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Diurnal hysteresis between evapotranspiration (ET) and vapor pressure deficit (VPD) was reported in many ecosystems, but justification for its onset and magnitude remains incomplete with biotic and abiotic factors invoked as possible explanations. To place these explanations within a holistic framework, the occurrence of hysteresis was theoretically assessed along a hierarchy of model systems where both abiotic and biotic components are sequentially added. Lysimeter evaporation (E) measurements and model calculations using the Penman equation were used to investigate the effect of the time lag between net radiation and VPD on the hysteresis in the absence of any biotic effects. Modulations from biotic effects on the ET-VPD hysteresis were then added using soil-plant-atmosphere models of different complexities applied to a grassland ecosystem. The results suggest that the hysteresis magnitude depends on the radiation-VPD lag, while the plant and soil water potentials are both key factors modulating the hysteretic ET-VPD relation as soil moisture declines. In particular, larger hysteresis magnitude is achieved at less negative leaf water potential, root water potential, and soil water potential. While plant hydraulic capacitance affects the leaf water potential-ET relation, it has negligible effects on the ET-VPD hysteresis. Therefore, the genesis and magnitude of the ET-VPD hysteresis are controlled directly by both abiotic factors such as soil water availability, biotic factors (leaf and root water potentials, which in turn depend on soil moisture), and the time lag between radiation and VPD.
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A number of hysteretic phenomena have been studied in hydrologic systems including (i) sorption and desorption in soil wetting/drying experiments [Mualem, 1974] and soil moisture spatial organization [Ivanov et al., 2010], (ii) hydrologic reservoir dynamics [O'Kane, 2005; O'Kane and Flynn, 2007] with switches and thresholds analogous to the so-called Preisach model [Richter, 1938], and (iii) hysteresis between ground heat flux and net radiation in ground surface energy balance [Sun et al., 2013]. In addition to these well documented phenomena, a diurnal hysteresis between evapotranspiration (ET) (or transpiration) and vapor pressure deficit (VPD) has also been observed in many different ecosystems (Figure 1) with abiotic and biotic factors both offered as plausible explanations [Wullschleger et al., 1998; Meinzer et al., 1999; O'Brien et al., 2004; Zeppel et al., 2004; Ewers et al., 2005; O'Grady et al., 2008] (Table 1). Such hysteresis is associated with a clockwise looping pattern when measured ET time series is plotted against the VPD time series over the course of a day as illustrated in Figure 1. Explaining the causes of this hysteretic phenomenon appears fraught with complex interactions between exogenous and endogenous factors to the plant system [Tuzet et al., 2003], because both lags between environmental drivers and storage compartments may cause this pattern. Some studies suggest that the hysteretic pattern is linked to stomatal closure under high VPD [Unsworth et al., 2004], while others attributed it to changes in soil hydraulic conductivity [O'Grady et al., 1999]. It was also conjectured that the magnitude of this hysteresis varies with soil moisture [Wullschleger et al., 1998], but other experiments suggested that this hysteresis was more evident under higher VPD regardless of soil moisture [O'Grady et al., 2008; Chen et al., 2011]. These examples illustrate that several interrelated mechanisms can cause hysteresis in the ET-VPD relation that continue to resist complete theoretical treatment, framing the compass of this work.
Table 1. Reported Hysteresis Between ET (or Tr) and VPDa
time lag of PAR and VPD; root, leaf, and soil water potential
From a mathematical perspective, the simplest model for hysteresis that is true to its ancient Greek word origin (hysteresis means lagging behind) is a phase angle difference between input and output time series. Such type of hysteresis is commonly referred to as “rate-dependent” hysteresis because censoring the input variations nullifies the hysteresis. For example, a phase angle difference between photosynthetically active radiation (PAR) and VPD will trigger asymmetric responses of stomatal conductance and transpiration during the morning and during the afternoon, thus causing a rate-dependent hysteresis in the ET-VPD relation. In contrast, irreversible thermodynamic changes or nonlinear regulation of a storage component leading to multiple internal states independent of an external forcing are associated with “rate-independent” hysteresis. The differences between soil water potential-soil moisture relations (or the soil water retention curves) during wetting and drying are examples of this rate-independent hysteretic behavior. We hypothesize that the ET-VPD relation emerges from combinations of rate-dependent and rate-independent processes, which in turn are controlled or mediated by both abiotic and biotic factors. Only by analyzing hysteresis in systems where such factors can be isolated, we can disentangle the drivers of the ET-VPD relation and unfold the mechanisms behind the hysteresis phenomenon.
