# Persistence analysis of velocity and temperature fluctuations in convective
surface layer turbulence

Subharthi Chowdhuri
Indian Institute of Tropical Meteorology
Ministry of Earth Sciences, India
<EMAIL_ADDRESS>
&Tamás Kalmár-Nagy
Department of Fluid Mechanics, Faculty of Mechanical Engineering
Budapest University of Technology and Economics, Budapest, Hungary
<EMAIL_ADDRESS>
&Tirtha Banerjee
Department of Civil and Environmental Engineering
University of California, Irvine, CA 92697, USA
<EMAIL_ADDRESS>

###### Abstract

Persistence is defined as the probability that the local value of a
fluctuating field remains at a particular state for a certain amount of time,
before being switched to another state. The concept of persistence has been
found to have many diverse practical applications, ranging from non-
equilibrium statistical mechanics to financial dynamics to distribution of
time scales in turbulent flows and many more. In this study, we carry out a
detailed analysis of the statistical characteristics of the persistence
probability density functions (PDFs) of velocity and temperature fluctuations
in the surface layer of a convective boundary layer, using a field-
experimental dataset. Our results demonstrate that for the time scales smaller
than the integral scales, the persistence PDFs of turbulent velocity and
temperature fluctuations display a clear power-law behaviour, associated with
self-similar eddy cascading mechanism. Moreover, we also show that the effects
of non-Gaussian temperature fluctuations act only at those scales which are
larger than the integral scales, where the persistence PDFs deviate from the
power-law and drop exponentially. Furthermore, the mean time scales of the
negative temperature fluctuation events persisting longer than the integral
scales are found to be approximately equal to twice the integral scale in
highly convective conditions. However, with stability this mean time scale
gradually decreases to almost being equal to the integral scale in the near
neutral conditions. Contrarily, for the long positive temperature fluctuation
events, the mean time scales remain roughly equal to the integral scales,
irrespective of stability.

_K_ eywords Convective turbulence $\cdot$ Integral scale $\cdot$ Persistence
$\cdot$ Power-law $\cdot$ Probability density function

##  1 Introduction

Let $f(t)$ denote a stochastic signal fluctuating in time governed by a
particular dynamics. The persistence is then the probability $P(t)$ that the
quantity $f(t)-\overline{f(t)}$ does not change sign up to the time $t$
(Majumdar,, 1999; Bray et al.,, 2013), where the overbar denotes the time
average. Despite its simple description, only for some specific systems, such
as those exhibiting fractional Brownian motions, the persistence probability
density functions (PDFs) could be analytically shown to decay as a power-law,
$P(t)\propto t^{-(1-H)}$ (Majumdar,, 1999; Molchan,, 1999; Aurzada and
Baumgarten,, 2013; Aurzada and Simon,, 2015). Here $H$ is the Hurst exponent
($0<H<1$), whose value when $0.5$ indicates simple Brownian motion. The power
law form of $P(t)$ dictates that as the $H$ values get larger the persistence
PDFs decrease more slowly, which seems consistent with the general notion that
a stochastic signal displays anti-persistent or persistent behaviour depending
on whether $0<H<1/2$ or $1/2<H<1$ (Wang et al.,, 2000). However, for other
complex systems, no theoretical solutions exist for the persistence PDFs and
these need to be computed empirically from the experimental data at hand
(Perlekar et al.,, 2011). Notwithstanding the theoretical challenges, the
concept of persistence has many practical applications, such as in the field
of biology where one can ask how long does it take for an epidemic to spread
(Sornette,, 2006), or in financial markets to assess when does a preferred
stock will cross a threshold price (Zheng,, 2002), or in the field of
geophysics to predict when will the next earthquake have a dangerously high
magnitude (Corral,, 2004). Note that, depending on the context, the
persistence could also be referred to as distributions of the first-passage
time, or survival probability distributions, or return-time distributions, or
the distributions of the inter-arrival times between the successive zero-
crossings (Grebenkov et al.,, 2020).

In turbulent flows, the interest in the concept of persistence or zero-
crossings grew with the analytical result from Rice, (1945), through which it
was possible to show that the frequency of the zero-crossings in a turbulent
signal was related to the Taylor microscale (Sreenivasan et al.,, 1983). This
connection was intriguing, because it implied the dissipation rate of the
turbulent kinetic energy could be directly computed from the zero-crossing
frequencies. Since then, several studies are carried out in wall-bounded
turbulent flows to verify this result and the agreements obtained with the
theoretical prediction are more or less satisfactory (Antonia et al.,, 1976;
Narayanan et al.,, 1977; Sreenivasan et al.,, 1983; Poggi and Katul,, 2010).
However, there has been a fair amount of disagreement among different
experiments regarding the statistical characteristics of the PDFs of the
inter-arrival times between the successive zero-crossings (hereafter, the
persistence PDFs). Narayanan et al., (1977) found that in a turbulent boundary
layer, the persistence PDFs of the velocity fluctuations were log-normal to a
good approximation. Later, Sreenivasan et al., (1983) and Kailasnath and
Sreenivasan, (1993) found the persistence PDFs of the velocity fluctuations
and momentum flux signals were double-exponential in nature. Their
interpretation of this behaviour was that the long intervals are a consequence
of the large-scale structures passing the sensor and the short intervals are a
consequence of the impinging small-scale motions superposed on the large-scale
structures. Subsequently, Bershadskii et al., (2004) showed that the
persistence PDFs of temperature fluctuations from a turbulent convection
experiment followed a power-law distribution, which indicates scale-free
behavior. In a follow up study, Sreenivasan and Bershadskii, (2006) commented
that when the temperature behaved like an active scalar in convective
turbulence, the persistence PDFs followed a power-law distribution. On the
other hand, when the temperature behaved like a passive scalar in shear-driven
turbulence, the persistence PDFs followed a log-normal distribution. Recently,
Kalmár-Nagy and Varga, (2019) noted that the persistence PDFs of velocity
fluctuations followed log-normal distribution in a turbulent flow around a
street canyon.

In atmospheric turbulence the investigation of the statistical properties of
the persistence PDFs of turbulent fluctuations is quite rare. Nevertheless,
there are a few limited studies available from the atmospheric surface layer
which report the persistence PDFs of velocity and scalar fluctuations (Yee et
al.,, 1993, 1995; Cava and Katul,, 2009; Cava et al.,, 2012). The atmospheric
surface layer is a generalization of the inertial layer of unstratified wall-
bounded flows by including the effect of buoyancy, where the effects of
surface roughness are no longer important and the modulations by the outer
eddies are not too strong (Barenblatt,, 1996; Davidson,, 2015). Yee et al.,
(1993, 1995) reported that the persistence PDFs of scalar concentrations
displayed a double power-law in a near-neutral surface layer. Later, Katul et
al., (1994) also observed the same, when they investigated the persistence
PDFs of the burst events in the sensible heat flux in a convective surface
layer. Pinto et al., (2014) have shown that the double power-law feature in a
distribution function is related to the presence of two sets of fractals with
two different fractal dimensions associated with two different scale free
processes. However, Narasimha et al., (2007) found the persistence PDFs of the
momentum flux events in a near-neutral surface layer followed an exponential
distribution, suggestive of a Poisson type process. Recently, Cava and Katul,
(2009) demonstrated that the persistence PDFs of the velocity and scalar
fluctuations in a canopy surface layer turbulence could be power-laws with
log-normal cut-offs or log-normal distributions depending on the measurement
height in the canopy. Although in due course, Cava et al., (2012) showed that
the persistence PDFs of the velocity and scalar fluctuations in canopy and
atmospheric surface layer turbulence could be modelled as a power-law
distribution with an exponential cut-off in convective conditions. Chamecki,
(2013) has lent support for this model by investigating the persistence PDFs
of velocity fluctuations above and within a cornfield canopy.

Therefore, from this brief review, it is apparent that there is a very little
consensus about the statistical characteristics of the persistence PDFs for
both laboratory and atmospheric turbulent flows. Nevertheless, in a convective
atmospheric surface layer, detailed understanding of the persistence
properties of velocity and temperature fluctuations is important since it
holds the key to explain the quadrant cycles of the heat and momentum fluxes.
This is because, the switching patterns of the heat and momentum fluxes from
one quadrant to the other are dependent on the zero-crossings of the component
signals, as described by their persistence PDFs. Thus, we define our
objectives for this study as:

  1. 1.

To carry out a detailed analysis to establish the statistical scaling
properties of the persistence PDFs of velocity and temperature fluctuations in
a convective surface layer.

  2. 2.

To empirically connect the statistical scaling properties of the persistence
PDFs with the turbulent structures in a convective surface layer.

The present study is organized in three different sections. In Section 2, we
provide the descriptions of the field-experimental dataset and methodology
used in this study, in Section 3 we present and discuss the results, and
lastly in Section 4 we conclude by presenting our findings and providing the
future research direction.

##  2 Materials and methods

In this study, the dataset being used is from the Surface Layer Turbulence and
Environmental Science Test (SLTEST) experiment. The SLTEST experiment ran
continuously for nine days from 26 May 2005 to 03 June 2005, over a flat and
homogeneous terrain at the Great Salt Lake desert in Utah, USA (40.14∘ N,
113.5∘ W), with the aerodynamic roughness length ($z_{0}$) being
$z_{0}\approx$ 5 mm (Metzger et al.,, 2007). During this experiment, nine
North-facing time synchronized CSAT3 sonic anemometers were mounted on a 30-m
mast, spaced logarithmically over an 18-fold range of heights, from 1.42 m to
25.7 m, with the sampling frequency being set at 20 Hz.

