The turbulent kinetic energy (TKE) budget is given by without invoking any special assumption: ∂e∂t+Uj∂e∂xj=δi3gTui′T′‾-ui′uj′‾∂Ui∂xj-∂uj′e‾∂xj-1ρ∂ui′p′‾∂xi-ϵ, where i and j are the usual tensor indices, which can take the values of 1, 2 and 3 to indicate x, y and z directions, respectively, and δi3 is the Kronecker delta. e=(1/2)(σu2+σv2+σw2)=(1/2)(u′2‾+v′2‾+w′2‾) is the TKE, U denotes mean longitudinal velocity; u′, v′ and w′ denote the fluctuations from mean for the longitudinal, transverse and vertical velocity components; g is acceleration due to gravity; T denotes mean potential temperature; T′ is the potential temperature fluctuation; p′ is the dynamic pressure perturbation; ρ is density of air. The first term on the left-hand side (LHS) denotes storage or TKE tendency. The second term on the LHS indicates advection of TKE by mean wind flow. The first term on the right-hand side (RHS) denotes buoyant production/destruction of TKE. The second term on the RHS denotes mechanical/shear production of TKE. The third term on RHS denotes turbulent transport of TKE and can also be called turbulent flux divergence. The fourth term on RHS denotes transport of TKE by pressure velocity correlation. ϵ is the dissipation of TKE.

Expanding the equations in terms of x, y and z coordinates, the full TKE budget can be written as Eq. () as shown in Appendix . Since it is difficult to keep track of the full equation due to the large number of terms, it would be easier to use a simple form of the TKE budget 0=-u′w′‾dUdz+gTw′T′‾-ϵ-Imbalance, where the “imbalance” is defined in Eq. (). Note that u′w′‾ and w′T′‾ denote vertical momentum flux and sensible heat flux, respectively. Also note that if the term imbalance is set to zero, one recovers the TKE budget for an idealized surface layer where the coordinate system is aligned with the mean wind, and a planar, homogeneous flow with zero subsidence is assumed. Since our objective in the current problem is to study the effect of heterogeneity, we cannot make these assumptions. Moreover, we are also constrained by being able to measure only at two single points in space quite far apart. Single point eddy covariance measurements cannot compute spatial gradients, and the pressure perturbations are not measured either. Thus, explicit computations of the imbalance terms are not possible. Due to the three-dimensional nature of the problem, it is also difficult to anticipate what degrees of assumptions are sufficient, so that some of the terms can be ignored safely.

Under these constraints, a strategy is needed to evaluate the TKE budget. The dominant mechanical production term, the buoyant production/destruction term and the dissipation term will be evaluated directly from the data. The residual of the TKE budget will be described as the imbalance as per Eq. () which would contain the effects of advection and transport terms. The advantage of using this strategy is that since the original TKE budget equation has to be closed, the errors in computing the production and dissipation terms can also be assumed to be inside the imbalance term. Imbalance=-u′w′‾dUdz+gTw′T′‾-ϵ.

To compute the mechanical production term, we momentarily assume that the TKE budget is well balanced and Monin–Obukhov similarity theory (MOST) is valid . This allows us to write dUdz=ϕm(ζ)u*κz, where ϕm is the stability correction function for momentum which varies with the stability parameter ζ=(z-d)/L and κ=0.4, the von Kármán constant. u*=u′w′‾2+v′w′‾2 is the friction velocity, z is the measurement height, and L=-u*3/(κ(g/T)w′T′‾) is the Obukhov length; d is zero plane displacement height, taken as 2/3 of canopy height. The standard MOST scaling relations for ϕm are used, i.e., ϕm=0.74+4.7ζ for stable (ζ>0) and ϕm=(1-16ζ)-1/4 for unstable (ζ<0) stratification .

