The leaf temperature model was based on the leaf energy balance equation (e.g. see Nobel, 2005):
- (Eqn 1)
where the rate of change in leaf temperature is determined by the balance between the radiative energy flux (Qin (W m−2)), the convective energy flux α(Tair – Tleaf) and the evaporative energy flux LE. α (W m−2 K−1) is the convective heat transfer coefficient; Tair, ambient temperature; L (J kg−1), latent heat for evaporation of water; E (kg m−2 s−1), transpiration rate; lleaf, leaf thickness; Cleaf (J m−3 K−1), specific thermal capacity of the leaf.
The radiative heat flux (Qin) consists of the direct, reflected and diffuse solar radiation absorbed by the leaf at visible wavelengths, the infrared (IR) radiation absorbed by the leaf and the IR radiation emitted by the leaf. The IR radiation absorbed by the leaf further consists of the IR radiation emitted by the atmosphere and the surroundings. This radiative heat flux term is written as:
- (Eqn 2)
where S (W m−2) is the solar radiation at visible wavelengths; ashort and aIR are the leaf absorptance at visible and IR wavelengths, respectively; eIR, leaf emissivity at IR wavelengths; σ, Stefan–Boltzman constant (5.67 × 10−8 W m−2 K−4); Tsky and Tsurf, effective radiative temperatures of the sky and the surroundings, respectively (Gates, 1968; Nobel, 2005). The multiplier ‘2’ in the last term accounts for the radiation being emitted from both leaf surfaces. We assumed that the leaf was some distance from the ground and, consequently, the leaf area for forced convection took both upper and lower surfaces into account. Solar direct radiation was incident only on the top surface, as was the radiation from the sky. Reflected solar radiation from the surroundings and IR radiation from the surfaces were incident upon the lower surface only.
For convective cooling, we calculated the heat transfer coefficient (α) on the basis of the relationship between the Nusselt number (Nu) and Reynolds number (Re) determined for a circular horizontal disc of the diameter of the leaf (see, for example, Monteith & Unsworth, 1990; Bird et al., 2002):
- (Eqn 3)
where k (0.026 W m−1 K−1 is the thermal conductivity of air; d, the diameter of the leaf. The diameter of the leaf was defined as the diameter of the largest circle that could be inscribed within the leaf margins. This measure provides the largest continuous width across a leaf from the windward edge, whilst also accounting for leaves of different shapes, for which a single width calculation is otherwise complicated (Schuepp, 1993). Following Monteith & Unsworth (1990) and Bird et al. (2002), the equations used to calculate the Nusselt number for laminar and turbulent forced convection were:
- (Eqn 4)
- (Eqn 5)
To account for a potentially more efficient heat transfer by convection from real leaves than from a circular disc (Schuepp, 1972; Grace & Wilson, 1976; Nobel, 2005), we selected a Reynolds number (Re) for transition to turbulent flow at the lower end of the range given for flat plates, namely 1 × 104 (Bird et al., 2002).
At very low wind speeds, the majority of convective cooling occurs via free or mixed convection (a combination of forced and free convection), rather than forced convection alone (Bird et al., 2002). To take this into account, we calculated the Grashof number (Gr) as:
- (Eqn 6)
where g (9.81 m s−2) is the gravitational acceleration; ρ, the density of air; β, the volumetric thermal expansion coefficient of air; ΔT, the temperature difference between ambient air and the leaf; d, the leaf diameter; μ, the viscosity of air. The Grashof number describes the relative importance of buoyancy forces relative to viscous forces in the flow, and the relative magnitudes of Gr and Re can be used as an indicator of the relative importance of free and forced convection in cooling an object (Bird et al., 2002). Here, when Gr was < 10 times larger than Re2, we used the equations for forced convection (laminar or turbulent, depending on Re). If GrRe−2 > 0.1, we replaced α from Eqn 1 with the heat transfer coefficient for mixed convection using the scheme presented in Bird et al. (2002):
- (Eqn 7)
where the equation for free convection determined for real leaves by Dixon & Grace (1983) is:
- (Eqn 8)
and Nuforced is calculated using Eqn 4. At low wind speeds, free convection thus enhances heat transfer relative to pure forced convection. If wind speed vanishes completely, Eqn 7 results in a heat transfer coefficient of pure free convection.
