ indicates unstable stratification and thus ζ decreases with increasing instability. z0m and z0h are the characteristic roughness lengths for momentum and heat transfer, respectively. ζ0m=z0m/L and ζ0h=z0h/L are the stability parameters associated with roughness lengths. Pr0=Km/Kh is the turbulent Prandtl number where Km and Kh are eddy diffusivities of momentum and heat, respectively. T* is a characteristic temperature scale, obtained from H and the characteristic velocity scale, i.e., H=-ρCpu*T*.

Combining Eqs. (), (), (), and (), one can write rH=Pr0κ2uln⁡z-dz0m-ψmζ,ζ0mln⁡z-dz0h-ψhζ,ζ0h, where ψm and ψh are the integral stability correction functions for momentum and heat, respectively. Following , they can be parameterized for unstable conditions as ψmζ,ζ0m=2ln⁡1+x1+x0+ln⁡1+x21+x02-2tan⁡-1x+2tan⁡-1x0,ψhζ,ζ0h=2ln⁡1+y1+y0, where x=(1-γmζ)1/4, x0=(1-γmζ0m)1/4, y=(1-γhζ)1/2, and y0=(1-γhζ0h)1/2. Different values for the parameters γm and γh are reported in the literature, and the ones suggested by are used, i.e., γm=γh=16. This formulation for rH given by Eq. () with some approximations (ζ0m=ζ0h=0) was first used by and is called the “reference parameterization” . The full form of Eq. () was used by with their only approximation being Pr0=1. Several other studies also used semi-empirical and empirical parameterizations and included the bulk Richardson number RiB given by RiB=gTaTa-Tsz-dU∥2, with U∥ the horizontal wind speed at the height that corresponds to the Ta measurement.

compiled different parameterizations of rH which we will test in the context of the canopy convector effect against our LES output. Table lists the details of the different parameterizations as compiled by . These parameterizations based on MOST , empirical (E) and semi-empirical (SE) assumptions can be classified into two categories. Formulations by , , and have assumed z0m≠z0h, which should be a more realistic assumption. On the other hand, formulations by , , and assumed z0m=z0h. Different parameters used in the empirical formulations are also listed in Table .

One important point to note is that only the formulation by uses the stability parameters associated with the roughness lengths ζ0m and ζ0h. Also note that all parameterizations assume a turbulent Prandtl number of unity; i.e., the diffusivities for momentum and heat are assumed to be the same. We shall later discuss the consequence of letting this parameter vary. Another important approximation necessary to evaluate all formulations in Table  is a prescription for the roughness lengths z0m and z0h. Effects of different roughness lengths will be investigated in the following section. However, a relation between the two roughness lengths (κB-1=ln⁡(z0m/z0h)) was proposed by and , where κB-1 is called an “excess resistance parameter”. suggested an average value of κB-1=2.0 , which will be used throughout this work.

Before moving on to the usage of LES, it warrants mentioning that the entire roughness length formulation (Eqs. – and Table ) is based on different variants of analytic approximation approaches to reduce the complexity of flow in and above the forest canopy to a 2-D surface equivalent. It is widely accepted that the MOST approach is not completely accurate close to the canopy . It was proposed that a mixing length driven approach can be applied . Nonetheless, large-scale models, which cannot vertically resolve the canopies, still use MOST, and it has been demonstrated to be relatively accurate. Thus from an operational perspective, the present formulation revisits the current leading approach for simplification of the physics in a parameterized way that can be used by coarse-resolution models.

