# Causal features in turbulent channel flow

Kosuke Osawa<EMAIL_ADDRESS>Javier Jiménez  School of Aeronautics,
Universidad Politécnica de Madrid, 28040 Madrid, Spain.

###### Abstract

The causal relevance of local flow conditions in wall-bounded turbulence is
analysed using ensembles of interventional experiments in which the effect of
perturbing the flow within a small cell is monitored at some future time. When
this is done using the relative amplification of the perturbation energy,
causality depends on the flow conditions within the cell before it is
perturbed, and can be used as a probe of the flow dynamics. The key scaling
parameter is the ambient shear, which is also the dominant diagnostic variable
for wall-attached perturbations. Away from the wall, the relevant variables
are the streamwise and wall-normal velocities. Causally significant cells are
associated with sweeps that carry the perturbation towards the stronger shear
near the wall, whereas irrelevant ones are associated with ejections that
carry it towards the weaker shear in the outer layers. Causally significant
and irrelevant cells are themselves organised into structures that share many
characteristics with classical sweeps and ejections, such as forming spanwise
pairs whose dimensions and geometry are similar to those of classical
quadrants. At the wall, this is consistent with causally significant
configurations in which a high-speed streak overtakes a low-speed one, and
causally irrelevant ones in which the two streaks pull apart from each other.
It is argued that this is probably associated with streak meandering.

###### keywords:

##  1 Introduction

Although turbulence is a high-dimensional chaotic system, it is often modelled
as a collection of compact and approximately autonomous coherent structures.
These are typically intermittent, emerging and vanishing with a lifetime and
frequency that depend on their nature and size, and are characterised both by
evolving relatively independently from their flow environment, and by having a
measurable influence on the rest of the flow (Jiménez, 2018a). As such, it is
important to clarify not only how they behave individually, but how are they
connected among themselves in space and in time.

Such causal connections would help us understand how turbulence works, both
from the fundamental point view and in practical applications connected with
flow control and prediction. For example, it is important to avoid introducing
in the initial conditions of numerical weather forecasting spurious
perturbations that would later amplify significantly (Rodwell & Wernli, 2023),
and identifying such highly influential events would help us improve
prediction accuracy. Another example is flow control, which intrinsically
tries to modify the future of the flow by altering its present state.
Understanding which structures are causally important and which ones have no
significant effect in the evolution of the flow would clearly help in
optimising this process.

Conversely, elucidating the connections between different flow regions, not
necessarily initially identified as coherent, may lead to the discovery of
novel coherent structures that describe turbulence better than the known ones,
or to previously overlooked connections between known structures that can be
incorporated into better flow models (Jiménez, 2020b, 2023).

For example, quasi-streamwise rollers, streamwise-velocity streaks and wall-
normal velocity bursts are believed to be essential for maintaining wall-
bounded turbulence. The most common hypothesis is that there is a self-
sustaining process (SSP) in which at least two of these structures mutually
induce each other (Jiménez & Moin, 1991; Hamilton et al., 1995; Waleffe,
1997), but the details are incomplete. For example, recent evidence suggests
that bursts are able to sustain a cycle by themselves (Jiménez, 2018a), while
streaks are byproducts rather than actors in the SSP (Jiménez, 2022). Even
apparently straightforward connections, such as the generation of the streaks
by bursts (Kim et al., 1971) are only incompletely understood, because the two
phenomena have very different length scales (Jiménez, 2018a). Establishing the
causality relations between these different structures would throw light on
whether they are indeed connected, on the sequence in which they are linked
and on whether some component is missing from the model.

With the goal of minimising bias, our strategy is to exclusively characterise
flow regions in terms of their influence on the future of the flow, without
necessarily relating them to previously known coherent structures. Only once a
particular flow template has been identified as highly causal or as especially
irrelevant will we try to classify it within existing theories, or to
recognise it as something new.

There two general approaches to causality. The first one is observational and
non-intrusive, and is often the only option when the system is hard to
replicate (e.g. astrophysics), difficult to experiment with (e.g. some social
sciences), or simply too large to easily simulate. Unfortunately, it is
generally believed that observation is not enough to unambiguously establish
cause and effect, because correlation does not imply causation (Granger, 1969;
Pearl, 2009) but, even in those cases, a careful consideration of the temporal
evolution of the system may lead to the identification of causal histories
when they cross neighbourhoods of particular interest, typically extreme
events (Angrist et al., 1996). A related approach is the operator
representation of turbulence time series, examples of which are Froyland &
Padberg (2009); Kaiser et al. (2014); Schmid et al. (2018); Brunton et al.
(2020); Fernex et al. (2021); Taira & Nair (2022); Jiménez (2023); Souza
(2023), among others. Another example is the analysis of data series from
wall-bounded turbulence by Lozano-Durán et al. (2020) using tools of transfer
entropy, or the improved version in which their applicability to subgrid
modelling and flow control was demonstrated by Lozano-Durán & Arranz (2022).