The main objective here is therefore to propose a theoretical explanation for the hysteretic ET-VPD behavior that accommodates both abiotic and biotic factors. To achieve this goal, a hierarchy of mathematical models and data sets are employed, starting from an abstract representation of hysteretic loops and then subsequently including physical processes and biological components. In this way, the different causes of hysteresis are addressed sequentially, allowing their interactions to be disentangled. The manuscript is structured as follows: A mathematical representation for a “generic” system is first employed so as to demonstrate how a phase angle difference between input and output time series leads to hysteresis (Figure 2a). As a precursor to the ET-VPD hysteresis, the evaporation (E)-VPD case for saturated bare soils is considered and used as a prototypical example for hysteresis induced by phase angle differences (Figure 2b). The Penman equation is employed to show how the time lags between radiation and VPD explain the E-VPD hysteresis observed from lysimeter measurements. As an intermediate stage to include biotic factors, the Penman equation is replaced by Dalton equation for ET assuming the canopy is perfectly coupled with the atmosphere, where maximum canopy conductance is reduced linearly with decreasing root-zone soil moisture (Figure 2c). It is shown that the generic role of a phase angle difference between input and output can be transformed into an ordinary differential equation whose form resembles that of a root-zone soil water balance that includes interplay between storage, ET, and recharge. Hence, soil moisture storage can introduce a rate-dependent ET-VPD hysteresis analogous to phase angle difference, at least within the confines of a linearized soil-plant water balance. The role of nonlinearities in soil-plant hydrodynamics and variability in the main meteorological drivers on the ET-VPD hysteresis is finally explored using a detailed soil-plant-atmosphere continuum (SPAC) model (Figure 2d), in which the root water uptake is coupled with plant water transport, storage and evapotranspiration. Numerical experiments are conducted using the SPAC model to investigate the main factors controlling the onset and magnitude of the hysteresis as a function of various abiotic and biotic factors. Based on the SPAC model, plant hydraulic capacitance is taken into account to investigate its possible effects on the ET-VPD hysteresis in large plants (Figure 2e).
2 Materials and Methods
2.1 A Mathematical Representation of Hysteresis: Phase Angles and Storage Effects
Rate-dependent hysteresis implies a time lag between a time-evolving input variable x(t) and an output variable y(t), illustrated here via sinusoidal functions separated by a constant phase angle difference Φ. This input-output system (Figure 2a) is given as
where ω is the angular frequency.
The system exhibits a hysteresis due to a finite Φ, which results in a “time lag” between the two time series (Figure 3a and 3b). The value of Φ controls the magnitude of the hysteresis as illustrated in Figures 3c and 3d, and the hysteresis vanishes when Φ= 0. Consider again the output series of equation (2), expanded here so that
Noting that time from equation (1), y can be linked to x by eliminating t resulting in
The first term of equation (4) is the linear dependence of instantaneous y on x, while the second term illustrates the mathematical origin of the hysteresis. Hysteresis here originates from the term asin(x), which can take on two different states depending on x. Now, to link this type of phase-angle generated hysteresis with storage or capacitance encountered in the soil-plant system, the expansion in equation (3) can be recasted as an ordinary differential equation (ODE) in y by eliminating time as
The mathematical form of this first-order linear nonhomogeneous ODE is common to many ET-storage models as discussed elsewhere [Palmroth et al., 2010]. Its general solution is given as
where A is an integration constant and v is an integration variable. Note that y(t) depends on the history of x(t) instead of its instantaneous value, which is a necessary condition for hysteresis. Hence, it can be surmised that for a nonzero constant Φ, phase-angle differences, time lags, and storage all produce equivalent looping patterns associated with rate-dependent hysteresis. With this theoretical background, the E-VPD and ET-VPD hysteresis can now be explored by employing a hierarchy of mathematical models that include increasingly detailed physical, biological, and hydrological processes that impact the forcing and history of y(t).
