12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

LOGARITHMIC SCALING OF HIGHER-ORDER TEMPERATURE
MOMENTS IN THE ATMOSPHERIC SURFACE LAYER

Kelly Y. Huang
Dept of Civil and Environmental Engineering
University of Notre Dame
Notre Dame, IN, 46556, United States
yhuang28@nd.edu

Matt K. Fu
GALCIT
CaltechPasadena, CA 91125, USA
mkfu@caltech.edu

Clayton P. Byers
Dept of Engineering
Trinity College
Hartford, CT, 06106, United States
clayton.byers@trincoll.edu

Gabriel G. Katul
Dept of Civil and Environmental Engineering
Duke University
Durham, NC, 27708, United Statesgaby@duke.edu

ABSTRACT

ior, which has known analytical expressions for the probability
distribution function and subsequent statistics.
By assuming a Gaussian distribution for u′ and that theeddies are non-interacting, Meneveau & Marusic (2013) wereable to develop a generalized logarithmic-law relation for thehigh-order statistics of u′ in the inertial sublayer given by

A generalized logarithmic law for high-order moments ofpassive scalars is proposed. This law is analogous to the generalized log law that has been proposed for high-order momentsof the turbulent longitudinal velocity and is derived by combining the random sweeping decorrelation hypothesis with aspectral model informed by the attached eddy hypothesis. The
proposed theory predicts that the high-order moments of passive scalar fluctuations within the inertial sublayer will vary
logarithmically with distance from a boundary (z). The proposed theory is evaluated using highly resolved time-series
measurements of temperature and streamwise velocity fluctuations obtained in the first meter of the atmospheric surface
layer (ASL) under near-neutral thermal stratification. The logarithmic dependence with z within the inertial sublayer is observed in both the air temperature and velocity moments, withgood agreement to the predictions from the proposed theory.
Surprisingly, the proposed theory appears to be as, if not more,
valid for transported passive scalars than for the longitudinalvelocity.


z

+ 1/p
u′2p
= B p − A p ln
,
δ

(1)

where p ≥ 1 is the moment order, B p is a flow-dependent
constant, A p = A1 [(2p − 1)!!]1/p , A1 ≈ 1.25 is the TownsendPerry constant (see Smits et al., 2011; Marusic et al., 2013;
Meneveau & Marusic, 2013), !! is the double factorial, z is
the wall-normal distance, δ is an outer length scale such as theboundary layer height, and the + superscript indicates normalization by the friction velocity u∗ . The scaling law of equation
1 was recovered through simpler arguments that take advantage of (i) RSDH to generalize the behavior of high-order spectra and (ii) the consequences of the attached eddy model on theconventional spectrum of u′ (Katul et al., 2016). In arriving
at equation 1, high-order spectra were related via RSDH to amodel spectrum for u′ characterized by a k−1 scaling regime ofwavenumber k at large scales (i.e. attached eddies) and k−5/3
for inertial scales (i.e. detached eddies) up to the Kolmogorov
micro-scale. Integration of this model spectrum and its combination with RSDH then recovered equation 1 and linked A p
and B p explicitly to the Kolmogorov constant and Re for Gaussian statistics. While RSDH has been investigated extensively
in terms of the turbulent velocity, particularly u′ , significantly
less attention has been paid to the behavior of passive scalars
such as temperature in near-neutral stratification. Previous numerical studies have indicated that in isotropic turbulence, passive scalar fields adhere to the RSDH as well as, if not better
than, the velocity field (Yeung & Sawford, 2002).
Here, a new theory is proposed based on RSDH and thesimilarity between temperature and streamwise momentum forthe behavior of passive scalar moments in the inertial sublayer.
Following the methodology of Huang & Katul (2022), we de-

