Did You Know Trees Feel the Pull of the Moon? Scientists Link Sap Flow to Lunar Tides

Authors: (1) Le Mouel Jean-Louis, UMR7154, Universite de Paris, Paris, France; (2) Gibert Dominique, UMR5276, Laboratoire de geologie de Lyon Terre, plan ´ etes et environnement (LGL-TPE), Lyon, France; (3) Boule Jean-Baptiste, UMR7196, Museum National d’Histoire Naturelle, Paris, France; (4) Zuddas Pierpaolo, UMR7619, Sorbonne Universite, Paris, France; (5) de Bremond d’Ars Jean, UMR6118, Universite de Rennes, G ´ eosciences Rennes, Rennes, France; (6) Courtillot Vincent, UMR7154, Universite de Paris, Paris, France; (7) Lopes Fernando, UMR7154, Universite de Paris, Paris, France; (8) Geze Marc, UMR7245, Museum National d’Histoire Naturelle, Paris,France; (9) Maineult Alexis, UMR7619, Sorbonne Universite, Paris, France. Table of Links Abstract and 1 Introduction 2 Back to the Gibert et al. (2006) experiment and introducing the new setup 2.1 Recall of the Gibert et al. (2006) experiment 2.2 The new set up 3 The Singular Spectrum Analysis Method 4 Data analysis and model 5 Discussion and References Abstract In this paper, we revisit old electrical measurements conducted on a poplar tree in 2003 (Gibert et al. 2006), which showed the presence of a diurnal electrical signal, attributed to an electrokinetic phenomenon correlated with sap flow, even in winter. We reanalyze these data using the singular spectrum analysis method and demonstrate that the electrical signal measured at various locations (roots, trunk, and branches) on the poplar tree can be decomposed to over 80% into a sum of 7 pseudo-periods, all linked to luni-solar tides. The precision of the extracted periods exceeds that of models. To validate these old measurements, we have reproduced the 2003 protocol since 2018 at the Jardin des Plantes in Paris, on 3 oak trees and 3 hornbeams. The electrical signals today exhibit the same characteristics as those of 2003. Tidal forces could be a driving force of sap flow in trees. 1 Introduction Trees, like the majority of plants on our planet’s surface, can be broadly schematized as having a root system connected to branches through a trunk. Channels, such as phloem or xylem, originating from the rhizoderm and extending to the leaves, play a crucial role in facilitating proper irrigation and supplying nutrients for the plant’s growth. The fluid, varying in complexity based on its location within the plant, functions as its circulatory system and is commonly referred to as sap. A significant and ongoing debate revolves around the question of how this sap moves within the plant. In the literature, three main hypotheses are typically discussed to explain this movement. The first one involves capillarity (eg. [1–5]), the second revolves around osmotic pressure (eg. [6–10]) and the third, frequently debated hypothesis is that of evapotranspiration (eg. [11–14]). This idea originates from climatologist Thornthwaite [15], who sought to identify various geophysical phenomena contributing to the transfer of liquid water from the Earth’s surface to the atmosphere, ultimately classifying different climates. Plant evapotranspiration is recognized as one of these phenomena, alongside snow sublimation and the evaporation of free water. In their study of electrical signals measured on a standing poplar over the course of a year, Gibert et al. [16] demonstrated two important findings. Firstly, the recorded electrokinetic signals were associated with the sap flow in the tree (estimated by Ganier probes), and secondly, a diurnal oscillation, albeit modest, persisted even during the winter season. For several decades in geophysics, it has been established and observed that changes in velocity in an established flow or the addition of a saline front lead to measurable variations in electrical potential. This electrokinetic phenomenon is commonly referred to as spontaneous potential (SP, eg. [17–19]). In the natural environment, diurnal oscillations are not solely attributable to thermal fluctuations resulting from the alternation of day and night. Instead, they are generally linked to terrestrial gravitational tides caused by the combined influences of celestial bodies such as the Moon and the Sun (cf. [20]). Our planet experiences disruptions during its revolution and rotation, with the Moon and the Sun playing pivotal roles. These celestial interactions lead to phenomena like the precession of the equinoxes, which occurs over a period of approximately 26,000 years (for further details, see, eg. [21, 22]). In a broader context, mobile masses on the Earth’s surface oscillate under the influence of these luni-solar forces, with coastal tides being a primary manifestation. These forces also impact and vertically oscillate water confined in deep aquifers (eg. [23–27]), potentially influencing processes such as water-rock interactions (eg. [28,29]). Consequently, these interactions may play a role in providing nutrients to plants (eg [30]). We have demonstrated, over long periods exceeding 200 years, how the global series of volcanic eruptions activity (cf. [31]), the average sea levels evolution (cf. [32]) and the dynamics of the trees population within a Tibetan juniper forest (cf. [33]), could be decomposed into a series of cyclical patterns, all linked to the gravitational forces, this time originating from the Jovian planets. The first two phenomena being related to vertical movements of fluids. As previously explained, tidal phenomena exhibit a multi-scale nature in time, spanning from a few hours to thousands of years, and are marked by extensive spatial dimensions. In this presented study, utilizing both historical observations and contemporary data, our objective is to extend the conclusions drawn by Gibert et al. ( [16]) and provide further clarity on the driving mechanisms reflected in the electrical signals of trees. We replicated the poplar electrical monitoring experiment conducted by Gibert et al., this time on hornbeams, oaks, a sequoia, a walnut, and a cypress. Since 2019, we have established a tree observatory at the Museum National d’Histoire Naturelle (Paris, France), where we measure electrical potentials, temperature variations, and tree seismicities. In Section 2, we will recall the protocol used by Gibert et al. ( [16]), as well as the one we have implemented over the past 5 years. In Section 3, we will present the temporal analysis method that we employed for our electrical data. In Section 4, we will present a few observations along with the results of our analysis, drawing from both the historical data from Gibert et al. ( [16]) and the new observations. We will conclude in Section 5 with an in-depth discussion of this work. 2 Back to the Gibert et al. (2006) experiment and introducing the new setup 2.1 Recall of the Gibert et al. (2006) experiment Here, we briefly recall, and present in Figure 1, the acquisition protocol that was used in 2002-2003 for the Remungol poplar tree experiment. On that occasion, 32 stainless steel electrodes were inserted into the tree and 2 non-polarizable electrodes ( [34]) were buried in the soil., (i) 2 non-polarizable Petiau ( [34]) electrodes in the ground (number 101 and 102),. (ii) 5 stainless steel electrodes on the visible roots of the poplar tree (number 01 to 04), (iii) 2 stainless steel sets of electrode crowns, one positioned at a height of 1 meter from the ground and surrounding the trunk (number 1 to 8) and the other positioned at 3.4 meters from the ground (number 11 to 18), (iv) 2 stainless steel sets of 5 electrodes each on the north-facing side of the trunk, one between 0.9 and 3.4 meters (number 30 to 34) the other, from 5.5 meters to 10.5 meters (number 21 to 26) The measuring instrument employed is a Keithley 2701 digital multimeter with an input impedance greater than 100 MΩ, equipped with a relay matrix having 40 measurement channels controlled by acquisition software. Electric potential measurements are taken at all electrodes at a sampling interval of 1 minute. A UT time base was used, synchronized in real-time with the Frankfurt atomic clock. Both the computer and the multimeter are powered by a backup generator, mitigating interruptions due to brief electrical power line failures. In Figure 2, we present an example of recorded data from both the roots, the trunk, and the branches during the period from October 1, 2002, to the end of June 2003. On the x-axis, which represents time, each tick corresponds to the first 4 day of each month. Subsequently, we will present and analyze this time range, which is the longest with minimal gaps, covering transitions through each of the seasons. As illustrated in this figure, the electrical potentials, both in the roots, trunk, and branches, can vary by several hundred millivolts over the 320 days depicted. An activity (early) appears to commence in January 2023, reaching its peak during the summer of the same year. Clearly, the signals are not stationary, neither in the strict sense nor in the broader sense. Peaks of polarization are observed at times, and there is a gap in the data. These symptoms led us to choose Singular Spectrum Analysis as the method for analysis and extraction, which we will discuss later (Section 3). Henceforth, when referring to the data from Gibert et al. ( [16]), we will use the acronym RP (Remungol Poplar). 2.2 The new set up In order to better understand the physics and physiology of trees, which were partially discussed in Gibert et al. ( [16]), we have chosen to revisit the RP experiment and extend it by establishing a geophysical observatory for living trees at the Museum National d’Histoire Naturelle (MNHNo, Paris, France). At MNHNo, we have currently instrumented 1 sequoia, 1 walnut, 1 cypress, 3 oaks, and 2 hornbeams: 2 conifers and 6 deciduous trees. The potentials are measured using stainless steel electrodes on both generators on the north side of the trunks and on crowns at various heights (from 1 to 9 meters) as well as on the branches of the trees. For the reference electrodes, we consistently use non-polarizable Petiau-type electrodes buried at a depth of approximately 1 meter. The trees are divided into 2 groups, with each group sharing the same non-polarizable electrode as a reference.We have added thermal measurements (Pt-100 and Pt-1000 probes) that record the temperature both inside and outside the trees, at various heights, as well as in the soil alongside the reference electrodes. We also monitor seismic activity as well as the variations in the inclination of trees (trunk and branches) over time. All measurements have been acquired and sampled at one-second intervals since 2019 using Gantner modular acquisition systems[1]. These data acquisition now have significantly better performance than the Keitley 2701 used in 2003 : a high-impedance (> 100 MΩ) and a dynamic range of 24 bits capable of sampling up to 20 kHz per channel. In Figure 3, we present a collage of photos showcasing our installations. Figure 3a shows the installation of the last electrode at 5 meters above the ground on the north side of the sequoia trunk. The white boxes on the trunk contain the Gantner acquisition system, as seen in Figure 3c. In Figure 3b, the installation of the first inclinometer on a branch at 3.5 meters above the ground is shown. On Figure 3c, the Gantner acquisition system (blue box at the top right) is accompanied by 3 electrical acquisition modules (at the bottom). Figure 3d displays a night photo of the acquisition boxes for the 3 oaks and 2 hornbeams. In Figure 3e, there is an example of combined electrical/thermal measurement in the sequoia at the same location. Finally, Figure 3f shows the canopy of the walnut tree captured between two research buildings. Only the electrical measurements will be addressed in this study. In Figure 4, examples of raw electrical measurements recorded between April 11, 2023, and May 11, 2023, are presented. The data are acquisitions from the 2 hornbeams at the top, the 3 oaks in the center, and a comparison between an oak and a hornbeam at the bottom. All 5 trees share the same reference no-polarizable electrode (Petiau), and the distance between the furthest individuals is approximately 10 meters. As observed, the order of magnitude of the potentials corresponds to those recorded in RP in 2003. Additionally, although diurnal oscillation is present in all individuals, within the same species, the variations in electrical amplitudes can range from 1 to 4 from one individual to another (see Figure 4 in the center). We will delve into these data in more detail later on. [1] https://www.gantner-instruments.com/ 3 The Singular Spectrum Analysis Method The data, for which we have provided some examples (see Figures 2 and 2), correspond to a class of signals that are non-stationary in the strict sense and piecewise continuous. We are not in optimal conditions for their analysis in the Fourier sense (see [35,36]). This is why we opted for the Singular Spectrum Analysis (SSA), which has historically been developed, in the field of paleoclimatology, to analyze this type of signals (see [37,38]). We will present a brief overview of this method, and all the calculation details can be found in the reference work by Golyandina et al. ( [39]). SSA can be summarized in four steps. Let us consider a discrete time series (XN) of length N (N > 2): For further details regarding the properties of square matrices with constant values along ascending diagonals, such as Hankel or Toeplitz matrices, we invite the reader to explore the work of Lemmerling and Van Huffel ( [40]). Step 2: Decomposition in Singular Values Singular Value Decomposition (SVD, cf. [41]) of nonzero trajectory matrix X (dimensions L × K ) takes the following shape: where d is the rank of X (d = rank X = max{i|λi > 0}). SVD allows one to write X as a sum of d unitary matrices, defined in a univocal way. we obtain for the transposed trajectory matrix: which corresponds to a representation of the K lagged vectors in the orthogonal basis (V1, . . . , Vd). One sees why SVD is a very good choice for the analysis of the embedding matrix, since it provides two different