Improving the stem heat balance method for determining sap-flow in wheat

Agricultural and Forest Meteorology 186 (2014) 34–42

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Improving the stem heat balance method for determining sap-flow in wheat Matthias Langensiepen a,∗ , Moritz Kupisch a , Alexander Graf b , Marius Schmidt b , Frank Ewert a a Crop Science Group, Institute of Crop Science and Natural Resource Protection, Faculty of Agriculture, University of Bonn, Katzenburgweg 5, 53115 Bonn, Germany b Agrosphere, Institute of Bio- and Geosciences, Jülich Research Center, 52425 Jülich, Germany

a r t i c l e

i n f o

Article history: Received 10 April 2013 Received in revised form 18 August 2013 Accepted 21 November 2013 Keywords: Sap flow Triticum aestivum Stem heat balance Numerical simulation Eddy correlation Validation

a b s t r a c t A novel micro-sensor for measuring sap-flow in thin plant stems designed by Dynamax Inc. based on the heat-balance theory was applied in wheat (Triticum aestivum) grown under ambient field conditions. The sensor measures axial and radial temperature changes in a constantly heated and thermally insulated stem section. The temperatures are altered by sap-flow activity and this information is used to solve the stem energy balance equation with respect to its convective heat flow residual which indicates sap-flow. Results from four different field experiments show that the majority of heat energy input was diverted to radial heat flow, leaving only little energy partitioned to convective heat flow. Determinations of gravimetric sap-flow were extremely noisy in consequence, rendering the method unreliable for practical application. Temperature differences across the heater consistently correlated with fluctuating net-radiation however, which motivated us to establish an empirical method for determining gravimetric sap-flow based on this temperature information alone. Numerical simulations showed that gravimetric sap-flow and temperature difference are nearly linearly and positive correlated within an observed sap flow range between 0 and 1.7 g h−1 , beyond which the relation became non-linear and even inverse at higher velocities. It remains to be tested whether such higher fluxes can be reached in practice and we provide a solution for these cases. Statistical noise overrode the error introduced by assuming a linear relation between sap flow and temperature difference within the range between 0 and 1.7 g h−1 . The resulting factors were determined under stable sap flow conditions greater than 1 g h−1 and used for generating daily cycles of sap flow using temperature information alone. The approach was successfully validated in 2011 and 2012 against independent measurements of latent heat flux conducted in closed and dense wheat fields using the eddy-covariance technique. We thereby improved the application of the new micro-sensor in wheat. Suggestions for further enhancements of the method are discussed. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Thermoelectric methods for measuring sap flow in plants are frequently applied in studies of plant-water relations (Cohen, 1994; Smith and Allen, 1996). The constant power stem heat balance method is particularly recommended for application when sap flow is low (Cohen, 1994) or stem diameters are smaller than 10 mm (Senock and Ham, 1993). Automated temperature measurements across defined axial and radial dimensions of a stem section are taken in this method for determining the partitioning of input energy from a circular heater into convective heat which indicates sap-flow and conductive heat (Sakuratani, 1981; Baker

∗ Corresponding author. Tel.: +49 228732871. E-mail address: [email protected] (M. Langensiepen). 0168-1923/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.agrformet.2013.11.007

and van Bavel, 1987). It has been successfully applied in a large variety of plant species (Dynamax Inc. Houston, Texas, USA) including soybean (Sauer et al., 2007), maize (Cohen al., 1993), cotton (Dugas, 1990), coffee (Meinzer et al., 1992), grapewine (Escalona et al., 2000), maple (Wullschleger et al., 1998), peach (Massai and Remorini, 2000), beech (Steppe and Lemeur, 2004), oak (Katul et al., 1997) and tropical trees (Meinzer et al., 1993). Dynamax Inc. (2009) offers stem heat balance sensor models for different stems sizes and has recently introduced a new microsensor to facilitate sap-flow measurements in thin stems with diameters ranging between 2.1 and 5 mm (SGA2 and SGA3 models, Dynamax, 2009). We used this sensor type for determining sapflow in commercially grown winter wheat (Triticum aestivum) and tested the results against independent measurements of latent heat flux. When initially applied under sunny midday weather conditions, 8% of the available input energy was partitioned to conductive

