The relative importance of ejections and sweeps to momentum transfer in the atmospheric boundary layer

Abstract

Using an incomplete third-order cumulant expansion method (ICEM) and standard second-order closure principles, we show that the imbalance in the stress contribution of sweeps and ejections to momentum transfer (ΔS

o
) can be predicted from measured profiles of the Reynolds stress and the longitudinal velocity standard deviation for different boundary-layer regions. The ICEM approximation is independently verified using flume data, atmospheric surface layer measurements above grass and ice-sheet surfaces, and within the canopy sublayer of maturing Loblolly pine and alpine hardwood forests. The model skill for discriminating whether sweeps or ejections dominate momentum transfer (e.g. the sign of ΔS

o
) agrees well with wind-tunnel measurements in the outer and surface layers, and flume measurements within the canopy sublayer for both sparse and dense vegetation. The broader impact of this work is that the “genesis” of the imbalance in ΔS

o
is primarily governed by how boundary conditions impact first and second moments.

Full text

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Boundary-Layer Meteorol (2006)
DOI 10.1007/s10546-006-9064-6
ORIGINAL ARTICLE
The relative importance of ejections and sweeps
to momentum transfer in the atmospheric
boundary layer
Gabriel Katul · Davide Poggi · Daniela Cava ·
John Finnigan
Received: 9 June 2005 / Accepted: 8 December 2005
© Springer Science+Business Media B.V. 2006
Abstract
dard second-order closure principles, we show that the imbalance in the stress contribution
of sweeps and ejections to momentum transfer (?So) can be predicted from measured pro-
files of the Reynolds stress and the longitudinal velocity standard deviation for different
boundary-layer regions. The ICEM approximation is independently verified using flume
data, atmospheric surface layer measurements above grass and ice-sheet surfaces, and within
the canopy sublayer of maturing Loblolly pine and alpine hardwood forests. The model skill
fordiscriminatingwhethersweepsorejectionsdominatemomentumtransfer(e.g.thesignof
?So) agrees well with wind-tunnel measurements in the outer and surface layers, and flume
measurements within the canopy sublayer for both sparse and dense vegetation. The broader
impact of this work is that the “genesis” of the imbalance in ?Sois primarily governed by
how boundary conditions impact first and second moments.
Using an incomplete third-order cumulant expansion method (ICEM) and stan-
G. Katul (B ) · D. Poggi
Nicholas School of the Environment and Earth Sciences,
Duke University Box 90328, Durham, NC, USA
e-mail: gaby@duke.edu
G. Katul
Department of Civil and Environmental Engineering,
Duke University, Durham, NC, USA
D. Poggi
Dipartimento di Idraulica Trasporti e Infrastrutture Civili, Politecnico di Torino,
Corso Duca degli Abruzzi, 24
10129, Torino, Italy
D. Cava
CNR - Institute of Atmospheric Sciences and Climate, Section of Lecce,
Polo Scientifico dell’Università
Strada Prov. Lecce-Monteroni km 1,200
73100 Lecce, Italy
J. Finnigan
CSIRO Marine and Atmospheric Research, FC Pye Laboratory, Black Mountain,
Canberra, ACT 2601, Australia

