# Persistence behaviour of heat and momentum fluxes in convective surface
layer turbulence

Subharthi Chowdhuri
Indian Institute of Tropical Meteorology
Ministry of Earth Sciences
Dr. Homi Bhaba Road, Pashan, Pune-411008, India
<EMAIL_ADDRESS>
&Thara V Prabha
Indian Institute of Tropical Meteorology
Ministry of Earth Sciences
Dr. Homi Bhaba Road, Pashan, Pune-411008, India
<EMAIL_ADDRESS>
&Tirtha Banerjee
Department of Civil and Environmental Engineering
University of California, Irvine, CA 92697, USA
<EMAIL_ADDRESS>

###### Abstract

The characterization of heat and momentum fluxes in wall-bounded turbulence is
of paramount importance for a plethora of applications, ranging from
engineering to Earth sciences. However, how the turbulent structures
associated with velocity and temperature fluctuations interact to produce the
emergent flux signatures, is not evident till date. In this work, we
investigate this fundamental issue by studying the switching patterns of
intermittently occurring turbulent fluctuations from one state to another, a
phenomenon called persistence. We discover that the persistence patterns for
heat and momentum fluxes are widely different. Moreover, we uncover power-law
scaling and length scales of turbulent motions that cause this behavior.
Furthermore, by separating the phases and amplitudes of flux events, we
explain the origin and differences between heat and momentum transfer
efficiencies in convective turbulence. Our findings provide new understanding
on the connection between flow organization and flux generation mechanisms,
two cornerstones of turbulence research.

_K_ eywords Convective turbulence $\cdot$ Heat and momentum fluxes $\cdot$
Persistence $\cdot$ Phase and amplitudes $\cdot$ Probability density function

##  1 Introduction

The ensemble averaged vertical turbulent fluxes of momentum or heat are
expressed as the covariance between the vertical and streamwise velocity
fluctuations ($\overline{u^{\prime}w^{\prime}}$) or between the vertical
velocity and temperature fluctuations ($\overline{w^{\prime}T^{\prime}}$). The
primes in the flux expressions denote the turbulent fluctuations in the
streamwise velocity ($u^{\prime}$), vertical velocity ($w^{\prime}$), or
temperature ($T^{\prime}$). Besides, the overline symbol indicates the average
over several ensemble members. For practical purposes, such ensemble average
is replaced by the average over time or space by applying the ergodic
hypothesis (Monin and Yaglom,, 1971). In turbulent flows, these fluxes
quantify the amount of heat or momentum being transported from (to) the
surface to (from) other locations. The sign convention dictates that, the
positive or negative values of the fluxes denote the direction of the
transport from or towards the surface. The estimation of these fluxes have
numerous usages, such as, in wall-bounded turbulent flows, the momentum
transport towards the wall is related to surface drag which determines the
power requirements and efficiencies in many engineering applications (e.g.,
Holmes et al.,, 2012). Additionally, in the geophysical context, these fluxes
quantify the surface-atmosphere momentum and heat exchanges, which eventually
drive the Earth’s climate (e.g., Baldocchi et al.,, 2001). Therefore, it is of
paramount importance to develop a comprehensive understanding of the turbulent
generation mechanisms of the heat and momentum fluxes. Since the flux
computation involves the product of two turbulent quantities, a fundamental
research question is, _how do the turbulent structures (of different time or
length scales) associated with the velocity and temperature fluctuations
interact to produce the momentum and heat flux signatures_?

The common method of describing the fluctuation characteristics and the
associated fluxes, corresponding to the different scales of the turbulent
motions, is the spectral approach (Taylor,, 1938). However, a landmark study
by Kline et al., (1967) noted that the transport of momentum in wall-bounded
shear flows was not a continuous process, as assumed by the spectral analysis.
Instead they found that, the intermittent ejections of low-momentum fluid
parcels from the wall ($u^{\prime}<0$ and $w^{\prime}>0$) accompanied with the
sweeps of high-momentum fluid towards the wall ($u^{\prime}>0$ and
$w^{\prime}<0$) were responsible for the generation of the momentum flux.
Subsequently, to diagnose the intermittent signatures of the ejection and
sweep motions and estimate their contributions towards the momentum flux, a
conditional sampling technique named quadrant analysis was introduced (Wallace
et al.,, 1972; Lu and Willmarth,, 1973; Wallace,, 2016). Along with the
utility of the quadrant analysis, a few studies additionally noted that the
ejection and sweep motions in wall-bounded turbulent flows occurred with a
range of different time scales as more often than not they switched
irregularly from one quadrant state to the other (Rao et al.,, 1971;
Alfredsson and Johansson,, 1984; Luchik and Tiederman,, 1987). While
evaluating the statistical characteristics of these time scales, Rao et al.,
(1971) and Alfredsson and Johansson, (1984) found that the mean time scales of
the burst events (sequence of ejections exceeding a certain threshold) in the
near-wall region were influenced by both the small and large scale motions.

Such results were intriguing, considering their relevance to the connection
between the turbulent structures of different time scales and the intermittent
generation of the momentum fluxes. Nevertheless, a detailed treatment of the
time scale distribution of the flux-carrying motions was severely lacking
(Narasimha,, 1990). To tackle this problem, Kailasnath and Sreenivasan, (1993)
demonstrated that the probability density functions (PDFs) of the time scales
of the turbulent motions contributing to the momentum flux can be
systematically studied by exploring the zero-crossing properties of the
instantaneous $u^{\prime}w^{\prime}$ signal. Their method was inspired by the
results of Sreenivasan et al., (1983) where the PDFs of the time intervals
between the zero-crossings of the turbulent fluctuations were used to probe
the distribution of time scales in a wall-bounded flow. Note that, in the
parlance of non-equilibrium statistical mechanics, the distribution of zero-
crossing time intervals in a stochastic signal is equivalent to the
persistence PDFs, where persistence is the probability $P(t)$ that the signal
does not change its sign up to the time $t$ (Majumdar,, 1999; Chowdhuri et
al., 2020a, ). Hereafter, we refer the zero-crossing PDFs as the persistence
PDFs, given its technical suitability (Chowdhuri et al., 2020a, ).

One of the crucial aspects of the study by Kailasnath and Sreenivasan, (1993)
was to investigate how the persistence PDFs of the individual velocity signals
compared to the persistence PDFs of their product which constituted the
momentum flux. They found that in the inertial layer of a flat-plate boundary
layer, the persistence PDFs of the momentum flux signal closely followed the
same PDFs of the vertical velocity fluctuations, with both displaying a nearly
identical single exponential function. This was in sharp contrast with the
streamwise velocity fluctuations, whose persistence PDFs displayed a double
exponential structure with two different exponents. They interpreted this
behaviour as both the small and large scales were relevant for the variations
in the streamwise velocity fluctuations. However, for the variations in the
wall-normal velocity and in the momentum flux, only the large scales were
primarily responsible, since their zero-crossing PDFs displayed a single
exponential function. Moreover, they also proposed a connection to associate
such behaviour with the properties of the attached eddies which populated the
inertial layer, consistent with the Townsend’s attached eddy hypothesis
(Townsend,, 1976).

In general, the findings of Kailasnath and Sreenivasan, (1993) shed light on
the fundamental issue of the relationship between the turbulent structures of
different scales and the associated intermittent flux signatures. However,
while evaluating the time scales of the momentum flux signals, they did not
consider the quadrant effects. We hypothesize that the conditional sampling of
the flux events can establish an important link between the momentum flux
partition among the four different quadrants and the time scales of the
turbulent flow. Such an analysis can identify the time scales associated with
individual quadrant motions that contribute towards momentum flux generation.
To the best of our knowledge, there has been no subsequent studies to
scrutinize this aspect. Although until now, we have discussed the
characteristics of the momentum flux signal in turbulent shear flows, it can
be noted that even in buoyancy-driven turbulent flows, the instantaneous heat
flux signals ($w^{\prime}T^{\prime}$) display similar intermittent signatures
(Shang et al.,, 2003). Therefore, one can contemplate that, such investigation
will be pertinent for a convective atmospheric surface layer (ASL) flow, where
both the shear and buoyancy play significant roles to maintain the turbulence.

The ASL is a generalization of the inertial layer of wall-bounded shear flows
by including the effect of buoyancy (Davidson,, 2015). The vertical extent of
the ASL is approximately up to the lowest 10% of the convective boundary layer
(excluding the roughness sublayer) where the fluxes are nearly constant with
height (Wyngaard,, 1992). During the early days of ASL research, Haugen et
al., (1971) observed that in convective conditions the time traces of the
instantaneous $w^{\prime}T^{\prime}$ signals displayed intermittently
occurring long sequences of positive heat flux events. They further noted that
the time-averaged characteristics of the heat flux and their transport
efficiencies (expressed through the correlation coefficient between
$w^{\prime}$ and $T^{\prime}$) were primarily determined by these long
persistent events. On the other hand, for the instantaneous
$u^{\prime}w^{\prime}$ signals they found that the long sequences of the flux-
carrying motions either transported momentum in the upward or in the downward
direction. In a highly-convective ASL flow (characterized by strong thermal
stratification), the amplitudes of the large persistent positive and negative
momentum flux events were almost similar to each other. This caused the time-
averaged momentum flux to become quite small due to the near-perfect
cancellation of such positive and negative values. However, with the decrease
in the strength of the thermal stratification (near-neutral ASL), the negative
amplitudes associated with the long sequences overwhelmed the positive ones.
Therefore, the momentum transport efficiency increased as the ASL approached
the near-neutral conditions. Haugen et al., (1971) postulated that such
changes associated with thermal stratification were related to the changes in
the topology of coherent structures from cellular patterns to horizontal
rolls. Nevertheless, their conclusions were based on visual inspections of a
few 15-min records, and of a qualitative nature.

