Spectral Short-circuiting and Wake Production within the Canopy Trunk Space of an Alpine Hardwood Forest

Boundary-Layer Meteorol (2008) 126:415–431DOI 10.1007/s10546-007-9246-x

ORIGINAL PAPER

Spectral Short-circuiting and Wake Production withinthe Canopy Trunk Space of an Alpine Hardwood Forest

Daniela Cava · Gabriel G. Katul

Received: 20 June 2007 / Accepted: 13 November 2007 / Published online: 8 December 2007© Springer Science+Business Media B.V. 2007

Abstract Using synchronous multi-level high frequency velocity measurements, theturbulence spectra within the trunk space of an alpine hardwood forest were analysed. Thespectral short-circuiting of the energy cascade for each velocity component was well repro-duced by a simplified spectral model that retained return-to-isotropy and component-wisework done by turbulence against the drag and wake production. However, the use of an aniso-tropic drag coefficient was necessary to reproduce these measured component-wise spectra.The degree of anisotropy in the vertical drag was shown to vary with the element Reynoldsnumber. The wake production frequency in the measured spectra was shown to be consistentwith the vortex shedding frequency at constant Strouhal number given by fvs = 0.21u/d ,where d can be related to the stem diameter at breast height (dbh) and u is the local meanvelocity. The energetic scales, determined from the inflection point instability at the canopy–atmosphere interface, appear to persist into the trunk space when Cduacr hc

β� 1, where Cdu

is the longitudinal drag coefficient, acr is the crown-layer leaf area density, hc is the canopyheight, and β is the dimensionless momentum absorption at the canopy top.

Keywords Canopy turbulence · Foliage drag · Spectral short-circuiting ·Strouhal instabilities · Trunk space · Wake production · Von Karman streets

1 Introduction

The spectral properties of turbulence within the trunk space of forested ecosystems are nowreceiving significant attention because of the proliferating number of applications requiring

D. Cava (B)CNR - Institute of Atmosphere Sciences and Climate, National Research Council (CNR), Lecce, Italye-mail: [email protected]

G. G. KatulNicholas School of the Environment and Earth Sciences, Duke University, Durham, NC, USAe-mail: [email protected]

G. G. KatulDepartment of Civil and Environmental Engineering, Duke University, Durham, NC, USA

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416 D. Cava, G. G. Katul

these properties. For example, dispersal and release of spore and pollen in the under-story(e.g. Raynor et al. 1974; Sawford and Guest 1991; Wilson 2000; Aylor and Flesch 2001;Kuparinen 2006), transport of respired CO2 from the forest floor (e.g., Juang et al. 2006) orrelease of NO and its conversion to O3 (e.g. Kaplan et al. 1988; Backwin et al. 1990), andspread of odour and development of chemo-tactic algorithms (e.g. Vergassola et al. 2007)all require some knowledge of time or length scales ‘encoded’ in the turbulent energy spec-trum. While all these sample applications deserve significant research attention, examiningthe main mechanisms by which turbulent eddies are produced, dissipated, and transportedacross various wavenumbers (K ) remains a legitimate fundamental problem in its own right.

In the canopy sub-layer (i.e. the layer dynamically influenced by the canopy structureand by its characteristic length scales) dominant eddies are linked to a hydrodynamic insta-bility process produced by the inflection point in the mean velocity profile in the uppercanopy. These eddies have integral length scale of order the canopy height and their energycorresponds to the main spectral peak. Moreover two additional processes, absent in theatmospheric surface layer (ASL), are known to modify the turbulent kinetic energy (TKE)spectrum in the canopy sub-layer: (1) the work that the mean flow exerts against the foliagedrag (WD) thereby producing TKE by wakes (hereafter referred to as WKE), and (2) thespectral short-circuiting of the energy cascade that represents the same process but acts onturbulent eddies rather than on the mean flow (Finnigan 2000; Poggi and Katul 2006). Thesetwo effects primarily modify the TKE energy cascade within the so-called inertial sub-range(ISR) exhibiting the well-celebrated K −5/3 scaling. Hence, progress on these two processesmay offer new clues on the problem of turbulence itself.

Theoretical (Wilson 1988; Finnigan 2000; Poggi and Katul 2006) and laboratoryexperiments have already explored some aspects of these two spectral modifications, butfield support remains lacking, especially for the latter (see summary review in Table 1).Finnigan (2000) noted that the WKE of leaves is essentially ‘invisible’ to standard sonicanemometry measurements given their large averaging path length (≈0.10–0.15 m) with re-spect to leaf dimensions. Moreover, the unavoidable multiplicity of length scales responsiblefor wake production inside forested canopies can frustrate the unambiguous detection ofthese two spectral mechanisms in field experiments (e.g., see Table 1). This multiplicity inlength scales tends to spectrally broaden the wake generation and reduce the scale separationbetween shear- and wake-production, even blurring them over a range of frequencies (or K ).However, within the trunk space of forested canopies without significant under-story foliage,variability in wake-producing length scales may be much less significant when compared tothe crown space. Tree stems have diameters comparable to or larger than the sonic anemo-meter pathlength and often populate trunk spaces. It follows that (i) a wake production lengthscale (d) may be dominant and resolvable by standard sonic anemometry, and (ii) this d issufficiently smaller than large-scale eddies (of order the canopy height) in the trunk space sothat the separation between shear production and wake production scales is sufficiently largefor unambiguous identification of the latter. Finally, given that the element Reynolds numbersin the trunk space is not too large in dense canopies, the frequency of vortex shedding ( fvs)may be predicted from d and the local mean longitudinal velocity (u) assuming a constantStrouhal number (St = fvsd/u) (e.g. Cantwell and Coles 1983; Tritton 1988; Williamsonand Brown 1998).

