Introduction

In the average, transpiration accounts for 61% of the evapotranspiration (Schlesinger and Jasechko 2014). It means, that most of water evaporated from the ecosystems has to pass through the plant and that its precise amount is regulated by the vegetation. Therefore, the evaporation models have developed from purely physical approach (i.e., Penman 1948) to the ones that implemented the principles of turbulent diffusion theory and plant stomatal regulation (Monteith 1965). In coniferous and other stands, with strong coupling between the canopy and the atmosphere, g c (see Table 1 for the list of abbreviations) has a major effect on transpiration (Jarvis and McNaughton 1986; Meinzer et al. 1997; Phillips and Oren 1998; Hernandez-Santana et al. 2016). If evapotranspiration can be measured, actual stomatal conductance can then be easily calculated using an inverted Penman–Monteith equation (Martin et al. 1997; Cienciala et al. 1997; Granier et al. 2007; Whitley et al. 2009; Braun et al. 2010; Wang et al. 2014; Fu et al. 2016). This equation combined with sap flow measurements provides a powerful tool for estimating canopy conductance for both whole-tree and forest stand levels with results comparable to leaf level measurements with subsequent upscaling (Ewers et al. 2007b).

Table 1 A summary of abbreviations and symbols used in the equations and throughout the text
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The aim of calculating stomatal conductance with the P–M equation is to determine to what extent g c is dependent on environmental variables such as solar radiation and air vapor pressure deficit (Lohammar et al. 1980; Oguntunde et al. 2007), which may be combined for example with air temperature (Jarvis 1976; Stewart 1988; Sommer et al. 2002), soil moisture (Stewart 1988; Granier and Loustau 1994; Harris et al. 2004), and eventually xylem architecture (Zhang et al. 2012). A range of functions exist for these models, from a simple linear approach (Oguntunde et al. 2007) using more complicated non-linear equations relating two (Lohammar et al. 1980) or more (Jarvis 1976) environmental variables, to complex models incorporating photosynthetic and biochemical regulation to measure stomatal conductance (Ball et al. 1987; Collatz et al. 1991; Tardieu and Davies 1993; Leuning et al. 1995; Jarvis and Davies 1998; Berry et al. 2010; Buckley et al. 2012; Mirfenderesgi et al. 2016; Xu et al. 2016). However, parameterizing the last mentioned models which include photosynthesis has become increasingly difficult due to varying degrees of model sophistication and limited data availability for most field studies (Ward et al. 2008). For this reason, empirical models are commonly selected to focus on two primary environmental drivers: R g and D. In conditions of limited soil water content, soil water deficit may be added as a third driver.

There are two main types of the empirical (phenomenological) models of the stomatal conductance. First group is represented by the Stewart (1988) and the Lohammar model (1980). These models predict stomatal responses solely on the base of environmental factors. This approach is still widely used because it is simple and its modular structure makes it easy to incorporate in the larger models, and because its responses to the environmental factors can be separated and visualized (Egea et al. 2011; Buckley and Mott 2013). For its simplicity this kinds of models are also used to derive stomatal conductances from the sap flow measurements (Lu et al. 1995; Cienciala et al. 1997; Whitehead 1998; Oren et al. 1999; Ewers and Oren 2000; Ewers et al. 2007a; García-Santos et al. 2009; Wang et al. 2014; Fu et al. 2016). In such cases, the simple models are often preferred over the more recent Ball–Berry model and its modifications, which are otherwise widely used in canopy and global circulation models (Verhoef and Egea 2014). The main complication for the use of Ball–Berry model in derivation of g c from the sap flow is that it requires the knowledge of net assimilation. Net assimilation can be directly measured (which data are not always available with sap flow measurements) and then plotted against measured g c or modeled from biochemistry of photosynthesis but it presents an extra effort that does not need to be done when using the simpler Lohammar model. For the sake of simplicity and to make our work comparable to similar studies (i.e., Oren et al. 1999; Ewers and Oren 2000; Ewers et al. 2007a; García-Santos et al. 2009; Wang et al. 2014; Fu et al. 2016) we decided to use those kinds of empirical models that do not include the photosynthesis.