The results of the LES simulations are presented in Fig.  as temporally and spatially averaged vertical profiles for all five stability classes, where the lightest cyan shade indicates near neutral and the most magenta shade indicates free convective conditions. Panel (a) shows the mean wind speed (U), panel (b) shows the standard deviation of longitudinal velocity fluctuations (σu), and panel (c) shows the friction velocity (which can be taken as a measure of turbulent intensity) (u*) at every level for each simulation: u*=(u′w′‾2+v′w′‾2)1/4. In the second row, panel (d) shows profiles of temporally and spatially averaged potential temperature (T), panel (e) shows the kinematic sensible heat flux (w′T′‾), and panel (f) shows the Prandtl number Pr0=Km/Kh. The profiles (except Pr0) are shown in their dimensional form to clearly illustrate the differences between the different stability conditions. The simulation results closely follow the results presented in and . It is interesting to observe that the magnitude of velocity, the velocity fluctuations, and the turbulent intensity decrease gradually from the near neutral to free convective conditions, i.e., with increasing instability. The potential temperature also reduces with increasing instability at all heights. On the other hand, the sensible heat flux appears to increase with increasing stability, especially more above the forest (z/hc=1 indicates the canopy top). These results are physically consistent. The near neutral case is dominated by mechanical shear driven turbulence – given by the highest mean velocity. The free convection case is fully buoyancy driven and the motion is fully upwards – as is evident from the near zero mean horizontal velocity. For the same reasons, the turbulent intensity and friction velocity follow the same pattern. The strongly unstable cases have the highest heat fluxes, which is also physically consistent. Note that the canopy top sensible heat flux is similar to the imposed value of 0.18 K m s-1 that was used to drive the simulations.

Figure shows temporally and spatially averaged vertical profiles for the different stability conditions with the same color coding as Fig. . We investigate the vertical profile of rH in order to assess the uncertainty that arises from varying the reference height for the air temperature under varying stability. The temperature of the canopy top is taken as the surface temperature (Ts) and thus results are shown from above the canopy top, i.e., z/hc=1. Panel (a) shows the difference of surface and air temperature (Ts-Ta(z)). Panel (b) shows the stability parameter ζ at every level computed as ζ=(z-d)/L as explained in Sect. . Panel (c) plots canopy aerodynamic resistance to heat transfer (rH) at every level computed from Eq. (). As is evident from panel (c), the aerodynamic resistance reduces with increasing instability, confirming the hypothesis constructed earlier and thus clearly demonstrating the canopy convector effect (CCE). As noted by , with increasing instability, “convective updraughts developing at side walls of roughness elements extend upwards and provide extra resistances to the mean flow. Then the mean flow interacts with both solid obstacles and their virtual extensions (updraughts), which results in the increased roughness length”. This increased roughness can be recognized as the aerodynamic roughness. For the same physical roughness of the canopy, an increase in instability increases this aerodynamic roughness and, in turn, reduces rH. The low aerodynamic resistance effectively allows larger eddies to form above the forest canopy which are more efficient to dissipate the sensible heat by promoting buoyancy. This description refers to a more general phenomenon as opposed to the description by which identifies the higher physical roughness of the canopy compared to the desert and is thus a more site specific description. Nevertheless, it is acknowledged that the more generic description presented here can be reconciled with the explanation from by noting that increased physical roughness can also result in increased aerodynamic roughness. Also incidentally, RY10 reported a value of rH≈16 for the Yatir forest which is of similar order of magnitude as what is found in panel (c) of Fig. . One important point to note in Fig.  is the magnitude of the Prandtl number, which is almost fixed to about 0.335 above the canopy. This can be reconciled with the theoretical prediction of the variation of Pr0 with stability by . For stability ranges 1≤-ζ≤10, Pr0 is also estimated to be approximately 0.33, consistent with the stability ranges plotted in Fig. . The variation of Pr0 with stability is discussed further in Appendix A.

