acta geophysica - duke university782 d. poggi and g.g. katul situated on complex topography, this...

Acta Geophysica vol. 56, no. 3, pp. 778-800

DOI: 10.2478/s11600-008-0029-7

© 2008 Institute of Geophysics, Polish Academy of Sciences

Micro- and macro-dispersive fluxes in canopy flows

Davide POGGI 1 and Gabriel G. KATUL

2

1Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino, Italy

e-mail: [email protected] 2Nicholas School of the Environment and Earth Sciences, and

Department of Civil and Environmental Engineering, Duke University, Durham, NC, USA

e-mail: [email protected]

A b s t r a c t

Resolving every detail of the three-dimensional canopy morpholo-gy and its underlying topography remains untenable when modeling high Reynolds number geophysical flows. How to represent the effects of such a complex morphological variability and any concomittant topographic variability into one-dimensional bulk flow representation remains a fun-damental challenge to be confronted in canopy turbulence research.

Theoretically, planar averaging to the scale of interest should be applied to the time-averaged mean momentum balance; however, such averaging gives rise to covariance or dispersive terms produced by spa-tial correlations of time-averaged quantities that remain ‘unclosed’ or re-quire parameterization. When the averaging scale is commensurate with few canopy heights, these covariances can be labeled as ‘micro-dispersive’ stresses. When averaging is intended to eliminate low-wavenumber topographic variations, we refer to these covariances as ‘macro-dispersive’ terms. Two flume experiments were used to explore the magnitude and sign of both micro- and macro-dispersive fluxes rela-tive to their conventional Reynolds stresses counterparts: a rod-canopy with variable roughness density and a dense rod canopy situated on gen-tle hilly terrain. When compared to the conventional momentum flux, the micro-dispersive fluxes in the lowest layers of sparse canopies can be significant (~50%). For dense canopies, the dispersive terms remain neg-ligible when compared to the conventional momentum fluxes throughout.

MICRO- AND MACRO-DISPERSIVE FLUXES IN CANOPY FLOWS

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For the macro-dispersive fluxes, model calculations suggest that these terms can be neglected relative to the Reynolds stresses for a deep cano-py situated on a narrow hill. For the region in which topographic varia-tions can interact with the pressure, both model calculations and flume experiments suggest that the macro-dispersive fluxes cannot be neg-lected, and their value can be 20% of the typical Reynolds stresses.

Key words: canopy turbulence, dispersive fluxes, double averaging, flow on hills, micro-dispersion, macro-dispersion, planar averaging.

1. INTRODUCTION

Flow within and above canopies, whether be it vegetated or urban, are inhe-rently three-dimensional and influenced by both – the intrinsically variable canopy morphology and possibly underlying topographic variations. How such a variability is represented in transport problems remains a subject of active research. Naturally, resolving every nuance and detail of the canopy morphology remains untenable in high Reynolds number transport problems pertaining to applications such as biosphere-atmosphere exchange of trace gases, seed and pollen dispersal by wind, or urban pollution. In these appli-cations, the problem is often casted as how to include the effects of such a three-dimensional canopy morphology and topographic variations into one-dimensional bulk flow properties most pertinent to momentum, heat, and mass transport (Katul and Albertson 1998, 1999, Hsieh et al. 2000, MacDo-nald 2000, Siqueira et al. 2000, Nathan et al. 2002, Soons et al. 2004, Katul et al. 2005, Nathan and Katul 2005, Cava et al. 2006, Juang et al. 2006). The planar averaging of the main variables to the scale of interest (i.e., upscaling) is often employed in such problems. Such averaging must account for the in-herently variable three-dimensional canopy morphology, and the less appre-ciated topographic effects. In the case of urban canopies, this upscaling is further complicated by the hierarchical structure of the canopy morphology (street or canyon scale, neighbourhood scale, city scale, and regional scale) as discussed in reviews elsewhere (Arnfield 2003, Britter and Hanna 2003, Belcher 2005, Kanda 2006) and possibly topographic variations.

When planar averaging is applied to the time-averaged mean momentum balance, covariance terms arising from the spatial correlation of time-averaged quantities are produced (Raupach and Shaw 1982, Finnigan 2000, Nikora et al. 2001). These spatial covariances, often referred to as dispersive fluxes (or stresses), are among the least understood terms in the momentum equation for canopy flows though they are now receiving more attention (see Table 1 for recent studies). This is not surprising as these terms have tradi-tionally evaded direct measurements until the last two decades or so, follow-ing pioneering wind tunnel experiments in the mid-eighties (Raupach et al. 1986, Kaimal and Finnigan 1994).

