To quantify the compound effects of air temperature and precipitation regimes on canopy temperature and the potential of irrigation to reduce the occurrence of crop heat stress, we developed a mechanistic model describing the coupled canopy energy and water balances and their interactions with the water balance of the rooting zone (see the model structure in Fig. 1 and the Supplement for details and symbols). The model allows us to explore how plant traits and physiological responses to growing conditions interact with air temperature and soil water availability in defining canopy temperature, while relying on parameters with clear physiological meanings (Table S2 in the Supplement).

To limit parameter and computational requirements, a minimalist approach was used, lumping the canopy in a big leaf model (Amthor, 1994; Jarvis and McNaughton, 1986; Bonan, 2019) and the soil water dynamics in a bucket-filling model, with instantaneous losses via runoff and percolation below the rooting zone (e.g., Milly, 1994; Rodriguez-Iturbe et al., 1999). These simplifications are expected to have minor repercussions on our conclusions (see Sect. S5 in the Supplement).

As detailed in the Supplement, combining the canopy water and energy balance, the canopy temperature, Tc, can be obtained as follows: 1Tc=Ta+Q↓+Bn, ref↓-λgv, cDcpgH, c+λgv, css+4εcσTa31-exp-Kbl,dLAI, where Ta is the air temperature, Q↓ is the net absorbed shortwave radiation, Bn, ref↓ is the net absorbed longwave radiation at Ta (isothermal radiation), and D is the atmospheric vapor pressure deficit. gv, c and gH, c are the total canopy conductances to water vapor and heat, respectively, which include stomatal and aerodynamic conductances, λ, cp, εc, σ, and Kbl,d are constants (Table S1), ss is the slope of the vapor pressure vs. temperature curve, dependent on Ta, and LAI is the leaf area index.

We explicitly included the dependence of stomatal conductance on environmental conditions and plant physiology, exploiting an optimality principle, namely that plants are assumed to maximize carbon uptake over a given period, subject to limited water availability (Mäkelä et al., 1996; Eqs. S9–S11 in the Supplement). We chose this approach because it is simple, yet based on an evolutionary principle, and has led to promising results (Buckley et al., 2017; Eller et al., 2020). Many stomatal optimization models based on water use efficiency assume that photosynthesis is limited either by RuBisCO (ribulose-1,5-bisphosphate carboxylase-oxygenase) or electron transport rate. To avoid this a priori assumption, we approximated the original Farquhar et al. (1980) model for the photosynthetic rate with a hyperbolic function that includes both limitations while retaining the same physiological parameters (Vico et al., 2013). This model was further developed here to account for the effects of the leaf boundary layer conductance and day respiration in addition to the key stomatal and non-stomatal effects of limited water availability on marginal water use efficiency and metabolic activity (Zhou et al., 2013; Manzoni et al., 2011; Vico and Porporato, 2008; see Sect. S1.2.1 for details). The results obtained with an alternative, empirical model of canopy conductance parameterized with eddy covariance data (Eqs. S30–S32; Novick et al., 2016) further support our mechanistic approach. But, they also highlight the need to explicitly represent canopy gas exchanges to capture the dependence of canopy temperature on air temperature, unless site-specific and crop-specific data are available to determine the canopy conductance empirically (Fig. S9). Finally, aerodynamic conductances to heat and vapor were determined based on wind velocity, U, and leaf width via well-established, semi-empirical relations describing heat and mass transport inside the leaf boundary layer and to the bulk atmosphere (Sects. S1.2.2 and S1.2.3).

