The Role of Wake Production on the Scaling Laws of Scalar Concentration Fluctuation Spectra Inside Dense Canopies

Transcription

1 Boundary-Layer Meteorol (211) 139:83 95 DOI 1.17/s ARTICLE The Role of Wake Production on the Scaling Laws of Scalar Concentration Fluctuation Spectra Inside Dense Canopies D. Poggi G. G. Katul B. Vidakovic Received: 26 July 21 / Accepted: 2 December 21 / Published online: 28 December 21 Springer Science+Business Media B.V. 21 Abstract The scalar concentration fluctuations within a plane parallel-to-the-ground surface were measured inside a model canopy composed of densely arrayed rods using the laser-induced fluorescence technique. Two-dimensional scalar concentration spectra were computed and were shown to exhibit an approximate 3 power-law scaling at wavenumbers larger than those associated with wake production during quiescent instances when von Karman vortex streets dominated the flow. However, during instances when sweeps disrupted the flow, the spectral exponents increased above 3. The 3 power-law for these concentration fluctuation spectra measurements was shown to be consistent with a simplified spectral budget for locally homogeneous and isotropic turbulence augmented with a relaxation time scale similarity argument that assumed a constant enstrophy injection rate and wake generation mechanism. Hence, the origin of this 3 power-law scaling here differs from the well-known 3 power-law result for the so-called inertial diffusive range derived for the scalar concentration spectrum at small Prandtl numbers. Keywords Canopy turbulence Laser-induced fluorescence technique Scalar concentration spectra Wake production D. Poggi (B) G. G. Katul Dipartimento di Idraulica, Trasporti ed Infrastrutture Civile, Politecnico di Torino, Torino, Italy davide.poggi@polito.it G. G. Katul Nicholas School of the Environment and Earth Sciences, Duke University, Durham, NC, USA G. G. Katul Department of Civil and Environmental Engineering, Duke University, Durham, NC, USA B. Vidakovic Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA , USA

2 84 D. Poggi et al. 1 Introduction Well above rough surfaces, three canonical regimes characterise the spectrum of turbulent kinetic energy (TKE): the energetic region in which TKE is produced, the standard inertial subrange region (hereafter referred to as ISR) characterising three-dimensional turbulence at high Reynolds numbers (Kolmogorov 1941 hereafter, K41), and the viscous dissipation region. However, inside canopies, the work that the flow exercises against the foliage drag produces TKE by wakes as well as a spectral short-circuiting of the energy cascade that alters these three TKE spectral regimes (Finnigan 2; Poggiand Katul 26; Poggiet al. 28). What is the impact of these canopy-related processes on the energy cascade of passive scalars remains the subject of research now receiving attention in canopy turbulence. This interest, in part, is being driven by biosphere atmosphere studies (Finnigan 2) of gas exchange (e.g. CO 2 and water vapour transport), ecological studies of seed and pollen spread pertinent to geneflow (Nathan et al. 22), and air quality studies including the transport of ammonia (e.g. ammonium nitrate and ammonium sulfate aerosols that adversely affect human health) from the soil to the atmosphere overlaying the canopy (Stutton et al. 1995), all requiring basic understanding of the canonical length scales affecting scalar concentration fluctuations within the canopy. A survey of field experiments already suggests that the spectra of turbulent flow variables inside canopies are steeper than the ISR predicted 5/3 (Finnigan 2; Cava and Katul 28), though the genesis of this spectral steepening remains elusive and has resisted a complete theoretical treatment. Previous flume experiments, including flow visualisation experiments, have already shown that the organised vortical motion deep within a rod canopy originates from quasi two-dimensional classical von Karman vortex streets (Poggi et al. 24b). Moreover, one-dimensional spectra (along the longitudinal direction) of the measured vertical velocity component (w) were shown to collapse for both dense and sparse rod canopies with a unique secondary peak at a dimensionless frequency fd r /u =.21, where d r is the rod diameter and u is the mean longitudinal velocity component (Poggi et al. 24b). The collapse of these w spectra at a dimensionless frequency =.21 further suggests that the classical Strouhal number (St = fd r /u) linking the frequency of periodic vortices to the mean velocity and the characteristic length scale of the obstacle (here, d r ) is the appropriate frequency at which vortices are produced. The spatial coherency, periodicity, and the alternating character of these vortices forms the basis of the present investigation. Based on these preliminary results, we proceed to explore the consequence of such vortical structures on the scaling laws of the scalar concentration fluctuation spectra. Our working hypothesis is that the spectral steepening from the 5/3 scaling (for dimensionless frequencies >.21) often observed may well be connected to the long durations of these quiescent phases inside the canopy, where the flow is dominated by von Karman streets. The von Karman streets are quasi two-dimensional eddies and exhibit an apparent inverse cascade. By inverse cascade, we are referring to the fact that these eddies are produced at d r and tend to grow in size spatially (before colliding with adjacent rods) rather than break-up into smaller eddies (as in the ISR due to vortex stretching). Another corollary hypothesis here is that if a constant enstrophy injection rate is assumed during such quiescent periods, the predictions of novel power laws for the scalar spectra can be made and tested with the measurements reported herein. Hereafter, x, y, andz define the longitudinal (along the mean flow), lateral, and vertical coordinates, and u and w are the turbulent velocity excursions along the x and z directions, respectively.

