# Level-crossings reveal organized coherent structures in a turbulent time
series

Subharthi Chowdhuri<EMAIL_ADDRESS>Department of Civil and Environmental
Engineering, University of California, Irvine, CA 92697, USA  Tirtha Banerjee
Department of Civil and Environmental Engineering, University of California,
Irvine, CA 92697, USA

###### Abstract

In turbulent flows, energy production is associated with highly organized
structures, known as coherent structures. Since these structures are three-
dimensional, their detection remains challenging in the most common situation,
when single-point temporal measurements are considered. While previous
research on coherent structure detection from time series employs a
thresholding approach, the thresholds are ad-hoc and vary significantly from
one study to another. To eliminate this subjective bias, we introduce the
level-crossing method and show how specific features of a turbulent time
series associated with coherent structures can be objectively identified,
without assigning a prior any arbitrary threshold. By using two wall-bounded
turbulence time series datasets, we successfully extract through level-
crossing analysis the impacts of coherent structures on turbulent dynamics,
and therefore, open an alternative avenue in experimental turbulence research.
By utilizing this framework further we identify a new metric, characterized by
a statistical asymmetry between peaks and troughs of a turbulent signal, to
quantify inner-outer interaction in wall turbulence. Moreover, a connection is
established between extreme value statistics and level-crossing analysis,
thereby allowing additional possibilities to study extreme events in other
dynamical systems.

††preprint:

##  I Introduction

Coherent structures in turbulent flows, ranging from astrophysical to
engineered, to atmospheric systems, are best described by their phenomenology,
such as: (a) their characteristic scales are comparable to the integral scales
(Kaftori, Hetsroni, and Banerjee, 1995); (b) they induce non-Gaussian
fluctuations in the turbulent variables (Majda and Kramer, 1999); and (c) they
have large contributions to turbulent fluxes and kinetic energy (Jiménez,
2018). Geometrically, these structures are three-dimensional and can take
various shapes based on the types of turbulent flow. Examples include,
granular patterns in astrophysical flows (Chian  _et al._ , 2014); hairpin
structures in neutral wall-bounded flows (Adrian, 2007); and counter-rotating
roll vortices in atmospheric turbulence (Young  _et al._ , 2002).
Notwithstanding their significance in drag reduction (Marusic  _et al._ ,
2021), consideration of coherent structures are also important for weather and
climate models since disregarding those can cause significant uncertainties in
turbulence parameterization (Salesky and Anderson, 2020).

Despite these structures can be visually recognized from three-dimensional
numerical simulations, smoke visualization experiments, particle velocimetry
measurements, or satellite images, it remains challenging to detect them from
the most common form of turbulent experiments where the variables are measured
at a single point in time. Previous studies on coherent structure detection
from turbulent time series employ a thresholding approach, where the
thresholds are set either in the temporal or spectral domain.

Regarding spectral domain, Perry and Chong (1982) demonstrated how the spectra
of streamwise velocity fluctuations in time encoded the information about
hairpin eddy structures by displaying a $-1$ spectral power law. Therefore, by
choosing an appropriate cut-off wavelength ($\lambda$), the effect of hairpin
eddy structures on the velocity statistics could be inferred from a single
point time series (Marusic and Monty, 2019). Apart from hairpin eddies, the
contributions from very large scale motions (VLSMs) on velocity statistics are
detected by setting $\lambda$ comparable to the boundary layer depth
(Balakumar and Adrian, 2007). By contrast, in the temporal domain, thresholds
are applied directly on the time series and historically were chosen in a
manner so that the frequency of the detected structures matched with the smoke
visualization experiments (Antonia, 1981).

However, the thresholding procedure in temporal domain suffers from
subjectivity as the threshold values differ significantly from one study to
another (Subramanian  _et al._ , 1982). Additionally, the rationale behind
their choices also varies, since some studies consider the thresholds where
the probability density functions (PDFs) of the time series differ from a
Gaussian, whereas others choose them from a quadrant perspective (Alfonsi,
2006). Conversely, the thresholds in the spectral domain often require
information about certain parameters (such as boundary layer height) whose
measurements are rarely available (Liu and Zheng, 2021). To eliminate these
difficulties, we introduce a level-crossing technique through which the
coherent structures can be detected from time series without assigning a prior
any tunable thresholds or external parameters.

Although a handful of previous research, such as the ones by Tardu and Bauer
(2015) and Poggi and Katul (2010), have used level-crossing analysis to study
the Reynolds stress production and dissipation of turbulence kinetic energy in
wall turbulence, we demonstrate that this approach could be generalized
further to detect coherent structures. In a level-crossing method (Blake and
Lindsey, 1973), one seeks a statistical description of time scales
$t_{p}|_{\alpha}$ up to which a stochastic variable $f(t)$ remains larger or
smaller than $\overline{f}\pm(\alpha\times\sigma_{f})$, where $t$ is time,
$\overline{f}$ is the temporal mean, $\sigma_{f}$ is the standard deviation,
and $\alpha$ is a given threshold. A brief review of level-crossing approach
is provided by Friedrich  _et al._ (2011), which, in other words, is a
generalization of the zero-crossing or persistence analysis where $\alpha$
level is set at zero (Majumdar, 1999; Tardu and Bauer, 2015). For many
different turbulent flows, the PDFs ($P(t_{p}|_{\alpha=0})$) of
$t_{p}|_{\alpha=0}$ are power-laws with an exponential cut-off (Perlekar  _et
al._ , 2011; Chowdhuri, Kalmár-Nagy, and Banerjee, 2020; Heisel  _et al._ ,
2022). On the one hand, the exponential cutoff represents a Poisson
distribution, associated with $t_{p}$ values larger than the integral scales
$\gamma$ (Cava  _et al._ , 2012). On the other hand, Blake and Lindsey (1973)
shows $P(t_{p}|_{\alpha})$ becomes a Poisson distribution when $\alpha$ values
are substantially large.

Several points are now considered. First, the Poisson distribution is
associated with a stochastic process for which the autocorrelation function
stays zero at all time scales (Majumdar, 1999). Second, in a turbulent time
series, the measurements become weakly correlated with each other at scales
larger than $\gamma$, since the autocorrelation functions drop to zero (Cava
_et al._ , 2012). Third, the characteristic scales of coherent structures are
comparable to $\gamma$ (Kaftori, Hetsroni, and Banerjee, 1995). Fourth, in a
randomly-shuffled (RS) signal, the autocorrelation functions cease to exist
(Lancaster  _et al._ , 2018). By combining all these aspects, we hypothesize
the thresholds to detect coherent structures could be objectively determined
as that particular $\alpha$ for which $P(t_{p}|_{\alpha})$ of the original
signal matches with its RS counterpart. Intrigued by this possibility, we ask:
(1) By changing $\alpha$ what turbulent flow features are revealed? (2) Do the
detected coherent structures from the critical $\alpha$ value obey the flow
physics? (3) Can we identify the organizational aspects of coherent structures
through the level-crossing approach?

