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Vibration behavior of plantation-grown Scots pine trees in response to wind excitation
Agricultural and Forest Meteorology 150 (2010) 984–993
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Agricultural and Forest Meteorology journal homepage: www.elsevier.com/locate/agrformet
Vibration behavior of plantation-grown Scots pine trees in response to wind excitation D. Schindler a,∗ , R. Vogt b , H. Fugmann a , M. Rodriguez c,d , J. Schönborn a , H. Mayer a a
Meteorological Institute, Albert-Ludwigs-University Freiburg, 79085 Freiburg, Germany Department Geosciences, Institute for Meteorology, Climatology and Remote Sensing, University of Basel, 4010 Basel, Switzerland Department of Mechanics, LadHyX, Ecole Polytechnique-CNRS, 91128 Palaiseau, France d UMR547 PIAF, INRA, Univ Blaise Pascal, F-63100 Clermont Ferrand, France b c
a r t i c l e
i n f o
Article history: Received 24 September 2009 Received in revised form 11 March 2010 Accepted 16 March 2010 Keywords: Wind load Tree sway Modes of vibration Bi-orthogonal decomposition Pinus sylvestris
a b s t r a c t This paper deals with the dynamic responses of three plantation-grown Scots pine trees to turbulent wind loading. Although individual tree movement was complicated and irregular in the analyzed wind speed range (hourly mean wind speeds at canopy top below 4 m s−1 ), results from the cross-wavelet transform and wavelet coherence calculations demonstrate that wind-induced tree displacement strongly covaried with wind loads exerted on the trees by coherent structures. These wind loads caused largest tree displacements in the along wind direction as well as in the across wind direction. The absorption of energy available from wind loads in the range of the damped natural sway frequencies of the sample trees’ stems was of minor importance for tree displacement. The application of the bi-orthogonal decomposition provided insight into the resulting modal deformations of the sample trees’ stems. Based on measured stem displacement data, the mean mode shapes for the first eight stem vibration modes (four mean mode shapes in along wind direction and four mean mode shapes in across wind direction) could be identified. The shapes of these modes were quite similar to the normal mode shapes of a clamped-free beam. For the analyzed wind speed range it is shown that most of the energy absorbed by the trees from the wind is dissipated by tree sway in the first mode. Higher vibration modes localized on the sample trees’ stems were only of minor importance for the trees’ responses to wind. © 2010 Elsevier B.V. All rights reserved.
1. Introduction One important field in wind–tree interaction research is the analysis of the vibration behavior of conifers in response to wind loading. The analysis of their wind-induced vibrations was subject to numerous field (e.g. Holbo et al., 1980; Mayer, 1987; Gardiner, 1994, 1995; Peltola, 1996; Rudnicki et al., 2001; Schindler, 2008) and modeling studies (e.g. Gardiner, 1992; Kerzenmacher and Gardiner, 1998; Saunderson et al., 1999) as well as to studies combining both the experimental and the modeling approach (e.g. Flesch and Wilson, 1999; Spatz et al., 2007; Sellier et al., 2008). Continuously distributed elastic structures like trees will vibrate in resonance at certain characteristic frequencies when excited by the wind. Associated with these characteristic frequencies are characteristic forms (modes) of vibration amplitude. Under free vibrations situations these forms are called normal modes. Mode characteristics are functions of mass and stiffness distributions
∗ Corresponding author at: Meteorological Institute, University of Freiburg, D79085 Freiburg, Germany. Tel.: +49 761 2033588; fax: +49 761 2033586. E-mail address: [email protected] (D. Schindler). 0168-1923/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2010.03.003
within the structure, and may be excited by temporal and spatial variations of the wind load (Simiu and Scanlan, 1996; Thomson and Dahleh, 1998). In previous studies it was shown that monopodial conifers respond well to turbulent wind loading with stem vibrations in the range of the damped natural sway frequency of their fundamental mode (Holbo et al., 1980; Mayer, 1987; Gardiner, 1992, 1994; Peltola et al., 1993; Peltola, 1996; Flesch and Wilson, 1999; Schindler, 2008). It is also known that bending vibrations of the stem of coniferous forest trees are the dominating dynamic response to wind excitation (Mayer, 1987). The amplitude of these vibrations is a function of the wind load. That implies that wind loads surpassing the wind load resistance of the stem of conifers cause fatal wind damage. For example, Schelhaas et al. (2003) demonstrated that in the period 1950–2000 storms accounted for the largest part of total damage in European forests among the abiotic disturbances. Quine and Gardiner (2007) wrote a comprehensive summary on implications of wind damage on forest ecology and forestry. Most of the abovementioned studies focused on the response of the stem to wind excitation in the fundamental mode. There are only a few studies (e.g. Sellier and Fourcaud, 2005; Sellier
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et al., 2006, 2008; Spatz et al., 2007; Moore and Maguire, 2008; Rodriguez et al., 2008) in which higher order vibration of the stem and maybe the branches of coniferous trees was analyzed. In these studies it was shown that multimodal vibration of the aerial parts of trees controls their response to some external excitation to a great extent. For individual trees of a number of different species James et al. (2006) showed that the dynamic sway of the branches greatly influences whole tree sway. There are a few further studies (Spatz et al., 2007; Moore and Maguire, 2008; Rodriguez et al., 2008) that showed that structural damping or multiple resonance damping is an effective component of whole tree movement damping. Further processes known to counter wind-induced tree displacement are aerodynamic drag of the crowns (Mayhead, 1973; Amiro, 1990; Rudnicki et al., 2004; Sellier and Fourcaud, 2005) and viscous damping of the wood (Milne, 1991; Wood, 1995). Within forest stands friction between neighboring trees is a source of tree movement damping (Milne, 1991; Rudnicki et al., 2008). de Langre (2008) recently wrote a comprehensive review on mechanical interactions between wind and plants. Since there are only few quantitative results for higher order vibration of forest-grown coniferous trees in response to wind excitation, the objectives of this study were to analyze (1) the windinduced dynamic sway behavior of three Scots pine trees with a focus on their response to intermittent, coherent structures, (2) the characteristics of their stem vibration modes. The relevance of coherent structures and vibration modes for wind-induced tree displacement is then discussed.
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Fig. 1. Study area and spatial arrangement of the ten (T1–T10) adjacent sample trees. The black circles indicate the mean crown radii of the sample trees at zeroplane displacement height zd (z/h = 0.78).
2. Materials and methods 2.1. Measurement site The study area and the measurement setup described hereafter correspond to the study area and measurement setup presented in Schindler (2008). The study area was located about 25 km southwest of Freiburg (Southwest Germany) in the flat southern Upper Rhine Valley (47◦ 56 04 N, 7◦ 36 02 E, altitude range above sea level: 198–202 m) within a planted Scots pine (Pinus sylvestris L.) forest (Fig. 1). Wind-induced stem displacement of ten (T1–T10) adjacent Scots pine trees (inset in Fig. 1) as well as near-surface airflow properties were monitored from October 2005 to April 2006. During the measurement period the Scots pine forest at the measurement site had a mean stand density of 800 trees ha−1 and a mean stand height (h) of 14.5 m. The mean stem diameter at breast height (dbh) was 17.5 cm and the plant area index (PAI) was 1.9. Based on wind speed measurements (Robinson, 1961), the zero-plane displacement height (zd ) representative for the Scots pine forest at the measurement site was estimated to be 11.3 m (z/h = 0.78). In this paper results from the responses of the three sample trees T1, T2, and T5, which responded well to the measured airflow characteristics, are presented. The heights (ht) of T1, T2, and T5 were 14.6 m, 15.2 m, and 15.1 m. Further characteristics of the samples trees are given in Table 1. Fig. 2 provides schematic side views (to improve clarity the x-axis was doubled) of the mean shapes of T1, T2, and T5. These illustrate that the crowns were rather short and located above z/h = 0.5. As a rule of thumb, the crowns’ mean maxi-
Fig. 2. Schematic side views (to improve clarity the x-axis was doubled) of the mean shapes of the three sample trees (a) T1, (b) T2, and (c) T5.
mum lateral extent was less than 10% of the respective sample tree height. 2.2. Airflow and stem displacement measurements Eight ultrasonic anemometers (USA-1, METEK, Germany) mounted on a 28 m high triangular lattice tower were used to measure (sampling rate: 20 Hz) airflow characteristics within and above the Scots pine forest (Schindler, 2008). Wind velocity was measured at 2.2 m (z1 /h = 0.15), 7.3 m (z2 /h = 0.5), 8.7 m (z3 /h = 0.6), 10.9 m (z4 /h = 0.75), 13.1 m (z5 /h = 0.9), 15.2 m (z6 /h = 1.05), 21.8 m
Table 1 Characteristics of the three sample trees T1, T2, and T5. Tree
Tree height ht (m)
Diameter 1.3 m dbh (cm)
Slenderness ht/dbh
Crown projected area a (m2 )
Largest mean crown radius cr (m)
Crown length l (m)
T1 T2 T5
14.6 15.2 15.1
15.9 18.3 19.2
92 83 79
5.8 7.1 6.3
1.4 1.5 1.2
5.7 6.1 6.3
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From WnAB (c) the smoothed global cross-wavelet spectrum was obtained as (Torrence and Webster, 1999): 1 AB Wn (c) N N−1
GCWS(c) =
(2)
n=0
where N is the number of data points. GCWS is a measure of common power between two time series. Furthermore, the wavelet-squared coherence Rn2 (‘wavelet coherency’) was calculated according to Torrence and Webster (1999): Fig. 3. Deformation patterns associated with the first four normal modes (M1 –M4 ) of a clamped-free beam, hc1 = 1/7ht, hc2 = 3/7ht, and hc3 = 5/7ht are the clinometer mounting heights, ht is the respective sample tree height.
