Boundary-Layer Meteorol (2010) 136:219–233
DOI 10.1007/s10546-010-9503-2
ARTICLE
Evaluation of the Turbulent Kinetic Energy Dissipation
Rate Inside Canopies by Zero- and Level-Crossing
Density Methods
D. Poggi · G. G. Katul
Received: 10 September 2009 / Accepted: 26 April 2010 / Published online: 16 May 2010
© Springer Science+Business Media B.V. 2010
Abstract Inferring the vertical variation of the mean turbulent kinetic energy dissipation
rate (ε) inside dense canopies remains a basic research problem to be confronted. Using
detailed laser Doppler anemometry (LDA) measurements collected within a densely arrayed
rod canopy, traditional and newly proposed methods to infer ε profiles are compared. The
traditional methods for estimating ε at a given layer include isotropic relationships applied to
the viscous dissipation scales that are resolved by LDA measurements, higher order structure
function methods, and residuals of the turbulent kinetic energy budget in which production
and transport terms are all independently inferred. The newly proposed method extends earlier approaches based on zero-crossing statistics, which were shown to be promising in a
number of laboratory flows. The extension to account for an arbitrary threshold (hereafter
referred to as the level-crossing method) instead of zero-crossing minimizes the effects of
instrument noise on the inferred ε. While none of the ε methods employed here can be titled
as ‘measured’, these methods differ in their underlying assumptions and simplifications.
Above the canopy, where a balance between production and dissipation rate of turbulent
kinetic energy is expected, the agreement among all the methods is reasonably good. In
the lower-to-middle layers of the canopy, all the methods agree except for those based on a
structure-function inference of ε. This departure can be attributed to the lack of a well-defined
inertial subrange in these layers. In the upper canopy layers, the disagreements between the
methods are largest. Even the higher order structure-function methods disagree with each
other when ε is inferred from third- and fifth-order moments. However, for all layers within
the canopy, the proposed zero- and threshold-crossing methods agree well with estimates of
D. Poggi
Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino, Italy
G. G. Katul (B)
Nicholas School of the Environment, Duke University, Box 90328, Durham, NC 27708-0328, USA
e-mail: [email protected]
G. G. Katul
Department of Civil and Environmental Engineering, Pratt School of Engineering, Duke University,
Durham, NC 27708, USA
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ε derived from the isotropic relationship applied to the viscous dissipation range. Finally, the
advantages of introducing thresholds to minimize two types of instrument noises, additive
and multiplicative, are briefly discussed.
Keywords Canopy turbulence · Rice’s formula · Telegraphic approximation · Turbulent
kinetic energy dissipation rate · Zero-crossing statistics
1 Introduction
Two reviews on the structure of turbulence within the canopy sub-layer (CSL), one published
about a decade ago (Finnigan 2000) and another published some three decades ago (Raupach
and Thom 1981), virtually share the same message about the difficulty in estimating the
mean turbulent kinetic energy dissipation rate (ε) inside canopies. A more robust estimation
of ε profiles within vegetated canopies is now needed in wide ranging applications such as
Lagrangian trajectory analysis of passive scalars (Rodean 1996; Poggi et al. 2006) and heavy
particle dispersion (Wilson 2000; Nathan et al. 2002; Katul et al. 2005), relaxation time scale
formulation in higher order closure modeling (Katul et al. 2004; Poggi et al. 2004b; Juang
et al. 2008), or in formulating subgrid models for large-eddy simulation (Moeng and Sullivan
1994; Sullivan et al. 1994; Patton et al. 1998). There are two reasons why the determination
of ε remains a major research obstacle for canopy flows. Firstly, the work that the mean flow
and turbulence exercise against the foliage drag produces turbulent kinetic energy (TKE) by
wakes and by spectral short-circuiting of the energy cascade over a broad range of scales, and
secondly the means by which this extra TKE is transferred and dissipated has resisted complete theoretical treatments (Finnigan 2000; Poggi and Katul 2006; Cava and Katul 2008).
On the experimental side, measuring ε inside canopies remains a formidable challenge, especially using standard sonic anemometry or hot-film probes (Kaimal and Finnigan 1994). Its
indirect inference from structure function scaling laws or as a residual of the TKE budget
cannot be rigorous given that the numerous assumptions employed by these two approaches
are routinely violated inside dense canopies. The inference of ε from isotropic relationships
applied to the finest length scales commensurate with the Kolmogorov dissipation scale (η)
requires the computation of spatial gradients at even smaller spatial scales when compared
to the previous two approaches. These fine-scale measurements are known to be difficult and
sensitive to instrumentation filtering and noise contamination even when conducted in the
most idealized of flow conditions.