3 Wake Production on the Scaling Laws 85 2 Scaling Analysis As a starting point for deriving the concentration fluctuation spectra inside a canopy, the budget for the scalar fluctuation spectrum φ cc for a locally homogeneous and isotropic turbulence at small scales is considered. This simplified budget is given by (Corrsin 1964; Hill 1978; Poggi and Katul 26; Danaila and Antonia 29) φ cc (t, k) = W c (t, k) 2χ c k 2 φ cc (t, k), (1) t where W c (t, k) is the transfer of scalar variance via interactions between turbulent velocity and scalar concentration fluctuations, χ c is the molecular diffusivity of scalar c, k is the spatial wavenumber, t is time, and p= p= φ cc (t, p)dp = σc 2 (t) is the time-evolving spatial scalar variance. By small scales here, we refer to eddy sizes that are smaller than the eddies containing much of the TKE. In general, W c = S cc (t, k)/ k, wheres cc (t, k) represents the scalar variance flux transfer term from wavenumbers <k to wavenumbers >k (PoggiandKatul 26; Danaila and Antonia 29). This variance flux transfer term does not produce or dissipate scalar variance (i.e. p= p= W c (p)dp = ) and can be modelled via a simplified scaling argument (Poggi and Katul 26; Danaila and Antonia 29)inwhichS cc (= kφ cc /τ(k)) is linked to the available scalar energy (= kφ cc ) and a wavenumber dependent TKE relaxation time scale τ(k). With this simplification for W c, the spectral budget reduces to ( kφcc (t, k) φ cc (t, k) = ) 2χ c k 2 φ cc (t, k). (2) t k τ(k) At steady state, the general solution of Equation (2) as a function of an arbitrary τ(k) can be derived and is given by (Danaila and Antonia 29) [ ( ( ) )] τ(k) d k φ cc (k) = C 1 exp + 2χ c kτ(k), (3) k dk τ(k) where C 1 is an integration constant. One common model for τ(k) is given by (Katul and Chu 1998; Poggi and Katul 26; Danaila and Antonia 29) 1/2 τ(k) = 1 p=k, (4) p= p2 φ tke (p)dp where φ tke is the TKE spectrum. This representation assumes that turbulent eddies at wavenumber k are strained by larger scale eddies characterised by wavenumbers <k. Based on this approach, if the TKE spectrum exhibits a power law of the form φ tke (k) = ak m (with m < 3), the resulting relaxation time scale is also a power law of the form (Danaila and Antonia 29) ( ) 3 m 1/2 τ(k) = ak 3 m. (5) Upon replacing this power-law model for τ(k) into Equation (3) and simplifying, we obtain (Danaila and Antonia 29) φ cc (k) = C 1 k (m 5)/2 exp ( 4χ c(3 m) 1/2 a 1/2 (1 + m) k(1+m)/2 ), (6)