D. POGGI and G.G. KATUL 780

Table 1 Micro-dispersive fluxes from various experiments and simulations

Study Comments on Dispersive Fluxes

Wind Tunnel and Flume Experiments

Raupach et al. 1986 Wind tunnel experiments for dense canopies (rods). Dispersive fluxes are negligible.

Bohm et al. 2000 Wind tunnel study with sparse vegetation. Dispersive fluxes for momentum can be significant in the lower canopy layers.

Cheng and Castro 2002 Wind tunnel experiments for dense urban canopies (blocks). Dispersive fluxes are small and negligible when compared to the turbulent momentum fluxes.

Poggi et al. 2004b Flume experiments with canopy (rods) density varying from sparse to dense. Dispersive fluxes can be large for sparse canopies, especially in the lower canopy layers, reaching more than 30% of the momentum turbulent flux, but negligible for dense canopies. These conclu-sions were valid for two high Reynolds numbers.

Pokrajac et al. 2007 Open channel flow near 2-dimensional rough beds (square roughness with uniform spacing resembling ‘k-type’ flow – a type of flow in which the obstacles are at their densest arrangement to maximize the drag on individual roughness elements but minimize sheltering effects by minimizing wake interferences). Dispersive fluxes can be large near the interface.

Field Experiments

Christen and Vogt 2004 Cork oak plantation with tree density of 76 trees ha–1 and 8 measurement stations all positioned at z/ hc = 0.18 from the ground. Dispersive fluxes for momentum were about 15% of turbulent momentum flux, and for heat, they are < 5% of the turbulent sensible heat flux.

Simulations (LES or DNS)

Coceal et al. 2006 Direct numerical simulations of turbulent flow over regular arrays of urban-like, cubical obstacles. Within the arrays, significant dispersive stresses were reported whereas immediately above the obstacles, the Reynolds stresses continue to dominate.

Martilli and Santiago 2007

RANS simulations over an array of cubes. Dispersive fluxes are similar in magnitude to the turbulent fluxes inside the obstacles.

To date, however, much of the (limited) experiments focused on quanti-

fying dispersive fluxes arising from an averaging operation intended to elim-inate variations due to the canopy structure (see Table 1), and hereafter, these dispersive fluxes are refered to as ‘micro-dispersive’ fluxes. In upscal-


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ing beyond these ‘micro-scales’, planar averaging must be sufficiently large to eliminate variations due to the largest length scale that may impact the flow (e.g., from canyon to neighbourhood, or neighbourhood to city scale, or stand-level to ecosystem scale). Hence, beyond the variations in canopy structure, topographic variations can also play a significant local role wheth-er be it in urban areas or vegetated ecosystems. In the case of a canopy

Fig. 1. Multiscale view of planar averaging scales across a hierarchy of scales. Bottom figure is micro-dispersion, and middle figure is macro-dispersion. The top figure shows the grid cell of meso-scale models that must account for both – micro- and macro-dispersion. See color version of this figure in electronic edition.


situated on complex topography, this largest length scale may be dictated by how topography impacts the bulk spatial flow patterns (Raupach et al. 1992, Raupach and Finnigan 1997, Belcher and Hunt 1998, Wilson et al. 1998, Finnigan and Belcher 2004, Katul et al. 2006, McLean and Nikora 2006, Poggi and Katul 2007b, Poggi et al. 2007). When averaging over such larger length scales (see Fig. 1), the resultant dispersive terms are labeled as ‘ma-cro-dispersive fluxes’ to distinguish them from the micro-dispersive terms. These macro-dispersive fluxes can play a significant role from local-scale, employed to quantify scalar and particle transport, to meso-scale models, used in meteorological or air quality forecasting (Fig. 1). The need for their quantification cannot be overstated – few vegetated or urban canopies are si-tuated on flat terrain; and more often than not, topographic variations may be more significant than canopy height or roughness density variations. Hence, given this ‘knowledge gap’, the main objective of this work is to present ex-perimental and model calculations on the magnitude of micro- and macro-dispersive fluxes. The combined effect of complex terrain variations and fine-scale heterogeneous canopy morphology remains a basic challenge to be confronted, and practical problems must account for both given the domain of interest. However, the general treatment of these combined issues is well beyond the scope of a single study.

Here, two ‘elementary’ cases are considered for exploring micro- and macro-dispersive fluxes: a rod-canopy with variable roughness density and a dense rod canopy situated on gentle hilly terrain. The results from these two cases are not intended to provide ‘finality’ to issues pertaining to dispersive fluxes; rather they are intended to provide bench-mark results in terms of order of magnitude and patterns so as to begin progress on this problem.