2.2 The E-VPD Hysteresis as a Prototypical Case of a Phase Angle Effect
For wet-surface evaporation, a time lag between net radiation and VPD can lead to an E-VPD hysteresis precisely because E is approximately in phase with net radiation. To illustrate this, consider the Penman equation for wet surface evaporation given as
where λ is latent heat of vaporization, Δ is the slope of saturation vapor pressure-temperature curve, Rn is net radiation, Gs is soil heat flux, ρ is air density, cp is air specific heat, γ is psychrometric constant given as , where pa is atmospheric pressure and was set to 99.6 kPa (the same value was applied throughout this paper), and ga is the aerodynamic conductance given by
where zr is the reference height, d is zero plane displacement height (assumed zero for bare soil), zo is the momentum roughness height, zov is the vapor roughness height, u is the mean wind speed at zr, and κ is von Kármán's constant. Thermal stratification effects on ga are neglected due to the fact that (i) zr is usually small, on the order of 1 m in such studies, and (ii) the sensible heat flux is also small over wet bare soil so that turbulence is sustained by mechanical production of turbulent kinetic energy instead of buoyant production. Equation (7) is generally used to interpret lysimeter studies with no plants. Here, published data from a large (=6 m diameter) weighting lysimeter is used [Katul and Parlange, 1992].
For surfaces that are decoupled from the atmosphere (i.e., the aerodynamic conductance is small) [see Jarvis and McNaughton, 1986], according to equation (7)E becomes directly proportional to the net radiation. In this extreme case, evaporation and radiation are predicted to be linearly correlated with VPD losing control of E.
2.3 ET-VPD Hysteresis as a Prototypical Case of Storage
To introduce biotic factors on the ET-VPD hysteresis, a simplified hydrologic balance model whose form resembles equation (5) is now developed (see Figure 4a). The root-zone soil moisture balance is described by
where V is the volume of soil in the root zone per ground area and was set to 0.3 m3 m−2, Qi is the water input rate per ground area to the soil surrounding the roots, and ET is the water output rate per ground area. Note that plant hydraulic capacitance is momentarily neglected, thus restricting the analysis to small plants (e.g., grasses and forbs).
When Qi is assumed to be proportional to the difference of soil water content (instead of soil water potential for simplicity) between adjacent bulk soil and the soil in the root zone, it can be expressed as
where ks is an effective mass transport coefficient related to soil hydraulic conductivity and soil water retention curve, and is set to be independent on soil water content, and θa is the soil water content of the bulk soil. Without loss of generality, θa is assumed to be at field capacity, in turn equal to 80% of the saturated soil water content (θs = 0.5 m3 m−3).
Assuming the canopy is perfectly coupled with the atmosphere, ET can be computed as a function of canopy conductance (gc) and VPD through Dalton's law,
where gc only depends on soil moisture to eliminate effects from other meteorological variables and is given by
where gcmax is the maximum canopy conductance, which was set to 0.6 mol m−2 s−1. Here, let gc depend on soil moisture linearly to simplify the problem,
It is worth noticing that equation (11) emphasizes the vegetation controls on ET, because the canopy is assumed to be perfectly coupled to the atmosphere (i.e., aerodynamic resistance is negligible compared to stomatal conductance) [see Jarvis and McNaughton, 1986]. While this is not the case in general, we selected this formulation to contrast the hysteretic patterns caused by soil-vegetation-atmosphere interactions with the ones obtained in a purely abiotic system. This assumption also allows to analytically recast the problem as a simple storage problem (equation (5)). In fact, obtaining θ from equations (11)–(13) and combining equation (10), equation (9) can be recasted into
where ε is a constant given by . This differential equation describes ET (output) and VPD (input) as discussed in Figure 2c and resembles equation (5) except that the bracketed coefficient is time dependent instead of a constant as in equation (5). An analytical solution is difficult without a priori imposing a time series of VPD. Simulation for the water balance of this linearized soil-plant system was conducted during a period without rain with a VPD time series described as
which is a periodic function with maximum and minimum values of 2.5 and 0.5 kPa, respectively.
Up to this point, the multiple meteorological forcings and nonlinearities describing the water pathway within the soil-plant system have been ignored for analytical tractability. These effects on the ET-VPD hysteresis are considered last via a detailed model and measurements collected within a grass-covered surface.