We focus our attention on wall-bounded turbulence, since in such flows the
properties of coherent structures are well-established (Jiménez, 2018). We use
two hot-wire temporal datasets, collected from a zero-pressure gradient
turbulent boundary layer generated in the Melbourne wind tunnel (Marusic,
2020). During our presentation, we arrange the article in three different
sections. In Section II, we provide brief descriptions of the experimental
datasets and methodology, in Section III we introduce the results and discuss
them, and lastly in Section IV we summarize the key takeaways and provide the
scope for further research.

##  II Dataset and methodology

Corresponding to both hot-wire datasets, the friction Reynolds numbers ($Re$)
are of the order of $10^{4}$ as illustrated in Baars  _et al._ (2015). The
wall-normal heights are normalized by friction velocity ($u_{*}$) and
kinematic viscosity ($\nu$) and denoted as $y^{+}$, where + refers to wall-
scaling. We restrict $y^{+}\leq 10^{4}$, up to which the flow is fully
turbulent (Iacobello  _et al._ , 2023). Out of the two datasets, one of them
were sampled at a frequency ($f_{s}$) of 20 kHz (T1 dataset), while the other
at 44 kHz (T2 dataset). Note that for both datasets, one probe is fixed
(reference probe) while the others traverse across heights and are
synchronized with the reference probe (Baars  _et al._ , 2015). Moreover, for
the T1 dataset, the time series of streamwise velocity were collected over
three acquisition cycles, with each of $120$-s duration. Therefore, the
results presented for the T1 dataset are ensemble-averaged over these three
measurement cycles. However, for the T2 dataset, only a single cycle of
$360$-s duration was used. Regarding our purposes, we consider the streamwise
velocity fluctuations ($u^{\prime}$) after subtracting the temporal mean
($\overline{u}$). Subsequently, on these $u^{\prime}$ time-series we apply
level-crossing and event-synchronization analysis, whose rationale are
discussed below.

###  II.1 Level-crossing analysis

Figure 1: (a) A segment of $u^{\prime}$ time series and it’s telegraphic
approximations (TA) at different threshold levels ($\alpha$) are shown from
$y^{+}=$ 66.84 of T1 dataset. The level-crossing time scales at different
$\alpha$ values are denoted as $t_{p}|_{\alpha}$. (b) The PDFs of
$t^{+}_{p}|_{\alpha}$ are shown for $\alpha=0,2,-2$, corresponding to the
original and randomly-shuffled (RS) signals. (c) The energy spectrum of the
time series at $y^{+}=$ 66.84 is compared with the TA series at different
$\alpha$ levels. The power-laws $-1$ and $-5/3$ are shown in dash-dotted gray
lines. (d) Normalized mean time scales
($\overline{t_{p}|_{\alpha}}/\gamma_{u}$) are plotted against $\alpha$ values,
by considering all the heights from the T1 dataset. The gray shaded colors
indicate different heights as per the color-bar denoting $\log_{10}(y^{+})$ in
Fig. 3. The horizontal blue dash-dotted line indicates
$\overline{t_{p}|_{\alpha}}=\gamma_{u}$.

To demonstrate the philosophy behind level-crossing analysis, we use a segment
of a $u^{\prime}$ time series (normalized by its standard deviation
$\sigma_{u}$) from the T1 dataset at height $y^{+}=66.84$ (Fig.1a).
Corresponding to this time series, one can generate its telegraphic
approximations (TA) by denoting the values above a threshold to be 1 and 0
otherwise (Sreenivasan and Bershadskii, 2006). In the bottom panels of Fig.1a,
we show three TA sequences at threshold levels $\alpha=0,2,-2$. One can
clearly see as the threshold levels are increased, the timescales
$t_{p}|_{\alpha}$ of the TA patterns become substantially large. In fact, if
$t_{p}$ values become comparable to the integral scales of $u^{\prime}$
($\gamma_{u}$), one would expect the TA patterns associated with those
$\alpha$ levels to resemble a random configuration.

This indeed appears to be the case when one compares the PDFs of
$t_{p}|_{\alpha}$ for the three $\alpha$ values between the original and
randomly-shuffled (RS) signals (Fig.1b). As opposed to $\alpha=0$, for
$\alpha=2,-2$, $P(t_{p}|_{\alpha})$ of $u^{\prime}$ signal has an excellent
agreement with its RS counterpart. This can be further confirmed through the
q-q plots, where the $t_{p}|_{\alpha}$ values between the original and RS
signals follow a straight line with 45∘ slope for $\alpha=-2,2$, thereby
indicating they are both sampled from similar distributions (not shown here).

It is also interesting to investigate how the energy spectrum of the TA
patterns change as $\alpha$ is varied systematically. Sreenivasan and
Bershadskii (2006) showed that the energy spectrum of TA patterns
corresponding to $\alpha=0$ level, preserve the information about the spectral
power laws albeit with some change depending on the Reynolds number of the
flow. In Fig.1c we show the energy spectra of the TA patterns with different
$\alpha$ values and compare the same with the original $u^{\prime}$ signal at
$y^{+}=66.84$ (green line with circular markers). One can see at $\alpha=0$
level (black line with circles), the energy spectrum shows a $-1$ spectral
scaling at smaller frequencies similar to original signal, but at larger
frequencies the $-5/3$ scaling law appears to be a little different. However,
at large enough $\alpha$ values (indicated by deep red or blue colors), the
scaling laws disappear from the TA energy spectra and they nearly attain a
flat shape as expected for a random signal.

More importantly, for all the available heights from the T1 dataset, if one
plots the mean time scales ($\overline{t_{p}|_{\alpha}}$) against the $\alpha$
values, then $\overline{t_{p}|_{\alpha}}$ exceeds $\gamma_{u}$ considerably as
$\alpha$ increases (Fig.1d). Therefore, one can conclude that by increasing
$\alpha$ a critical value is reached, using which one could study certain flow
features whose characteristic scales are comparable to $\gamma_{u}$.

###  II.2 Event-Synchronization analysis

Figure 2: Shannon entropies of the synchronized event lengths are plotted
against different time lags ($\Delta t^{+}$) for the (a) T1 and (b) T2
datasets, where the values are normalized by the entropies of the synchronized
event lengths computed for the full signal.

By conducting an event-synchronization analysis, one seeks to describe how the
positive and negative patterns (with respect to $\alpha=0$ level) in a
turbulent signal are coupled with each other across different wall-normal
heights. This information is important to establish how the non-local
influences impact the organization of turbulent events in wall-bounded flows.