2. THEORY

2.1 Governing equations

The Navier-Stokes (NS) and continuity equations for the instantaneous flow variables are given by

3 and 0 ,i j i i ii

j i j i

u u u p u p pug vt x x xj x t x

δ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂

+ = − + + =⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

where t is time, xi (x1 = x, x2 = y, x3 = z) are the longitudinal, lateral, and ver-tical directions, respectively, ui (u1 = u, u2 = v, u3 = w) are the instantaneous velocity components along xi , p is the static pressure normalized by the mean fluid density ρ, ν is the kinematic viscosity, g is the gravitational acce-leration constant and δi3 is the Kronecker delta (non-zero when I = 3). De-composing the flow variables into temporal averages (denoted by overbar) with turbulent excursions defined from their time-averaged states (i.e.,


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j j jϕ ϕ ϕ′= + ), and upon time-averaging the resultant equation, we obtain the standard mean momentum equation

, where .i i ij i jj ij i j

j i j j i

u u p u uu u u vt x x x x x

τ τ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

′ ′+ = + = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

2.2 Canopy induced micro-dispersive terms

As earlier stated, the complex and three-dimensional geometry of canopy elements cannot all be precisely resolved, and in canopy turbulence, it is common to average the equations over some “minimal” control area, such as few canopy heights in the case of a forest or the area between rods in the case of a canopy composed of a regular array of rods. When flow variables are decomposed into planar averages (denoted by angular brackets) with spa-tial turbulent excursions defined from the planar averaged quantities using the usual convention (Raupach and Shaw 1982), ,j j jϕ ϕ ϕ′′= + the double-averaged momentum equation for non-stratified flows becomes

,i i ijj sf c

j i j

u u pu H F

t x x xτ∂ ∂ ∂ ′∂

+ = − + +∂ ∂ ∂ ∂

(1)

where Hsf is the Heaviside step function (unity if the averaging plane inter-sects the canopy, zero outside), and

,i jij i j i j

j i

u uu u u u v

x xτ

∂ ∂⎛ ⎞′ ′ ′ ′′ ′′= − − + +⎜ ⎟∂ ∂⎝ ⎠

(2)

and

2

.ic

i j i

p uF vx x x′′ ′′∂ ∂

= − +∂ ∂ ∂

All double primed terms arise because horizontal averaging and differen-tiation do not commute, given the multiply-connected air spaces within the canopy (Raupach and Shaw 1982). Hence, they represent the explicit effects of the vegetation on the spatial and temporal statistics of the flow. These terms contain the drag force Fc , which is composed of a form and viscous drag terms. The stress tensor ijτ′ also contains the conventional turbulent and viscous stresses and the dispersive flux term i ju u′′ ′′ resulting from spatial correlations of the velocity components within the averaging volume. Ana-logous to the standard Reynolds stresses, i ju u′′ ′′ is not a priori known and have to be empirically modeled. The above equations can be further simplified by noting that advection may be neglected on flat terrain and for stationary and planar-homogeneous high Reynolds number flows, the classical equa-tions are recovered (Raupach and Thom 1981, Finnigan 2000), and given by


,i jsf c

i j

p u uH F

x x′ ′∂ ∂

= +∂ ∂

(3)

where ic sf

i c

u upF Hx L′′∂

= − =∂

is the common parametrization of the ele-

ment drag force. Here i iφ φ φ= is the magnitude of some arbitrary vector flow variable, Lc = (Cd a)–1 is the canopy adjustment length scale, Cd is the drag coefficient, and a is the canopy density.

2.3 Topographically induced macro-dispersive terms

When considering flow over complex topography, it is convenient to de-compose the mean longitudinal momentum balance equation into a back-ground equilibrium state and a perturbation induced by topographic variations (Jackson and Hunt 1975). For canopy flows on gentle cosine hills, this background equilibrium state can be unambiguously defined using a spatial averaging of pressure and mean velocity along the streamline coordi-nate (defined next) and over one topographic wavelength (Poggi and Katul 2007a, b, Poggi et al. 2007) given as

( ) ( ) ( ) ( )d , , , , ( ) , .b bwavelengh

u u s u x z U x z u x z p p z p x z= +∆ = +∆ = +∆∫

For a stationary and high Reynolds number flow, the background momentum equation is given by

2

b b bsf

c

p UHx z L

τ∂ ∂= +

∂ ∂ (4)

and the topographically perturbed mean momentum and continuity equations are given by

