Terrestrial Field Dissipation Studies. Purpose, Design, and Interpretation
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These experiments were conducted at the Hahn-Meitner Institute in Berlin in and the results were published in several papers  ,  , . I will use several IR images of these real heat fields throughout this text to further illustrate the HFD methodology. IR imaging provides a salient three-dimensional visualization of the HFD method. Tangential and radial views of the heat field generated around a linear heater are shown in Fig. S1 Supplementary material , both under conditions of zero flow and ascending sap. A tangential view of the heat field demonstrates two dimensions: axial and tangential which differ even for conditions of zero flow Fig.
S1c , as thermal conductivity is higher in the axial than in the tangential direction. The anisotropy of sapwood produces a heat field in a symmetrical ellipse form under zero flow whereas in an isotropic medium, the heat field would be in the form of a circle. Moving sap deforms the elliptical heat field in the tangential stem section Fig. S1d as well as in a radial direction along sapwood depth, where water movement occurred Fig.
- 3D dimension of the HFD method.
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This illustrates that the assumption of a perfect linear heater with equal temperature  can, in reality, only be valid under zero flow Fig. Under moving sap conditions, the temperature of a heater cannot be radially even as more heat is stored in the heartwood shown as the hottest area in Fig. S1f , whereas the coldest zone in a heater occurs in sapwood depths with the highest flow rates.
Effectively, a radial view of a heat field near the same heater will differ for trees with different proportions of sapwood and heartwood depths. Combining radial and tangential views of the heat field near a linear heater under moving sap conditions makes it possible to visualize the HFD method as a 3D image Fig. Thus, for zero flow conditions K -values can be easy determined as intersection of dT as or dT s-a with Y axis of a K -diagram.
Each temperature difference has a daily loop with higher values in the morning and lower values in the evening. K is equal to 1.
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- Revisiting the Heat Field Deformation (HFD) method for measuring sap flow.
K -diagrams are also useful for explaining the analysis at the heart of the HFD method: using independent readings indicating the sensitivity of measured temperature gradients, and alternating them out switching when one gradient becomes non-sensitive. The maximum levels for both dT as and dT sym serve as threshold values, where the switching between the sensitivity of the measured temperature gradients takes place. This transition zone is rather narrow for most species. Ideally, dT sym should be constant and should not decrease after approaching its maximum value, which is never the case in real conditions.
It was found that both temperature gradients, dT sym and dT s-a , are directly and highly proportional under very low flow rates up to their maximum value and they differ between each other on K- value Fig.
Additionally, dT s-a remains almost constant with further progressively increasing flow rates, whereas dT sym begins to decrease after reaching its maximum value in the morning Fig. To illustrate the HFD method in clearer terms, Fig. In this way, the advantages of both thermocouple arrangements around a linear heater can be exploited without the limitations occurring when each of them are used separately.
From the example of single day records of temperature gradients shown via the K -diagram and presented in Fig. It is important to note that during a daily cycle, we monitor hysteresis loops of observed temperature differences caused by daily variations of tissue temperatures and water content. Using the ratio of temperature gradients is valuable for sap flow calculation as one value of the ratio suits both morning and evening values of the temperature differences. Two frontal tangential images are shown in Fig. IR image in Fig. Under these conditions, dT sym is equal to zero because both upper and lower temperatures measured in axial directions are the same.
Two other temperature differences, dT as and dT s-a , are equal to each other in the absolute value K- value. I will provide several examples later in the text to demonstrate how different factors influence K- value. K -value is stable and is fixed at Y-axis, reflecting intrinsic and extrinsic conditions for a heat conductance.
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It thus illustrates the condition when both heat conduction and convection take place in the sapwood near a heater. Thermoisotherms look like asymmetrical elongated and deformed ellipses with the heater centre situated close to their lower focus.
The absolute values of the aside temperature differences dT as and dT s-a are no longer equal to each other. When the HFD sensor is installed in a plant organ stem, root, branch , well insulated from being influenced by surrounding variables, and the power supply is switched on, the heat field around a heater will be established within several minutes. Under well maintained power supply and insulation during sap flow monitoring, K- value will be stably fixed on the Y axis of a K -diagram for each measured tangential section of the plant organ Fig.