Before explaining this analysis it is prudent to mention briefly about the
probe arrangements for the T1 and T2 datasets. For both datasets, one probe is
fixed at a location (either in the inner or outer layer), whereas the other
probes travel across $y^{+}$ while at all times being synchronized with the
reference probe. Specifically, for the T1 dataset, the reference probe is
located at $y^{+}=4.3$, and for T2 dataset it is at $y^{+}=474$ where the
outer peaks appear in the turbulence spectra (Baars  _et al._ , 2015).

For event synchronization analysis, we consider a joint distribution between
the positive and negative patterns corresponding to the velocity signals from
the reference probe ($u^{\prime}_{\rm ref}$) and from a traveling probe
situated at any particular height ($u^{\prime}$). This joint distribution is
studied in terms of a binary sequence whose values are 1 when $u^{\prime}_{\rm
ref}$ and $u^{\prime}$ are simultaneously positive or negative. On the other
hand, when the signs mismatch, the sequence attains zero. We refer to this as
an overlap binary sequence and compute its time scales ($t_{p}$) based on the
duration where it stays at 1 or 0. To quantify the synchronization, Shannon
entropies of the overlap event lengths ($N_{p}$) are considered and compared
with a RS sequence (by taking a ratio), which is supposedly devoid of any
coupling effect. Mathematically, Shannon entropy of the overlap event lengths
is bounded within, $0\leq H^{x_{\rm ref},x}_{\rm n}(N_{p})\leq 1$, where
$x_{\rm ref}$ is the reference signal, $x$ is the signal from the travelling
probe, and 1 (0) indicates no (complete) synchronization between the two
signals.

To incorporate the effect of turbulent scales, the aforementioned procedure is
operated on $\Delta u_{\rm ref}$ and $\Delta u$ signals, where $\Delta u$
denotes velocity differences at a time lag $\Delta t$. The time lags are
normalized with wall-scaling and denoted as $\Delta t^{+}$. The synchronized
entropy values at any $\Delta t^{+}$ are scaled with the entropy values for
the full signal ($H^{u^{\prime}_{\rm ref},u^{\prime}}_{\rm n}(N_{p})$). From
Fig. 2 one can see that irrespective of the datasets or $y^{+}$ values, the
scaled entropies decrease with increasing scales and approach unity at $\Delta
t^{+}=10^{3}$, which physically represents the time scales of the outer- layer
structures (Baars  _et al._ , 2015). This implies at scales comparable to
outer-layer structures the synchronized entropies of the velocity differences
become equal to the full signal values. Therefore, one could infer that the
positive and negative patterns in the $u^{\prime}$ signals at any $y^{+}$
value carry the signatures of the structures residing in the outer-layer.
Further implications of this phenomenon are discussed in Section III.

##  III Results and discussion

###  III.1 Level crossings and extreme values

Figure 3: From the T1 dataset (a) vertical profiles of scaled Shannon
entropies, corresponding to the event lengths at different $\alpha$ levels
($N_{p}|_{\alpha}$) are shown; (b) $\mathcal{H}^{u^{\prime}}_{\rm
n}(N_{p}|_{\alpha})$ values are plotted for different $y^{+}$ values (see the
color bar), and $\alpha^{N}_{\rm th}$ and $\alpha^{P}_{\rm th}$ are identified
where $\mathcal{H}{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})=0.8$; (c) PDFs of
$\hat{u}/\alpha_{\rm th}$ are shown, where $\hat{u}=u^{\prime}/\sigma_{u}$;
(d) time fractions ($\mathcal{T}_{f}$) occupied by the samples exceeding
$\alpha_{th}$ and their comparison with a Gaussian distribution
($\mathcal{T}_{f,\rm G}$) is shown; (e) PDFs of $|u^{\prime}_{\rm
th}|/\overline{|u^{\prime}_{\rm th}|}$ are shown, where $u^{\prime}_{\rm th}$
are the samples exceeding $\alpha_{\rm th}$; and (f) Shannon entropies of the
synchronized event lengths are plotted against different time lags ($\Delta
t^{+}$) and $\Delta u$ denotes velocity increments.

To address the research questions, we begin with the probability distributions
of event lengths ($P(N_{p}|_{\alpha})$) as $\alpha$ is varied. We consider
$N_{p}|_{\alpha}$ since it is a discrete variable and represented through
probability mass functions whose computation is insensitive to binning. Note
that $N_{p}|_{\alpha}$ and $t_{p}|_{\alpha}$ are interchangeable through
$t_{p}|_{\alpha}=N_{p}|_{\alpha}/f_{s}$. To characterize $P(N_{p}|_{\alpha})$,
we consider its Shannon entropy compared with a RS sequence of $u^{\prime}$.
The entropy is denoted as $H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})$, whose
mathematical expression is provided in Eq. (1) of Appendix A. Furthermore,
$H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})$ is bounded between $0\leq
H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})\leq 1$ with 1 indicating a random
configuration. From Fig. 3a, one observes that the effect of changing $\alpha$
either from positive or negative side on $H^{u^{\prime}}_{\rm
n}(N_{p}|_{\alpha})$ is asymmetric. The vertical profiles of
$H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha\geq 0})$ show an inflection point
around $y^{+}\approx 70$, while when $\alpha$ is approached from the negative
side an another inflection point appears at $y^{+}\approx 12$. The position
$y^{+}=70$ indicates the location where the outer layer begins (Bae and Lee,
2021), whereas at $y^{+}=12$ the inner-layer structures are active (Baars  _et
al._ , 2015). As shown later in Section III.3, this asymmetrical progression
is related to inner-outer interaction in wall turbulence.

However, these inflection points disappear with increasing $\alpha$. In fact,
$H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})$ tend towards unity at large $\alpha$
values (Fig. 3b). This apparent randomness in $N_{p}$ is associated with the
fact that with increasing $\alpha$, $t_{p}|_{\alpha}$ become statistically
comparable to $\gamma_{u}$ (Fig. 1d). For accuracy purposes, we consider those
$\alpha$ values as the critical ones where $H^{u^{\prime}}_{\rm
n}(N_{p}|_{\alpha})$ crosses 0.8 (see Appendix A). These values are denoted as
$\alpha^{P}_{\rm th}$ and $\alpha^{N}_{\rm th}$ (Fig. 3b), respectively, and
any difference between them is correlated to the skewness of $u^{\prime}$
($\mathcal{S}(u^{\prime})$, Fig. 5b).

