**a. Theoretical framework**

The transpiration is a process occurring at leaf level. However, “*the single-layer model (…) is adequate to illustrate the nature of the meteorological feedback processes which operate in much the way we describe, although they must be represented in a more complex fashion in more complex models of canopy processes*” (McNaughton and Jarvis, 1991). Hence, the theoretical upscaling framework of transpiration from leaves to canopy is left to the cited literature: here we adopt the “big-leaf” approach to estimate the hedgerow olive orchard transpiration under the following form:

$$\lambda T=\frac{{\epsilon }A+\varrho {c}_{p}D{g}_{a}/\gamma }{{\epsilon }+1+{g}_{a}/{g}_{c}}$$

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where *λ* is the latent heat of vaporization of water (J kg− 1), *A* is the available energy (*R**n**-G*), with *R**n* the net all-wave radiation above the stand (J m− 2 s− 1), *G* the heat flux to soils (J m− 2 s− 1), *ρ* the density of dry air (kg m− 3), *c**p* the specific heat of air at constant pressure (J kg− 1 K− 1), *D* the atmospheric vapour pressure deficit (kPa), *ε = Δ/γ* with *Δ* the rate of change of saturation water vapor pressure with temperature (kPa K− 1) and *γ* the psychrometric constant (kPa K− 1), *g**a* the aerodynamic conductance (m s− 1) and *g**c* the canopy conductance (m s− 1). Since the olive orchard was drip irrigated, the evaporation component is low than 10% of evapotranspiration (Bonacela et al., 2001; Egea et al., 2016) and was here neglected.

In Eq. (1) it is supposed (Monteith, 1965) that the energy conservation principle is applied to the zero-plane displacement (the height of the plane, *d* (m), considered as origin, *z* = 0, for the coordinate system in which the model is formulated or, in other words, the level of the base of the roughness elements). Thus, the energy balance is a boundary condition applied on a plane *z(0)* where the energy is entirely available, i.e. *z(0) = d + z**0*, with *z**0* the roughness length for momentum transfer.

The model expressed by Eq. (1) introduces two conductances expressing the transport of water from the canopy to the atmosphere: (i) the canopy conductance, *g**c*, which describes the diffusion of vapour due to stomatal regulation of the canopy thought as a big leaf; (ii) the aerodynamic conductance, *g**a*, between the zero-plane displacement and a reference plane over the canopy, which accounts for the convective transport of vapour toward the atmosphere.

In this study, since the evaporative surface is heterogeneous and the architecture of such a hedgerow canopy is complex and has impacts on the convective transport of water vapour, we make the following hypothesis:

i the canopy is a semi-porous medium where the mechanism of vapor diffusion is strongly influenced by both the architecture of the canopy and the stomatal regulation (Perrier, 1975a; Daudet et al., 1999, among others);

ii the energy conservation boundary condition is applied to the top of the canopy, so that

iii the aerodynamic resistance is experimented between this plane and the reference surface;

iv the canopy conductance indicated as diffusion canopy conductance, *g**cd*, (Perrier, 1975b; Katerji and Rana, 2011), can be written as:

$$\frac{1}{{g}_{cd}}={r}_{c}+{r}_{0}$$

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with *r**0* resistance depending on the canopy structure (Appendix I) and *r**c* resistance depending on the mean stomatal regulation of leaves.

Under these hypotheses, introducing Eq. (2) in Eq. (1), the canopy conductance of Eq. (1) becomes “*g**cd*“ and maintains the same meaning of the big leaf approach by Monteith (1965), while the aerodynamic conductance is expressed as (Perrier, 1975a; Rana et al., 1994; Katerji and Rana, 2011)

$${g}_{a}=\frac{{k}^{2}u\left(z\right)}{ln\left(\frac{z-d}{{z}_{0}}\right)ln\left(\frac{z-d}{{h}_{c}-d}\right)}$$

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*z* (m) is the height of the reference plane, *d* (m) is the zero-plane displacement height of the measurement surface, *z**0* (m) is the roughness length for momentum transfer, *h**c* (m) is the mean crop height, *k* (0.4) is the von Karman constant; *u(z)* (m s− 1) is the wind speed at the reference height above the canopy. The above relationship is valid in neutral atmospheric conditions, and it assumes a logarithmic profile of wind speed and air temperature in the convective boundary layer and on the mixing length turbulent model. Note that in the expression of *g**a*, the roughness length to the diffusion of water vapour and heat is expressed by *(h**c**-d)*.