geometrical descriptions. This step is called regrouping the eigen triplets (λ, U and V). In the limit case m = d, (12) becomes exactly (6), and we find again the unit matrices. Next, how can one associate pairs of eigen triplets? This means separating the additive components of a time series. One must first consider the concept of separability. Thus, one reconstructs the time series with length N from the matrices of step 3. If one applies the diagonal mean to unit matrices, then the series one obtains are called elementary series. Note that one can naturally extend the SSA of real signals to complex signals. One only has to replace all transposed marks with complex conjugates. As mentioned above, step 3 is the most difficult part. We have chosen one approach among many others: iterative SSA. Since the relation (6) is linear, we can iterate the decomposition. We start with a small value of L (we are looking for the longest period) that we increase until obtaining a quasi-Hankel matrix (step 3 and 3). We then extract the corresponding lowest frequency component that it subtracted from the original signal. We increase again the value of L to find the next component (shortest period). The algorithm stops when no pseudo-cycle can be detected or extracted. In this way, we scan the series from low to high frequencies. 4 Data analysis and model In a first step, we decompose the 2003 RP data using SSA for each of the 32 electrodes (see Figure 2). We observe that, in addition to diurnal and semi-diurnal oscillations ( [16]), more than 70% of the data variability is carried by five major earth tides (see Table 1 and [42, 43]). Examples of extraction results are shown in Figure 5. For pedagogical reasons, we cannot overlay 32 curves on the same graph. The signals, once extracted by SSA, are regular enough for the calculation of a Fourier transform to make sense, in order to extract the nominal oscillation period of each of them. Presentation of some lunar-solar tides detected and extracted from the SP signals In electrodes E1, E8, E32, and E34 (refer to our schematic in Figure 1), we have detected and extracted pseudo-cycles associated with the purely solar tide P1(refer to our Table 1), which we present in Figure 5 (top-right). In Figure 6 (topright), we show their respective Fourier spectra to determine the periodicities of these oscillations. These periodicities appear to be remarkably close, if not identical, to the ones obtained through calculations (refer to equations 01a and 01b), and we have indicated their theoretical values in the top-right corner of Figure 04b. Regarding the amplitudes, we observe a behavior that can only be attributed to the tree itself, as the variations in amplitudes are different at the same height (∼1m) within the same crown (electrodes E1 and E8). We also note that these amplitudes evolve and seem to anticipate both solstices by approximately one month in all cases. We change the set of four electrodes for Figures 5 (top-left) and 6 (top-left) to demonstrate the generality of our findings. The second component we have detected and extracted is a purely lunar tide, the K1 tide. Once again, as shown in the period spectra in Figure 6 (top-left), the values of these periods are very close to the expected theoretical value of 23.93 hours. Similarly to what we observed for the S1 tide, the amplitudes of the K1 tide also appear to vary with the dates of solstices and equinoxes. The only difference here is that these variations are in phase with the mentioned dates. And as we observed for the previous tide, the potential increases from spring to reach its maximum in summer. The others components, Figures 5 (middle-left/right and bottom-left/right) and 6 (middle-left/right and bottom-left/right), present in sequence the waveforms and period spectra of the following lunar-solar tides: S1, the fortnightly Mf , and K2. Similar to the examples we have described before, the waveforms of these lunar-solar tides shown in Figures 5 and 6 are clearly modulated throughout the year, with a narrowing around the Winter solstice and a maximum amplitude shortly before the Summer solstice. These waveforms may not be in phase across different parts of the Remungol poplar tree (roots, trunk, branches), but in all cases, their calculated periods are very close to the expected values (< 0.01%). For each of the 34 electrodes, it was possible to detect and extract between 7 luni-solar tides, which on average account for approximately 70% of the raw variance for each signal. A mechanism Let us then envision briefly an electrokinetic mechanism. Suppose the tree contains channels, taken as regular cylinders reaching going up to the top of the tree, in which the sap circulates. This circulation is not yet well understood (see [5, 44–49]). Let a channel of section S of undetermined height. We will just make an order of magnitude computation of the pressure change in the channel due to the presence of a tidal forces of the Sun or (and) the Moon. Let us take the case of the Moon. g being the gravity of the Earth (in fact an acceleration in m.s −2 , and we neglect here the effect of the Earth’s rotation; g = 9.81m.s −2 at the Earth’s surface). The vertical component the Moon tide, always an attraction, is on the Earth’s surface given by the following relation, σ f is the conductivity of the fluid (sap), ε is the conductivity of the fluid (sap), ζ is the ζ-potential (here between the fluid and the wood wall of the channel) and η the viscosity of the fluid. In fact we know very little about the value of C relevant to the trees. The value of the electric field can be of the order of 10 mV/m only if C is of the order of 10V/Pa. Such value is large compared to the values found in literature, generally relevant to rocks. A generalization across and within species As we have already mentioned, tidal waves are waves with large geographical extensions; this means that they can be considered constant above MNHNo. We also decomposed the signals from our oaks and hornbeams using SSA (see Figure 4), and we only present the lunar-solar tide k1 (see Figure 7). The first observation here is that, regardless of the tree species and the individual within that same species, the cycles of earth-tides are in phase. The amplitudes during the months of April and May 2023, for all individuals, are on the order of ± 10 to 20 mV. These values are consistent with those of electrode E18 of RP (also on the North-facing trunk) during the same months (see Figure 5, top left, yellow curve). Thus, regardless of the period, both today and 20 years ago, measurements of electrical potentials on the trunks of three different species of deciduous trees (Poplar, Hornbeams, and Oaks) reveal, on one hand, comparable amplitudes of these potentials. On the other hand, the pseudo-cycles that compose them, such as the K1 tide shown here, are in phase, independent of the individual. These electrical potentials, at least as shown here for the diurnal K1, have been associated with the sap ascent through an electrokinetic effect (see [16]). The fact that all signals for this same potential are in phase and of the same order of magnitude, despite the differences in individuals (different sizes, different canopies, different physiology), leads us to believe that the observations we have made and the mechanism we propose are robust. For a significant part, the displacement of sap is due, as with all fluid masses outside the Earth, to the forcing effect of terrestrial tides. 