2.4 Field Measurements
To explore abiotic and biotic controls on the ET-VPD hysteresis with water storage confined to the soil system, data collected in a grass-covered forest clearing within the Blackwood Division of the Duke forest near Durham, North Carolina, USA, are used. A C3 grass (Festuca arundinaria) is the primary component of this grassland, together with other C4 grasses [Novick et al., 2004]. The data collection commenced on 23 May 1997 and was terminated on 14 June 1997 as described elsewhere [Lai and Katul, 2000]. The leaf area index (LAI) was about 1.5 m2 m−2 during that period. The air temperature (Ta), air relative humidity (Rh), Rn, u, and Latent Heat (LE) were measured at 3.3 m above the ground surface, with an averaging interval of 20 min (LE was obtained from an eddy covariance flux system sampling at 10 Hz). The precipitation (P) was measured near ground surface. Photosynthetically active radiation (PAR) was assumed to linearly correlate with Rn. The soil heat flux Gs was measured at 0.05 m depth and soil water content (θ) was measured at 0.05, 0.1, 0.15, 0.2, 0.28, 0.33, 0.38 and 0.45 m depth. Heat flux and soil moisture measurements were averaged over 20 min intervals, consistent with other measurements.
2.5 A Soil-Plant-Atmosphere Water Transport Model
To investigate the main abiotic and biotic factors contributing to ET-VPD hysteresis for the grassland experiment, a soil-plant-atmosphere continuum model is developed for rain-free period. The model builds on earlier works [Williams et al., 1996; Siqueira et al., 2008; Drewry et al., 2010; Manzoni et al., 2013; Volpe et al., 2013] by adopting a series representation of the pathway for water transport from soil to leaf: water transport from the soil to root, assumed to be governed by soil hydraulic properties, and water transport from the root to the leaf, assumed to be governed by xylem hydraulic properties [Manzoni et al., 2013]. The model considers steady state water transport in the plant (i.e., plant capacitance is ignored). The capacitance of angiosperm tree leaves being in the range of 500–800 mmol m−2 MPa−1 [e.g., Sack et al., 2003; Domec et al., 2009], and the capacitance of grass leaf being about twice that of trees (personal communication with Jean-Christophe Domec), implies that capacitance of grass is estimated to be 0.011–0.017 cm given a leaf area index (LAI) of 1.5 m2 m−2 (typical at the study site, see section 'Field Measurements') and leaf water potential of −4 MPa (extremely dry for leaf). This capacitance accounts for less than 10% of diurnally averaged ET (an average value of 0.2 cm d−1 of the study period) and therefore, as a first approximation, can be neglected (at least compared to belowground water stores). This approximation is relaxed in a more complete version of the SPAC model described in section 'A SPAC Model Including Plant Hydraulic Capacitance'. In addition, the plant cover is assumed to be sufficiently dense so that transpiration (Tr) is reasonably approximated by evapotranspiration (i.e., Tr = ET). In the scenario ignoring plant capacitance, all storage fluctuations originate from the soil system.
A multi-layered scheme is chosen to discretize the soil system (total simulation depth is 100 cm in this study) into finite sized layers (layer depth dz = 1 cm) and soil water content θ(z, t) is modeled via a one-dimensional Richard's equation modified to include the root-water sink term S and is given as
where q is the vertical water flux given by Darcy's law for unsaturated conditions,
where K(θ) and ψ(θ) are, respectively, determined from the hydraulic conductivity and a nonhysteretic soil water retention curve given as [Clapp and Hornberger, 1978]
where Ks is the soil hydraulic conductivity at saturation, ψs is the soil potential near saturation, θs is the near-saturated soil water content, and b is the exponent of the soil water retention curve. These soil hydraulic parameters were linearly interpolated at all layers using measurements described elsewhere [Lai and Katul, 2000] (see Table 2).
At intermediate depths, the parameters were linearly interpolated.
Silt clay loam
The root water uptake (i.e., the main sink component S in the modified Richard's equation) of each layer is determined by the soil root conductance and water potential difference between bulk soil and root (a more realistic assumption than in equation (10)) and is given as
where ψr is root water potential and gsr(z, t) is the soil root conductance. The gsr is determined by combining the soil-to-root surface conductance (gs-r) and the root conductance (gr) as
The gs-r is computed by a simplified soil-root conductance model [Daly et al., 2004] incorporating root zone depth of each layer (Zi), root area index of each soil layer (RAI(z)) and soil hydraulic conductivity of each layer (Ki) expressed as
where Lsr is the effective length of water travelling from soil to root surface obtained as
where Zi is set as half of the depth of discretized layer, i.e., Zi = dz/2, r is root radius, and RAI is determined by distributing the total root area index (RAI) into different layers, based on a root density profile Df(z),
The vertical distribution of root density is assumed to follow [Jackson et al., 1996]
where β is an empirical parameter that depends on root depth and plant type, and z, as before, is the soil depth.