The model was written using Matlab 7.7 (The MathWorks Inc., Natick, MA, USA). In the model, Eqn 1 was solved numerically using the Runge–Kutta four method and a time step of 0.093 s. The driver of leaf temperature fluctuations was a predescribed wind speed regime (adapted from Vogel, 2005, described later in this paragraph). At each time step, the heat transfer coefficient (Eqn 3) was determined using the appropriate equation (Eqn 4, 5 or 7), depending on the wind speed and the temperature difference between the leaf and ambient air at the previous time step. The incoming solar radiation (both direct and diffuse), ambient temperature and the effective temperature of the sky and the surroundings were set constant and made to represent clear sky conditions on a summer’s day in a desert (see Table 1). The heat capacity for each species was calculated as the mass fraction-weighted average based on the measured water content and dry density of each leaf (see Measured leaf traits, below). The heat capacity for dry matter was set at 1.3 MJ m−3 K−1 (Simpson & TenWolde, 1999; Jayalakshmy & Philip, 2010). The short-term variation in the wind speed was reconstructed from the 9-min wind speed regime measured by Vogel (2005) using a heated thermistor at the top of an oak canopy. This wind regime is comparable with the wind speed regimes recorded around Australian desert shrubs during a hot summer’s day (A. Leigh and N. Booth, unpublished data). The leaf temperature was initially set to ambient and the model was run for several consecutive 9-min wind speed cycles. The model always equilibrated during the first 9-min cycle. Leaves of the same diameter, irrespective of thickness, equilibrated at the same average temperature, with large leaves equilibrating at higher temperatures than small leaves. For presentation and calculations, we omitted the first wind speed cycle to remove the effects of equilibration.
Table 1. Model parameters, their values and sources, estimated for summertime desert conditions
|Solar radiation direct at visible wavelengths||450 W m−2 (c. 2100 μmol m−2 s−1 PAR)||Campbell & Norman (1998) Monteith & Unsworth (1990)|
|Solar radiation diffuse||100 W m−2||Monteith & Unsworth (1990)|
|Ambient temperature||46°C||Average maximum temperature in Death Valley during July (The Weather Channel)|
|Effective radiative temperature of the sky||−20°C||Nobel (2005)|
|Effective radiative temperature of the surroundings||70°C||Nobel (1988)|
|Reflectance of the surroundings||0.35||Sandy soil (Nobel, 2005)|
|Leaf absorptance at visible wavelengths||Measured for each species||See text and Table 2|
|Leaf absorptance at IR wavelengths||0.95||Nobel (2005)|
|Total absorbed radiative energy by the leaf||1184 W m−2||Compare Nobel (2005) 1229 W m−2|
|Leaf emissivity at IR||0.95||Nobel (2005)|
|Leaf thickness||Measured for each species||See Table 2|
|Leaf transpiration rate for L. tridentata with autumn plant water potential of − 2 to − 3 MPa||5 mmol m−2 s−1||Medeiros & Pockman (2010)|
|Leaf diameter||Measured for each species||See Table 2|
|Leaf heat capacity||Calculated based on measurements||See text and Table 2|
To validate the performance of our model, we conducted two different tests. First, we tested the accuracy of the numerical solution method against an analytical solution for Eqn 1; second, we tested the model with field measurements of real leaves of an arid zone species (details in Supporting Information Notes S1). The tests found the numerical solution to only slightly underestimate the amplitude of temperature variation relative to the analytical solution, and the model captured the amplitude and nuances of leaf temperature variation very well.