4 86 D. Poggi et al. where the integration constant C 1 can be determined by noting that N = χ c p= p= p 2 φ cc (p)dp, (7) where N is the scalar variance dissipation rate. This condition yields ( ) N 1 C 1 = χ ( ) = N (3 m) 1/2 c k= k= k (m 1)/2 exp 4χ c(3 m) 1/2 a 1/2 (1+m) k(1+m)/2 dk 2a 1/2. (8) With this estimate of C 1, the scalar spectrum is given by φ cc (k) = N (3 m) 1/2 2a 1/2 k (m 5)/2 exp ( 4χ c(3 m) 1/2 ) a 1/2 (1 + m) k(1+m)/2. (9) Note that the canonical form of this solution to φ cc resembles a stretched-exponential (Laherrere and Sornette 1998) with a power-law decay at low wavenumbers with censoring by an exponential cut-off due to molecular effects at high wavenumbers. According to this solution, the molecular effects become pronounced (i.e. reducing φ cc by 1% or more) at wavenumber k s given by 2/(1+m) k s = log(s) 4χ c (3 m) 1/2 a 1/2 (1+m), (1) where s.9. Naturally, the assumption that φ tke (k) = ak m extends well beyond k s is questionable, especially if the ratio of the fluid viscosity (ν) toχ c is of order unity, or if k s >(ɛ/ν 3 ) 1/4,whereɛ is the mean turbulent kinetic energy dissipation rate. Not withstanding this argument, a number of features about the relationship between the TKE and the scalar spectral exponents are revealed by this solution when k k s. For this range of wavenumbers, φ cc (k) C 1 k α = C 1 k (m 5)/2. This relationship between α = (m 5)/2andm, also derived in Danaila and Antonia (29), is now explored for the following case studies. 2.1 Three-Dimensional Turbulence and the Inertial Subrange According to the classical ISR theory earlier mentioned for locally homogeneous and isotropic turbulence, φ tke = C o ɛ 2/3 k 5/3 for very high Reynolds number, where C o is the Kolmogorov constant. Hence, for such a TKE spectrum, a = C o ɛ 2/3 and m = 5/3, resulting in C 1 = C c N ɛ 1/3 with C c = (3C o ) 1/2 and α = 5/3, which are consistent with the K41 scalar spectrum within the ISR. Also, when C o =.5, the numerical value of the spectral scalar constant C c =.82 is in good agreement with reported values for temperature fluctuations measured at high Reynolds numbers in the atmospheric surface layer (Hsieh and Katul 1997). 2.2 Two-Dimensional Turbulence and the Inverse Cascade According to the Kraichnan Batchelor ( KB) theory (Kraichnan 1967; Kraichnan and Montgomery 198), a distinctive feature of forced two-dimensional turbulence is that when energy is injected into the system at wavenumber k i, the spectra φ tke ɛ 2/3 k 5/3 for k < k i (inverse energy cascade), and φ tke (k) β 2/3 k 3 for k > k i (direct enstrophy cascade),