Upon considering $P(u^{\prime})$, one can see the samples that exceed these
critical values reside in the PDF tails (Fig. 3c). For visualization purposes,
before plotting $P(u^{\prime})$, we scale the positive and negative
$u^{\prime}$ values with $\alpha^{P}_{\rm th}\sigma_{u}$ and $\alpha^{N}_{\rm
th}\sigma_{u}$, respectively. Under this scaling, the values beyond $\pm 1$ in
Fig. 3c, indicate those critical $u^{\prime}$ samples exceeding either
$\alpha^{P}_{\rm th}\sigma_{u}$ (red-shaded regions) or $\alpha^{N}_{\rm
th}\sigma_{u}$ (blue-shaded regions). Specifically, from Fig. 3d, the time
fractions ($\mathcal{T}_{f}$) associated with these critical samples
($u^{\prime}_{\rm th}$) are nearly 1-3% of the total sample length, and their
values differ significantly from the ones obtained through a Gaussian
distribution of $u^{\prime}$ ($\mathcal{T}_{f,\rm G}$). Accordingly,
$P(|u^{\prime}_{\rm th}|/\overline{|u^{\prime}_{\rm th}|})$ follow an
exponential distribution (Fig. 3e), compliant with the theory of extreme value
statistics (Kinnison, 1983; Comtet, Leboeuf, and Majumdar, 2007). Note that we
consider the absolute values of $u^{\prime}_{\rm th}$, since their PDFs remain
same irrespective of the sign.

###  III.2 Identifying coherent structures

Next, we establish that $u^{\prime}_{\rm th}$ carry the signatures of the
outer-layer coherent structures. In wall turbulence, the presence of hairpin
structures organize the streamwise velocity field into alternating high- and
low-speed streaks (Adrian, 2007). This, in turn, induces positive and negative
fluctuations in $u^{\prime}$ signals. Through an event synchronization
analysis (see Section II.2), we identify how well these positive and negative
patterns are coupled with each other across $y^{+}$. This is achieved through
a scale-wise analysis of the Shannon entropies of overlap event lengths,
normalized with a RS sequence devoid of any synchronization ($H^{\Delta u_{\rm
ref},\Delta u}_{\rm n}(N_{p})$, $\Delta u=u^{\prime}(t+\Delta
t)-u^{\prime}(t)$). Note that $H^{\Delta u_{\rm ref},\Delta u}_{\rm n}(N_{p})$
is bounded between 0 to 1, where 1 (0) indicates no (complete)
synchronization. From Fig. 3f, one observes that the positive and negative
events across all $y^{+}$ values are most strongly coupled at scales $\Delta
t^{+}\approx 1000$ ($\Delta t^{+}$ is the normalized time lag), representing
the outer-layer structures (Baars  _et al._ , 2015). This signifies the events
in $u^{\prime}$, occurring at heights deep within the inner layer, preserve
information about the outer-layer structures.

Figure 4: The contours of event amplitudes ($\langle A^{+}_{uu}\rangle$) are
plotted separately for the (a) l- and (b) s-type events from T1 dataset. The
time scales of these events are denoted as $t_{p}|_{l}/\gamma_{u}$ and
$t_{p}|_{s}/\gamma_{u}$, where $\gamma_{u}$ is the integral time scale. (c)
Fractional contributions of l-type events to the variance
($\mathcal{V}_{f}|_{l}$) and occupation time ($\mathcal{T}_{f}|_{l}$) are
shown. The PDFs of (d) $t_{p}|_{l}/\gamma_{u}$ and (e) $t_{p}|_{s}/\gamma_{u}$
are shown from T1 dataset. (f) Shannon entropy ratios of $N_{p}$ corresponding
to l- and s-type events are plotted.

To extract that information, we conditionally sample the events based on
whether they contain the samples satisfying $u^{\prime}\geq\alpha^{P}_{\rm
th}\sigma_{u}$ and $u^{\prime}\leq\alpha^{N}_{\rm th}\sigma_{u}$ (large, or
l-type events) or not (small, or s-type events). This concept is graphically
illustrated in Appendix B. The time scales of l- and s-type events are denoted
as $t_{p}|_{l}/\gamma_{u}$ and $t_{p}|_{s}/\gamma_{u}$, respectively. In Figs.
4a–b, the contributions of l- or s-type events ($\langle A^{+}_{uu}\rangle$,
see Eq. (2) in Appendix B) against their time scales to streamwise velocity
variance ($\sigma_{u}^{2}$) are plotted separately. Quite remarkably, most of
the contributions of l-type events to $\sigma_{u}^{2}$ come from the heights
in and around $y^{+}=474$, where the influence of the outer-layer structures
are the strongest (Baars  _et al._ , 2015). Conversely, s-type events
contribute the most at heights $y^{+}=12$, where the inner-layer structures
reside (Baars  _et al._ , 2015). Precisely, at heights $y^{+}\geq 70$, the
total contributions of l-type events to the velocity variance remain between
40-60%, although they only occupy $\approx$ 20% of the time (Fig. 4c). These
contributions compare well with those from VLSMs in wall turbulence (Balakumar
and Adrian, 2007).

Moreover, the PDFs of $t_{p}|_{l}/\gamma_{u}$ and $t_{p}|_{s}/\gamma_{u}$
appear to be quite different (Figs. 4d–e). Specifically,
$P(t_{p}|_{l}/\gamma_{u})$ follows a log-normal distribution (verified with
q-q plots), while $P(t_{p}|_{s}/\gamma_{u})$ is a power-law of exponent $-1.6$
with an exponential cut off at scales comparable to $\gamma_{u}$. Sreenivasan
and Bershadskii (2006) demonstrated that the log-normal distribution describes
the size distributions of the dissipative structures, while we associate it
with the l-type event sizes. Furthermore, by considering the Shannon entropies
of event lengths, l-type events are more organized than the s-type ones (since
$H^{u^{\prime}}_{\rm n}(N_{p}|_{l})\ll H^{u^{\prime}}_{\rm n}(N_{p}|_{s})$) as
$y^{+}$ approaches the outer layer (Fig. 4f). These outcomes confirm that the
detected extremes in $u^{\prime}$ carry the signatures of the outer-layer
structures and are further utilized to infer about the velocity scales and
inner-outer interaction in wall turbulence. Although in Figs. 4a–b and d–e, T1
dataset is considered, similar findings are obtained for T2 dataset also (see
Fig. S1 in Supplementary material).

###  III.3 Connections to the turbulent dynamics

Figure 5: (a) Velocity ($\alpha_{\rm th}\sigma_{u}/u_{*}$) and mean time
scales ($T^{+}_{\rm th}$) at the threshold levels are shown for the T1
dataset. (b) Scatter plots between $x^{N}_{\rm th}/x^{P}_{\rm th}$
($x=\alpha,T$) and $\mathcal{S}(u^{\prime})$ are plotted. For the T1 dataset,
ratios of (c) mean time scales ($T^{N}_{\rm th}/T^{P}_{\rm th}$) and (d) their
standard deviations ($\sigma^{N}_{\rm th}/\sigma^{P}_{\rm th}$) are compared
with Gaussian rank surrogate, phase-randomized surrogate, and with a Fourier-
filtered signal for which the outer-layer influences are removed; (e) the mean
level-crossing time scales at different $\alpha/\alpha_{\rm th}$-levels are
compared with a phase-randomized signal, where $R|_{\alpha/\alpha_{\rm th}}$
denotes the ratios between the two; and (f) third-order structure function
skewness ($D_{uuu}/{(D_{uu})}^{3/2}$) are compared between the original (gray
shaded lines) and a conditionally-shuffled signal (colored lines).