BACKGROUND
A stationary and planar-homogeneous turbulent boundary
layer flow at high Reynolds number (Re) is considered herewith w′ , u′ and θv′ defining the wall-normal velocity, streamwise velocity, and virtual potential temperature fluctuations,
respectively, and primed quantities are turbulent fluctuations
from the mean state that is indicated by overline.
p These assumptions ensure that the friction velocity u∗ = −u′ w′ andthe temperature scale θ∗ = −w′ θ ′ /u∗ are independent of distance z from the boundary.
The random sweeping decorrelation hypothesis (RSDH)
is a kinematic model predicated on the assumption that smallscale eddies are swept by the larger, energetic eddies without
any dynamic distortion (Tennekes, 1975). Hence, motion of
the smaller scales, particularly in the inertial range, are due tothe kinetic energy of the large scales. A consequence of thiskinematic assumption is the tendency towards Gaussian behav-

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
allowing the higher order statistical moments and spectra to be
expressed as a function of the second (p = 1) moment resultingin(p)Eθ (k) = α(p)θ ′2

p−1

Eθ (k),(4)

where α(p) = p2 × 1 × 3 × 5 × . . . × (2p − 3) is a consequenceof Gaussian statistics for p > 1.
The high-order moments for temperature θ ′2p =R ∞ (p)′p 2 are now interpreted with the RSDH (see0 Eθ (k) dk + θequation 4) to yield
Figure 1. Illustration of an idealized piece-wise temperature
spectrum and its scaling regimes as a function of wavenumber k. The k2 behavior (analogous to the Saffman spectrum)is the minimum required for dEθ /dk → 0 as k → 0. For analytical tractability, the Kolmogorov k−5/3 inertial scaling is
extended to the Batchelor scale with a cutoff compensating forthe energy contribution not resolved here by the expected exponential decay due to molecular effects.

h i p−1 Z ∞2θ ′2p = α(p) θ ′2
Eθ (k) dk + θ ′p .
2

It can be shown for Gaussian statistics that the θ ′p term iszero for odd p and diminishes rapidly for even p so that its contribution is only significant for p = 2 with a value of (3/2)θ ′2p .
To arrive at an analytic expression, an idealized shape forEθ (k) is prescribed and shown in figure 1 based on Huang &Katul (2022) and given by the piecewise function

rive an analogous logarithmic law for the high-order statistics
of temperature fluctuations θ ′2p by combining the predictionsof RSDH with those from the attached eddy hypothesis in near
neutral stability conditions. The predicted behavior is foundto be in agreement with highly resolved measurements of temperature obtained in the atmospheric surface layer under nearneutral conditions. Further, the deviations between the temperature statistics and their corresponding theoretical predictions
are found to be significantly smaller than those found for thecollocated streamwise velocity.

−1/2

Z ∞
0
Z 1/z
1/δ

Z +∞
0

(p)

Eθ (k) dk.

Eθ (k) dk = θ ′2 =C1 θ∗2 k−1 dk +

Z 1/δ
0

Z 1/λB
1/z

C1 θ∗2 δ 3 k2 dk+Cθ εθ −2/3 k−5/3 dk,

and allowing the temperature variance to ultimately be expressed as(2)" 2/3 #
z
+
θ ′2
11
3
λB
≡ θ ′2 = C1−
−C1 ln
,
2
6
2 z
δ
θ∗

where B p,θ is a flow-dependent offset, A p,θ is the slope, analogous to the Townsend-Perry constant in equation 1, and the+ superscript again indicates normalization by the friction velocity and scalar wall flux.
For a θ ′ time series, the integral of the pth order spectrum,(p)Eθ (k), is related to statistical moments of θ ′ through2(6)

where λB = Scm η is the Batchelor scale, close to the Kolmogorov microscale η in near-neutral atmospheric flows because the molecular Schmidt number (Scm ) is close to unity fortemperature and many other scalars. Assuming that productionof turbulent kinetic energy and temperature variance are eachbalanced by their respective dissipation within the inertial sublayer (i.e., ε = u3∗ /κz and εθ = u∗ θ∗2 /κz, and κ ≈ 0.41), thespectral model in equation 6 can be integrated, giving