(z7 /h = 1.5), and 26.8 m (z8 /h = 1.85). Based on the measured wind velocity series, mean wind speed at canopy height (z/h = 1.0) Uh was approximated by cubic spline interpolation. The measurement tower was set up in a small canopy gap (mean gap diameter: 6 m) to the east of the sample trees. The Institute of Meteorology, Climatology and Remote Sensing of the University of Basel provided the software (author: Andreas Christen) used to store airflow data on a commercial PC. In order to determine stem displacement in response to wind loading, three biaxial clinometers (model 902-45, Applied Geomechanics, USA) were mounted on each of the three sample trees. The mounting heights of the clinometers corresponded to hc1 = 1/7ht, hc2 = 3/7ht, and hc3 = 5/7ht. These mounting heights were assumed to be approximately the heights of the antinodal points of vibration of the 4th normal mode of a clamped-free beam. Fig. 3 illustrates the deformation patterns associated with the first four normal modes (M1 –M4 ) of a clamped-free beam (Simiu and Scanlan, 1996). The clinometers measured the angular position of the sample trees’ stems and were sampled at a frequency of 10 Hz by a CR5000 micrologger (Campbell Scientific, USA). In order to minimize sensor heating, all clinometers were pointed northward. Schindler (2008) described the conversion of angular position (deg) measurements into stem displacement (m) in x- (dx ) and y-direction (dy ) along the stem. Displacement of the sample trees between z = 0 m to the top height was estimated for 0.2 m intervals using cubic spline interpolation. A wavelet filter was applied to reduce instrument noise in the clinometer signals.
Rn2 (c) =
|c −1 WnAB (c)|2 −1 c |WnA (c)|2 c −1 |WnB (c)|2
(3)
where · indicates smoothing in both time and scale. Rn2 provides a measure of the intensity (0 ≤ Rn2 ≤ 1) of covariance of A and B (Jevrejeva et al., 2003). Rn2 (c) was used to calculate the global wavelet coherence spectrum GWCS (Torrence and Webster, 1999): GWCS(c) =
Rn2 (c)
(4)
n
All wavelet calculations were realized with wavelet software provided by C. Torrence and G. Compo. The software is available at URL: http://atoc.colorado.edu/research/wavelets/. 2.4. Bi-orthogonal decomposition The bi-orthogonal decomposition (BOD) was used to capture and to analyze the main modal features of the sample trees’ oscillations using the stem displacement records. Basically, the BOD can be used to decompose a m-degree of freedom signal into m modes in the time-space domain (Aubry et al., 1991; Hémon and Santi, 2003). In order to apply the BOD to the stem displacement data, the matrix D = [dx0 , . . ., dx15 , dy0 , . . ., dy15 ], which contained hourly time series of dx and dy from z = 0 m to the top height of a sample tree, was defined. For each sample tree, stem displacement values in xand y-direction were used from 16 heights (1 m segments). D was decomposed according to D=
∞ √
˛k k k
(5)
k=1
2.3. Wavelet analysis In order to analyze the intermittent nature of large, coherent eddies (Finnigan, 2000) and their effects on displacement of the sample trees, a wavelet analysis (e.g. Torrence and Compo, 1998; Jevrejeva et al., 2003; Grinsted et al., 2004) was carried out. The wavelet analysis makes it possible to expand time series into time frequency space and detect localized, intermittent structures (Grinsted et al., 2004). The covariance of time series of airflow and stem displacement was studied by smoothed cross-wavelet spectra calculated as (Torrence and Compo, 1998): WnAB (c) = WnA (c)WnB∗ (c)
(1)
where A and B are time series, WnA and WnB are their wavelet transforms (based on the Morlet wavelet), n is the time index, c is the wavelet scale, and the asterisk denotes complex conjugate. WnAB (c) was calculated for 200 scales which ranged up to 100 s. The wavelet scale was converted to the equivalent Fourier period, which is for the Morlet wavelet almost equal to the equivalent Fourier period (Torrence and Compo, 1998).
where k is the mode number, k are temporal functions (chronos), k are spatial functions (topos), and the weight factor of the struc√ tures (k , k ) is the real scalar value ˛k . k and k form a set of normalized orthogonal functions. Each BOD mode contains information on the time characteristics and on the space characteristics of the stem displacement data. The eigenvalue ˛k is a measure of the relative signal energy contained in the kth BOD mode. It was used to rank the spatio-temporal modes in descending order. Referring to Aubry et al. (1991), k are the eigenfunctions of the spatial correlation operator of D with ˛k , k are the eigenfunctions of the temporal correlation operator of D with ˛k . Since stem displacement was recorded discretely, vectors replaced the temporal and spatial functions. Correlation matrices replaced the correlation operators. The spatial correlation matrix of the signal was computed as S = DT D
(6)
and topos were determined from the following eigenvalue problem: S = ˛
(7)
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Fig. 4. Mean bivariate histograms of the sample trees’ responses to wind loading at zero-plane displacement height zd (z/h = 0.78) in the four wind speed classes Uh1 (0 ≤ Uh ≤ 1 m s−1 (a)), Uh2 (1 < Uh ≤ 2 m s−1 (b)), Uh3 (2 < Uh ≤ 3 m s−1 (c)), Uh4 (Uh > 3 m s−1 (d)), Uh being the wind speed at canopy height (m s−1 ). The small black rectangle on the right side of the upper left box indicates the position of the sonic anemometer profile mounted to the measurement tower.
Chronos were derived from projecting the bi-orthogonal decomposition on topos and taking advantage of the orthogonality property of the modes: 1 k = √ Dk ˛k
(8)
A more detailed description of the computational details associated with the application of the BOD can be found in Py et al. (2005, 2006). 3. Results and discussion 3.1. Bivariate histograms Hourly runs of tree response data were analyzed for seven windy days (December 16, 2005; December 19, 2005; December 31, 2005; February 8, 2006; February 18, 2006; March 3, 2006; March 9, 2006) with near-neutral stability. Atmospheric stability was quantified by the absolute hourly values of the stability index (z − zd )/L (L is the Obukhov-length). Altogether 151 h for which (z − zd )/L was equal or less than 0.05 were available for the analysis of tree responses to wind loading. Hourly values of mean wind speed at canopy top ranged from 0.4 m s−1 to close to 4.0 m s−1 . On all other days during the measurement campaign wind speed was too low to excite pronounced tree responses. Since the sample trees’ responses on wind excitation are a function of Uh (Mayer, 1987; Gardiner, 1995), the hourly mean values of Uh were divided into the four wind speed classes Uh1 (0 ≤ Uh ≤ 1 m s−1 ; n = 13), Uh2 (1 < Uh ≤ 2 m s−1 ; n = 45), Uh3 (2 < Uh ≤ 3 m s−1 ; n = 60), and Uh4 (Uh > 3 m s−1 ; n = 33), n being the number of hourly values. During the chosen hours wind was blowing predominantly from
south to southwest. The prevailing wind directions in Uh1 –Uh4 were 210◦ ± 41◦ (mean ± standard deviation), 189◦ ± 25◦ , 193◦ ± 12◦ , and 207◦ ± 10◦ . In order to get an impression on the wind-induced movement of the sample trees, a top view of the sample trees’ responses to wind loading at zd is given in Fig. 4. For each sample tree, mean twodimensional density functions were calculated for each of the wind speed classes Uh1 –Uh4 . It is noticeable that mean tree displacement increased from Uh1 (Fig. 4a) to Uh4 (Fig. 4d) in x-direction as well as in y-direction. The bivariate histograms are of near-circular shape without a clearly dominating major axis along the prevailing wind direction. Stem displacement in the along wind direction did not far exceed stem displacement in the across wind direction. With increasing wind speed absolute values of stem displacement progressively differed between the sample trees. T1 showed greatest stem displacements, which might be the result of additional space due to the small canopy gap east of T1 where the measurement tower was set up. One reason for the near-circular shapes of the bivariate histograms were phase shifts between dx and dy which prevented tree sway back and forth along characteristic lines. Instead, the sample trees mostly swayed on irregular orbits, which formed the mean shape of the bivariate histograms. For the wind speed range comparable to the wind speed range observed in this study complicated tree response to wind loading is also known from Amtmann (1986) and Peltola (1996). For mean wind speed at canopy top between 5.0 m s−1 and 7.9 m s−1 the sway patterns of ten adjacent lodge pine trees presented by Rudnicki et al. (2003) were also circular and elliptical in shape. In contrast, more regular tree response to wind loading is known from Gardiner (1995). His results demonstrate that the along wind displacement of two
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Fig. 5. Normalized global cross-wavelet spectra (GCWS) calculated between (a) fluctuations of the streamwise wind vector component u and the fluctuations of the stem displacement component in y-direction dy , (b) u and the fluctuations of the stem displacement component in x-direction dx , (c) fluctuations of the across wind vector component v and dy , (d) v and dx over the hourly run from 13:00 to 14:00 CET on March 3, 2006.