Here, we explore an alternative method to estimate ε within and above canopies, which
we call the ‘threshold-crossing method’. This approach extends an earlier method that is
based on measuring zero-crossing density and relating this zero-crossing density to the mean
squared gradients of the velocity using a number of assumptions (Sreenivasan et al. 1983).
The zero-crossing density Nd (0) method (Sreenivasan et al. 1983) employs Rice’s formula
(Rice 1945), which states that
q̇ 2
Nd (0) = ,
(1)
4πq 2
where q̇ = dq/dt and q are stationary Gaussian distributed and statistically independent processes with q = 0, where the overbar indicates time averaging. Shortly after the publication
of Rice’s formula, it was originally suggested that when q represents a longitudinal turbulent
velocity component, ε can be estimated from Nd (0) given its links with the longitudinal
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Dissipation and Level-Crossing
221
velocity Taylor microscale λ. This linkage is developed and discussed in Sect. 3.2 below.
A number of investigations have already shown that ε or the Taylor microscale λ estimated
from Nd (0) agree well with independent measurements in several laboratory flows that do
not diverge appreciably from Gaussian (Sreenivasan et al. 1983; Kailasnath and Sreenivasan
1993). For canopy turbulence applications, the premise that ε may be estimated from Nd (0) is
rather ‘seductive’ given that, (i) gradient measurements are not required, (ii) no assumptions
are made about scaling laws in the structure functions, and (iii) no simplifications need to
be adopted in the TKE budget for which ε has to be computed as a residual. However, canopy turbulence is strongly inhomogeneous and non-Gaussian, and hence, the use of Rice’s
formula may be highly questionable.
Here, a number of methods to estimate ε profiles inside and above a canopy composed
of densely arrayed rods in a flume are compared, where the velocity time series used in
the ε estimation was measured using laser Doppler anemometry. We also extend Rice’s
formula to include ‘threshold-crossing’ in lieu of zero-crossing thereby generalizing the zerocrossing approach to any arbitrary threshold. This latter approach minimizes the effects of
noise on crossing density statistics, which we explore via simulations. A comparison between
ε profiles estimated from higher order structure functions, the residual in TKE budgets where
the production and transport terms are independently estimated, the isotropic relationships
applied to mean squared gradients computed in the viscous dissipation range, Nd (0) and
Nd (Tc ) for various Tc thresholds, is presented.
Throughout, both meteorological and index notation are used with x 1 = x, x2 = y and
x3 = z representing the longitudinal, lateral, and vertical directions respectively, and with
u 1 = u, u 2 = v, and u 3 = w representing the longitudinal, lateral, and vertical velocity components respectively, with z = 0 being the ground. Primed quantities represent instantaneous
turbulent excursions from mean states, the latter being represented by time averaging above
the canopy, and time and planar averaging inside the canopy (Raupach and Shaw 1982).
For notational simplicity, both averaging operators are represented by the overbar, and the
root-mean-squared value of an arbitrary variable q is defined as σq = q 2 .
2 Experimental Set-Up
Much of the experimental set-up is described elsewhere (Poggi et al. 2004a,c,d), but for
completeness, a brief review is presented. The experiment was conducted in an 18 m long,
0.90-m wide, and 1 m deep re-circulating rectangular flume having glass sidewalls to permit
optical access. The canopy was composed of stainless steel cylinders, 120 mm tall (= Hc )
and 4 mm in diameter (= dr ), which were arranged in a regular pattern along a 9-m long test
section. The rod density was 1072 rods m−2 and resulted in a drag coefficient comparable
to drag coefficients reported for crops and densely forested canopies (Katul et al. 2004).
The stationary flow rate employed here resulted in a friction velocity (= u ∗ ) of 0.098 m s−1
based on the measured shear stress at the canopy top and based on the uniform flow depth
Hw of 0.60 m. The canopy Reynolds numbers (Re∗ = u ∗ Hc /ν) was about 12,000. The
u and w series were collected at 2500–3000 Hz for a sampling duration of 300 s per level
above the channel bottom using two-component laser Doppler anemometry (LDA). The sampling frequency of the LDA was sufficiently high to resolve the entire inertial sub-range and
more than half a decade of the viscous dissipation range above the canopy. Given the planar
non-homogeneity in the single-point statistics within the canopy, as may occur when considering the flow statistics near and away from a rod, 11 planar sampling measurement locations
were employed to compute the spatial average of temporally-averaged flow statistics. These
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Fig. 1 Measured profiles of normalized mean velocity Ū /u ∗ (upper left), normalized standard deviations
σu /u ∗ and σw /u ∗ (top-right), skewness Skq (lower left), kurtosis or flatness factor K u q (lower right). Closed
circles are for the vertical velocity components and open circles are for the longitudinal velocity components.