We construct a velocity scale $(\alpha_{\rm th}\sigma_{u})$ for the outer-
layer structures and plot their profiles against $y^{+}$ in Fig. 5a. For
$70<y^{+}<10^{3}$ (i.e., log-layer), this scale attains a near-constant value
of $\approx 5u_{*}$. It remains interesting to see whether this velocity
scale, as obtained from the critical $\alpha$ values, can better collapse the
turbulence statistics among different experiments. This exercise is, however,
out of scope of the present study. On the other hand, to quantify the
influence of outer-layer structures on turbulence organization from
$u^{\prime}$ time series, we consider the mean time scales at
$\alpha=\alpha_{\rm th}$ level ($T_{\rm th}=\overline{t_{p}|_{\alpha_{\rm
th}}}$). The wall-normalized mean time scales for the positive and negative
side are denoted as $T^{P+}_{\rm th}$ and $T^{N+}_{\rm th}$, respectively,
with their behaviors being very different. For instance, in the log-layer,
$T^{P+}_{\rm th}$ increases as ${(y^{+})}^{1/2}$, while $T^{N+}_{\rm th}$ is
nearly constant at $10^{3}$. This increase of $T^{P+}_{\rm th}$ can be
explained by considering how the hairpin structures merge progressively to
form VLSMs (Adrian, 2007), whose characteristic scales ($\approx 10^{3}$ wall
units, (Harun and Lotfy, 2018)) match with $T^{N+}_{\rm th}$ values.

Unlike $\alpha^{N}_{\rm th}/\alpha^{P}_{\rm th}$, the difference between
$T^{P}_{\rm th}$ and $T^{N}_{\rm th}$ is anti-correlated to
$\mathcal{S}(u^{\prime})$ (Fig. 5b). In fact, for both datasets, the largest
values of $T^{N}_{\rm th}/T^{P}_{\rm th}$ are obtained when the skewness of
$u^{\prime}$ is nearly zero. Instead, we propose that the non-unity values of
$T^{N}_{\rm th}/T^{P}_{\rm th}$ are caused by coherent structures and would
disappear for a phase-randomized (PR) surrogate, since randomizing the Fourier
phases destroys any organizational aspects associated with coherent structures
(Tobias and Cattaneo, 2008). Clearly, from Fig. 5c, $T^{N}_{\rm th}/T^{P}_{\rm
th}$ approaches 1 for a PR time series (see red dash-dotted line). We use an
IAAFT (iteratively adjusted amplitude Fourier transform) model for PR
purposes, which preserves the signal PDFs and autocorrelation functions
(Lancaster  _et al._ , 2018). In fact, if only 10-50% of the Fourier phases
are randomized that itself has a significant effect on $T^{N}_{\rm
th}/T^{P}_{\rm th}$ (shown as dash-dotted lines with lighter red shades).
Contrarily, if $P(u^{\prime})$ are transformed to Gaussian while maintaining
the temporal structure (otherwise known as Gaussian rank surrogate (Bogachev,
Eichner, and Bunde, 2007)), then $T^{N}_{\rm th}/T^{P}_{\rm th}$ overlaps with
the original (gray dash-dotted line indicates the Gaussian rank surrogate in
Fig. 5c). Hence, the temporal organization of the signal sets the values of
$T^{N}_{\rm th}/T^{P}_{\rm th}$. Accordingly, if one removes the outer-layer
influences by choosing a Fourier cut-off filter at $\lambda^{+}=7000$ and
apply inverse Fourier transform (Baars  _et al._ , 2015), it changes
$T^{N}_{\rm th}/T^{P}_{\rm th}$ considerably (see blue dash-dotted line in
Fig. 5c). By repeating the analysis on the ratios of the standard deviations
or any other higher-order statistics (for instance, skewness and kurtosis) of
$t_{p}|_{\alpha_{\rm th}}$, the outcome remains the same. However, for
illustration purposes, we only show the results in Fig. 5d corresponding to
the standard deviations of $t_{p}|_{\alpha_{\rm th}}$ ($\sigma^{N}_{\rm
th}/\sigma^{P}_{\rm th}$). Thus, the statistical asymmetry between
$t_{p}|_{\alpha^{P}_{\rm th}}$ and $t_{p}|_{\alpha^{N}_{\rm th}}$ quantifies
inner-outer interaction in wall turbulence, as an alternative to the amplitude
modulation coefficient proposed by Mathis, Hutchins, and Marusic (2009).

Additionally, a PR procedure destroys non-linear dependencies in a signal
(Lancaster  _et al._ , 2018), which indicates that the non-unity values of
$T^{N}_{\rm th}/T^{P}_{\rm th}$ are related to non-linear dynamics. This is at
odds with persistence or zero-crossing analysis, where the time scale
statistics depend only on the autocorrelation functions accounting for the
signals’ linear structure (Majumdar, 1999; Poggi and Katul, 2009). Therefore,
the level-crossing statistics unveil hidden non-linearities in a stochastic
signal. To establish this feature more convincingly, in Fig. 5e, we show how
the mean time scales change between the original and PR signal, as $\alpha$ is
varied systematically. This is quantified through a timescale ratio,
$R|_{\alpha/\alpha_{\rm th}}$, as illustrated in Eq. (3) of Appendix C.
Further this ratio $R|_{\alpha/\alpha_{\rm th}}$ deviates from unity, strong
non-linear dependencies regulate the timescale statistics.

Apparently, for heights within the inner layer, non-linear dependencies have
the strongest effects on $\overline{t_{p}|_{\alpha}}$ at $\alpha_{\rm
th}$-level. More importantly, this non-linearity influences
$\overline{t_{p}|_{\alpha}}$ the most when the threshold is approached from
the negative side ($\alpha^{N}_{\rm th}$). Since $\alpha^{N}_{\rm th}$ carry
the signatures of the low-speed streaks ($u^{\prime}<0$), this implies that
the outer-layer influences on the inner-layer dynamics are governed through a
non-linear interaction associated with low-speed streaks. This mechanism was
earlier hypothesized by Schoppa and Hussain (2002), but our results
demonstrate it for the first time through an experimental dataset.
Interestingly, such non-linear effects on $\overline{t_{p}|_{\alpha}}$ become
irrelevant when the absolute values of $u^{\prime}$ are considered (see
Appendix C).