Here, measured spectra of the three velocity components in the trunk space of an alpinehardwood forest were used to explore how the short-circuiting of the energy cascade andwake production modify the ISR. The theoretical framework for this analysis was alreadypresented in Finnigan (2000) but is revised here to include the lateral velocity component

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Spectral Short-circuiting and Wake Production 417

Table 1 Partial list of canopy turbulence experiments reporting short-circuiting and/or wake production intheir velocity spectra

Study Short-circuiting Wake production

Wind-tunnel and flume experiments

Seginer et al. (1976) Yes Yes, at St = 0.21Raupach et al. (1986) Yes Yes, for w (including data just above the canopy)Poggi et al. (2004) Yes Yes, at St = 0.21

Atmospheric Surface Layer/Tower wakeBarthlott and Fiedler (2003) Yes Yes, mainly for w and v. They note that the frequency of

tower wakes is independent of stability.Forested canopies

Amiro (1990) Yes No, except a secondary peak in w was observed at thelowest level in a black-spruce forest canopy

Baldocchi and Meyers (1988) Yes YesBlanken et al. (1998) Yes NoGardiner (1994) Yes NoKruijt et al. (2000) Yes No, except for a possible value at z/hc = 0.27 at

Cuieiras forest.Liu et al. (2001) Yes NoLauniainen et al. (2007) Yes No, except a clear secondary peak in w observed at the

lowest level in a pine forest canopyVillani et al. (2003) Yes No, except for stable runs

and an anisotropic drag coefficient. Furthermore, the hypothesis that the frequency at whichwake production occurs can be predicted from a constant St is also explored.

Unless otherwise stated, the following nomenclature is used: the canopy crown refers tothe layer populated by foliage, the trunk-space refers to the layer where stems and somedead branches mainly impose drag on the flow, and the forest floor refers to the lowermostlayer just above the litter. The longitudinal, lateral, and vertical coordinates are x , y, and z,respectively, and the instantaneous velocity components along these three directions are u,v, and w, respectively. Overbar indicates time (and space) averaging (Raupach and Shaw1982).

2 Experimental set-up

Much of the experimental set-up is described in Marcolla et al. (2003) and Cava et al. (2006)and only the salient features are repeated. The experimental site is an uneven-aged mixedconiferous forest in Lavarone, Italy (45.96◦ N, 11.28◦ E; 1,300 m asl). The canopy is about28 m tall (= hc) and is primarily composed of Abies alba (70%), Fagus sylvatica (15%),and Picea abies (15%). The tree crown commences at 10–12 m above the forest floor. Themaximum leaf area index (LAI), when expressed as half of the total leaf surface area per unitground area, is 9.6 m2 m−2, and more than 90% of this leaf area is concentrated in the crownspace (see Fig. 1). The tree density (ntd) averaged 1,300 stems ha−1 (or 0.13 stems m−2) fortrees with diameter at breast height (dbh) exceeding 0.075 m. The dbh ranged from 0.075 to0.7 m. In the under-story, some suppressed beeches form a discontinuous second layer from0 to 4 m (see Fig. 1).

The data were collected as part of an intensive field campaign conducted between Juliandays 222 and 251 in year 2000. The tower was equipped with five sonic anemometerssituated at 33, 25, 17.5, 11 and 4 m from the forest floor (see Fig. 1). The turbulent velocitycomponents and sonic anemometry temperature were sampled at 20 Hz and half-hourly mean

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Fig. 1 Leaf area densitydistribution adjacent to the tower(black line) and locations of sonicanemometers (diamond). Thedashed line represents the meancanopy height and the dot linerepresents the crown base

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.2

0.4

0.6

0.8

1

1.2

a (m−1)

z/h c

Crown Space

Trunk Space

Fig. 2 Conceptual diagram showing departures from ISR scaling due to short-circuiting and wake production(after Finnigan 2000). The vortex shedding visualization is from Taneda (1965)

values were computed for each level following two coordinate rotations to ensure v = w = 0.The atmospheric stability conditions were classified based on hc/L , where L is the Obukhovlength measured at the canopy top. Since stable stratification is known to dampen eddy sizesand thereby diminishing the spectral spread between shear-produced and wake-producedturbulence, only runs with −hc/L > 0 were used here.

3 Theory

Figure 2, revised from Finnigan (2000), schematically shows how the short-circuiting andwake production alter the standard ISR scaling within the trunk space. Models for thevelocity spectra must be able to (i) describe how this short-circuiting results in steepen-ing of the spectral exponent beyond ISR scaling (i.e. K −5/3) and (ii) estimate the energeticscales commensurate with wake production.