Equation () allows us to compute the mechanical production term in Eq. () as -u′w′‾dUdz=ϕmu*3(κz). The buoyancy term can directly be computed from the EC measurements as well. To compute the dissipation term ϵ, we use the scaling relation of second-order structure function Duu=u(x+r)-u(x)2‾ in the inertial subrange Duu(r)=Cuϵ2/3r2/3, where Cu≈2 and r is the spatial lag in the longitudinal direction, which can be computed by multiplying the sampling time interval with the mean longitudinal velocity, assuming that Taylor's frozen turbulence hypothesis is valid (r=uΔt). The range of r where this relation is valid is found to be between 0.2 to 2 m, and ϵ is found by regression of Eq. (). Note that the computation of ϵ is independent of any assumptions used to compute the production terms.

The measurements were conducted in Yatir Forest and the surrounding shrubland in Israel between 18 and 30 August 2015 as part of the “Climate feedbacks and benefits of semi-arid forests” (CliFF) campaign, a joint collaboration between Karlsruhe Institute of Technology (KIT), Germany, and the Weizmann Institute, Israel. Figure  gives an idea of the locations of the EC towers. Tower 1 (lat 31.375728, long 35.024262) was located in the semiarid shrubland 620 m above sea level and tower 2 (lat 31.345315, long 35.052224) was located inside the forest 660 m above sea level. The linear distance between the two locations was measured to be 4.3 km, and as can be observed from Fig. , there is a distinct surface heterogeneity between the two sites. The climate of the area is in between Mediterranean and semiarid, with a mean annual precipitation of about 285 mm . Note that the measurement sites reported in this work are different from those in . The trees in the forest were mostly Aleppo pine (Pinus halepensis), with an average height of 10 m with negligible height variation. The surrounding land was sparsely populated by small shrubs, and in the dry season, when the measurements were conducted, was mostly free of vegetation. Thus, it is referred to as “desert” for easy distinction . The measurement height for the forest was 19 m above ground (9 m above the canopy height). Note that with this height selection, the measurements were conducted above the roughness sub-layer, which ends at approximately 2 times the canopy height . A mast was used over the desert and the measurement height was 9 m until 23 August, after which it was changed to 15 m for the remaining period. In this zone of the atmospheric surface layer, the longitudinal and crosswise velocity variances decrease logarithmically with height and the vertical velocity variance shows independence from height . High-frequency turbulent data were collected at 20 Hz and 30 min averaging periods were used for both sites. After conducting quality control of the data following , a planar fit coordinate rotation is applied to the velocity components since the data are collected on a sloped ground. The coordinate rotation following ensures that the cross stream velocity component v is zero and corrects the tilting of the anemometer with respect to the local streamlines. Moreover, a different set of coordinate rotation is applied for the desert data after 23 August.

In addition, two Doppler lidars were used at the two locations which measured vertical velocities . The Doppler lidars used were StreamLine systems from HaloPhotonics. They were operated in a vertical stare mode most of the time (interrupted every half hour for less than 90 s). Technical specifications and instrument settings of the Doppler lidars are given in Table 1. The Doppler lidar at tower 1 was not working from 19 August 2015, 15:00 UTC, until 21 August 2015, 10:30 UTC and very briefly on the 23 August 2015 around 10:00 UTC due to power cuts.

Time series of mean speed (ms-1), mean vertical velocity (W, ms-1) (after applying coordinate rotation), friction velocity (u*, ms-1) and mean near-surface air (potential) temperature (T, K) for the measurement period are shown in Fig. . Figure  shows time series of longitudinal velocity variance (u′u′‾, m2s-2), vertical velocity variance (w′w′‾, m2s-2), momentum flux (u′w′‾, m2s-2) and sensible heat flux (w′T′‾, Kms-1). The black line indicates desert, and the red line indicates forest. As noted, the desert is associated with a higher wind speed because of a lower amount of friction on the desert surface. The higher vertical velocity over the desert indicates the presence of stronger updrafts, which would be explained by higher buoyancy-driven turbulence. The friction velocity (u*) over the forest is much higher compared to the desert, especially in the daytime, which is expected because of higher surface roughness over the forest. u*, above both the forest and the desert, shows a strong diurnal cycle. However, there seems to be a prominent increase in u* over the desert after 23 August. This can be attributed to the raising of the tower height. Moreover, the gentle topography around the desert could result in the strong vertical updrafts above the desert. Interestingly, the near-surface air temperatures over both the forest and the desert show a strong diurnal cycle and their differences are about 5 K on average during daytime and almost zero at night.