The gr is determined by root area index and root permeability (Pr) as
For steady state conditions, the total root water uptake flux is
When neglecting the xylem hydraulic capacitance, ET also equals xylem flow, which is modeled ignoring gravitational force and is given as
where ψl is leaf water potential and gp is plant xylem conductance. The gp is described by a vulnerability curve [Sperry et al., 1998; Daly et al., 2004] of the form
where gpmax is the maximum conductance per ground area, and c1 and d1 are parameters of the vulnerability curve.
Water released into the atmosphere from leaves (i.e., ET) is modeled by a Penman-Monteith approach modified for canopy conductance (gc) given as
The gc is modeled using Jarvis model [Jarvis, 1976], where effects from VPD, PAR, and ψl are included and are given by
where gcmax is the maximum canopy conductance, f1 is the reduction function for reduced PAR given by
where k is a light sensitivity parameter; f2 is a reduction function due to increased VPD and is given by [Oren et al., 1999]
The upper limiting value of f2 is set to 1 to avoid approaching to infinity when VPD is close to 0.
The effect of ψl on canopy conductance is described by
where c2 and d2 are the parameters determining the shape of the curve. As a point of departure from the original approach of Jarvis , air temperature adjustment functions were not included, because temperature has minor control on stomatal conductance [Mott and Peak, 2010] and is usually overshadowed by VPD effects.
For steady state water transport and assuming ET1 = ET2 = ET3, equations (16)–(34) form a closed set of equations with three unknowns, i.e., ET, ψl and ψr. Table 3 summarizes all model parameters used in the model calculations.
Table 3. Summary of the Parameters for the SPAC Modela
Note that the SPAC model has a different structure from the simple soil-plant water balance model, and the parameters do not necessarily have the same value and unit.
2.6 A SPAC Model Including Plant Hydraulic Capacitance
The SPAC model described in section 'A Soil-Plant-Atmosphere Water Transport Model' neglects plant hydraulic capacitance and is therefore limited to the description of small plants. However, plant capacitance has been suggested as one possible factor in generating the ET-VPD hysteresis in trees, because it allows faster transpiration in the morning when storage in stems is still large and VPD is low, but slows down transpiration in the afternoon when VPD is highest [Goldstein et al., 1998; Unsworth et al., 2004; Bohrer et al., 2005]. When plant capacitance is accounted for, two storage compartments may generate hysteretic behavior: the soil moisture and plant water pools. To disentangle these different capacitive effects, the controls on hysteresis by plant capacitance (C) are explicitly accounted for by modifying equation (28) as
where C is treated as a constant, a reasonable approximation as long as the leaves do not loose turgor [Sack et al., 2003].
2.7 Numerical Simulations Using the SPAC Model
Numerical runs were conducted to investigate the effects of the PAR (in phase with Rn)–VPD time lag, and soil and plant water status on the ET-VPD hysteresis using the SPAC model, while assuming plant capacitance is negligible. These runs were conducted with an initial soil water moisture set at 80% of θs (i.e., close to field capacity). The model calculations were carried out to mimic a dry-down cycle at constant LAI. The meteorological conditions for these runs were constructed using periodic meteorological variables of Rn, PAR, u, Ta, and Rh (Figure 5). As measurements at the same site show that the time lag between PAR and VPD follows a normal distribution (with mean μ = 2.9 h, and standard deviation σ = 1.3 h) (Figure S1), the periodic meteorological variables were set so that the time lag between PAR and VPD ranges from 1 to 7 h at one hour intervals. These simulations allow two scenarios to be tested under no plant capacitance assumption. The first scenario is intended to investigate the effects of the time lag between PAR and VPD on the ET-VPD hysteresis, and the other scenario is intended to investigate the onset and evolution of the hysteresis along with the progressive depletion of soil water content at a given time lag between PAR and VPD. Numerical experiments were also conducted considering different levels of plant hydraulic capacitance (using equation (35)), with the time lag between Rn and VPD of 3 h so as to investigate the joint effect of soil and plant capacitance on the ET-VPD hysteresis.