The normalizing variables are the friction velocity at the canopy top (u ∗ ) and the canopy height (Hc )
11 locations were not uniformly spaced. They were selected such that sampling locations
were more densely placed in regions where the flow statistics exhibit highest spatial variability. In this work, instead of measuring across a large number of positions at each level above
the ground, a single run was employed but chosen such that the measured local temporal
statistics of this run were representative of the horizontally-averaged temporal statistics. The
description of this analysis and the more spatially expansive data for the same model canopy
and flow conditions can be found in Poggi et al. (2004c). In a separate study, it was shown
that dispersive terms arising from planar averaging inside the canopy can be neglected for
dense canopies (Poggi et al. 2004a; Poggi and Katul 2008a,b), and hence the impact of dispersive terms arising from spatial averaging will not be considered. The entire water depth
was sampled uniformly every 10 mm, but we focus here on the CSL region, extending up to
2.6Hc .
Figure 1 illustrates a number of canonical features about the measured bulk flow properties of CSL turbulence for this flume experiment. The mean velocity Ū inside the canopy
is generally small but finite with a strong inflection point near the canopy top resembling a
mixing layer rather than a boundary layer (Raupach et al. 1996). Moreover, the measured
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223
Ū /u ∗ ≈ 3.3 at z/Hc = 1 is consistent with a wide range of field experiments conducted
for dense canopies (Raupach et al. 1996; Katul and Albertson 1998; Finnigan 2000). The
normalized square root of the longitudinal and vertical velocity variances, σu /u ∗ and σw /u ∗ ,
are damped inside the canopy though they remain significant even in the vicinity of z = 0.
The longitudinal and vertical velocity skewness and flatness factor profiles, computed from
Skq = q 3 /σq3 and K u q = q 4 /σq4 for q = u, w, respectively are also shown. Inside the
canopy, the flow statistics are clearly non-Gaussian, with positively skewed longitudinal
velocity and negatively skewed vertical velocity consistent with arguments that organized
sweeps dominate momentum transfer vis-à-vis ejections (Poggi et al. 2004b; Poggi and Katul
2007). Moreover, with intermittency often defined as 3/K u q , it is clear that the flow inside
the canopy is highly intermittent but almost Gaussian close to the canopy top. Hence, from
Fig. 1, it is evident that one of the key assumptions for the use of Rice’s formula is violated
for canopy flows, given the strong departure from Gaussian assumptions in u and w.
3 Methods of Analysis
The various methods used to compute the ε profiles are reviewed next and then compared
against this dataset.
3.1 Traditional Methods
The traditional methods for estimating ε include using (1) isotropic relationships for velocity
measurements that resolve the kinetic energy dissipation rate spectrum, (2) spectra or structure function scaling laws, and (3) the residual of the TKE budget. A brief description of how
the LDA velocity measurements were used for each method is provided next.
3.1.1 Dissipation Estimates from Isotropic Relationships
The mean turbulent kinetic energy dissipation rate can be estimated from the isotropic relationship (Hinze 1959),
∂u ε = 15ν
∂x
2
,
(2)
where x is the longitudinal distance, and ∂(.)/∂ x should be computed on spatial scales smaller
than η. Using Taylor’s frozen turbulence hypothesis (Taylor 1938; Wyngaard and Clifford
1977; Hsieh and Katul 1997), the spatial gradients in Eq. 2 can be determined from temporal
gradients so that
ε = 15ν
1
U
2
∂u ∂t
2
.
(3)
This method is particularly well-suited to LDA and hot-wire velocity measurements since the
sampling frequency is sufficiently high to resolve the viscous dissipation frequency range. So
far, this estimation cannot be successfully applied to full-scale flow using sonic anemometry
data because the sampling frequency and path-averaging cut-offs do not allow the dissipation
range to be properly resolved.
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3.1.2 Structure-Function Approach
The velocity structure functions are widely used to infer ε in atmospheric turbulence (Kaimal
and Finnigan 1994). In the case of u, the pth-order structure function velocity difference along
the longitudinal direction x for separation distance r is given as:
D p (r ) = |[u(x + r ) − u(x)]| p .