In the classical process for deriving stomatal conductance from sap flow measurements, g c is calculated using the inverted form of the P–M equation followed by regression analyses usually performed between individual environmental variables and g c. One disadvantage of this approach is that it requires using an enveloping curve for D and excluding data when D rates are low to avoid unrealistically high values of g c (Phillips and Oren 1998; Ewers and Oren 2000; García-Santos et al. 2009). This paper proposes an improved way of modeling canopy sap flow in a forest stand through direct parameterization of a P–M equation. Above mentioned traditional approach first (1) calculates the g c from inverted P–M equation, than (2) it rejects all data where g c seem unrealistic (especially at low D), (3) it searches for dependence between g c and weather conditions using regression analysis and suitable (i.e., Jarvis–Stewart of Lohammar) model, (4) substitutes that sub-model into the P–M equation and calculates the sap flow. In contrast, proposed approach enables to perform modeling of sap flow in one step and it does not require data rejection when D levels are low. The g c and stomatal response to R g and D can be computed in a second step from the Lohammar equation. We also propose a few modifications to the Lohammar equation. Because some data suggested, that stomata response to D may not be always exponential (i.e., Jones 2014, p. 140, Fig. 6.10; Bourne et al. 2015, Fig. 2), we introduced an arc tangential relationship allowing for more precise descriptions of stomata behavior at low D. An additional parameter describing minimal stomatal conductance (similarly to the parameters of ‘b’ or ‘g 0’ in Ball–Berry model and its modifications) was then introduced permitting calculation of nocturnal sap flow. The aim of this study is to improve how sap flow and stomatal conductance are modeled from Penman–Monteith equation, to test the applicability of proposed approach for modeling sap flow in a forest stand and to compare the results with those derived from using the more traditional approach. The analysis allows describing the response of g c of the particular tree species to various environmental drivers (i.e., R g or D). Sap flow calculated by this method may also be used for simulation of water use of the forests when it was not measured or to fill the gaps in the sap flow measurements.

Materials and methods

Site description

The study was conducted in an open P. canariensis forest at its upper distribution limit at 2070 m a.s.l. in the Teide National Park (28°18′21.5″N, 16°34′5.8″W), Tenerife, Canary Islands, Spain. At the time of the study (2008), the trees were 50–60 years old with a mean canopy height of 10.3 m. The stand density was 291 trees ha−1, with a basal area of 16.7 m2 ha−1. The diameter at breast height (DBH) averaged 25.4 ± 9.3 cm (Fig. 1), and the plant area index (LAI) was 3.6 (Brito et al. 2014). The field site is characterized by a Mediterranean climate with alternating warm and dry periods from June to September, and a cold and wet period from October to May. During the period 1971–1999, mean annual precipitation recorded at the Portillo weather station located approximately 300 m southeast of the study site was 368 mm, falling mostly during the winter months (December–February; 190 mm), while summer was the driest season (June–August; 3 mm). The mean annual air temperature was 10.7 °C. Daily means of global radiation, vapor pressure deficit, soil water potential and daily sums of precipitation within the investigated period are at shown in Fig. 2. The geologic substrate is of volcanic origin, and the soil at the study site is classified as a leptosol.

Fig. 1
figure 1

Diameter distribution of the pine trees at the research site, their number per ha (black bars) and the number and distribution of the sample trees used for sap flow measurements (gray bars)

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Fig. 2
figure 2

Daily means of global radiation (R g), vapor pressure deficit (D, a), daily means of soil water potential (SWP) and daily sums of precipitation (b). Wet and dry periods are highlighted with a gray background. Dotted line on the b indicates values of SWP used to parameterize the model (9) after the rain event in the transient period

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Field measurements

Sap flow was measured from 25 January to 10 September 2008. Ten sample trees with diameter at breast height (DBH) from 21 to 44 cm were selected for continuous sap flow measurements. Tree sap flow was measured by the tissue heat balance method with constant heating power (Čermák et al. 1973; Kučera et al. 1977) manufactured by EMS Brno, Czech Republic (type P4.1, heating power 0.63 W). Electrode lengths of 70 and 80 mm were used to cover the conductive sapwood depth. Thirty-minute averages from sap flow values measured in 1 min intervals were stored on a data logger. Scaling up of sap flow from the sample trees to the stand level was based on the DBH of the sample trees and diameter distribution of the trees in the forest stand (Čermák et al. 2004), as well as for the study site, described in detail by Brito et al. (2014).