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Boundary-Layer Meteorol (2006)
Keywords
Ejections and sweeps · Momentum transfer
Canopy turbulence · Closure modelling · Cumulant expansions ·
1. Introduction
The interest in ejections and sweeps dates back to early experiments by Kline et al. (1967)
who demonstrated via flow visualization that fluid motion near a wall is “far from being
completely chaotic in nature” revealing a definite “sequence of ordered motion”. To quan-
tify the impact on momentum transfer of such coherent motion, often called the bursting
or ejection-sweep cycle, several techniques were proposed and tested (Robinson 1991). A
popular technique, known as conditional sampling and quadrant analysis, was introduced by
Lu and Willmarth (1973) to detect signatures of such coherent motion in component velocity
time series and to derive quantitative information about them (see reviews in Antonia 1981;
also Cantwell 1981). Building on earlier work by Frenkiel and Klebanoff (1967), Nakagawa
and Nezu (1977) and Raupach (1981) employed a two-dimensional Gram-Charlier cumu-
lant expansion method (CEM) for the joint probability density function (JPDF) of turbulent
longitudinal (u?) and vertical (w?) velocity components to link non-Gaussian JPDF with the
statistical properties of the ejection-sweep cycle. In particular, the CEM approach used by
Nakagawa and Nezu (1977) was able to analytically couple the imbalance in stress contri-
bution of sweeps and ejections to turbulent diffusion processes through the third (or mixed)
moments. Such a linkage was timely because advances in second-order closure modelling
required better parameterization of the turbulent diffusion processes (Hanjalic and Launder
1972; Mellor 1973; Donaldson 1973; Lumley 1978). Hence, it is no surprise that the imbal-
ance in the stress contribution of sweeps and ejections and non-Gaussian JPDF received
significant experimental and theoretical attention (e.g. Durst et al. 1992; Jovanovic et al.
1993; Durst and Jovanovic 1995).
Withintheatmosphericboundarylayer(ABL),theasymmetryintheJPDFalsomotivated
model developments such as top-down/ bottom-up diffusion (e.g. Weil 1990; Sorbjan 1999;
Patton et al. 2003) or structural models for the triple moments that require a priori knowl-
edge as to whether sweeps or ejections dominate the momentum transfer (Nagano and
Tagawa 1990).
Here, we investigate whether the profiles of first and second moments in the ABL are
sufficient to predict a priori whether ejections or sweeps dominate momentum transfer. If
successful,theimplicationofthisworkisthatthe“genesis”ofthisimbalancecanbeexplained
by how boundary conditions impact lower-order moments.
2. Theory
The ejection–sweep cycle is typically quantified via quadrant analysis and conditional sam-
plingmethodsreviewedinAntonia(1981).Quadrantanalysisreferstothejointscatteracross
four quadrants defined by a Cartesian plane whose abscissa is u?and ordinate is w?. The four
quadrantsareconnectedwithfourmodesofmomentumtransfer:eventsinquadrantII(u?< 0,
w?> 0) and quadrant IV (u?> 0 and w?< 0) are called ejections and sweeps respectively;
events in quadrant I (u?> 0 and w?> 0) and quadrant III (u?< 0 and w?< 0) are called
outward and inward interactions, respectively.
Nakagawa and Nezu (1977) and Raupach (1981) defined the imbalance in the contribu-
tion of sweeps and ejections to momentum transfer using the difference in stress fraction
contributions of quadrant II and quadrant IV:

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Boundary-Layer Meteorol (2006)
?So=u?w?IV− u?w?I I
u?w?
(1)
where u?w?is the momentum flux andu?w?IV
IVandII,respectively.ThedefinitioninEq.(1)ensuresthat?Soisboundedbetween−1and
1 (assuming |u?w?| > 0). Based on this definition, sweeps dominate the momentum transfer
when ?So> 0, and conversely. Because ?Sobecomes ill-defined when |u?w?| → 0, our
analysis is restricted to cases when |u?w?| is finite. This restriction implies that the purely
convective boundary layer will not be treated here.
Using the Gram-Charlier series expansion of the JPDF (Kampé de Fèriet 1966), truncated
to the third order, Raupach (1981) demonstrated that
?
where,
?1
C2= −
?
Katul et al. (1997a, b) noted that for the range of skewness values encountered in the atmo-
sphericboundarylayer(includingcanopyturbulence),thecontributionofthe1
to Eq. (2) is small, which lead Katul et al. (1997a, b) to further simplify Eq. (2) to give:
u?w?
andu?w?I I
u?w?are the stress fractions in quadrants
?So=
1 + Ruw
Ruw
√2π
2C1
(1 + Ruw)2+
C2
1 + Ruw
?
(2)
C1= (1 + Ruw)
6(M03− M30) +1
6(2 − Ruw)(M03− M30) +1
2(M12− M21)
?
,
?1
2(M12− M21)
?
,
and Ruw=u?w?
σuσw, Mji=u?jw?i
σj
uσiw, σs=
s?2and where u?j??j=2→ u?2, etc...
6(M03− M30)
?So≈
1
2Ruw
√2π
(M21− M12)
(3)
where,
M21=w?u?u?
M12=w?w?u?
σwσ2
u
,
(4a)
σuσ2
w
.
(4b)
We refer to Eq. (3) as an incomplete cumulant expansion (ICEM) because of the elimination
of the velocity skewness.
We tested predictions from Eq. (3) against quadrant analysis using laboratory and field
measurements collected over a wide range of atmospheric conditions (Fig. 1). The flume
data are collected within and above a dense canopy composed of cylindrical rods (Poggi et
al. 2004a, b); the field datasets include surface-layer measurements above tall grass (Katul et
al. 1997a) and ice surfaces (Cava et al. 2001, 2005); and the forest canopy data include mea-
surements within a maturing pine stand (Katul and Albertson 1998) and an alpine hardwood
forest (Cava et al. 2006). The measurements within the Antarctic surface layer above ice also
include runs sampled at different heights for planar homogeneous flows, flows perturbed by
katabatic winds, and flows perturbed by a ridge. When taken together, this ensemble dataset
populates the entire plausible values of ?So(i.e. −1 to 1) as shown in Figure 1.