(4)
According to the Kolmogorov’s (1941) second similarity hypothesis (hereafter referred to as
K41), D p (r ) in the inertial subrange depends on ε and r only and is given by
D p (r ) = C p (εr ) p/3 ,
(5)
where C p is a similarity constant. For p = 3, C p = 4/5 can be determined analytically
for locally homogeneous and isotropic turbulence using the von Karman-Howarth equation (Monin and Yaglom 1975). Moreover, any intermittency corrections modifying the p/3
scaling become less significant for p = 3. Hence, ε (labelled hereafter as ε Dp ) can be determined from the intercept when regressing log[D p (r )] upon log(r ). Naturally, this similarity
hypothesis or the use of the von Karman-Howarth equation fails when the wake production
and short-circuiting of the energy cascade injects new length scales (Finnigan 2000; Poggi
and Katul 2006; Cava and Katul 2008; Poggi et al. 2008). To illustrate this point, Fig. 2
presents variations of D p (r ) as a function of r for u and w inside the canopy (z/Hc = 0.5)
and above the canopy (z/Hc = 2.3). A clear inertial subrange exists for the above-canopy
cases (> two decades). However, inferring ε from the measured D p (r ) is clearly problematic inside the canopy. There appears to be a limited range of scales where the scaling laws
agree with K41 predictions (approximately 1/2 decade), and, for the purposes of ε profile
comparisons, these are the ranges used to infer ε inside the canopy from structure-function
methods. Two higher order moments were also employed in the estimation of ε(= ε D3 and
ε D5 for p = 3 and 5) to ensure the outcome is not sensitive to the choice of the analyzing
moment. Although second-order structure functions may converge faster than third-order
structure functions as shown by Chamecki and Dias (2004), we used p = 3 here since (a) the
exponent p/3 = 1 is exact and linear in r and the dissipation rate, (b) C p can be determined
analytically for locally homogeneous and isotropic turbulence. Moreover, p = 5 was used
to ensure the outcome is not sensitive to the choice of the analyzing moment to address the
issue of statistical convergence.
The separation distances (r ) in Fig. 2 are normalized by the rod diameter and the Strouhal
number, St ≈ 0.2, to emphasize the scale at which wake production and the generation of
Von Karman streets occur inside the canopy (Poggi et al. 2004c, 2006; Poggi and Katul 2006;
Cava and Katul 2008). Note that, inside the canopy, the ‘apparent’ inertial subrange scaling
in Fig. 2 commences immediately after wake-production scales. These apparent inertial subrange scales cannot be locally isotropic as stipulated by K41 given that not many cascading
steps occur to ‘wipe’ out the anisotropy introduced by the wake-production mechanism.
3.1.3 The Turbulent Kinetic Energy Budget
In a stationary and planar homogeneous flow in the absence of subsidence, and upon neglecting the dispersive and the pressure transport terms, the TKE budget reduces to:
εd = Ps + Pw + PT ,
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(6)
Dissipation and Level-Crossing
225
Fig. 2 Examples of measured higher order structure functions (D3 (r ), diamond, and D5 (r ), circles) for the
longitudinal velocity time series as a function of normalized separation distance r inferred from Taylor’s frozen
turbulence hypothesis for four dimensionless distances from the channel bottom (z/Hc ). The normalization
of r is based on the rod diameter (dr ) and the dimensionless Strouhal number (St = 0.21) at which wake
production occurs. The solid lines show the K41 scaling, the vertical solid lines in the top panels are when r
approaches the size of the von Karman vortices injected in the flow field by the rods
where Ps is the shear production, Pw is the wake production, and PT represents the transport
terms. From the observations used herein, we estimated these terms as follows:
Ps = −u w ∂U
,
∂z
3
Pw = 2Cd aU ,
PT =
∂(w e )
∂z
,
(7a)
(7b)
(7c)
with Cd accounting for sheltering effects and local Reynolds number variations with height,
and is given elsewhere (Poggi et al. 2004b), a is the frontal area index (Poggi et al. 2004c), e
is the instantaneous turbulent kinetic energy, and e = u i u i /2 is the turbulent kinetic energy.
By estimating Ps , Pw , and PT from the measured profiles of u w , U , and w e , εd can then
be computed.
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D. Poggi, G. G. Katul
Fig. 3 The variation of the Taylor microscale λ(Tc ) as a function of the threshold Tc for various normalized
distances from the channel bottom (z/Hc ). External panels comparison between measured (open circles) and
2
modelled values inferred from λ(0) e−TC /2 (solid dot-line), where λ(0) was computed from the measured
zero-crossing statistics. Middle panel the one-to-one comparison between modelled and measured λ(TC ) for
all z/Hc along with the correlation coefficient (R)
3.2 Proposed Zero- and Level-Crossing Methods
For a time series sampled at discrete times ti (i = 1, 2, 3, . . . , N ) and at a rate dt = ti+1 − ti ,
the times at which a threshold-crossings occurs are defined by the indicator function I (Tc , ti )
given as
1 if u (ti ) − Tc u (ti+1 ) − Tc < 0
,
(8)
I (TC , ti ) =
0 otherwise
where TC = Sh/σu is a non-dimensional threshold (see Fig. 3).