During the daytime convective periods, the standard practice is to compute the
turbulent statistics in the atmospheric surface layer over a 30-min period
(Panosfsky and Dutton,, 1984; Kaimal and Finnigan,, 1994; Foken and Napo,,
2008). Therefore, the data from all the nine sonic anemometers were restricted
to rain-free daytime convective periods and subsequently being divided into
30-min sub-periods, containing the 20-Hz measurements of the three wind
components and the sonic temperature (Chowdhuri et al., 2019b, ). To select
the 30-min periods for the persistence analysis, we followed the detailed data
selection methods as outlined in Chowdhuri et al., 2019a . Note that, we
rotated the coordinate systems of all the nine sonic anemometers in the
streamwise direction by applying the double-rotation method of Kaimal and
Finnigan, (1994) for each 30-min period.

A total of 261 combinations of the stability ratios ($-\zeta=z/L$, where $L$
is the Obukhov length) were possible for the selected 30-min periods from the
convective conditions ($-L>0$). The stability ratio $-z/L$ is the ratio
between the turbulent kinetic energy generated due to buoyancy and due to
shear, with the Obukhov length ($L$) being defined as,

$L=-\frac{u_{*}^{3}T_{0}}{k_{v}gH_{0}},$ (1)

where $T_{0}$ is the surface air temperature, $g$ is the acceleration due to
gravity (9.8 m s-2), $H_{0}$ is the surface kinematic heat flux, $k_{v}$ is
the von Kármán constant (0.4), and $u_{*}$ is the friction velocity. It is to
be noted that, these are the same set of runs used by Chowdhuri et al., 2019a
for their study of turbulence anisotropy. The entire range of $-\zeta$ (12
$\leq-\zeta\leq$ 0.07) was divided into six stability classes and considered
for the persistence analysis (see table 1). For each 30-min run, the turbulent
fluctuations of the three velocity components in the streamwise
($u^{\prime}$), cross-stream ($v^{\prime}$), and vertical ($w^{\prime}$)
directions, along with the fluctuations in the sonic temperature
($T^{\prime}$) were computed by removing the linear trend from the 30-min
period associated with the respective variables (Donateo et al.,, 2017).

Table 1: The six different stability classes formed from the ratio $-\zeta=z/L$ in an unstable atmospheric surface layer flow, where $z$ is the height above the surface and $L$ is the Obukhov length. The ratios span from highly convective ($-\zeta>2$) to near neutral ($0<-\zeta<0.2$) conditions. The number of 30-min runs and associated heights with each of the stability classes are given. The total number of zero-crossings (# ZC) in $u^{\prime}$, $v^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals associated with each stability class are also provided. Stability class | Number of | Heights | $u^{\prime}$ | $v^{\prime}$ | $w^{\prime}$ | $T^{\prime}$
---|---|---|---|---|---|---
30-min runs |  | (# ZC) | (# ZC) | (# ZC) | (# ZC)
$-\zeta\ >$ 2 | 55 |  $z=$ 6.1, 8.7, 12.5, 17.9, 25.7 m | 116459 | 86576 | 142751 | 127633
1 $<\ -\zeta\ <$ 2 | 53 |  $z=$ 3, 4.3, 6.1, 8.7, 12.5, 17.9, 25.7 m | 119511 | 92236 | 189131 | 138785
0.6 $<\ -\zeta\ <$ 1 | 41 |  $z=$ 2.1, 3, 4.3, 6.1, 8.7, 12.5, 17.9 m | 97575 | 73884 | 184652 | 115173
0.4 $<\ -\zeta\ <$ 0.6 | 34 |  $z=$ 1.4, 2.1, 3, 4.3, 6.1, 8.7 m | 90551 | 70957 | 193302 | 107356
0.2 $<\ -\zeta\ <$ 0.4 | 44 |  $z=$ 1.4, 2.1, 3, 4.3, 6.1 m | 128538 | 98123 | 293816 | 154774
0 $<\ -\zeta\ <$ 0.2 | 34 |  $z=$ 1.4, 2.1, 3 m | 114383 | 95228 | 285961 | 143470

A graphical illustration of the persistence phenomenon is provided in figure
1, where a 120-s long section of $u^{\prime}$ signal is shown for a particular
30-min run, corresponding to a $-\zeta=10.6$. Figure 1 shows that the
$u^{\prime}$ signal displays persistent positive or negative (i.e. above or
below the mean) values for a particular amount of time, denoted as $t_{p}$.
Note that, the persistence time $t_{p}$ can also be interpreted as the inter-
arrival time between the subsequent zero-crossings where the $u^{\prime}$
signal changes its sign. The zero-crossings are identified by using the
telegraphic approximation (TA) of the $u^{\prime}$ signal as,

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

and locating the points where the TA series changes its value from 0 to 1 or
vice-versa (see figure 1b). The associated probability that the $u^{\prime}$
signal stays positive or negative for $t_{p}$ amount of time can be evaluated
by constructing the persistence PDFs using standard statistical procedures
(see Appendix A). Moreover, the same concept can also be extended to other
turbulent fluctuations such as $v^{\prime}$, $w^{\prime}$, and $T^{\prime}$.

For any stability class as outlined in table 1, the number of persistent
events (corresponding to each $t_{p}$ values) in $u^{\prime}$, $v^{\prime}$,
$w^{\prime}$, and $T^{\prime}$ signals can be represented by the number of
zero-crossings. In table 1, we provide the number of zero-crossings or
persistent events by considering all the 30-min runs from a particular
stability class. Typically, we encounter in the order of 105 number of zero-
crossings for each signal from every stability class (see table 1). Therefore,
the persistence PDFs of velocity and temperature fluctuations for each of
these six stability classes are constructed over these large number of
ensemble events, to ensure their statistical robustness. In the subsequent
section, we discuss the properties of these persistence PDFs, corresponding to
these six stability classes.

Figure 1: A 120-s long section of a time series of $u^{\prime}$ from a highly
convective surface layer corresponding to a $-\zeta=$ 10.6 are shown for (a)
actual values and (b) its telegraphic approximation (TA), where $u^{\prime}>0$
is denoted as 1 and $u^{\prime}<0$ is denoted as 0. The red horizontal line
denotes the position of zero and the red-crosses show the points where the
$u^{\prime}$ signal changes its sign from positive to negative or vice-versa
(zero-crossings). To provide an example, two particular regions of
$u^{\prime}$ signal are highlighted where the positive and negative values
persist for a time $t_{p}$ (around 30-50 s).

##  3 Results and discussion

Before describing the features of the persistence PDFs, it is worthwhile to
discuss about the presentation of the results. Earlier studies by Narayanan et
al., (1977), Sreenivasan et al., (1983), and Kailasnath and Sreenivasan,
(1993) on turbulent boundary layer flows have presented the results on
persistence time scales after converting those to spatial length scales by
employing Taylor’s frozen turbulence hypothesis. It is important to note that
although they presented results on the inter-arrival times between the two
successive zero-crossings of the turbulent signals, their convention is
equivalent to persistence time $t_{p}$ (see figure 1). They interpreted this
spatial length scale as a representative of the eddy length scale of the flow,
given the connection between the mean value of the zero-crossings with the
Taylor microscale (Antonia et al.,, 1976; Sreenivasan et al.,, 1983;
Kailasnath and Sreenivasan,, 1993).

In the convective surface layer turbulence, it is a common practice to
normalize the spatial length scales in the streamwise direction by the height
above the surface (Kaimal et al.,, 1972; Kader et al.,, 1989; Kader and
Yaglom,, 1991; McNaughton and Laubach,, 2000; Banerjee and Katul,, 2013;
Ghannam et al.,, 2018). This is related to the assumption that the turbulent
structures in a convective surface layer are self-smilar with height (Monin
and Yaglom,, 1971; Tennekes and Lumley,, 1972; Kader and Yaglom,, 1991; Malhi
et al.,, 2004; McNaughton, 2004a, ; McNaughton, 2004b, ). Thus, before
computing the persistence PDFs, it seems apropos to convert the persistence
time $t_{p}$ to a streamwise length ($t_{p}\overline{u}$, where $\overline{u}$
is the mean horizontal wind speed) by applying Taylor’s frozen turbulence
hypothesis and subsequently normalizing the same with $z$. Note that, for all
our selected runs $\sigma_{u}/\overline{u}$ (where $\sigma_{u}$ is the
standard deviation of the streamwise velocity) was less than 0.2, so the
Taylor’s frozen turbulence hypothesis could be assumed to be valid (Willis and
Deardorff,, 1976). The normalized variable $(t_{p}\overline{u})/z$ has a large
range, given the minimum of $t_{p}$ is restricted by the sampling interval of
0.05 s (20-Hz sampling frequency) and the maximum of $t_{p}$ could go as large
as in the order of 102 s (see figure 1). A well-suited strategy to evaluate
the PDFs of such variables is to take the logarithmic transformation and then
binning the transformed variables in the logarithmic space (Christensen and
Moloney,, 2005; Newman,, 2005; Pueyo,, 2006; Sims et al.,, 2007; Benhamou,,
2007; White et al.,, 2008; Dorval,, 2011; Newberry and Savage,, 2019). More
details about the effect of binning strategy on the persistence PDFs is
provided in the Appendix A.