( ) ,bb sf c

u u pUU u w H Fx z z x z

τ∂∆ ∂∆ ∂∆ ∂∆⎛ ⎞∂+ ∆ + ∆ + = − − ∆⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(5)

0 ,u w

x z∂∆ ∂∆

+ =∂ ∂

(6)

where subscript b and ∆ indicate background state and perturbation from the background state, respectively, ∆Fc = (1/ Lc) ( )2

bu U u∆ + ∆ is the nonli-

near drag due to the canopy elements and u w u wτ ′ ′ ′′ ′′∆ = − − . Note that τb is not derived from a spatial average but it is imposed to satisfy the back-ground budget equation (4) When double-averaging these equations to cha-racterize the bulk effects on the flow due to the joint canopy and topographic variability, the ‘macro-scale’ dispersive fluxes arise. For simplicity, define


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φ φ∆ = ∆ and, as was done before for the NS equations in the microscale dispersion case, consider the spatial averaging over length scales commensu-rate with the hill wavelength (hereafter referred to by double lines) to give

.sf cu u w u H Fx z z

τ∂∆ ∆ ∂∆ ∆ ∂∆+ = − ∆

∂ ∂ ∂ (7)

Equation (7) can be rewritten as

, where .sf cu u H F u w u w w ux z

τ τ′∂∆ ∆ ∂∆ ′ ′ ′ ′′ ′′= + ∆ ∆ = −∆ −∆ +∆ ∆∂ ∂

The above ‘macro-scale’ averaged equation resembles eqs. (3) and (4) in ca-nonical form but differs in the interpretation of τ . The τ now contains the memory of both the Reynolds stress (time) and the micro-dispersive terms (earlier described), and a new macro-dispersive terms arising because topo-graphic variations perturb the flow from its background state and these per-turbations are correlated in space. We note that both, u w′ ′∆ and u w′′ ′′∆ , are generally very small, bu w′ ′ being very close, at least for gentle hills, to the spatial average of u w′ ′ . If this is true, the macro-dispersive stress is the

leading term contributing to τ ′∆ . The importance of these macro-dispersive term is evaluated experimen-

tally and using simplified first-order closure models to assess their magni-tude outside the domain of the experiments. In these calculations, we focus on conditions where micro-dispersive terms are small when compared to the Reynolds stress, as may be anticipated for dense canopies (see Table 1 for experimental evidence) and as we show here.

3. EXPERIMENTS

Two separate experiments were carried out to explore the magnitude of mi-cro- and macro-dispersive fluxes. For micro-dispersive fluxes, the experi-ment was intended to illustrate the role of canopy density variations, while for the macro-scale dispersive fluxes it was intended to explore the spatial variations of the velocity perturbations inside dense canopies on a cosine hill where the hill elevation was commensurate with the canopy height. The data from this experiment was then used to explore the basic assumptions em-ployed by analytical first-order closure models for velocity perturbations.

3.1 Micro-dispersion experiments

Much of the experimental setup is described elsewhere (Poggi et al. 2003, 2004a, b, c); however, a brief review of the salient features is provided here for completeness. The experiment was conducted at the Hydraulics Labora-


tory, DITIC Politecnico di Torino, in a recirculating rectangular flume (18 m long, 0.90 m wide and 1 m deep) at a flow rate Qr ≈ 120 l /s. The water depth hw was maintained at a uniform value of 0.6 m.

The model canopy was composed of an array of vertical steel cylinders, 12 cm tall (= h) and 4 mm in diameter (= dr). The cylinders were arranged in a regular pattern along the 9 m long and 0.9 m wide test section. Data were collected for five canopy roughness densities: 67, 134, 268, 536, and 1072 rods /m2

(hereafter referred to as D1, D2, D3, D4 and D5) equivalent to an

element area index a (frontal area per unit volume) of 0.27, 0.53, 1.07, 2.13, and 4.27 mm, respectively. Note that the densest frontal area index leads to a drag coefficient comparable to drag estimates reported for dense vegetated ecosystems with leaf area index ranging from 3.5 to 6.0 (Katul and Albert-son 1998, Katul and Chang 1999, Katul et al. 2004). Hereafter, according Poggi et al. (2004), we will refer to D4 and D5 as dense canopies and to D1-D3 as sparse canopies.