It is interesting to study whether the different parameterizations capture the correct behavior of rH at different heights across stability. To compute rH variations, the LES generated profiles of mean velocity u, sensible heat flux, air temperature, and Prandtl number (thus the diffusivities) are used where all of them have z variations. The friction velocity u* and the roughness lengths are fixed. Figure  plots the variation of rH with height as obtained from the LES (black “+” markers), and the predicted rH from different parameterizations for the different stability cases – near neutral (column 1), weakly unstable (column 2), mildly unstable (column 3), and strongly unstable (column 4). The top row compares the parameterizations by (blue line), (red line), (black line), and (pink line), which assume z0m≠z0h. The bottom panel compares the parameterizations by (blue dashed line), (red dashed line), (black dashed line), and (pink dashed line). It should be noted that a single value of roughness z0m=0.6hc has been chosen for all cases by trial and error to obtain a “good” comparison in Fig. . As observed, none of the parameterizations can capture the correct height variations of rH, except the one by for more unstable cases. However, all parameterizations seem to do a decent job close to the canopy top. This clearly indicates that one single value for z0m as suggested by these parameterizations is inadequate. To study whether the different parameterization schemes can capture the canopy convector effect, rH computed from each method is plotted for different heights for the different stability classes in Fig. . The title of each panel describes which parameterization is plotted and the color shades starting from cyan to purple indicate increasing instability. As is evident, only the parameterization by captures the canopy convector effect for weaker instabilities. The parameterization by also displays the signatures of the CCE, however weakly. The other formulations cannot capture the correct trend of the CCE at all. Thus at this stage it is clear that the formulation, based on MOST and distinguishing between two different roughness lengths, is the most promising candidate for parameterizing rH compared to the other formulations which apply some form of approximation or do not apply MOST.

Until this stage, the momentum roughness length has been prescribed by trial and error, and it warrants a more detailed investigation. To explore the effect of different roughness lengths, the parameterization by is computed across a wide range of z0m and compared with the LES outputs for the different stability classes. As observed, an increase in z0m with increasing instability captures the height variation better than a single roughness length for all stability classes, further providing support for the notion put forward by . Hence the formulation by can be modified to include the effects of stratification on several parameters. suggested a stability dependent zero-plane displacement length as well as a stability dependent z0m based on dimensional analysis, given by ds=d1+0.56hc-L1/3, and z0ms=z0m1+1.15hc-L1/3, where ds and z0ms are the stability dependent zero-plane displacement length and roughness lengths for momentum, respectively, d and z0m being their neutral counterparts. d=(2/3)hc can be assumed as usual. The neutral z0m can be assumed to be related to LAI as given by . According to the relation used by , for an LAI of 5, z0m=0.12hc can be obtained (which is almost constant for a wide range of canopy drag coefficients and LAI). Moreover, if one uses the correct stability dependent Prandtl number Pr0(ζ) instead of setting it to unity, an improved parameterization based on can be written as rH=Pr0(ζ)κ2uln⁡z-dsz0ms-ψmζ,ζ0msln⁡z-dsz0hs-ψhζ,ζ0hs.

Note that z0ms and z0hs are still related by the same relation κB-1=ln⁡(z0ms/z0hs) with κB-1=2.0 as discussed earlier and ζ=(z-ds)/L and ζ0ms=z0ms/L, ζ0hs=z0hs/L. If a Prandtl number of unity is still assumed but the roughness lengths are assumed to be varying with stability as given by Eq. () with a neutral value of z0m=0.2hc, the formulation by is found to display the correct behavior of canopy convector effect with stability as observed in Fig. . Panel (a) shows the variation of rH with height across stability according to the improved formulation as given by Eq. (). Panel (b) shows similar variations of rH computed from the LES repeated again for comparison. The profile for the near neutral case crosses over the more highly unstable cases at heights around 6hc; however, the general behavior of the CCE is captured well. On the other hand, if the full complexity of Eq. () is used including a stability dependent Prandtl number (discussed in Appendix ) but using the canopy top surface value of the sensible heat flux for all computations involved, the variation of modeled rH is shown in panel (a) of Fig. . rH computed from the LES results using the surface value of the heat flux is shown in panel (b). This assumption of a constant sensible heat flux in the canopy sub layer or the atmospheric surface layer is a more realistic one than a monotonically reducing sensible heat flux with height as shown in panel (e) of Fig. . In fact, the surface layer is defined as a constant flux layer . The reducing flux profiles in LES are common features of large-eddy simulations since the top boundary of the LES domain is assumed stress free . Figure  correctly captures the order of magnitude of the rH observed from the simulations, and also captures the CCE correctly. However, it is acknowledged that the exact profiles of the observed rH can not be captured. However, these different comparisons highlight the uncertainties involved in the parameterization of rH.