Since then, numerous studies have documented that akin to laboratory flows,
the heat and momentum transport processes in a convective ASL are intermittent
in nature, which occur with a range of different time scales (Narasimha and
Kailas,, 1990; Duncan and Schuepp,, 1992; Caramori et al.,, 1994; Katul et
al.,, 1994; Narasimha et al.,, 2007). Additionally, several other researchers
have demonstrated that the time-averaged transport characteristics of the heat
and momentum in a convective ASL flow are strongly dependent on the strength
of the thermal stratification (De Bruin et al.,, 1993; Li and Bou-Zeid,, 2011;
Chowdhuri and Prabha,, 2019). Specifically, these studies show that in a
highly-convective ASL the heat and momentum transports are decoupled from each
other, whereas they are strongly coupled in near-neutral conditions. Recently,
Salesky et al., (2017) have illustrated that, with thermal stratification
(stability), the change in the topology of the coherent structures in a
convective flow influences the transport efficiencies of the heat and
momentum.

Broadly, the characteristics of the time-averaged heat and momentum fluxes in
a convective ASL flow could be perceived as a well-researched problem.
However, since the times of Haugen et al., (1971), a grand challenge still
remains to connect the intermittent behaviour of these fluxes with the
different scales of the turbulent eddies which influence their component
signals. Undoubtedly, such investigation is of fundamental interest, to
improve our understanding of the generation mechanisms of the heat and
momentum fluxes. This problem also has practical implications regarding the
development of next-generation transport models of convective turbulence. To
address this issue, we ask the following questions:

  1. 1.

What type of event signatures are generated in the momentum and heat flux
signals, due to the presence of turbulent eddies of different time scales?

  2. 2.

What is the relation between these event signatures at different time scales
with the heat and momentum transport efficiencies in a convective ASL flow?

  3. 3.

How does the change in the thermal stratification modulate the event
signatures at different time scales?

In this article, we attempt to answer these questions through persistence
analysis of the turbulent heat and momentum fluxes in a convective ASL flow.
The persistence analysis is well-suited for this purpose, since it deals with
the time intervals between the zero-crossings of an intermittent turbulent
signal, eventually being related to the time scales of the turbulent
structures. During our presentation, we arrange the paper in three different
sections. In Section 2, we provide brief descriptions of the field-
experimental dataset and methodology, in Section 3 we introduce the results
and discuss them, and lastly in Section 4 we summarize the key takeaways and
provide the scope for further research.

##  2 Data and Methodology

In this study, we have used the dataset from the Surface Layer Turbulence and
Environmental Science Test (SLTEST) experiment. The details about the SLTEST
experiment and the set-up of the instruments are provided in Chowdhuri et al.,
2020b  and Chowdhuri et al., 2020a . In this experiment, nine North-facing
time synchronized CSAT3 sonic anemometers were deployed on a 30-m mast, spaced
logarithmically from 1.42 m to 25.7 m, with the sampling frequency being set
at 20 Hz. We followed the same data processing steps as outlined in Chowdhuri
et al., 2020a  to select the 30-min periods from the daytime convective
conditions. Total 261 combinations of the stability ratios ($-\zeta=z/L$,
where $z$ is the measurement height and $L$ is the Obukhov length) were
possible for these selected 30-min periods. The entire range of $-\zeta$ (12
$\leq-\zeta\leq$ 0.07) was divided into six stability classes and considered
for the persistence analysis of the turbulent heat and momentum fluxes. These
are the same set of stability classes used by Chowdhuri et al., 2020b  and
Chowdhuri et al., 2020a  for their study of turbulence anisotropy and
persistence behaviour of turbulent velocity and temperature fluctuations (see
their respective tables 2 and 1). Note that, for the sake of brevity, the same
information is not repeated here.

Figure 1: A 120-s long section of a time series of (a) $u^{\prime}$, (b)
$w^{\prime}$, (c) $T^{\prime}$, (d) $u^{\prime}w^{\prime}$, and (e)
$w^{\prime}T^{\prime}$ from a highly convective surface layer corresponding to
a $-\zeta=$ 10.6 are shown. The actual values are displayed as solid black
lines which correspond to the left hand side of the $y$ axis. Similarly, the
associated telegraphic approximations (TA) are displayed as solid blue lines,
which correspond to the right hand side of the $y$ axis.

A graphical demonstration of the persistence phenomenon is provided in figure
1, where a 120-s long section of streamwise velocity fluctuations
($u^{\prime}$), vertical velocity fluctuations ($w^{\prime}$), temperature
fluctuations ($T^{\prime}$), and their products representing the instantaneous
streamwise momentum ($u^{\prime}w^{\prime}$) and heat fluxes
($w^{\prime}T^{\prime}$) are shown for a particular 30-min run, corresponding
to $-\zeta=10.6$. The associated telegraphic approximations (TA) of these time
series are presented at the right hand side of the $y$ axis in figure 1,
displayed in blue (see Section 3.1 for additional details). A visual
inspection of figure 1 suggests that the $w^{\prime}$ signals switch more
often from the positive (negative) to the negative (positive) turbulent
states, as compared to the $u^{\prime}$ or the $T^{\prime}$ signals.
Nevertheless, the streamwise momentum ($u^{\prime}w^{\prime}$) and the heat
flux ($w^{\prime}T^{\prime}$) signals involve both the combinations of the
$w^{\prime}$ and $u^{\prime}$ or $w^{\prime}$ and $T^{\prime}$ signals.
Therefore, the switching tendencies of these turbulent fluxes (figures 1d and
e) must be related to the persistence signatures of their component signals.
This issue is discussed further in Section 3.1.

As described by Chowdhuri et al., 2020a , the persistence time ($t_{p}$) is
defined as the time up to which a fluctuating turbulent signal stays positive
or negative before being switched to the other state. In addition to that, the
associated probability density function (PDF) of $t_{p}$ describes its
statistical characteristics, which in turn are related to the turbulent
structures in a convective flow (Chowdhuri et al., 2020a, ). We apply these
same concepts in the present study to investigate the persistence behaviour of
the instantaneous heat and momentum flux signals. Typically, we encounter in
the order of 105 number of zero-crossings (the points where the TA series
changes from 0 to 1 or 1 to 0) for the $u^{\prime}$, $w^{\prime}$,
$T^{\prime}$, $u^{\prime}w^{\prime}$, and $w^{\prime}T^{\prime}$ signals for
every six stability classes (Chowdhuri et al., 2020a, ). Therefore, the
persistence PDFs of the $u^{\prime}w^{\prime}$ and $w^{\prime}T^{\prime}$
signals for each of these six stability classes are constructed over these
large number of ensemble events, to ensure their statistical robustness. Note
that, the persistence PDFs are computed via logarithmic binning and subsequent
transformation to the linear space using a change of variable, as illustrated
by Chowdhuri et al., 2020a . In the following section, we discuss the
properties of the persistence PDFs of the streamwise momentum and the heat
flux signals, corresponding to these six stability classes.

##  3 Results and discussion

We begin by presenting the results of the persistence PDFs of the
instantaneous momentum and heat flux signals ($u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$), including a comparison with the persistence of the
component signals themselves ($u^{\prime}$, $w^{\prime}$, and $T^{\prime}$).
Subsequently, we discuss the effect of separation into four different
quadrants on the persistence PDFs of the flux signals. Furthermore, we explore
the role of the intermittent flux events of different persistence time scales
towards the heat and momentum transport efficiencies. In order to achieve such
objective, we introduce a novel approach to separate the phases and amplitudes
of the component signals associated with such events of various time scales.
Additionally, we also provide plausible physical explanations of the obtained
results during the course of our presentation.

###  3.1 Persistence PDFs of heat and momentum fluxes

Prior to embarking on a detailed analysis regarding the persistence PDFs of
the heat and momentum flux signals, perhaps it is prudent to establish a
phenomenological connection between the persistence properties of the fluxes
and their component signals. In order to derive such a heuristic relation, we
turn our attention towards the telegraphic approximations (TA) of the
turbulent signals. Specifically, we ask the question _if the TA representation
of the component signals are known, then what would be the equivalent TA
representation of their product?_

####  3.1.1 Association between the persistence time scales of the flux and
its components

The TA representation of a turbulent signal can be expressed as,

$(x^{\prime})_{\rm TA}=\frac{1}{2}(\frac{x^{\prime}(t)}{|x^{\prime}(t)|}+1),$
(1)

where $x$ can be either $u$, $w$, or $T$. From the available measurements if
we know the TA representation of the two signals, lets say $u^{\prime}$ and
$w^{\prime}$, then the equivalent TA representation of their product
$u^{\prime}w^{\prime}$ can be written as,

$(u^{\prime}w^{\prime})_{\rm TA}=1-|u^{\prime}_{\rm TA}-w^{\prime}_{\rm TA}|.$
(2)

Note that, the same expression is true for the product of the $w^{\prime}$ and
$T^{\prime}$ signals as well. However, for the illustration purpose we will
limit our discussion to the product of the $u^{\prime}$ and $w^{\prime}$
signals only, although the similar logic can be extended to any two signals.