Multiple runs were used to measure the velocity by two-component Laser Doppler Anemometry (LDA) employed in forward scattering mode. A measurement run consists of sampling the u and w time series at a single location (between or above the rods). For each run, the sampling duration

Fig. 2. Setup showing the 11 sampling points relative to the rods for the micro-dispersive flux analysis. The spatial averages are based on individual weights αi for each area as shown. The sampling locations illustrating sample vertical variations in dispersive terms (Fig. 4) are highlighted as s1 to s4. Note that

11

1

1ii

α=

≈∑ . The sampling

locations in the vertical are also shown. See color version of this figure in electronic edition.


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and frequency were 300 s and 2500-3000 Hz, respectively. Given the interest in micro-dispersion, u′′ and w′′ were sampled at 11 measurement locations, shown in Fig. 2. These 11 locations were unevenly spaced between the rods and chosen so that sampling was densest within regions in which the flow statistics exhibited the highest spatial variability. At each of these 11 planar positions, 15 runs in the vertical were sampled and analyzed for each of the five roughness densities. Using the vertically-averaged velocity Uo, v across the entire water depth hw , the resulting bulk Reynolds number Reb (= Uo, v hw /ν ) was in excess of 170,000 which ensures fully-developed turbulent flows.

3.2 Macro-dispersion experiments

The details of the experimental setup for the topography, canopy configura-tion, and coordinate systems employed for the velocity sampling are re-viewed elsewhere (Poggi and Katul 2007a, b, Poggi et al. 2007) but an overview is again provided for completeness. Velocity and pressure mea-surements were conducted in the same recirculating flume facility presented above (Fig. 2). The hill topography was constructed using four modules of a wavy stainless steel wall, each representing a cosine hill with a shape func-tion given by ( ) ( )/ 2cos π ,f x H kx= + where x is the longitudinal distance, H = 0.08 m is the hill height, k = π / (2L) is the hill wavelength with L = 0.8 m (Fig. 3). The model canopy was composed of vertical stainless steel cylinders (hc = 10 cm and diameter dr = 0.4 cm) arranged on the wavy surface using a density of 1000 rods /m2. The vertical distribution of the rods frontal area was designed to resemble the leaf area density of a tall hardwood forest at maximum leaf area index (Poggi et al. 2007). Much of the ‘density’ was concentrated in the top third of the canopy but almost constant (and small) in the bottom two-thirds.

The acquisition of the velocity necessitates an appropriate coordinate system be a priori defined. This reference coordinate system can be external-ly (and arbitrarily) imposed (e.g., rectangular Cartesian system), related to the geometry of the surface (e.g., terrain following systems), or allowed to adjust according to the flow dynamics (e.g., streamline coordinates). As dis-cussed in Finnigan and Belcher (2004) the latter choice is the most preferred for flow over hills and is adopted here. This coordinate system reduces to terrain-following near the ground and to rectangular Cartesian well above the hill thereby retaining advantages of both coordinate systems in the ap-propriate regions. To within an error of order (H/L)2, the rectangular (X and Z) and the displaced coordinate systems can be explicitly related by

( )

( )

/ 2sin e ,

/ 2cos e ,

kZ

kZ

x X H kX

z Z H kX

−

−

= +

= −


where H , L, and k are schematically shown in Fig. 3. Because (H/L)2 is suf-ficiently small here ( = 0.01), the above equation is used to relate these two coordinate systems without any further adjustments.

The velocity time series were acquired above the third hill module using the same LDA described above. The LDA measurements were performed at 0.40 m from the lateral wall at ten longitudinal positions to cover one hill module, and along a large number of vertical positions (∼_ 35) in the dis-placed coordinate system. The LDA measurements were conducted at fully-developed turbulent flow conditions characterized by a bulk Reynolds num-ber Reb > 1.3×105. These velocity data were then used to estimate the macro-dispersive terms and assess the skill of the first-order closure model to re-produce them. After evaluating the model, it is used to explore the macro-dispersive fluxes outside the domain of the experiment (i.e., a wide range of L/ Lc and h / Lc).

Fig. 3. Photographs of the experimental setup showing the recirculating flume (used in both micro- and macro-dispersive experiments), the four hill modules, the coordi-nate system and hill/canopy attributes, and the canopy composed of rods on hills. See color version of this figure in electronic edition.