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heat. Average partitioning was typically less than 2%. Since convective flux is calculated as the residual of the stem heat balance equation, slight changes in conductive heat caused drastic fluctuations in calculated sap flow. We hypothesize that these fluctuations do not originate from thermal noise which would result from insufficient shielding of the measuring device against ambient air or loose contacts between sensors and stems, but are instead an inherent property of the micro-sensor method when applied to wheat. Reported partitioning of heat input energy to convective heat flux is normally far higher in other plant species than observed in this study and typically ranges between 30% and 60% under midday conditions, depending on soil water availability (Senock et al., 1996; Herzog et al., 1997; Gerdes et al., 1994; Kjelgaard et al., 1997). In a numerical exercise, Baker and Nieber (1989) examined the effects of sap flow and vascular bundle distribution on the heat field deformation around the stem heater. They concluded that the theoretical assumptions of the heat balance method are more applicable to dicotyledonous plants with vascular tissues located at the stem periphery, than to monocotyledonous plants in which they are distributed across the stem cross-sectional area. However, this is not the case for wheat and other grass species of the monocotyledonous Poaceae family in which vascular tissues are located within the hollow stem perimeter, as it is also the case in dicotyledonous plants. Thermal disequilibrium (i.e. heat storage), an error source when the stem heat balance method is applied to plant species with medullary and vascular parenchyma (Grime and Sinclair, 1999; Steppe et al., 2005), cannot occur in these plants because the heat storage capacity of stem air is negligible. In contrast to other observations reported in the literature (Senock and Ham, 1993) we noted marked diurnal patterns of temporal changes in temperature differences across the heater during all days which closely correlated with diurnal patterns in canopy net radiation which drives transpiration. The aim of this study was to examine and improve the heat balance technique for application in thin, hollow wheat stems. In particular we test two hypotheses: (1) The low values of convective heat are caused by the thermal properties of the thin, hollow wheat tillers and (2) the observed changes in temperature differences across the heater are caused by variations in sap flow. 2. Theory The installation of a heat-balance sensor around a hollow, thin wheat internode is illustrated in Fig. 1. Heat input from a circular heater (Qh ) is conducted outwards (Qr ) and inwards from the heater surfaces, the rate depending on the thermal properties of the adjacent materials (see Smith and Allen (1996) for further details). Convective sap flow (Jw ) in the thin internode wall insufficiently cools down the heating element which is why most heat is emitted radially (Qr ) into the sheath material and inwards through the stem wall into the internode air. Heat convection in the trapped stem air can, however, only insufficiently cool the heater due to its low specific heat (Table A1). The major portion of available heat (Qh ) is thus conducted outwards into the sheath material as radial heat (Qr ), leaving only little energy partitioned to convective heat (Qf ) which indicates sap flow (Fig. 2). The mathematical solution of the heat balance equation for determining sap flow in plants is only briefly summarized here since it has been covered extensively in the literature (Sakuratani, 1981; Baker and van Bavel, 1987; Ishida et al., 1991). Assuming a steady state condition for stem temperature during a small time interval, water flux Jw (g s−1 ) through vascular tissues can be quantified with (Senock and Ham, 1993) Jw =

Qh − Qv − Qr Cw (Td − Tu )

(1)

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Fig. 1. Illustration of a heat-balance sensor installation on a hollow wheat internode. Heat emitted from a circular heating element is partitioned into conductive heat (Qr ) and convective heat (Qr ) which indicates sap flow. Only one pair of thermocouples is used for measuring downstream (Td ) and upstream (Tu ) temperatures of the sapconducting internode wall which is why, by design, axial heat conduction (Qv ) is set to zero. The low specific heat of the entrapped air prevents significant radial heat emission into the internodium. Heat exchange with the canopy air is prevented by using cork, coated foam, and silver foil insulation material. We additionally wrapped a layer of plastic foil around the device to prevent intrusion of rainwater.

where Qh is the constant energy input into a stem section generated by a circular heater, Qr the radial heat loss determined in the insulating sheath material of the sensor determined during nighttimes and set constant during the day, and Qv the apical and basal heat conduction along the stem axis quantified with Fourier’s equation (all expressed in W). As recommended by the manufacturer (Dynamax, 2009), thermal conductivity of a hollow tiller is set to a constant value (0.28 W m−1 K−1 ) and sheath conductance determined under zero flow conditions during night time. Td and Tu are the stem temperatures (K) measured at defined distances downstream and upstream from the heater, respectively, and Cw is the specific heat of water (4.18 J g−1 K−1 ). In contrast to larger heat balance sensors distributed by Dynamax, in which two pairs of thermocouples are embedded in the inner sensor wall for measuring Td and Tu , the new sensor SGA3 tested in this study uses only one pair of thermocouples for this purpose. By design, Qv was set to zero and Qr compensates for any missing energy in the heat balance (Dynamax, 2009). Qv is thus lumped into a residual term that is estimated from the output of the radial thermopile. A typical daily cycle is shown in Fig. 2 which is characterized by strong fluctuations of these variables. The marked diurnal

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Fig. 2. Daily cycle of net radiation (A), stem-temperature difference across a circular heater (B), stem energy-balance components (C), and sap flow in mature wheat tillers (D) calculated following the Dynamax method for SGA3 gauges. Qh is the stem heat input power and Qr , Qf , and Qv are the radial, convective, and conductive heat fluxes across the stem heater, respectively. Data within the dashed intervals were used on this particular day for correlating sap flow with temperature differences between the downstream (Td ) and upstream (Tu ) sensors.