5 Discussion From a series of electrical SP data recorded on a Poplar tree in 2003 (see [16]), we propose to address the question of sap flow mechanism in trees. Today, for this mechanism, three sources are mentioned, which are not necessarily exclusive: capillarity, osmotic pressure, and evapotranspiration. We have discussed and critiqued the possibility of these mechanism in our introduction. In a recent study (see [33]), we were able to measure the extent to which gravitational forces acting on Earth, through variations in its rotation speed and tilt, can influence the growth rates of tree rings in a Tibetan juniper forest for over 1000 years by modulating the incoming solar radiation. The growth rates of tree rings are linked to photosynthesis and, therefore, are also related to the flow of sap circulating in the xylem and phloem, which transport nutrients throughout the plant. At first order, like all fluids inside and on the surface of the Earth, the movement of this sap, regardless of its density, must be partially or entirely driven by lunar-solar tides, especially during the periods (¡18.6 years) that are of particular interest in this paper. Gibert et al. ( [33]) emphasized that the sap flow and the recorded electrical signal were most likely linked by an electrokinetic phenomena. Therefore, in order to explain the movement of sap from a physics perspective, we are dealing here with a simple harmonic pumping phenomenon (see [17, 18]). As illustrated in Figure 2, the continuously recorded electrical signals in trees are non-stationary signals in the strict sense. This inherently prevents the direct use of a Fourier transform to determine the periodicities (i.e., lunar-solar tides) that may compose them. Furthermore, measurement vagary sometimes lead to interruptions in the recording system, resulting in discontinuous data. That is why we chose the Singular Spectrum Analysis (SSA) method to analyze the measurements of the Remungol poplar. This method helps limit possible errors in the interpretation of traditional spectra. In Figures 5 and 6, we presented the five main pseudo-periodicities extracted from the electrical potential data of the poplar. All of them correspond very precisely to periods of lunar-solar tides (P1, K1, S1, Mf , and K2, cf. our Table 1). The periods that we have extracted from the electrical signals, which serve as proxies for sap flow, are within 0.01% of the calculated lunar-solar tidal periods. This level of precision is remarkable and can typically only be achieved by expensive gravimeters costing tens of thousands of euros. The sum of the extracted components, which we believe to be tidal components, generally accounts for more than 70% of the total variance in the recorded electrical signals. If we examine in more detail the signals recorded in different areas of the Poplar tree (root, branch, trunk) for a given tide, we observe that the waveforms associated with these tides are not modulated in the same way. These amplitude modulations can be completely distinct from each other, as is the case for the tide and the electrodes E1, E8, E32, and E34 (cf. Figures 4). However, they can also be the same but phased in time. For example, we can observe that the patterns of the signals recorded by the electrodes E7 and E12 for the tide are the same but shifted by approximately 2 weeks (see Figures 5). Undoubtedly, the differences in amplitudes and phases observed in the recorded signals of the Poplar tree are likely to be related to its physiological response. Based on these observations, we embarked on a mathematical resolution of the equations associated with the phenomenon that we believe to be responsible for the sap flow, aiming to evaluate the electrokinetic coefficient associated with the Poplar tree, similar to what is done in geophysics for rocks (see equation 17). We obtained a value of 10 V/Pa, which is generally ten times higher than the values measured in rocks. However, it should be noted that trees are not rocks, as we can by the picture of the xylem geometry in Figure 01, sap flow is not disrupted by any tortuosity, unlike in rocks. Furthermore, it is important to emphasize that sap is not water. The question of the differences in amplitude modulations between different areas of the Poplar tree, as well as the question of the generality of our surprising results, led us to reproduce the same experiment in 2019 at the National Museum of Natural History, but this time on different tree species. As we mentioned, we have three oak trees, two hornbeams, one sequoia, one cypress, and a walnut tree. We have only presented the observations we record on the oak and hornbeam trees for two reasons. The first reason is that, we have multiple individuals of the same species, allowing us to make intra-species comparisons. The second reason is that, these five trees are in the same location, sharing the same reference electrode. This allows us to consider inter-species comparisons, given that the spreading of gravitational waves is such that all Parisian trees will be affected simultaneously and with the same amplitude by earth tides. As shown in Figures 4 and 7, the amplitudes and their variations differ significantly from one individual to another, which we could not estimate in the Remungol experiment. As we can see in Figure 13a and in more detail in Figure 7 (once the tide is extracted), on the one hand, the diurnal components of the three oak trees do not have the same electrical amplitudes (at the same electrode height), and on the other hand, longer-period oscillations appear on 2 of them (see Figure 4). We make the same observations for the hornbeams. However, what is confirmed by Figures 13c and 14c is that the components corresponding to the diurnal tide are perfectly in phase, from one individual to another, regardless of the species Therefore, our hypothesis of the lunar-solar tide as the main driving force of sap flow is confirmed both by the richness of the observations and their incredible precision. 