Environmental variables such as global radiation, air temperature, air humidity, precipitation and wind speed were taken from a meteorological station [uEMSet 99 (EMS Brno)] installed on an open plot close to the study site. Soil water potential between 0 and −1.1 MPa, which was the limit of the equipment, was measured with three gypsum blocks at depths of 25–30 cm (sensors GB2, Delmhorst, Inc., USA; datalogger ModuLog 1029, EMS Brno, Czech Republic). All environmental data were recorded with a 30-min resolution.

Data processing

New approach: derivation of canopy conductance by direct parameterization of the Penman–Monteith equation

The analysis of canopy transpiration (E p, mm h−1) was made based on the P–M equation with the simplification that for long periods, stand transpiration equals sap flow. The general form of the P–M equation is:

$$\lambda E_{\text{p}} = \frac{{\Delta \left( {R_{\text{n}} - G} \right) + \rho \times c_{\text{p}} \times D \times g_{\text{a}} }}{{\Delta + \gamma \left( {1 + {\raise0.5ex\hbox{$\scriptstyle {g_{\text{a}} }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {g_{\text{c}} }$}}} \right)}},$$
(1)

and the aerodynamic conductance is:

$$g_{\text{a}} = \frac{{k^{2} u}}{{\ln^{2} \left[ {{\raise0.5ex\hbox{$\scriptstyle {\left( {z - z_{\text{d}} } \right)}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {z_{0} }$}}} \right]}},$$
(2)

All above listed variables were directly measured or calculated. The only unknown variable was canopy conductance g c.

Parameterization of the P–M equation is based on the assumption that canopy conductance g c depends on solar radiation and D, according to a suitable formula. We used a modified version of the Lohammar equation (1980). The original form of this equation supposes that stomatal opening is caused by solar radiation and that stomatal closure results from high evaporative demands. Consequently, this proposes that stomatal closure is complete at night. However, calculating transpiration response to night D values called for an additional parameter (g min), similarly as in the Ball–Berry model (1987). Some measured data suggested, that opening of stomata in response to the D may not always follow strictly exponential manner (i.e., Jones 2014, p. 140, Fig. 6.10; Bourne et al. 2015). Therefore, alongside with the exponential form of the model, we also applied arctangent equation which gave us more flexibility when describing stomatal behavior under conditions of low D. The original form of the Lohammar model was:

$$g_{\text{c}} = \frac{{R_{\text{g}} }}{{R_{\text{g}} + R_{0} }} \times \frac{{g_{ \text{max} } }}{1 + a \times D},$$
(3)

and the proposed modification is:

$$g_{\text{c}} = g_{ \text{min} } + \frac{{R_{\text{g}} }}{{R_{\text{g}} + R_{0} }} \times g_{ \text{lim} } \times \left( {0.5 - \frac{1}{\pi }{\text{arctg}}\left( {\frac{D}{a} - b} \right)} \right),$$
(4)

The resulting form of the modified Penman–Monteith equation used for the parameterization was:

$$E = \frac{{{{\left( {\Delta \times \left( {R_{\text{n}} - G} \right) + \rho c_{\text{p}} Dg_{\text{a}} } \right)} \mathord{\left/ {\vphantom {{\left( {\Delta \times \left( {R_{\text{n}} - G} \right) + \rho c_{\text{p}} Dg_{\text{a}} } \right)} \lambda }} \right. \kern-0pt} \lambda }}}{{\Delta + \gamma \times \left( {1 + \frac {g_{\text{a}}}{{g_{\text{min} } + \frac{{R_{\text{g}} + 1}}{{R_{\text{g}} + R_{0} }} \times g_{\text{lim}} \left( {0.5 - \frac{1}{\pi }{\text{arctg}}\left( {\frac{D}{a} - b} \right)} \right)}}} \right)}},$$
(5)

The parameters a, b, R 0, g min and g lim were optimized by minimizing the residual sum of squares of the observed and modeled conductances. Calculations were performed using non-linear multivariate analysis with Mini32 statistical software (EMS Brno, Czech Republic).