pattern in Td − Tu (Fig. 2B) indicates that convective heat transport (Qf ) must have taken place to a considerable extent. Applying the recommended calculation procedure (Dynamax, 2009) for determining Qf does not confirm this observation in practice (Fig. 2C). It diverts the major portion of heat supply (Qh ) to radial conduction (Qr ). Since Qv is zero for the tested sensor model (SGA3) and Qr compensates for any missing energy in the heat balance (Eq. (1)), only little remaining energy is diverted to Qf . Temporal changes in calculated Jw (Fig. 2D) are thus not identical with those of measured net radiation (Fig. 2A). High calculated Jw values during early mornings (Fig. 2D) are artifacts of the power saving procedure during nighttime and are normally excluded from the diurnal cycle. Dynamax (2009) also recommends the application of low and high-pass filters to restrict Qf to dimensions defined by the ratio between Qf and Qh as well as model-specific minimum and maximum temperature differences across the stem heater. Applying this procedure for filtering sensor outputs invalidated all measurements. Based on repeated similar observations made during two field seasons, we conclude that the Dynamax method has serious limitations with respect to its use with wheat (see also Fig. 9). To allow for its practical application, the method hence needed to be improved. Thermocouples measuring Td and Tu are in physical contact with the thin sap conducting wall of the hollow stem above and below the heater. Since, according to the theory of the stem heat balance method (Baker and Nieber, 1989), the vertical heat field across the heated stem deforms in proportion to convective heat transport by the ascending sap, temporal changes in Td − Tu indicate changes in gravimetric sap-flow. Changes in Jw are inversely proportional to (Td − Tu )2 as can be theoretically explained by taking the partial derivative of Eq. (1) with respect to Td − Tu : ∂Jw Qh − Qr = −  ∂(Td − Tu ) Cw (Td − Tu )2

(2)

Senock and Ham (1993) caution that since Jw in this equation is proportional to Qh − Qr , Jw is especially susceptible to errors when Qh − Qr is large and Td − Tu is below 1 ◦ C. They also analyzed the

relation between Jw and Td − Tu for small stemmed plants using Glycince max as an example and found that they are positively correlated within a sap-flow range between 0.2 and 10 g h−1 and negatively correlated at higher sap-flow rates up to 50 g h−1 . The only available information in the literature about sap flow rates in wheat tillers are given by Senock et al. (1996) which ranged between 0 and 5 g h−1 in their field experiment, but these values have never been verified. Based on the results of the aforementioned studies of Senock and Ham (1993) and Baker and Nieber (1989), we assumed that for wheat, Jw and Td − Tu must be positively correlated in this range. We analyzed this assumption with a numerical experiment conducted on an idealized wheat stem assumed to consist of a thin water pipe representing the xylem, two adjacent cell walls and an inner air space (Fig. A1). The arrangement of the different sensor materials around the hollow wheat stem and their heat physical properties were defined as closely as possible to the original setup. Heat transport in the different sensor materials and the wheat stem has been calculated with the standard heat convection-diffusion equation (Details about the calculation procedures and simulation setup are given in Appendix A1). Results of the simulations are shown in Fig. 3 for a hollow wheat stem with a typical diameter of 3.5 mm. Jw and Td − Tu are positively and almost linearly correlated within the range of sap-flow rates found in this study (0–1.7 g h−1 ) for both, steady-state (curve C) and dynamically changing temperature conditions of the system (curve B). It is unlikely, however, that a steady-state condition is ever reached, as ambient temperature fluctuations, heat-exchange with the air, and temperature drift in the sap-conducting pathway between roots and shoots (Cohen, 1994) cause temperature perturbations in the heat-balance system. Yet, it is interesting to note that curve C becomes strongly non-linear beyond Jw = 1.4 g h−1 and even exhibits a negative correlation between Jw and Td − Tu at sap flow rates greater than Jw = 2.8 g h−1 . The transition between positive and negative correlation between Jw and Td − Tu is shifted to Jw = 4 g h−1 under non-steady conditions (curve B) and remains

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We tested the validity of this approach against independent measurements of latent heat fluxes from a mature dense and closed wheat canopy (LAI > 4.1) using the eddy covariance method as a reference, assuming that intermediate plant water storage due to hydraulic tiller capacitance is negligible (Wallace and Biscoe, 1983) and soil evaporation is negligibly small. 3. Material and methods 3.1. Field experiments

Fig. 3. Relation between temperature difference across a stem-heater (Td − Tu ) and sap flow Jw experimentally determined for soybean (A) and wheat (D), and simulated for wheat under dynamic (B) and stationary (C) temperature states of the sap-flow system. Stem diameters were 3.5 mm and heating power 0.05 Watt in all cases. The range of sap velocities was defined using information from Senock et al. (1996).