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Authors: (1) Le Mouel Jean-Louis, UMR7154, Universite de Paris, Paris, France; (2) Gibert Dominique, UMR5276, Laboratoire de geologie de Lyon Terre, plan ´ etes et environnement (LGL-TPE), Lyon, France; (3) Boule Jean-Baptiste, UMR7196, Museum National d’Histoire Naturelle, Paris, France; (4) Zuddas Pierpaolo, UMR7619, Sorbonne Universite, Paris, France; (5) de Bremond d’Ars Jean, UMR6118, Universite de Rennes, G ´ eosciences Rennes, Rennes, France; (6) Courtillot Vincent, UMR7154, Universite de Paris, Paris, France; (7) Lopes Fernando, UMR7154, Universite de Paris, Paris, France; (8) Geze Marc, UMR7245, Museum National d’Histoire Naturelle, Paris,France; (9) Maineult Alexis, UMR7619, Sorbonne Universite, Paris, France. Authors: Authors: (1) Le Mouel Jean-Louis, UMR7154, Universite de Paris, Paris, France; (2) Gibert Dominique, UMR5276, Laboratoire de geologie de Lyon Terre, plan ´ etes et environnement (LGL-TPE), Lyon, France; (3) Boule Jean-Baptiste, UMR7196, Museum National d’Histoire Naturelle, Paris, France; (4) Zuddas Pierpaolo, UMR7619, Sorbonne Universite, Paris, France; (5) de Bremond d’Ars Jean, UMR6118, Universite de Rennes, G ´ eosciences Rennes, Rennes, France; (6) Courtillot Vincent, UMR7154, Universite de Paris, Paris, France; (7) Lopes Fernando, UMR7154, Universite de Paris, Paris, France; (8) Geze Marc, UMR7245, Museum National d’Histoire Naturelle, Paris,France; (9) Maineult Alexis, UMR7619, Sorbonne Universite, Paris, France. Table of Links Abstract and 1 Introduction 2 Back to the Gibert et al. (2006) experiment and introducing the new setup 2.1 Recall of the Gibert et al. (2006) experiment 2.2 The new set up 3 The Singular Spectrum Analysis Method 4 Data analysis and model 5 Discussion and References Abstract and 1 Introduction 2 Back to the Gibert et al. (2006) experiment and introducing the new setup 2.1 Recall of the Gibert et al. (2006) experiment 2.2 The new set up 3 The Singular Spectrum Analysis Method 4 Data analysis and model 5 Discussion and References Abstract In this paper, we revisit old electrical measurements conducted on a poplar tree in 2003 (Gibert et al. 2006), which showed the presence of a diurnal electrical signal, attributed to an electrokinetic phenomenon correlated with sap flow, even in winter. We reanalyze these data using the singular spectrum analysis method and demonstrate that the electrical signal measured at various locations (roots, trunk, and branches) on the poplar tree can be decomposed to over 80% into a sum of 7 pseudo-periods, all linked to luni-solar tides. The precision of the extracted periods exceeds that of models. To validate these old measurements, we have reproduced the 2003 protocol since 2018 at the Jardin des Plantes in Paris, on 3 oak trees and 3 hornbeams. The electrical signals today exhibit the same characteristics as those of 2003. Tidal forces could be a driving force of sap flow in trees. 1 Introduction Trees, like the majority of plants on our planet’s surface, can be broadly schematized as having a root system connected to branches through a trunk. Channels, such as phloem or xylem, originating from the rhizoderm and extending to the leaves, play a crucial role in facilitating proper irrigation and supplying nutrients for the plant’s growth. The fluid, varying in complexity based on its location within the plant, functions as its circulatory system and is commonly referred to as sap. A significant and ongoing debate revolves around the question of how this sap moves within the plant. In the literature, three main hypotheses are typically discussed to explain this movement. The first one involves capillarity (eg. [1–5]), the second revolves around osmotic pressure (eg. [6–10]) and the third, frequently debated hypothesis is that of evapotranspiration (eg. [11–14]). This idea originates from climatologist Thornthwaite [15], who sought to identify various geophysical phenomena contributing to the transfer of liquid water from the Earth’s surface to the atmosphere, ultimately classifying different climates. Plant evapotranspiration is recognized as one of these phenomena, alongside snow sublimation and the evaporation of free water. In their study of electrical signals measured on a standing poplar over the course of a year, Gibert et al. [16] demonstrated two important findings. Firstly, the recorded electrokinetic signals were associated with the sap flow in the tree (estimated by Ganier probes), and secondly, a diurnal oscillation, albeit modest, persisted even during the winter season. For several decades in geophysics, it has been established and observed that changes in velocity in an established flow or the addition of a saline front lead to measurable variations in electrical potential. This electrokinetic phenomenon is commonly referred to as spontaneous potential (SP, eg. [17–19]). In the natural environment, diurnal oscillations are not solely attributable to thermal fluctuations resulting from the alternation of day and night. Instead, they are generally linked to terrestrial gravitational tides caused by the combined influences of celestial bodies such as the Moon and the Sun (cf. [20]). Our planet experiences disruptions during its revolution and rotation, with the Moon and the Sun playing pivotal roles. These celestial interactions lead to phenomena like the precession of the equinoxes, which occurs over a period of approximately 26,000 years (for further details, see, eg. [21, 22]). In a broader context, mobile masses on the Earth’s surface oscillate under the influence of these luni-solar forces, with coastal tides being a primary manifestation. These forces also impact and vertically oscillate water confined in deep aquifers (eg. [23–27]), potentially influencing processes such as water-rock interactions (eg. [28,29]). Consequently, these interactions may play a role in providing nutrients to plants (eg [30]). We have demonstrated, over long periods exceeding 200 years, how the global series of volcanic eruptions activity (cf. [31]), the average sea levels evolution (cf. [32]) and the dynamics of the trees population within a Tibetan juniper forest (cf. [33]), could be decomposed into a series of cyclical patterns, all linked to the gravitational forces, this time originating from the Jovian planets. The first two phenomena being related to vertical movements of fluids. As previously explained, tidal phenomena exhibit a multi-scale nature in time, spanning from a few hours to thousands of years, and are marked by extensive spatial dimensions. In this presented study, utilizing both historical observations and contemporary data, our objective is to extend the conclusions drawn by Gibert et al. ( [16]) and provide further clarity on the driving mechanisms reflected in the electrical signals of trees. We replicated the poplar electrical monitoring experiment conducted by Gibert et al., this time on hornbeams, oaks, a sequoia, a walnut, and a cypress. Since 2019, we have established a tree observatory at the Museum National d’Histoire Naturelle (Paris, France), where we measure electrical potentials, temperature variations, and tree seismicities. In Section 2, we will recall the protocol used by Gibert et al. ( [16]), as well as the one we have implemented over the past 5 years. In Section 3, we will present the temporal analysis method that we employed for our electrical data. In Section 4, we will present a few observations along with the results of our analysis, drawing from both the historical data from Gibert et al. ( [16]) and the new observations. We will conclude in Section 5 with an in-depth discussion of this work. 2 Back to the Gibert et