5 Wake Production on the Scaling Laws 87 where k = 2π/l is the wavenumber, l is an eddy size, and β is an enstrophy injection rate (assumed to be constant), respectively. The validity of this scaling relationship relies on turbulence being well developed (e.g. R λ > 1) and constantly forced throughout the fluid volume so that ɛ and β remain constant (Kellay and Goldburg 22). These scaling predictions have been confirmed for grid turbulence in a flowing soap film (Couder et al. 1989; Martin et al. 1998; Rutgers 1998; Kellay and Goldburg 22). Based on the analytical result here between α and m, the inverse cascade is characterised by m 3, and the predicted α 1 consistent (in the limit) with the Batchelor spectrum (Batchelor 1959) for passive scalars in quasi two-dimensional turbulence. The α 1 is also consistent with a wealth of numerical simulation results and laboratory experiments for the inverse cascade portion of two-dimensional turbulence (Wells et al. 27). In short, Equations (3) and(4) can reproduce the key spectral power-law features for three-dimensional turbulence and, in the limit, for two-dimensional turbulence provided φ tke possess a clear power-law behaviour. 2.3 Canopy Turbulence and Wake Generation In the case of canopy turbulence, φ tke may no longer possess a clear scaling law due to the work that the flow exercises against the foliage drag (C d A l,wherec d is a dimensionless drag coefficient and A l is the leaf area density or frontal area density). Hence, the inference of τ(k) from φ tke becomes difficult because the canonical form of φ tke is not a priori known and difficult to predict in such a situation (Poggi and Katul 26). Nonetheless, it may be more convenient to derive τ(k) from simplified dimensional considerations rather than from φ tke (Corrsin 1964) via Equation (4). Dimensional considerations would suggest that the relevant processes governing τ(k) should, at minimum, include the wake generation mechanism (C d A l u 3 ), the enstropy injection rate (β), and wavenumber k. Moreover, if we assume that the enstrophy injection rate is independent of k, then these variables can be uniquely combined to yield: β τ(k) = C 2 C d A l u 3 k 2, (11) where C 2 is a similarity constant. Upon replacing this representation of τ(k) in Equation (3) and simplifying yields, φ cc (k) = C 2 k 3 ( 2χ c C 3 β/(c d A l u 3 ) ). (12) This solution should hold for the range of scales affected by the wake generation mechanism, which is likely to commence around eddy sizes as large as 5d r (i.e. the scale at which enstrophy is injected into the flow by the rods) to scales commensurate with.1d r provided this scale is much larger than the Kolmogorov microscale (η = (ν 3 /ɛ) (1/4) ), where ν is the kinematic viscosity. Finally, while the resulting τ(k) from this similarity argument still exhibits a power-law scaling (k 2 ), reconciling this power-law result with Equation (5) would have yielded a TKE spectrum that should increase with increasing k rather than decay, a finding that suggests that the TKE spectrum for this range of scales does not possess a clear power-law behaviour as already shown elsewhere (Poggi and Katul 26). We should also note here that the finite u component inside the canopy and a small χ c would lead to φ cc k 3, implying that the term 2χ c C 3 β/(c d A l u 3 ) can be ignored. In the discussion section, an order of magnitude estimate is provided in support of neglecting this adjustment to the 3 exponent.

6 88 D. Poggi et al. The exponents of the scalar concentration fluctuation spatial spectra (in k x and k y )are analysed next to assess their isotropy and their numerical values (i.e. 3, 5/3, or 1) inside dense canopies using observations from the experiments. 3 Experimental Facilities While the flume configuration, the rod canopy, the sensors used to acquire the velocity and scalar concentration time series, and the data processing are presented elsewhere (Poggi et al. 22, 24a,b, 28; Poggi and Katul 26), the salient features are briefly reviewed. The experiment was carried out in a large rectangular constant head recirculating channel, 18 m long,.9 m wide, and 1 m deep with glass side walls to permit the passage of laser light. The canopy was composed of vertical stainless steel cylinders.12 m tall (= h)and.4 m in diameter (= d r ) arrayed in a regular square pattern at n p = 172 rods m 2,wheren p is the canopy density defined as the number of rods per unit ground area. Using the h, d r,andn p, the effective frontal area index is.81 m 2 m 2. The drag coefficient C d.3 for this large value of n p was comparable to drag coefficients reported for agricultural crops and dense forests, which often range from.15 to.3 (Katul et al. 24). 3.1 Velocity Measurements The longitudinal and vertical velocity time series were measured using two-component laser Doppler anemometry (LDA). Further details about the LDA configuration and signal processing can be found elsewhere (Poggi et al. 24a,b). The LDA sampling frequency and duration were 3 Hz and 3 s respectively, and were deemed appropriate to resolve all the three canonical regions of the TKE spectra, including the viscous dissipation region, above the canopy. The measurements were carried out when the water depth (= h w ) attained a uniform and steady-state value of.6 m. The velocity sampling location was chosen such that the measured local temporal statistics were representative of the horizontally-averaged temporal statistics, following an analysis of the more spatially expansive data for the model canopy. The entire water depth was sampled at.1 m vertical increments. 3.2 Scalar Concentration Measurements The local instantaneous dye concentration in a plane parallel to the channel bottom was measured using the laser-induced fluorescence (LIF) technique. The concentration measurements were conducted by (i) injecting Rhodamine 6G as a tracer, (ii) providing a horizontal light sheet between two lines of rods using a lens system, and (iii) recording a time sequence of images. The light source was provided by a 3 mw continuous fixed wavelength ionargon laser (Melles Griot mod. 543-A-A3), and the images were recorded at 25 Hz using a colour CCD video camera (Poggi and Katul 26; Poggi et al. 26). Digital movies with a spatial resolution of.17 m ( pixels) were collected at two heights: z/h =.2 and z/h =.5. Three 2- min video sequences for each of the two depths were then used to compute the instantaneous two-dimensional planar concentration. The velocity and concentration measurements were collected at two bulk canopy Reynolds numbers, Re (= u h/ν) of 6 and 12, where u = ( u w ) 1/2 is the friction velocity measured at z/h = 1, primed quantities are turbulent excursions, the overbar indicates time averaging over the sampling duration.