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4. RESULTS

4.1 Evaluation of micro-dispersive terms

The micro-dispersive fluxes can be estimated from the data as follows: Let αi be the area-weight assigned to each of the 11 measurement locations shown

in Fig. 2 (i.e., 11

1

1ii

α=

=∑ ). The relevant flow statistics were computed as

11 11 11

1 1 1

11

1

, , ,

, , .

i i i i i ii i i

i i i i i i i i ii

u w u w u u w w

u u u w w w u w u w

α α α

α

= = =

=

′ ′ ′ ′= = =

′′ ′′ ′′ ′′ ′′ ′′= − = − =

∑ ∑ ∑

∑

Figure 4 illustrates the vertical structure of the constitutive terms of the dispersive fluxes across the five different roughness densities. The vertical variations of u′′ (Fig. 4a) and w′′ (Fig. 4b) are presented as a function of normalized height for planar positions situated just downstream from a rod (s1), just upstream from the consecutive rod (s2), in between the rods (s3), and near the center bounded by the corner rods (s4) as shown in Fig. 3 for all five canopy densities. At s1, the spatial excursions are, as expected, slower longitudinal velocity and a concomitant downward fluid movement near the canopy top given the immediate effect of the rod to slowdown the flow rela-tive to the spatial average, as expected from a laterally homogeneous 2D continuity. At s2, the longitudinal spatial excursion is, by and large, negative but the vertical velocity excursion is upwards. At s3, the spatial excursions are small and show no particular vertical structure, suggesting that the im-mediate effect of the rod on the spatial patterns of the bulk flow was dissi-pated. In the center of the domain (s4), the spatial excursions diverge from the cell-averaged values in an opposite manner to s1, as expected. Finally, note that the positive w′′ in the lowest level of the canopy is consistent with findings from other studies (Nepf and Koch 1999). More important, these spatial patterns appear coherent across all five canopy densities, but their overall contribution to the mean momentum budget significantly differs.

The relative importance of these dispersive fluxes, when compared to the conventional momentum flux, is shown in Fig. 5 for all five a values. For sparse canopies, the dispersive flux in the lowest layers ( z / h < 0.5) can be significant when compared to the mean momentum flux (Fig. 5). For the top 1/3 of the canopy, however, the dispersive flux becomes at most about 20% of the total stress, but with an opposite sign (as expected from the analysis in Fig. 4). For dense canopies, the dispersive fluxes are negligible when com-pared to the convential momentum fluxes throughout. These results do not appear to be sensitive to the bulk Reynolds number, as discussed elsewhere


Fig. 4. Variations of u′′ (a) and variations of w′′ (b) as a function of normalized height (z / hc) for s1 (top left), s2 (top right), s3 (bottom left), and s4 (bottom right). The planar positions of s1, s2, s3, and s4 relative to the rods are shown in Fig. 3. All velocity variables are normalized by the friction velocity, u

* . See color version

of this figure in electronic edition.

(a)

(b)


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Fig. 5. Comparison between the profiles of the dispersive fluxes and the standard Reynolds stresses for all 5 canopy densities. Note that the micro-dispersive fluxes are large only for the sparsest and in the mid to lower canopy levels. See color ver-sion of this figure in electronic edition.

(Poggi et al. 2004b). The findings here are also in agreement with Bohm et al. (2000)'s sparse canopy wind tunnel measurements and the dense urban canopy measurements reported by Cheng and Castro (2002) and the rod ca-nopy wind tunnel experiments of Raupach et al. (1986). Finally, note that al-so the gradient of the dispersive flux, crucial in the momentum equation, is significant when compared to the mean momentum flux.

4.2 Evaluation of macro-dispersive terms

To evaluate the macro-dispersive fluxes, the experimental measurements de-scribed above along with the analytical solution proposed by Finnigan and Belcher (2004), hereafter referred to as FB04 (Finnigan and Belcher 2004, Katul et al. 2006), are employed. A brief review of FB04 within the inner and canopy layers is presented in the Appendix. The simplified nonlinear so-lution of FB04 models the flow field in terms of ∆u and ∆w, which represent the constitutive terms of the macro-dispersive fluxes. The experi-ments are mainly intended to explore the canonical form of ∆u, ∆w and

w u∆ ∆ across the hill and to compare these measurements with predictions

by FB04. Next, FB04 is used to estimate the relative importance of w u∆ ∆ to the conventional Reynolds stress for a wide range of topographic and ca-nopy conditions. Figure 6a shows the spatial structure of the measured per-turbed flow within the canopy and in the inner layer. Just above the canopy, and near the hill-top, an over-speeding zone is observed. Within this region, ∆u reaches its maximum value, which is coincident with the location of max-imum ∆w (and hence, locally, a significant contribution to w u∆ ∆ ). The spa-tial extent of this region is similar to the region noted within the inner layer


Fig. 6a. Measured variations in ∆u, ∆w, and ∆u, ∆w across the hill (in m /s). The dashed and dash-dot lines are the inner layer and the canopy layer depth, respectively. The location of the test sections, with respect to the hill, are also presented as vertical lines. Note the large contribution of the region at the base of the hill above the canopy to the macro-dispersive flux. See color version of this figure in electronic edition.