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Fig. 3 Idealized diagram explaining key attributes of wake production, Strouhal instabilities, and Von Karmanstreets (VKS) along with LIF measurements showing the VKS signature inside a canopy composed of denselyarrayed rods within a flume

3.1 Wake Generation and Vortex Shedding

Vortex shedding behind isolated cylinders (or stems in the trunk space) is often analysedusing a constant St that relates fvs , d , and u using

St = fvs d

u≈ 0.21,

for a wide range of element Reynolds numbers Re = du/ν ∈ [360 − 2 × 105

], where ν is

the kinematic viscosity of air (≈ 1.5 × 10−5 m2 s−1). In the trunk space, the values of themean longitudinal velocity and wake generation length scale are, respectively, u ≈ 0.1 m s−1

and d ≈ 0.1 m resulting in Re ≈ 660. Hence, the order of magnitude of the Reynolds numberin the trunk space (≈ 103) is generally well within the range where St remains constant (seeTable 1 in Fey et al. 19981). This Re, however, is larger than the Re often linked with thesustained spatial coherency of classical Von Karman Streets (Re < 200) behind cylinders(a sample is shown in Fig. 3), though the Strouhal instability mechanism remains similar. VonKarman streets are successions of eddies that are produced close to a cylinder (or bluff object)that then break away alternatively from both sides of the cylinder (see Fig. 3). For moder-ate Re (<200), these vortices are emitted regularly, are symmetrically staggered laterally inthe longitudinal plane, and rotate in opposite directions (see Fig. 3 for a sample arrange-ment). This staggered arrangement and counter-rotating vortex motion is often used in visualidentification of Von Karman streets. Following their generation at higher Re (>400), theseVon Karman vortices may quickly loose their spatial coherency (i.e. lateral symmetrical stag-gering and counter-rotating motion) in the trunk space and form a turbulent wake. Taneda’s(1965) seminal experiments demonstrate that not only laminar wakes but also turbulent wakesshow a strong tendency to self-organize into a Von Karman street configuration (see Fig. 3).These visualization experiments also suggest that the stability of the Von Karman vortexstreets arrangement against artificial disturbances appears to be even enhanced for the turbu-lent case. Although the fluid motion within these vortices becomes irregular with increasing

1 In Fey et al. (1998), it was shown that St = St∗ + m√Re

, were St∗ and m varied with the Re range as

follows: For 360<Re<1,300, 1,300<Re<5,000, and 5,000<Re<2 ×105, St∗= 0.22, 0.21, and 0.18, respec-tively, and m = −0.44, +0.34, and +2.2, respectively. Nonetheless, these Re variations result in a near-constantSt ∈ [0.18 − 0.22] for 360 <Re< 2 × 105.

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Re (up to 100,000), their shedding occurs with quite precise periodicity well describedby fvs .

Some support for the existence of canopy Von Karman streets within the trunk spacecomes from wake spectra reported by Seginer et al. (1976) and from laser-induced fluo-rescent (LIF) flow visualization experiments reported in Poggi et al. (2004) and Poggi andKatul (2006). The latter two studies demonstrate that vortex motions resembling Von Karmanstreets are un-ambiguously established in the deeper layers of a model canopy composedof densely arrayed rods in a flume. Figure 3 shows one snapshot example of the LIFmeasured dye concentration in the x–y plane situated at z/hc = 0.2 (set-up and LIFdescribed in Poggi et al. 2006). Clearly, the signature of the initial phases of Von Karmanstreets vortices are evident here though these Von Karman streets cannot remain coher-ent for three reasons: (1) the Reynolds number is too high so that turbulence spreads intoregions between them and disrupts their regularity, (2) sweeping events penetrating the canopyperiodically interrupts them, and (3) once these Von Karman streets initially form and spatiallygrow (even during a quiescent phase), they are disrupted by collisions with adjacent cylinders.

Figure 4a–c present the ensemble-averaged normalized spectra for the three velocity com-ponents measured at the two levels within the trunk space along with the remaining threelevels for reference. These spectra were computed from runs of about 55 min (i.e. 216 datapoints per run). Each run has been divided into eight sections with 50% overlap after eachsection was windowed using Hamming’s method; then these eight spectra were ensembledaveraged at each frequency to obtain the spectrum relative to the single run. While the spectraabove the canopy follow their usual ASL form in the ISR (e.g., Su et al. 2004), an ambiguousenergy peak emerges within the trunk space (z/hc = 0.14) and just below the crown space(z/hc = 0.39). This energy peak occurs at wavenumbers well predicted by St = 0.21 whensetting d = 0.2 m. Other laboratory studies and tower distortion studies for the ASL, alsopresented in Table 1, support the finding that wake production frequencies are well predictedby a constant St = 0.21. The d = 0.2 m here is a reasonable choice when consideringthat dbh varied from 0.075 to 0.7 m in this stand. Moreover, this energetic wavenumber(= St/d) is the same for all the three velocity components and for these two trunk-spacelevels (z/hc = 0.14 and 0.39). Note that d/St ≈ 1 m is actually smaller than the averagetree-to-tree spacing (estimated as 1/

√ntd ≈ 2.8 m) ensuring that at least two shedding occur

before a vortex collision with adjacent trees.Given the large variations in wind direction within the trunk space in a typical 30-min

period, vortex shedding is likely to originate from multiple stem sizes surrounding the tower.Hence, a portion of the spectral ‘broadening’, about a decade around d = 0.2 m in the ensem-ble spectra shown in Fig. 4, may be explained by variability in dbh. Recall that Von Karman’sanalysis employs linear stability theory for a two-dimensional inviscid flow. Hence, variationsin fvs linearly scale with variations in dbh, the latter varied by about a decade in this stand.