nearly linear until 1.7 g h−1 , as it is also the case with the linear relation D determined in this study (see Fig. 6). Curve A has been digitalized from the publication of Senock and Ham (1993) on sapflow in Soybean stems which had the same diameter of 3.5 mm and received the same heating power input as in the numerical analysis on convective heat transport in wheat (curves B and C). Jw and Td − Tu are positively correlated even in this case, although the stem architectures of both plant species differ fundamentally. In summary, all numerical simulations and practical observations shown above verify the assumption that, for small stems and low sap-flow rates up to 1.7 g h−1 , Jw and Td − Tu are positively and almost linearly correlated. This seems to be particularly true for hollow wheat stems in which the xylem elements are embedded in thin lignin and cellulose cells layers at the periphery which facilitate an efficient heat transport between the sensor and the ascending sap. This is not the case in larger-stemmed plants in which much thicker cell layers act as efficient thermal buffers and Jw and Td − Tu are inversely related in consequence. Yet, as it is the case in thick stems, Jw and Td − Tu in small-stemmed plants are becoming also non-linear and even inversely related when sap-flow increases beyond 1.7 g h−1 . The reason is an increasing imbalance between heat-supply and heat-uptake by the sap when gravimetric sap-flow increases. This observation is confirmed by numerical simulations of Baker and Nieber (1989) who demonstrated that the rate of heat-uptake is decreasing when sap-velocity increases. Whether sap-velocities in wheat tillers can reach 5 g h−1 under field conditions, as reported by Senock et al. (1996), remains to be validated. The non-linearity of the relation between Jw and Td − Tu would certainly need to be considered in these situations. Since velocities falling in this range were not observed in this study, we assumed a positive linear relation between both variables within a range between 0 and 1.7 g h−1 , as suggested by the numerical simulations. Based on these practical and theoretical considerations, we correlated Td − Tu with Jw for Td − Tu > 1 ◦ C, as recommended by Senock and Ham (1993), and when Jw was relatively stable over periods of 30 min. Empirically determined values of the resulting fitting factor k were averaged for each day and used for establishing daily cycles of sap flow using (Td − Tu ) alone, thereby assuming that k remains constant over the course of a day: Jw,t = k(Td,t − Tu,t ) where t is the time of measurement during a day.

(3)

The validation experiments were conducted from June 3rd to July 5th 2011 and May 16th to July 17th 2012 in a winter wheat field of 6.9 ha size located near Merzenhausen in Germany (50◦ 55 N, 6◦ 17 E). The canopy was fully closed and leaf area index ranged between 4.1 and 5.7 during these times. We could therefore assume that the contribution of latent heat flux from soil evaporation to total latent heat flux was negligibly small and the eddy covariance station hence predominantly measured transpiration. Management followed standard optimum agronomic practice and the crop did not experience any significant water stress due to frequent rainfalls. Sap flow and eddy-covariance measurements where carried in the center of a wheat field. We counted an average 590 tillers per square-meter during the periods of measurements and used this number for converting average sap flow per tiller (g h−1 ) to an area based measure (mm h−1 ) for comparison reasons. 3.2. Sap flow measurements Observed average tiller diameters ranged between 3.4 and 5.2 mm during each measurement series in 2011 and 2012. To cover this range, 8 SGA3 (2.75–4 mm range) and 8 SGA5 (5–7 mm range) sensors were installed in 2011. Outputs from the second set were often characterized by high fluctuations in Td − Tu which resulted from insufficient contacts between the thermocouples and the tillers. This is the reason we excluded this set from the analysis and exclusively used the SGA3 sensor type in subsequent experiments. Gaps in the sheath joints were closed with plastic rubber provided by the manufacturer for this purpose and the device was properly shielded against the canopy air using different insulation materials (see Fig. 1). 16 sensors were operated in 2012 at the same location near the eddy tower. Additional 24 sensors were installed in 2011 and 2012 in a commercial winter wheat field of 1.5 ha size at Selhausen (50◦ 52 N, 6◦ 26 E), about 10 km apart from the validation experiment. Fetch and field size prevented the installation of an eddy station in this field. The data of the sap-flow systems were used in addition to those of the validation site for analyzing the nature of the k factor in Eq. (3). The sensors were operated strictly according to the manufacturer’s guidelines (Dynamax, 2009). Frequent field visits were performed to keep the delicate installations operating and correct any harm caused by mechanical stress or mouse damage. Signals were averaged over 5 min intervals with a CR1000 data logger and two AM 16/32 multiplexers (Campbell Scientific, Logan, Utah). Sensor heat inputs were controlled by voltage regulators linked to the CR1000. Measurement and control systems were obtained from Dynamax (Houston, Texas, USA) and consisted of Flow 32A unit and Flow 32B expansion kits. Voltage was supplied by conventional car batteries. Raw outputs of the thermocouples were post-processed and averaged into 30 min intervals with a computer script written in the R language (R Core Team, 2012). The script was validated against standard computations normally carried out with the datalogger using the Campbell Scientific programming language. Eq. (1) was solved for each individual sap-flow sensor reading separately and