7 Wake Production on the Scaling Laws 89 4Results Before presenting the φ cc and their exponents, an overview of the bulk flow properties is warranted. 4.1 The Bulk Flow Field Figure 1 presents the LDA measured profiles of u, the normalised velocity component, standard deviations (σ u = (u 2 ) 1/2, σ w = (w 2 ) 1/2 ), the dissipation rate ɛ, and the Taylor microscale Reynolds number (R λ )forthetwore runs. Unlike boundary-layer flows, u is finite throughout the canopy and is characterised by an inflection point near the canopy top (z/h = 1) resembling mixing layers rather than boundary layers (Raupach et al. 1996). The fact that the instabilities near the canopy top are analogous to mixing layers (i.e. Kelvin Helmholtz instabilities) has important consequences to the frequency of the occurrence of sweeps, particularly to their mean return time interval. It is this (long) return interval of sweeps that allows the flow deep inside the canopy to establish a quiescent phase dominated by von Karman streets. The return interval of sweeps is generally longer than what would be expected for a typical boundary layer with a similar aerodynamic roughness length (Raupach et al. 1996). In terms of bulk second moments, the value of σ w /u 1.1 rather than 1.25 at z/h = 1 also suggests that the flow near the canopy top is analogous to a mixing layer (Raupach et al. 1996). In general, the standard deviations of the velocity components (σ u, σ w ) appear comparable to each other deeper in the canopy at a given Re, suggesting that the isotropic approximation in the scalar spectral budget is not contradicted by the order of year and field data (Katul and Chang 1999; Finnigan 2; Poggi et al. 24a). While u and the normalised velocity variances are attenuated inside the canopy by factors of 2 3, they do remain significant at the levels where the LIF measurements were conducted (z/h =.2,.5)., was.3m or about.25h. This distance was much larger than d r so that when an apparent inverse cascade is spawned by the production of von Karman vortices and these vortices begin to grow in size (rather than break up as in three-dimensional turbulence due to vortex stretching), they have ample distance to develop before colliding with the adjacent rod. This development time ( 5d r /u(z)) is comparable to or faster than the mean return interval of sweeps (.3h/u h ), where u h = u(h) is the mean velocity at the canopy top (Raupach et al. 1996; Katul et al. 1998). Note that the ratio of the development time to the mean return interval of sweeps is 15 (d r /h) (u h /u(z)),which is on the order of unity for the arrangement here (z/h =.2 and.5). While Re is high for both runs, the canopy flow here is characterised by two other important Reynolds numbers that remain low (but finite) at z/h =.2 and.5. The first is the element Reynolds numbers (Re e = u d r /ν). The finite yet small (Re e = 2 4) is needed for producing organized von Karman streets at a near-constant St =.2 (Fey et al. 1998). The second is the Taylor microscale Reynolds number (R λ ). The R λ profiles, computed from the ɛ and σ u profiles shown in Fig. 1, are indeed small in the deeper layers of the canopy and appear insensitive to variations in Re.ThesmallR λ for the two runs here also suggests that once von Karman streets are produced, their rapid distortion or dissipation by canopy turbulence is small. It is for this reason that the von Karman streets remain spatially coherent after being spawned from the rod canopy with an energy injection scale l i 5d r. Based on this injection scale and the value of n p, the rod spacing (=.3 m) is larger than l i =.2 m thereby allowing the locally produced von Karman streets to grow before The distance between the rods, n 1/2 p