3.2 Spectral Short-circuiting

As shown schematically in Fig. 2, eddies characterized by wavenumbers d/St � K −1 � hc

lose TKE to WKE and heat thereby violating one of the basic assumptions of ‘inertial’ energycascade. For this range of wavenumbers, models for the three individual velocity spectrawithin the trunk space must account for at least two basic mechanisms: the continual extrac-tion of energy by the drag elements and the loss or gain of energy by the pressure-velocityinteraction—the main mechanism responsible for local isotropy in the absence of canopydrag.

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100

10−2

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

101

10−3

10−2

10−1

100

101

(f/fvs

)

100

10−2

(f/fvs

)

100

10−2

(f/fvs

)

f S u (

f)

z/hc=1.18z/hc=0.89z/hc=0.62z/hc=0.39z/hc=0.14

a)

−2/3

f S v (

f)

b)

−2/3

Top

Crown

TrunkSpace

f S w

(f)

c)

−2/3

Fig. 4 Variations of ensemble-averaged normalized energy spectra for unstable conditions at all five heightsas a function of normalized frequency f for the longitudinal velocity (a), lateral velocity (b), and verticalvelocity (c). The ISR scaling Si ∝ f −5/3 is also shown for reference. The f is normalized by fvs = 0.21u/dso that the energy at f/ fvs = 1 represents wake production by Strouhal type instabilities (vertical dottedlines). The spectra at the five levels are shifted to permit comparisons

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The longitudinal, lateral, and vertical velocity spectra can be described by a simplifiedmodel (Finnigan 2000)

d

d K

(α−1

1 ε1/3 K 5/3Su(K ))

= −c1ε

Q

[Su(K ) − 1

3(Su(K ) + Sv(K ) + Sw(K ))

]

−2CduauSu(K ), (1a)d

d K

(α−1

2 ε1/3 K 5/3Sv(K ))

= −c1ε

Q

[Sv(K ) − 1

3(Su(K ) + Sv(K ) + Sw(K ))

]

−CdvauSv(K ), (1b)d

d K

(α−1

3 ε1/3 K 5/3Sw(K ))

= −c1ε

Q

[Sw(K ) − 1

3(Su(K ) + Sv(K ) + Sw(K ))

]

−CdwauSw(K ), (1c)

where K is the scalar wavenumber along the longitudinal direction (K = fu , where f is

the frequency in Hz and u is the longitudinal mean velocity in m s−1), Su , Sv , and Sw arethe longitudinal, lateral, and vertical velocity spectra, ε is the mean TKE dissipation rate

via the usual inertial eddy cascade, Q = 12

(u′2 + v′2 + w′2

)is the TKE, u′, v′, and w′ are

the instantaneous turbulent longitudinal, lateral, and vertical velocities, respectively, Cdu ,Cdv , and Cdw are the drag coefficients along the longitudinal, lateral, and vertical direc-tions, respectively, u is the mean longitudinal velocity, α1 = 0.5, α2 = α3 = 0.66 are theKolmogorov constants for the longitudinal, lateral and vertical directions, respectively,c1 = 1.8 (see Ayotte et al. 1999) and a is the local leaf area density. The first term onthe right-hand side of Eq. 1 is the energy re-distribution term among components, modelledusing standard return-to-isotropy principles, while the second term is the short-circuiting ofthe energy cascade by the drag elements, approximated using a two-term binomial expansionafter Finnigan (2000). Variants to this approximation are further discussed in the appendix.

An important simplification introduced by Wilson (1988) and later adopted by Finnigan(2000) is that ε/WD � 1 and ε maintains a near-constant isotropic value during the cascade,where WD is the work done by turbulence against the drag elements and represent the lossof energy to heat and WKE, given by

WD ≈ (au)(

2Cduu′u′ + Cdvv′v′ + Cdww′w′)

. (2)

To solve the system in Eq. 1, the three energy components must be known or specified ata reference wavenumber (Kref ). Here, Kref ≈ (0.5hc)

−1 to ensure that the calculationscommence at a K larger than the wavenumbers expected for shear production, and we usedmeasured Su(Kref ), Sv(Kref ), and Sw(Kref ) to initialise the numerical integration. Thenumerical integration was terminated before K = (d/St)−1, or the wavenumber at whichwake production is expected to occur. With 0.5hc ≈ 14 m and noting from Fig. 4 thatd/St ≈ 1 m, no more than one decade of scales experiencing spectral circuiting is to beexpected in the trunk space.

4 Results and Discussion

The theoretical framework developed above for the energy short-circuiting is tested usingthe Lavarone data. Necessary conditions that permit the determination of whether energeticlength scales developed at the canopy–atmosphere interface persist in the trunk space are

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0 30 60 90 120

0 30 60 90 120

0 30 60 90 120

−0.5

0

0.5

−0.5

0

0.5

−0.5

0

0.5

u’ (

m s−1

)v’

(m

s−1)

Time (seconds)

w’

(m s−1

)

a)

b)

c)

filtered

filtered

filtered

original

original

original

Fig. 5 Original and filtered (one decade around the fvs ) velocity time series (m s−1) showing the signatureof Strouhal instabilities. The filtered velocity is used to drive trajectory calculations in Fig. 6

also presented. But first, possible links between wake production, vortex shedding, constantSt , and Von Karman streets are further analyzed.