The alternative is interventional causality, in which the system is directly
modified and the consequences observed. This offers more control over what is
being analysed, and safer inferences (Pearl, 2009), but presumes a
sufficiently cheap way of modifying the system. Essentially, in dynamical
system notation, non-interventional methods provide information about the
behaviour of the system while it moves within its attractor, while
interventional ones give additional information about the system by observing
what happens outside it.

Turbulence, which is expensive to simulate and hard to modify experimentally,
was for a long time considered to be in the group of phenomena that could only
be observed, but the increased speed of computers, as well as better
experimental techniques, slowly eroded that difficulty (Jiménez & Moin, 1991;
Jiménez & Pinelli, 1999). More recently, fast GPUs have speeded the numerical
simulation of realistic turbulent flows to the point of allowing the practical
simulation of artificially modified flow ensembles that can be considered
interventional (Vela-Martín & Jiménez, 2021). They have opened the possibility
of Monte Carlo studies in which the consequences of ‘randomly’ modified flows
are examined.

Examples of this approach are Jiménez (2018b, 2020b), who introduced localised
perturbations in two-dimensional turbulence in order to determine which parts
of the flow result in significant perturbation growth or decay after a certain
time. This allowed the identification of causally significant and irrelevant
flow structures, including the relatively unexpected relevance of vortex
dipoles rather than individual vortices, and eventually led to new models for
the two-dimensional energy cascade (Jiménez, 2021). Encinar & Jiménez (2023)
extended the technique to three-dimensional homogeneous isotropic turbulence
and demonstrated that causal events are in that case characterised by either
high kinetic energy or high dissipation rate, depending on the spacial scale
of the initial perturbation, and that strong strain, rather than high
vorticity, is the main prerequisite for perturbation growth. In these two
cases, it is interesting that some of the significant structures were not the
classically expected ones, underlining the ability of Monte Carlo
interventional experiments to mitigate the bias of conventional wisdom.

In this study, we adopt the interventional approach, following the basic
methodology in Jiménez (2018b). Spatially localized perturbations are imposed
on a fully developed turbulent channel flow, and their influence is measured
by their ability to alter the future evolution of the flow.

Numerical experiments that track the development of perturbation ensembles in
wall turbulence are not a new, probably starting with the computation by Keefe
et al. (1992) of the Lyapunov spectrum in a low-Reynolds number channel. On a
similar subject, Nikitin (2008, 2018) investigated the Reynolds number scaling
of the leading Lyapunov exponent of a turbulent channel. Lyapunov analyses do
not typically control the form of the initial perturbation, leaving the system
to choose the most unstable direction in state space, and taking precautions
to avoid nonlinearities, but Cherubini et al. (2010) and Farano et al. (2017)
turned the problem around by searching for weakly or fully nonlinear
perturbations that optimally grow in energy after a given target time. They
work on an initially stationary flow with a turbulent profile and, when they
constrain the initial energy of the perturbation, they obtain optimals that
are localised in physical space. More recently Ciola et al. (2023) extended
the analysis to snapshots of real turbulence, finding that, for properly
chosen target times, the optimal precursor is an early stage of an Orr burst.
However, due to the difficulty of convergence over long times, the result only
applied to short delays of the order of a few tens of viscous units. Moreover,
the solution is only optimal for the snapshot at which it is applied, making
it difficult to generalise the result.

The choice of the size of the initial perturbations is important, and data
assimilation experiments have been conducted to estimate the minimum size
below which perturbations are enslaved to their environment. The first was
probably Yoshida et al. (2005) in isotropic three-dimensional turbulence, who
showed that randomised scales smaller than 30 Kolmogorov (1941) viscous
lengths are regenerated if continuously assimilated to larger structures. Wang
& Zaki (2022) conducted similar experiments in channel turbulence. They
replace some layers with white noise and showed how they synchronised with the
original flow when assimilated through their boundaries. The maximum
synchronisation thickness is approximately 30 viscous lengths (approximately
12 Kolmogorov units) for layers attached to the wall, and twice the Taylor
microscale for layers away from it. However, since Encinar & Jiménez (2023)
found that freely evolving perturbations grow even below the assimilation
limit, the result in wall-bounded turbulence remains uncertain.