al. (2006) experiment and introducing the new setup 2.1 Recall of the Gibert et al. (2006) experiment Here, we briefly recall, and present in Figure 1, the acquisition protocol that was used in 2002-2003 for the Remungol poplar tree experiment. On that occasion, 32 stainless steel electrodes were inserted into the tree and 2 non-polarizable electrodes ( [34]) were buried in the soil., (i) 2 non-polarizable Petiau ( [34]) electrodes in the ground (number 101 and 102),. (ii) 5 stainless steel electrodes on the visible roots of the poplar tree (number 01 to 04), (iii) 2 stainless steel sets of electrode crowns, one positioned at a height of 1 meter from the ground and surrounding the trunk (number 1 to 8) and the other positioned at 3.4 meters from the ground (number 11 to 18), (iv) 2 stainless steel sets of 5 electrodes each on the north-facing side of the trunk, one between 0.9 and 3.4 meters (number 30 to 34) the other, from 5.5 meters to 10.5 meters (number 21 to 26) The measuring instrument employed is a Keithley 2701 digital multimeter with an input impedance greater than 100 MΩ, equipped with a relay matrix having 40 measurement channels controlled by acquisition software. Electric potential measurements are taken at all electrodes at a sampling interval of 1 minute. A UT time base was used, synchronized in real-time with the Frankfurt atomic clock. Both the computer and the multimeter are powered by a backup generator, mitigating interruptions due to brief electrical power line failures. In Figure 2, we present an example of recorded data from both the roots, the trunk, and the branches during the period from October 1, 2002, to the end of June 2003. On the x-axis, which represents time, each tick corresponds to the first 4 day of each month. Subsequently, we will present and analyze this time range, which is the longest with minimal gaps, covering transitions through each of the seasons. As illustrated in this figure, the electrical potentials, both in the roots, trunk, and branches, can vary by several hundred millivolts over the 320 days depicted. An activity (early) appears to commence in January 2023, reaching its peak during the summer of the same year. Clearly, the signals are not stationary, neither in the strict sense nor in the broader sense. Peaks of polarization are observed at times, and there is a gap in the data. These symptoms led us to choose Singular Spectrum Analysis as the method for analysis and extraction, which we will discuss later (Section 3). Henceforth, when referring to the data from Gibert et al. ( [16]), we will use the acronym RP (Remungol Poplar). RP 2.2 The new set up In order to better understand the physics and physiology of trees, which were partially discussed in Gibert et al. ( [16]), we have chosen to revisit the RP experiment and extend it by establishing a geophysical observatory for living trees at the Museum National d’Histoire Naturelle ( MNHNo, Paris, France). MNHNo, At MNHNo, we have currently instrumented 1 sequoia, 1 walnut, 1 cypress, 3 oaks, and 2 hornbeams: 2 conifers and 6 deciduous trees. The potentials are measured using stainless steel electrodes on both generators on the north side of the trunks and on crowns at various heights (from 1 to 9 meters) as well as on the branches of the trees. For the reference electrodes, we consistently use non-polarizable Petiau-type electrodes buried at a depth of approximately 1 meter. The trees are divided into 2 groups, with each group sharing the same non-polarizable electrode as a reference.We have added thermal measurements (Pt-100 and Pt-1000 probes) that record the temperature both inside and outside the trees, at various heights, as well as in the soil alongside the reference electrodes. We also monitor seismic activity as well as the variations in the inclination of trees (trunk and branches) over time. All measurements have been acquired and sampled at one-second intervals since 2019 using Gantner modular acquisition systems[1]. These data acquisition now have significantly better performance than the Keitley 2701 used in 2003 : a high-impedance (> 100 MΩ) and a dynamic range of 24 bits capable of sampling up to 20 kHz per channel. In Figure 3, we present a collage of photos showcasing our installations. Figure 3a shows the installation of the last electrode at 5 meters above the ground on the north side of the sequoia trunk. The white boxes on the trunk contain the Gantner acquisition system, as seen in Figure 3c. In Figure 3b, the installation of the first inclinometer on a branch at 3.5 meters above the ground is shown. On Figure 3c, the Gantner acquisition system (blue box at the top right) is accompanied by 3 electrical acquisition modules (at the bottom). Figure 3d displays a night photo of the acquisition boxes for the 3 oaks and 2 hornbeams. In Figure 3e, there is an example of combined electrical/thermal measurement in the sequoia at the same location. Finally, Figure 3f shows the canopy of the walnut tree captured between two research buildings. Only the electrical measurements will be addressed in this study. In Figure 4, examples of raw electrical measurements recorded between April 11, 2023, and May 11, 2023, are presented. The data are acquisitions from the 2 hornbeams at the top, the 3 oaks in the center, and a comparison between an oak and a hornbeam at the bottom. All 5 trees share the same reference no-polarizable electrode (Petiau), and the distance between the furthest individuals is approximately 10 meters. As observed, the order of magnitude of the potentials corresponds to those recorded in RP in 2003. Additionally, although diurnal oscillation is present in all individuals, within the same species, the variations in electrical amplitudes can range from 1 to 4 from one individual to another (see Figure 4 in the center). We will delve into these data in more detail later on. [1] https://www.gantner-instruments.com/ 3 The Singular Spectrum Analysis Method The data, for which we have provided some examples (see Figures 2 and 2), correspond to a class of signals that are non-stationary in the strict sense and piecewise continuous. We are not in optimal conditions for their analysis in the Fourier sense (see [35,36]). This is why we opted for the Singular Spectrum Analysis (SSA), which has historically been developed, in the field of paleoclimatology, to analyze this type of signals (see [37,38]). We will present a brief overview of this method, and all the calculation details can be found in the reference work by Golyandina et al. ( [39]). SSA can be summarized in four steps. Let us consider a discrete time series (XN) of length N (N > 2): For further details regarding the properties of square matrices with constant values along ascending diagonals, such as Hankel or Toeplitz matrices, we invite the reader to explore the work of Lemmerling and Van Huffel ( [40]). Step 2: Decomposition in Singular Values Singular Value Decomposition ( SVD, cf. [41]) of nonzero trajectory matrix X (dimensions L × K ) takes the following shape: Step 2: Decomposition in Singular Values