4.1 Vortex Shedding

Figure 4 reported the spectral energy peak at the vortex shedding frequency fvs ≈ 0.2 ud ,

where d = 0.2 m. Using this fvs estimate, the three velocity components were band-passfiltered to include one decade of Fourier modes centred around fvs and their reconstructedtime series (referred to as u f and v f ) are shown in Fig. 5 along with the original series. Usingu f and v f in Fig. 5, concomitant trajectories around the mean state were computed using

dx

dt= u f (t), (3a)

dy

dt= v f (t), (3b)

and a sub-sample of these trajectories is shown in Fig. 6 taken at an element Reynoldsnumber of 6,600. It is clear that these oscillations appear consistent with Von Karman streetinstabilities reported by Taneda (1965). The Von Karman streets developing immediatelyafter the instability eventually loose their spatial coherency to become a turbulent wake. Thehypothetical line in Fig. 6 also shows the expected trajectory if the main streamline alongthe coherent portion of the Von Karman Streets is followed. A sample trajectory from Eq. 3is shown for comparison. The analysis here is suggestive that the wake production inside thetrunk space appears to originate from vortex motion resembling Von Karman streets.

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0-0.1

0.1

0.2

0

1 2 3 4

x (m)

y (m

)

v fu f

0-0.1

-0.05

0

0.05

0.1

-0.1

-0.05

0

0.05

0.1

10 20 30

Time (seconds)

40 50 60 80

0a)

b)

10 20 30 40 50 60 80

5 6 7 8 9

Fig. 6 Top: Von Karman streets observed by Taneda (1965) behind a flat plate at an element Reynolds numberof 6,600. The wake here is simply an extension of the boundary layer formed on the plate (i.e. no significanteddy shedding). Middle: A hypothetical line tracing the expected trajectory along the organized phase ofthe Von Karman streets (delineated by vertical dashed lines). Bottom: Sample trajectories (x,y) around themean state when the instantaneous velocity perturbations are the filtered velocity (u f , v f ) in Fig. 5. Note thesimilarities between middle and bottom panels

4.2 Energy Short-circuiting

Figure 7 compares the measured and modelled energy spectra for z/hc = 0.14 and z/hc =0.39 for the lowest and highest Re (bounding all possible values). In these model calcula-tions, ε/Q = βhc/σw with β = 0.2. This relationship has been obtained by Poggi et al.(2006) for dense canopy when z/hc < 0.75, and it has been tested by Cava et al. (2006) forthe same dataset here. Furthermore, Cdu was set to a constant (= 0.25) identical to the valuereported in Marcolla et al. (2003) for low wind speeds but Cdv and Cdw were used as fittingparameters to match the measured spectra at the wavenumber just prior to wake generation.The latter wavenumber was taken as the minimum wavenumber before an increase in energywas detected with increasing wavenumber for d/St � K −1 � hc/2.

In terms of degrees of freedom, the Su budget has no fitting parameters because Cdu wasalready determined separately from the mean longitudinal momentum budget (described inMarcolla et al. 2003). For Sv and Sw budgets, each has one fitting parameter (Cdv and Cdw).The optimised Cdv appeared to be 10 times larger than Cdu irrespective of Re but the opti-mised Cdw varied anywhere from 15 to 40 times Cdu depending on u (for z/hc = 0.14). Infact, Fig. 8 shows the variation of Cdw/Cdu as a function of Re for all the runs (at both levels).With increasing Re, the apparent anisotropy in Cdw/Cdu diminishes but remains significantlylarger than unity (at both levels). Whether this is indicative of actual anisotropy in the formdrag or simply an outcome of model assumptions is difficult to discern without independent

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10−2

10−1

100

101

10−4

10−3

10−2

10−1

k (m−1)

kS(k

)

z/hc=0.14

10−2

10−1

100

101

10−4

10−3

10−2

10−1

k (m−1)

kS(k

)

z/hc=0.39

10−2

10−1

100

101

10−4

10−3

10−2

10−1

k (m−1)

kS(k

)

10−2

10−1

100

101

10−4

10−3

10−2

10−1

k (m−1)

kS(k

)

a) b)

c) d)

Cdw

=40

Re =2067

Cdw

=30

Re =3000

Cdw

=20

Re =2133

Cdw

=25

Re =1867

Fig. 7 Comparison between measured (Su in *; Sv in o; Sw in +) and modelled (lines) energy spectra insidethe trunk space for z/hc = 0.14 (left) and z/hc = 0.39 (right) and for a wide range of RE. The ISR scalingis also shown as in Fig. 4

Fig. 8 Cdw/Cdu versus Re forall the runs at z/hc = 0.14 (stars)and z/hc = 0.39 (diamonds).Horizontal bars represent onestandard deviation

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 60000

5

10

15

20

25

30

35

40

45

50

Re

Cd w

/ C

d u

estimates of the anisotropy in ε and an assessment of the return-to-isotropy spectral closuremodel. However, we note from Fig. 8 that the regression model Cdw/Cdu ≈ 70916 Re−1+10(at z/hc = 0.14) well reproduced the data and is suggestive that Cdw → Cdv (i.e. both dragcoefficients are 10 times larger than Cdu thereby hinting at possible isotropy in the planeperpendicular to the mean flow direction) as Re increases.