The longitudinal velocity variance u′u′‾ over the forest and the desert show similar variations over time. The vertical velocity variance w′w′‾ over the forest is higher than its desert counterpart; however, after 23 August, the levels of w′w′‾ over the desert increase as well and become similar to the forest. This is due to changing the tower height. As the vertical profiles of w′w′‾ are different between the desert and the forest (due to roughness length differences), the observed differences between w′w′‾ are a function of observation height. At 15 m above the desert and 19 m above the forest floor, high enough to be in the “constant flux layer”, the vertical profiles of TKE (u′u′‾+w′w′‾) converge. However, when observed at a lower elevation and below the constant flux layer, the data show clear differences in w′w′‾.

The vertical momentum flux u′w′‾ over the forest is much higher compared to the desert, which is also expected because of the higher surface roughness of the forest, making it a much more efficient momentum sink compared to the desert. Note that the shear transport of momentum flux is still much more effective over the forest compared to the desert because of roughness effects even though the mean quantities can be higher over the desert. The sensible heat flux w′T′‾ over the forest is also higher, as discussed before, due to the canopy convector effect.

Figure shows the time series of the components of the TKE budget as discussed in Sect. . The first row shows mechanical production of TKE (PMech, m2s-3); the second row shows buoyant production of TKE (PBuoy, m2s-3); the third row shows full TKE production (PTKE, m2s-3), which is the sum of mechanical and buoyant TKE production. The fourth row shows dissipation of TKE (ϵ, m2s-3) and the fifth row shows an imbalance of TKE (Imb, m2s-3). The black line indicates desert, and the red line indicates forest. As noted in Fig. , the production of turbulence is mostly by mechanical or shear forcing because of the roughness of the forest, whereas mechanical production of TKE over the desert is very small and does not have a strong diurnal cycle like the forest, although it increases slightly after 23 August. On the other hand, TKE production over the desert is mostly carried by buoyancy. Buoyant TKE production is slightly larger over the forest. The buoyant TKE production over the desert is also higher after 23 August. Given the moderate temperature difference between the desert and the forest, the difference in their corresponding buoyant TKE production is interesting. It also indicates that mechanical forcing and not buoyancy makes a difference (mechanical production is higher by approximately 1 order of magnitude than buoyant production) in the turbulence generation over the desert and the forest. The diurnal cycle of the TKE dissipation ϵ is interesting as well. The dissipation of TKE seems to be higher above the forest as well compared to the desert.

A smaller TKE dissipation is recorded when the measurement location is further from the ground and above the roughness sub-layer. One strong argument for observed changes after 23 August being tower-height effects rather than a change in any large-scale forcing is that changes in the desert are observed only after the 23 August, while the forest observations maintain rather consistent dynamics.

The diurnal cycles of the TKE imbalance computed by Eq. () are also very interesting. The imbalance over the forest is often positive over the daytime, while over the desert it is often negative, highlighting the difference in turbulent transport and advection over the two different regimes. Also note that the positive imbalance for the forest and negative imbalance for the desert almost have a phase (anti-)synchronization, indicting that the turbulence above the forest and the desert are responsive to one another and that they are part of a coupled system, indicating again the role of the secondary circulations.