To further investigate the influence of these outer-layer structures on the
energy cascading process, we consider a $u^{\prime}$ time series where only
the values exceeding $\alpha^{P}_{\rm th}$ and $\alpha^{N}_{\rm th}$ are
randomly-shuffled while the others are kept intact. This operation selectively
destroys the turbulence organization associated with outer-layer structures.
We subsequently calculate the third-order structure function skewness
($D_{uuu}/{(D_{uu})}^{3/2}$), as its non-zero values are related to the
turbulence kinetic energy (TKE) cascading from large to small scales (Liu  _et
al._ , 2023). If $D_{uuu}/{(D_{uu})}^{3/2}$ are compared between the original
and conditionally-shuffled signal, at scales smaller than $\gamma_{u}$,
$D_{uuu}/{(D_{uu})}^{3/2}$ of the conditionally-shuffled signals decreases
significantly with increasing $y^{+}$ (Fig. 5f). Since within the inner-layer
the TKE is also carried by the inner-layer structures,
$D_{uuu}/{(D_{uu})}^{3/2}$ values remain slightly larger for the
conditionally-shuffled signal. Apart from $D_{uuu}/{(D_{uu})}^{3/2}$
approaching zero, this conditional-shuffling procedure destroys the inertial
subrange scaling in second-order structure functions (Fig. S2 in Supplementary
material). Therefore, we establish the impact of outer-layer coherent
structures on the energy cascade in wall turbulence. It is important to note
that these outcomes from Fig. 5 remain unchanged whether T1 or T2 datasets are
considered (Fig. S3 in Supplementary material).

##  IV Conclusion

To summarize, our method of coherent structure detection is entirely data
driven and the inferences being obtained from the two datasets match with the
existing knowledge of wall-bounded flows. In particular, this detection scheme
does not require any external inputs or arbitrary thresholds, thereby making
it an attractive choice in experimental turbulence research. This flexibility
offers a great advantage in case of atmospheric flows, since coherent
structures in such high-$Re$ flows scale with boundary layer height whose
measurements are rarely available. Moreover, through level-crossing approach,
we provide compelling evidence that the inner-outer interaction in wall
turbulence can be quantified by only considering the statistical asymmetry
between the peaks and troughs of a turbulent signal. For future research
endeavors, it would be interesting to compare this asymmetry parameter among
different experiments in wall-bounded turbulence, spanning both internal and
external flows. On the interdisciplinary front, the level-crossing framework
can be used to detect extremes in other dynamical systems (hydrology, stock
markets, etc.), or to generate training datasets for state-of-the-art machine
learning models which often fail to predict the extreme occurrences (Ding  _et
al._ , 2019).

## Conflict of Interest Statement

The authors have no conflicts to disclose.

## Author’s contributions

SC and TB designed and conceptualized this study. SC wrote the manuscript and
prepared the figures, while TB provided comments and corrections.

## Supplementary Material

The Supplementary figures relevant to this article are provided in a separate
document.

## Acknowledgements

SC and TB acknowledge the funding support from the University of California
Office of the President (UCOP) grant LFR-20-653572 (UC Lab-Fees); the National
Science Foundation (NSF) grants NSF-AGS-PDM-2146520 (CAREER), NSF-OISE-2114740
(AccelNet) and NSF-CPS-2209695 ; the United States Department of Agriculture
(USDA) grant 2021-67022-35908 (NIFA); and a cost reimbursable agreement with
the USDA Forest Service 20-CR-11242306-072.

## Availability of data

The data that support the findings of this study are openly available at
https://doi.org/10.26188/5e919e62e0dac.

##  Appendix A Statistical robustness of event entropy curves

Figure 6: (a) The Shannon entropy curves of event lengths are plotted with
respect to the scaled threshold $\alpha/\alpha_{\rm th}$. This scaling ensures
that the Shannon entropy curves converge towards the 0.8 value. (b) The number
of zero-crossings ($\mathcal{Z}$) are plotted against $\alpha/\alpha_{\rm
th}$. The blue horizontal dash-dotted line indicates the number
$\mathcal{Z}=10^{3}$. (c) The cumulative distribution functions of the event
lengths are shown for different levels of $\alpha/\alpha_{\rm th}$. (d) For
$y^{+}=66.84$, the Shannon entropy curve of event lengths (black line) is
compared with a randomly-shuffled model of 50 realizations (gray shaded
lines). (e) For $y^{+}=4.3$, the Shannon entropy curves are compared between
individual ensembles of the measured time series (gray shaded lines) and the
averaged one (black line). (f) The impacts of sampling frequencies ($f_{s}$)
and the length of the time series ($N$) on the entropy curve are investigated
by systematically varying $f_{s}$ and $N$ (see the legend).

We begin by plotting the scaled Shannon entropies of the event lengths (with
respect to a RS signal) where $\alpha$ values are normalized with either
$\alpha^{P}_{\rm th}$ or $\alpha^{N}_{\rm th}$ (depending on the sign),
denoted together as $\alpha_{\rm th}$. Owing to how $\alpha_{\rm th}$ is
defined, this normalization ensures that the scaled Shannon entropy curves
collapse at 0.8 for all the $y^{+}$ values (Fig. 6a). However, it raises a
question of why we consider 0.8 as our choice instead of 1.

This choice is influenced by the statistical accuracy associated with
$H^{u^{\prime}}_{\rm n}$ values. If we consider the mathematical expression of
$H^{u^{\prime}}_{\rm n}$,

$\mathcal{H}^{u^{\prime}}_{\rm
n}(N_{p})=\frac{\sum_{i=1}^{\mathcal{Z}}P(N^{\rm RS}_{p,i})\ln[P(N^{\rm
RS}_{p,i})]}{\sum_{i=1}^{\mathcal{Z}}P(N_{p,i})\ln[P(N_{p,i})]},$ (1)

where $\mathcal{Z}$ is the number of times the signal crosses $\alpha$-level,
we can clearly see the estimation of $H^{u^{\prime}}_{\rm n}$ is dependent on
$\mathcal{Z}$. Our intuition suggests that as $\alpha$ increases, the number
of level-crossings would decrease given the rareness in the occurrences of
large values in the signal. In Fig. 6b, we plot the number of level-crossings
against $\alpha/\alpha_{\rm th}$ values. As one may note, $\mathcal{Z}$ values
decrease beyond 1000 when the $\alpha_{\rm th}$ level is crossed. Since 1000
is a large number to ensure the estimates are statistically robust, we
consider the $H^{u^{\prime}}_{\rm n}$ values to be 0.8. This can be further
confirmed by plotting the cumulative distribution functions (CDFs) of event
lengths. For visualization purposes, we only show the results corresponding to
the $u^{\prime}$ signal at $y^{+}=66.84$. Quite clearly, the CDFs display
abrupt jumps as $\alpha$ becomes larger than $\alpha_{\rm th}$, due to the
lesser number of samples being used to compute their distributions (Fig. 6c).