Stomatal conductance (g s) may be derived from canopy conductance (g c) using the following equation:

$$g_{\text{c}} = g_{\text{s}} \times {\text{LAI,}}$$
(6)

The time lag between R g, D (which determine foliar transpiration) and stem sap flow were estimated performing time series cross-correlation analysis. The respective environmental variable was shifted in 30-min step intervals and correlated to sap flow until the highest R 2 was reached. Only the lagged environmental data were used for parameterization of the P–M equation.

Classical approach

Results of the new approach were compared to a classically used method (Granier and Loustau, 1994). Here, the canopy conductance was calculated from the inverted form of the Penman–Monteith equation:

$$\frac{1}{{g_{\text{c}} }} = \left[ {\frac{s}{\gamma }\left( {\frac{{R_{\text{g}} - \lambda E}}{\lambda E}} \right) - 1} \right]\frac{1}{{g_{\text{a}} }} + \frac{{\rho_{\text{a}} c_{\text{p}} }}{E\lambda \gamma }D,$$
(7)

Subsequently, canopy conductance was modeled using both a modified Jarvis–Stewart approach (Jarvis 1976; Granier and Loustau 1994) and a modified Lohammar model (Eq. 4) which was previously used for the direct parameterization of the P–M equation. The Jarvis–Stewart model was used in the form:

$$g_{\text{c}} = a\left( {\frac{{R_{\text{g}} }}{{R_{\text{g}} + b}}} \right)\left( {\frac{1 - cD}{1 + dD}} \right) + g_{\text{min} } ,$$
(8)

where a, b, c, d and g min are estimated parameters based on the standard minimal sum of squares criterion. Modeled canopy conductance was used for modeling the canopy sap flow by applying the P–M approach. For parameterizations of Eqs. (4) and (8) and classical approach we used data when D was higher than 600 Pa (Ewers and Oren 2000). These parameters estimated for high D were also used to model sap flow when the D was low.

Effect of soil water availability on sap flow

Three time periods characterized by contrasting soil water availability were apparent in the study period. The first, characterized by non-limiting water availability [‘wet’ period; mean soil water potential (SWP) was −0.02 MPa and it was always higher than −0.05 MPa within this period], lasted from the beginning of the year to the end of April. The second, the transient period, lasted from May to the middle of June. The SWP ranged between −0.05 and −1.1 MPa most of the time. The last, dry period, occurred from the middle of June until the end of the investigated period. The SWP in this period was lower than −1.1 MPa, which was the lowest value that could be measured by the used equipment (Fig. 2). To avoid any adverse effects of soil water availability on the stand transpiration, initial parameterization of the proposed model was carried out for the period of non-limiting water access, from 27 February to 5 April, 2008. All comparisons between the proposed and classical approaches were drawn from this same period.

We later modified Eq. (4) in a way that enabled us to use the soil water potential data to describe the effects of soil water availability on stomatal conductance:

$$g_{\text{c}} = g_{ \text{min} } + \frac{{R_{\text{g}} }}{{R_{\text{g}} + R_{0} }} \times g_{ \text{lim} } \times \left( {0.5 - \frac{1}{\pi }{\text{arctg}}\left( {\frac{D}{a} - b} \right)} \right) \times (c \times \ln ({\text{SWP}}) + d),$$
(9)

wherein SWP (MPa) represents the absolute value of soil water potential and c, d are parameter estimates. These calculations were performed in 24 h resolution within a period from 15 February to 1 September, 2008. In a 2-week long period from 25 May to 10 June, following a rainfall event, gypsum block sensors indicated an increase in soil water potential which, however, was not entirely reflected in tree sap flow (Figs. 2b, 3). This prompted us to use linear interpolation for the SWP between the beginning and end of this period, from 25 May to 12 June and to use only these data for the model parameterization.