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Boundary-Layer Meteorol (2006)
-1-0.8-0.6 -0.4-0.200.20.40.60.81
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Measured ∆So
ICEM Modeled ∆So
Flume (rods)
Grass, ASL
Pine Forest, 6 levels
Hardwood Forest, 5 levels
Ice, ASL


Fig. 1 Comparison between measured (i.e. quadrant analysis) and ICEM modelled ?So. The open circles
are from a flume experiment comprised of cylindrical rods (dense canopy); the diamonds are data taken from
six levels within a maturing Loblolly pine forest; the stars are data from a surface-layer experiment above
a tall grass surface; the diamond are data taken from five levels within an alpine hardwood forest; and the
triangles are data taken in an Antarctic surface layer and include runs in which the surface layer is perturbed
by the presence of a ridge. The field experiments include a wide range of atmospheric stability conditions.
The 1:1 line is shown for reference. Regressing the modelled (ˆ y =ICEM) upon the measured (ˆ x =quadrant
analysis)?Sousingallthedata(n = 1340 points)yields ˆ y = 1.0ˆ x+0.031withacoefficientofdetermination
R2= 0.92
Despite the simplifications in its derivation (i.e. truncation of cumulants beyond order 3
and elimination of the two velocity skewness values), Eq. (3) reproduces the measured ?So
surprisingly well (R2= 0.92, regression slope = 1.0, regression intercept = 0.03; sample
size n = 1,341). In addition to the comparisons in Fig. 1, Eq. (3) was independently tested
by Katul et al. (1997a) in the atmospheric surface layer above a bare soil surface for a wide
rangeofstabilityconditions,Katuletal.(1997b)nearthecanopytopofamaturingpineforest
and a southern hardwood canopy also for a wide range of atmospheric stability conditions,
and more recently by Fer et al. (2004) for under-ice boundary-layer flows. All three studies
reported R2> 0.9 in their comparisons.
Next, we consider whether Eq. (3) can be used to link ?Soto second-order flow statistics
thereby addressing the study objectives. Upon replacing the moments in Eq. (4) back into
Eq. (3), we obtain
?So≈
1
2√2π u?w?
?
w?u?u?
σu
−w?w?u?
σw
?
.
(5)

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Boundary-Layer Meteorol (2006)
Standard gradient-diffusion closure for the triple moments results in (e.g. Wilson and
Shaw 1977):
?du?
dxi
??
characteristic length for the triple moments. For a one-dimensional case, w?u?u?and w?u?w?
are given by
u?
iu?
ju?
k= −qλ
ju?
k
+du?
iu?k
dxj
+
du?
dxk
iu?
j
?
whereq =
u?u?+ v?v?+ w?w??
isacharacteristicturbulentvelocityandλisanunknown
w?u?u?= −qλdu?u?
w?u?w?= −2qλdu?w?
dz
,
(6a)
dz
.
(6b)
Upon combining Equations (6) and (5), we obtain (Poggi et al. 2004b)
?So≈
−qλ
2u?w?√2π
?
1
σu
dσ2
dz
u
−
2
σw
du?w?
dz
?
.
(7)
Again, it is not our intention to investigate the magnitudes of ?Sopredicted by Eq. (7)
because such an investigation requires a rigorous determination of λ in Eq. (6) and accurate
measurements of the profiles of the second-order moments. However, Eq. (7) does offer a
desirable predictive skill: the sign of ?So. That is, solely from the relative importance of the
profiles of second-order statistics, it is possible to distinguish whether momentum transfer
will be dominated by sweeps or ejections, the objective of this study.
3. Results and discussion
We consider next all three layers of the ABL: the outer region, the dynamic sub-layer, and
the canopy sublayer (sparse and dense), and we evaluate whether changes indσu
explain the sign of ?So(measured by quadrant analysis). For the purposes of the following
discussion, we introduce four length scales: z—the height from the ground surface, d—the
zero-displacement height, h—the canopy height, δ—the ABL height and consider the four
cases below.
Case (1)—The Outer Region. In the outer region, roughly the region defined by(z−d)
0.4, Raupach’s (1981) wind-tunnel experiments suggest that u?w?< 0,
du?w?
dz
> 0 for both rough and smooth boundary layers (Fig. 2). Based on the sign of these
three quantities in Fig. 2, Eq. (7) unambiguously predicts ?So< 0 (i.e. ejections dominate)
consistent with the measured ?Soby quadrant analysis.
Case (2)—The constant stress layer. In the neutral atmospheric surface layer (or dynamic
sublayer), defined as the constant stress region (see region bounded by 0.1 <(z−d)
Fig.2),surface-layersimilaritytheorypredictsthatσu
and via Eq. (7), ?So= 0 also in agreement with the measurements (Fig. 2).
The fact that ejections and sweeps equally contribute to momentum transfer is also in
agreement with other surface-layer measurements above grass and bare soil surfaces (Katul
dzanddu?w?
dz
δ
>
dσ2
dz
u
< 0, and
δ
< 0.2 in
u∗≈2.0;u?w?
u2∗
= −1,
dσ2
dz=0,du?w?
u
dz
=0