The canopy conductances affect and are affected by the soil water balance and water transport along the SPAC. On the one hand, soil water potential influences leaf water potential and, hence, leaf physiological activities (stomatal conductance, metabolic rates, and marginal water use efficiency). On the other hand, stomatal conductance and atmospheric water demand drive the rate of canopy water losses and, hence, the decline of soil water content. We represented the soil water content as soil saturation, s (0≤s≤1; hereafter soil moisture), linked to soil water potential, ψs, via texture-dependent soil water retention curves (Eq. S24). A bucket-filling model was used to describe the soil moisture dynamics, with precipitation and irrigation as input and evapotranspiration, deep percolation below the rooting zone and surface runoff as losses but neglecting the root structure, the time needed for the water to be redistributed within the soil, and lateral soil water movements (Sect. S1.3.1; Vico and Porporato, 2010). The soil water balance was coupled to a minimalist description of water transport through the SPAC to determine the leaf water potential. The SPAC was modeled as a series of conductances from the soil, through the plant, to the atmosphere (Sect. S1.3.2; Manzoni et al., 2013).

These model components provide conductances and boundary conditions to apply Eq. (1) and quantify how canopy temperature, Tc, changes with environmental conditions and management, namely air temperature and humidity, wind velocity, incoming solar radiation, precipitation and irrigation applications, if any. The model needs to be solved iteratively (Fig. 1). At each time step (1 d; see Sect. 2.3), the model considers the previous soil moisture and current atmospheric conditions. The previous canopy temperature and water potential are used as initial guesses for the numerical integration. First, the model determines the canopy boundary layer and aerodynamic bulk conductances and water supply and demand. Then, the canopy water potential ψc is determined iteratively by equating water supply and demand. After convergence is reached on ψc, the canopy energy balance is used to determine Tc iteratively. Finally, the soil water balance is updated with inputs and losses cumulated over the time step.

Based on Tc, we derived the following two metrics representing the potential for heat stress damage: (i) Tc, mean, the mean canopy temperature during a specific period (anthesis; see Sect. 2.3), and (ii) PCHS, the fraction of days during such a period when Tc exceeded the crop-specific threshold Tth, above which detrimental effects of crop heat stress are likely. PCHS is thus a measure of the duration of the detrimental conditions, while Tc, mean quantifies the level of detrimental conditions.

While the model is of general applicability, we focused on the case of wheat (Triticum aestivum) – a staple crop with relatively low tolerance to high temperatures when compared with other crops (Sanchez et al., 2014) – grown at 45∘ N. All the model parameters are summarized in Table S2.

We restricted our analyses to anthesis, when wheat is most vulnerable to heat (Porter and Gawith, 1999) and water (Daryanto et al., 2017) stress. Anthesis was assumed to last 21 d (Mäkinen et al., 2018), starting on day 140 of the year, i.e., 20 May (in line with observations and simulations at the latitude selected; Semenov et al., 2014; Bogard et al., 2011). For simplicity, the timing and length of anthesis were kept constant under all climatic scenarios, regardless of irrigation applications.

The model is capable of simulating the diurnal course of the key variables, but, for simplicity, we focused on the central part of the day, when incoming shortwave radiation at the top of the canopy Q0↓ and air temperature Ta are at or near their daily maxima and Tc is expected to peak. Wind velocity U was assumed to be at the lowest end of its realistic range, and Q0↓ to be that of clear sky conditions, thus providing the maximum expected Tc and a conservative estimate of the frequency of occurrence of potentially damaging temperatures.

Measured environmental conditions relative to a specific location could be used to force the model. Yet, here we employed synthetically generated environmental conditions, varying their parameters to systematically explore several climate scenarios. Daily precipitation was idealized as a marked Poisson process (Rodriguez-Iturbe et al., 1999), i.e., exponentially distributed interarrival times, with average frequency λp. Event depth was also assumed to be exponentially distributed, with average αp (Sect. S1.4.2). The variability of Ta around its long-term average μTa was described via an Ornstein–Uhlenbeck process (Sect. S1.4.3; Benth and Benth, 2007). In line with the focus on the warmest part of the day, Ta is interpreted as the maximum daily air temperature. Finally, U, Q0↓, and RH (relative humidity) were assumed to be constant during the simulations (Table S2), whereas air water vapor pressure, ea, and vapor pressure deficit, D, were calculated based on Ta (Campbell and Norman, 1998).