Fig. 3
figure 3

Daily sums of measured and modeled sap flow. Two models were used. One (Eq. 5, model without SWP) was parameterized on the period with the unlimited soil water availability (from 27 February to 5 April, 2008) and these parameters were used to simulate the sap flow during entire season. Second one (Eq. 9, model including SWP) included soil water potential into the simulation

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Comparison among the sub-models for dependence of stomatal conductance on environmental variables and between classical and proposed approach

Three kinds of models for canopy conductance dependence on environmental variables (such as R g, D) were applied during the process of direct parameterization. The Jarvis–Stewart model (8), both the original and the modified Lohammar Eqs. (3, 4), and the three with the applied (or not applied) added parameter of minimal canopy conductance (g min)—altogether provided six variants. These equations were included in the process of direct parameterization of the P–M equation to calculate sap flow and canopy conductance. Also, all these sub-models were used to calculate sap flow and g c by classical approach. Values of modeled sap flow and canopy conductance were compared to the sap flow measured in the field and to a canopy conductance calculated from an inverted P–M Eq. (7). All simulations intended for comparison among the models were performed for the period of non-limiting water access, from 27 February to 5 April, 2008, and therefore, no submodel for the soil water potential was applied. It is well known that the occurrence of low D (high air humidity) leads to unrealistically high values of g c calculated by the classical approach. For this reason Ewers and Oren (2000) recommended to remove all data where D is lower than 600 Pa from the analysis. However, in our case this would mean rejection of 818 out of the measured 1872 values. Therefore, we did two comparisons on two different datasets. First one covered entire wet period and included all data, including those when D was lower than 600 Pa. The second dataset contained only 20 full days when D was higher than 600 Pa.

Results

Measured sap flow

Daily maximum levels of sap flow for individual sample trees were between 10 and 67 kg day−1 with maximal values of stand sap flow 0.89 mm day−1 (Fig. 3). Sap flow lagged behind evapotranspiration demands of atmosphere, R g and D. Estimated time lags between the sap flow and climate variables were 60 min for R g and 30 min for D.

Modeling sap flow and stomatal conductance: direct parameterization

Our proposed model successfully predicted diurnal courses of stand sap flow during the wet period, yielding R 2 = 0.96 and having the lowest AIC (Table 2), as soil moisture is not a limiting factor during this part of the season. However, the model parameterization for the wet season overestimated daily sums of sap flow during the transient and dry periods, beginning in May (Fig. 3). Differences between measured and modeled sap flow may be perceived as quantitative indicator of water deficit for transpiration. The model provided an acceptable description of tree sap flow on a diurnal basis (Figs. 4, 5), and even took into account night flows. Discrepancies between measured and modeled values appeared after rainfall, when it was apparent that the sap flow based model could not accurately predict evaporation of intercepted water.

Table 2 Index for determining linear regression (R 2) between measured sap flow (or canopy conductance) and those simulated by the model. Akaike’s information criterion (AIC) for various models. Simulated values of sap flow and stomatal conductance comes from following models: the Jarvis–Stewart model (Eq. 8), the modified Lohammar model (Eq. 4), the original Lohammar model (Eq. 3) and values of sap flow and canopy conductance calculated from an inverted P–M equation (Eq. 7). Models incorporating additional parameters of minimal stomatal conductance are labeled ‘g min’
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Fig. 4
figure 4

Examples of diurnal curves of measured sap flow (black line), modeled values using the proposed approach (red line), and the classical approach (green line), (a, d). Typical diurnal courses of stomatal conductance (b, e) estimated using the classical approach from Eq. (7) (green line) and using the proposed approach (red line). Corresponding global radiation (R g) and vapor pressure deficit (D) (c, f) for days with high (a–c) and low D (d–f) (color figure online)

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Fig. 5
figure 5

Measured and modeled values of sap flow using the proposed (a) and the classical approach (b) during the wet period (n = 955)

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Parameterization of the P–M equation was further complicated by the occurrence of night sap flow. The Lohammar equation, in its original form (3), accounts for the full closure of the stomata and no cuticular transpiration, therefore, for zero transpiration at night. Introducing the parameter g min into the equation allowed us to estimate night flows. This minimal nocturnal stomatal conductance (g min) was calculated as 0.00020 m s−1. The highest stand canopy conductance of 0.0033 m s−1 and highest average conductance of 0.00107 m s−1 were calculated during the wet period.