Sitka spruce trees clearly exceeded the across wind displacement. Despite the rather symmetric mean two-dimensional density functions, in the following paragraphs dy will be treated as the stem displacement component approximately aligned in the along wind direction. dx is the across wind stem displacement component. 3.2. Wavelet analysis Wavelet analysis was carried out for the hourly run from 13:00 to 14:00 CET on March 3, 2006. During this run, mean wind speed
at canopy top Uh equaled to 3.7 m s−1 , which was one of the highest hourly mean wind speed values observed during the measurement period. In Fig. 5 normalized (by their maxima in the examined range of scales) results from GCWS-calculations between the fluctuations of the streamwise (u ), the crossflow wind vector component (v ) (u and v calculated from the airflow measurements at z6 /h = 1.05), and the fluctuations of stem displacement components (dx , dy ) (dx and dy calculated from the stem displacement at zd ) of T1, T2, and T5 are presented. The GCWS-calculations including u and dy (Fig. 5a) revealed two distinct peaks in common power. One peak in the range of the natural sway periods of the sample trees (<4 s), another
Fig. 6. Global wavelet coherence spectra (GWCS) calculated between (a) fluctuations of the streamwise wind vector component u and the fluctuations of the stem displacement component in y-direction dy , (b) u and the fluctuations of the stem displacement component in x-direction dx , (c) fluctuations of the across wind vector component v and dy , (d) v and dx over the hourly run from 13:00 to 14:00 CET on March 3, 2006.
D. Schindler et al. / Agricultural and Forest Meteorology 150 (2010) 984–993
peak at 47.2 s. The equivalent Fourier frequency associated with the second peak is 0.02 Hz. The results from the GCWS-calculations confirm the results from the dy -wavelet variance spectra calculations reported by Schindler (2008) in a quantitative way. He qualitatively showed the coincidence of peaks in u - and dy -wavelet variance spectra. The GCWS-peaks in the range of the natural sway periods of the sample trees result from swaying back and forth along the ydirection. The GCWS-peaks at 47.2 s are associated with the arrival of coherent structures in the streamwise direction. Similar results were obtained for u and dx (Fig. 5b). The global cross-wavelet spectra calculated for T1, T2, and T5 also show distinct peaks in two very similar ranges as were observed for u and dy . The first sharp peaks reflect the duration of the natural sway periods of the sample trees in x-direction. The second broader peaks are also the consequence of excitation of the sample trees by wind loads related to coherent structures. The periods of these peaks were a little shorter (T1, T2: 42.6 s; T5: 44.2 s) than the period calculated from the time series of u and dy . The results from GCWS-calculations between v and fluctuations of stem displacement in x- and y-direction (Fig. 5c, d) demonstrate that there was high common power only in the range of the natural sway periods of the sample trees. Due to the absence of larger scale organized structures from the across wind direction (Finnigan, 2000), the global cross-wavelet spectra did not show further pronounced peaks in common power at periods longer than the natural sway periods of the sample trees. In contrast to that, global wavelet coherence spectra (GWCS) did not show peaks in the range of periods that corresponded to the natural sway periods of the sample trees (Fig. 6), meaning that the localized covariation in time frequency space between wind loading and tree response was rather weak. GWCS calculated between u and dy (Fig. 6a) as well as GWCS calculated between u and dx (Fig. 6b) showed distinct maxima (larger than 0.5) at periods greater than 40 s. This provides evidence that wind loads associated with coherent structures, along wind stem displacement, but also across wind stem displacement covaried in this range of periods. GWCS calculated between v and stem displacement in y(Fig. 6c) and x-direction (Fig. 6d) was generally lower than corresponding GWCS in the along wind direction. Although GWCS of v and dx was at a maximum at 33.8 s for all sample trees, the absolute values of these GWCS-maxima were clearly lower than the GWCSmaxima between u and dx as well as between u and dy . Over all Fourier periods wavelet coherence between v and dy was low which means that their covariation was small. Based on the obtained wavelet results the mean spacing and the occurrence frequency of coherent structures in the streamwise direction were analyzed. For that, mean streamwise coherent structure separation (u ) estimated from the u time series and normalized by canopy height h was calculated according to (e.g. Raupach et al., 1996): TUc u = h Nh
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Although the sample trees responded well to turbulent wind loading in the range of their fundamental sway frequency, the results from the wavelet analyses demonstrate that wind loads associated with the arrival of coherent structures in the streamwise direction showed the strongest covariation with stem displacement. These wind loads not only strongly covaried with stem displacement in the along wind direction but also with stem displacement in the across wind direction. The sample trees’ responses in the range of their fundamental sway frequency seem to be a result from their damped harmonic oscillator-like vibration behavior as stated by Gardiner (1995). He presented similar results for the vibration behavior of spruce trees and stated that trees do not resonate with turbulent wind components in the range of their natural sway frequency but strongly react on impulsive wind loading during the passage of gusts. This leads to the conclusion that energy absorption from turbulent wind loading in the frequency range close to the fundamental stem sway frequency (f1 ) is of minor importance in the analyzed range of wind speeds. 3.3. Fourier analysis In order to determine the frequencies that are associated with the vibrational modes of the sample trees, the Fourier transform was applied to the fluctuations of stem displacement in y- and x-direction. Periodicities in tree response were examined in the fre-
(9)
where T is the total time of the data series in seconds, Uc is the convection velocity (assumption that Uc = 1.8Uh ; Raupach et al., 1996), N is the number of coherent structures detected by the Mexican hat wavelet detection function. Schindler (2008) reported N = 73 for the processed hourly run which resulted in u /h = 22.7. This value is outside the cloud of u /h data points presented by Brunet and Irvine (2000) when compared to w /h (mean streamwise coherent structure separation estimated from the vertical wind vector component w). For the analyzed hourly run u /w equaled to 6.5 whereas the mean ratio u /w (slope of a regression line) given by Brunet and Irvine (2000) equaled to 3.07 (r2 = 0.51, r2 being the coefficient of determination).
Fig. 7. Energy spectra (fSdy (f)) of the fluctuations of along wind stem displacement dy of the sample trees T1, T2, and T5 at the clinometer mounting heights (a) hc3 , (b) hc2 , and (c) hc1 over hourly runs assigned to Uh4 (Uh > 3 m s−1 ).
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D. Schindler et al. / Agricultural and Forest Meteorology 150 (2010) 984–993 Table 2 Frequencies of the first (f1 ) to the fourth (f4 ) peak in Fourier energy spectra of fluctuations of stem displacement d in y- (fSdy (f)) and x-direction (fSdx (f)) that are associated with the first (M1 ), second (M2 ), third (M3 ), and fourth (M4 ) vibrational mode of the sample trees T1, T2, and T5. Frequency (Hz)
f1 f2 f3 f4
dy
dx
T1
T2
T5
T1
T2
T5
0.29 0.99 1.77 3.32
0.34 0.99 2.12 3.63
0.34 0.99 2.12 3.80
0.31 1.03 1.85 2.77
0.34 0.99 2.12 3.63
0.34 1.08 2.03 3.63
cided with the values of f1 –f4 in x-direction. For T1 and T5 they differed to some extent. With the exception of fSdx (f) calculated at hc1 , the shapes of the Fourier spectra calculated based on the dy - and dx -series were quite similar. Due to small fluctuations in stem displacement in x-direction at hc1 , instrument noise might have prevented a drop in spectral energy of dx at higher frequencies. Since in this study only stem displacement was measured, it can only be speculated on the sources of sway frequency differences in the two stem displacement directions as well as on the sources of higher order stem vibration. As already mentioned, it is known that vibration mode characteristics are functions of mass and stiffness distributions within a structure, and may be excited by temporal and spatial variations of the wind load. For young maritime pines Sellier and Fourcaud (2005) explained anisotropic sway frequency differences by morphological differences of their sample trees.