Direct numerical simulations of a turbulent channel up toReτ ≤ 4088 for Pr = 0.7 − 1 revealed qualitative agreement
between the streamwise velocity variance and passive scalarvariance (Pirozzoli et al., 2016). Additionally, a k−1 scalingof temperature spectra in the atmospheric surface layer has
been found with good agreement in multiple stability conditions (Pond et al., 1966; Katul et al., 1995; Li et al., 2016) andthe current measurements here. Combining these results with
Katul et al. (2016) and Huang & Katul (2022) allows for thetheoretical development of a logarithmic law in the high-ordereven moments of the temperature variance given by(θ ′p − θ ′p )2 = θ ′2p − θ ′p =if k ∈ [0, δ −1 ]if k ∈ [δ −1 , z−1 ]if k ∈ [z−1 , λB−1 ]if k > λB−1 ,


2 3 2

C1 θ∗ δ kC θ 2 k−1
1 ∗Eθ (k) =−2/3 −5/3

C
k

θ εθ


0

THEORY


z

+ 1/p
θ ′2p
= B p,θ − A p,θ ln
,
δ

(5)

0


so that A1,θ = C1 and B1,θ = C1

11
3
6 −2



λB
z

(7)

2/3 
. Note thevalue of Cθ is related to C1 through matching the piecewisefunction at k = 1/z. Equation 5 can then be written as
h i p−1 h z i.
θ ′2p = α(p) θ ′2θ∗2 B1,θ − A1,θ ln
δ

(3)


Finally, substituting θ ′2 = θ∗2 B1,θ − A1,θ ln

Under the assumptions of RSDH, the probability distributionfunction (PDF) of the time series will be normally distributed,

2p

z
δ

(8)


, normaliz-ing by θ∗ , and raising to a power of 1/p provides the sought2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
resulth z i+(θ ′2p )1/p = α(p)1/p B1,θ − A1,θ ln
.
δ

(9)

This expression has the same form as equation 2 with A p,θ =α(p)1/p A1,θ and B p,θ = α(p)1/p B1,θ .

As part of the Idealized horizontal Planar Array study for
Quantifying Surface heterogeneity (IPAQS) (Morrison et al.,2021) that took place at the Surface Layer Turbulence andEnvironmental Science Test (SLTEST) facility in the western deserts of Utah, USA, measurements of longitudinal velocity and temperature were acquired at z = 0.0625, 0.125,
0.25, 0.5, and 1.0 m above the ground during a three-day intensive sampling period (18 - 20 June 2018). The SLTEST facility is a unique field site with near-canonical boundary conditions (its aerodynamic roughness ranges from sub-millimeterto less than 6 mm) and predictable wind patterns, making it
an ideal location for probing high Re turbulent boundary layer
flows (Klewicki et al., 1998; Metzger & Klewicki, 2001). The
equivalent sand grain roughness for these measurements wasestimated to be 2.5 mm using the relation for a zero-pressuregradient neutral boundary layer (Huang et al., 2021a), meaning lowest station height corresponds to 25 times that of theequivalent sand grain roughness.
Nano-scale thermal anemometry probes (NSTAPs) operated in constant-current anemometry mode and their cold-wirevariants (TNSTAPs) were used to capture velocity and temperature measurements, respectively, at each height with a sampling frequency of 100 Hz. The sensing elements of thesenano-scale sensors are platinum wire ribbons 2 µm in width,
100 nm in thickness, 60 µm in length for the NSTAP and 200µm in length for the TNSTAP. The size of these sensors provide high spatial resolution with minimized end-conduction
effects and temporal filtering (Hultmark et al., 2011; Arwatz
et al., 2015). Relevant stresses and heat fluxes are calculated
from data taken by a nearby triaxial sonic anemometer (Campbell Scientific CSAT3) at approximately 10 m west of the measurement tower at z = 2 m. Further details regarding the experimental set-up and calibration procedure can be found elsewhere (Huang et al., 2021a).
Trends associated with the varying free stream velocity
were subtracted from the raw time series (Hutchins et al.,2012). Four 30-min records are examined in the current study
and summarized in Table 1 after the following data qualifications:u∗(ms−1 )w′ θv′(Kms−1 )

ζ

Rf

2000
2030
2200
2230

7.55
6.56
6.60
7.34

0.28
0.22
0.24
0.29

-0.017-0.018-0.024-0.029

0.025
0.053
0.056
0.038

0.014
0.040
0.035
0.016

number R f is defined as

Rf =(g/θv )w′ θv′.
w′ u′ (du/dz)(11)

3. Stationarity of u and turbulence intensity u′ /u were observed, as assessed by the reverse arrangement test andthe runs test with a 95% confidence interval (Bendat &Piersol, 2011).
The four runs satisfying these conditions were near mid-night(local time) when the stable boundary layer depth has equilibrated to its near-neutral value. Taylor’s hypothesis was used
to convert temporal differences in the velocity and temperature
signals to spatial differences in the streamwise direction.