Modeling sap flow and stomatal conductance: the classical approach

The classical approach, which applies an inverted P–M equation for calculating diurnal courses of stomatal conductance with subsequent regression analysis between calculated g c and environmental variables (Eqs. 7, 8), also allowed for the estimation of stand water use in a diurnal resolution. Calculations of canopy conductance upon low D were subjected to an error typical for this method: when D was low, g c unrealistically increased (Fig. 4e). The error in estimating long-term water use (i.e., during the wet period) was similar to the method of direct parameterization if only the periods of high D were considered. Correlation between measured and modeled sap flow was slightly less than it was for the model derived via direct parameterization (Table 2).

Comparison of proposed and classical approach and of different models for dependence of stomatal conductance on environmental factors

All models acceptably described diurnal courses of sap flow and stomatal conductance when D was high, with R 2 of linear regression between measured and predicted values >0.94 and >0.87 for sap flow and stomatal conductance, respectively (Table 2). However, when low D was included, only the proposed approach was able to predict the sap flow with success. The classical approach yielded unrealistically high values of g c (Fig. 4e) and there was no correlation between g c and environmental factors (R g and D). Therefore, we were not able to model a sap flow when all data were used in a classical approach. A modified Lohammar equation incorporating parameter of minimal stomatal conductance (g min) proved the most effective correlating with measured values, yielding highest R 2 and having the lowest AIC of all models. Absolute values of sap flow and stomatal conductance were generally more accurate when predicted by models incorporating parameter g min, which were able to account for the effect of night sap flow.

Effect of soil water availability

Sap flow dropped by 95% during the dry period (Fig. 3). The transient period showed a decrease in water use and lasted 2 months, from the end of April to the end of June. Daily sums of sap flow were successfully predicted in all wet, transient and dry seasons. Interpolation of SWP from 25 May to 12 June, after the single rain event during transient period, helped to model the decline in sap flow. Otherwise the model would overestimate measured sap flow and in this period would reach up to the values predicted from Eq. (5) (Fig. 3). The R 2 of a linear regression between measured and modeled sap flow was 0.93.

Discussion

Measured sap flow

Low values of daily sums of stand sap flow reflected both harsh tree line conditions at the altitude of roughly 2200 m a.s.l. (Fernández-Palacios and de Nicolás 1995) and low stand density. Stand transpiration in one studied forest stand was lower than average at 0.80 mm day−1 (with a maximum of 1.85 mm day−1) in a lower situated stand of a P. canariensis (Luis et al. 2005). Stomatal conductance was also lower than in previous studies for lower altitudes (Wieser et al. 2002; Luis et al. 2005). A decrease in summer months of sap flow and stomatal conductance, which occurred during the studied period, was typical only for the higher altitudes, as summer drought at lower altitudes is often mitigated by a high frequency of clouds that do not occur at high altitudes in the summer (Gieger and Leuschner 2004; Luis et al. 2005; Wieser et al. 2006).

The estimated time lag of 1 h between the diurnal course of global radiation and stem sap flow, on the lower end of a commonly used range of 0–4 h (Martin et al. 1997; Čermák et al. 2007; Whitley et al. 2009; Ward et al. 2013) was typical for the tree species with reduced (more sparse) crowns, and therefore, smaller water storage capacity (Anfodillo et al. 1998). Response to changes in D during the day were faster than to R g as once the pool of the easily accessible stored water is depleted, the hydraulic signals are transmitted throughout the plant at a very high speed (Malone 1993). A lag of sap flow compared to the potential evapotranspiration was also observed when the canopy was wet from rain. Such a lag time may reach up to 2 h (Langensiepen et al. 2009). In this case, a simple P–M based model was not able to distinguish between evaporation and transpiration, and therefore, could not accurately predict sap flow.