3.1 Phase Angle Difference Between Solar Radiation and VPD
Figure 6a shows the diurnal courses of Rn, VPD, and model-calculated E using the Penman equation for wet surfaces at constant wind speed (1.5 m s−1) and other meteorological drivers in the research site, while Figure 6b shows the diurnal courses of Rs, VPD, and the wet bare soil E measured by a large (=6 m diameter) weighting lysimeter [Katul and Parlange, 1992]. The results in Figures 6a and 6b demonstrate that E is almost in phase with Rn (or short wave radiation Rs) for both model calculation and measurement, because the energy term (i.e., the first one of equation (7)) is generally the dominant component of E, contributing more than 75%. In the extreme case of perfectly uncoupled canopy, in which E equals the equilibrium evaporation [Jarvis and McNaughton, 1986], E and Rn would be perfectly in phase and VPD would lose its control on E. Hence, as long as the aerodynamic conductance is larger than zero, any time lags between Rn (or Rs) and VPD result in a time lag between E and VPD. As a result, both calculations and measurements exhibit looping patterns readily explained by phase angle differences associated with rate-dependent hysteresis described in section 'A Mathematical Representation of Hysteresis: Phase Angles and Storage Effects'(Figure 6c). Hence, the phase angle difference between radiation and VPD is the main “abiotic” control of these hysteretic loops.
3.2 Soil Water Storage Effects on ET-VPD Hysteresis
Using equation (14), the effect of water storage on ET-VPD hysteresis is now assessed momentarily in absence of phase shifts between irradiance and VPD. A hysteresis between ET and VPD (Figure 7a) is generated when ks is set to 0 cm d−1 as this scenario tracks a monotonic dry-down process. However, similar hysteresis patterns are also evident when considering water input from adjacent soil (Figure 7b), where ks is set to 0.432 cm d−1. These results indicate that soil moisture storage produces an ET-VPD hysteresis that is transient but similar in all other aspects to the time lag between radiation and VPD in the wet bare soil evaporation case. Therefore, observed hysteretic loops in the ET-VPD relation are likely due to the compound effect of soil moisture changes and lags in the environmental drivers, as assessed with the complete model in section 'Compound Biotic and Abiotic Drivers of the ET-VPD Hysteresis Using the SPAC Model'.
3.3 Performance of the SPAC Model
Prior to presenting model simulations and conclusions from these model runs, an evaluation of the SPAC model was conducted for the grassland ecosystem case study. Using the time series of the measured meteorological variables (P, Rn, Gs, Ta, Rh, and u) at the site as forcing terms, the SPAC model reproduced the dynamics of θ and ET (Figures S2a and S2b) thereby lending some confidence in its ability to estimate plant water status. In particular, the modeled diurnal variations in ψl and ψr (Figure S2c), which are the main control variables of water flow in the soil-plant system, appear consistent with previous studies [Tuzet et al., 2003; Daly et al., 2004]. Most relevant to the study objectives here is the fact that the SPAC model correctly reproduces the diurnal hysteresis between ET and VPD (Figures 8a–8c) for various soil moisture states, and can thus be used to explore the onset and magnitude of the ET-VPD hysteresis (Figure 8d). Effects such as soil-root separation during drying and, more importantly, hydraulics of water flow at the interface and within the rooting system that produce variable root pressure are ignored [Amenu and Kumar, 2008; Siqueira et al., 2008]. In addition, soil hydraulic parameters may also bring some uncertainties. As a result of these simplifications and uncertainties, minor differences between soil moisture measurements and model estimations emerged. Despite these data-model differences, the model captures the general characteristics of soil moisture dynamics at all the layers within the root zone.
3.4 Compound Biotic and Abiotic Drivers of the ET-VPD Hysteresis Using the SPAC Model
Consistent with previous results, the SPAC model shows that a time lag between net radiation and VPD generates hysteresis and that this lag affects the hysteresis magnitude (Figure 9). A maximum magnitude value is reached at the time lag of 5 h, regardless of soil moisture content (Figure 9a) or plant water potential (Figure 9b).