Under the above assumptions, the actual transpiration for the hedgerow olive orchard can be written as

$$\lambda T=\frac{{\epsilon }A}{{\epsilon }+1+{g}_{a}/{g}_{cd}}+\frac{\rho {c}_{p}D{g}_{a}/\gamma }{\epsilon +1+{g}_{a}/{g}_{cd}}$$

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Two terms appear in this expression: the first one is the "radiative term", representing the contribution of the radiative energy to T, the second one, called "convective term", is due to both the natural and forced convection in the low atmosphere above the canopy (Monteith, 1965; McNaughton and Jarvis, 1986; Rana et al., 1994).

To establish the boundary values of *g**a*, the limits of *λT* for *g**a**→∞* and *g**a**→0* must be calculated for the full and null coupling between canopy and atmosphere, as (McNaughton and Jarvis, 1986):

The term (5a) is known as “equilibrium transpiration”, because the canopy is completely decoupled from the atmosphere and the transpiration is addressed only by the available energy at the canopy reference surface; the term (5b) is known as “imposed transpiration”, because the canopy is fully connected to the atmosphere and the transpiration is addressed by the stomatal regulation at a given thermodynamic condition of air expressed by vapour pressure deficit *D*. Between these two boundary values for *g**a*, a decoupled factor *Ω*, having values in the range 0–1, can be defined (McNaughton and Jarvis, 1986) to take into account all actual possibilities, as:

$${\Omega }=\frac{\epsilon +1}{\epsilon +1+{g}_{a}/{g}_{cd}}$$

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therefore, the transpiration can be determined in “elastic” way as:

$$\lambda T={\Omega }\lambda {T}_{eq}+\left(1-{\Omega }\right)\lambda {T}_{imp}$$

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Figure 1 reports some examples of relationship between *λT* and *g**cd* for some values of wind speed *v*, at given thermodynamic condition of the atmosphere for a typical clear spring morning in Mediterranean region. A particular value of *g**cd*, known as “critical conductance”, *g**** (firstly described by Daudet and Perrier, 1968, in an unfamiliar and rarely cited study of several decades ago) divides the graph in two zones, A and B; in the zone A (*g**cd* *> g****) the transpiration is an increasing function of the wind speed; in the zone B (*g**cd* *< g****) the transpiration decreases with the wind speed; the latter phenomenon is less evident but has strong impacts on the coupling of tree orchards (Daudet et al., 1999). To give an expression to the critical conductance *g**** it is necessary to look for the characteristic value of *g**cd* that annuls the derivative \(\partial \lambda T/\partial {g}_{a}\)(Rana et al. 1994; 1997) i.e.:

$${g}^{*}=\frac{\varDelta A}{\left(\epsilon +1\right)D\rho {c}_{p}}$$

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It is useful to underline the dependence of *g**** on climatological factors only and that it is quasi-linearly correlated to the Monteith’s (1986) “climatic or isothermal conductance”

$${g}_{i}=\frac{\gamma A}{\varrho {c}_{p}D}$$

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If we substitute the expression (8) into (4), it is possible to write the decoupling framework of *λT* under the form:

\(\frac{\lambda T}{A}={\Omega }\frac{\epsilon }{\epsilon +1}+\left(1-{\Omega }\right)\frac{\epsilon }{\epsilon +1}\) \(\frac{{g}_{cd}}{{g}^{*}}\) (10)

The above expression provides a clear interpretation for the coupling process between canopy and atmosphere (McNaughton and Jarvis, 1986):

1. When *Ω = 1* (canopy isolated by atmosphere), the transpiration is an almost fixed constant fraction of available energy (Priestley and Taylor, 1972) at equilibrium between the crop in given water conditions and the atmosphere.

2. When *Ω = 0* (canopy fully coupled to atmosphere), the conditions are imposed by the atmosphere and the crop proceeds at transpiration rates depending on the weather (*g****) and is effectively controlled by the stomata at instantaneous time scale (*g**cd*).