The present study targets the nonlinear evolution of localised perturbations
applied to instantaneous snapshots of turbulent channels at a moderate but
non-trivial Reynolds number, over times of the order of an eddy turnover. A
Monte Carlo search is used to apply the analysis across snapshots, and across
as many combinations of perturbation location, size and target time as
practicable. The basic assumption is that causality depends of the local state
of the neighbourhood at which the perturbation is applied, and the details of
this dependence are extracted from the database of numerical experiments using
standard methods of data analysis.

The organisation of the paper is as follows. The numerical setup and the
definition of the initial perturbations are described in §2. How their
evolution can be used to determine causality is discussed in §3–4, and the
relation between causal structures and the surrounding flow field is discussed
in §5. Conclusions are offered in §6.

##  2 Numerical setup

$U_{b}h/\nu$ |  $Re_{\tau}$ |  $L_{x}\times L_{y}\times L_{z}$ |  $N_{x}\times N_{y}\times N_{z}$ |  $\Delta x^{+}$ |  $\Delta y^{+}_{\min}$ |  $\Delta y^{+}_{\max}$ |  $\Delta z^{+}$
---|---|---|---|---|---|---|---
11180 |  600.9 |  $\pi h\times h\times\pi h$ |  $128\times 192\times 256$ |  14.7 |  0.46 |  6.51 |  7.4
Table 1: Computational parameters. $L_{i}$ is the domain size along the $i$-th
direction, $h$ is the ‘half-channel’ height, equivalent to the domain height
in open channels, and $U_{b}$ is the bulk velocity. The grid dimensions,
$N_{i}$, and effective resolutions, $\Delta{x_{i}}$, are expressed in terms of
Fourier modes.

(a) (b)

Figure 1: (a) Velocity fluctuation intensities. Symbols, present open channel
at $Re_{\tau}=601$; dashed, open channel at $Re_{\tau}=541$ (Pirozzoli, 2023);
solid, full channel at $Re_{\tau}=547$ (del Álamo & Jiménez, 2003). Black:
$u^{\prime}$; red, $v^{\prime}$; blue, $w^{\prime}$. (b) Premultiplied
spanwise spectrum of the turbulent kinetic energy, normalised with
$u_{\tau}^{2}$. Contours are logarithmically equispaced from
$k_{z}E_{KK}/u_{\tau}^{2}=0.56$ to 2.8. The vertical solid line is the present
computational box. The vertical thinner line is $\lambda_{z}=2\pi/k_{z}=h$,
and the horizontal line is $y^{+}=300$. Filled contours are the present
simulation; lines are del Álamo & Jiménez (2003).

To save computational resources, we analyse simulations of a pressure-driven
turbulent open channel flow in a doubly periodic domain, between a no-slip
wall at $y=0$ and an impermeable free-slip one at $y=h$. The streamwise, wall-
normal and spanwise directions are $x,y$ and $z$, respectively, and the
corresponding velocities are $u,v$ and $w$, although position and velocities
are occasionally denoted by their components
$\boldsymbol{x}=\\{x_{j}\\},j=1\ldots 3$. The domain size is $L_{x}\times
L_{y}\times L_{z}=\pi h\times h\times\pi h$, and the Reynolds number is
$Re_{\tau}=u_{\tau}h/\nu=600.9$. The ‘+’ superscript denotes wall units,
normalised with the friction velocity $u_{\tau}$ and with the kinematic
viscosity $\nu$. Capital letters, as in $U(y)$, denote variables averaged over
the simulation ensemble and over wall-parallel planes, lower case ones are
fluctuations with respect to this average, and primes are root-mean-squared
fluctuation intensities. Repeated indices, including squares, imply summations
unless otherwise noted. The simulation code is standard dealiased Fourier
spectral along $x$ and $z$, as in Kim et al. (1987), but uses seven-points-
stencil compact finite differences for the wall-normal derivatives, as in
Hoyas & Jiménez (2006). Time marching is semi-implicit third-order Runge–Kutta
(Spalart et al., 1991), and the mass flux is kept constant. The numerical
$y$-grid is stretched at the no-slip wall with a hyperbolic tangent. See table
1 for other numerical parameters.

Figure 1(a) compares the resulting fluctuation profiles with existing data
from regular and open channels. It was shown by Lozano-Durán & Jiménez (2014)
that a computational box with $L_{z}/h=\pi$ reproduces well the statistics of
regular channels, and figure 1 shows that the same is true for open ones. In
particular, figure 1(b) shows that the spanwise kinetic energy spectrum fits
well within the computational box. On the other hand, the figure shows that
open and full channels only agree below $y/h\approx 0.5$, above which the
effect of ‘splatting’ at the top wall is particularly visible in the
fluctuations of the cross-flow velocities (Perot & Moin, 1995). We will only
use the range $y^{+}\lesssim 300$ for the rest of the paper. It is also clear
from the figure that the energy at long wavelengths above $y^{+}=100$ is
higher than in del Álamo & Jiménez (2003). This is due to the short
computational box, which inhibits the instability of the streaks (Abe et al.,
2018), and results in two pairs of large streamwise streaks and rollers that
dominate the flow.