Therefore, we begin with discussing the properties of the persistence PDFs of
velocity and temperature fluctuations in a convective surface layer by
presenting the empirical results scaled with $z$. Subsequently, to develop a
physical understanding of these persistence PDFs, we present comprehensive
evidence to underpin their scaling properties and relate them with the
turbulent flow structures.

###  3.1 Persistence PDFs of velocity and temperature fluctuations

In figure 2, we show the persistence PDFs of velocity ($u^{\prime}$ and
$w^{\prime}$) and temperature fluctuation ($T^{\prime}$) signals, plotted
against $(t_{p}\overline{u})/z$ for the six different stability classes
outlined in table 1. These persistence PDFs ($P[(t_{p}\overline{u})/z]$) are
computed after logarithmically binning the ensemble of values of
$(t_{p}\overline{u})/z$ from a particular stability class and then using (16)
to take into account the effect of the variable bin-width owing to the
logarithmic transformation (see Appendix A). The log-log representation is
used in figure 2, such that the power-law functions on such plots would be
represented by straight line segments.

Similar to Bershadskii et al., (2004) and Chamecki, (2013), the persistence
PDFs are computed separately for the positive and negative fluctuations (shown
as blue and red coloured markers in figure 2) for $u^{\prime}$, $w^{\prime}$,
and $T^{\prime}$ signals, and compared with the persistence PDFs for the total
fluctuations (combining both positive and negative) shown in grey coloured
markers. Note that, we focus on the $u^{\prime}$, $w^{\prime}$, and
$T^{\prime}$ signals since unravelling the characteristics of their
persistence PDFs have implication towards understanding the genesis of
ejection and sweep cycles in the streamwise momentum ($u^{\prime}w^{\prime}$)
and heat flux ($w^{\prime}T^{\prime}$) signals. Nevertheless, the persistence
PDFs of the $v^{\prime}$ signals are shown in figure S1 of the supplementary
material and they display similar characteristic shape as the $u^{\prime}$
signals for all the six stability classes.

Figure 2: The persistence PDFs of the normalized streamwise sizes
$(t_{p}\overline{u})/z$ corresponding to the positive (red) and negative
(blue) fluctuations in the streamwise velocity ($u^{\prime}$, inverted
triangles), temperature ($T^{\prime}$, circles), and vertical velocity
($w^{\prime}$, squares) are shown separately for the six different stability
classes as indicated in the right most corner. The grey coloured inverted
triangle, circle, and square markers on all the panels show the persistence
PDFs computed after considering both the positive and negative fluctuations
together. The PDFs of $w^{\prime}$, $T^{\prime}$, and $u^{\prime}$ are shifted
vertically by two decades for visualization purpose and the legends
corresponding to each panel are provided at the right-most bottom corner. The
thick black lines on all the panels show the power-laws with their respective
slopes being mentioned on the panel (a). The dashed black line denotes the
value $(t_{p}\overline{u})/z=$ 1.

From figure 2a, one can notice that under highly convective stratification
regimes, an extensive straight line segment is observed in the persistence PDF
of $u^{\prime}$ signal, with a cutoff at $(t_{p}\overline{u})/z\approx 10$.
Moreover, the persistence PDFs of $T^{\prime}$ and $w^{\prime}$ signals also
display a straight line segment for the small values of
$(t_{p}\overline{u})/z$, although their extent remains shorter than the
$u^{\prime}$ signal, with cutoffs at $(t_{p}\overline{u})/z\approx 4$ and
$(t_{p}\overline{u})/z\approx 1$ respectively (figure 2a). In the log-log
representation, a straight line segment is reminiscent of a power-law
function, thus indicating the persistence PDFs of $u^{\prime}$, $w^{\prime}$,
and $T^{\prime}$ signals behave in a power-law fashion up to a certain
threshold of $(t_{p}\overline{u})/z$. Note that, the exponents of these power-
law functions in figure 2a are different for the $u^{\prime}$, $T^{\prime}$
and $w^{\prime}$ signals, being equal to $1.6$, $1.4$, and $1.25$
respectively. These exponents have been estimated from a linear regression on
the log-log plots in figure 7, which we will revisit in Section 3.3 while
discussing the physical significance of this power-law behaviour. Since power-
laws are scale invariant, a different scaling of the persistence time $t_{p}$
as employed in figure 8 would not affect their exponents.

To judge the effect of stability on the exponents of these power-law
functions, we compared the exponents obtained from the highly convective
stability (figure 2a) with the other five stability classes (figures 2b to
2f). We note that for the $w^{\prime}$ signal, with the change in stability,
the extent of this power-law region gradually shrinks, becoming almost non-
existent in the near neutral stability (figures 2a to 2f). On the contrary, no
discernible change is being observed in the power-law behaviour of the
$u^{\prime}$ signals with stability. However, for the $T^{\prime}$ signals
there is a modest change in the extent of the power-law region with stability,
although not as clearly visible as in the $w^{\prime}$ signal. Along with
that, the threshold streamwise sizes up to which the power-law behaviour holds
are also not similar for these three signals. For the $u^{\prime}$ signal, at
streamwise sizes greater than about $10z$, the departure from the power law is
noted. Whereas for the $w^{\prime}$ signal, the threshold streamwise size is
much closer to $z$. On the other hand, for the $T^{\prime}$ signal the
threshold streamwise size is approximately in between of the sizes
corresponding to $u^{\prime}$ and $w^{\prime}$ signals.

Additionally, we note that for the $u^{\prime}$ and $w^{\prime}$ signals, the
persistence PDFs of positive and negative fluctuations almost coincide with
each other for all the values of $(t_{p}\overline{u})/z$. Nevertheless, the
same is not true for the $T^{\prime}$ signal. It could be noticed that for the
highly convective stability class ($-\zeta>2$, figure 2a) the persistence PDFs
show a slight disparity between the positive and negative $T^{\prime}$ signals
at the larger values of $(t_{p}\overline{u})/z$. However, this difference
gradually disappears with the change in stability from highly convective to
near neutral (figures 2a to 2f). Therefore, one may ask _for highly convective
stability, what causes this discrepancy between the persistence PDFs
associated with positive and negative temperature fluctuations?_

####  3.1.1 Linkage between the persistence PDFs and asymmetric distribution

To investigate the aforementioned question, it is useful to consider a
premultiplied form of the persistence PDFs such as,
$(t_{p}\overline{u})/z\times P[(t_{p}\overline{u})/z]$. From phenomenological
arguments, we will show that such form of the persistence PDF is equivalent to
a time-fraction distribution associated with $(t_{p}\overline{u})/z$, by
directly connecting the premultiplied persistence PDF with the PDF of the
corresponding signal itself. We emphasize that such a connection is not
possible to establish from the persistence PDFs alone (shown in figure 2),
without considering its premultiplied form.

Therefore, to prove such linkage, let us consider we have an $N$-member
ensemble of $(t_{p}\overline{u})/z$ values corresponding to a signal in a
particular state. Note that, in the present context, there are only two
possible states where the signal could either be above or below the mean
(positive or negative fluctuations). Since the persistence time is computed
based on the number of points lying between successive zero-crossings
multiplied by the sampling interval (see figure 1), we can write,

$\sum\limits_{i=1}^{Q}{[(t_{p}\overline{u})/z]_{i}n_{i}}\propto T_{f},$ (3)

where $Q$ is the number of unique members in the $N$-member ensemble of
$(t_{p}\overline{u})/z$, $n_{i}$ is the frequency of occurrences of these
unique members, and $T_{f}$ is the total time-fraction spent by the signal in
that particular state. The right hand side of (3) stems from the fact that the
left hand side of the same equation can be rearranged as,

$\displaystyle\begin{split}\sum\limits_{i=1}^{Q}{[(t_{p}\overline{u})/z]_{i}n_{i}}&=N\sum\limits_{i=1}^{Q}{[(t_{p}\overline{u})/z]_{i}P[(t_{p}\overline{u})/z]}\\\
&=N\overline{(t_{p}\overline{u})/z},\end{split}$ (4)

where $P[(t_{p}\overline{u})/z]=(n_{i}/N)$ is the probability of selecting a
unique member from the $N$-member ensemble. Given the sum of all the possible
$t_{p}$ values is equal to the total time spent by the signal in a particular
state and $N$, $\overline{u}$, and $z$ are constants for a particular period,
the quantity $N\overline{(t_{p}\overline{u})/z}$ in (4) is equivalent to
$T_{f}$ with some proportionality constants. Thus, from (3) and (4), we can
rewrite (3) in the integral form as,

$\int_{(\frac{t_{p}\overline{u}}{z})_{\rm
min}}^{(\frac{t_{p}\overline{u}}{z})_{\rm max}}(\frac{t_{p}\overline{u}}{z})\
P[(\frac{t_{p}\overline{u}}{z})]\ d{(\frac{t_{p}\overline{u}}{z})}\propto
T_{f}.$ (5)