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the resulting sap flux computations were averaged for each measuring location. A graphical user interface with four graphic panels was programmed in R for displaying daily time series of Qf scaled to its maximum and minimum limits for detecting highest convective flux rates (left upper window), heat-balance components Qh , Qr , and Qf (left lower window), Jw (right upper window), and Td − Tu (right lower window). A graphic picker was used for drawing rectangles around regions during which Jw and Td − Tu were stable over 30 min. The rectangles were then displayed in all graphic windows for the chosen time interval, allowing for visual comparisons between the shown time series. Jw and Td − Tu were averaged for each time interval, the corresponding k subsequently calculated, displayed, averaged over the day and transferred into a data input file which was used for calculating Jw based on information of Td − Tu alone. We admit that this procedure is laborious. Time requirements for carrying out sap-flow calculation were still smaller than in the case of the manual Excel-sheet method of the manufacturer. Attempts to fully automate sap-flow calculations based on the aforementioned protocol failed, because the effects of frequently changing weather conditions on sap-flow variables could not be generalized. The low resolution of Qf prevented the determination of k for days with frequent midday cloud occurrences which caused strong transient fluctuations in sap-flow. The values were interpolated in such cases. 3.3. Meteorological measurements A tower was placed in the field center near the sap flow installations and equipped with meteorological instruments. Temperature, humidity (HMP45C, Vaisala, Inc., Woburn, MA, USA), global and net radiation (SP-LITE and NR-LITE, Kipp and Zonen, Delft, Netherlands) were continuously monitored 2 m above soil level and stored at 30 min time intervals. Wind components and sonic temperature were measured at the same height with a three dimensional sonic anemometer (CSAT3, Campbell Scientific, Inc., Logan, UT, USA). Water vapor density was determined with an open-path infrared gas analyzer (IRGA, model LI7500, LI-COR Inc., Lincoln, NE, USA). Flux related measurements were taken with a sampling frequency of 20 Hz. A standard data logger was used for processing and storing all readings (CR5000, Campbell Scientific, Inc., Logan, UT, USA). To provide data suitable for validating the improved method, a series of post-processing steps were performed using the software TK3 (Mauder and Foken, 2011) which includes the source flux area model of Kormann and Meixner (2001) The results of these tests were consolidated in an overall flag system according to the scheme of the Spoleto agreement for the CarboEurope-IP (Mauder and Foken, 2011). Based on this scheme, each 30-min flux of latent (LE) and sensible (H) heat was classified as high, moderate, or low quality data. Only the fluxes of latent energy flagged with ‘high quality’ passing all quality controls were used for further analysis. To exclude fluxes affected by neighboring fields and an adjacent street, only measurements with a cumulated source contribution of the target field of more than 80% were used for further analysis.

Fig. 4. Correlation between measured net radiation and stem temperature difference across a constant power stem heater (Td − Tu ).

flow. The close association between both variables demonstrates that changes in Td − Tu are proportional to gravimetric sap-flow Jw which is driven by transpiration. Sap velocities Jw within the range between 0 and 1.7 g h−1 are linearly, positively correlated with Td − Tu (Fig. 5) under midday conditions which confirm the results of the numerical simulations discussed above. Transient adaptations of sap-flow to frequently occurring cloud obstructions of the sky, normally occurring under temperate climate conditions and not considered in Eq. (3), introduced scatter in the relation between both variables. We can only speculate about the reasons for the positive temperature offset around 0.3 ◦ C under zero flow conditions observed in both years which may be caused by miniature imperfections of temperature sensor placements in the micro-sensor or temperature drift in the sap stream. In spite of these uncertainties, both linear correlations demonstrate that a single factor k could be determined for establishing daily cycles of Jw based on information about Td − Tu alone, thereby avoiding the negative influence of high energy partitioning of Qh to Qr on the determination of Jw . Analyzing different sets of k revealed that this factor increases only from 0.37 to 0.55 during the season (Fig. 6) in comparison to results from a similar study conducted on soybean (Senock et al., 1996) in which k varied within a much larger range between 0.2 and 3.2 for the same range of sap velocities between 0 and 1.7 g h−1 and the same heater output of 0.05 W. The increases in k can be

4. Results Net radiation and average differences between downstream and upstream stem temperatures (Td − Tu ) measured during full daily cycles are strongly correlated (Fig. 4). The scatter in this relation is likely caused by transient plant hydraulic regulation processes which occur during high fluctuations of the physical environment (Boyer and Kramer, 1995). Soil water availability was not limiting and also not severely variable due to frequent rainfalls during the 2011 and 2012 seasons. The major portion of absorbed solar radiation was therefore partitioned into latent heat which drives sap

Fig. 5. Correlation between measured temperature differences across a constant power stem heater (Td − Tu ) and sap flow calculated with the standard procedures heat balance method (Dynamax, 2009) under midday conditions for 2011 (dashed line; y = 0.61x − 0.17; SD = 0.21; r2 = 0.62; N = 546) and 2012 (solid line, y = 0.61x − 0.22; SD = 0.29; r2 = 0.56; N = 423).