We also tested how modelled ε/WD varied with Re, where WD is integrated over the entirespectra (Fig. 9). Recall that one of the primary assumptions in Eq. 1 is that ε/WD � 1. With

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1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 60000.15

0.2

0.25

0.3

Re

ε / W

D

Fig. 9 Same as Fig. 7 but for ε/Wd

increasing Re, Fig. 9 suggests that more of the total amount of TKE flowing to dissipationscales occurs via WD when compared to the flux of energy cascading at ε. For the higherReynolds number range, an approximate ‘plateau’ is reached at a value of about 0.21, which isin good agreement with the value (= 0.2) reported by Shaw and Seginer (1985) and Finnigan(2000). Despite the simplified estimate of ε, ε/WD here agrees with other studies. Finally,with increasing Re, the ε in the wavenumber range d/St < K −1 < 0.5hc may approach itsisotropic state suggestive that part of the anisotropy in the drag may be linked to anisotropyin ε. In the Appendix, we also show that the error in representing

(|u| ui ui − u3)

by uu′i u

′i

decreases with increasing Re when Re < 1,000. This dependency is also suggestive that thebinomial expansion approximation may contribute to the apparent anisotropy in the drag, butcannot solely explain it given that the Re in Figs. 7 and 8 exceeds 1,000.

4.3 Shear Production Length Scale

The small ε/WD has an important implication to the coupling between the turbulencegenerated at the forest–atmosphere interface and the turbulence in the trunk space. To illus-trate, consider the eddy de-correlation or relaxation time scale, given by

τ =12 u′

i u′i

ε + WD. (4)

With ε/WD � 1, and WD ∼ 12 Cdauu′

i u′i (see Appendix for the 1/2 factor), results in

τ =12 u′

i u′i

WD≈ 1

Cdau. (5)

Using Taylor’s frozen turbulence hypothesis, Ld = τ u and defines a characteristic eddy sizethat ‘adjusts’ to the canopy drag, estimated as

Ld ∼ 1

Cda. (6)

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This length scale can be compared to the shear length scale produced by the inflection-pointinstability near the canopy top, defined as (Raupach et al. 1996; Finnigan 2000)

Ls = u

∂ u/∂z. (7)

For a near-exponential mean velocity profile attenuated at a rate β = u∗U

,

Ls = hc

β. (8)

Hence, LsLd

≈ hcβ

Cda ≈ Cdahcuu∗

Now, if Ls/Ld � 1 (say of order 10), then eddies produced by inflectional instabilities in themean flow near the canopy–atmosphere interface remain coherent within the trunk space. Forthe Lavarone site, using a Cd ≈ 0.2, a ≈ L AI/δ being representative of the crown where δ

is the crown thickness (≈ 0.6hc), and β ≈ 0.3 results in

Ls/Ld ≈ 0.2 × L AI

0.6hchc

1

0.3≈ 1.11L AI > 10,

for LAI = 9.6 m2 m−2. Hence, based on the above argument, Ls maintains its coherencyalmost all the way into the trunk space and dictates the main energetic length scale there.Figure 4 supports this argument given that the energetic peak at z/hc = 0.89, 0.62, 0.39remains coherent throughout. These findings are consistent with the conclusions reportedin Villani et al. (2003). Based on the analysis here, if Ls/Ld � 1 (as is the case for theVillani et al. 2003), then eddies generated near the canopy top are likely to survive into thetrunk space. However, at z/hc = 0.14, an intermittent boundary layer may originate fromthe forest floor and the increased role of wake production distorts this peak.

5 Summary and Conclusions

The spectrum of turbulence within the trunk space for unstable atmospheric conditionsexhibited three properties: (i) the energetic scale of the large eddies appear to be determinedby the inflection point instability when Cdahc

uu∗ � 1, (ii) the spectral short-circuiting of

the energy cascade for each velocity component is well reproduced by a simplified spectralmodel that retains the return-to-isotropy and the component-wise work done by turbulenceagainst the drag and wake production, and (iii) the wake production frequency is well mod-eled from an obstacle length scale d=dbh, the local mean velocity u, and a constant Strouhalnumber (i.e. fvs = 0.21u/d).

For reproducing the measured departure from inertial subrange scaling using this simpli-fied model of the short-circuiting, an anisotropic drag coefficient was necessary. The degreeof anisotropy in the drag was also shown to vary with the element Reynolds number. Thereasons for the strong anisotropy and its dependence on the element Reynolds number remainspeculative. Possible explanations range from errors in the binomial approximation for thework done by turbulence, which approximates the

(|u| ui ui − u3)

by uu′i u

′i , to anisotropy in

ε that decays with elements Reynolds number. Future work will explore how stable stratifi-cation modifies this emerging picture for the trunk space.