RESULTS
Before evaluating the model predictions, this uniquedataset can be used to first assess the validity of the key assumptions underpinning the proposed model. These are (i) the
tendency towards Gaussian statistics, (ii) existence of inertial
range behavior, and (iii) spectral behavior consistent with theattached eddy model. The Gaussian behavior of the velocity
and temperature fluctuations can be assessed using the PDFs
of the measurements. The PDFs for each quantity under neutral stability conditions are shown in figure 2 at a representative
measurement station corresponding to z =0.125 m. The PDFs
are normalized by their standard deviation σu and σθ , respectively and their skewness Su′ and Sθ ′ and kurtosis Ku′ and Kθ ′are shown. The PDFs of both temperature and velocity exhibit near-Gaussian behavior, with the most evident deviations
being super-Gaussian in the positive tail.
The deviations of these velocity and temperature PDFs
from Gaussian behavior are highlighted in figure 3 for thesame station at z = 0.125 m. There, each of the PDFs fromfigure 2 is normalized by the Gaussian distribution such thatdeviations from Gaussian behavior are indicated in proportion
to unity. While the velocity PDF, PDF(u′ /σu ), is found to
be within ±10% of a Gaussian distribution for u′ /σu ≤ |2|,
the scalar PDF exhibits a wider region of agreement, such that
PDF(θ ′ /σθ ) is within ±10% for −3 < θ ′ /σθ < 2. However,the differences between the velocity and temperature PDFs
are largest in the tails of the distribution, where deviations
from Gaussian behavior are approximately twice as large foru′ /σu ≥ 4 compared to θ ′ /σθ ≥ 4. Though the exact behavior and statistics differ with station height, the tendency for the

1. The incoming wind direction aligned with the sensors, asverified by the sonic anemometer.
2. Near-neutral stability was reached, with both the MoninObukhov stability parameter |ζ | = |z/L| ≈ 0 and the flux
Richardson number R f ≈ 0. The Obukhov length L isdefined asu3∗,κ g/θv w′ θv′u(ms−1 )

Table 1. Mean flow properties relative to the triaxial sonicanemometer positioned at z = 2 m for the 30-min periods analyzed (local time LT = UTC + 6 h) on 21/06/2018. The boundary layer height δ was estimated to be 60 m (Huang et al.,2021b), and R f was estimated using both the sonic anemometer data and the mean velocity profile given by the NSTAPs
evaluated at z = 2 m.

EXPERIMENTAL METHODOLOGY & DATA

L=−

LT

(10)

where g is the acceleration due to gravity, and θv is themean virtual potential temperature. The flux Richardson3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
PDF(u′ /σu )PDF(θ ′ /σθ )

100

U+
100

u′2

+

9

θ+

θ ′2

+

9

344

25

8

8
342

10−2

Sθ ′ Kθ ′0.07 3.04

Su′ Ku′0.21 3.24

10−4
-5

0
′
u /σu

5

-5

0
′
θ /σθ

10−2
10−4
5

15

PDFexp /PDFnorm

104

5

103

104

5

z+

Figure 4. Mean and variance profiles for velocity (left) andtemperature (right) as a function of wall distance and normalized by inner units. Mean quantities (U + and θ + ) are given+in black, while variances for velocity (u′2 ) and temperature+(θ ′2 ) are denoted in blue and orange, respectively.