Comparison between direct parameterization and the classical approach

The classical approach for calculating g c and transpiration (or sap flow) using the P–M equation consists of several steps. In the first step, g c is calculated from Eq. (7). The second step includes regression between R g, D and g c. Here, it is often necessary to introduce an enveloping curve into the regression. Utilization of the inverted form of P–M equation, in combination with the exponential equation characterizing the dependence of g c on D, causes an unrealistic increase of the g c on low D (Fig. 4e). It happens because low sap flow is divided by low D, which brings large error into the calculation. To avoid this phenomenon, Ewers and Oren (2000) recommend excluding all data where D is lower than 600 Pa. This often affects considerable parts of a dataset and in our case we would reject 44% of the data from the period of models comparison. The proposed approach using parameterization of the P–M equation with simultaneous derivation of all parameters, using an arc tangential relationship for D allowed us to predict g c even on low D, and thus allowed us to predict sap flow for any time of day. Furthermore, the one-step calculation process proved to be the most consistent and resulted in the highest R 2 between observed and predicted sap flow and in lowest AIC of the model (Table 2).

These two approaches have, however, common limitations. First of them is that they are not able to distinguish between transpiration and evaporation. When the rain or dew occurs intercepted water is evaporated first from the needle surfaces before the transpiration follows. Evaporation consumes part of the radiation energy and increases the lag between potential evapotranspiration and sap flow. Second issue relates to the hysteresis pattern between sap flow and potential evapotranspiration (Oren et al. 2001; Matheny et al. 2014a, b). Trees store considerable amount of water in the needles, branches, stem and roots (Phillips et al. 2003; Warren et al. 2005; Čermák et al. 2007; Oliva Carrasco et al. 2015; Urban et al. 2015; Mirfenderesgi et al. 2016). Resistances to the xylem water transport make it easier to use some parts of the stored water for transpiration instead of withdrawing it directly from soil (Sack and Holbrook 2006). Therefore, gradient of water potential develops along the pathway of xylem water transport and time lag occurs between transpiration and sap flow. Water stored in the tree is refilled during the periods of low evapotranspiration demands. It brings further uncertainty into the modeling of sap flow from weather data. One solution would be to introduce tree water potential or water storage into the simulation, like in the resistance–capacitance circuit (Sperry et al. 1998; Steppe et al. 2006; Bonan et al. 2014) or porous media models (Bohrer et al. 2005; Chuang et al. 2006; Janott et al. 2011; Mirfenderesgi et al. 2016). These models are able to numerically solve for the water potential and they do not require it as a mandatory input variable because continuous measurement of water potential in conifers is a significant challenge and these data are often not available. Water potential can be estimated also indirectly, from the dendrometers readings (Perämäki et al. 2001) or from changes in a stem water storage measured by soil moisture sensors working on principle of time domain reflectometry (TDR) (Irvine and Grace 1997) or frequency domain reflectometry (FDR) (Matheny et al. 2015; Carrasco et al. 2015). Even though these indirect approaches have been developed, such data are available only in a few studies. Therefore, with limited data availability, parameterizing of the P–M equation is still a viable option to calculate g c and to model stand sap flow.

Different models for dependence of g c on environmental variables

Stomatal conductance models relate stomatal aperture to global radiation and vapor pressure deficit. While R g opens the stomata, D closes the stomata (Fig. 6). Strong correlation between g s and D lead to the assumption that a functional relationship exists (Meinzer and Grantz 1990; Aphalo and Jarvis 1991; Addington et al. 2004). Both the Jarvis–Stewart model incorporating the added parameter g min (Eq. 8) and the modified Lohammar equation (Eq. 4) acceptably described dependence of stomatal conductance on environmental variables with only negligible differences in R 2 of the models within a parameterized period.