At a first glance the relationship provided in Eq. (2) may seem counter-
intuitive, as one may naively expect the TA representation of the product of
the two signals will be equal to the product of the TA’s of the individual
components. Nevertheless, expressing the TA representation of the product in
such way will be incorrect. This is because, if both the signals are negative
(individual TA’s equal to 0) the products are positive and hence their TA
representations are equal to 1. To demonstrate this, in figure S1
(supplementary material) we provide a typical example of the original TA
approximated time series of $u^{\prime}w^{\prime}$ and $w^{\prime}T^{\prime}$
for a highly-convective stability, corresponding to $-\zeta=10.6$. We compare
the original $(u^{\prime}w^{\prime})_{\rm TA}$ and
$(w^{\prime}T^{\prime})_{\rm TA}$ with Eq. (2) and with the product
$u^{\prime}_{\rm TA}\times w^{\prime}_{\rm TA}$. The result clearly shows that
the expression $u^{\prime}_{\rm TA}\times w^{\prime}_{\rm TA}$ does not
capture the original TA signatures of $(u^{\prime}w^{\prime})_{\rm TA}$ and
$(w^{\prime}T^{\prime})_{\rm TA}$, whereas Eq. (2) detains that information
perfectly well.

From the definition of persistence, the time scale $t_{p}$ of any fluctuating
signal ($x^{\prime}$) can be expressed as,

$t|(\Delta x^{\prime}_{\rm TA}=0)\implies t_{p},$ (3)

where $\Delta x^{\prime}_{\rm TA}$ is the difference between the successive TA
values of $x^{\prime}_{\rm TA}$. To connect the persistence time scales of the
component signals with their product, we can transform Eq. (2) in the
difference form as,

$|\Delta(u^{\prime}w^{\prime})_{\rm TA}|=\Big{|}|\Delta u^{\prime}_{\rm
TA}|-|\Delta w^{\prime}_{\rm TA}|\Big{|}.$ (4)

From Eq. (4), one can infer that the zero-crossings of
$(u^{\prime}w^{\prime})_{\rm TA}$ will be located at those points where,
$|\Delta u^{\prime}_{\rm TA}|\neq|\Delta w^{\prime}_{\rm TA}|$. Therefore, the
persistence time scales of $u^{\prime}w^{\prime}$ will be related to the
persistence time of that component signal which switches its states more
rapidly than the other one. However, there is also an another alternate
scenario for a hypothetical case where the $u^{\prime}$ and $w^{\prime}$
signals are perfectly correlated with each other. In such scenario, $|\Delta
u^{\prime}_{\rm TA}|$ will be equal to $|\Delta w^{\prime}_{\rm TA}|$ at all
time scales, which will result in an infinite persistence of their product
$u^{\prime}w^{\prime}$.

Nevertheless, for all practical purposes, we can anticipate that the
persistence PDFs of the products of the turbulent signals will be closer to
the persistence PDFs of the individual signals at small time scales. The
reason is the probability of occurrence of the events persisting for short
times will not be much different for the $u^{\prime}$, $w^{\prime}$, and
$u^{\prime}w^{\prime}$ signals, since all of them rapidly fluctuate. On the
other hand, at large time scales, the persistence PDFs of the products are
expected to be more closer to that signal which has a lower chance of
encountering such long sequences. Note that, similar inference can also be
drawn for the product between $w^{\prime}$ and $T^{\prime}$. Next, we describe
how the insights gained from the aforementioned analysis compare with the
observations of the persistence PDFs of heat and momentum fluxes in a
convective ASL flow.

####  3.1.2 Features of the flux persistence PDFs

Figure 2 shows the persistence PDFs of the instantaneous streamwise momentum
($u^{\prime}w^{\prime}$) and heat fluxes ($w^{\prime}T^{\prime}$), compared
with their component signals ($u^{\prime}$ and $w^{\prime}$ or $w^{\prime}$
and $T^{\prime}$) for the six stability classes in a convective ASL flow. In
figure 2, the persistence time scales are converted to streamwise lengths
($t_{p}\overline{u}$, where $\overline{u}$ is the mean wind speed) by applying
the Taylor’s frozen turbulence hypothesis and scaled with $z$. Such scaling
stems from the supposition that the eddies near the surface linearly grow with
$z$ (Tennekes and Lumley,, 1972; Davidson,, 2015). Hereafter, we denote the
converted streamwise lengths related to persistence as $\ell_{p}$. Apart from
that, the coloured dashed arrows in figure 2 indicate the normalized integral
length scales of $u^{\prime}$ ($\Lambda_{u}/z$, red dashed arrows),
$w^{\prime}$ ($\Lambda_{w}/z$, blue dashed arrows), and $T^{\prime}$ signals
($\Lambda_{T}/z$, pink dashed arrows) computed from the exponential fits to
the respective auto-correlation functions (Chowdhuri et al., 2020a, , see
their figure 5 for the auto-correlation plots).

Figure 2: The persistence PDFs of the normalized sizes $\ell_{p}/z$
corresponding to the streamwise velocity fluctuations ($u^{\prime}$, red
circles), temperature fluctuations ($T^{\prime}$, pink circles), vertical
velocity fluctuations ($w^{\prime}$, blue squares), streamwise momentum flux
($u^{\prime}w^{\prime}$, black inverted triangles), and the kinematic heat
flux ($w^{\prime}T^{\prime}$, grey inverted triangles) are shown separately
for the six different stability classes. The persistence PDFs of $w^{\prime}$
are plotted twice, since both the momentum and heat flux signals involve the
vertical velocity fluctuations. The panels corresponding to these six
stability classes are arranged from the top-left to the bottom-right as, (a)
$-\zeta>2$, (b) $1<-\zeta<2$, (c) $0.6<-\zeta<1$, (d) $0.4<-\zeta<0.6$, (e)
$0.2<-\zeta<0.4$, and (f) $0<-\zeta<0.2$. Note that, the persistence PDFs for
all the variables are computed after considering both the positive and
negative fluctuations together. For visualization purpose, the persistence
PDFs of $u^{\prime}$, $u^{\prime}w^{\prime}$, and the associated $w^{\prime}$
signals are shifted vertically upwards by two decades. The descriptions of the
markers are provided in the legend, placed at the right-most side of the
figure. The coloured arrows show the positions of the normalized integral
length scales, corresponding to $w^{\prime}$ ($\Lambda_{w}/z$, blue arrows),
$T^{\prime}$ ($\Lambda_{T}/z$, pink arrows), and $u^{\prime}$
($\Lambda_{u}/z$, red arrows) signals. The regions on all the panels
corresponding to $\ell_{p}/\Lambda_{w}\leq 1$ are marked in grey.

One can notice from figure 2 that, irrespective of the stability classes, for
length scales smaller than the integral scale of $w^{\prime}$
($\ell_{p}\leq\Lambda_{w}$), the persistence PDFs of momentum or heat nearly
collapse on the persistence PDFs of the individual component signals such as,
$u^{\prime}$ and $w^{\prime}$ or $T^{\prime}$ and $w^{\prime}$. However, for
$\ell_{p}>\Lambda_{w}$ the persistence PDFs of the turbulent fluxes more
closely follow the PDFs associated with the $w^{\prime}$ signal. The reason
behind this observed phenomenon is twofold and in accord with the inferences
drawn from Eq. (4). First, the integral scales of the $u^{\prime}$ and
$T^{\prime}$ signals remain significantly larger than the $w^{\prime}$ signal.
Second, Chowdhuri et al., 2020a  have demonstrated that the persistence PDFs
follow a power-law distribution up to the scales comparable with the integral
scale before they start to drop off exponentially. Combining these two
rationales, it is imperative that at larger time scales, greater than
$\Lambda_{w}$, the persistence PDFs of $w^{\prime}$ would fall off faster than
the $u^{\prime}$ or $T^{\prime}$ signals, given $\Lambda_{w}<\Lambda_{u,T}$.
As a consequence, the flux persistence PDFs of $u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$ follow the same PDFs of the $w^{\prime}$ signals at
scales larger than $\Lambda_{w}$, whereas nearly matching with both the
individual signals for the scales $\ell_{p}\leq\Lambda_{w}$.

In summary, this analysis show that the persistence PDFs of the turbulent
fluxes have two distinct characteristics which can be separated based on the
integral scale of $w^{\prime}$. At scales smaller than $\Lambda_{w}$, the
persistence PDFs of $u^{\prime}w^{\prime}$ and $w^{\prime}T^{\prime}$ are
influenced by the power-law regions of their component PDFs. Contrarily, at
scales larger than $\Lambda_{w}$, the persistence PDFs of the turbulent fluxes
follow the $w^{\prime}$ signal quite well. This is because the persistence
PDFs of $w^{\prime}$ decrease faster than the $u^{\prime}$ or $T^{\prime}$
signals.

To explore this further, we note that the turbulent transports of heat and
momentum are accomplished through events from four different quadrants, namely
the down-gradient and counter-gradient quadrants (Wallace,, 2016). In a
convective ASL, the transport efficiencies of the heat and momentum are
intimately related to how the total fluxes are partitioned between the down-
gradient and counter-gradient quadrants (Li and Bou-Zeid,, 2011). Therefore,
as a first step to connect the persistence behaviour of the fluxes at
different scales with their transport characteristics, it is important to
decompose the heat and momentum flux signals into four different quadrants. We
present the results associated with this aspect in Section 3.1.3.

####  3.1.3 The quadrant characterization of the flux persistence PDFs

Figure 3 displays the persistence PDFs of $u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$ signals decomposed into four different quadrants.
Additionally, we also compare the persistence PDFs from four quadrants with
the total persistence of the heat and momentum signals (see the respective
grey circles and inverted triangles in figure 3). Such comparison is required
to identify at what scales which quadrant events dominate the persistence
behaviour of the flux signals.