Fig. 6b. Same as Fig. 6a but for quantities modeled by FB04. See color version of this figure in electronic edition.


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Fig. 7. Modeled dispersive fluxes for a deep canopy on a shallow hill. Dispersive fluxes can be large as the interactive pressure zone is approached. Note that the expe-rimental setup (dot) is outside the domain of validity of FB04 (shown as dotted lines). See color version of this figure in electronic edition.

above the hill with no canopy (Poggi et al. 2007). Just above the canopy, and near the hill-bottom, a slow-down region is also evident. Within this region, ∆u reaches its minimum value; however, unlike the over-speeding region, ∆w reaches its minimum value at a higher depth. Hence, the minimum in ∆u and ∆w are not precisely in phase. In the lower canopy region, the most striking feature is the large and near-maximum ∆u in a thin layer situated on the upwind (and part of the downwind) side near the hill surface. However, in this region, ∆w is almost negligible and this overspeeding need not signif-icantly contribute to w u∆ ∆ .

The FB04 model captures these essential zones of large negative and positive dispersive terms and is suggestive that the spatial coherency of these terms can be predicted from such an analysis – though the precise one-to-one agreement remains illusive. It should be noted that the lack of one-to-one agreement is not surprising because the FB04 model validity is restricted to gentle narrow hills and deep canopy, while the experiment here is outside this domain, as shown in Fig. 7. Notwithstanding this limitation, Fig. 7


shows the modeled dispersive fluxes across a wide range of canopy and hill configuration both within and outside the domain of validity of FB04 (the experiment is shown as one dot outside this domain). Key observations from this model analysis include the following:

1. For a deep canopy situated on a narrow hill, the macro-dispersive terms can be neglected relative to the conventional Reynolds stresses

2. For the region in which the topographic variations interact with the pressure, the macro-dispersive fluxes cannot be neglected – their modeled value may be up to 20% of the typical Reynolds stress.

3. For a given set of canopy attributes (i.e., h / Lc fixed), as L / Lc de-creases, the relative importance of the macro-dispersive terms tends to increase.

4. For a given L / Lc , increasing h / Lc leads to a non-monotonic change in the macro-dispersive fluxes when normalized by their Reynolds stresses counterpart.

5. While the experiment is outside the domain of validity of FB04, the measured macro-dispersive fluxes relative to the Reynolds stress are on the order of 15%, while the FB04 modeled dispersive fluxes for the same L / Lc and h / Lc are about 20%, a reasonable agreement giv-en all the assumptions (and uncertainties) in both estimates.

The summary in Fig. 7 provides some guidance as to when dispersive fluxes can be neglected relative to the conventional Reynolds stresses de-pending on canopy and hill configuration. This is only a necessary first step for incorporating their effect in meso-scale models (e.g., Fig. 1).

5. CONCLUSIONS

Two flume experiments were used to explore the magnitude and sign of mi-cro- and macro-dispersive fluxes relative to their standard Reynolds stresses counterparts: a rod-canopy with variable roughness density and a dense rod canopy situated on gentle hilly terrain. The results from these two cases can be summarized as follows:

For the micro-dispersive fluxes, vertical variations of u′′ and w′′ appear to be large adjacent to the rods. The excursions in this area are a slower longitudinal velocity and a concomittant downward flu-id movement near the top. These spatial patterns appear coherent across all five canopy densities, but their overall contribution to the mean momentum budget differs. When compared to the convention-al momentum flux, the dispersive fluxes in the lowest layers (z / h < 0.5) can be significant (> 50%) when compared to the mean turbulent momentum flux for the sparsest canopy case. For dense ca-


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nopies, the dispersive fluxes are negligible when compared to the conventional momentum fluxes throughout.

For macro-dispersive fluxes, these terms can be neglected relative to the Reynolds stresses for a deep canopy situated on a narrow hill. In the region in which topographic variations can interact with the pres-sure, the macro-dispersive fluxes cannot be neglected, and their modeled value may be larger than 20% of the typical Reynolds stresses. For a given set of canopy attributes (i.e., h / Lc fixed), we found that as L / Lc decreases, the relative importance of the macro-dispersive terms tends to increase. However, for a given L / Lc , in-creasing h / Lc leads to a non-monotonic change in the importance of macro-dispersive fluxes.