The overall density of threshold- and zero-crossings is given by (Bershadskii et al. 2004;
Sreenivasan and Bershadskii 2006a,b; Cava and Katul 2009; Poggi and Katul 2009),
N −1
I (Tc , ti )
Nd (Tc ) = i=1
,
(9a)
N −1
N −1
I (0, ti )
.
(9b)
Nd (0) = i=1
N −1
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Dissipation and Level-Crossing
227
As earlier noted, for a zero mean Gaussian process having a variance σ 2 and an autocorrelation function f (r ), the expected number of crossings of a level TC per unit of time is
given as
Nd (TC ) =
1 21 −T 2 /2
− f (0) e C ,
π
(10)
which recovers the zero-crossing formulation when TC = 0, to yield (Sreenivasan et al. 1983):
Nd (0) =
1 21
− f (0) .
π
(11)
These formulations can be used to estimate ε based on the longitudinal velocity Taylor
microscale. As discussed in Liepmann (1949), the Taylor microscale can be estimate from a
parabolic extrapolation of the curvature of f (r ) at r = 0, given by (Tennekes and Lumley
1972; Pope 2000)
− 1
2
1 λ = − f (0)
.
2
(12)
It follows from these expressions that λ can be estimated from threshold- and zerocrossings as
√
2
2
λ(Tc ) =
(13a)
[Nd (TC )]−1 e −TC /2 ,
π
√
2
λ(0) =
(13b)
Nd (0)−1 .
π
Hence, when a threshold Tc is imposed, λ(Tc ) = λ(0)e −TC /2 . We comparedestimates
of λ
2
based on the inferred Nd (Tc ) directly from the time series and based on λ(0)e −TC /2 , where
only Nd (0) was
inferred
from the time series as well. We found that the assumed ‘threshold’
−TC2 /2
to the zero-crossing estimate of λ is quite reasonable for various z/Hc
correction e
and Tc values as evident from Fig. 3.
To link λ to ε, we note that, for isotropic flow,
∂u
∂x
2
=
2
2σu2
ε
,
=
2
λ
15ν
(14)
thereby allowing ε to be written as
εTC =
15π 2
2
ν σu2 [Nd (TC )]2 e TC ,
2
(15)
15π 2
ν σu2 Nd2 (0).
2
(16)
and
εzc =
The above two formulations have a number of advantages: they are simple to implement and
require only event-counting, no derivative estimation and no inertial subrange identification
for structure function computations. What is not known is how robust are these estimates of
ε to the assumptions in Rice’s formula inside canopies.
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D. Poggi, G. G. Katul
Fig. 4 Comparison between the traditional and newly proposed methods to estimate the normalized TKE
dissipation rate (ε) profile as a function of normalized height (z/Hc ). The εi are based on isotropic relationships, εd is based on the residual in the TKE budget, ε D3 and ε D5 are based on structure–function methods
with p = 3 and p = 5 respectively, and εzc is based on the zero-crossing method. The normalizing variables
are canopy height (Hc ) and friction velocity at the canopy top (u ∗ )
4 Results
Figure 4 presents comparisons between the various methods used to estimate ε profiles for
the experiment here. Good agreement between all methods is noted when z/Hc > 1.2. For
the middle to lower layers inside the canopy (0.05 < z/Hc < 0.6), best agreements are
noted between εi , εd , and εzc , with structure–function based estimates of ε underestimating these three methods (for both p = 3 and p = 5). In the upper layers of the canopy
(0.7 < z/Hc < 1), the disagreements are not small near the canopy top. The estimates εi and
εzc agree with each other but the structure–function based estimates diverge from each other.
The structure–function based estimate of ε inferred from p = 3 generally underestimates εi
and εzc , though the opposite is true for p = 5, though given the lack of scaling in Fig. 2,
especially for p = 5, this disagreement is not entirely surprising. Finally, εd estimates in
the upper canopy region match with the structure–function based estimates for p = 5 but
we note a departure with the other estimates that may be attributed to approximations in the
evaluation of TKE (Eq. 6).
It is clear that each method has its deficiencies. It is conceivable that the isotropic assumption at the finest scales may be less valid inside canopies given that wake production is large
in this vicinity. Wake production occurs at scales commensurate with dr and some anisotropy
from this production may still persist even at scales comparable to η. However, if this is the
case, then structure–function based estimates are even less reliable than εi . We repeated the
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Dissipation and Level-Crossing
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Fig. 5 The effects of varying the threshold Tc on the threshold-crossing estimates of εT C . For reference, εi
are also shown
same calculations for the threshold-based approach for a number of Tc values and the εT C
results appear to be robust to variations in Tc (Fig. 5).