Figure 3: Persistence PDFs of the streamwise momentum ($u^{\prime}w^{\prime}$,
inverted triangles), and the heat flux ($w^{\prime}T^{\prime}$, circles)
signals, considered separately for the four different quadrants. The panels
corresponding to the six stability classes are arranged from the top-left to
the bottom-right as, (a) $-\zeta>2$, (b) $1<-\zeta<2$, (c) $0.6<-\zeta<1$, (d)
$0.4<-\zeta<0.6$, (e) $0.2<-\zeta<0.4$, and (f) $0<-\zeta<0.2$. For the
visualization purpose, the persistence PDFs of the $u^{\prime}w^{\prime}$
signals associated with all the four different quadrants are shifted
vertically upwards by three decades. The thick black lines show the power-laws
with their respective slopes being mentioned on the panel (a). The grey
inverted triangles and circles denote the $u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$ persistence PDFs, computed after considering both the
positive and negative values together. The descriptions of the markers are
provided in the legend, placed at the right-most side of the figure. The
vertical dashed arrows in black indicate the position
$\ell_{p}/\Lambda_{w}=1$. The regions on all the panels corresponding to
$\ell_{p}/\Lambda_{w}>1$ are marked in grey.

An apparent observation from figure 3 is, for the $u^{\prime}w^{\prime}$
signal, the persistence PDFs corresponding to all the four quadrants and their
total are in excellent agreement with each other, irrespective of the
stability conditions. Nevertheless, there is a faint indication that the
$u^{\prime}w^{\prime}$ persistence PDFs concurrent with the outward-
interaction quadrants slightly deviate from the rest of the three quadrants.
On the other hand, for the $w^{\prime}T^{\prime}$ signal, approximately at
scales $\ell_{p}>\Lambda_{w}$, the persistence PDFs corresponding to the down-
gradient and counter-gradient quadrants remain largely separated from each
other. To be precise, for the highly-convective stability class ($-\zeta>2$),
the quadrant partition occurs at scales $\ell_{p}\approx 0.3\Lambda_{w}$,
instead of at $\Lambda_{w}$. However, with the decrease in $-\zeta$ the
threshold indeed becomes closer to $\Lambda_{w}$. Inspite of such dependency,
the threshold at $\Lambda_{w}$ serves well to differentiate the transport
characteristics between the large and small scale flux events (see Section
3.3). More importantly, the reason behind the quadrant separation for the heat
flux events is tied to the fact that the probabilities of encountering long
sequences of counter-gradient activities remain significantly low.
Interestingly, the discrepancy between the quadrants is quite prominent for
the highly-convective stability (figure 3a), but becomes inconspicuous in the
near-neutral conditions (figure 3f). Moreover, it is also noticeable that
irrespective of stability the total persistence PDFs of the
$w^{\prime}T^{\prime}$ signals are in accord with the down-gradient quadrants.

Along with the aforementioned features, one can notice that up to the scale
$\ell_{p}\leq\Lambda_{w}$, the persistence PDFs of the heat and momentum flux
signals collapse onto each other following a power-law behaviour. Note that,
such power-law signatures in the flux persistence PDFs are quite extensive for
the highly-convective stability (figure 3a). Therefore, the data from such
stability conditions have been used for the estimation of its exponent. Since
we plot the persistence PDFs in log-log plots where power-law appears as a
straight line, the exponent is determined through linear regression for scales
$0.01\leq\ell_{p}\leq\Lambda_{w}$. For both the $u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$ signals, we obtain a best fit exponent of $-1.4$ with
$R^{2}>0.95$. Moreover, to assess the effect of stability on the power-law
exponent, the same power-law curves as obtained for the highly-convective
stability are compared with the other five stability classes (figures 3b to
3f). Such comparison shows that, even though there is no discernible change in
the slope of the power-law, its extent gradually gets shorter as the ASL
approaches the near-neutral stability (figures 3a to 3f). Since the
measurements from the near-neutral stability class belong to the lowest three
SLTEST levels, the shrinkage in the power-law regime is related to
insufficient sampling of the small scale eddies at 20-Hz sampling frequency
(Chowdhuri et al., 2020a, ). On the other hand, at scales
$\ell_{p}>\Lambda_{w}$, the deviation from the power-law behaviour becomes
notable for both the flux persistence PDFs where they follow an exponential
distribution, hallmark of a Poisson process (Chowdhuri et al., 2020b, ).

In a nutshell, figure 3 elucidates the effects of quadrant separation on the
flux persistence PDFs. An emergent result is the quadrant segregation does not
have any appreciable effect on the persistence PDFs of the momentum flux
($u^{\prime}w^{\prime}$) signals at all the scales of motions. Conversely, for
the heat flux ($w^{\prime}T^{\prime}$) signals, the quadrant partition only
has significance at scales $\ell_{p}>\Lambda_{w}$. Physically, this difference
in quadrant behaviour between the $u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$ signals is related to transport asymmetry associated
with the down-gradient and counter-gradient quadrants. Such interpretation is
feasible, since the contrast between the persistence PDFs of positive and
negative values corresponding to any stochastic signal is related to the
skewness of that signal, as demonstrated by Chowdhuri et al., 2020a . It is
worth noting that, in a highly-convective ASL, the PDFs of the
$w^{\prime}T^{\prime}$ signals are significantly asymmetric between the
positive and negative values due to the non-Gaussian character of the
temperature fluctuations. On the contrary, the PDFs of the
$u^{\prime}w^{\prime}$ signals are more symmetric because both the streamwise
and vertical velocity fluctuations are close-to-Gaussian in nature (Chowdhuri
et al., 2020b, , see their figure 5). Additionally, for scales
$\ell_{p}>\Lambda_{w}$, both the flux persistence PDFs follow a Poisson
distribution. Previous studies have surmised such phenomenon as a signature of
random deformation of the coherent structures due to the presence of the
ground (Cava et al.,, 2012; Chowdhuri et al., 2020a, ).

Nonetheless, for scales $\ell_{p}\leq\Lambda_{w}$, the persistence PDFs of
both the fluxes collapse in a power-law fashion. The exponents of this power-
law remain same for the $u^{\prime}w^{\prime}$ and $w^{\prime}T^{\prime}$
signals, being equal to $-1.4$. The analysis of Chowdhuri et al., 2020a  has
revealed that the power-law regimes of the persistence PDFs for the velocity
and temperature signals are related to the self-similar Richardson cascade
which constitute the inertial subrange of the turbulence spectrum. Besides,
Chowdhuri et al., 2020a  reported the respective power-law exponents to be
equal to $-1.6$, $-1.25$, and $-1.4$ for the $u^{\prime}$, $w^{\prime}$, and
$T^{\prime}$ signals. They linked the difference in the exponents to the
disparity in the small-scale intermittency, expressed through the framework of
self-organized criticality. Interestingly, the power-law exponents
corresponding to the flux signals are $-1.4$ irrespective of their type and
coincidentally being equal to the same for the $T^{\prime}$ signal. Further
explanation behind this coincidence is elusive at present.

Hitherto, from the detailed discussion of the flux persistence PDFs, one could
sense that the persistence properties of the momentum and heat flux signals
exhibit significantly different behaviour for scales smaller or larger than
the integral scale of the vertical velocity. Since $\Lambda_{w}$ is of the
same order as $z$ (Chowdhuri et al., 2020a, ), such discrepancy reflects
distinct attributes of turbulent transport associated with the detached and
attached eddies, characterized by length scales smaller and larger than $z$
(Marusic and Monty,, 2019). However, it is important to remember that the
persistence analysis only provides information on the distribution of time or
length scales of the flux events but not on the flux values themselves
(Bershadskii et al.,, 2004). Therefore, such analysis alone is insufficient to
deduce the effects of the flux events, distinguished by their length scales
greater or lesser than $\Lambda_{w}$, on the heat and momentum transport
efficiencies. In Section 3.2, we introduce a methodology where the transport
characteristics are studied separately for the heat and momentum flux events
with length scales $\ell_{p}>\Lambda_{w}$ and $\ell_{p}\leq\Lambda_{w}$.

###  3.2 Polar representation of the quadrant planes

In general, the heat and momentum transport characteristics allied to the
turbulent motions are expressed through quadrant analysis, where each point
plotted on the $u^{\prime}$-$w^{\prime}$ or $T^{\prime}$-$w^{\prime}$ quadrant
plane corresponds to a certain flux event. An example of such quadrant plot is
provided in figure S2 (supplementary material) to explain the concepts. The
usual practice in this analysis is to report the momentum or heat flux
fractions and time fractions from each quadrant and assess the relative
importance of the various turbulent motions, associated with different flow
structures (Wallace,, 2016). Additionally, it is also worthwhile to mention
that, the transport efficiencies of the heat and momentum are intimately
linked to the partition of the fluxes among these four different quadrants (Li
and Bou-Zeid,, 2011; Chowdhuri and Prabha,, 2019).

However, on a rudimentary level, the existence of the heat and momentum flux
depends upon the strength of the coupling between the two turbulent signals as
described by the governing Navier-Stokes equations. In order to understand the
cause of the coupling, it is useful to envisage the quadrant plane as a phase
space by drawing analogies with non-linear dynamical systems. From this
perspective, each point in the quadrant plane is related to a particular state
of flux generation, designated with two parameters which are the phase angles
and amplitudes. To accomplish such an objective, the polar representation of
the quadrant plane is a well-suited approach (Mahrt and Paumier,, 1984). One
should note that, in this approach the amplitudes are not the intensities of
the flux themselves, since the flux values are a combination of both the
phases and amplitudes (see Eq. (7)).