**b. The aerodynamic conductance**

The value of *Ω* may be submitted to wide variations depending on the aerodynamic roughness of the transpiring unit (see Eqs. 3 and 6). Furthermore, correct estimations of the aerodynamic conductance, which determine the degree of coupling, are difficult to obtain because wind speed is usually measured at short distance from the surface, whereas gradients of temperature and humidity often persist for large distance above the reference level; McNaughton and Jarvis (1983) estimated that about one-third of the conductance across the surface layer is located above the usual instrument height.

Therefore, the aerodynamic characteristics at the interface canopy-atmosphere inside the fully developed boundary layer, through the roughness and zero plane-displacement lengths and the wind speed at the reference height, must be accurately determined to have correct *Ω* values.

Here, since the canopy structure of the olive stand is in rows of closed adjacent trees, the roughness length of momentum transfer was calculated following Alfieri et al. (2019), who identified a sigmoidal relationship between *z**0* and wind direction:

$${z}_{0}={\xi }_{min}+\frac{{\xi }_{max}-{\xi }_{min}}{1+exp\left[-\beta \left(\omega -{\omega }_{0}\right)\right]}$$

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where *ω* is the relative wind direction, defined as 0° when the wind direction is parallel to the row direction, i.e., north to south, and 90° when the wind direction is perpendicular to the row. *ξ**min* *= 0.1642*, *ξ**max* *= 0.3107*, *β = 0.1270*, *ω**0* *= 24.52* are fitting parameters of the sigmoid function, representing the minimum and maximum *z**0*, the slope and the offset in *ω*, respectively. The zero- plane displacement height was set *d = 0.67 h**c*.

Since *Ω* is referred to the canopy site, and the weather variables were here measured in a separate meteorological station (see next section), Rana and Katerji (2009) were followed to upscale variables from the station to the canopy (supplementary material 1).

**c. The canopy stomatal conductance**

Although here the analysis was made at canopy level, and the canopy conductance was derived by the inversion of the Penman-Monteith model (Eq. 1) and conscious that the canopy conductance is not a purely physiological variable (i.a., Rochette et al., 1991), an investigation by using independent measurements at leaf scale can support the correctness of the found *g**cd*, at least regarding the order of magnitude. The theoretical development of the leaf stomatal conductance scaling up to the canopy is left to the huge literature on the subject (see for example the physical approach proposed by Baldocchi et al., 1991, among many others); here we estimate a mean stomatal canopy conductance (*g**sc*, m s− 1) as (Szeicz and Long, 1969):

$${g}_{sc}=\frac{R{T}_{l}}{{P}_{a}}\stackrel{-}{{G}_{s}}LAI$$

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with \(\stackrel{-}{{G}_{s}}\) (mol m− 2 s− 1) the mean conductance measured at leaf level, *LAI* leaf area index, *R* (8.314 J mol− 1 K− 1) the gas constant, *T**l* (K) the leaf temperature and *P**a* the atmospheric pressure (set at standard value = 101325 Pa). The *LAI* values were estimated as detailed in Supplementary material 2.

**d. The site, the crop and ancillary measurements**

The study was realized in three years (2019, 2020 and 2021) focusing on the most evaporative demanding periods in the site (Katerji et al., 2017), i.e. from 1 July to 31 August. The olive orchard (cv. *Arbosana*) is located at the University of Bari experimental farm at Valenzano, southern Italy (41° 01’ N; 16° 45’ E; 110 m a.s.l.), on a sandy clay soil (sand, 630 g kg− 1; silt, 160 g kg− 1; clay, 210 g kg− 1) classified as a Typic Haploxeralf (USDA) or Chromi-Cutanic Luvisol (FAO). The site is characterised by typical Mediterranean climate with a long-term average (1988–2018) annual rainfall of 560 mm, two third concentrated from autumn to winter, and a long-term average annual temperature of 15.6°C. The olive grove has been planted in early summer 2006; the self-rooted trees were trained according to the central leader system and spaced 4.0 m × 1.5 m (1,667 trees ha− 1) with a North–South rows orientation, according to the SHD cropping system. Trees were 1.75 ± 0.46 m high. Routine cultural nutrition, soil management, pests and diseases control practices were set up as described by Camposeo and Godini (2010). Lots of 180 m2 surface, 60 m apart, with 35 trees in each one, were submitted to two irrigation regimes: full irrigated (FI) versus regulated deficit irrigation (RDI), the last applicated throughout the pit hardening phase, when the tree is least sensitive to water deficit; during this phenological phase, irrigation was interrupted for about one month (19/07–20/08/2019; 15/07–18/08/2020; 14/7–14/08/2021). Irrigation was scheduled following the evapotranspiration method, by restoring 100% of crop evapotranspiration lost in each irrigation interval, as recommended by the FAO56 guideline. The plots were irrigated by a dripline equipped with 2.5 L h− 1 emitters, 0.6 m apart.