It is important to take into account whether the entropy curves when compared
with a RS signal change if different realizations of random sequences are
used. We test this by generating 50 different realizations of RS sequences and
compute the entropy curves for each of such realizations. In Fig. 6d we show
such comparisons using $u^{\prime}$ signal at $y^{+}=66.84$ as the test case.
No difference is noted in the results. Moreover, in the figures discussed in
the main text, we show only the ensemble-averaged results by combining all the
three measurement cycles over which the turbulent time series were collected
at each $y^{+}$ value (Baars  _et al._ , 2015). In Fig. 6e, we compare the
entropy curves for each ensemble member with the averaged one. We consider the
$u^{\prime}$ signal at $y^{+}=4.3$ from the T1 dataset, since at this height
the number of ensemble members remains the largest (120). It can be seen that
the ensemble-averaged and individual entropy curves almost overlap with no
major differences (Fig. 6e).

As a last measure, we investigate the influence of the length of the time
series ($N$) and sampling frequencies ($f_{s}$) on the Shannon entropy curves.
We artificially change the sampling frequencies by block averaging the
$u^{\prime}$ signal values and by doing so we reduce the sampling frequencies
as low as 0.05 times the original. Although the entropy curves do change under
this operation, their overall shapes remain the same and therefore only appear
as a scaled version of the original (Fig. 6f). This change mainly occurs since
by block averaging we alter the standard deviations of the signal and thus the
$\alpha$ levels. Potentially it is also possible to increase the sampling
frequencies by incorporating an interpolation model, namely piecewise cubic
Hermite interpolating polynomial. By utilizing this model, we increase the
sampling frequencies two and four times the original, and study its effects on
the entropy curves. Similar as before, the curves preserve their shapes and
scale according to the $f_{s}$ values (not shown). On the other hand, if we
sub-sample the time series at different lengths compared to the original,
$H^{u^{\prime}}_{\rm n}$ remains nearly the same even when sub-sampling
reduces the original signal length by 95% (Fig. 6f). Hence, we conclude that
the estimation of the Shannon entropy curves are statistically robust, placing
confidence in the computed $\alpha$ values used later to detect coherent
structures.

##  Appendix B l- and s-type events

Figure 7: The concept of l- and s-type events are illustrated through a
segment of a $u^{\prime}$ time-series at $y^{+}=66.84$ from T1 dataset. The
thresholds $\alpha^{P}_{\rm th}$ and $\alpha^{N}_{\rm th}$ identified from the
entropy curves (see Fig. 3b) are shown as horizontal red and blue dash-dotted
lines, respectively. The red-colored events (l-type events) contain at least
one of these thresholds, whereas the blue-colored ones (s-type events) do not
contain any of these. The time scales associated with l- and s-type events are
denoted as $t_{p}|_{l}$ and $t_{p}|_{s}$, respectively.

In this section we illustrate the concepts of l- and s-type events and
establish their importance in turbulent dynamics. For this purpose, we use the
same segment of the $u^{\prime}$ time series at $y^{+}=66.84$, as done
earlier. In Fig. 7, the three dash-dotted horizontal lines indicate $\alpha=0$
(black), $\alpha^{P}_{\rm th}$ (red), and $\alpha^{N}_{\rm th}$ (blue) levels.
The l-type events are defined as those positive or negative blocks where at
least one of the $u^{\prime}$ samples satisfy the relation
$u^{\prime}\geq\alpha^{P}_{\rm th}\sigma_{u}$ and
$u^{\prime}\leq\alpha^{N}_{\rm th}\sigma_{u}$. On the other hand, s-type
events are those which do not satisfy the above condition. To distinguish the
l-type events from s-type ones, we use red-(blue) shaded regions to indicate
the l-(s) type events. The time scales associated with l-and s-type events are
denoted as $t_{p}|_{l}$ and $t_{p}|_{s}$ respectively, as shown in Fig. 7.
These time scales are subsequently normalized with $\gamma_{u}$, which is the
integral scale of the $u^{\prime}$ signal.

Although while demarcating between the l- and s-type events we used
$\alpha_{\rm th}$, it is possible to do the same with any $\alpha$ values. For
instance, if the $\alpha$ values are chosen to be very small then nearly all
the positive and negative events satisfy the condition of the l-type events,
and therefore, they become almost indistinguishable from the unconditioned
ones (i.e., the original zero-crossing events). Conversely, if the $\alpha$
values are too large then the number of l-type events decrease substantially
and overshadowed by the s-type events. Accordingly, it is interesting to
consider how the statistics of l- and s-type events change when the $\alpha$
values are varied systematically. We focus on the PDFs of
$t_{p}|_{l}/\gamma_{u}$ and $t_{p}|_{s}/\gamma_{u}$, and the event
contributions to the velocity variance. The contributions from a particular
event (either l- or s- type) to the velocity variance is defined as,

$\langle A^{+}_{uu}\rangle=\frac{1}{T\times
u_{*}^{2}}{\int_{t}^{t+(t_{p}|_{l,s})}{u^{\prime}}^{2}(t)\,dt},$ (2)

where $T$ is the total signal duration. Note that the contributions are scaled
with the friction velocity and further divided by the logarithmic bin-width so
the estimations remain nearly independent of the bin choice.

Figure 8: For the l- and s-type events, ((a),(d)) PDFs of time-scales
($P(t_{p}|_{x}/\gamma_{u}$, where $x=\rm l,s$) and the event amplitude
($\langle A^{+}_{uu}\rangle$) curves ((b), (e)) are shown, with systematically
changing the $\alpha/\alpha_{\rm th}$ values (see the color bar). The thick
black lines denote the curves at $\alpha/\alpha_{\rm th}=1$. The pink dash-
dotted lines indicate a power-law of slope $-1$ and the red lines indicate the
location where the time scales ($t_{p}|_{x}$) equal to $\gamma_{u}$. (c)
Fractional contributions of l-type or s-type events to the variance
($\mathcal{V}_{f}$) and occupation time ($\mathcal{T}_{f}$) are shown for
various $\alpha/\alpha_{\rm th}$ values. The vertical and horizontal black
lines indicate $\alpha/\alpha_{\rm th}=1$ and
$\mathcal{T}_{f},\mathcal{V}_{f}=0.5$. (f) The intermittency coefficient
($\mathcal{I}$) corresponding to l- and s-type events are shown against
$\alpha/\alpha_{\rm th}$.

For the same $u^{\prime}$ signal as used in Fig. 7, in Figs. 8a and d, we show
how the PDFs of $t_{p}|_{l}/\gamma_{u}$ and $t_{p}|_{s}/\gamma_{u}$ change as
$\alpha$ is varied. Specific to the l-type events, the PDFs at small $\alpha$
values are equivalent to the zero-crossing PDFs of $u^{\prime}$ signal, but as
$\alpha$ increases the power-law exponent changes gradually from $-1.6$ to
$-1$, with eventually attaining a log-normal distribution. On the other hand,
the s-type events approach the zero-crossing PDFs at larger $\alpha$ values,
notwithstanding their evolution remains very different from the l-type ones.
In particular, the distributions of $t_{p}|_{s}/\gamma_{u}$ differ
significantly from $t_{p}|_{l}/\gamma_{u}$.