Since (5) holds true for both positive and negative fluctuations, we may
rewrite (5) as,

$\int_{(\frac{t_{p}\overline{u}}{z})_{\rm
min}}^{(\frac{t_{p}\overline{u}}{z})_{\rm
max}}\Big{(}\Big{[}\frac{t_{p}\overline{u}}{z}P(\frac{t_{p}\overline{u}}{z})\Big{]}_{\rm
N}-\Big{[}\frac{t_{p}\overline{u}}{z}P(\frac{t_{p}\overline{u}}{z})\Big{]}_{\rm
P}\Big{)}\ d{(\frac{t_{p}\overline{u}}{z})}\propto\Delta T_{f},$ (6)

where the subscripts P and N refer to the positive and negative fluctuations
and $\Delta T_{f}$ is the difference in the time fractions associated with
those. For a particular signal, the difference in time fractions associated
with positive and negative fluctuations can be estimated from the PDF of that
signal as,

$\Big{[}\int_{-\infty}^{0}p(\hat{x})d\hat{x}-\int_{0}^{\infty}p(\hat{x})d\hat{x}\Big{]}=\Delta
T_{f},$ (7)

where $p(\hat{x})$ is the PDF of the signal $x$ normalized as
$\hat{x}=x^{\prime}/\sigma_{x}$, such that $\hat{x}$ is a standard variable
with zero mean and unit standard deviation. Through cumulant expansion of
Nakagawa and Nezu, (1977), $p(\hat{x})$ can be expressed as,

$\displaystyle\begin{split}&p(\hat{x})=G(\hat{x})\Big{[}1+\frac{\overline{{\hat{x}}^{3}}}{6}({\hat{x}}^{3}-3\hat{x})+\frac{1}{24}({\overline{{\hat{x}}^{4}}}-3)({\hat{x}}^{4}-6{\hat{x}}^{2}+3)\Big{]}\\\
&\quad\hskip
56.9055ptG(\hat{x})=\frac{1}{\sqrt{2\pi}}\exp(-\frac{{\hat{x}}^{2}}{2}),\end{split}$
(8)

where $G(\hat{x})$ is the standard Gaussian distribution and
$\overline{{\hat{x}}^{3}}$ and $\overline{{\hat{x}}^{4}}$ are the skewness and
kurtosis of the signal $x$. Katul et al., 1997a  showed that if the kurtosis
is ignored in (8), then the left hand side of (7) can be analytically
evaluated as,

$\displaystyle\begin{split}&\int_{-\infty}^{0}p(\hat{x})d\hat{x}=\frac{6+(\sqrt{2/\pi})\
\overline{{\hat{x}}^{3}}}{12}\\\
&\quad\quad\int_{0}^{\infty}p(\hat{x})d\hat{x}=\frac{6-(\sqrt{2/\pi})\
\overline{{\hat{x}}^{3}}}{12}.\end{split}$ (9)

By replacing the result from (9) in (7), we can write (6) as,

$\int_{(\frac{t_{p}\overline{u}}{z})_{\rm
min}}^{(\frac{t_{p}\overline{u}}{z})_{\rm
max}}\Big{(}\Big{[}\frac{t_{p}\overline{u}}{z}P(\frac{t_{p}\overline{u}}{z})\Big{]}_{\rm
N}-\Big{[}\frac{t_{p}\overline{u}}{z}P(\frac{t_{p}\overline{u}}{z})\Big{]}_{\rm
P}\Big{)}\
d{(\frac{t_{p}\overline{u}}{z})}\propto\frac{\overline{{\hat{x}}^{3}}}{3\sqrt{2\pi}}.$
(10)

Hence, from (6)–(10) we conclude that the difference between the positive and
negative fluctuations in the premultiplied form of the persistence PDFs is
directly related to the non-Gaussian characteristics of the signal,
approximated by its skewness. In addition to this, we also note that by
performing a change of variables, Dorval, (2011) has shown that the
premultiplied PDF of a stochastic variable $x$ is equivalent to the PDF of
$\log(x)$, known as the logarithmic PDF (see Appendix A for further details).

####  3.1.2 Premultiplied or logarithmic persistence PDFs of velocity and
temperature fluctuations

Figure 3 shows the logarithmic PDFs of $(t_{p}\overline{u})/z$ for the same
six different stability classes shown in figure 2. These PDFs are computed
after taking the logarithm of $(t_{p}\overline{u})/z$ in the base 10 and
dividing the fraction of samples by the logarithmic bin-width
$d\log_{10}[(t_{p}\overline{u})/z]$. Similar to figure 2, in figure 3 the
logarithmic persistence PDFs are shown separately for the positive and
negative fluctuations and compared with the total fluctuations (combining both
positive and negative), corresponding to the $u^{\prime}$, $w^{\prime}$, and
$T^{\prime}$ signals. Note that, owing to premultiplication, the power-law
sections of the premultiplied PDFs are comparatively flattened, whereas an
exaggeration occurs at the larger values of $(t_{p}\overline{u})/z$ (for a
clear demonstration see figure 8 in Appendix A).

Figure 3: Same as in figure 2 but for the logarithmic persistence PDFs of the
normalized streamwise sizes $(t_{p}\overline{u})/z$. The logarithmic PDFs of
$w^{\prime}$, $T^{\prime}$, and $u^{\prime}$ are shifted vertically by a
decade for visualization purpose and the legends corresponding to each panel
are provided at the right-most bottom corner. The coloured arrows on all the
panels show the position of the normalized integral length scales,
corresponding to $w^{\prime}$ (blue arrows), $T^{\prime}$ (pink arrows), and
$u^{\prime}$ (red arrows) signals.

From figure 3a, we notice that in highly convective stability the
premultiplied PDFs of the positive and negative $T^{\prime}$ signals are
clearly separated at larger values of $(t_{p}\overline{u})/z>4$ (shown by the
divergence between the red and blue circles in figure 3a). However, they agree
with each other at the smaller values of $(t_{p}\overline{u})/z$. Using (10),
we can conclude that this difference between positive and negative
$T^{\prime}$ signals at the large values of $(t_{p}\overline{u})/z$ is related
to the non-Gaussian characteristics of the temperature fluctuations, expressed
by its skewness. Chowdhuri et al., 2019a  noted that the skewness and kurtosis
of the temperature fluctuations are strongly non-Gaussian ($\approx$ 1.5 and 5
respectively) in the local free-convection limit ($-\zeta>$ 1). This
observation is in agreement with the previous studies in the convective
surface layer (Chu et al.,, 1996; Liu et al.,, 2011; Garai and Kleissl,, 2013;
Lyu et al.,, 2018). Our results show that this non-Gaussianity in the
$T^{\prime}$ signal is only realised at large sizes of persistent temperature
patterns, approximately greater than 4 times the measurement height.

It is also interesting to note that, the total time fractions at these large
sizes are governed by the negative fluctuation patterns alone (figure 3a).
Katul et al., 1997a  have shown that the difference in the PDFs between the
positive and negative $T^{\prime}$ signals could be attributed to the
asymmetry between the warm-updraft and cold-downdraft motions, under the
assumption that the contributions from the counter-gradient quadrants could be
ignored. Therefore, this finding is consistent with Adrian et al., (1986),
where they showed from laboratory experiments that in highly-unstable
conditions the temperature fluctuation patterns are governed by the more
frequent cold-downdrafts bringing well-mixed air from aloft, interspersed with
intermittent occurrences of the warm-updraft motions rising from the ground.

However, this clear disparity between the premultiplied PDFs of positive and
negative $T^{\prime}$ signals, gradually disappears as the near neutral
stability is approached (figures 3a to 3f). This is congruous with the close
to Gaussian characteristics of the $T^{\prime}$ signal in the near neutral
stability (Chowdhuri et al.,, 2015; Chowdhuri and Prabha,, 2019). Apart from
that, for $u^{\prime}$ and $w^{\prime}$ signals no difference is observed
between the premultiplied PDFs of positive and negative fluctuations,
irrespective of the stability classes. The reason for this is the PDFs of
$u^{\prime}$ and $w^{\prime}$ signals remain very nearly Gaussian for all the
stability conditions in an atmospheric surface layer (Chu et al.,, 1996;
Chowdhuri et al., 2019a, ).

In summary, for convective surface layer turbulence the persistence PDFs of
the velocity and temperature fluctuations follow a power-law function up to a
certain threshold size. This threshold size and the power-law exponents are
not similar for the $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals. For
the $u^{\prime}$ signals the threshold size is an order of magnitude larger
than $z$, whereas for the $w^{\prime}$ signals it is approximately equal to
$z$. On the other hand, the threshold size for the $T^{\prime}$ signals is
somewhere in between of $u^{\prime}$ and $w^{\prime}$ signals. It is also
remarkable to note that in a convective surface layer the non-Gaussian effects
in the $T^{\prime}$ signals are only perceived at those sizes which are larger
than this threshold size, where the power-law behaviour ceases to exist.
Therefore, it is imperative to ask in convective surface layer turbulence,

  1. 1.

What statistical properties of velocity and temperature fluctuations give rise
to the power-law behaviour punctuated with a cut-off in their persistence
PDFs?

  2. 2.

How are those statistical properties connected with the structures of the
turbulent flow?