Fig. 8. Energy spectra (fSdx (f)) of the fluctuations of across wind stem displacement dx of the sample trees T1, T2, and T5 at the clinometer mounting heights (a) hc3 , (b) hc2 , and (c) hc1 over hourly runs assigned to Uh4 (Uh > 3 m s−1 ).
quency domain by the Fourier energy spectra fSdy (f), and fSdx (f), S being the spectral energy, f is the frequency. In Figs. 7 and 8 mean log fSdy (f) and mean log fSdx (f) calculated at hc3 (Figs. 7a and 8a), hc2 (Figs. 7b and 8b), and hc1 (Figs. 7c and 8c) in Uh4 are plotted versus log f. Only the displacement series from Uh4 were chosen for this analysis because Uh4 encompassed the wind speed range for which the sample trees’ dynamic responses to wind loading were most pronounced. Several characteristic spectral peaks are present in mean log fSdy (f) and mean log fSdx (f). The most prominent peaks appear in the frequency range between 0.25 and 0.40 Hz. The frequencies associated with these peaks were the same frequencies as obtained by means of the wavelet analysis. They were interpreted as the damped natural sway frequencies (f1 ) associated with the first vibrational modes (M1 ) of the sample trees’ stems. With increasing frequency three further minor peaks could be identified. These peaks indicate higher mode vibration and were interpreted as the frequencies f2 , f3 , and f4 associated with the second (M2 ), third (M3 ), and fourth (M4 ) vibration modes of the stems. Since the same number of spectral peaks was not clearly identifiable at all displacement measurement heights, the values of f1 –f4 were determined from those displacement spectra in which they were evident. Their values are given in Table 2. The values of f1 are proportional to dbh/ht2 and close to the f1 -values that are predicted for pines when using the formula f1 = 0.0948 + 2.65519·dbh/ht2 given by Moore and Maguire (2004). For T2 the values of f1 –f4 in y-direction coin-
Fig. 9. Mean shapes of the first four stem vibration modes (M1 –M4 ) of (a) T1, (b) T2, and (c) T5 obtained from stem displacement measurements in y-direction. The amplitude is arbitrarily scaled.
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3.4. Bi-orthogonal decomposition The application of the bi-orthogonal decomposition enabled the extraction of characteristic mode shapes associated with f1 –f4 . For each of the peaks (four in along wind direction and four in across wind direction) in the presented Fourier displacement spectra, modal forms of vibration amplitude distributed along the stems of the sample trees could be obtained. Over the whole analyzed wind speed range, M1 and M2 showed almost exactly the same modal shapes in y- and x-direction. Because the characteristics of M3 and M4 got better with increasing Uh , mean BOD-mode shapes are presented for hourly mean wind speeds contained in Uh4 . Figs. 9 and 10 show the mean shapes of M1 –M4 of T1 (Figs. 9a and 10a), T2 (Figs. 9b and 10b), and T5 (Figs. 9c and 10c) resulting from wind-induced stem displacement in y- and x-direction. Since topos are eigenvectors they are shown with arbitrary amplitude. In both stem displacement directions, the results for M1 –M4 are very similar among the sample trees. The shapes of M1 and M2 show reasonable agreement with the mode shapes presented by Thomson and Dahleh (1998) or Hodges and Pierce (2006) for a clamped-free beam. However, the shapes of M3 and M4 differ somewhat from the mode shapes of a clamped-free beam: M3 has less whereas M4 has more deformation at the stem base than expected. The deformations of M1 and M2 are also consistent with the shapes of the first (I) and second (I ) bending modes of the stem of a young maritime pine presented by Rodriguez et al. (2008). The graphs in Fig. 11 illustrate the sums of relative energy contained in M1 for both stem displacement directions as a function of Uh . Even at low wind speeds (Uh < 1 m s−1 ), relative energy conFig. 11. Signal energy (%) contained in the first vibration mode (M1 ), for (a) displacement of the sample trees T1, T2, and T5 in y-direction (dy ) and (b) displacement of T1, T2, and T5 in x-direction (dx ) as a function of wind speed at canopy top (Uh ).
Fig. 10. Mean shapes of the first four stem vibration modes (M1 –M4 ) of (a) T1, (b) T2, and (c) T5 obtained from stem displacement measurements in x-direction. The amplitude is arbitrarily scaled.
tained in M1 dominated the stem displacement signal of T1, T2, and T5. The relative energy content of M1 was at least 89% in ydirection (Fig. 11a) and at least 60% (Fig. 11b) in x-direction over the presented wind speed range. At lower wind speed there were a few hourly intervals, in which the relative signal energy contained in M1 was comparatively low. If this was the case, relative signal energy contained in M2 increased. Hence, the sums of relative energy contained in M1 and M2 amounted to at least 97% over all analyzed hourly runs. The large amount of relative energy contained in M1 is consistent with the results from the Fourier analysis and findings from previous studies (e.g. Holbo et al., 1980; Mayer, 1987). M1 normally dominates the response of monopodial conifer trees to dynamic wind loading. The relative signal energy contents of M3 and M4 were very low (<3%) and started to fall back to zero with increasing wind speed. These results imply that beginning with the lowest wind speeds, higher order stem vibration (especially in along wind direction) was not an important mechanism for the dissipation of energy of the near-surface airflow transferred into tree movement. In Fig. 12 the ratio q of the eigenvalues of M1 in the x-direction to the eigenvalues of M1 in the y-direction is presented as a function of Uh . At the lowest wind speeds there is some scatter in the plot. Especially for T1, relative energy contained in M1 of dy was greater during a few hours than relative energy contained in M1 of dx (q < 1). Yet, during a few hours, q was greater than one for all sample trees. Beginning with Uh = 2.5 m s−1 all hourly q-values were close to one. The anisotropic signal energy contents of M1 converged with increasing wind speed. Since the effects of the soil and the root system on tree sway dynamics were not investigated in this study, the question of their relevance for the vibration behavior of the sample trees cannot be answered. Nevertheless, it is assumed that in the analyzed wind
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ther field studies is also seen in the field of tree–tree interactions in high winds. For lodgepole pine trees Rudnicki et al. (2008) showed that at higher wind speeds damping due to crown collisions influences tree sway dynamics. They reported a decrease in fundamental sway frequency through dissipation of kinetic energy when adjacent trees experienced intense crown collisions. This implies that – as has recently been done by Py et al. (2005, 2006) for crop canopies – future field studies should also focus on the response behavior of large groups of neighboring forest trees. The analysis of their interactions is crucial for an understanding of crown collision dynamics and energy dissipation within forest trees especially when it comes to the reduction of wind-induced damage. Acknowledgements
Fig. 12. Ratio q of the eigenvalues of the first mode of stem displacement (M1 ) in x-direction to eigenvalues of M1 in y-direction of the sample trees T1, T2, and T5 as a function of wind speed at canopy top (Uh ).
The German Research Foundation (DFG SCHI 868/1) and the Department of the Environment of the federal state of BadenWuerttemberg (RESTER-UniFR) supported this work. References
speed range and under the prevailing soil water conditions (soil water content clearly below field capacity) their effect on tree vibrations was of minor importance. 4. Conclusions The results from the wavelet analyses suggest that stem displacement and wind loading exerted by coherent structures strongly covaried whereas covariation was weak in the range of the fundamental sway period of the sample trees’ stems. From this, it is deduced that wind-induced displacement of the Scots pines was primarily a function of impulsive wind loads associated with coherent structures. This argumentation is consistent with the conclusions of Gardiner (1995) and Schindler (2008) who stated that the periodicity of coherent structure arrival is more important for large coniferous forest tree displacements than the energy available in the range of the damped natural sway frequency of trees. Moreover, in the analyzed hourly run the arrival frequency of coherent structures was far off the damped natural sway frequencies of the samples tree. From this, it is concluded that no frequency lock-in mechanism, through which the sample trees and the wind fluctuations interact in the range of the trees’ natural sway frequencies, significantly altered the flow dynamics at the measurement site. Py et al. (2006) recently reported a frequency lock-in mechanism of coherent structures on plant motion of crop (alfalfa, wheat) canopies. They showed that wind-induced coherent motion of the crop canopies occurred near the natural vibration frequency of the crop plants. By means of the bi-orthogonal decomposition it was pointed out that even at very low wind speeds most of the stem displacement signal energy was contained in the first stem vibration mode, especially in the along wind direction. The resistance to wind loading of the slender plantation-grown sample trees with their small, short crowns relies mainly on sway of their stems in the first mode. How multiple resonance damping and damping effects of the foliage contribute to the damping of stem vibration needs to be further investigated, for example, by conducting studies in which windinduced branch dynamics are also monitored. Another question that needs to be addressed in the future relates to the response behavior of forest trees to (potentially) damagecausing wind loads. So far not much is known about wind–tree interactions under strong wind conditions. At higher wind speeds (higher than measured in this study) when impulsive wind loading becomes more frequent and possibly in phase with tree motion, resonance between impulsive wind loading and tree motion might enhance the increasing wind load (Gardiner, 1995). Need for fur-
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D. Schindler et al. / Agricultural and Forest Meteorology 150 (2010) 984–993 son, E.A., Miyanishi, K. (Eds.), Plant Disturbance Ecology—The Process and the Response. Elsevier, Amsterdam, pp. 103–155. Raupach, M.R., Finnigan, J.J., Brunet, Y., 1996. Coherent eddies and turbulence in vegetation canopies: the mixing-layer analogy. Bound. -Layer Meteor. 78, 351–382. Robinson, S.M., 1961. A method of machine computation of wind profile parameters. In: Lettau, H. (Ed.), Studies of the Three-Dimensional Structure of the Planetary Boundary-layer. University of Wisconsin, Madison, pp. 63–70. Rodriguez, M., de Langre, E., Mulia, B., 2008. A scaling law for the effect of architecture and allometry on tree vibration modes suggests a biological tuning to modal compartmentalization. Am. J. Bot. 95, 1523–1537. Rudnicki, M., Lieffers, V.J., Silins, U., 2003. Stand structure governs the crown collisions of lodgepole pine. Can. J. For. Res. 33, 1238–1244. Rudnicki, M., Mitchell, S.J., Novak, M.D., 2004. Wind tunnel measurements of crown streamlining and drag relationships for three conifer species. Can. J. For. Res. 34, 666–676. Rudnicki, M., Silins, U., Lieffers, V.J., Josi, G., 2001. Measure of simultaneous tree sways and estimation of crown interactions among a group of trees. Trees 15, 83–90. Rudnicki, M., Meyer, T.H., Lieffers, V.J., Silins, U., Webb, V.A., 2008. The periodic motion of lodgepole pine trees as affected by collisions with neighbors. Trees 22, 475–482. Saunderson, S.E.T., England, A.H., Baker, C.J., 1999. A dynamic model of the behavior of Sitka spruce in high winds. J. Theor. Biol. 200, 249–259.