Fig. 6
figure 6

Response of relative canopy conductance to global radiation (a) and canopy conductance to vapor pressure deficit (b) during the wet period estimated using the modified Lohammar equation

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The mechanism driving stomatal opening in the morning due to R g led to an assumption about stomatal movements only during daytime conditions (Shimazaki et al. 2007). However, evidence of night sap flow response to ΔD suggests that large portions of water are lost through cuticular transpiration (Burghardt and Riederer 2003), stomatal closure is incomplete or that stomatal movements occur at night (Caird et al. 2007; Kavanagh et al. 2007; Zeppel et al. 2011). Caird et al. (2007) summarized a large body of evidence showing that g s changes at night were based upon various driving factors, including D. We, therefore, incorporated the parameter g min characterizing night stomatal conductance into Eq. (4). Inclusion of this parameter improved the correlation (Table 2). The parameter describing minimal stomatal conductance is not a new idea. Similar parameter was used, for example, in the Ball–Berry model. There is a difference between the g min in our study and g 0 in the Ball–Berry model. While g min describes a true minimal stomatal conductance, g 0 is a stomatal conductance at light compensation point of photosynthesis. Estimation of the g min from the sap flow data is further complicated by the changes in water content in plant tissues. Therefore, the exact amount of water transpiration at night could not be accurately described from the sap flux measurements as we were unable to separate plant refilling from actual transpiration to the atmosphere (Wang et al. 2012). The hysteresis pattern observed in the diurnal sap flux response to changing D (Oren et al. 2001) indicated that refilling occurred. More complex models like FETCH2 (Mirfenderesgi et al. 2016) which recently occurred would be needed to identify to what degree contributes aboveground tree water storage to the tree water uptake and transpiration.

Effect of soil moisture

Soil water content is the other driving mechanism of stomatal aperture (Granier and Loustau 1994; Zweifel et al. 2009; Brito et al. 2014). We defined a submodel describing the effect of actual soil water potential on plant sap flow. This modification allowed for modeling sap flow on a daily basis in wet, dry and transient periods of a season. However, a more thorough understanding of the soil water potential in rooting zones (or of the predawn tree water potential) and plant ecophysiological adaptation to imminent water stress is necessary. In our case study, sap flow did not scale with measured soil water potential in a short (2-week long) period after the rain event which occurred in the middle of the transient period, when measured soil water potential increased from −0.5 MPa to almost zero, while sap flow increased only slightly (Figs. 2, 3). Decoupling of the sap flow from measured soil water potential may occur for two reasons. The first may be that it is linked to the position of soil water sensors, which may not be able to describe true water availability for the tree root system (as rooting depth can easily reach 15 m, while the sensors were installed in 30 cm depth (Luis et al. 2005; Brito et al. 2015)). The second relates to a tree’s ecophysiological adaptation to water stress (i.e., needle shedding, which was observed at the site (Brito et al. 2014), an accumulation of ABA, or embolism of xylem tracheids). Therefore, it is advisable to monitor predawn tree water potential, as an indicator of a tree water status, and to monitor long-term tree adaptation to water stress.

Conclusions

  1. 1.

    Direct parameterization of the Penman–Monteith equation with an incorporated modified Lohammar model is a suitable approach to derive diurnal courses of stomatal conductance of the forest stand. Closer correlation between measured and calculated sap flow was found using the proposed approach by applying the direct parameterization than was observed using the classical calculation method.

  2. 2.

    Direct parameterization of the P–M equation allowed for calculation of canopy conductance upon low D and avoided unrealistic increases of computed g c, typical for the classical approach of calculation. Furthermore, the direct parameterization approach does not require computing the enveloping curve and thus avoids subjective selection of the data by the user. Moreover, direct parameterization allows for calculation of g c and modeling sap flow to occur in one step. This makes it possible to quickly and automatically repeat the calculation, i.e., for different time periods.

  3. 3.

    The proposed modification of the Lohammar et al. (1980) equation successfully described dependence of canopy conductance on measured environmental variables (global radiation and vapor pressure deficit). Its modified form allowed for more precise descriptions of g c, especially upon low D, then the commonly used three-parameter function. Including a parameter describing minimal stomatal aperture allowed for modeling of night sap flow.

Author contribution statement

JK contributed to the work by devising of the methodology, data analysis and work on the manuscript. MJ and PB performed the field work and contributed to the writing of the manuscript. JU contributed to the data analysis and writing of the manuscript.