Figure 4: The graphical illustrations of the (a) $u^{\prime}$-$w^{\prime}$ and
(b) $T^{\prime}$-$w^{\prime}$ quadrant planes are provided. The fluctuations
in $u$, $w$, and $T$ are normalized by their respective standard deviations
($\sigma_{u}$, $\sigma_{w}$, and $\sigma_{T}$) and denoted as $\hat{x}$
($\hat{x}=x^{\prime}/\sigma_{x}$, where $x$ can be $u$, $w$, or $T$). The four
different quadrants are marked on the respective panels. In a polar co-
ordinate representation, each point on the quadrant plane ({$\hat{u},\hat{w}$}
or {$\hat{T},\hat{w}$}) can be described by their amplitudes
($r_{\hat{w}\hat{u}}$ or $r_{\hat{w}\hat{T}}$) and the phase angles
($\theta_{\hat{w}\hat{u}}$ or $\theta_{\hat{w}\hat{T}}$). The phase angles can
take values between $-\pi$ to $\pi$ depending on the position of the
quadrants, as displayed on the panels. The grey shaded regions in the quadrant
planes denote the down-gradient quadrants, whereas the yellow shaded regions
denote the counter-gradient quadrants.

To explain such approach, figure 4 graphically illustrates the concept of
quadrant planes of $u^{\prime}$-$w^{\prime}$ and $T^{\prime}$-$w^{\prime}$
from the perspective of a polar reference frame. Following the standard
practice, the $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals in figure 4
are normalized by their respective standard deviations ($\sigma_{u}$,
$\sigma_{w}$, and $\sigma_{T}$) and denoted as $\hat{x}$
($\hat{x}=x^{\prime}/\sigma_{x}$, where $x$ can be $u$, $w$, or $T$). Such
normalization is necessary since it removes the issue regarding the difference
in the units, while computing the phase angles and amplitudes (defined below).
In the polar reference frame, each point on the $\hat{x}$-$\hat{w}$ (where $x$
can be $u$ or $T$) quadrant plane can be associated with an amplitude
$r_{\hat{w}\hat{x}}$ and phase angle $\theta_{\hat{w}\hat{x}}$, expressed as,

$\displaystyle\theta_{\hat{w}\hat{x}}=\arctan{(\hat{w}/\hat{x})}$ (5)
$\displaystyle r_{\hat{w}\hat{x}}=\sqrt{\hat{w}^{2}+\hat{x}^{2}}.$ (6)

The lengths of the phase vectors (shown as arrows) on figure 4 indicate the
values of $r_{\hat{w}\hat{x}}$, whereas $\theta_{\hat{w}\hat{x}}$ are the
angles subtended by these vectors with the $x$ axes. The values of the phase
angles vary between $-\pi$ to $\pi$ and their ranges are related to the four
different quadrants as demonstrated in table 1. For identification purposes,
the $\theta_{\hat{w}\hat{x}}$ ranges which denote the down-gradient (counter-
gradient) quadrants are marked as grey (yellow) shaded regions in figure 4.

Table 1: The four quadrants of $\hat{u}$-$\hat{w}$ and $\hat{T}$-$\hat{w}$ and the associated distribution of phase angles in a convective ASL Phase angle |  $\hat{u}$-$\hat{w}$ quadrant | Quadrant name |  $\hat{T}$-$\hat{w}$ quadrant | Quadrant name
---|---|---|---|---
$0\leq\theta_{\hat{w}\hat{u}},\theta_{\hat{w}\hat{T}}\leq\pi/2$ |  $\hat{u}>$ 0, $\hat{w}>$ 0 | Outward-interaction |  $\hat{w}>$ 0, $\hat{T}>$ 0 | Warm-updraft
$\pi/2\leq\theta_{\hat{w}\hat{u}},\theta_{\hat{w}\hat{T}}\leq\pi$ |  $\hat{u}<$ 0, $\hat{w}>$ 0 | Ejection |  $\hat{w}>$ 0, $\hat{T}<$ 0 | Cold-updraft
$-\pi\leq\theta_{\hat{w}\hat{u}},\theta_{\hat{w}\hat{T}}\leq-\pi/2$ |  $\hat{u}<$ 0, $\hat{w}<$ 0 | Inward-interaction |  $\hat{w}<$ 0, $\hat{T}<$ 0 | Cold-downdraft
$-\pi/2\leq\theta_{\hat{w}\hat{u}},\theta_{\hat{w}\hat{T}}\leq 0$ |  $\hat{u}>$ 0, $\hat{w}<$ 0 | Sweep |  $\hat{w}<$ 0, $\hat{T}>$ 0 | Warm-downdraft

In the polar co-ordinate system, the normalized instantaneous flux
($\hat{u}\hat{w}$ or $\hat{w}\hat{T}$) associated with each point is expressed
as $(r_{\hat{w}\hat{x}}^{2}\sin{2\theta_{\hat{w}\hat{x}}})/2$. This is
because,

$\hat{x}\hat{w}=r_{\hat{w}\hat{x}}\cos{(\theta_{\hat{w}\hat{x}})}\times
r_{\hat{w}\hat{x}}\sin{(\theta_{\hat{w}\hat{x}})}\implies\frac{1}{2}r_{\hat{w}\hat{x}}^{2}\sin{(2\theta_{\hat{w}\hat{x}})},$
(7)

given
$\sin{(2\theta_{\hat{w}\hat{x}})}=2\sin{(\theta_{\hat{w}\hat{x}})}\cos{(\theta_{\hat{w}\hat{x}})}$
and $x$ can be either $u$ or $T$.

Subsequently, the averaged normalized flux over all the four quadrants can be
written as,

$\overline{\hat{x}\hat{w}}=\frac{1}{2}\overline{r_{\hat{w}\hat{x}}^{2}\sin{2\theta_{\hat{w}\hat{x}}}}.$
(8)

The quantity on the left hand side of Eq. (8) is the correlation coefficient
($R_{wx}$) which indicates the transport efficiency of the momentum or the
heat fluxes. Apart from that, since $r_{\hat{w}\hat{x}}^{2}$ is a positive
definite quantity, the distribution of the fluxes among the four quadrants is
primarily decided by the phase angle $\theta_{\hat{w}\hat{x}}$. Therefore, to
simplify the expression in Eq. (8) we may replace the $r_{\hat{w}\hat{x}}^{2}$
values of every point in the quadrant plane by a single averaged value,
$\overline{r_{\hat{w}\hat{x}}^{2}}$. Note that,

$\overline{r_{\hat{w}\hat{x}}^{2}}=\overline{{\big{(}{\frac{x^{\prime}}{\sigma_{x}}}\big{)}}^{2}+{\big{(}{\frac{w^{\prime}}{\sigma_{w}}}\big{)}}^{2}}\implies\overline{{\big{(}{\frac{x^{\prime}}{\sigma_{x}}}\big{)}}^{2}}+\overline{{\big{(}{\frac{w^{\prime}}{\sigma_{w}}}\big{)}}^{2}}.$
(9)

Since $\overline{{({{x^{\prime}}/{\sigma_{x}}})}^{2}}$ and
$\overline{{({{w^{\prime}}/{\sigma_{w}}})}^{2}}$ are both equal to 1, from Eq.
(9) it indicates $\overline{r_{\hat{w}\hat{x}}^{2}}=2$. Consequently, this
enables us to simplify Eq. (8) as,

$R_{wx}=\frac{\overline{r_{\hat{w}\hat{x}}^{2}}}{2}\times\overline{\sin{2\theta_{\hat{w}\hat{x}}}}\implies\overline{\sin{2\theta_{\hat{w}\hat{x}}}}$
(10)

by substituting $\overline{r_{\hat{w}\hat{x}}^{2}}=2$. If the two signals are
strictly phase-locked, then the phase angle $\theta_{\hat{w}\hat{x}}$ can take
only two values which are $\pm\ \pi/4$ and $\pm\ 3\pi/4$. By replacing those
in Eq. (10), one can deduce the correlation coefficients ($R_{wx}$) for such
occasions would be perfectly $\pm\ 1$.

To scrutinize this further, from the theory of probability one can modify Eq.
(10) as,

$R_{wx}=\int_{-\pi}^{\pi}P(\theta_{\hat{w}\hat{x}})\sin{(2\theta_{\hat{w}\hat{x}})}\
d\theta_{\hat{w}\hat{x}},$ (11)

where $P(\theta_{\hat{w}\hat{x}})$ is the PDF of the phase angle
$\theta_{\hat{w}\hat{x}}$, with the constraint,

$\int_{-\pi}^{\pi}P(\theta_{\hat{w}\hat{x}})\ d\theta_{\hat{w}\hat{x}}=1.$
(12)

Let us consider an another case, where the $\theta_{\hat{w}\hat{x}}$ values
are uniformly distributed over the quadrant plane, such that
$P(\theta_{\hat{w}\hat{x}})=1/(2\pi)$. This scenario would arise if the phase
vectors were oriented in random directions with no phase-locking whatsoever.
In that situation, Eq. (11) would be simplified as,

$R_{wx}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(2\theta_{\hat{w}\hat{x}})}\
d\theta_{\hat{w}\hat{x}}.$ (13)

Since $\sin{(2\theta_{\hat{w}\hat{x}})}$ is an odd function, from elemental
calculus we know that the output of the integration in Eq. (13) is 0. This
implies a completely inefficient transport of the turbulent fluxes. As a
result, the heat and momentum transport efficiency can be judged by
investigating the departure of the respective phase angle PDFs from a uniform
distribution. An implicit assumption in such argument is, the role played by
the amplitudes can be considered to be independent of the phase angles.