Acknowledgements The authors thank D. Poggi for providing the Von Karman streets image used inFig. 3, and A. Cescatti for providing the profile of the leaf area density used in Fig. 1. Support from

123


‘Cooperazione Italia-USA su Scienza e Tecnologia dei Cambiamenti Climatici, Anno 2006–2008’, the Na-tional Science Foundation (NSF-EAR 0628342 and 0635787), the US Department of Energy (DOE) throughthe Office of Biological and Environmental Research (BER) Terrestrial Carbon Processes (TCP) program(Grants # 10509-0152, DE-FG02-00ER53015, and DE-FG02-95ER62083), and the Binational AgriculturalResearch and Development (BARD) program (Grant #. IS3861-06) is acknowledged.

Appendix: Various expansions of WD

An expression for the rate at which turbulent fluctuations do work (per unit density) againstan isotropic drag and lose TKE to WKE and heat can be derived from:

WD = Cda[|u| ui ui − u3] , (9a)

where |u| = √ui ui . The above expression is simply the difference between the work done

against the aerodynamic drag by the total flow and the mean flow, respectively. Using atwo-term binomial expansion around u, Finnigan (2000) showed that for low-intensity tur-bulence

WD ≈ 3

4Cdau

[u′

i u′i

]. (9b)

Other expressions can also be derived for low-intensity turbulence. For example, in verylow-intensity turbulence,

|u| ≈ u, ui ui = u2 + u′2 + v′2 + w′2, so that

WD = Cdau[u′2 + v′2 + w′2

]. (9c)

Another (conservative) approximation that may be adopted is in the treatment of the triplecorrelation. If |u| ui ui ≈ |u|ui ui , then

|u| ui ui ≈ |u|ui ui = |u|(

u2 + u′i u

′i

)≈

(u2 + u′

i u′i

)3/2 ≈((

u2 + u′i u

′i

)1/2)3

. (10a)

Hence,

|u| ui ui ≈((

u2 + u′i u

′i

)1/2)3

≈(

u

(

1 + 1

2

u′i u

′i

u2

))3

≈ u3

(

1 + 3

2

u′i u

′i

u2

)

. (10b)

Here, the quantity δ = u′i u′

iu2 � 1 so that the expansion of (1 + δ)n ≈ 1 + nδ. This

approximation results in

WD = 3

2Cdau

[u′2 + v′2 + w′2

]. (11)

This estimate may be an upper limit because |u| ui ui < |u|ui ui .Using the trunk-space velocity measurements, Fig. 9 compares measured

(|u| ui ui − u3)

with uu′i u

′i . While the scatter is large, it is clear that uu′

i u′i explains a significant portion

(r2 = 0.32) of the variation of(|u| ui ui − u3

). The regression slope in Fig. 10 is 0.5 also

suggesting that WD ≈ 12 Cdauu′

i u′ appears to be the optimal model for this dataset.

More relevant to Fig. 8 is the fact that the error, given as

ϕ =∣∣∣∣∣

(|u| ui ui − u3) − uu′

i u′i(|u| ui ui − u3

)

∣∣∣∣∣, (12)

123


10−3

10−2

10−1

10−3

10−2

10−1

WD measured

WD

mod

eled

3/4

3/2

1

0.5 (data fit)

Fig. 10 Comparison between measured(|u| ui ui − u3

)and uu′

i u′i . The grey lines correspond to slopes = 3/4

(dotted), 1.0 (continuous), and 3/2 (dashed). The dotted black line is the regression fit (slope = 0.5, R2 = 0.32)

102

103

10−2

10−1

100

101

Re

φ

Fig. 11 The dependence of ϕ on Re. The dotted black line corresponds to the data fit for Re<1000(φ = (0.97 - 0.0005 Re), R2 = 0.32)

varies with Re for Re<1000. This error, presented in Fig. 11, appears to diminish withincreasing Re suggesting that some of the anisotropy in Fig. 8 may be linked to the binomialapproximation.

123


References

Amiro BD (1990) Drag coefficients and turbulence spectra within three boreal forest canopies. Boundary-Layer Meteorol 52:227–246

Aylor DE, Flesch TK (2001) Estimating spore release rates using a Lagrangian stochastic simulation model.J Appl Meteorol 40:1196–1208

Ayotte KW, Finnigan JJ, Raupach RM (1999) A second-order closure for neutrally stratified vegetative canopyflow. Boundary-Layer Meteorol 90:189–216

Bakwin PS, Wofsy SC, Fan S-M, Keller M, Trumbore SE, Dacosta JM (1990) Emission of nitric oxide (NO)from tropical forest soils and exchange of NO between the forest canopy and atmospheric boundarylayers. J Geophys Res 95(D10):16755–16764

Baldocchi DD, Meyers TP (1988) A spectral and lag correlation analysis of turbulence in a deciduous forestcanopy. Boundary-Layer Meteorol 45:31–58

Barthlott C, Fiedler F (2003) Turbulence structure in the wake region of a meteorological tower. Boundary-Layer Meteorol 108:175–190

Blanken PD, Black TA, Neumann H, Den Hartog G, Yang PC, Nesic Z, Staebler R, Chen W, NovakMD (1998) Turbulent flux measurements above and below the overstory of a boreal aspen forest.Boundary-Layer Meteorol 89:109–140

Cantwell B, Coles D (1983) An experimental study of entrainment and transport in the turbulent near wakeof a circular cylinder. J Fluid Mech 136:321–374

Cava D, Katul GG, Scrimieri A, Poggi D, Cescatti A, Giostra U (2006) Buoyancy and the sensible heat fluxbudget within dense canopies. Boundary-Layer Meteorol 118:217–240