The SPAC model shows that when soil moisture content is used as an index, it controls the magnitude of the hysteresis in an indirect manner, where the area of the hysteresis, Ahys depends on θ linearly below a moisture threshold of about 0.175 m3 m−3 in the root zone (Figure 10a). Furthermore, Ahys is near its maximum value at this threshold. Hence, the SPAC model indicates that soil moisture content affects the ET-VPD hysteresis under water stress defined by θ < 0.175 m3 m−3. At higher soil moisture, the shape of the hysteretic loop is thus primarily controlled by the time lag between radiation and VPD. This finding may fingerprint how Ahys can be used to detect the onset of soil moisture stress on ET. As an alternative way of characterizing soil water status, soil water potential was also tested as the possible factor of the hysteresis magnitude. Results show that soil water potential controls the Ahys in a linear manner (Figure 10b). In particular, the hysteresis is more evident under less negative soil water potential but the response of Ahys to the soil water potential exhibits no particular threshold as is the case with soil moisture. The SPAC model suggests that plant water potential is also a superior indicator for the magnitude of the hysteresis (Figure 10c) and that the leaf water potential controls Ahys in a linear manner. It should be noted that root water potential correlates well with leaf water potential (data not shown here), and therefore, the correlations shown in Figure 10c would remain also when plotting Ahys as a function of root water potential.
The plant hydraulic capacitance of trees was speculated to lead to a hysteresis between leaf water potential and transpiration in some experiments, where the magnitude of the hysteresis increased with the magnitude of the hydraulic capacitance [Hinckley et al., 1978; Waring and Running, 1978]. We first run the SPAC model without the direct effect of plant capacitance to simulate a grassland system. With this setup, the model also generates similar relations between leaf potential and ET (Figure 11a) as reported for trees. The presence of this hysteretic behavior is thus caused by water storage effects in soil, which acts as a nonlinear capacitor. As shown theoretically in section 'A Mathematical Representation of Hysteresis: Phase Angles and Storage Effects' and via the SPAC model calculations, capacitance (or storage effects) produces hysteresis that resembles those caused by phase angle differences or time lags between periodic series.
To assess the role of plant hydraulic capacitance, we employed the same SPAC model, but artificially added a capacitance term. With this setup, the leaf water potential-ET relation did not change significantly as long as the capacitance was lower than about 800 mmol m−2 MPa−1, which means that the role the grass capacitance is negligible. Further increasing capacitance by one order of magnitude (to 8000 mmol m−2 MPa−1, which is close to the maximum capacitance in Sack et al. ) shows that the capacitance leads to much wider leaf water potential-ET hysteretic loops (Figure 11a). However, the capacitance has little effects on the relation between hysteresis magnitude and leaf potential (Figure 11b) in our numerical runs. The capacitance is found to cause a leaf potential lag behind that without capacitance (data not shown). This phenomenon results from the regulation of the plant water pool and illustrates why a wider hysteresis between leaf water potential and ET hysteretic loops (Figure 11a) emerges under higher capacitance level. These results also indicate that the capacitance regulates the phase of diurnal time series of leaf potential, but has little effects on the ET-VPD hysteresis.
Here, it is shown that the E-VPD and the ET-VPD loops are primarily generated by phase angle differences between output (ET or E) and input (VPD) time series, which are prototypical examples of rate-dependent hysteresis. The time lag between Rn or PAR and VPD is shown to generate the main E-VPD hysteresis, verified by both model calculations based on Penman equation and lysimeter measurements in wet condition. Interestingly, when larger scales are considered, the lag between radiation and VPD can be ascribed to the dynamics of storage compartments as well–heat and water vapor within the evolving atmospheric boundary layer during the day and entrainment of heat and moisture from the capping inversion [Porporato, 2009]. In addition to the time lag between the environmental drivers of ET, soil moisture storage effects also generate a hysteresis between ET and VPD that is qualitatively similar to the one caused by time lags. In fact, it was shown that storage effects can be made mathematically equivalent to phase angle effects (or time lag). While soil moisture has been shown to alter the shape of the hysteretic loops [Tuzet et al., 2003], the fact that it might cause hysteresis independently of time lags between radiation and VPD had not been noticed.
The SPAC model supports the finding that a time lag between radiation and VPD is one primary reason for the onset of the hysteresis. Model calculations also suggest that leaf and root water potentials exert significant control on the magnitude of the hysteresis. These results imply that plant water status is a key factor controlling the magnitude of the hysteresis. The reason why plant water status explains the magnitude of the hysteresis is that plant water status not only controls ET by sensing the atmospheric demand but also regulates the internal state of the system by sensing the soil water potential. However, plant water status has been rarely reported in field experiments as a factor contributing to the ET-VPD hysteresis perhaps due to the difficulty in measuring leaf water potential during the course of a day over extended dry downs.