A follow-up question then is: what are the advantages of employing Tc over zero-crossing
statistics? It should be clear that zero-crossing statistics are sensitive to contamination by
instrumentation noise, and a threshold imposed by Tc may minimize their impact on ε. To
illustrate this point, Fig. 6 considers a longitudinal velocity time series that was synthetically
‘infected’ with white noise in an additive manner in one case, and in a state-dependent manner in another. For the latter case, this noise is assumed to be proportional to the amplitude of
the velocity. The σ of the noise is on the order of σu . The intensity of the synthetic noise was
varied by changing the number of added noisy points to the time series, and for both types
of noise infection and noise intensity, introducing a threshold Tc significantly minimized the
impact on the inference of ε when compared to ε inferences made only with zero-crossing
statistics. The infected series results in ε estimates higher than the un-infected series when
no thresholding is employed. The agreement between ε inferred from the un-infected and
thresholded series infected with noise is far superior.
5 Conclusions
Using detailed laser Doppler anemometry measurements collected inside a densely arrayed
rod canopy, a number of methods to infer ε profiles were compared. These methods included
traditional approaches such as the TKE budget, structure–function methods, and isotropic
relationships, and newly proposed methods such as threshold or zero-crossing statistics.
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D. Poggi, G. G. Katul
Fig. 6 Sensitivity analysis of ε to contamination by instrumentation noise. Left panels longitudinal velocity
time series synthetically ‘infected’ with additive white noise (upper panel) and state dependent-noise (bottom
panel). Central panels number of threshold-crossing events as a function of the threshold level and noise
intensity. The closed symbols are for the original time series, empty symbols for noisy series. Right panels
comparison between ε computed from zero- and threshold-crossing methods, hereafter labelled as ε ZC and
εT s , respectively, for a noisy time series
Zero- and threshold-crossing methods offer a number of advantages over traditional methods, though these methods have rarely been explored in the canopy sublayer. It is conceivable
that the lack of ‘popularity’ of these methods inside canopies stems from assumptions inherent in Rice’s formula. Application of Rice’s formula requires that the stochastic process
and its derivative be simultaneously Gaussian and independent of each other. Both of those
assumptions are severely violated inside dense canopies, where finite skewness and large
flatness factors are defining syndromes of the flow, and where velocity gradients are heavytailed. We showed that despite the restrictive assumptions of Rice’s formula, the ε inferred
from zero- or threshold-crossing methods agrees well with predictions from isotropic relationships. Above the canopy, all the methods agree well with each other even though the
flow maintains its non-Gaussian properties. This robustness may be due to the fact that zerocrossing statistics are preserved in a telegraphic approximation (TA) of the velocity series.
All the non-Gaussianity was shown to be encoded in the fraction of time the process resides
above the threshold (or zero) in the TA series. We also showed that the use of a threshold may
be a practical way to minimize the impact of instrument noise (additive or multiplicative) on
zero-crossing statistics.
Appendix
We should note that zero-crossing statistics are preserved in the so-call telegraphic approximation (TA) of a time series u (ti ), which is defined as (see also Fig. 7)
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Dissipation and Level-Crossing
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Fig. 7 The linkages between the time series of the longitudinal velocity excursions (u), their telegraphic
approximation (TA), threshold-crossing statistics, and non-Gaussian properties. Top left panel the normalized
u inside the canopy at z/Hc = 0.5 along with its TA counterpart for TC = 0; note that TA can only be unity or
zero. The bottom-left panel illustrates how the zero-crossings and threshold (= Tc ) crossings are preserved in
the TA series (only the values I = 1 are shown). Top-right the profiles of the fraction of time the TA process
takes on the value of unity (Tc ) across the CSL for various TC values directly measured (open circles)
and predicted from the Sku profile in Fig. 1. Bottom right comparisons between CEM-predicted (lines) and
measured (open circles) relationship between (Tc ) versus Sku for various TC values
T A(TC , ti ) =
1 u (ti ) − TC > 0
.
0 otherwise
(17)
Non-Gaussian statistics such as skewness are entirely encoded and preserved in the first
moment of T A(TC , ti ). To illustrate this ‘preservation’ property, consider a cumulant expansion (CEM) of the probability density function p(û), where û = u /σu and as a case study,
all cumulants beyond order 3 are discarded. This cumulant expansion leads to (Nakagawa
and Nezu 1977; Raupach 1981; Katul et al. 1997, 2006):
2 1
û
1
p(û) ≈ √ exp −
(18)
1 + Skû (û 3 − 3û) .