8 9 D. Poggi et al. U (ms -1 ) uw/u * 2, σu /u *, σ w /u * z/h c z/h c Re * =6 Re * = z/ h du/dz (s -1 ) R λ z/h Fig. 1 Variations in the mean velocity component u (top-left) for the two bulk Reynolds numbers, the second moments (top-right) of the flow normalised by u at the canopy top (z/h = 1) the vertical derivative of U (bottom left), and the Taylor microscale Reynolds number R λ (bottom right). The mean velocity statistics in the top-left panel were not normalised by u to graphically highlight the impact of the bulk Reynolds number (Re = U b h w /ν) differences across the two experiments, where U b is the depth-averaged velocity magnitude across the water depth (h w ). Here, z/h = 1 is the canopy top. The LIF measurements were conducted at z/h =.2,.5 colliding with the downstream rods. Because the LIF images cover only one rod spacing in the spanwise direction, the focus is primarily on the potential onset of a 3 exponent in the scalar concentration spectrum. The 3 power law is expected to hold at scales smaller than l i =.2 m (injection scale) and to be terminated at scales larger than.17 m, which is the spatial resolution of the acquired images. 4.2 Instantaneous Scalar Spectra in Space and the Dynamics of Their Scaling Exponent The time sequences of concentration images were analysed using a two-dimensional fast Fourier transform. The φ cc, computed for each image using the squared amplitudes of the Fourier coefficients, were determined in the streamwise and transverse directions. The scaling exponent α in each of these directions (hereafter referred to as α s and α T for streamwise and transverse directions, respectively) were determined from a regression analysis applied to spatial scales ranging from.25d r to 2.5d r (see Fig. 2). The upper limit is smaller than the injection scale l i and the lower limit is chosen to be larger than twice the spatial resolution of the acquired images. The outcome of this analysis is a time series of α s and α T computed for each image (e.g. see Fig. 2), at each z/h and for each of the two Re.Bothα s and α T are presented to highlight the degree of isotropy in the scaling exponents. Figure 3 presents this time series for the lower Re (for illustration), and also shown are the ensemble-averaged φ cc

9 Wake Production on the Scaling Laws 91 (a) (b) Φ cc (t,k) Φ cc (t,k) k/d R k/d R α T (a) (b) t (sec) Fig. 2 Top panels Two instantaneous images collected at z/h =.2andRe = 6, one during a quiescent period showing the von Karman streets, the other during a mild sweeping event where the canonical features of the organised von Karman streets are disturbed. The rods are shown as white circles and the mean flow direction is from bottom to top. Middle panels The instantaneous scalar spectra (φ cc (t)) corresponding to these two images as a function of normalised wavenumbers. The rod diameter d r is the normalising variable for the wavenumbers. The 5/3and 3 scalings are also shown as solid lines. Wake energy is injected at k/d r =.2. The range of scales (in red) used in the regression analysis to determine α are shown (k/d r = ). Note that the power-law scaling is closer to 3 in the left panel (dominated by von Karman streets) while it is closer to 1.67 in the right panel (dominated by finer scale turbulence). Bottom panels A 25-s time sequence of the computed transverse exponent α T for z/h =.2 andre = 6. The instances corresponding to these two images are shown as vertical dashed lines for the two z/h values. From Fig. 3, α 3 in both streamwise and transverse directions, much of the time consistent with the 3 scaling derived here. In the canopy experiment, the sweeping motion from aloft periodically disrupts this mechanism (see Fig. 2, top panels), which after some initial phase, re-establishes itself between two successive sweeps. Increasing Re increased the frequency of sweep disturbances, known to linearly scale with u/h as earlier discussed. The increase in the sweep-induced disturbance increases the variance of α. Instances where the dye was partially or completely washed out by a strong sweeping event often leads to spurious exponents. These effects are best captured in Fig. 4, which shows the probability density function p(α) of the spectral exponents α s and