As baseline pedoclimatic conditions, we considered a sandy loam soil, an average precipitation frequency λp of 0.2 d-1, an average event depth αp of 8.2 mm (corresponding to an average annual precipitation total of 600 mm), a long-term average air temperature μTa of 25 ∘C, an air temperature standard deviation of 3.6 ∘C, an air relative humidity RH of 40 %, a wind velocity U of 4 ms-1, and a net incoming shortwave radiation Q0↓ of 800 Wm-2. We also explored additional pedoclimatic conditions. Specifically, we considered more extreme precipitation scenarios, comprising increasing precipitation from increasing precipitation frequency, and a constant average annual precipitation total, but with more intermittent precipitation, with a reduced average precipitation frequency (λp=0.07 d-1) and increased average event depth (αp=23.5 mm). Long-term average air temperatures μTa of 20 and 30 ∘C were also explored. Separate sensitivity analyses were run for the standard deviation of air temperature (Fig. S6), soil texture (Fig. S7), and U, Q0↓, and RH (Fig. S8).

For the irrigated case, a demand-based (water) stress-avoidance irrigation was considered whereby an irrigation application is triggered whenever soil water potential reached the intervention point, ψ̃s (Vico and Porporato, 2011). To ensure well-watered conditions, ψ̃s was set to -0.07 MPa, i.e., just above the incipient water stress for wheat (-0.1 MPa; Kalapos et al., 1996). Each irrigation application restored a preset target soil water potential, ψ^s, set at -0.01 MPa. The difference between the intervention point and the target soil water potential is large enough to allow the use of a traditional irrigation technology (e.g., sprinkler systems or surface irrigation; see Vico and Porporato, 2011 and references therein).

Finally, the crop- and developmental-stage-specific temperature threshold above which detrimental effects of crop heat stress are likely, Tth, was set equal to the maximum baseline (i.e., cardinal) temperature during anthesis. Tth is a large source of large uncertainty when aiming at defining the occurrence of crop heat stress and its consequences on the crop and final yield (Siebert et al., 2017; Wanjura et al., 1992). Even within a specific developmental stage, there is a large variability in reported baseline and optimal temperatures because of differences in variety, growing conditions, and experimental approach. Furthermore, a crop's baseline and optimal temperatures are often defined based on air temperature, although plants respond to canopy or even organ temperature. As shown below, the differences between air and canopy temperatures can be large, particularly under limited plant water availability. To make the comparison between Tc and Tth meaningful, we considered a maximum baseline temperature obtained under well-watered conditions and low D and set Tth equal to 30 ∘C (Saini and Aspinall, 1982). This value is similar to those obtained in other experiments focusing on wheat (Porter and Gawith, 1999).

The simulated canopy temperatures were not normally distributed, according to the Anderson–Darling test (p<0.05). To test if median Tc, mean and PCHS differed across scenarios, we employed the Mood test. And, to test the difference in their variances, we used the Brown–Forsythe test. The test results are summarized in Tables S3–S8. Differences are commented on when p<0.05.

We quantified the compound effect on canopy temperature from the following environmental conditions: air temperature, soil water availability, incoming shortwave radiation, wind velocity, relative humidity, soil texture, and irrigation. Our model is an improvement with respect to existing approaches which simulate canopy temperature in agricultural systems and rely on empirical corrections of values determined by means of the energy balance under extreme conditions (Fang et al., 2014; Webber et al., 2016). Lacking adequate modeling tools has limited our ability to effectively quantify the likelihood and extent of potential heat damage to crops and the potential improvements by irrigation.