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Fig. 6. Seasonal changes of the ratio between wheat sap flow and temperature gradient across the stem heater obtained at two different sites near Jülich, Germany.

explained by stem diameter growth and associated increases in vascular transport capacities (Taneda and Tateno, 2004). Numerical simulations (Appendix A1) confirm, that increases in k are related to stem diameter growth within the operational range of the SGA3 sensor between 3 and 4 mm (Fig. 7). They also show that, for same stem diameters, k increases disproportionally with increasing gravimetric sap-flow, invalidating thereby the assumption taken in equation 3 that k remains constant over the course of a day. This practically implies that k should be more appropriately determined for both, discrete stem diameters and sap-velocities Jw when Qf is stable over short periods of time. Repeated observations could then be turned into three-dimensional lookup-tables for determining k based on observed combinations of stem diameters and sap-velocities. Statistical noise and low resolution of Qf at sap-flow rates below 1 g h−1 prevented a differentiation between different k values for different sap-velocities in this study. We need to stress in this context that wheat gravimetric sap-flow in this study reached peak values of 1.7 g h−1 in four independent wheat trials carried out during two seasons under a wide range of weather conditions including hot and dry days. These observations are staying in strong contrast to peak values of 5.1 g h−1 reported by Senock et al. (1996). The establishment of diameter and velocity variant k values would certainly be necessary in these cases, as demonstrated by the

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numerical simulations shown in Fig. 7. However, since the range of sap-velocities observed in our study was far lower and the error introduced by using a single k factor in this range (Fig. 7) overridden by statistical noise, we choose to determine k on a daily time basis. We tested the validity of sap flow calculations carried out with this approach against independent measurements of latent heat flux using the eddy covariance technique. Measured gravimetric sap flow in the tillers was converted to an area based transpiration unit for this purpose (mm h−1 ) using information about average tiller density. We also assumed that soil evaporation is negligible in the dense closed wheat canopies and dew evaporation is below 0.2 mm d−1 (Jacobs et al., 1994). The source of sensible heat flux is almost exclusively canopy transpiration under these conditions which we express in units of mm h−1 for comparison reasons by dividing measured latent heat flux by the latent heat of the air. The energy balance closure of the station was frequently high, with slopes of the regression of turbulent fluxes vs. available energy being higher than 0.9 on 26 out of 43 measuring days in 2011 and 42 out 62 days in 2012. Fig. 8 shows a day with energy balance closure of 0.97 with two marked peaks in wheat transpiration determined by the eddy covariance method. Transpiration calculated with the original method (Eq. (1)) roughly followed this temporal pattern during the morning and matched the occurrences of the two peaks, however only in relative terms. The high partitioning of input heat to radial conduction prevented an appropriate resolution in the axial convection term which would be necessary for obtaining a good match with transpiration inferred from the eddy covariance method. The application of our method drastically improved the determination of transpiration using the same sensor configuration. Transpiration determined with the sap flow method was slightly lagging behind evapotranspiration inferred from eddy covariance measurements during morning hours. We assume that this small difference is caused by dew evaporation from plant surfaces (Jacobs et al., 1994). The validity of our approach is proven by high correlations between eddy-covariance and sap flow measurements of crop transpiration in 2011 (ET = 1.0 × T + 0; SD = 0.07; r2 = 0.79, N = 789) and 2012 (ET = 1.02 × T + 0; SD = 0.05; r2 = 0.94, N = 738) shown in Fig. 9. In contrast, crop transpiration determined with the original Dynamax sap flow method was only loosely correlated with eddy-covariance determinations of evapotranspiration in 2011 (ET = 0.13 × T + 0.35; SD = 0.22; r2 = 0.16, N = 789) and 2012 (ET = 0.26 × T + 0.34; SD = 0.21; r2 = 0.06, N = 738).

5. Discussion

Fig. 7. Simulated relations between stem diameters and k for different sap velocities in wheat stems. The range of sap velocities was defined using information from Senock et al. (1996), but actually restricted to an upper level of 1.7 g h−1 experimentally determined in this study.

The assumption underlying the stem heat balance method is that changes in Td − Tu depend on the partitioning of heat input energy (Qh ) into convective heat transport along the stem axis (Qf ) and radial heat conduction (Qr ) into the sensor sheath (Baker and van Bavel, 1987). Stem heat storage has been identified as a significant factor influencing this interrelation (Cohen et al., 1993; Grime and Sinclair, 1999; Steppe et al., 2005). Although theoretical (Kucera et al., 1977; Steppe et al., 2005) and operational (Ishida et al., 1991) methods are available for quantifying this effect in practice, they are currently not applied in standard systems. For example, using a sensor for small-stemmed roses with slightly larger dimensions than in our study (SGA10, Dynamax Inc.), Rose and Rose (1998) found that transpiration determined with the stem heat balance method without a capacitance term is within an average 15% range of gravimetric transpiration losses of potted rose plants. The performance of this gauge was so well that calculated sap-flux closely traced oscillations of gravimetric water loss between 2 and 69 g per hour (Rose et al., 1994). However,