The results of the SPAC model provide some explanation as to why earlier studies (Table 1) differed in their causal explanation of the onset and magnitude of the hysteresis loop. Moreover, the effects of soil moisture on the hysteresis are consistent with a former study [O'Grady et al., 2008], in which the hysteresis did not depend on θ when θ was large. Wullschleger et al.  also suggested that the hysteresis is more evident in wet soils, and this finding can now be explained by the dominance of the PAR-VPD lag in such conditions. That is, soil water content may not be a good indicator for the ET-VPD hysteresis magnitude for such conditions and its effects may be masked by another significant factor (e.g., time lag of PAR and VPD). Different from the findings here and the empirical evidence (Figure 8) but using a conceptually similar model, Tuzet et al.  showed that the area of the hysteretic loop increases as soil moisture declines. These different behaviors may be caused by differences in the sensitivity of stomatal conductance to leaf water potential—for instance, stomata that close readily as leaf water potential declines cause smaller hysteretic loops [Tuzet et al., 2003]. A threshold-like dependence on soil moisture similar to the Ahys-θ relation, has been reported for canopy-level stomatal conductance [Juang et al., 2007], which implies some causal connection between soil moisture control on stomatal conductance and the ET-VPD hysteresis. This mechanistic connection and the high sensitivity of hysteresis on stomatal control as highlighted by this and previous studies deserve further investigation. The effects of plant capacitance on the hysteresis magnitude are negligible in our numerical runs for a grassland conducted here; however, artificially increasing the plant capacitance leads to looping patterns resembling those generated by the nonlinear soil capacitance.
The relation between the hysteresis magnitude and plant water potential provides a possible way to detect plant water stress. Provided the same VPD range and the same time lag between Rn and VPD, a lower ET-VPD hysteresis magnitude indicates higher plant water stress.
Both biotic factors (leaf water potential and root water potential) and abiotic factors (soil water status and time lag between PAR and VPD originating from boundary layer processes) contribute to the ET-VPD hysteresis and control its magnitude. Leaf (or root) water potential controls the hysteresis magnitude in a linear manner, so is the effect of soil water potential. Soil moisture content controls the hysteresis magnitude in an indirect manner. Where soil moisture is depleted below an ecosystem-specific threshold, it influences hysteresis magnitude. However, soil moisture does not alter the hysteresis magnitude above this soil moisture threshold thereby explaining why well-watered conditions are most conducive to wide hysteretic loops. The comparisons between effects of plant water potential, soil water potential, and soil moisture content illustrate that both plant and soil water potentials are better indicators of the hysteresis than the soil moisture. In addition, plant components introducing hydraulic buffers (e.g., hydraulic capacitance) amplify the magnitude of the hysteresis between leaf water potential and ET, but this effect is negligible in the ET-VPD relations of the studied grassland. These findings enable hydrologists, ecologists, and plant physiologists to disentangle the complex mechanisms causing the hysteretic ET-VPD relation using techniques and theories developed in dynamical systems theory.
The authors thank R. Oren for constructive comments and valuable suggestions, JC Domec for the assistance in determining the grass capacitance, D. Way for pointing out the lack of responsiveness of stomata to air temperature, and the Editors and reviewers, who provided constructive criticisms. This work was supported, in part, by the United States Department of Energy (DOE) through the Terrestrial Ecosystem Science program (DE-SC0006967), the US Department of Agriculture (USDA 2011-67003-30222), the National Science Foundation (NSF-EAR-10-13339 and NSF-AGS-1102227), and the Binational Agricultural Research and Development Fund (IS-4374-11C). Q. Zhang and D. Yang were supported by the National Natural Science Funds for Distinguished Young Scholar (project 51025931) and National Natural Science Funds (projects 51139002 and 51209117). Q. Zhang is also grateful to the China Scholarship Council for the financial support of a 9 month study at Duke University. S. Manzoni acknowledges support through an excellence grant from the Faculty of Natural Resources and Agricultural Sciences and the vice chancellor of the Swedish University of Agricultural Sciences.