The aforementioned method based on the polar representation of the quadrant
planes offers a unique opportunity to dissect the role of phase coupling
towards the transport efficiencies of heat and momentum. It is therefore of
practical interest to investigate this aspect corresponding to the flux events
with length scales larger or smaller than $\Lambda_{w}$. The intention behind
such separation criterion emerges from the flux persistence PDFs as presented
in figures 2 and 3. To briefly recapitulate, at length scales smaller than the
integral scale of $w^{\prime}$, the persistence PDFs of the heat and momentum
flux events from any quadrant follow an identical power-law distribution. Such
a phenomenon demonstrates the effect of the detached eddies from the inertial
subrange of the turbulence spectrum. On the contrary, at length scales larger
than $\Lambda_{w}$ (reminiscent of the attached eddies), the persistence PDFs
of the heat flux events are clearly dominated by the down-gradient motions and
follow an exponential distribution. However, for the momentum flux events,
even at these large length scales, the quadrant effect is negligible on their
persistence PDFs, albeit the nature of the distribution remains exponential.
To get further insights, in figure 5, we present results to study the
distributions of the phase angles and amplitudes in relation to the flux
events, conditionally sampled based on the threshold $\ell_{p}=\Lambda_{w}$.

Figure 5 shows the PDFs of the phase angles ($\theta_{\hat{w}\hat{x}}$) and
amplitudes ($r_{\hat{w}\hat{x}}$) associated with the small and large scale
flux events, delineated accordingly with respect to their persistence time
scales $\ell_{p}/\Lambda_{w}\leq 1$ and $\ell_{p}/\Lambda_{w}>1$. Note that,
both of these PDFs are computed by collecting all the points which reside
within the ensemble of flux events from a particular stability class and
persisting for times as prescribed before. Hereafter, we denote these
conditional PDFs of the phase angles as
$P[\theta_{\hat{w}\hat{x}}|(\ell_{p}\leq\Lambda_{w})]$ and
$P[\theta_{\hat{w}\hat{x}}|(\ell_{p}>\Lambda_{w})]$ respectively.
Additionally, $P[\theta_{\hat{w}\hat{x}}|(\ell_{p}\leq\Lambda_{w})]$ and
$P[\theta_{\hat{w}\hat{x}}|(\ell_{p}>\Lambda_{w})]$ individually satisfy the
constraint in Eq. (12), so that when integrated over the limit $-\pi$ to $\pi$
the result is unity. Other than that, the grey and yellow shaded regions in
figures 5a, b, d, and e identify the ranges of the phase angles which
correspond to the regions ascertained to the down-gradient and counter-
gradient quadrants respectively (see table 1 and figure 4). One may bear in
mind that the areas occupied by the phase angle PDFs in these shaded regions
are proportional to the fraction of time spent by the signal in that quadrant
state. This is because,

$(T_{f})_{\rm X}=\int_{-\pi}^{\pi}P(\theta_{\hat{w}\hat{x}})I_{\rm
X}(\theta_{\hat{w}\hat{x}})\ d\theta_{\hat{w}\hat{x}},$ (14)

where $(T_{f})_{\rm X}$ is time fraction spent in quadrant X (X could be any
one of the four quadrants) and $I_{\rm X}(\theta_{\hat{w}\hat{x}})$ is an
identity function which is unity when $\theta_{\hat{w}\hat{x}}$ lies within
quadrant X or zero elsewise. It is noteworthy that in a hypothetical
framework, if the two signals are completely phase-locked (phase difference of
$0$ or $\pi$) the PDFs of the phase angles would be a superposition of two
Dirac-delta functions at $\theta_{\hat{w}\hat{x}}$ values of $\pm\ \pi/4$ and
$\pm\ 3\pi/4$. However, for all practical purposes such case is not possible
since it would imply an infinite persistence of the flux signals as they would
never cross the zero. That being said, we thus expect the phase angle PDFs to
be distributed in a particular way over all the possible
$\theta_{\hat{w}\hat{x}}$.

Figure 5: The PDFs of $\theta_{\hat{w}\hat{u}}$ are shown for the momentum
flux events of length scales (a) $\ell_{p}>\Lambda_{w}$ or (b)
$\ell_{p}\leq\Lambda_{w}$. Similarly, the PDFs of $\theta_{\hat{w}\hat{T}}$
are shown for the heat flux events corresponding to (d) $\ell_{p}>\Lambda_{w}$
and (e) $\ell_{p}\leq\Lambda_{w}$. The associated PDFs of the amplitudes are
shown in panels (c) and (f) for $r_{\hat{w}\hat{u}}$ and $r_{\hat{w}\hat{T}}$
respectively. The dash-dotted and the dashed lines in the panels (c) and (f)
describe the respective momentum or heat flux events with length scales
$\ell_{p}\leq\Lambda_{w}$ and $\ell_{p}>\Lambda_{w}$ (see the legend in panel
(c)). The description of the various markers indicating the six different
stability classes is provided in the legend of panel (a). The grey and yellow
shaded regions correspond to the down-gradient and counter-gradient quadrants
respectively, as displayed in figure 4.

From figure 5a, the PDFs of the phase angles associated with the large scale
momentum flux events display a prominent peak related to the ejection motions
(the grey shaded region on the right), irrespective of the stability classes.
However, for the negative values of the phase angles, the PDFs have two
distinct peaks in the grey and yellow shaded regions with approximately
similar values. From Eq. (14), this suggests that for the large scale momentum
flux events, there is almost an equal tendency for the phase vectors to either
reside within the sweep or within the inward-interaction quadrants.
Nevertheless, a minute change in such behaviour is observed for the near-
neutral stability class. In those conditions, the PDF peak values within the
sweep quadrants slightly exceed the values associated with the inward-
interaction quadrants. On the other hand, for all the stability classes, the
PDFs of the phase angles associated with the large scale heat flux events
exhibit a bi-modal behaviour with two distinguished peaks corresponding to the
warm-updraft and cold-downdraft motions (the right and left grey shaded
regions in figure 5d). Be that as it may, with the decrease in the thermal
stratification, the PDF values of
$\theta_{\hat{w}\hat{T}}|(\ell_{p}>\Lambda_{w})$ corresponding to the counter-
gradient quadrants (yellow shaded regions in figure 5d) show considerable
increase. All of these features of the phase angle PDFs associated with the
large scale heat and momentum flux events are in sync with the findings of
Haugen et al., (1971). They observed that in a convective ASL flow, the heat
flux signals mainly exhibited large persistent positive events, whereas for
the momentum flux long sequences of both negative and positive activities were
common.

From figure 5, one could further notice that the shapes of the PDFs of the
phase angles differ significantly between the large ($\ell_{p}>\Lambda_{w}$)
and small ($\ell_{p}\leq\Lambda_{w}$) scale flux events (figures 5a, b, d, and
e). For the small scale momentum flux events, an immediate observation is the
PDFs of the phase angles is nearly an inverted version of the PDFs associated
with the large scale momentum flux events (figure 5b). In figure 5b, the
maximum values of the PDFs for all the stability classes are located at around
0, with troughs replacing the peaks at approximately the same positions as in
figure 5a. Conversely, for the small scale heat flux events (figure 5e), the
behaviour of the phase angle PDFs is not completely opposite to that of the
large scale events (figure 5d), but the distinction between the two modes is
definitely more vague.

In addition to the phase angles, we can also investigate the amplitude PDFs
($r_{\hat{w}\hat{x}}$) for both large and small scale heat and momentum flux
events (figures 5c and f). Such information would be important to evaluate
whether there are any specific amplitudes which occur most often. From figures
5c and f, we note that the PDFs of the amplitudes are uni-modal in character,
with a shift in their peak positions for the large scale heat or momentum flux
events (approximately from 0.8 to 1.2). Apart from that, the PDFs of the
amplitudes collapse sufficiently well for all the stability classes,
corresponding to the small scale flux events. On the other hand, for the large
scale flux events, such collapse is relatively poor for the small values of
the amplitudes, located to the left of the peak position.