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Agricultural and Forest Meteorology journal homepage: www.elsevier.com/locate/agrformet
Vibration behavior of plantation-grown Scots pine trees in response to wind excitation D. Schindler a,∗ , R. Vogt b , H. Fugmann a , M. Rodriguez c,d , J. Schönborn a , H. Mayer a a
Meteorological Institute, Albert-Ludwigs-University Freiburg, 79085 Freiburg, Germany Department Geosciences, Institute for Meteorology, Climatology and Remote Sensing, University of Basel, 4010 Basel, Switzerland Department of Mechanics, LadHyX, Ecole Polytechnique-CNRS, 91128 Palaiseau, France d UMR547 PIAF, INRA, Univ Blaise Pascal, F-63100 Clermont Ferrand, France b c
a r t i c l e
i n f o
Article history: Received 24 September 2009 Received in revised form 11 March 2010 Accepted 16 March 2010 Keywords: Wind load Tree sway Modes of vibration Bi-orthogonal decomposition Pinus sylvestris
a b s t r a c t This paper deals with the dynamic responses of three plantation-grown Scots pine trees to turbulent wind loading. Although individual tree movement was complicated and irregular in the analyzed wind speed range (hourly mean wind speeds at canopy top below 4 m s−1 ), results from the cross-wavelet transform and wavelet coherence calculations demonstrate that wind-induced tree displacement strongly covaried with wind loads exerted on the trees by coherent structures. These wind loads caused largest tree displacements in the along wind direction as well as in the across wind direction. The absorption of energy available from wind loads in the range of the damped natural sway frequencies of the sample trees’ stems was of minor importance for tree displacement. The application of the bi-orthogonal decomposition provided insight into the resulting modal deformations of the sample trees’ stems. Based on measured stem displacement data, the mean mode shapes for the first eight stem vibration modes (four mean mode shapes in along wind direction and four mean mode shapes in across wind direction) could be identified. The shapes of these modes were quite similar to the normal mode shapes of a clamped-free beam. For the analyzed wind speed range it is shown that most of the energy absorbed by the trees from the wind is dissipated by tree sway in the first mode. Higher vibration modes localized on the sample trees’ stems were only of minor importance for the trees’ responses to wind. © 2010 Elsevier B.V. All rights reserved.
1. Introduction One important field in wind–tree interaction research is the analysis of the vibration behavior of conifers in response to wind loading. The analysis of their wind-induced vibrations was subject to numerous field (e.g. Holbo et al., 1980; Mayer, 1987; Gardiner, 1994, 1995; Peltola, 1996; Rudnicki et al., 2001; Schindler, 2008) and modeling studies (e.g. Gardiner, 1992; Kerzenmacher and Gardiner, 1998; Saunderson et al., 1999) as well as to studies combining both the experimental and the modeling approach (e.g. Flesch and Wilson, 1999; Spatz et al., 2007; Sellier et al., 2008). Continuously distributed elastic structures like trees will vibrate in resonance at certain characteristic frequencies when excited by the wind. Associated with these characteristic frequencies are characteristic forms (modes) of vibration amplitude. Under free vibrations situations these forms are called normal modes. Mode characteristics are functions of mass and stiffness distributions
∗ Corresponding author at: Meteorological Institute, University of Freiburg, D79085 Freiburg, Germany. Tel.: +49 761 2033588; fax: +49 761 2033586. E-mail address: [email protected] (D. Schindler). 0168-1923/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2010.03.003
within the structure, and may be excited by temporal and spatial variations of the wind load (Simiu and Scanlan, 1996; Thomson and Dahleh, 1998). In previous studies it was shown that monopodial conifers respond well to turbulent wind loading with stem vibrations in the range of the damped natural sway frequency of their fundamental mode (Holbo et al., 1980; Mayer, 1987; Gardiner, 1992, 1994; Peltola et al., 1993; Peltola, 1996; Flesch and Wilson, 1999; Schindler, 2008). It is also known that bending vibrations of the stem of coniferous forest trees are the dominating dynamic response to wind excitation (Mayer, 1987). The amplitude of these vibrations is a function of the wind load. That implies that wind loads surpassing the wind load resistance of the stem of conifers cause fatal wind damage. For example, Schelhaas et al. (2003) demonstrated that in the period 1950–2000 storms accounted for the largest part of total damage in European forests among the abiotic disturbances. Quine and Gardiner (2007) wrote a comprehensive summary on implications of wind damage on forest ecology and forestry. Most of the abovementioned studies focused on the response of the stem to wind excitation in the fundamental mode. There are only a few studies (e.g. Sellier and Fourcaud, 2005; Sellier
D. Schindler et al. / Agricultural and Forest Meteorology 150 (2010) 984–993
et al., 2006, 2008; Spatz et al., 2007; Moore and Maguire, 2008; Rodriguez et al., 2008) in which higher order vibration of the stem and maybe the branches of coniferous trees was analyzed. In these studies it was shown that multimodal vibration of the aerial parts of trees controls their response to some external excitation to a great extent. For individual trees of a number of different species James et al. (2006) showed that the dynamic sway of the branches greatly influences whole tree sway. There are a few further studies (Spatz et al., 2007; Moore and Maguire, 2008; Rodriguez et al., 2008) that showed that structural damping or multiple resonance damping is an effective component of whole tree movement damping. Further processes known to counter wind-induced tree displacement are aerodynamic drag of the crowns (Mayhead, 1973; Amiro, 1990; Rudnicki et al., 2004; Sellier and Fourcaud, 2005) and viscous damping of the wood (Milne, 1991; Wood, 1995). Within forest stands friction between neighboring trees is a source of tree movement damping (Milne, 1991; Rudnicki et al., 2008). de Langre (2008) recently wrote a comprehensive review on mechanical interactions between wind and plants. Since there are only few quantitative results for higher order vibration of forest-grown coniferous trees in response to wind excitation, the objectives of this study were to analyze (1) the windinduced dynamic sway behavior of three Scots pine trees with a focus on their response to intermittent, coherent structures, (2) the characteristics of their stem vibration modes. The relevance of coherent structures and vibration modes for wind-induced tree displacement is then discussed.
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Fig. 1. Study area and spatial arrangement of the ten (T1–T10) adjacent sample trees. The black circles indicate the mean crown radii of the sample trees at zeroplane displacement height zd (z/h = 0.78).
2. Materials and methods 2.1. Measurement site The study area and the measurement setup described hereafter correspond to the study area and measurement setup presented in Schindler (2008). The study area was located about 25 km southwest of Freiburg (Southwest Germany) in the flat southern Upper Rhine Valley (47◦ 56 04 N, 7◦ 36 02 E, altitude range above sea level: 198–202 m) within a planted Scots pine (Pinus sylvestris L.) forest (Fig. 1). Wind-induced stem displacement of ten (T1–T10) adjacent Scots pine trees (inset in Fig. 1) as well as near-surface airflow properties were monitored from October 2005 to April 2006. During the measurement period the Scots pine forest at the measurement site had a mean stand density of 800 trees ha−1 and a mean stand height (h) of 14.5 m. The mean stem diameter at breast height (dbh) was 17.5 cm and the plant area index (PAI) was 1.9. Based on wind speed measurements (Robinson, 1961), the zero-plane displacement height (zd ) representative for the Scots pine forest at the measurement site was estimated to be 11.3 m (z/h = 0.78). In this paper results from the responses of the three sample trees T1, T2, and T5, which responded well to the measured airflow characteristics, are presented. The heights (ht) of T1, T2, and T5 were 14.6 m, 15.2 m, and 15.1 m. Further characteristics of the samples trees are given in Table 1. Fig. 2 provides schematic side views (to improve clarity the x-axis was doubled) of the mean shapes of T1, T2, and T5. These illustrate that the crowns were rather short and located above z/h = 0.5. As a rule of thumb, the crowns’ mean maxi-
Fig. 2. Schematic side views (to improve clarity the x-axis was doubled) of the mean shapes of the three sample trees (a) T1, (b) T2, and (c) T5.