logarithmic behavior. Following Vallikivi et al. (2015), whofound logarithmic behavior above z+ = zu∗ /ν > 400, the lowest measurement station to the ground at z+ ≈ 850 should bewell within the log-layer. This assumption is further supportedby the mean and variance profiles for the velocity and temperature shown in figure 4, where logarithmic behavior is foundin the region 103 < z+ < 104 . Here, A1,u and A1,θ are estimated as 0.9 and 0.93, respectively, by a least squares fittingof the data to equation 1. This value for the velocity comparesreasonably well the A1 ≈ 1.25 found by Meneveau & Marusic(2013).
Finally, the streamwise wavenumber spectra of the velocity and temperature fluctuations can be compared to the idealized model, each of which are shown in figure 5. In bothquantities, distinct k−1 and k−5/3 scaling regions are identified, consistent with the assumptions of the simplified model.
However, due to the lack of convergence and stationarity of
the flow, no data are available in the low wave-number region
where the k2 scaling is assumed. The expected area under themodeled k2 region is identically equal to 31 C1 in normalized
variables. Compared to the integrated normalized spectrum of
figure 5 for each case, this comes out to between 4% and 7.5%of the total integrated area. As this only affects the offset B1,θin equation 9, the overall agreement between the model spectrum and the dataset is acceptable for the purposes here.

32
16
8
4
u’/σ uθ ’/σ θGaussian
±10%

1/2

103
z+

64

1/4

6
338

temperature fluctuations towards more Gaussian behavior relative to the collocated velocity is a consistent feature across allz.
The super-Gaussian behavior in the positive tails differsfrom the observations of Meneveau & Marusic (2013) andSamie et al. (2018) who observed sub-Gaussian behavior forboundary layer measurements up to Reτ = 20, 000, and is apotentially indicative of effects stemming from the roughness
sublayer (Heisel et al., 2020). Recent studies such as Marusic
et al. (2013) and Wei et al. (2005) suggest that the extent of
the buffer layer could be Reynolds number dependent, extending up to z+ ≈ 3(δ u∗ /ν)1/2 ≈ 3 × 103 , with ν being kinematic
viscosity, even in smooth-wall conditions. This criteria places
the lowest two measurement heights within the buffer region
where sweeps tend to dominate (Heisel et al., 2020), potentially explaining the observed increase in Su′ and Ku′ .

1

7
340

6

Figure 2. Measured probability density functions (open symbols) of the velocity (left) and temperature (right) fluctuations
at z = 0.125 m. The solid black line indicates a Gaussian distribution.

2

7

20

DISCUSSION
Given the strength of the underlying assumptions andthe similarity between the statistical behavior of temperatureand velocity signals, the predicative capability of the proposed RSDH model in capturing the behavior of the highorder moments can be assessed. Because the temperature exhibits a distribution that more closely follows Gaussian behavior, the RSDH predictions for high-moments should hold+for (θ ′2p )1/p . A comparison between the high-order moments and the RSHD predictions can be seen in figure 6 forboth the velocity and temperature measurements. In each case,the slope and offset of the p = 1 statistics (i.e., the variances)
are determined from integration of the spectral RSDH models
given by Katul et al. (2016) and the present study (see equation
7) rather than fitting a regression to the variance profiles. Similarly, the RSDH predictions for the high-order moments are
determined from the p = 1 case through equation 9 that can beapplied analogously to both velocity and temperature. Figure
6 shows that for both temperature and velocity, there appears-5 -4 -3 -2 -1 0 1 2 3 4 5u’/σ u , θ ’/σ θ
Figure 3. Deviations of measured PDFs from Gaussian behavior. PDFs of measured velocity (blue) and temperature(orange) are normalized by a Gaussian distribution given √by PDFnorm = exp −x2 /2 / 2π, where x = u′ /σu , θ ’/σθ .
Dashed black line at unity indicates Gaussian behavior witha gray band showing the region of ±10% deviation. Values
less than and greater than 1 indicate sub-Gaussian and superGaussian behavior, respectively.