This makes it possible to compensate for differences in thermal conductivities in different species and different levels of applied energy. These daily R pulsations start from the same point specific for a measured place K -value on the Y axis if conditions for zero flow are met in corresponding nights. Thus, K -value within R accounts for corrective changes of the initial measurement conditions, leading to changes in recorded temperature differences. It has been shown experimentally  ,  and confirmed during later verification on stem segments  that R is highly proportional to sap flow rates, Q eqn.
An empirical strategy was chosen due to the complex nature of the 3D heat conduction and convection in living trees. Because the intrinsic and extrinsic aspects of each place of measurement are directly reflected in the corresponding K- value, it was assumed that K- value can serve as a correction for a fixed value of thermal diffusivity D for which the nominal value of D 2. However, the nominal value, D nom , may differ from that suggested by Marshall .
Recent gravimetrical verification of the HFD measurements may indirectly support this, when Steppe et al.
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There are, however, other reasons for the discrepancies found by the mentioned authors, mainly due to measurements taken from only one HFD sensor per stem. I do agree, however, with Fuchs et al. This will help to determine the correct D nom for the HFD method and to confirm or disprove the universality of D nom for different species.
In order to study the influence of sensor geometry on R in 2D tangential stem sections, I conducted a similar experiment in two large trees: a conifer tree Pseudotsuga menziesii and a diffuse porous tree Laurus azorica. A network of 23 thermometers was installed around a heater in the tree stem with a step of grid equal to 5 mm starting from the heater at the distance Z tg equal to 5 mm in tangential direction and at the distance Z ax equal to 10 mm in axial direction. Absolute temperatures were further used for reconstructing the temperature gradient in different directions from the heater, and the influence of sensor geometry on the measured temperature gradients was analyzed.
Results were similar also for Laurus tree. However, the mentioned authors of other heat based systems used exclusively axial arrangements of thermometers around a heater, and thus, their approach cannot be directly integrated into the HFD methodology. Finally, for each measured tangential section of a plant organ, the so-called sap flow per section, q i g cm -1 h -1 , which represents a preliminary 2D surrogate of sap flux density, is calculated as a product of D nom , R and DCF eqn.
Each tangential section has its own K-value and R, both creating its specific SFS corresponding to particular conditions of measurements.
The more such tangential sections in sapwood are analyzed along their radial depth, the more precise sap flow radial profiles can be determined. After numerous experiments on hundreds of plants and more than 60 species it was inferred that the same empirical formula could be applied for different species. Whenever it was possible, sap flow measurements using the HFD were compared with measurements simultaneously conducted with known recognized methods such as tissue heat balance THB -  ,  , thermal dissipation TD -  or Heat-Ratio Method HRM -  and results were in a close range.
It is interesting to underline the identity between sap flow per section, SFS, have been measured and calculated by two very different methods: HFD and THB, which have a solid theoretical basis . Results of comparative studies at the stem level in several species indicate a good correlation between both methods for medium and high flow rates unpublished data and .
Both low night time re-saturating flows, as well as reverse flows which cannot be detected by the THB method, are important in that they provide valuable information about plant water status. While the same sensor configuration is used for recording the reverse flow, flow direction and thus, changed significance of temperature gradients, as dT as changes responding to dT s-a , and vice versa should be taken into consideration  , .
When recorded diurnal values of dT sym become negative, the eqn. It is not recommended, however, to use eqn. Infra-red images of such heat field in stem section of a live tree resembled a trembling drop of mercury. Changing the formula does not influence the results at this stage see Fig. S3 in Supplementary material.
Terrestrial Field Dissipation Studies:Purpose Design & Interpretation
This figure demonstrates altering the sap flow direction in a lateral root of a Quercus suber tree at the end of a dry season. Before rainfall, stable negative values of dT sym up to After the two first rains DOY and , the magnitude of negative dT sym substantially decreased. Using the eqn. Such switching threshold values for dT sym are rather individual for each specific case, but usually do not exceed As it was mentioned above, for a full picture of how sap movement influences the deformation of a heat field around a linear heater, the third dimension i.
As we have seen from the IR images, a heater temperature is uneven positioned in the radial direction see Fig. S1 in Supplementary material under conditions of moving sap as heat field deformation occurred only in sapwood SW. To further confirm that L sw has influence on the measured temperature gradients, a simple experiment was conducted with periodical movements of the heater along the stem xylem radius, thereby artificially introducing different relative proportions between heater length in SW and HW Fig. S4 in Supplementary material.