The role of environmental conditions is mediated by plant physiology and its response to conditions. Indeed, losses via evapotranspiration dominated the soil water balance in all the climatic scenarios explored (see Sect. S3.1). But, despite the complex mechanisms behind canopy temperature, the resulting pattern was relatively simple. Canopy temperature increased from cooler temperatures and wetter soils to warmer and drier conditions (Fig. 3). Under well-watered conditions, some thermoregulation occurred, cooling down or warming up the canopy, depending on air temperature, to maintain the canopy near-optimal temperature for photosynthesis (Michaletz et al., 2016). This thermoregulation capability was lost when low water availability limited evaporative cooling. The differences in canopy and air temperatures provided by the model are in line with experimental observations and other model results, thus lending support to our approach. In wheat, for example, field observations and model results showed that daily maximum or mid-day canopy temperature was 2 to 10 ∘C warmer than air under water stress and from 1 to 2 ∘C warmer to up to 6 ∘C cooler than air temperature under well-watered conditions (Pinter et al., 1990; Rashid et al., 1999; Jensen et al., 1990; Howell et al., 1986; Ehrler et al., 1978; Balota et al., 2008; Neukam et al., 2016; Webber et al., 2016; Schittenhelm et al., 2014; Webber et al., 2018; Mon et al., 2016). Our simulations led to canopies being 2 to 10 ∘C warmer than air under water stress and to a cooling effect of 1 to 2 ∘C under warm but well-watered conditions. Differences between model results and observations can be ascribed to cultivar-specific traits, specific approach to measuring canopy temperature, measurement timing and position (within or just above the canopy), and environmental conditions (e.g., solar radiation and soil texture). Some of these aspects can be accounted for by the model, by adjusting the parameters to the specific crop and variety, and environmental conditions.

The difference between canopy and air temperature was higher than, and independent of, air temperature when soil water potential was below a critical value (Fig. 3). This threshold-like response mirrors that of stomatal closure and plant transpiration reduction with water stress (for wheat; e.g., Sadras and Milroy, 1996; Shen et al., 2002; Wang et al., 2008; Wu et al., 2011; Kalapos et al., 1996). Yet, no threshold for stomatal closure was imposed a priori in the model. The emerging threshold of soil water potential (-0.14 MPa) is comparable with the soil water potential corresponding to incipient stomatal closure in some experiments (-0.1 MPa; Kalapos et al., 1996) but higher than those of others (between -0.27 and -0.35 MPa, depending on the cultivar; Wang et al., 2008) and lower than the value often assumed to correspond to well-watered conditions (-0.03 MPa; Ali et al., 1999; Laio et al., 2001).

Climate change is expected to alter both air temperature and precipitation regimes, with further increases in average and extremely high air temperatures and, in some regions, scarcer or more intermittent precipitation, i.e., longer dry spells (IPCC, 2013). Co-occurring dry and hot extremes are becoming increasingly frequent (Alizadeh et al., 2020; Zscheischler and Seneviratne, 2017). We showed that these compound changes can increase canopy temperature and its variability (Figs. 4 and 5).

For set air temperature conditions, even with same average precipitation totals, less frequent but larger precipitation events increased the median of and variance in canopy temperature, and the fraction of days during which the temperature threshold for potential heat damage was exceeded (Fig. 5). Larger, less frequent precipitation events result in enhanced losses via runoff and percolation below the rooting zone, thus reducing plant water availability. The ensuing (longer) dry down can lead to lower soil moisture levels, potentially enhancing canopy temperature. It is thus important to consider not only seasonal precipitation totals but also their timing. Indeed, reductions in the number of rainy days have already reduced crop yield and could even override the benefits of increased total precipitation (Ram, 2016). For a set precipitation regime, an increase in long-term average air temperature resulted not only in a higher mean canopy temperature during anthesis, as expected (Eq. 1), but also in a larger variability in such a mean (Figs. 4 and 5). These complex, compound effects show that it is necessary to explicitly consider not just the means but also the timing of and variability in air temperature and precipitation, and their joint effects, when quantifying the potential of climate change to cause crop heat stress. Hence, models accounting in full for the stochasticity of environmental conditions are needed.