mum lateral extent was less than 10% of the respective sample tree height. 2.2. Airflow and stem displacement measurements Eight ultrasonic anemometers (USA-1, METEK, Germany) mounted on a 28 m high triangular lattice tower were used to measure (sampling rate: 20 Hz) airflow characteristics within and above the Scots pine forest (Schindler, 2008). Wind velocity was measured at 2.2 m (z1 /h = 0.15), 7.3 m (z2 /h = 0.5), 8.7 m (z3 /h = 0.6), 10.9 m (z4 /h = 0.75), 13.1 m (z5 /h = 0.9), 15.2 m (z6 /h = 1.05), 21.8 m
Table 1 Characteristics of the three sample trees T1, T2, and T5. Tree
Tree height ht (m)
Diameter 1.3 m dbh (cm)
Slenderness ht/dbh
Crown projected area a (m2 )
Largest mean crown radius cr (m)
Crown length l (m)
T1 T2 T5
14.6 15.2 15.1
15.9 18.3 19.2
92 83 79
5.8 7.1 6.3
1.4 1.5 1.2
5.7 6.1 6.3
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From WnAB (c) the smoothed global cross-wavelet spectrum was obtained as (Torrence and Webster, 1999): 1 AB Wn (c) N N−1
GCWS(c) =
(2)
n=0
where N is the number of data points. GCWS is a measure of common power between two time series. Furthermore, the wavelet-squared coherence Rn2 (‘wavelet coherency’) was calculated according to Torrence and Webster (1999): Fig. 3. Deformation patterns associated with the first four normal modes (M1 –M4 ) of a clamped-free beam, hc1 = 1/7ht, hc2 = 3/7ht, and hc3 = 5/7ht are the clinometer mounting heights, ht is the respective sample tree height.
(z7 /h = 1.5), and 26.8 m (z8 /h = 1.85). Based on the measured wind velocity series, mean wind speed at canopy height (z/h = 1.0) Uh was approximated by cubic spline interpolation. The measurement tower was set up in a small canopy gap (mean gap diameter: 6 m) to the east of the sample trees. The Institute of Meteorology, Climatology and Remote Sensing of the University of Basel provided the software (author: Andreas Christen) used to store airflow data on a commercial PC. In order to determine stem displacement in response to wind loading, three biaxial clinometers (model 902-45, Applied Geomechanics, USA) were mounted on each of the three sample trees. The mounting heights of the clinometers corresponded to hc1 = 1/7ht, hc2 = 3/7ht, and hc3 = 5/7ht. These mounting heights were assumed to be approximately the heights of the antinodal points of vibration of the 4th normal mode of a clamped-free beam. Fig. 3 illustrates the deformation patterns associated with the first four normal modes (M1 –M4 ) of a clamped-free beam (Simiu and Scanlan, 1996). The clinometers measured the angular position of the sample trees’ stems and were sampled at a frequency of 10 Hz by a CR5000 micrologger (Campbell Scientific, USA). In order to minimize sensor heating, all clinometers were pointed northward. Schindler (2008) described the conversion of angular position (deg) measurements into stem displacement (m) in x- (dx ) and y-direction (dy ) along the stem. Displacement of the sample trees between z = 0 m to the top height was estimated for 0.2 m intervals using cubic spline interpolation. A wavelet filter was applied to reduce instrument noise in the clinometer signals.
Rn2 (c) =
|c −1 WnAB (c)|2 −1 c |WnA (c)|2 c −1 |WnB (c)|2
(3)
where · indicates smoothing in both time and scale. Rn2 provides a measure of the intensity (0 ≤ Rn2 ≤ 1) of covariance of A and B (Jevrejeva et al., 2003). Rn2 (c) was used to calculate the global wavelet coherence spectrum GWCS (Torrence and Webster, 1999): GWCS(c) =
Rn2 (c)
(4)
n
All wavelet calculations were realized with wavelet software provided by C. Torrence and G. Compo. The software is available at URL: http://atoc.colorado.edu/research/wavelets/. 2.4. Bi-orthogonal decomposition The bi-orthogonal decomposition (BOD) was used to capture and to analyze the main modal features of the sample trees’ oscillations using the stem displacement records. Basically, the BOD can be used to decompose a m-degree of freedom signal into m modes in the time-space domain (Aubry et al., 1991; Hémon and Santi, 2003). In order to apply the BOD to the stem displacement data, the matrix D = [dx0 , . . ., dx15 , dy0 , . . ., dy15 ], which contained hourly time series of dx and dy from z = 0 m to the top height of a sample tree, was defined. For each sample tree, stem displacement values in xand y-direction were used from 16 heights (1 m segments). D was decomposed according to D=
∞ √
˛k k k
(5)
k=1
2.3. Wavelet analysis In order to analyze the intermittent nature of large, coherent eddies (Finnigan, 2000) and their effects on displacement of the sample trees, a wavelet analysis (e.g. Torrence and Compo, 1998; Jevrejeva et al., 2003; Grinsted et al., 2004) was carried out. The wavelet analysis makes it possible to expand time series into time frequency space and detect localized, intermittent structures (Grinsted et al., 2004). The covariance of time series of airflow and stem displacement was studied by smoothed cross-wavelet spectra calculated as (Torrence and Compo, 1998): WnAB (c) = WnA (c)WnB∗ (c)
(1)
where A and B are time series, WnA and WnB are their wavelet transforms (based on the Morlet wavelet), n is the time index, c is the wavelet scale, and the asterisk denotes complex conjugate. WnAB (c) was calculated for 200 scales which ranged up to 100 s. The wavelet scale was converted to the equivalent Fourier period, which is for the Morlet wavelet almost equal to the equivalent Fourier period (Torrence and Compo, 1998).
where k is the mode number, k are temporal functions (chronos), k are spatial functions (topos), and the weight factor of the struc√ tures (k , k ) is the real scalar value ˛k . k and k form a set of normalized orthogonal functions. Each BOD mode contains information on the time characteristics and on the space characteristics of the stem displacement data. The eigenvalue ˛k is a measure of the relative signal energy contained in the kth BOD mode. It was used to rank the spatio-temporal modes in descending order. Referring to Aubry et al. (1991), k are the eigenfunctions of the spatial correlation operator of D with ˛k , k are the eigenfunctions of the temporal correlation operator of D with ˛k . Since stem displacement was recorded discretely, vectors replaced the temporal and spatial functions. Correlation matrices replaced the correlation operators. The spatial correlation matrix of the signal was computed as S = DT D
(6)
and topos were determined from the following eigenvalue problem: S = ˛
(7)
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Fig. 4. Mean bivariate histograms of the sample trees’ responses to wind loading at zero-plane displacement height zd (z/h = 0.78) in the four wind speed classes Uh1 (0 ≤ Uh ≤ 1 m s−1 (a)), Uh2 (1 < Uh ≤ 2 m s−1 (b)), Uh3 (2 < Uh ≤ 3 m s−1 (c)), Uh4 (Uh > 3 m s−1 (d)), Uh being the wind speed at canopy height (m s−1 ). The small black rectangle on the right side of the upper left box indicates the position of the sonic anemometer profile mounted to the measurement tower.
Chronos were derived from projecting the bi-orthogonal decomposition on topos and taking advantage of the orthogonality property of the modes: 1 k = √ Dk ˛k
(8)
A more detailed description of the computational details associated with the application of the BOD can be found in Py et al. (2005, 2006). 3. Results and discussion 3.1. Bivariate histograms Hourly runs of tree response data were analyzed for seven windy days (December 16, 2005; December 19, 2005; December 31, 2005; February 8, 2006; February 18, 2006; March 3, 2006; March 9, 2006) with near-neutral stability. Atmospheric stability was quantified by the absolute hourly values of the stability index (z − zd )/L (L is the Obukhov-length). Altogether 151 h for which (z − zd )/L was equal or less than 0.05 were available for the analysis of tree responses to wind loading. Hourly values of mean wind speed at canopy top ranged from 0.4 m s−1 to close to 4.0 m s−1 . On all other days during the measurement campaign wind speed was too low to excite pronounced tree responses. Since the sample trees’ responses on wind excitation are a function of Uh (Mayer, 1987; Gardiner, 1995), the hourly mean values of Uh were divided into the four wind speed classes Uh1 (0 ≤ Uh ≤ 1 m s−1 ; n = 13), Uh2 (1 < Uh ≤ 2 m s−1 ; n = 45), Uh3 (2 < Uh ≤ 3 m s−1 ; n = 60), and Uh4 (Uh > 3 m s−1 ; n = 33), n being the number of hourly values. During the chosen hours wind was blowing predominantly from
south to southwest. The prevailing wind directions in Uh1 –Uh4 were 210◦ ± 41◦ (mean ± standard deviation), 189◦ ± 25◦ , 193◦ ± 12◦ , and 207◦ ± 10◦ . In order to get an impression on the wind-induced movement of the sample trees, a top view of the sample trees’ responses to wind loading at zd is given in Fig. 4. For each sample tree, mean twodimensional density functions were calculated for each of the wind speed classes Uh1 –Uh4 . It is noticeable that mean tree displacement increased from Uh1 (Fig. 4a) to Uh4 (Fig. 4d) in x-direction as well as in y-direction. The bivariate histograms are of near-circular shape without a clearly dominating major axis along the prevailing wind direction. Stem displacement in the along wind direction did not far exceed stem displacement in the across wind direction. With increasing wind speed absolute values of stem displacement progressively differed between the sample trees. T1 showed greatest stem displacements, which might be the result of additional space due to the small canopy gap east of T1 where the measurement tower was set up. One reason for the near-circular shapes of the bivariate histograms were phase shifts between dx and dy which prevented tree sway back and forth along characteristic lines. Instead, the sample trees mostly swayed on irregular orbits, which formed the mean shape of the bivariate histograms. For the wind speed range comparable to the wind speed range observed in this study complicated tree response to wind loading is also known from Amtmann (1986) and Peltola (1996). For mean wind speed at canopy top between 5.0 m s−1 and 7.9 m s−1 the sway patterns of ten adjacent lodge pine trees presented by Rudnicki et al. (2003) were also circular and elliptical in shape. In contrast, more regular tree response to wind loading is known from Gardiner (1995). His results demonstrate that the along wind displacement of two
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Fig. 5. Normalized global cross-wavelet spectra (GCWS) calculated between (a) fluctuations of the streamwise wind vector component u and the fluctuations of the stem displacement component in y-direction dy , (b) u and the fluctuations of the stem displacement component in x-direction dx , (c) fluctuations of the across wind vector component v and dy , (d) v and dx over the hourly run from 13:00 to 14:00 CET on March 3, 2006.