Air temperature (*T**air*, °C) and vapour pressure deficit (*D*, kPa) through air relative humidity, global radiation (*R**g*, MJ m− 2 s− 1) and precipitation (*P*, mm), wind speed (u, ms− 1) and wind direction (degree) were collected at a standard agrometeorological station 120 m far from the experimental field.

*R* *n* and *G*, both in Wm− 2, was calculated as follows (Rana and Katerji, 2009, among others):

$$Rn=\left(1-\alpha \right){R}_{g}-\sigma \left(\frac{{T}_{max}^{4}+{T}_{min}^{4}}{2}\right)\left(0.34-0.15\sqrt{{e}_{a}}\right)\left(1.35\frac{{R}_{g}}{{R}_{g0}}-0.35\right)$$

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where *α* albedo of the crop directly determined on the orchard as mean of hourly daytime values (0.27) from January to December 2021; *T**max* and *T**min* (°K) are maximal and minimal air temperatures; *e**a* (kPa) is the actual vapour pressure; and *R**g0* (MJ m− 2) is the calculated clear-sky radiation. After a local calibration of twelve months (January-December 2021), *G* was considered as a constant at daily scale and equal to 0.09 *R**n*.

The degree of drought for each year under investigation was evaluated by the standard precipitation index (SPI, Naresh Kumar et al., 2009).

The Canopy conductance *g**cd* to calculate *Ω* (Eq. 6) was obtained by inverting Eq. (1)

$${g}_{cd}=\frac{\lambda T{g}_{a}}{\epsilon A+\varrho {c}_{p}D{g}_{a}/\gamma +\left(\epsilon +1\right)\lambda T}$$

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Here actual transpiration was measured by sap flow thermal dissipation method (TDM, Granier, 1985, 1987). *λT* was determined by the sap flow density, *J**s0* (g m− 2 s− 1) was measured in a set of selected plants and corrected for the underestimation induced by the wounds and other errors as described in Supplementary material 3.

Soil water content in volume (θ, m3 m− 3) was measured by capacitive probes (5TM, Decagon Devices Inc., USA). For each treatment, three points were monitored following the protocol described in Campi et al. (2020). At each point, two capacitive probes were installed horizontally into the soil profile and transversely to the row, at -0.12 and − 0.37 m from the soil surface to intercept the dynamics of θ below the dripping lines. One set of probes was installed between the tree rows. All sensors were connected to data-loggers (Tecno.el srl, Italy); integrated soil water content daily was determined for the soil profile (0.5 m) by integrating the three values measured at each depth, since each probe was supposed to detect the water content in a 0.25 m soil layer (Campi et al., 2019). θ measurements from the three points were pooled to obtain a single average value for each treatment.

Soil water availability was described through the relative extractable water (REW, unitless) calculated using the average soil water content across positions around the tree and soil layers (Granier, 1987; Tognetti et al., 2009):

$$REW=\frac{\theta -{\theta }_{min}}{{\theta }_{max}-{\theta }_{min}}$$

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where θ is the actual soil water content in the root zone, θmin the minimum soil water content observed during the experiment, and θmax is the maximum soil water content in the area (e.g., at field capacity).

Stomatal conductance (*G*s, molH2O m− 2 s− 1) was measured (2, 23, 31 July and 29 August 2019) on two healthy well light-exposed leaf per tree (on the West and East side) selected in the middle part of the canopy, by using a portable open gas-exchange system, fitted with a LED light source (LI-6400XT, LI-COR, Lincoln, NE, USA). At each measurement and canopy side, light intensity was maintained constant across the two treatments setting the LED light source at the natural irradiance detected near the leaf. For each treatment, *G**s* measurements were performed in the three plants where transpiration was measured by TDM and other two trees similar for dimension, vigour and health state, chosen in correspondence with the soil moisture probes. The data were subjected to one-way ANOVA using SAS/STAT 9.2 software package (SAS/STAT, 2010).