More broadly, there are a number of reasons why exploring micro- and macro-dispersive fluxes is a timely topic. The rapid progress in canopy LIDARS (light detection and ranging) are now permitting unprecedent view of canopy and topographic variability (Lefsky 2005a, b). Hence, the ability to quantify the statistical attributes of the canopy morphology is progressing rapidly. What is clearly missing is how to link such variability with the flow dynamics. Dispersive terms provide the theoretical under-pinning of such linkages, and hence, exploring their magnitude and importance should be a logical first step. The advent of Particle Image Velocity (PIV) meters, Laser Induced Fluorescence (LIF) measurement techniques, LDA, and high fre-quency imagery aquision camera are now permitting basic experiments to be carried out on these dispersive terms so that linkages between the statistical structure of canopy morphologies and the flow dynamics can be explored with unprecedent resolution. Such experiments may provide the bench-mark data sets and phenomenological models for linking dispersive fluxes to the statistical attributes of canopy morphology. The computational resources are also expanding to levels that permit high resolution LES to be carried out and be used to explore the structure and contribution of such dispersive flux-es for both laboratory settings (for evaluation purposes) and more complex settings as evidenced in Table 1. It is the combination of all three that is like-ly to yield most progress in this emerging area.

Acknowledgemen t s . This research was supported, in part, by the National Science Foundation (NSF-EAR 06-35787 and NSF-EAR-06-28432), the United States – Israel Binational Agricultural Research and De-velopment (Research Grant No. IS3861-06) from BARD) and the US De-partment of Energy (DOE) through the office of Biological and Environmental Research (BER), and Terrestrial Carbon Processes (TCP) program (Grants Nos. 10509-0152, DE-FG02-00ER53015, and DE-FG02-95ER62083).


A P P E N D I X

THE FINNIGAN-BELCHER (FB04) FORMULATION FOR FLOW INSIDE CANOPIES ON A COSINE HILL

Inner layer

For flow above a cosine hill, the solution for the perturbed longitudinal ve-locity above the canopy is given by

( ) ( )( )( )

( )( )( )

2 2 i0 1 0

00 0

1 ln e, 1 cos Re ,ln ln

kxi

b Ih z A Bu x z U U kx

d z z d z z⎧ ⎫⎡ ⎤ ⎡ ⎤+⎪ ⎪∆ = − −⎨ ⎬⎢ ⎥ ⎢ ⎥+ +⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

where UI0 = (H k / 2) (U20 / U

2I ) is a dimensionless scaling for the longitudinal

velocity in the inner region, ( )( )0 0 2 / ,iB K i k L z d h= + where 0K is the modified Bessel function of zeroth order, and U0 is the characteristic veloci-ty in the region well above the canopy top, UI = Ub (hi) is a characteristic ve-locity in the flow at the inner layer height hi estimated by solving

( )21 0 02 ln ,v ih L k h z z and d are the momentum roughness length and zero-

plane displacement height, respectively, and kv is the von Karman constant. Using the mean continuity equation, the solution for the perturbed vertical velocity above the hill is

( )

03 1

2i

1 1 1

1 1

( , ) ( ) ( , ) ,

( )( , ) 2 ln ln ( ) sin ( ) Re e ,( )

2 ( ) ,

I

v

i i kx

o

i

U kw x z A x u Y x zk

h h i d zY x z z z d d z kx A Bd z z kL

B K i k L d z h

∗∆ = +

+⎡ ⎤ ⎡ ⎤= + − + −⎢ ⎥ ⎢ ⎥+ ⎣ ⎦⎣ ⎦= +

where K1 is the modified Bessel function of the first kind. The constant 3A can be evaluated by imposing the appropriate boundary conditions (Poggi et al. 2007). In the case of a tall canopy covering the surface, A3(x) is the parame-ter that allows matching ∆w (x, z) inside and above the canopy.

Canopy sub-layer A unique function describing ∆u (x, z) within the entire canopy sub-layer can be derived by the following phenomenological expression (Finnigan and Belcher 2004):

( ) ( )21 2

22 22, .cz L

c b c b c c bp pu x z U L s g n U L A U e U

x xβ∂∆ ∂∆⎛ ⎞∆ = − − + −⎜ ⎟∂ ∂⎝ ⎠

(A1)

The first and second terms on the right side represent the non-linear response of the flow field to, respectively, the pressure gradient induced by the hill shape and the turbulent transfer of momentum above the canopy (via Uc). The vertical velocity component can be evaluated using the continuity equation.


797

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Received 4 September 2007 Accepted 16 April 2008

acta geophysica - duke university782 d. poggi and g.g. katul situated on complex topography, this...

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