10 92 D. Poggi et al. Fig. 3 The time series of the computed scaling exponents α s (streamwise direction) and α T (transverse direction) for each LIF image at the two heights (z/h =.2,.5) and at Re = 6. The ensemble averaged scalar spectra φ cc (k) as a function of normalised spatial wavenumber k are also shown. The ensemble averaging is performed across all sequence of images. The rod diameter d r is the normalising variable for the wavenumber. The 5/3and 3 scalings are also shown as solid lines. Wake energy is injected at k/d r =.2. The range of scales used in the regression analysis to determine α in the top and middle panels span more than one decade (k/d r = ) α S α T z/h c =.2 t (sec) z/h c =.5 t (sec) α T α S < Φ cc > < Φ cc > k/d r k/d r α T for the two Re and the two z/h. Forz/h =.2, the mode of p(α T ) 3 irrespective of Re, while the mode of p(α S ) 2.7. The difference between the p(α T ) and p(α S ) modes became smaller for z/h =.5 but this difference was not appreciably affected by Re. The impact of increased Re was most evident in the spread in p(α) at z/h =.5, as expected, given the increases in the frequency of sweeps. The frequency of washed out dye (corresponding to sweeping events) was captured by the frequency of α <1.5 (a spurious scaling). In fact, doubling Re roughly doubles the spread in exponents shown in Fig Discussion and Conclusions A novel scaling relationship for the scalar concentration fluctuation spectra inside dense canopies was proposed using a simplified scalar spectral budget. It was tested with detailed flume experiments using a densely arrayed rod canopy. The solution to this budget resulted in a 3 scaling exponent that commences at scales smaller than the energy injection scale by wakes. One of the main assumptions in this derivation was that the relaxation time scale of TKE depends on a wake production and an enstrophy injection rate that are independent of k, the latter being qualitatively analogous to the inverse cascade in two-dimensional turbulence. Thegenesis of the 3 power law here substantially differs from another theoretical result leading to the same scaling law for the so-called inertial-diffusive range (Gibson 1968). This spectrum can be readily derived from dimensional arguments for the case when the Prandtl number (Pr = ν/χ c ) is very small. Experimental support for the 3 powerlaw

11 Wake Production on the Scaling Laws 93 z/h c = z/h c = pdf(α T ),pdf(α S ) Re * = pdf(α T ),pdf(α S ) Re * = α T,α S α T,α S Fig. 4 The probability density function p(α) for each of the two heights and two Re. The curves for α T are dashed and for α s are solid. The best-fit Gaussian parabola is also shown for reference in the inertial-diffusive regime at Pr =.2 was reported using temperature measurements within a mercury fluid (Rusk and Sesonske 1966; Gibson 1968). Based on these arguments, φ cc (k) depends only on N, χ c,andk leading to (Gibson 1968) φ cc (k) N χ c k 3, (13) For the set-up here, the χ c for Rhodamine 6G is on the order of m 2 s 1, while ν is on the order of m 2 s 1. Hence, Pr > 3 and the observed 3 power law cannot be linked to the existence of an inertial-diffusive range. In fact, the finite u component in the deeper layers of the canopy and the small χ c are the main culprits for why a 3 and not a steeper exponent was observed in the LIF concentration measurements here. Equation (12) predicts an exponent steeper than 3 by2χ c C 3 β/(c d A l u 3 ). This anomalous adjustment to the 3 power law scales as 2χ c /(5d r u) at wavenumbers commensurate with wake production. From Fig. 1, with u =.1ms 1 and a d r =.4 m, this anomalous adjustment to the 3 exponent is 1 4 and can be ignored. Whether this scaling law here actually holds for scalar transport in real forested canopies is currently being explored. A number of difficulties may preclude its onset in field experiments due to the fact that (i) the wake generation mechanism occurs at multiple spatial scales (trunks, branches, etc...), (ii) the density stratification inevitably exists inside dense canopies, and (iii) the many scalars of interest (listed in the introduction here) are not generally passive but are biologically active. Acknowledgements We thank K.S. Lee and S. Thompson for helpful comments and assistance in the image analysis. G. Katul acknowledges support from the FulBright-Italy fellows program, the National Science Foundation (NSF-EAR , , and NSF-ATM , ) and the Binational

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