By turning our attention towards event contributions, one can see that with
increasing $\alpha$ values the $\langle A^{+}_{uu}\rangle$ curves of l-type
events attain their peaks at scales considerably larger than the integral
scales (Fig. 8b). By contrast, the peaks of the $\langle A^{+}_{uu}\rangle$
curves corresponding to s-type events are always smaller than the integral
scales (Fig. 8e). In fact, for small $\alpha$ values, their peaks occur at
scales significantly lesser than $\gamma_{u}$. Therefore, it is plausible that
by choosing an appropriate $\alpha$ one might separate the features of small-
scale turbulence by conditionally sampling only the s-type events. This is,
however, a topic for further research.

By integrating $\langle A_{uu}\rangle$ curves over all the possible time
scales and dividing by the velocity variance, yields fractional contribution
to $\sigma_{u}^{2}$ ($\mathcal{V}_{f}$) for either of the event types.
Similarly, by summing up all the possible time scales and dividing by $T$,
yields the occupation time fractions of l- and s-type events
($\mathcal{T}_{f}$). In Fig. 8c, we show how $\mathcal{T}_{f}$ and
$\mathcal{V}_{f}$ vary for the l-type and s-type events against
$\alpha/\alpha_{\rm th}$. At $\alpha_{\rm th}$ level, we see that the l-type
events nearly contribute 50% to the velocity variance while occupying 20% of
the time. On the other hand, s-type events occupy 80% of the time while
contributing the same to $\sigma_{u}^{2}$. This information can also be
studied in terms of an intermittency index ($\mathcal{I}$), defined as a ratio
between $\mathcal{V}_{f}$ and $\mathcal{T}_{f}$.

If $\mathcal{I}$ values are further scaled with the ones obtained from the
unconditioned events ($\mathcal{I}_{f}$), then $\mathcal{I}_{f}\to 1$ when
$\alpha$ is either too large or small, depending on s- or l-type events
respectively. When $\mathcal{I}_{f}$ is plotted against $\alpha/\alpha_{\rm
th}$, a clear demarcation is noticed between l- and s-type events in how they
approach the unit values (Fig. 8f). We hypothesize this asymmetrical
progression is related to the time-irreversible dynamics of wall-bounded flows
(Iacobello  _et al._ , 2023).

##  Appendix C Sign-indefinite velocity signal

Figure 9: (a) Vertical profiles of the Shannon entropy curves for the absolute
values of the $u^{\prime}$ signals ($|u^{\prime}|$) at different
$\alpha$-levels are shown for the T1 dataset. (b)
$\mathcal{H}^{|u^{\prime}|}_{\rm n}(N_{p}|_{\alpha})$ values are plotted for
different $y^{+}$ values, and $\alpha_{\rm th}$ is identified where
$\mathcal{H}^{|u^{\prime}|}_{\rm n}(N_{p}|_{\alpha})=0.8$. (c) The mean level-
crossing time scales at different $\alpha/\alpha_{\rm th}$-levels are compared
with a phase-randomized signal, where $R|_{\alpha/\alpha_{\rm th}}$ denotes
the ratios between the two. (d) The mean time scale computed at $\alpha_{\rm
th}$-level ($T_{\rm th}$) are compared between the original and absolute
values of the $u^{\prime}$ signals.

Some earlier studies used thresholds on the time series values to detect
coherent structures and suggested that the same could be applied
interchangeably on either the original or absolute values of the signal
(Narasimha  _et al._ , 2007). We, however, show that considering absolute
values of the velocity signals instead of the original affects how the events
are organized in the temporal space.

To begin with, we show how the Shannon entropy curves of the event lengths
would behave when the $\alpha$ levels are applied on the absolute values of
the $u^{\prime}$ signal (Fig. 9a). Note that it is not possible to set
$\alpha=0$ in case of absolute values since no crossings would be obtained in
that case. Therefore, the smallest $\alpha$ levels are chosen as slightly
larger than 0. By doing so, one observes that up to certain $\alpha$ values
the vertical profiles of $H^{|u^{\prime}|}_{\rm n}(N_{p}|_{\alpha})$ behave
identically as $H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})$ in Fig. 3a, when
$\alpha$ is approached from the positive side. In fact, similar to
$H^{u^{\prime}}_{\rm n}(N_{p}|_{\alpha})$, an inflection point in
$H^{|u^{\prime}|}_{\rm n}(N_{p}|_{\alpha})$ is observed at $y^{+}=70$.

A note is necessary here regarding the estimation of the critical $\alpha$
value ($\alpha_{\rm th}$) for the $|u^{\prime}|$ signal. The scaled entropy
curves of $|u^{\prime}|$ signals form an U shape, and because of that the 0.8
value can be reached either at small or large $\alpha$ levels (Fig. 9b). At
small $\alpha$ levels, the events have large time scales for the absolute
signal, since the number of crossings are limited. However, we choose the
critical $\alpha$ levels ($\alpha_{\rm th}$) from the larger side, in
accordance with the original signal.

However, the biggest difference between the original and absolute signal
occurs when one compares the mean time scales with the phase-randomized (PR)
surrogates. This comparison is quantified through a ratio defined as,

$R|_{\alpha/\alpha_{\rm
th}}=\frac{\overline{t_{p}|_{\alpha}}}{[{\overline{t_{p}|_{\alpha}}}]_{\rm
PR}}.$ (3)

Unlike $u^{\prime}$, for the absolute signals, $R|_{\alpha/\alpha_{\rm th}}$
stays almost near to unity for any $\alpha/\alpha_{\rm th}$ values (Fig. 9c).
This indicates, contrary to Fig. 5e, the effect of non-linear dynamics on the
temporal arrangement of the samples exceeding $\alpha_{\rm th}$ disappears by
taking the absolute values. We can further confirm this phenomenon by
comparing the vertical profiles of $T_{\rm th}/\gamma_{u}$ between
$u^{\prime}$ and $|u^{\prime}|$ signals.

For $|u^{\prime}|$, the mean time scales at $\alpha_{\rm th}$-level remain
closer to $T^{P}_{\rm th}$ instead of $T^{N}_{\rm th}$, where $T^{P}_{\rm th}$
and $T^{N}_{\rm th}$ values are obtained from the original $u^{\prime}$ signal
(Fig. 9d). More importantly, $T_{\rm th}$ of the absolute signal is nearly
insensitive when the Fourier phases are randomized. Since PR destroys the
organization of coherent structures, this indifference suggests that the
events detected from the absolute signals may not obey the turbulent flow
physics.

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