Sitka spruce trees clearly exceeded the across wind displacement. Despite the rather symmetric mean two-dimensional density functions, in the following paragraphs dy will be treated as the stem displacement component approximately aligned in the along wind direction. dx is the across wind stem displacement component. 3.2. Wavelet analysis Wavelet analysis was carried out for the hourly run from 13:00 to 14:00 CET on March 3, 2006. During this run, mean wind speed
at canopy top Uh equaled to 3.7 m s−1 , which was one of the highest hourly mean wind speed values observed during the measurement period. In Fig. 5 normalized (by their maxima in the examined range of scales) results from GCWS-calculations between the fluctuations of the streamwise (u ), the crossflow wind vector component (v ) (u and v calculated from the airflow measurements at z6 /h = 1.05), and the fluctuations of stem displacement components (dx , dy ) (dx and dy calculated from the stem displacement at zd ) of T1, T2, and T5 are presented. The GCWS-calculations including u and dy (Fig. 5a) revealed two distinct peaks in common power. One peak in the range of the natural sway periods of the sample trees (<4 s), another
Fig. 6. Global wavelet coherence spectra (GWCS) calculated between (a) fluctuations of the streamwise wind vector component u and the fluctuations of the stem displacement component in y-direction dy , (b) u and the fluctuations of the stem displacement component in x-direction dx , (c) fluctuations of the across wind vector component v and dy , (d) v and dx over the hourly run from 13:00 to 14:00 CET on March 3, 2006.
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peak at 47.2 s. The equivalent Fourier frequency associated with the second peak is 0.02 Hz. The results from the GCWS-calculations confirm the results from the dy -wavelet variance spectra calculations reported by Schindler (2008) in a quantitative way. He qualitatively showed the coincidence of peaks in u - and dy -wavelet variance spectra. The GCWS-peaks in the range of the natural sway periods of the sample trees result from swaying back and forth along the ydirection. The GCWS-peaks at 47.2 s are associated with the arrival of coherent structures in the streamwise direction. Similar results were obtained for u and dx (Fig. 5b). The global cross-wavelet spectra calculated for T1, T2, and T5 also show distinct peaks in two very similar ranges as were observed for u and dy . The first sharp peaks reflect the duration of the natural sway periods of the sample trees in x-direction. The second broader peaks are also the consequence of excitation of the sample trees by wind loads related to coherent structures. The periods of these peaks were a little shorter (T1, T2: 42.6 s; T5: 44.2 s) than the period calculated from the time series of u and dy . The results from GCWS-calculations between v and fluctuations of stem displacement in x- and y-direction (Fig. 5c, d) demonstrate that there was high common power only in the range of the natural sway periods of the sample trees. Due to the absence of larger scale organized structures from the across wind direction (Finnigan, 2000), the global cross-wavelet spectra did not show further pronounced peaks in common power at periods longer than the natural sway periods of the sample trees. In contrast to that, global wavelet coherence spectra (GWCS) did not show peaks in the range of periods that corresponded to the natural sway periods of the sample trees (Fig. 6), meaning that the localized covariation in time frequency space between wind loading and tree response was rather weak. GWCS calculated between u and dy (Fig. 6a) as well as GWCS calculated between u and dx (Fig. 6b) showed distinct maxima (larger than 0.5) at periods greater than 40 s. This provides evidence that wind loads associated with coherent structures, along wind stem displacement, but also across wind stem displacement covaried in this range of periods. GWCS calculated between v and stem displacement in y(Fig. 6c) and x-direction (Fig. 6d) was generally lower than corresponding GWCS in the along wind direction. Although GWCS of v and dx was at a maximum at 33.8 s for all sample trees, the absolute values of these GWCS-maxima were clearly lower than the GWCSmaxima between u and dx as well as between u and dy . Over all Fourier periods wavelet coherence between v and dy was low which means that their covariation was small. Based on the obtained wavelet results the mean spacing and the occurrence frequency of coherent structures in the streamwise direction were analyzed. For that, mean streamwise coherent structure separation (u ) estimated from the u time series and normalized by canopy height h was calculated according to (e.g. Raupach et al., 1996): TUc u = h Nh
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Although the sample trees responded well to turbulent wind loading in the range of their fundamental sway frequency, the results from the wavelet analyses demonstrate that wind loads associated with the arrival of coherent structures in the streamwise direction showed the strongest covariation with stem displacement. These wind loads not only strongly covaried with stem displacement in the along wind direction but also with stem displacement in the across wind direction. The sample trees’ responses in the range of their fundamental sway frequency seem to be a result from their damped harmonic oscillator-like vibration behavior as stated by Gardiner (1995). He presented similar results for the vibration behavior of spruce trees and stated that trees do not resonate with turbulent wind components in the range of their natural sway frequency but strongly react on impulsive wind loading during the passage of gusts. This leads to the conclusion that energy absorption from turbulent wind loading in the frequency range close to the fundamental stem sway frequency (f1 ) is of minor importance in the analyzed range of wind speeds. 3.3. Fourier analysis In order to determine the frequencies that are associated with the vibrational modes of the sample trees, the Fourier transform was applied to the fluctuations of stem displacement in y- and x-direction. Periodicities in tree response were examined in the fre-
(9)
where T is the total time of the data series in seconds, Uc is the convection velocity (assumption that Uc = 1.8Uh ; Raupach et al., 1996), N is the number of coherent structures detected by the Mexican hat wavelet detection function. Schindler (2008) reported N = 73 for the processed hourly run which resulted in u /h = 22.7. This value is outside the cloud of u /h data points presented by Brunet and Irvine (2000) when compared to w /h (mean streamwise coherent structure separation estimated from the vertical wind vector component w). For the analyzed hourly run u /w equaled to 6.5 whereas the mean ratio u /w (slope of a regression line) given by Brunet and Irvine (2000) equaled to 3.07 (r2 = 0.51, r2 being the coefficient of determination).
Fig. 7. Energy spectra (fSdy (f)) of the fluctuations of along wind stem displacement dy of the sample trees T1, T2, and T5 at the clinometer mounting heights (a) hc3 , (b) hc2 , and (c) hc1 over hourly runs assigned to Uh4 (Uh > 3 m s−1 ).
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D. Schindler et al. / Agricultural and Forest Meteorology 150 (2010) 984–993 Table 2 Frequencies of the first (f1 ) to the fourth (f4 ) peak in Fourier energy spectra of fluctuations of stem displacement d in y- (fSdy (f)) and x-direction (fSdx (f)) that are associated with the first (M1 ), second (M2 ), third (M3 ), and fourth (M4 ) vibrational mode of the sample trees T1, T2, and T5. Frequency (Hz)
f1 f2 f3 f4
dy
dx
T1
T2
T5
T1
T2
T5
0.29 0.99 1.77 3.32
0.34 0.99 2.12 3.63
0.34 0.99 2.12 3.80
0.31 1.03 1.85 2.77
0.34 0.99 2.12 3.63
0.34 1.08 2.03 3.63
cided with the values of f1 –f4 in x-direction. For T1 and T5 they differed to some extent. With the exception of fSdx (f) calculated at hc1 , the shapes of the Fourier spectra calculated based on the dy - and dx -series were quite similar. Due to small fluctuations in stem displacement in x-direction at hc1 , instrument noise might have prevented a drop in spectral energy of dx at higher frequencies. Since in this study only stem displacement was measured, it can only be speculated on the sources of sway frequency differences in the two stem displacement directions as well as on the sources of higher order stem vibration. As already mentioned, it is known that vibration mode characteristics are functions of mass and stiffness distributions within a structure, and may be excited by temporal and spatial variations of the wind load. For young maritime pines Sellier and Fourcaud (2005) explained anisotropic sway frequency differences by morphological differences of their sample trees.