In the subsequent sections, we will explore these questions by analysing
surrogate datasets and employing an alternate scaling of persistence time
scales of velocity and temperature fluctuations.

###  3.2 Scrutiny of persistence PDFs through surrogate data

The persistence PDFs are related to the inter-arrival time between the
successive zero-crossings of a time series. If these zero-crossings were
randomly located being independent of each other, we would have expected the
persistence PDFs to follow a Poisson distribution, which is exponential in
nature (Majumdar,, 1999; Santhanam and Kantz,, 2008). However, in a convective
surface layer the persistence PDFs of the turbulent velocity and temperature
fluctuations show a power-law structure. This indicates the zero-crossings of
these signals are not independent but correlated with each other. Moreover,
for the $T^{\prime}$ signals the power-law behaviour extends up to a certain
streamwise size ($\approx 4z$) beyond which the non-Gaussian effects dominate
the characteristics of the persistence PDFs.

Therefore, to gain more insight regarding the statistical characteristics of
the persistence PDFs of $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals
in convective turbulence, we generated two different types of surrogate
datasets, such as:

  1. 1.

The temporal correlation of the original time series is preserved by keeping
the auto-correlation function unchanged, but modifying the PDF of the time
series to be Gaussian.

  2. 2.

The temporal correlation of the original time series is destroyed by random
shuffling, but keeping the PDF of the time series unchanged.

The normal procedure to generate the first type of the surrogate dataset is
taking the Fourier transform of the time series and keeping the Fourier
amplitudes same but randomizing the associated Fourier phases, with inverse
Fourier transform being applied to the modified Fourier coefficients (Poggi
and Katul,, 2009; Lancaster et al.,, 2018). Since the Fourier amplitudes of
the surrogate dataset are kept intact during the process of Fourier phase-
randomization, this procedure preserves the Fourier spectrum and hence the
auto-correlation function of the time series. On the other hand, the second
type of the surrogate dataset is generated by randomly permuting (shuffling)
the original time series such that the temporal correlations are destroyed
completely.

####  3.2.1 Phase-randomization and randomization experiments

For our purpose, we performed the phase-randomization and randomization
experiments by varying the strength of the randomization, to investigate their
gradual effects on the behaviour of the persistence PDFs. A similar procedure
was suggested by Maiwald et al., (2008) to investigate the effect of the non-
linearity on a stochastic time series by increasing the strength of the phase-
randomization in a step-wise manner. In this context, the randomization
strength implies the percentage of the Fourier phases or the time series
values which have been shuffled randomly to generate the surrogate datasets
(see Appendix B for further details). Figure 4 shows the typical results from
these two experiments for the highly convective stability class. Note that,
similar results have been found for the other five stability classes as well,
which are not shown here. For the illustration purpose, the logarithmic
persistence PDFs of positive and negative $T^{\prime}$ signals are shown at
various randomization strengths from 0% (original time series) to 100%
(completely randomized) for phase-randomization (figure 4a) and randomization
experiments (figure 4b) respectively (see figures S2 and S3 in the
supplementary material for the effect on $u^{\prime}$ and $w^{\prime}$
signals). The logarithmic representation is chosen, since in this
representation the non-Gaussian characteristics of the $T^{\prime}$ signals
are better revealed while preserving the power-law structure of the
persistence PDFs (figures 3 and 8).

Figure 4: The logarithmic persistence PDFs of the normalized streamwise sizes
$(t_{p}\overline{u})/z$ corresponding to the positive (red) and negative
(blue) fluctuations in the temperature ($T^{\prime}$) signal from the highly
convective stability class ($-\zeta>$ 2) are shown, for two sets of
experiments such as: (a) Fourier phase-randomization (PR) and (b) temporal
randomization (R). The original logarithmic persistence PDFs are shown at the
bottom of both the panels and the rest are shifted vertically by three decades
where the original temperature signals are gradually randomized either in
their Fourier phases or temporal values, starting from 20 to 100%.

From figure 4a we note that the power-law structure of the $T^{\prime}$
persistence PDFs are preserved for all the randomization strengths of the
phase-randomization experiments, although the separation between the positive
and negative $T^{\prime}$ persistence PDFs become indistinguishable at 20%
randomization strength. This is related to the fact that in phase-
randomization experiments, the surrogate time series of temperature
fluctuations approach an almost Gaussian distribution at even 20%
randomization strength (see figure S4a in the supplementary material). On the
contrary, from figure 4b we notice that the power-law structure of the
$T^{\prime}$ persistence PDFs gradually disappear as the strength of the
randomization is increased to 100%. Nevertheless, in figure 4b the difference
between the positive and negative $T^{\prime}$ persistence PDFs remain
preserved, irrespective of the randomization strength. The reason for this is,
the non-Gaussian PDFs of the temperature fluctuations remain unchanged in the
randomization experiments (see figure S4b in the supplementary material).
Similar results are also obtained for the persistence PDFs of $u^{\prime}$ and
$w^{\prime}$ signals (see figures S2 and S3 in the supplementary material).

Figure 5: The auto-correlation functions are plotted against the normalized
time lags ($\tau\overline{u}/z$) for $u^{\prime}$ (red circles), $w^{\prime}$
(blue squares), and $T^{\prime}$ (pink inverted triangles) signals for the six
different stability classes as indicated at the right top most corner of each
panel. The exponential functions are fitted to the auto-correlation functions
to determine the integral length scale associated with $u^{\prime}$,
$w^{\prime}$, and $T^{\prime}$ signals. The legends shown at the right of each
panel describe the markers and the equations for the exponential fits.

To summarize the results from figure 4, we can conclude the power-law
behaviour of the persistence PDFs is related to the temporal coherence in the
time series. In a turbulent time series, the temporal coherence can be
described by the integral time scale, defined as the time up to which the
signal remains auto-correlated with itself (Tennekes and Lumley,, 1972; Kaimal
and Finnigan,, 1994; Wyngaard,, 2010).

####  3.2.2 Auto-correlation functions and integral scales

Figure 5 shows the auto-correlation functions ($R_{xx}(\tau)$, where $x$ can
be either $u^{\prime}$, $w^{\prime}$, or $T^{\prime}$ signals and $\tau$ is
the time lag) of the $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals,
plotted against the lags for the six different stability classes. Note that in
figure 5, the time lags are converted to a streamwise length
($\tau\overline{u}$) by using Taylor’s frozen turbulence hypothesis and
normalized with $z$. Apart from that, the auto-correlation functions are
ensemble averaged over the set of 30-min runs from a particular stability
class (see table 1) to ensure the results are statistically robust. These
auto-correlations can be represented with an exponentially decaying function
of the form,

$R_{xx}(\frac{\tau\overline{u}}{z})=\exp(-K\frac{\tau\overline{u}}{z})\implies\exp(-\frac{\ell_{x}}{\Lambda_{x}}),$
(11)

where $x$ can be either $u^{\prime}$, $w^{\prime}$, or $T^{\prime}$,
$\ell_{x}=\tau\overline{u}$, and $\Lambda_{x}=z/K$, known as the integral
length scale (Kaimal and Finnigan,, 1994).

From figure 5a, we note that the coefficients $K$ for the $T^{\prime}$ and
$w^{\prime}$ signals are closer to each other ($K=0.57$ and $0.79$
respectively), whereas the $K$ value for the $u^{\prime}$ signal remains much
lower ($K=0.15$). Using (11), this implies in the highly convective stability
the integral length scales of the $T^{\prime}$ and $w^{\prime}$ signals
($\Lambda_{x}=z/K$) almost coincide with each other, while the integral length
scale of the $u^{\prime}$ signal remains the largest. However, as the
stability changes from highly convective to near neutral the coefficients $K$
of the $T^{\prime}$ signals become closer to the $u^{\prime}$ signals,
implying the $\Lambda_{T}$ values approach $\Lambda_{u}$ (figure 5a to 5f). On
the other hand, the $\Lambda_{w}$ values remain approximately equal to $z$
irrespective of the stability classes ($K=0.79$ to $0.90$).

These results are in accord with the previous studies in surface layer
turbulence, where it has been noted that in highly convective stability the
$\Lambda_{T}$ values approach $\Lambda_{w}$, whereas in the near neutral
stability they are much closer to $\Lambda_{u}$ (Kader et al.,, 1989; Katul et
al., 1997b, ; Li et al.,, 2012). Apart from that, the approximate equality of
$\Lambda_{w}$ with $z$ supports the argument that the eddies which contribute
to the $w^{\prime}$ signal are attached to the ground (McNaughton et al.,,
2007; Banerjee and Katul,, 2013; Banerjee et al.,, 2015; Chowdhuri et al.,
2019b, ). If these integral length scales obtained from figure 5
($\Lambda_{u}$, $\Lambda_{w}$, and $\Lambda_{T}$) are replaced in the
persistence length scales ($t_{p}\overline{u}$) in figure 3, we notice that
the deviation from the power-law occurs at those scales larger than the
integral scales, for the $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals.
The integral length scales are comparable to the peak wavelength of the
turbulence energy spectrum (Kaimal and Finnigan,, 1994). This is demonstrated
in figure S5 (see supplementary material) for the highly convective stability
class ($-\zeta>2$), where the scaled peak wavenumbers ($\kappa z$, where
$\kappa$ is the streamwise wavenumber) of the $u^{\prime}$, $w^{\prime}$, and
$T^{\prime}$ spectra match with $z/\Lambda_{x}$ ($x$ can be either
$u^{\prime}$, $w^{\prime}$, or $T^{\prime}$ signals). Therefore, it indicates
the power-law behaviour of the persistence PDFs are connected to the eddies
from the inertial subrange of the turbulence spectrum (sizes smaller than the
integral scales). On the other hand, the departure from the power-law
behaviour in the persistence PDFs occurs at those scales comparable to the
energy containing scale of eddies (sizes larger than the integral scales). To
further explore this connection with the turbulent flow structures, it is
appropriate to represent the persistence PDFs by scaling their persistence
times with the integral scales. We present the results regarding those in the
subsequent section.