Figure is used to better understand the nature of turbulent transport between the desert and the forest. Panel (a) depicts the TKE imbalance over the desert vs. the net production of TKE over the forest. As observed, there is a significant correlation (0.5) between them, indicating that the advection and transport of TKE by flux divergence and pressure fluctuations reach downstream by means of the secondary circulations and produce TKE over the forest. On the other hand, the reverse is not true, as observed in panel (b) of Fig. . There is little correlation between the imbalance of TKE over the forest and the production of TKE over the desert (0.14). As observed in panel (c), the production over the desert is also well correlated with the production over the forest (0.3) as both the desert and the forest are subject to the same forcing. However, the TKE production over the desert is not that well correlated with the TKE imbalance over the desert as seen in panel (d). Thus, while there should be some cross correlation in panel (a) because of desert production, that is not the only effect. The nonlocal large-scale motions contribute to the transport over the desert (without significantly altering TKE production over the desert) which in turn cause TKE production above the forest because of the higher mechanical forcing.

Thus, it can be stated that at least in the canopy sub-layer and in the atmospheric surface layer, the effects of secondary circulations are transported from over the desert towards the forest following the background wind direction, and it is not the other way around. It is worth noting here that the term “secondary circulation” has been used somewhat loosely here and contain the effects of horizontal transport as well, since partitioning the imbalance term is not possible within the scope of this campaign. In the case of transport from the forest towards the desert, it is more likely that horizontal advection is the main mechanism. The nature of the full extent of the secondary circulation are a part of a much larger flow pattern and are not fully captured by the eddy covariance towers, which only capture the fine-scale turbulence. To reveal the full nature of the secondary circulations, one can look at lidar observations as shown in Appendix  as well as large eddy simulations .

Figure shows the time series of the triple moments w′w′u′‾, w′w′w′‾ and w′w′T′‾ in the first three rows. The vertical velocity skewness term w′w′w′‾ (second row) is of importance as it appears in the transport term of the TKE budget (Eq. ) and is a measure of non-Gaussian turbulence, which indicates the presence of nonlocal coherent motions such as sweeps and ejections. Note that the vertical velocity skewness is often negative above the canopy, which is consistent with the generic feature of canopy turbulence . The daytime vertical velocity skewness over the desert is often positive, indicating again the presence of nonlocal coherent structures active over the desert. The measure of skewness increases over the desert after 23 August, which is also due to the height change. The other two terms w′w′u′‾ and w′w′T′‾ are also associated with turbulent transport of momentum and heat as evident from their respective budget equations . ∂w′u′‾∂t=0=-w′w′‾∂U‾∂z-∂w′w′u′‾∂z+gT‾u′T′‾-1ρu′∂p′∂z‾, and ∂w′T′‾∂t=0=-w′w′‾∂T‾∂z-∂w′w′T′‾∂z+gT‾T′T′‾-1ρT′∂p′∂z‾.

Moreover, the triple moments have been shown to be directly correlated with the relative contributions of nonlocal events such as sweeps and ejections . Note that momentum transport term w′w′u′‾ is also opposite in sign for the desert and the forest, and it shows a strong diurnal cycle. After 23 August, an increase in momentum transport is noted for the desert. However, the diurnal cycle of the heat transport term w′w′T′‾ is not as strong as its momentum counterpart, but it is often found to be larger over the desert compared to the forest, consistent with the findings from the TKE budget that show heat is transported from over the desert towards the forest. It is, however, important to note that the structures are representative of the fine-scale turbulence and not directly representative of the large-scale circulation structures spanning the whole boundary layer. The fourth and fifth rows of Fig.  show the time series of the integral timescale of horizontal (Inu) and vertical (Inw) velocity components in seconds. Inu and Inw for every half hour time period are computed by integrating the normalized autocorrelation function of u and w until the first zero crossing . They can be interpreted as the characteristic timescale of the most energetic eddies in each direction. As noted in Fig. , timescales in the horizontal directions are larger compared to the vertical direction. More interesting is the observation that the integral timescales for the eddies above the desert are larger than those above the forest, which also increase after 23 August. This is another indicator of buoyant production of turbulence, which generates larger eddies than shear production.