Fey U, Konig M, Eckelmann H (1998) A new Strouhal–Reynolds-number relationship for the circular cylinderin the range 47<Re<23105. Phys Fluids 10:1547–1549

Finnigan J (2000) Turbulence in plant canopies. Ann Rev Fluid Mech 32:519–571Gardiner BA (1994) Wind and wind forces in a plantation Spruce forest. Boundary-Layer Meteorol 67:

161–186Juang J-Y, Katul GG, Siqueira MB, Stoy PC, Palmroth S, McCarthy HR, Kim HS, Oren R (2006) Modeling

nighttime ecosystem respiration from measured CO2 concentration and air temperature profiles usinginverse methods. J Geophys Res 111:D08S05 doi:10.1029/2005JD005976

Kaplan WA, Wofsy SC, Keller M, Da Costa JM (1988) Emission of NO and deposition of O3 in a tropicalforest system. J Geophys Res, 93(D2):1389–1395

Kruijt B, Malhi Y, Lloyd J, Norbre AD, Miranda AC, Pereira MGP, Culf A, Grace J (2000) Turbulencestatistics above and within two Amazon Rain Forest canopies. Boundary-Layer Meteorol 94:297–331

Kuparinen A (2006) Mechanistic models for wind dispersal. Trend Plant Sci 11:296–301Launiainen S, Vesala T, Mölder M, Mammarella I, Smolander S, Rannik Ü, Kolari P, Hari P, Lindroth A, Katul

GG (2007) Vertical variability and effect of stability on turbulence characteristics down to the floor of apine forest. Tellus B 59(5):919–936

Liu S, Liu H, Xu M, Leclerc MY, Zhu TY, Jin CJ, Hong ZX, Li J, Liu HZ (2001) Turbulence spectra anddissipation rates above and within a forest canopy. Boundary-Layer Meteorol 98:83–102

Marcolla B, Pitacco A, Cescatti A (2003) Canopy architecture and turbulence structure in a coniferous forest.Boundary-Layer Meteorol 108(1):39–59

Poggi D, Porporato A, Ridolfi L, Albertson JD, Katul GG (2004) The effect of vegetation density on canopysub-layer turbulence. Boundary-Layer Meteorol 111:565–587

Poggi D, Katul GG, Albertson JD (2006) Scalar dispersion within a model canopy: Measurements andthree-dimensional Lagrangian models. Adv Water Res 29:326–335

Poggi D, Katul GG (2006) Two-dimensional scalar spectra in the deeper layers of a dense and uniform modelcanopy. Boundary-Layer Meteorol 121:267–281

Raupach MR, Shaw RH (1982) Averaging procedures for flow within vegetation canopies. Boundary-LayerMeteorol 22:79–90

Raupach MR, Coppin PA, Legg BJ (1986) Experiments on scalar dispersion within a model plant canopy, Part1: the turbulent structure. Boundary-Layer Meteorol 35:21–52

Raupach MR, Finnigan JJ, Brunet Y (1996) Coherent eddies and turbulence in vegetation canopies: the mixinglayer analogy. Boundary-Layer Meteorol 78:351–382

Raynor GS, Hayes J, Ogden EC (1974) Particulate dispersion into and within a forest. Boundary Layer Mete-orol 7:429–456

Sawford BL, Guest FM (1991) Lagrangian statistical simulation of the turbulent motion of heavy particles.Boundary-Layer Meteorol 54:147–166

Shaw RH, Seginer I (1985) The dissipation of turbulence in plant canopies. Extended abstract: Am. Meteorol.Soc. 7th Symp. On Turbulence and Diffusion Abstracts, pp 200–203

123

http://dx.doi.org/10.1029/2005JD005976


Seginer I, Mulhearn PJ, Bradley EF, Finnigan JJ (1976) Turbulent flow in a model plant canopy. Boundary-Layer Meteorol 10:423–453

Su HB, Schmid HP, Grimond CSB, Vogel CS, Oliphant AJ (2004) Spectral characteristics and correctionof long-term eddy-covariance measurements over two mixed hardwood forests in non-flat terrain.Boundary-Layer Meteorol 110:213–253

Taneda S (1965) Experimental investigation of vortex streets. J Phys Soci Japan 20:1714–1721Tritton DH (1988) Physical fluid dynamics. Oxford Science Publications, 519 ppVergassola M, Villermaux E, Shraiman B (2007) Infotaxis as a strategy for searching without gradients. Nature

445:406–409Villani MG, Schmid HP, Su HB, Hutton JL, Vogel CS (2003) Turbulence statistics measurements in a northern

hardwood forest. Boundary-Layer Meteorol 108:343–364Williamson CHK, Brown GLK (1998) A series in 1/

√Re to represent the Strouhal-Reynolds number relation-

ship of the cylinder wake. J Fluid Struct 12:1073–1085Wilson JD (1988) A second-order closure model for flow through vegetated canopies. Boundary-Layer

Meteorol 42:371–392Wilson JD (2000) Trajectory models for heavy particles in atmospheric turbulence: Comparison with obser-

vations. J Appl Meteorol 39:1894–1912

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Documents

Spectral Short-circuiting and Wake Production within the Canopy Trunk Space of an Alpine Hardwood Forest