###  3.3 Scaling the persistence time by the integral scales

We begin with discussing the logarithmic persistence PDFs, since in that
representation the disparity between the PDFs corresponding to the positive
and negative fluctuations at larger persistence scales are highlighted more
clearly (see figure 3 in Section 3.1). The persistence time $t_{p}$ of
$u^{\prime}$, $w^{\prime}$, or $T^{\prime}$ signals are scaled with the
integral time scale ($\Gamma$) before computing the logarithmic persistence
PDFs. Note that, from the application of Taylor’s frozen turbulence hypothesis
this is equivalent to scaling the persistence length with integral scales
$\Lambda$.

####  3.3.1 Logarithmic persistence PDFs

Figure 6 shows the logarithmic persistence PDFs with the persistence times
normalized by $\Gamma$ corresponding to $u^{\prime}$, $w^{\prime}$, or
$T^{\prime}$ signals, for all the six stability classes (table 1). For all the
three signals in figure 6, a bend could be clearly observed in the logarithmic
persistence PDFs at the time scales larger than the integral scales,
indicating the clear deviation from the power-law. Apart from that, one can
notice from figure 6a (highly convective stability) that the difference
between the positive and negative persistence PDFs for the $T^{\prime}$
signals occurs exactly at those time scales which are larger than the integral
time scales of temperature (see the grey marked region in figure 6a). This
discrepancy gradually disappears in near neutral stability (see figures 6a to
6f), implying the non-Gaussian characteristics of the $T^{\prime}$ signal are
definitely related to the energy containing scales of motions.

However, the persistence PDFs deviate from the power-law form at scales larger
than the integral scales. Chamecki, (2013) showed for canopy surface layer
turbulence, the persistence PDFs of velocity fluctuations at large time scales
behave exponentially, the hallmark of a random Poisson type process. Cava et
al., (2012) hypothesized this as a consequence of random deformation of the
coherent structures, giving rise to several sub-structures with independent
arrival times. Be that as it may, Chamecki, (2013) also commented that this
exponential nature of the persistence PDFs could be better studied by
considering the cumulative distribution functions (CDF), given their smoothed
nature at large time scales.

Figure 6: The logarithmic persistence PDFs are shown for the persistence times
normalized by the integral time scales ($\Gamma$) associated with $w^{\prime}$
(squares), $T^{\prime}$ (circles), and $u^{\prime}$ (inverted triangles)
signals. The logarithmic PDFs are shifted vertically for the visualization
purpose. The panels on the right show the cumulative distribution functions of
the $T^{\prime}$ signals ($F_{T^{\prime}}(t_{p}/\Gamma_{T})$) plotted for
$t_{p}/\Gamma_{T}>1$ (marked as the grey regions on the left panels) on a log-
linear plot. This representation is chosen to determine the slope of the
exponential functions (represented as straight lines, see (13)) fitted
separately for the positive and negative temperature fluctuations persisting
for times larger than the integral time scales. The description of the markers
are shown in the legend, placed in a corner of the bottom panel (f) at the
left hand side.

The CDF [$F(t_{p}/\Gamma)$] is defined as,

$F(t_{p}/\Gamma)=\int_{{(t_{p}/\Gamma)}_{\rm
max}}^{(t_{p}/\Gamma)}P(t_{p}/\Gamma)\,d{(t_{p}/\Gamma)},$ (12)

which denotes the probability that the normalized persistence time has a value
smaller than or equal to $t_{p}/\Gamma$. From (12), it follows that for an
exponential distribution (Poisson process) both the PDFs and CDFs have the
same form, albeit with different proportionality constants. Additionally, the
CDFs also have an inherent advantage of being bin-independent with a smooth
convergence towards 1 (Newman,, 2005; White et al.,, 2008). At the right hand
side of figure 6 (see the side panels), the CDFs corresponding to the positive
and negative $T^{\prime}$ signals are shown for the time scales larger than
the integral scale $\Gamma_{T}$ [$(t_{p}/\Gamma_{T})>1$] in a log-linear co-
ordinate system, corresponding to all the six stability classes (the CDFs of
$u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals for the whole range of
$t_{p}/\Gamma$ are provided in figure S6). The exponential decay of the CDFs,

$F(t_{p}/\Gamma)\propto\exp\Big{[}-\lambda(t_{p}/\Gamma)\Big{]},$ (13)

in such plots would appear as a straight line with the slope of $\lambda$.

For the highly convective stability (panel (a) at the right hand side of
figure 6), we note that the $\lambda$ values obtained from the slopes of the
straight line fits for negative and positive $T^{\prime}$ are 0.7 and 1.1
respectively. It implies the mean time scales ($\Gamma/\lambda$) related to
the long persistent events of negative $T^{\prime}$ are almost twice the
integral scale of temperature, whereas for the positive $T^{\prime}$ they are
almost equal to the integral scale. Furthermore, with the change in stability
from highly convective to near neutral, the $\lambda$ values for the long
events of positive $T^{\prime}$ remain fixed at 1.1, but for the negative
$T^{\prime}$ these values gradually increase from 0.7 to 1.1 at near neutral
stability (see the side panels (a) to (f) in figure 6). This indicates the
mean time scales of the long persistent events of negative temperature
fluctuations become closer to the integral scale of temperature as the near
neutral stability is approached, while the same remains unchanged for the
positive fluctuation patterns.

Since these long events are associated with the energy containing scales (see
Section 3.2), the disparity in the mean time scales (or length scales from
Taylor’s frozen turbulence hypothesis) between the warm and cold events could
be explained from the coherent structure perspective. The large-eddy
simulation (LES) results of Khanna and Brasseur, (1998) and Salesky et al.,
(2017) indicate that in highly convective stability the vertical velocity and
temperature fluctuations display a cellular organization pattern. This
explains the closeness of the integral scales of $w^{\prime}$ and $T^{\prime}$
in highly convective stability (figure 5a). Apart from that, their LES
simulations also show that the warm-updraft motions are concentrated at the
cell edges, while the cold-downdraft motions occupy a larger area at the cell
centre. This geometrical asymmetry between the warm-updraft and cold-downdraft
motions might be the reason why the long warm events have mean length scales
comparable to the integral scales, while the scales of the long cold events
are almost twice of that. However, in near neutral conditions the warm and
cold fluids are positioned on the long streaky patterns of streamwise
fluctuations having similar streamwise lengths, comparable to the integral
scales (Khanna and Brasseur,, 1998; Salesky et al.,, 2017). Given the
closeness of $\Lambda_{T}$ and $\Lambda_{u}$ in the near neutral stability
(figure 5f), this elucidates why in such conditions an equivalence is observed
between the mean length scales of long warm and cold events.

####  3.3.2 Persistence PDFs

Figure 7: The persistence PDFs are shown for the persistence times normalized
by the integral time scales ($\Gamma$) associated with $w^{\prime}$ (squares),
$T^{\prime}$ (circles), and $u^{\prime}$ (inverted triangles) signals. The
thick black lines on all the panels show the best fit power-laws with their
respective slopes being mentioned on the panel (a). The regions on all the
panels corresponding to $t_{p}/\Gamma>1$ are marked in grey. The description
of the markers are shown in the legend, placed at the right bottom corner.

So far in figure 6, we have focused on the characteristics of the long
persistent events larger than the integral scales (associated with energy
containing eddies) by investigating the logarithmic representation of the
persistence PDFs. However, it would also be advantageous to focus on the
power-law structure of the persistence PDFs at scales smaller than the
integral scales (associated with the eddies from the inertial subrange). Since
the power-law behaviour is best represented in the original PDFs, figure 7
shows the persistence PDFs of $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$
signals with $t_{p}$ scaled with $\Gamma$.

From figure 7, one can notice the cut-off scale, where the deviation from the
power-law behaviour begins, is located almost exactly at the integral scale
$\Gamma$ for all the three signals. Apart from that, compared to figure 2, the
cut-off behaviour of the power-law is quite sharp in figure 7 due to the
correct scaling of $t_{p}$ with $\Gamma$. Therefore, in figure 7a (highly
convective stability) the exponents of the power-law functions are determined
by performing a linear regression for the range 0.01 $\leq t_{p}/\Gamma\leq$ 1
on the log-log plots. The slopes (exponents) of the best fit lines are found
to be $-1.6$, $-1.4$, and $-1.25$ for $u^{\prime}$, $w^{\prime}$, and
$T^{\prime}$ signals respectively, with $R^{2}$ values more than 0.98. Note
that, the exponent 1.4 for the $T^{\prime}$ signal is very close to 1.37 as
reported by Bershadskii et al., (2004) from their turbulent convection
experiments. To assess the effect of stability on the power-law behaviour, the
exponents obtained from the highly convective stability are compared with the
other stability classes. The following features emerge, such as:

  1. 1.