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applying the same SGA10 sensor model in soybean, Gerdes et al. (1994) found that sap flow calculated with the heat balance equation was up to four times higher than measured soil water uptake. They suspected that a too narrow spacing of the heater was the major cause. Based on the heat-balance theory of Sakuratani (1981) and Baker and van Bavel (1987), Senock and Ham (1993) designed a U-shaped sensor with heater dimensions and thermocouple spacing adapted to stems smaller than 5 mm. Calculated sap flow in soybean was within 5% of gravimetric measurements and, more importantly, the technical design of the sensor facilitated a far better partitioning of input energy to convective heat flux than in our study. However, they also noticed that decreasing soil-water availability caused 30–45% underestimations of sap flow as compared to gravimetric measurements due to a change in proportions of the heat-balance components. Including the rate of change in stem heat storage in the energy balance equation did not improve these results. As far as we know, information about the application of the heat balance method in wheat is limited to a single publication of Senock et al. (1996) who applied their U-shaped heat-balance sensor in a FACE experiment conducted in Phoenix, Arizona. Laboratory tests showed that calculated sap flow was within 10% of gravimetric measurements, but this observation has never been verified in the field. Peak tiller sap flow rates reached 5.1 g h−1 which appear unusually high compared to ours which reached peak levels around 1.7 g h−1 (Fig. 5). Part of this discrepancy can be explained by differences in climate conditions and associated genotype specific responses to the contrasting conditions in physical environments. Unfortunately, Senock et al. (1996) don’t provide information about the partitioning of input energy applied to their wheat tillers into the remaining components of the heat balance equation as they did in another publication on measuring sap flow in small-stemmed soybean (Senock and Ham, 1993). In contrast to our observations, as shown in Fig. 2, they measured pronounced daily trends in radial conduction (Qr ) and axial convection (Qf ) in soybean, both, under laboratory and variable field conditions (Senock and Ham, 1993). Whether this was also the case in their wheat experiment is not mentioned in their publication (Senock et al., 1996). Lacking resolution of Qf in wheat and resulting severe errors in the determination of Jw under variable flow conditions with the Dynamax method were the reason for linking Jw and Td − Tu with an empirical factor k under stable sap-flow conditions and using the resulting value for establishing daily courses of Jw based on information about Td − Tu alone. Numerical simulations have shown that

Fig. 8. Typical daily courses of wheat transpiration determined with the improved method for interpreting signals of heat balance sensors, the original method, and eddy covariance measurements. Merzenhausen, Germany, June 9th, 2011.

this approach is principally wrong as k varies with both, stem diameter growth and sap velocities. The error resulting from applying a single k for calculating daily courses of Jw is small, however, for sap-flow rates below 1.7 g h−1 and anyway overridden by statistical noise observed during normal gage operation. The approach suggested in this paper (Eq. (3)) is thus an effective solution for improving the determination of sap flow in small, hollow stems with the currently available micro-sensors. It is consistent with the original theory of the Dynamax heat balance approach (Baker and van Bavel, 1987). Determinations of sap flow with this method were closely correlated with independent eddy-correlation measurements of latent heat flux carried out at the same location (Fig. 9). Based on comparisons with an error estimation scheme (Mauder and Foken, 2011), the random errors of these reference fluxes were between 10% and 20%. By avoiding artifacts from transient changes in stomatal responses to light (Shimazaki et al., 2007) and adjustments of the plant hydraulic system to rapid changes of the physical environment (Meinzer, 2002) we were able to establish a linear empirical correlation between (Td − Tu ) and Jw for sap-velocities up to 1.7 g h−1 which can be used for calculating Jw for an entire day (Figs. 8 and 9).

Fig. 9. Correlation between eddy covariance measurements of evapotranspiration and canopy transpiration determined with Eq. (3) (round symbols) and Eq. (1) (crosses) in fully closed wheat canopies (LAI 4.1–5.2). More than 60% of values obtained with Eq. (1) are outside of the shown range in both years.

M. Langensiepen et al. / Agricultural and Forest Meteorology 186 (2014) 34–42

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6. Conclusion Insufficient partitioning of input energy to convective heatflux severely limits the application of the Dynamax heat-balance method for measuring sap-flow in wheat. We attempted to overcome the resulting noise in the determination of gravimetric sap-flow Jw by introducing Eq. (3). Numerical simulations have shown that Jw and temperature difference Td − Tu are positively and nearly linearly correlated in small stemmed plants when gravimetric sap-flow is lower than 1.7 g h−1 . The validity of this approach has been successfully tested against independent measurements of latent-flux. The relation becomes strongly non-linear and inverse with increasing gravimetric sap-flow beyond 1.7 g h−1 however, principally invalidating the assumption that k remains constant for a wide range of sap-velocities. Three-dimensional lookup-tables for determining k based on observed combinations of stem diameters and sap-velocities could be established in such cases. It remains to be experimentally tested whether gravimetric sap-flow in wheat tillers can reach such high levels. We must also caution that the findings of the numerical simulations are based on an idealized model in which the thermal diffusivities of wheat stem materials have been only crudely approximated (see Appendix A1 and Table A1). Vandegehuchte and Steppe (2012) have recently shown that in trees, such approximations can result into errors of sap flow density calculations of 10% and more. This seems to be even truer for small scale heat-transfer within the delicate anatomy of the walls of hollow wheat stems. Improving the engineering design of the sap-flow gauges, better adapted to the anatomical and heatphysical properties of wheat stems, is a further option to improve the resolution of convective heat determined with the sap-flow gauge.