Crops are also faced with increasing air carbon dioxide (CO2) concentration. While this aspect of global change was not explored here, an increase in air CO2 concentration could reduce stomatal conductance and, thus, enhance canopy temperature when all the other conditions are the same. But, reduced stomatal conductance can also reduce the rate of soil water storage depletion and, thus, the maximum canopy temperature reached during a dry down. The net result of an increase in air CO2 concentration is expected to be small. Indeed, an air CO2 concentration of 200 to 220 ppm (parts per million) above ambient conditions increased canopy temperature only up to 1 ∘C in free air CO2 enrichment experiments and model simulations (Webber et al., 2018), and a weak reduction in yield loss to heat with enhanced CO2 is expected (Schauberger et al., 2017).

By reducing the occurrence and extent of water stress, irrigation could lower canopy temperature, and its variability, and the frequency of it exceeding the threshold for potential heat damage (Fig. 5). Irrigation can have positive effects on yields, not only by reducing water stress but also by reducing heat stress. Indeed, the canopy-to-air temperature difference is well correlated with the final yield (e.g., Blum, 1996; Reynolds et al., 1994; Thapa et al., 2018), except under extremely dry conditions (Schittenhelm et al., 2014). This temperature difference is often used for cultivar selection (Graß et al., 2020; Munns et al., 2010).

The extent of the reduction in canopy temperature and, hence, of the occurrence of potential heat stress even under stress-avoidance irrigation depended on the precipitation regime and long-term average air temperature. Irrigation was particularly effective in reducing canopy temperature and the duration of potentially damaging conditions at lower long-term average air temperature. And, for a set long-term average air temperature, irrigation was slightly more effective under more intermittent precipitation (Table 1). Yet, irrigation aiming at maintaining the plants under well-watered conditions could not completely remove the possibility that canopy temperature exceeded the temperature threshold for potential heat damage, except under the coolest air temperature scenario. Furthermore, the benefits of irrigation became smaller as air temperature increased. Irrigation could also have indirect effects on canopy temperature. At the regional scale, irrigation, by enhancing evaporation, can further reduce air temperature (e.g., Sacks et al., 2009; Lobell et al., 2008a) and canopy temperature, while lengthening developmental stages. These effects could be included by altering the air temperature regime (see Figs. 3 and 4 and Table 1 for the effects of average air temperature) and the duration of the anthesis.

The risk of canopy temperature exceeding the temperature threshold for potential heat damage under (water) stress avoidance irrigation can be interpreted as the potential heat stress attributable only to air temperature. This is because no limitation on evaporative cooling is expected under the imposed irrigation scenario, where the soil water potential triggering an irrigation application was less negative than the critical soil water potential emerging from Fig. 3. The reduction in the fraction of time in which canopy temperature is above the threshold for potential heat damage obtained via irrigation (Table 1) is a measure of the relative importance of air temperature and water stress in defining high canopy temperatures. In addition, for the most effective use of the available water resources against heat stress, the emerging threshold of soil water potential that limits water-stress-induced high canopy temperatures (Fig. 3) could be used to define a crop-specific irrigation intervention point for irrigation. Maintaining the soil water potential above that threshold would require additional water resources, while leading to marginal further cooling effects, i.e., little advantage in staving off heat stress.

Irrigation could not fully eliminate the negative effects of heatwaves and the warmer conditions expected in the future, but a widespread use of irrigation could directly or indirectly mitigate the effects of heatwaves (van der Velde et al., 2010). Nevertheless, even for air temperatures for which irrigation can reduce the potential for heat stress damage, and considering these regional effects, expanding irrigation to mitigate the effects of high canopy temperatures can be unadvisable or impossible due to physical or economic water scarcity (Rosa et al., 2020), already unsustainable exploitation of water resources (Wada et al., 2010), or the negative impacts of irrigation on soil salt content and nearby water bodies (Daliakopoulos et al., 2016; Scanlon et al., 2007). Other management approaches are, thus, needed to limit the potential for crop heat stress, particularly under high average air temperatures (Deryng et al., 2011; Lobell et al., 2008b). Examples are shifting to more heat-tolerant cultivars and species (Tack et al., 2016), altering the sowing date (Lobell et al., 2014; Mourtzinis et al., 2019), or migrating crops (Sloat et al., 2020) so that anthesis occurs when air temperature is, on average, lower.