For the $u^{\prime}$ signal, no change in the power-law behaviour is observed
with stability.

  2. 2.

For the $T^{\prime}$ signal, no discernible change can be observed, although
at near neutral stability a slight departure from $-1.4$ exponent might be
possible.

  3. 3.

For the $w^{\prime}$ signal, the extent of the power-law behaviour gradually
decreases with stability.

Since the power-law structure in the persistence PDFs is associated with the
$t_{p}$ values smaller than the integral time scales, it is thus
representative of the eddies from the inertial subrange of the turbulence
spectrum. This observation lends considerable support to the hypothesis of Yee
et al., (1995) and Cava et al., (2012) where they connected this power law
behaviour in the persistence PDFs with self-similar Richardson cascading
mechanism, given the implied scale invariance associated with power-law
distributions (Newman,, 2005; Verma et al.,, 2006). Accordingly, it also
explains the reduction in the extension of the power-law behaviour for the
$w^{\prime}$ signal, associated with the near neutral stability. From table 1,
it is clear that the near neutral stability class ($0<-\zeta<0.2$) corresponds
only to the lowest three levels of the SLTEST experiment ($z=$ 1.4, 2.1, 3 m).
The inertial subrange of the $w$ spectra starts approximately at $\kappa z=1$,
considering $\Lambda_{w}\approx z$. If this is converted to frequency
($f=\overline{u}/2\pi z$), then we would obtain the threshold at $\approx$ 1
Hz (assuming $\overline{u}\approx$ 10 m s-1 and $z=2$ m), indicating only a
decade wide subrange being resolved at 20-Hz sampling rate. Therefore, the
decrease in the power-law range for the $w^{\prime}$ signal could be
attributed to insufficient sampling of small scale eddies at 20-Hz frequency
for the lowest three SLTEST levels.

####  3.3.3 Physical explanation of persistence exponents

In general, the exponents of the power-laws in persistence PDFs are non-
trivial and difficult to compute analytically except for simple systems such
as fractional Brownian motions (Majumdar,, 1999; Aurzada and Baumgarten,,
2013; Aurzada and Simon,, 2015). Recently, to explain these power-law
exponents, Cava and Katul, (2009) and Cava et al., (2012) have proposed an
ambitious connection with the self-organized criticality (SOC) observed in the
sandpile model of Bak et al., (1987, 1988). This is inspired by the results
from Sreenivasan et al., (2004) where such a connection was proposed for
turbulent convection. By using the results from Jensen et al., (1989) and
Bershadskii et al., (2004), Cava et al., (2012) connects this power-law
exponents as,

$m=3-\gamma-\frac{\mu}{2},$ (14)

where $m$ is the spectral slope of the telegraphic approximated time series,
$\gamma$ is the power-law exponent, and $\mu$ is the intermittency
coefficient. Sreenivasan and Bershadskii, (2006) found $m$ to be equal to 4/3,
as opposed to the 5/3 spectral slope in the inertial subrange. If we accept
this premise of SOC and assume $m$ to be the same for $u^{\prime}$,
$w^{\prime}$, and $T^{\prime}$ signals, then from (14) we can comment that the
intermittency coefficients ($\mu$) are larger for the $w^{\prime}$ and
$T^{\prime}$ signals ($\gamma=$ 1.25 and 1.4 respectively) compared to the
$u^{\prime}$ signal ($\gamma=$ 1.6). This implies in a convective surface
layer, the $w^{\prime}$ and $T^{\prime}$ signals are more intermittent in
nature than the $u^{\prime}$ signal.

However, an alternate theoretical framework also exists where the persistence
PDFs are connected to the fractal dimensions of small-scale turbulence
(Catrakis and Dimotakis,, 1996; Dimotakis and Catrakis,, 1999; Catrakis,,
2000). In retrospect, this alternate framework had been proposed to provide a
new methodology to compute the fractal dimensions, without relying on the box-
counting methods being used in the studies by Sreenivasan and Meneveau, (1986)
and Scotti et al., (1995) (see Sreenivasan, (1991) and Catrakis, (2008) for a
review). Nonetheless, at present, both of these frameworks based on SOC and
fractals seem plausible to connect these power-law exponents with the physical
nature of small-scale turbulence, but further research is required to assess
their viability. We present our conclusions in the next section.

##  4 Conclusion

We report the statistical scaling properties of the persistence PDFs of
turbulent fluctuations in streamwise and vertical velocity ($u^{\prime}$ and
$w^{\prime}$) and temperature ($T^{\prime}$) from the SLTEST experimental
dataset, in a convective surface layer. The important results from this study
can be summarized as:

  1. 1.

The persistence PDFs of $u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals
display a power-law behaviour up to a certain threshold scale, punctuated with
an exponential cut-off. The power-law exponents are 1.6, 1.25, and 1.4 for the
$u^{\prime}$, $w^{\prime}$, and $T^{\prime}$ signals respectively, with no
significant change being observed with stability.

  2. 2.

From randomization experiments, it is shown that this power-law behaviour of
the persistence PDFs is linked to the temporal coherence in the time series.
This temporal coherence is represented by the integral scales, computed from
the auto-correlation functions.

  3. 3.

By normalizing the persistence time scales with the integral scales, it is
found that the power-law behaviour in the PDFs is related to those scales
which are smaller than the integral scales. Since power-laws are synonymous
with scale-invariance, it implies the effect of self-similar eddy cascading
mechanism (Richardson cascade) on the persistence PDFs.

  4. 4.

A premultiplied form of the persistence PDF is used to demonstrate that the
non-Gaussian effects of the temperature fluctuations act only at those scales
which are larger than the integral scales. This indicates the non-Gaussian
characteristics of the temperature fluctuations are associated with the energy
containing scales of motions.

  5. 5.

From the exponential fits in the persistence CDFs, it is illustrated that the
mean time scales of the negative $T^{\prime}$ events persisting longer than
the integral scales are approximately twice the size of the integral scales in
highly convective conditions. However, this mean time scale gradually
decreases to almost being equal to the integral scale in the near neutral
stability.

  6. 6.

On the other hand, for the long positive $T^{\prime}$ events, the mean time
scales remain roughly equal to the integral scales, irrespective of stability.
This discrepancy with the negative $T^{\prime}$ events is interpreted to be
associated with the change in the topology of the coherent structures from
cellular convection patterns in highly convective conditions to horizontal
streaks in near neutral stability.

Note that, this study is the first of its kind for a convective surface layer
turbulence, where the entire focus has been on establishing the statistical
characteristics of the persistence PDFs. We have convincingly demonstrated
that the persistence PDFs of velocity and temperature fluctuations in a
convective surface layer follow a power-law distribution followed by an
exponential cut-off. Subsequently, by scaling the persistence time scales with
the integral scales, this statistical property of the persistence PDFs has
been associated with the turbulent structures in a convective flow.
Apparently, this scaling has been formulated by generating surrogate datasets
which preserve or destroy the temporal correlations in a turbulent signal.

Apart from that, it is also important to emphasize that the findings obtained
from this study have direct bearings on the quadrant behaviour of the
turbulent momentum and heat fluxes. This is because the switching patterns
from one quadrant to the other are related to the combination of the zero-
crossings in the component signals which construct the fluxes ($u^{\prime}$
and $w^{\prime}$ or $w^{\prime}$ and $T^{\prime}$). Since persistence PDFs are
related to the time spent between the zero-crossings in a signal, the future
research questions are:

  1. 1.

How the persistence PDFs of the heat and momentum fluxes are related to the
persistence PDFs of the velocity and temperature fluctuations?

  2. 2.

How much of the flux variation can be described by the persistence properties
of the component signals?

## Acknowledgements

Indian Institute of Tropical Meteorology (IITM) is an autonomous institute
fully funded by the Ministry of Earth Sciences, Government of India. 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. Dr. Kalmár-Nagy acknowledges funding from the Higher
Education Excellence Program of the Ministry of Human Capacities in the frame
of the Water Science & Disaster Prevention research area of Budapest
University of Technology and Economics (BME FIKP-VÍZ) and by the National
Research, Development, and Innovation Fund (TUDFO/51757/2019-ITM, Thematic
Excellence Program). The authors would also like to thank Dr. KG McNaughton
for providing them the SLTEST dataset. The computer codes used in this study
are available to all the researchers by contacting the authors.

##  Appendix A The effect of binning on persistence PDFs

Figure 8: An example is provided from the highly convective stability class
($-\zeta>2$) to illustrate the effect of linear and logarithmic binning on the
persistence PDFs of $w^{\prime}$ (squares), $T^{\prime}$ (circles), and
$u^{\prime}$ (inverted triangles) signals, considering both the positive and
negative fluctuations. The persistence times are normalized by the integral
time scales ($\Gamma$) associated with $w^{\prime}$, $T^{\prime}$, and
$u^{\prime}$ signals. The black coloured markers indicate the persistence PDFs
constructed from linear binning, whereas the orange coloured markers denote
the same but from logarithmic binning. To convert from logarithmic to linear
space, the logarithmic persistence PDFs are premultiplied with the factor
${([t_{p}/\Gamma]\log_{e}10)}^{-1}$ obtained from (16) and are shown as the
red coloured markers. Note that all the three PDFs for $w^{\prime}$,
$T^{\prime}$, and $u^{\prime}$ signals are shifted vertically by three decades
for visualization purpose. The thick grey lines show the same power-laws as in
figures 2 and 7.