2
6
2π
+∞
The third-order CEM expansion here retains the necessary condition that −∞ p(û)d û = 1.
Moreover, for the range of û and Sku encountered in our experiment, p(û) > 0. The fraction
of time û > Tc , which is identical to the fractions of times T A(TC , ti ) = 1, is given by
∞
(Tc ) =
Tc
Sku Tc2 − 1
TC2
TC
1
p(ξ )dξ =
exp −
+ er f c √
√
2
2
6 2π
2
(19)
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D. Poggi, G. G. Katul
where er f c [ ] is the complementary error function. Hereafter, we refer to the above expression as the CEM-predicted (Tc )–Sku relationship. The classical zero-crossing (0) is recovered upon imposing TC = 0 and results in
∞
p(ξ )dξ =
(0) =
0
1
1
− √ Sku .
2 6 2π
(20)
It is clear from this primitive illustration that a one-to-one connection between the fraction of
time the TA series takes on unity values (a first-moment statistic) and departures from Gaussianity (i.e. finite skewness) in the actual velocity time series must exist. Stated differently,
(TC ) may capture the non-Gaussian properties embedded in the distributional properties
of the series. Figure 7 shows that, indeed for the velocity time series, the CEM-predicted
(TC )–Sku relationship well reproduces the measurements within and above the canopy and
across a wide range of thresholds. Hence, it is conceivable that Nd2 (TC ) may be less sensitive
to the Gaussian assumptions, given that their effects are already accounted for in (TC ),
though this assessment is simply a speculation, and not a rigorous proof.
References
Bershadskii A, Niemela JJ, Praskovsky A, Sreenivasan KR (2004) Clusterization and intermittency of temperature fluctuations in turbulent convection. Phys Rev E 69(5):1–5
Cava D, Katul GG (2008) Spectral short-circuiting and wake production within the canopy trunk space of an
alpine hardwood forest. Boundary-Layer Meteorol 126(3):415–431
Cava D, Katul GG (2009) The effects of thermal stratification on clustering properties of canopy turbulence.
Boundary-Layer Meteorol 130(3):307–325
Chamecki M, Dias NL (2004) The local isotropy hypothesis and the turbulent kinetic energy dissipation rate
in the atmospheric surface layer. Q J R Meteorol Soc 130:2733–2752
Finnigan J (2000) Turbulence in plant canopies. Ann Rev Fluid Mech 32:519–571
Hinze JO (1959) Turbulence. McGraw-Hill, New York, 790 pp
Hsieh CI, Katul GG (1997) Dissipation methods, Taylor’s hypothesis, and stability correction functions in the
atmospheric surface layer. J Geophys Res Atmos 102(D14):16391–16405
Juang JY, Katul GG, Siqueira MB, Stoy PC, McCarthy HR (2008) Investigating a hierarchy of Eulerian closure
models for scalar transfer inside forested canopies. Boundary-Layer Meteorol 128(1):1–32
Kailasnath P, Sreenivasan KR (1993) Zero crossings of velocity fluctuations in turbulent boundary layers.
Phys Fluids 5(11):2879–2885
Kaimal JC, Finnigan JJ (1994) Atmospheric boundary layer flows: their structure and measurement. Oxford
University Press, New York, 289 pp
Katul GG, Albertson JD (1998) An investigation of higher-order closure models for a forested canopy.
Boundary-Layer Meteorol 89(1):47–74
Katul G, Hsieh CI, Kuhn G, Ellsworth D, Nie DL (1997) Turbulent eddy motion at the forest–atmosphere
interface. J Geophys Res Atmos 102(D12):13409–13421
Katul GG, Mahrt L, Poggi D, Sanz C (2004) One- and two-equation models for canopy turbulence. BoundaryLayer Meteorol 113(1):81–109
Katul GG, Porporato A, Nathan R, Siqueira M, Soons MB, Poggi D, Horn HS, Levin SA (2005) Mechanistic
analytical models for long-distance seed dispersal by wind. Am Nat 166(3):368–381
Katul G, Poggi D, Cava D, Finnigan J (2006) The relative importance of ejections and sweeps to momentum
transfer in the atmospheric boundary layer. Boundary-Layer Meteorol 120(3):367–375
Liepmann HW (1949) Die anwendung eines satzes uber die nullstellen stochastischer funktionen auf turbulenzmessungen. Helv Phys Acta 22(2):119–126
Moeng CH, Sullivan PP (1994) A comparison of shear-driven and buoyancy-driven planetary boundary-layer
flows. J Atmos Sci 51(7):999–1022
Monin AS, Yaglom AM (1975) Statistical fluid mechanics. MIT Press, Cambridge, MA, 782 pp
Nakagawa H, Nezu I (1977) Prediction of contributions to Reynolds Stress from bursting events in openchannel flows. J Fluid Mech 80:99–128
123
Dissipation and Level-Crossing
233
Nathan R, Katul GG, Horn HS, Thomas SM, Oren R, Avissar R, Pacala SW, Levin SA (2002) Mechanisms
of long-distance dispersal of seeds by wind. Nature 418(6896):409–413
Patton EG, Shaw RH, Judd MJ, Raupach MR (1998) Large-eddy simulation of windbreak flow. BoundaryLayer Meteorol 87(2):275–306
Poggi D, Katul GG (2006) Two-dimensional scalar spectra in the deeper layers of a dense and uniform model
canopy. Boundary-Layer Meteorol 121(2):267–281
Poggi D, Katul G (2007) The ejection-sweep cycle over bare and forested gentle hills: a laboratory experiment.