SVD, where d is the rank of X (d = rank X = max{i|λi > 0}). SVD allows one to write X as a sum of d unitary matrices, defined in a univocal way. we obtain for the transposed trajectory matrix: which corresponds to a representation of the K lagged vectors in the orthogonal basis (V1, . . . , Vd). One sees why SVD is a very good choice for the analysis of the embedding matrix, since it provides two different geometrical descriptions. This step is called regrouping the eigen triplets (λ, U and V). In the limit case m = d, (12) becomes exactly (6), and we find again the unit matrices. Next, how can one associate pairs of eigen triplets? This means separating the additive components of a time series. One must first consider the concept of separability. Thus, one reconstructs the time series with length N from the matrices of step 3. If one applies the diagonal mean to unit matrices, then the series one obtains are called elementary series. Note that one can naturally extend the SSA of real signals to complex signals. One only has to replace all transposed marks with complex conjugates. As mentioned above, step 3 is the most difficult part. We have chosen one approach among many others: iterative SSA . Since the relation (6) is linear, we can iterate the decomposition. We start with a small value of L (we are looking for the longest period) that we increase until obtaining a quasi-Hankel matrix (step 3 and 3). We then extract the corresponding lowest frequency component that it subtracted from the original signal. We increase again the value of L to find the next component (shortest period). The algorithm stops when no pseudo-cycle can be detected or extracted. In this way, we scan the series from low to high frequencies. SSA 4 Data analysis and model In a first step, we decompose the 2003 RP data using SSA for each of the 32 electrodes (see Figure 2). We observe that, in addition to diurnal and semi-diurnal oscillations ( [16]), more than 70% of the data variability is carried by five major earth tides (see Table 1 and [42, 43]). Examples of extraction results are shown in Figure 5. For pedagogical reasons, we cannot overlay 32 curves on the same graph. The signals, once extracted by SSA , are regular enough for the calculation of a Fourier transform to make sense, in order to extract the nominal oscillation period of each of them. RP SSA SSA Presentation of some lunar-solar tides detected and extracted from the SP signals Presentation of some lunar-solar tides detected and extracted from the SP signals In electrodes E1, E8, E32, and E34 (refer to our schematic in Figure 1), we have detected and extracted pseudo-cycles associated with the purely solar tide P1(refer to our Table 1), which we present in Figure 5 (top-right). In Figure 6 (topright), we show their respective Fourier spectra to determine the periodicities of these oscillations. These periodicities appear to be remarkably close, if not identical, to the ones obtained through calculations (refer to equations 01a and 01b), and we have indicated their theoretical values in the top-right corner of Figure 04b. Regarding the amplitudes, we observe a behavior that can only be attributed to the tree itself, as the variations in amplitudes are different at the same height (∼1m) within the same crown (electrodes E1 and E8). We also note that these amplitudes evolve and seem to anticipate both solstices by approximately one month in all cases. We change the set of four electrodes for Figures 5 (top-left) and 6 (top-left) to demonstrate the generality of our findings. The second component we have detected and extracted is a purely lunar tide, the K1 tide. Once again, as shown in the period spectra in Figure 6 (top-left), the values of these periods are very close to the expected theoretical value of 23.93 hours. Similarly to what we observed for the S1 tide, the amplitudes of the K1 tide also appear to vary with the dates of solstices and equinoxes. The only difference here is that these variations are in phase with the mentioned dates. And as we observed for the previous tide, the potential increases from spring to reach its maximum in summer. The others components, Figures 5 (middle-left/right and bottom-left/right) and 6 (middle-left/right and bottom-left/right), present in sequence the waveforms and period spectra of the following lunar-solar tides: S1, the fortnightly Mf , and K2. Similar to the examples we have described before, the waveforms of these lunar-solar tides shown in Figures 5 and 6 are clearly modulated throughout the year, with a narrowing around the Winter solstice and a maximum amplitude shortly before the Summer solstice. These waveforms may not be in phase across different parts of the Remungol poplar tree (roots, trunk, branches), but in all cases, their calculated periods are very close to the expected values (< 0.01%). For each of the 34 electrodes, it was possible to detect and extract between 7 luni-solar tides, which on average account for approximately 70% of the raw variance for each signal. A mechanism A mechanism Let us then envision briefly an electrokinetic mechanism. Suppose the tree contains channels, taken as regular cylinders reaching going up to the top of the tree, in which the sap circulates. This circulation is not yet well understood (see [5, 44–49]). Let a channel of section S of undetermined height. We will just make an order of magnitude computation of the pressure change in the channel due to the presence of a tidal forces of the Sun or (and) the Moon. Let us take the case of the Moon. g being the gravity of the Earth (in fact an acceleration in m.s −2 , and we neglect here the effect of the Earth’s rotation; g = 9.81m.s −2 at the Earth’s surface). The vertical component the Moon tide, always an attraction, is on the Earth’s surface given by the following relation, σ f is the conductivity of the fluid (sap), ε is the conductivity of the fluid (sap), ζ is the ζ-potential (here between the fluid and the wood wall of the channel) and η the viscosity of the fluid. In fact we know very little about the value of C relevant to the trees. The value of the electric field can be of the order of 10 mV/m only if C is of the order of 10V/Pa. Such value is large compared to the values found in literature, generally relevant to rocks. A generalization across and within species A generalization across and within species As we have already mentioned, tidal waves are waves with large geographical extensions; this means that they can be considered constant above MNHNo . We also decomposed the signals from our oaks and hornbeams using SSA (see Figure 4), and we only present the lunar-solar tide k1 (see Figure 7). The first observation here is that, regardless of the tree species and the individual within that same species, the cycles of earth-tides are in phase. MNHNo The amplitudes during the months of April and May 2023, for all individuals, are on the order of ± 10 to 20 mV. These values are consistent with those of electrode E18 of RP (also on the North-facing trunk) during the same months (see Figure 5, top