Acknowledgments This project was financed by the Deutsche Forschungsgemeinschaft within the framework of the collaborative research center TR 32 “Patterns in soil-vegetation- atmosphere systems” TR 32/2 2011 3009725. We thank Prof. Dr. Karl Schneider from the Department of Geography, University of Cologne, Germany, for providing us with data of the eddy covariance station at Merzenhausen. We thank Dr. Ute Voigt of Bernt Messtechnik in Düsseldorf, the German distributor of Dynamax measuring equipment, for her continuous technical support during the measuring campaigns. We also thank the reviewers of this paper for their valuable suggestions which considerably improved its quality.

Appendix A1. Heat-transport within the sap-flow system was numerically simulated with the heat-transfer module of COMSOL (COMSOL, Inc., Burlington, USA). The physical dimensions and composition of the system could only be approximated since the exact engineering details of the commercial sensor were not available (Model SGA3, Dynamax Inc., Houston, USA). Principle information about the sapflow system was taken from Baker and van Bavel (1987), van Bavel and van Bavel (1994), and Dynamax (2009). Dimensions of the sensor components were measured with an electronic precision caliper (Model PMS 150M, Conrad Electronic, Hirschau, Germany). Their arrangement around the stem was set up according to visual inspection in the COMSOL model. Two point temperature sensors were located 11 mm upstream and 11 mm downstream from the axial center of the heat source, respectively. Fig. 1 provides a 2-D view of the gage model used in the simulations.

Fig. A1.

Anatomical pictures of horizontal wheat stem slices were analyzed to quantify the proportion and material composition of the peripheral tissues which occupy 20% of the cross sectional area and encircle the trapped stem air. The physical model constructed for this analysis (Fig. A1) is a simplified representation of the stem wall, assumed to consist of an outer layer (70% cellulose, 10% lignin and 20% water entrapped in cell tissues), a water pipe representing the xylem, and an inner layer (20% cellulose and 80% water entrapped in cell tissues). The proportions of the outer layer, water pipe, and inner layer within the total cross-sectional area of the stem wall were set to 40%, 20%, and 40%, respectively. Diameters of the wheat stem were varied between 3 and 4 mm to analyze the effect of growth on the quotient of Jw and Td − Tu . Heat output from the stem heater was quantified with: Q =

Ptot V

(A1)

where the power Ptot was set to 0.05 Watt, typically applied during practical operation in this study, and V calculated according to the physical dimension of the heating element which was assumed to consist of 50% heating silver paste and 50% polyester. Ignoring viscous heating and pressure work, heat transfer in the sap-flow system was calculated with the convection-diffusion equation cp

∂T + cp u · ∇ T = ∇ · (k∇ T ) + Q ∂t

(A2)

where  is the density of a material, cp its specific heat capacity, T the temperature, t the time, u the velocity,  the Nabla operator, k the thermal conductivity and Q the heat input calculated with Eq. (A1). The viscosity of water was set to 1002 Pa s and its ratio of specific heats to 1. Eq. (A2) is reduced to cp

∂T + ∇ · (−k∇ T ) = Q ∂t

(A3)

when gravimetric sap-flow is zero and heat is only transported by conduction. The approximate heat physical properties of the stem and different gage materials were selected from the engineering literature (Table A1). They were weighted when an element was constituted of different materials, such as the cell wall in the stem periphery. Heat transfer within the stem-gage system was calculated in stationary and dynamic modes to differentiate between steady-state and non-steady state temperature conditions. ∂T/∂t in Eqs. (A2) and (A3) was set to zero in the first case. Dynamic, stepwise simulations

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Table A1 Heat transfer and mechanical properties of wheat tiller and sap-flow sensor materials. Material

Specific heat capacity (J/(kg K))

Density (kg/m3 )

Thermal conductivity (W m−1 K−1 )

Acrylic plastica Aluminuma Airb Cellulosec Foamb Kaptond Ligninc , e Polyethylene LDb Polyesterf Silverf Waterb

1470 900 0.001006 0.0016 0.0013 1090 1200 3400 0.001 230 0.004183

1190 2700 1.205 1500 22 1420 1240 920 1455 10510 998.3

0.18 160 0.0257 0.05 0.03 0.12 0.1775 0.34 0.21 418 0.58

a b c d e f

Comsol. Engineering toolbox (2013). Hatakeyama and Hatakeyama (2004). Dupont (2013). Voitkevich et al. (2012). Engineersedge (2013).

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