The existence of an inertial subrange can be substantiatedby assessing whether the lower order statistics of the measurements, such as mean and variance, exhibit a canonical412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

10−2

2p=8
2p=6
2p=4
2p=2

20

(kz)−1(kz)−5/3

10−4
kz

100

0

102

10−3

10−2
z/δ

102

z = 0.0625 m
z = 0.125 m
z = 0.25 m
z=1m

100

+

(kz)−110−240(kz)−5/3

2p=8
2p=6
2p=4
2p=2

20

10−4
10−2

kz

100

102
0

10−3

10−2
z/δ

Figure 5. Normalized spectra of the longitudinal velocity(Eu (kz)/u2∗ ) and temperature (Eθ (kz)/θ∗2 ) are computed using the frozen turbulence hypothesis (Taylor, 1938) and shown
as a function of normalized wavenumber kz in the upper andlower plots, respectively. Different shades of blue and orange
indicate the respective spectra for velocity and temperature at
the different measurement stations with lighter shades corresponding to distances further from the ground. For each quantity, there is a clear break point between the k−1 () and thek−5/3 ( ) scaling regimes as indicated by the black verticaldotted line ( ) at k = 1/z. The k = 1/δ cutoff between the k2
and the k−1 scaling regimes are also shown for each measurement height and denoted by their respective colors noted in thelegend to the right of each plot.

Figure 6. Profiles of high-order moments of velocity (top)and temperature (bottom) as a function of normalized distancefrom the ground z/δ . Central moments for 2p = 2, 4, 6, and 8
are denoted by (◦), (▽), (◁), and (▷) symbols, respectively.
The dashed lines represent the predictions for high-order moments as computed using equation 1 for the velocity statistics(Katul et al., 2016) and equation 9 for the temperature statistics. High-moments for velocity (blue) and temperature (orange) are denoted with darker shades of their respective colors.

Ap4

to be good qualitative agreement between the measurements
and their respective spectral RSDH models. In each case, themodel is well within the statistical uncertainty, with the exception of the p = 2, partially due to the additive (3/2)θ ′2p
contribution as noted in the theory section of the manuscript.
Further, it is difficult to discern any significant differences in
the performance of the model between the two quantities.

2
0

2

4

2p

6

8

2

4

2p

6

8

10
8
6
4
2
0

Bp

10−2

Eθ (kz)/θ∗2

40

+

100

(u′2p )1/p

z = 0.0625 m
z = 0.125 m
z = 0.25 m
z = 0.5 m
z=1m(θ ′2p )1/p

Eu (kz)/u2∗

102

Figure 7. Coefficients A p (blue) and A p,θ (orange) as a function of moment order 2p, averaged across available datasets.
Square markers (□) denote slopes (A p ) and offsets (B p ) determined through a linear regression of the central moment profiles for 2p = 2, 4, 6, and 8. The dashed lines represent thecoefficients as predicted by integration of the spectral modeland RSDH predictions given by equations 1 and 9. Due totheir similarity, these RSDH predictions appear overlapping.

However, the differences between the temperature and velocity statistics become more apparent in figure 7, which directly compares the slopes (A p ) and offsets (B p ) obtained fromthe measurements and RSDH model. There, the slope (A p ) and
offset (B p ) values from the RSDH predictions are extracted
from the lines shown in figure 6 and plotted in the left and rightplots of figure 7, respectively, as a function of moment orderp. These predictions are compared with experimental values
obtained from a linear regression of the higher order momentswith the logarithmic wall distance. While the performance of
the RSDH model is comparable for velocity and temperature
for p ≤ 3, the agreement between the RSDH model and experimentally derived parameters is far better for temperature up top = 4 than for velocity. This discrepancy is expected to widen
for progressively larger values of p; however, such high-ordermoments are not analyzed here due to limited convergence ofthe statistics in the dataset.

CONCLUSIONSThe similarity between temperature and momentum isused to propose a new theory for the behavior of passive scalar
moments in the inertial sublayer based on RSDH. The predicted behavior from the new model is found to be in good
agreement with highly resolved measurements of temperature
obtained in the atmospheric surface layer under near-neutral
conditions with deviations significantly smaller from RSDH

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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behavior than their collocated streamwise velocity counterpart.
While the performance of the new model is encouraging, extending and evaluating this framework under non-neutral stability conditions and turbulent Pr ̸= 1 remains an area of future
investigation.

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