In summary, the results obtained so far point out that the statistical
characteristics of the phase angles associated with the heat and momentum flux
events remain significantly different from each other. Inevitably, this effect
is more clearly visible for the flux events with $\ell_{p}$ larger than the
integral scale of the vertical velocity. To connect the behaviour of the phase
angle PDFs with the transport efficiencies, from Eq. (8) we can write,

$R_{wx}=\frac{1}{2}\Big{[}\overline{(r_{\hat{w}\hat{x}}^{2}\sin{2\theta_{\hat{w}\hat{x}}})|(\ell_{p}>\Lambda_{w})+(r_{\hat{w}\hat{x}}^{2}\sin{2\theta_{\hat{w}\hat{x}}})|(\ell_{p}\leq\Lambda_{w})}\Big{]},$
(15)

where the quantity $(r_{\hat{w}\hat{x}}^{2}\sin{2\theta_{\hat{w}\hat{x}}})$ is
divided between the events with scales $\ell_{p}>\Lambda_{w}$ and
$\ell_{p}\leq\Lambda_{w}$. From Eq. (11), we can further expand Eq. (15) as,

$\begin{split}R_{wx}&=\frac{1}{2}\Big{[}\langle{r_{\hat{w}\hat{x}}^{2}|(\ell_{p}>\Lambda_{w})}\rangle\int_{-\pi}^{\pi}P[\theta_{\hat{w}\hat{x}}|(\ell_{p}>\Lambda_{w})]\sin{[2\theta_{\hat{w}\hat{x}}|(\ell_{p}>\Lambda_{w})]}\
d\theta_{\hat{w}\hat{x}}+\\\
&\langle{r_{\hat{w}\hat{x}}^{2}|(\ell_{p}\leq\Lambda_{w})}\rangle\int_{-\pi}^{\pi}P[\theta_{\hat{w}\hat{x}}|(\ell_{p}\leq\Lambda_{w})]\sin{[2\theta_{\hat{w}\hat{x}}|(\ell_{p}\leq\Lambda_{w})]}\
d\theta_{\hat{w}\hat{x}}\Big{]},\end{split}$ (16)

with the constraints,

$\displaystyle\int_{-\pi}^{\pi}P[\theta_{\hat{w}\hat{x}}|(\ell_{p}>\Lambda_{w})]\
d\theta_{\hat{w}\hat{x}}=1$ (17)
$\displaystyle\int_{-\pi}^{\pi}P[\theta_{\hat{w}\hat{x}}|(\ell_{p}\leq\Lambda_{w})]\
d\theta_{\hat{w}\hat{x}}=1,$ (18)

after conditioning on the flux events based on their $\ell_{p}$ (as shown in
figure 5). Moreover, in Eq. (16) the angle brackets denote the most probable
amplitudes over those events with length scales $\ell_{p}>\Lambda_{w}$ and
$\ell_{p}\leq\Lambda_{w}$, respectively. As mentioned earlier, such
ramifications rest on the assumption that the amplitude effects on the
transport efficiency are independent from the phase angle distribution. We can
thus replace those values as 1.2 and 0.8 respectively, obtained from the peak
positions of the amplitude PDFs (see figures 5c and f). Therefore, from Eqs.
(11), (13), and (16) it is clear that, in order to evaluate the contribution
of the flux events at different length scales to the heat and momentum
transport efficiencies, one needs to estimate the departure of
$P[\theta_{\hat{w}\hat{x}}|(\ell_{p}>\Lambda_{w})]$ and
$P[\theta_{\hat{w}\hat{x}}|(\ell_{p}\leq\Lambda_{w})]$ from a uniform
distribution. In Section 3.3 we present results to quantify such departure of
the phase angle PDFs.

###  3.3 Phase angle PDFs and transport efficiency

In order to determine how closely the distribution of the phase angles in
figure 5 resemble a uniform distribution, it is imperative to consider the
cumulative distribution functions (CDFs), instead of the PDFs. This is
because, the CDFs of a uniform distribution can be represented through a
linear function. The CDFs of the phase angles and the equivalent CDFs
corresponding to a uniform distribution can be written as,

$\displaystyle
F(\theta_{\hat{w}\hat{x}})=\int_{\pi}^{\theta_{\hat{w}\hat{x}}}P(\theta_{\hat{w}\hat{x}})\
d\theta_{\hat{w}\hat{x}}$ (19) $\displaystyle
F_{u}(\theta_{\hat{w}\hat{x}})=\frac{\pi-\theta_{\hat{w}\hat{x}}}{2\pi}$ (20)

where $F(\theta_{\hat{w}\hat{x}})$ is the empirical CDF, and
$F_{u}(\theta_{\hat{w}\hat{x}})$ is the CDF for the uniform distribution.
Figure 6 shows the CDFs of the phase angles for the large and small scale heat
and momentum flux events and compare those with the CDF of a uniform
distribution. An apparent observation from figures 6a and b is, irrespective
of stability the statistical characteristics of the CDFs do not seem to differ
much from a uniform distribution for both the large and small scale momentum
flux events. However, for the large scale heat flux events the CDFs differ
significantly from the uniform distribution (figure 6c), although such
contrast becomes less obvious for the small scale heat flux events (figure
6d).

Figure 6: The cumulative distribution functions (CDFs) of the phase angles
($\theta_{\hat{w}\hat{u}}$) for the momentum flux events with length scales
(a) $\ell_{p}>\Lambda_{w}$ and (b) $\ell_{p}\leq\Lambda_{w}$. Similarly, the
CDFs of the phase angles ($\theta_{\hat{w}\hat{T}}$) are shown for the heat
flux events corresponding to (c) $\ell_{p}>\Lambda_{w}$ and (d)
$\ell_{p}\leq\Lambda_{w}$. The thick black lines on all the panels denote the
CDFs of the phase angles if they were distributed uniformly over the range
$-\pi$ to $\pi$. The description of the various markers indicating the six
different stability classes is provided in the legend of panel (a). In panel
(e), the normalized Shannon entropy associated with the distribution of the
phase angles are shown, corresponding to the length scales
$\ell_{p}>\Lambda_{w}$ and $\ell_{p}\leq\Lambda_{w}$. The legend of panel (e)
describes the markers and the lines on the plot.

Notwithstanding the fact that the CDFs provide a perceptible measure to
inspect whether the distributions of the phase angles differ from a uniform
distribution, the quantification of such effect remains an issue. To provide a
convenient solution to that, we introduce the normalized Shannon entropy of
the phase angle distribution corresponding to the large and small scale flux
events. From the information theory (Banerjee et al.,, 2018), the normalized
Shannon entropy ($H_{N}$) of the phase angle distribution can be defined as,

$H_{N}(\theta_{\hat{w}\hat{x}})=-\frac{1}{\ln{(N_{\rm b})}}\sum_{i=1}^{N_{\rm
b}}P_{i}(\theta_{\hat{w}\hat{x}})\ln{[P_{i}(\theta_{\hat{w}\hat{x}})]},$ (21)

where $N_{\rm b}$ is the number of bins in which the $\theta_{\hat{w}\hat{x}}$
values are divided (60 in our case), and $P_{i}(\theta_{\hat{w}\hat{x}})$ is
the probability of occurrence of a particular binned value
$\theta_{\hat{w}\hat{x}}$. Note that, for a uniform distribution Eq. (21) will
be equal to 1, given $P_{i}(\theta_{\hat{w}\hat{x}})=1/N_{\rm b}$ for all the
bin indexes. As a consequence, the departure from unity in Eq. (21) is
regarded as a metric quantifying the discrepancy with a configuration where
the phase vectors are randomly oriented. From Eq. (11) we know that, in the
absence of amplitude dependency such configuration of the phase vectors does
not cause any transport of the turbulent fluxes. Therefore, within that
constraint, the quantity $H_{N}(\theta_{\hat{w}\hat{x}})$ can be considered as
a proxy for the transport efficiency.

Figure 6e shows the variation of $H_{N}(\theta_{\hat{w}\hat{x}})$, related to
both the small and large scale momentum and heat flux events, with the
stability ratio $-\zeta$. Before we discuss the relevant features of figure
6e, it should be noted that the entropies are not computed for single 30-min
runs, but for an ensemble of runs within a particular stability bin. This is
necessary to ensure that the PDFs of the phase angles are statistically
reliable and the estimations of the normalized Shannon entropies are robust.
Moreover, in order to document the variation over a substantial range of
$-\zeta$ while maintaining the statistical robustness of the results, we
divide the $-\zeta$ values into nine number of bins, where each bin contains
equal number of 30-min runs (which is 29 for our case, given a total of 261
runs).

From figure 6e one could notice that, for all the stability values, the
quantity $H_{N}$ of the phase angles remains almost equal to unity
($H_{N}\approx 0.98$), corresponding to both the small scale momentum and heat
flux events. Nonetheless, a considerable departure from unity ($H_{N}\approx
0.75$) is noted for the large scale heat flux events, in highly-convective
conditions ($-\zeta>1$). Furthermore, there is an apparent tendency that the
$H_{N}$ values approach unity ($H_{N}\approx 0.9$) for the large scale heat
flux events as the $-\zeta$ values decrease towards the near-neutral
stability. Regardless, for the large scale momentum flux events no significant
departure from unity is noted in its $H_{N}$ estimates ($H_{N}\approx 0.96$),
irrespective of stability.

To put the above described results into perspective, it is useful to discuss
them from the standpoint of turbulent structures and the relation with the
heat and momentum transport characteristics in a convective ASL flow. From
figure 6, it is evident that for the small scale heat and momentum flux events
($\ell_{p}\leq\Lambda_{w}$), the phase vectors are oriented in a close-to-
random manner with the phase angles being distributed in a quasi-uniform way.
From the persistence PDFs of the heat and momentum flux events, we know that
at scales $\ell_{p}\leq\Lambda_{w}$, they follow a power-law distribution,
related to the eddies from the inertial subrange (figures 2 and 3). According
to the Kolmogorov’s hypothesis, the turbulence associated with the eddies from
the inertial subrange of the spectrum is quasi-isotropic in nature and hardly
transport any heat or momentum (Tennekes and Lumley,, 1972; Wyngaard and
Coté,, 1972). Since the random orientation of the phase vectors denotes no
flux transport (see Eq. (11)), this complements the properties of turbulence
in the inertial subrange, corresponding to scales $\ell_{p}\leq\Lambda_{w}$.