Fig. 8. Energy spectra (fSdx (f)) of the fluctuations of across wind stem displacement dx of the sample trees T1, T2, and T5 at the clinometer mounting heights (a) hc3 , (b) hc2 , and (c) hc1 over hourly runs assigned to Uh4 (Uh > 3 m s−1 ).
quency domain by the Fourier energy spectra fSdy (f), and fSdx (f), S being the spectral energy, f is the frequency. In Figs. 7 and 8 mean log fSdy (f) and mean log fSdx (f) calculated at hc3 (Figs. 7a and 8a), hc2 (Figs. 7b and 8b), and hc1 (Figs. 7c and 8c) in Uh4 are plotted versus log f. Only the displacement series from Uh4 were chosen for this analysis because Uh4 encompassed the wind speed range for which the sample trees’ dynamic responses to wind loading were most pronounced. Several characteristic spectral peaks are present in mean log fSdy (f) and mean log fSdx (f). The most prominent peaks appear in the frequency range between 0.25 and 0.40 Hz. The frequencies associated with these peaks were the same frequencies as obtained by means of the wavelet analysis. They were interpreted as the damped natural sway frequencies (f1 ) associated with the first vibrational modes (M1 ) of the sample trees’ stems. With increasing frequency three further minor peaks could be identified. These peaks indicate higher mode vibration and were interpreted as the frequencies f2 , f3 , and f4 associated with the second (M2 ), third (M3 ), and fourth (M4 ) vibration modes of the stems. Since the same number of spectral peaks was not clearly identifiable at all displacement measurement heights, the values of f1 –f4 were determined from those displacement spectra in which they were evident. Their values are given in Table 2. The values of f1 are proportional to dbh/ht2 and close to the f1 -values that are predicted for pines when using the formula f1 = 0.0948 + 2.65519·dbh/ht2 given by Moore and Maguire (2004). For T2 the values of f1 –f4 in y-direction coin-
Fig. 9. Mean shapes of the first four stem vibration modes (M1 –M4 ) of (a) T1, (b) T2, and (c) T5 obtained from stem displacement measurements in y-direction. The amplitude is arbitrarily scaled.
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3.4. Bi-orthogonal decomposition The application of the bi-orthogonal decomposition enabled the extraction of characteristic mode shapes associated with f1 –f4 . For each of the peaks (four in along wind direction and four in across wind direction) in the presented Fourier displacement spectra, modal forms of vibration amplitude distributed along the stems of the sample trees could be obtained. Over the whole analyzed wind speed range, M1 and M2 showed almost exactly the same modal shapes in y- and x-direction. Because the characteristics of M3 and M4 got better with increasing Uh , mean BOD-mode shapes are presented for hourly mean wind speeds contained in Uh4 . Figs. 9 and 10 show the mean shapes of M1 –M4 of T1 (Figs. 9a and 10a), T2 (Figs. 9b and 10b), and T5 (Figs. 9c and 10c) resulting from wind-induced stem displacement in y- and x-direction. Since topos are eigenvectors they are shown with arbitrary amplitude. In both stem displacement directions, the results for M1 –M4 are very similar among the sample trees. The shapes of M1 and M2 show reasonable agreement with the mode shapes presented by Thomson and Dahleh (1998) or Hodges and Pierce (2006) for a clamped-free beam. However, the shapes of M3 and M4 differ somewhat from the mode shapes of a clamped-free beam: M3 has less whereas M4 has more deformation at the stem base than expected. The deformations of M1 and M2 are also consistent with the shapes of the first (I) and second (I ) bending modes of the stem of a young maritime pine presented by Rodriguez et al. (2008). The graphs in Fig. 11 illustrate the sums of relative energy contained in M1 for both stem displacement directions as a function of Uh . Even at low wind speeds (Uh < 1 m s−1 ), relative energy conFig. 11. Signal energy (%) contained in the first vibration mode (M1 ), for (a) displacement of the sample trees T1, T2, and T5 in y-direction (dy ) and (b) displacement of T1, T2, and T5 in x-direction (dx ) as a function of wind speed at canopy top (Uh ).
Fig. 10. Mean shapes of the first four stem vibration modes (M1 –M4 ) of (a) T1, (b) T2, and (c) T5 obtained from stem displacement measurements in x-direction. The amplitude is arbitrarily scaled.
tained in M1 dominated the stem displacement signal of T1, T2, and T5. The relative energy content of M1 was at least 89% in ydirection (Fig. 11a) and at least 60% (Fig. 11b) in x-direction over the presented wind speed range. At lower wind speed there were a few hourly intervals, in which the relative signal energy contained in M1 was comparatively low. If this was the case, relative signal energy contained in M2 increased. Hence, the sums of relative energy contained in M1 and M2 amounted to at least 97% over all analyzed hourly runs. The large amount of relative energy contained in M1 is consistent with the results from the Fourier analysis and findings from previous studies (e.g. Holbo et al., 1980; Mayer, 1987). M1 normally dominates the response of monopodial conifer trees to dynamic wind loading. The relative signal energy contents of M3 and M4 were very low (<3%) and started to fall back to zero with increasing wind speed. These results imply that beginning with the lowest wind speeds, higher order stem vibration (especially in along wind direction) was not an important mechanism for the dissipation of energy of the near-surface airflow transferred into tree movement. In Fig. 12 the ratio q of the eigenvalues of M1 in the x-direction to the eigenvalues of M1 in the y-direction is presented as a function of Uh . At the lowest wind speeds there is some scatter in the plot. Especially for T1, relative energy contained in M1 of dy was greater during a few hours than relative energy contained in M1 of dx (q < 1). Yet, during a few hours, q was greater than one for all sample trees. Beginning with Uh = 2.5 m s−1 all hourly q-values were close to one. The anisotropic signal energy contents of M1 converged with increasing wind speed. Since the effects of the soil and the root system on tree sway dynamics were not investigated in this study, the question of their relevance for the vibration behavior of the sample trees cannot be answered. Nevertheless, it is assumed that in the analyzed wind
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ther field studies is also seen in the field of tree–tree interactions in high winds. For lodgepole pine trees Rudnicki et al. (2008) showed that at higher wind speeds damping due to crown collisions influences tree sway dynamics. They reported a decrease in fundamental sway frequency through dissipation of kinetic energy when adjacent trees experienced intense crown collisions. This implies that – as has recently been done by Py et al. (2005, 2006) for crop canopies – future field studies should also focus on the response behavior of large groups of neighboring forest trees. The analysis of their interactions is crucial for an understanding of crown collision dynamics and energy dissipation within forest trees especially when it comes to the reduction of wind-induced damage. Acknowledgements
Fig. 12. Ratio q of the eigenvalues of the first mode of stem displacement (M1 ) in x-direction to eigenvalues of M1 in y-direction of the sample trees T1, T2, and T5 as a function of wind speed at canopy top (Uh ).
The German Research Foundation (DFG SCHI 868/1) and the Department of the Environment of the federal state of BadenWuerttemberg (RESTER-UniFR) supported this work. References
speed range and under the prevailing soil water conditions (soil water content clearly below field capacity) their effect on tree vibrations was of minor importance. 4. Conclusions The results from the wavelet analyses suggest that stem displacement and wind loading exerted by coherent structures strongly covaried whereas covariation was weak in the range of the fundamental sway period of the sample trees’ stems. From this, it is deduced that wind-induced displacement of the Scots pines was primarily a function of impulsive wind loads associated with coherent structures. This argumentation is consistent with the conclusions of Gardiner (1995) and Schindler (2008) who stated that the periodicity of coherent structure arrival is more important for large coniferous forest tree displacements than the energy available in the range of the damped natural sway frequency of trees. Moreover, in the analyzed hourly run the arrival frequency of coherent structures was far off the damped natural sway frequencies of the samples tree. From this, it is concluded that no frequency lock-in mechanism, through which the sample trees and the wind fluctuations interact in the range of the trees’ natural sway frequencies, significantly altered the flow dynamics at the measurement site. Py et al. (2006) recently reported a frequency lock-in mechanism of coherent structures on plant motion of crop (alfalfa, wheat) canopies. They showed that wind-induced coherent motion of the crop canopies occurred near the natural vibration frequency of the crop plants. By means of the bi-orthogonal decomposition it was pointed out that even at very low wind speeds most of the stem displacement signal energy was contained in the first stem vibration mode, especially in the along wind direction. The resistance to wind loading of the slender plantation-grown sample trees with their small, short crowns relies mainly on sway of their stems in the first mode. How multiple resonance damping and damping effects of the foliage contribute to the damping of stem vibration needs to be further investigated, for example, by conducting studies in which windinduced branch dynamics are also monitored. Another question that needs to be addressed in the future relates to the response behavior of forest trees to (potentially) damagecausing wind loads. So far not much is known about wind–tree interactions under strong wind conditions. At higher wind speeds (higher than measured in this study) when impulsive wind loading becomes more frequent and possibly in phase with tree motion, resonance between impulsive wind loading and tree motion might enhance the increasing wind load (Gardiner, 1995). Need for fur-
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