The common method for computing the PDFs of a stochastic variable is to bin
the data in the linear space and computing the fraction of samples lying
within each bin, divided by the bin-width (Panofsky and Brier,, 1958; Lumley,,
1970). However, in the context of a stochastic variable which varies over a
substantial range, sometimes many orders of magnitude, the linear binning may
not be a good strategy. The PDFs obtained from linear binning for such
variables are usually dominated by a few high probability bins at the lower
range and an overwhelming amount of many low probability bins at the higher
range, as illustrated by Dorval, (2011). A better strategy is to use
logarithmic binning for stochastic variables which have positively skewed
distributions, such as having high densities at the lower range and low
densities at the large range of values (Dorval,, 2011). Intuitively, we may
expect the same with the persistence time, having large occurrences associated
with smaller time scales, with occasional incidences persisting for a long
time.

In the logarithmic binning exercise, a logarithmic transformation is applied
to the associated variable and then the fractions of samples are computed in
each logarithmic bin, divided by the bin-width in the logarithmic space to
compute the PDFs. The logarithmic binning has been used extensively to deduce
the characteristics of the distributions associated with neuronal inter-spike
intervals in biophysical signals (Selinger et al.,, 2007; Dorval,, 2011), to
estimate the size distribution of the avalanches associated with sandpile
models (Christensen and Moloney,, 2005), identifying the lévy flight patterns
in animal displacements (Benhamou,, 2007; Sims et al.,, 2007), and in many
other practical cases (see Newman, (2005) and the references therein for a
detailed review on this topic).

However, the logarithmic bin-width is variable in the linear space and
increases as the values increase, thus making the number of samples in each
logarithmic bin dependent on the linear width of the bin (White et al.,,
2008). By employing the change of variables technique, Dorval, (2011) showed
that the PDFs constructed in the linear ($P_{\rm lin}$) and logarithmic
($P_{\rm log}$) spaces are related to each other as,

$P_{\rm log}(x)=(\log_{e}10)[xP_{\rm lin}(x)],$ (15)

where $x$ is the associated stochastic variable and the factor $\log_{e}10$
emerges as the bins of $x$ are constructed in the powers of 10. Note that, in
this study we also used the bins in the power of 10 while constructing the
persistence PDFs. Since the PDFs should be related to constant bin-width in
the linear space (Panofsky and Brier,, 1958; Sims et al.,, 2007), we can
rearrange (15) to obtain the actual PDFs from the logarithmic PDFs as,

$P(x)=[P_{\rm log}(x)]{(x\log_{e}10)}^{-1}.$ (16)

An alternate way to obtain the actual PDFs is to divide the fraction of
samples within each logarithmic bin by the equivalent linear width in the
logarithmic space, also known as normalized logarithmic binning (White et
al.,, 2008). However, both of these methods yield the same result, since the
PDFs obtained from the normalized logarithmic binning are equivalent to (16).

In figure 8, we show an example of persistence PDFs for $u^{\prime}$,
$w^{\prime}$, and $T^{\prime}$ signals (combining both the positive and
negative fluctuations) from a highly convective stability class ($-\zeta>2$)
to illustrate the effect of linear and logarithmic binning strategies. Before
constructing the PDFs, the persistence time $t_{p}$ is normalized by the
integral scale $\Gamma$ associated with $u^{\prime}$, $w^{\prime}$, and
$T^{\prime}$ signals, due to the reasons as described in Section 3. We note
that, the linear PDFs and the PDFs described by (16) are equivalent to each
other, although the PDFs obtained from (16) display a broader range of power-
law behaviour and substantially less noisy than the linear PDFs in the larger
range of $t_{p}/\Gamma$ values. On the other hand, the logarithmic PDFs are
premultiplied PDFs as shown in (15). Therefore, compared to the actual PDFs,
the logarithmic PDFs exhibit a flatter slope in the smaller range of
$t_{p}/\Gamma$ values, concomitant with a slower decrease in the larger range.

##  Appendix B Description of phase-randomization and randomization
experiments

To perform the phase-randomization experiment at varying strength of
randomization, we first choose a 30-min time series of $u^{\prime}$,
$w^{\prime}$, or $T^{\prime}$ signals (having 36000 points at sampling
frequency of 20-Hz) from a particular stability class, as outlined in table 1.
The step-by-step methodology is provided below, such as:

  1. 1.

The Fourier transformation of the 30-min time series is performed, generating
36000 complex Fourier coefficients. Out of the 36000 coefficients, half of
these are the complex conjugates of each other, implying the need to only
consider the Fourier phases of the first 18000 coefficients.

  2. 2.

These 18000 Fourier phases are separated into two parts along their midpoint
at the 9000-th point. From the left half of the 9000 Fourier phases an
$(\frac{x}{2})$% of the phases are randomly chosen, with the same being
repeated for the right half as well.

  3. 3.

These $(\frac{x}{2})$% of the randomly chosen phases from the left and right
halves are shuffled with each other. Therefore, total $x$% Fourier phases of
the 18000 coefficients are scrambled in this procedure while keeping their
amplitudes intact. The revised Fourier coefficients are computed by using the
same amplitude but the scrambled phases, and then their complex conjugates are
taken for the rest of the 18000 points.

  4. 4.

The inverse Fourier transform is undertaken for these revised Fourier
coefficients, to generate the phase-randomized time series at an $x$%
randomization strength. The randomization strength implies the percentage of
the Fourier phases which have been shuffled randomly to generate the surrogate
dataset.

Note that, the aforementioned procedure can be adopted for the randomization
experiment as well. To generate a randomized dataset at an $x$% randomization
strength, $(\frac{x}{2})$% of the time series values are randomly shuffled
between the left and right halves, along the midpoint of the time series at
18000-th point. This type of shuffling procedure is known as Hurst’s card
simulation, by drawing analogy with the shuffling patterns of a standard deck
employed in card playing games (Strecker,, 2004).

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## Supplementary material

Figure S1: Same as in figure 2, but for the cross-stream positive and negative
fluctuations ($v^{\prime}$) included. Figure S2: Same as in figure 4, but for
the positive and negative fluctuations in $u^{\prime}$ and $w^{\prime}$
signals also included from the highly convective stability class ($-\zeta$>2),
during the phase-randomization experiment. The panels show the signals are
gradually randomized in their Fourier phases, beginning from the original
signals to 100% phase-randomized signals, in the increments of 20%. Figure S3:
Same as in figure 4, but for the positive and negative fluctuations in
$u^{\prime}$ and $w^{\prime}$ signals also included from the highly convective
stability class ($-\zeta$>2), during the temporal-randomization experiment.
The panels show the signals are gradually randomized in their temporal order,
beginning from the original signals to 100% randomized signals, in the
increments of 20%. Figure S4: The PDFs of the normalized temperature
fluctuations ($T^{\prime}/\sigma_{T}$) are shown from the highly convective
stability class ($-\zeta>2$) for the (a) phase-randomization (PR) and (b)
temporal-randomization (R) experiments. The different markers for both the
panels as shown in the respective legends denote different degree of
randomization, where 0% indicates the original signal and 100% indicates
completely randomized. Note that for increasing degree of phase-randomization
(see panel (a)) the original non-Gaussian temperature PDFs approach the
Gaussian PDFs (shown as the thick black line), whereas the temporal-
randomization preserves the PDF characteristics (see panel (b)). Figure S5:
The log-log plots of the normalized spectral energy densities with respect to
the normalized streamwise wavenumbers ($\kappa z$) are shown for the
$u^{\prime}$ (red circles), $T^{\prime}$ (pink circles), and $w^{\prime}$
(blue circles) signals from the highly convective stability class
($-\zeta>2$), with each being displaced vertically by the same amount for
visualization purpose. Note that the spectral energy densities for each signal
are normalized by their respective variances, such that the area under each
curve is exactly 1. The fitted equations from the spectral model of Kaimal and
Finnigan, (1994) are shown for each of the signals as indicated in the legend
placed at the right-most corner. The dashed vertical lines indicate the values
of $z/\Lambda_{x}$ ($x$ can be either $u$ (red), $T$ (pink), or $w$ (blue)),
where $\Lambda_{x}$ is the integral length scale obtained from the exponential
fit to the auto-correlation functions shown in figure 5a. An excellent match
is observed between the respective spectral peak positions and the values
$z/\Lambda_{x}$. Figure S6: The cumulative distribution functions plotted
against the persistence times normalized by the integral time scales
($\Gamma$) associated with $w^{\prime}$ (squares), $T^{\prime}$ (circles), and
$u^{\prime}$ (inverted triangles) signals. Each panel corresponds to the six
different stability classes as indicated in the right most corner and the CDFs
are shifted vertically for the visualization purpose. The grey shaded regions
in all the panels denote the range $t_{p}/\Gamma>$ 1.