Boundary-Layer Meteorol 122(3):493–515
Poggi D, Katul GG (2008a) The effect of canopy roughness density on the constitutive components of the
dispersive stresses. Exp Fluids 45(1):111–121
Poggi D, Katul GG (2008b) Micro- and macro-dispersive fluxes in canopy flows. Acta Geophys 56(3):778–
799
Poggi D, Katul G (2009) Flume experiments on intermittency and zero-crossing properties of canopy turbulence. Phys Fluids 21(6):065103
Poggi D, Katul GG, Albertson JD (2004a) A note on the contribution of dispersive fluxes to momentum
transfer within canopies—research note. Boundary-Layer Meteorol 111(3):615–621
Poggi D, Katul GG, Albertson JD (2004b) Momentum transfer and turbulent kinetic energy budgets within a
dense model canopy. Boundary-Layer Meteorol 111(3):589–614
Poggi D, Porporato A, Ridolfi L, Albertson JD, Katul GG (2004c) The effect of vegetation density on canopy
sub-layer turbulence. Boundary-Layer Meteorol 111(3):565–587
Poggi D, Porporato A, Ridolfi L, Albertson JD, Katul GG (2004d) Interaction between large and small scales
in the canopy sublayer. Geophys Res Lett 31(5):4
Poggi D, Katul G, Albertson J (2006) Scalar dispersion within a model canopy: measurements and
three-dimensional Lagrangian models. Adv Water Res 29(2):326–335
Poggi D, Katul GG, Cassiani M (2008) On the anomalous behavior of the Lagrangian structure function
similarity constant inside dense canopies. Atmos Environ 42(18):4212–4231
Pope SB (2000) Turbulent flows. Cambidge University Press, UK, 771 pp
Raupach MR (1981) Conditional statistics of Reynolds stress in rough-wall and small-wall turbulent boundary
layers. J Fluid Mech 108:363–382
Raupach MR, Shaw RH (1982) Averaging procedures for flow within vegetation canopies. Boundary-Layer
Meteorol 22(1):79–90
Raupach MR, Thom AS (1981) Turbulence in and above plant canopies. Ann Rev Fluid Mech 13:97–129
Raupach MR, Finnigan JJ, Brunet Y (1996) Coherent eddies and turbulence in vegetation canopies: the
mixing-layer analogy. Boundary-Layer Meteorol 78(3–4):351–382
Rice SO (1945) Mathematical analysis of random noise. Bell Syst Tech J 24(1):46–156
Rodean H (1996) Stochastic Lagrangian models of turbulent diffusion. American Meteorological Society,
Boston, MA, 84 pp
Sreenivasan KR, Bershadskii A (2006a) Clustering properties in turbulent signals. J Stat Phys 125(5–6):
1145–1157
Sreenivasan KR, Bershadskii A (2006b) Finite-Reynolds-number effects in turbulence using logarithmic
expansions. J Fluid Mech 554:477–498
Sreenivasan KR, Prabhu A, Narasimha R (1983) Zero-crossings in turbulent signals. J Fluid Mech 137
(DEC):251–272
Sullivan PP, McWilliams JC, Moeng CH (1994) A subgrid-scale model for large-eddy simulation of planetary
boundary-layer flows. Boundary-Layer Meteorol 71(3):247–276
Taylor GI (1938) The spectrum of turbulence. Proc R Soc Lond A Math Phys Sci 164(A919):0476–0490
Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, Cambridge, 300 pp
Wilson JD (2000) Trajectory models for heavy particles in atmospheric turbulence: comparison with observations. J Appl Meteorol 39(11):1894–1912
Wyngaard JC, Clifford SF (1977) Taylor’s hypothesis and high frequency turbulence spectra. J Atmos Sci
34(6):922–929
123