left, yellow curve). Thus, regardless of the period, both today and 20 years ago, measurements of electrical potentials on the trunks of three different species of deciduous trees (Poplar, Hornbeams, and Oaks) reveal, on one hand, comparable amplitudes of these potentials. On the other hand, the pseudo-cycles that compose them, such as the K1 tide shown here, are in phase, independent of the individual. These electrical potentials, at least as shown here for the diurnal K1, have been associated with the sap ascent through an electrokinetic effect (see [16]). The fact that all signals for this same potential are in phase and of the same order of magnitude, despite the differences in individuals (different sizes, different canopies, different physiology), leads us to believe that the observations we have made and the mechanism we propose are robust. For a significant part, the displacement of sap is due, as with all fluid masses outside the Earth, to the forcing effect of terrestrial tides. RP 5 Discussion From a series of electrical SP data recorded on a Poplar tree in 2003 (see [16]), we propose to address the question of sap flow mechanism in trees. Today, for this mechanism, three sources are mentioned, which are not necessarily exclusive: capillarity, osmotic pressure, and evapotranspiration. We have discussed and critiqued the possibility of these mechanism in our introduction. In a recent study (see [33]), we were able to measure the extent to which gravitational forces acting on Earth, through variations in its rotation speed and tilt, can influence the growth rates of tree rings in a Tibetan juniper forest for over 1000 years by modulating the incoming solar radiation. The growth rates of tree rings are linked to photosynthesis and, therefore, are also related to the flow of sap circulating in the xylem and phloem, which transport nutrients throughout the plant. At first order, like all fluids inside and on the surface of the Earth, the movement of this sap, regardless of its density, must be partially or entirely driven by lunar-solar tides, especially during the periods (¡18.6 years) that are of particular interest in this paper. Gibert et al. ( [33]) emphasized that the sap flow and the recorded electrical signal were most likely linked by an electrokinetic phenomena. Therefore, in order to explain the movement of sap from a physics perspective, we are dealing here with a simple harmonic pumping phenomenon (see [17, 18]). As illustrated in Figure 2, the continuously recorded electrical signals in trees are non-stationary signals in the strict sense. This inherently prevents the direct use of a Fourier transform to determine the periodicities (i.e., lunar-solar tides) that may compose them. Furthermore, measurement vagary sometimes lead to interruptions in the recording system, resulting in discontinuous data. That is why we chose the Singular Spectrum Analysis (SSA) method to analyze the measurements of the Remungol poplar. This method helps limit possible errors in the interpretation of traditional spectra. In Figures 5 and 6, we presented the five main pseudo-periodicities extracted from the electrical potential data of the poplar. All of them correspond very precisely to periods of lunar-solar tides (P1, K1, S1, Mf , and K2, cf. our Table 1). The periods that we have extracted from the electrical signals, which serve as proxies for sap flow, are within 0.01% of the calculated lunar-solar tidal periods. This level of precision is remarkable and can typically only be achieved by expensive gravimeters costing tens of thousands of euros. The sum of the extracted components, which we believe to be tidal components, generally accounts for more than 70% of the total variance in the recorded electrical signals. If we examine in more detail the signals recorded in different areas of the Poplar tree (root, branch, trunk) for a given tide, we observe that the waveforms associated with these tides are not modulated in the same way. These amplitude modulations can be completely distinct from each other, as is the case for the tide and the electrodes E1, E8, E32, and E34 (cf. Figures 4). However, they can also be the same but phased in time. For example, we can observe that the patterns of the signals recorded by the electrodes E7 and E12 for the tide are the same but shifted by approximately 2 weeks (see Figures 5). Undoubtedly, the differences in amplitudes and phases observed in the recorded signals of the Poplar tree are likely to be related to its physiological response. Based on these observations, we embarked on a mathematical resolution of the equations associated with the phenomenon that we believe to be responsible for the sap flow, aiming to evaluate the electrokinetic coefficient associated with the Poplar tree, similar to what is done in geophysics for rocks (see equation 17). We obtained a value of 10 V/Pa, which is generally ten times higher than the values measured in rocks. However, it should be noted that trees are not rocks, as we can by the picture of the xylem geometry in Figure 01, sap flow is not disrupted by any tortuosity, unlike in rocks. Furthermore, it is important to emphasize that sap is not water. The question of the differences in amplitude modulations between different areas of the Poplar tree, as well as the question of the generality of our surprising results, led us to reproduce the same experiment in 2019 at the National Museum of Natural History, but this time on different tree species. As we mentioned, we have three oak trees, two hornbeams, one sequoia, one cypress, and a walnut tree. We have only presented the observations we record on the oak and hornbeam trees for two reasons. The first reason is that, we have multiple individuals of the same species, allowing us to make intra-species comparisons. The second reason is that, these five trees are in the same location, sharing the same reference electrode. This allows us to consider inter-species comparisons, given that the spreading of gravitational waves is such that all Parisian trees will be affected simultaneously and with the same amplitude by earth tides. As shown in Figures 4 and 7, the amplitudes and their variations differ significantly from one individual to another, which we could not estimate in the Remungol experiment. As we can see in Figure 13a and in more detail in Figure 7 (once the tide is extracted), on the one hand, the diurnal components of the three oak trees do not have the same electrical amplitudes (at the same electrode height), and on the other hand, longer-period oscillations appear on 2 of them (see Figure 4). We make the same observations for the hornbeams. However, what is confirmed by Figures 13c and 14c is that the components corresponding to the diurnal tide are perfectly in phase, from one individual to another, regardless of the species Therefore, our hypothesis of the lunar-solar tide as the main driving force of sap flow is confirmed both by the richness of the observations and their incredible precision. 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