To tease apart the effects of leaf size, thickness and thermal mass on leaf temperature during lulls in wind speed, we wrote Eqn 1 for a situation in which a leaf is in radiative equilibrium (no transpiration) and the only driving force for leaf temperature changes is convective heat transfer. Integrating this equation to obtain leaf temperature as a function of time results in an exponential function with a time constant τ that depends only on the leaf thickness (lleaf), specific thermal capacity of the leaf (Cleaf (J m−3 K−1)) and the convective heat transfer coefficient (α (W m−2 K−1)):
- (Eqn 9)
The time constant is essentially the ratio of the leaf thermal mass (numerator) to the boundary layer conductance (denominator); it determines the speed with which the leaf temperature responds to a step change in ambient temperature (deviation from equilibrium) via convective cooling. The heat transfer coefficient α, which represents the effects of the boundary layer on the time constant, depends on the leaf size (two-dimensional area) (see Eqns 3–8); the larger the leaf, the longer the time constant. By inserting Eqns 4 and 5 into the time constant, we can see that, for laminar forced convection, for example:
- (Eqn 10)
Here, d denotes the leaf diameter and v the wind speed. In the case of turbulent flow, the time constant depends more strongly on the wind speed than size (leaf diameter; ), whereas, under pure laminar free convection (Eqn 8), size dominates over the driving force for cooling, which becomes the temperature difference between ambient air and the leaf (). It should be noted that the heat transfer coefficient, and therefore also the time constant, is not in fact a constant, but changes with changing wind speed or leaf–air temperature difference. It should also be noted that, in Eqns 9 and 10, the specific thermal capacity does not change with leaf size.
Application of the model
The investigation was carried out in two stages. First, we addressed whether relatively small differences in leaf thickness (up to 0.8 mm) could have any notable influence on the damping of the amplitude of the leaf temperature response compared with the effects of leaf transpiration rate, absorptance, size and water content. Second, we determined whether increased leaf thickness, via an influence on the amplitude of the leaf temperature response, could prevent thermal damage for different desert species during a sudden drop in wind speed on a hot day under desert conditions (Table 1).
The first stage of the study used L. tridentata (Creosote Bush), a widespread species in the deserts of south-west USA and Mexico, to examine the effect of leaf thickness relative to transpiration (latent heat loss), absorptance (radiative load) and size (boundary layer) by simulating changes in each parameter. To estimate the effects of latent heat loss when soil and plant water potential are relatively favourable, we used a transpiration rate of 5 mmol m−2 s−1, appropriate for L. tridentata in late autumn (October) (Medeiros & Pockman, 2010). Because the water vapour concentration gradient from stomata to air outside a leaf increases with increasing leaf temperature, a drop in wind speed could lead to an increase in transpirational cooling. However, the extent of this effect, particularly for leaves close to ambient temperature under hot summer conditions, is slight, that is, a change in leaf temperature of < 0.05°C (Notes S2, Fig. S1). Therefore, in this experiment, the imposed transpiration rate was set to be constant. Absorptance was made to vary from the normal (measured) absorptance for L. tridentata of 0.8 to a hypothetical reflective counterpart with an absorptance of 0.3. Leaf size was made to vary from the normal (measured) L. tridentata diameter of 4 mm to a hypothetical large leaf, similar to a comparatively large-leaved American desert species, E. farinosa, at 40 mm. Leaf thickness was set to 1.0 mm, representing a moderately thick leaf, typical of many nonsucculent arid zone species (Wright & Westoby, 2002), with a hypothetical thin leaf set at 0.2 mm, representing more temperate species (Roderick et al., 2000; Wright & Westoby, 2002). In addition, given the high thermal mass of water, we examined the effect of changing water content, relative to the influence of transpiration, absorptance and size, whilst holding the thickness constant. For each case, we calculated the results for an L. tridentata leaf of high (0.85), normal (0.59) and low (0.35) water content.
In the second stage of the study, we used the model to examine the relative effects of thickness on the buffering against excursions to damaging leaf temperatures in Californian desert species with known damage thresholds (TS20) (Knight & Ackerly, 2002). As well as varying in thermal tolerance, these species also vary morphologically, especially in absorptive properties (Table 2). The environmental conditions were set to represent a hot summer’s day in the Mojave Desert: an ambient temperature of 46°C and a soil surface temperature of 70°C (Table 1). During southern American desert conditions in late summer, particularly when soil and plant water potentials become critically low, stomatal conductance in L. tridentata ceases altogether (Hamerlynck et al., 2000; Medeiros & Pockman, 2010). Under such conditions, leaves are particularly vulnerable to thermal damage; thus, the second part of the study assumed an absence of transpirational cooling. Here, we looked at the effect of thickness by first simulating the leaf temperature response to the wind speed regime for a leaf of normal (measured) thickness for each species (Table 2), and then reducing the thickness to a ‘thin’ 0.2 mm.