On the other hand, for the large scale heat flux events
($\ell_{p}>\Lambda_{w}$), associated with the energy containing motions, the
distribution of the phase angles differ significantly from the uniform
distribution in highly-convective stability. This indicates the vertical
velocity and temperature fluctuations are phase-locked to a certain degree in
a highly-convective ASL flow and hence related to efficient transport of heat
(see Eqs. 11 and 16). Such a contention is in agreement with the large-eddy
simulation studies, where the researchers have shown that both the vertical
velocity and temperature patterns overlay on each other in the form of
cellular structures (Khanna and Brasseur,, 1998; Salesky et al.,, 2017). This
configuration is efficient in transporting the heat flux, which agrees with
our assessment. Nevertheless, the phase angle PDFs between the $w^{\prime}$
and $T^{\prime}$ signals gradually resemble a uniform distribution as the
near-neutral stability is approached. This behaviour is in concurrence with
the observation that the heat-transport efficiency decreases as the turbulence
becomes more shear dominated (Li and Bou-Zeid,, 2011). Therefore, we can
conclude that the heat-transport efficiency in a convective ASL flow can be
explained by the departure of the phase-angles from a uniform distribution,
corresponding to the events with scales $\ell_{p}>\Lambda_{w}$.

Interestingly, an identical deduction cannot be made for the large scale
momentum flux events ($\ell_{p}>\Lambda_{w}$). This is because, for these
events, the phase angle PDFs remain similar to a uniform distribution for all
the $-\zeta$ values. From Eq. 16 it would imply that the momentum transport
remains inefficient irrespective of the strength of the thermal
stratification. However, such an implication does not agree with the
ubiquitous result that the momentum transport efficiency increases with the
decrease in $-\zeta$. Needless to say, this brings into consideration the role
of amplitudes in the momentum flux generation. While connecting the phase
angle PDFs directly with the correlation coefficient $R_{uw}$, it is assumed
that the amplitudes of all the phase vectors can be replaced with a single
value, while preserving their angles. Since such an assumption produces output
which is incompatible with the measurements, we thus infer that the amplitude
variations play a significant part to generate the momentum flux. Ignoring the
amplitude effect results in almost no transport of averaged momentum even
during the near-neutral conditions. This agrees with the observations of
Haugen et al., (1971), Högström and Bergström, (1996), and Narasimha et al.,
(2007) where they noted that in a near-neutral ASL flow, the averaged momentum
flux is mainly generated through burst like events associated with strong
gusts in the streamwise velocity fluctuations. We present our conclusions in
Section 4.

##  4 Conclusion

In this study, we provide a detailed account of the persistence properties of
turbulent heat and momentum fluxes as obtained from the SLTEST experimental
dataset, in a convective surface layer. Furthermore, we also establish a novel
linkage between the persistence of the flux events and the heat and momentum
transport characteristics. We develop such correspondence through a framework
based on the concept of phase space in non-linear dynamical systems.

On a larger scale, the ramifications of this research are directed towards
providing answers to the questions posed in the introduction. Keeping that in
mind, the important results from this paper can be noted as:

  1. 1.

The comparison of the persistence PDFs of heat and momentum fluxes
($u^{\prime}w^{\prime}$ and $w^{\prime}T^{\prime}$) with the component signals
($u^{\prime}$ and $w^{\prime}$ or $w^{\prime}$ and $T^{\prime}$) reveals that
at scales ($\ell_{p}$) smaller than the integral scale of the vertical
velocity ($\Lambda_{w}$), the persistence PDFs of the component signals and
their products are in excellent agreement with each other. For such scales,
the persistence PDFs of the products ($u^{\prime}w^{\prime}$ and
$w^{\prime}T^{\prime}$) follow an identical power-law distribution with an
exponent of $-1.4$, irrespective of the stability conditions. On the other
hand, for scales larger than the integral scale of the vertical velocity, both
the flux persistence PDFs nearly collapse on the persistence PDFs of the
$w^{\prime}$ signals and deviate significantly from the $u^{\prime}$ or
$T^{\prime}$ signals.

  2. 2.

The flux persistence PDFs are investigated separately by considering the
distribution of the time scales from the four different quadrants. We discover
that, for the momentum flux events, the persistence PDFs of the
$u^{\prime}w^{\prime}$ signals are indistinguishable for all the four
quadrants by collapsing onto one another. However, for the heat flux events,
the effect of quadrant separation on the $w^{\prime}T^{\prime}$ persistence
PDFs remains insignificant for scales smaller than the integral scale of the
vertical velocity. Contrarily, for $\ell_{p}>\Lambda_{w}$, the persistence
PDFs of the heat flux events are primarily governed by the down-gradient
quadrants; warm-updrafts and cold-downdrafts.

  3. 3.

For scales $\ell_{p}\leq\Lambda_{w}$, the persistence PDFs of both the flux
events show a similar characteristics in terms of quadrant behaviour and are
also akin to a power-law distribution. Such power-law behaviour is related to
the eddies from the inertial subrange of the turbulence spectrum.
Nevertheless, at scales $\ell_{p}>\Lambda_{w}$, the persistence PDFs differ
from the power-law distribution and drop off exponentially. Moreover, at such
time scales, the persistence PDFs of the $w^{\prime}T^{\prime}$ signals are
dominated by the organized heat flux events from the down-gradient quadrants.
Therefore, the investigation of flux persistence leads to a criterion to
separate two different eddy processes, based on the integral scale of
$w^{\prime}$. Since the integral length scale of $w^{\prime}$ is of the same
order as $z$, they reflect the properties of the attached and detached eddies,
based on the Townsend’s attached eddy model.

  4. 4.

To gain insight into the transport mechanisms related to these two eddy
processes, we scrutinize the effects of phase angles and amplitudes associated
with the flux component signals ({$u^{\prime}$,$w^{\prime}$} and
{$T^{\prime}$,$w^{\prime}$}). We derive a simple relation between the PDFs of
the phase angles and transport efficiencies by ignoring the variations in the
amplitudes. Under such assumption, the departure of the phase angle PDFs from
a uniform distribution is shown to be a necessary condition for the flux
transport. The results indicate that the phase angles of the component signals
related to the heat and momentum flux events with length scales
$\ell_{p}\leq\Lambda_{w}$ are almost uniformly distributed. Given these events
are associated with the detached eddies from the inertial subrange, this
agrees with the general notion that quasi-isotropic turbulence for such size
ranges hardly transport any flux.

  5. 5.

For the large scale heat flux events ($\ell_{p}>\Lambda_{w}$), the departure
of the phase angle PDFs from a uniform distribution is the strongest for the
highly-convective regime. However, with the change in stability towards the
near-neutral regime, the same PDFs resemble closely a uniform distribution.
Such variation explains the gradual reduction in the heat transport efficiency
as the near-neutral stability is approached.

  6. 6.

On the other hand, for the large scale momentum flux events, the phase angle
PDFs remain close to a uniform distribution irrespective of all the stability
classes. Such a behaviour would suggest, on an average there is nearly no
transport of momentum in a convective ASL flow. This result is antithetical to
the observation that the momentum transport efficiency increases with the
decrease in strength of the thermal stratification. In order to explain the
contradiction, the amplitude effects need to be taken into account for the
momentum transport.

To sum up, we analyze the heat and momentum transport characteristics of two
different eddy processes, identified through the statistical properties of the
flux persistence PDFs in a convective ASL flow. We demonstrate that the heat
transport efficiency in convective conditions is related to the phase angle
distributions, associated with the events which persist for times larger than
the integral scale of vertical velocity. On the contrary, for the momentum
transport efficiency, the phase information between the streamwise and
vertical velocity fluctuations remains largely irrelevant. More importantly,
if the amplitude effects are disregarded then it results in a situation with
nearly no transport of streamwise momentum. For practical purposes, these
results offer a unique perspective towards modelling the heat and momentum
transport processes in a convective ASL flow.

## Acknowledgements

Indian Institute of Tropical Meteorology (IITM) is an autonomous institute
fully funded by the Ministry of Earth Sciences, Government of India. The
authors S Chowdhuri and T.V.Prabha gratefully acknowledge the Director, IITM,
Dr. Kalmár-Nagy, and Mr. Siddharth Kumar for the constant encouragement and
stimulating discussions during the research. The author T. Banerjee
acknowledges the funding support from the University of California Laboratory
Fees Research Program funded by the UC Office of the President (UCOP), grant
ID LFR-20-653572. Additional support was provided by the new faculty start up
grant provided by the Department of Civil and Environmental Engineering, and
the Henry Samueli School of Engineering, University of California, Irvine. The
authors are also immensely grateful to Dr. Keith G McNaughton for kindly
providing them the SLTEST dataset for this research. The computer codes used
in this study are available to all the researchers by contacting the
corresponding author.

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## Supplementary material

Figure S1: A typical example is shown for the original TA approximated time
series of (a) $u^{\prime}w^{\prime}$ and (b) $w^{\prime}T^{\prime}$ for a
highly-convective stability, corresponding to $-\zeta=10.6$. In both the
panels, the original $(x^{\prime}w^{\prime})_{\rm TA}$ (thick black lines) are
compared with the formulations $1-|x^{\prime}_{\rm TA}-w^{\prime}_{\rm TA}|$
(dashed red lines) and $x^{\prime}_{\rm TA}\times w^{\prime}_{\rm TA}$ (dashed
blue lines), where $x$ can be either $u$ or $T$. It can be noted that, the
expression $1-|x^{\prime}_{\rm TA}-w^{\prime}_{\rm TA}|$ captures all the
signature of $(x^{\prime}w^{\prime})_{\rm TA}$ perfectly well. Figure S2: An
example is shown to illustrate the concepts of (a) $\hat{u}$-$\hat{w}$ and (b)
$\hat{T}$-$\hat{w}$ quadrant planes. Note that, the symbol
$\hat{x}=x^{\prime}/\sigma_{x}$, where $x$ can be $u$, $w$, or $T$. The points
shown in black belong to a 30-min run from highly-convective stability,
corresponding to $-\zeta=10.6$. The four different quadrants for each of the
planes are marked in the respective panels.