The original code was ported to CUDA by Vela-Martín et al. (2021) for the
efficient simulation of high-Reynolds number channel turbulence in GPU
clusters. It has been adapted to a single GPU for the present experiments, but
the original reference should be consulted for full details.

###  2.1 The initial perturbations

$l_{\mbox{\scriptsize\it cell}}^{+}$ |  $y_{\mbox{\scriptsize\it cell}}^{+}$ (set 1) |  $y_{\mbox{\scriptsize\it cell}}^{+}$ (set 2) | Symbol
---|---|---|---
25 |  0, 12.5, 37.6, 62.6, 87.6, 113, 138, 163, 188, 213, 238, 288 |  0, 138 |
50 |  0, 25.0, 50.0, 75.0, 100, 125, 150, 175, 200, 225, 275 |  NA |
75 |  0, 12.6, 37.6, 62.6, 87.6, 113, 138, 163, 188, 213, 263 |  0, 38, 113, 188, 263 |
100 |  0, 25.0, 50.0, 75.0, 100, 125, 150, 175, 200, 250 |  NA |
150 |  0, 25.0, 50.0, 75.0, 100, 125, 150, 175, 225 |  0,75,225 |
Table 2: Parameters of the perturbation cells. See text for details.

Figure 2:  Schematic of the numerical experiment. Green, isosurface of the
turbulent kinetic energy for the reference flow at $t=0$,
$|\boldsymbol{u}_{\mbox{\scriptsize\it ref}}|^{+}=4.5$. Colour intensity
encode the distance from the wall; red, perturbation kinetic energy at some
later time, $|\boldsymbol{u}_{\mbox{\scriptsize\it
ref}}(T)-\boldsymbol{u}_{\mbox{\scriptsize\it mod}}(T)|^{+}=0.17$, for a
causally significant perturbation ; blue, same for a causally irrelevant
perturbation.

As mentioned in the introduction, the interventional identification of
causality follows Jiménez (2018b, 2020b). The idea is to apply a spatially
localized initial perturbation to existing turbulence, after which the flow is
allowed to develop naturally (see figure 2). The effect is measured after some
time. Unlike the sensitivity analysis of the mean velocity profile in Farano
et al. (2017), each causality experiment is the response to a particular
perturbation on a particular location of a given flow snapshot, and the
numerical experiment has to be repeated many times for different snapshots and
perturbations. The goal is to create a data base of responses from which to
extract the characteristics that make a particular flow location influential
for the future behaviour of turbulence (i.e., causally significant). To ensure
independence, the 40 initial reference snapshots used for our experiments are
separated by at least 1.4 turnovers (defined as $h/u_{\tau}$).

Perturbations modify the flow within a cubical cell of side
$l_{\mbox{\scriptsize\it cell}}$ centred at $\boldsymbol{x}_{c}$. Although
there are countless choices for the form of the disturbance, and even if
experience shows that the manner in which the flow is disturbed influences the
outcome of the experiment (Jiménez, 2020b; Encinar & Jiménez, 2023), cost
considerations limit us to a single perturbation scheme. Specifically, the
flow is modified by removing the velocity fluctuations within the cell,
overwriting the velocity field with its $y$-dependent cell average. Defining
the cell average of a variable $f$ as

$\langle{f}\rangle_{c}(y)=l_{\mbox{\scriptsize\it
cell}}^{-2}\int^{x_{c}+l_{\mbox{\scriptsize\it
cell}}/2}_{x_{c}-l_{\mbox{\scriptsize\it
cell}}/2}\int^{z_{c}+l_{\mbox{\scriptsize\it
cell}}/2}_{z_{c}-l_{\mbox{\scriptsize\it
cell}}/2}{f(x,y,z){\,\rm{d}}x{\,\rm{d}}z},$ (1)

the perturbed velocity $\boldsymbol{u}_{\mbox{\scriptsize\it mod}}$ is

$\boldsymbol{u}_{\mbox{\scriptsize\it
mod}}=\begin{cases}\langle{\boldsymbol{u}_{\mbox{\scriptsize\it
ref}}}\rangle_{c}&\mbox{when}\;|x_{j}-x_{cj}|\leq l_{\mbox{\scriptsize\it
cell}}/2,\\\ \boldsymbol{u}_{\mbox{\scriptsize\it
ref}}&\mbox{otherwise.}\end{cases}$ (2)

where $\boldsymbol{u}_{\mbox{\scriptsize\it ref}}\/$ is the unperturbed flow,
and an extra pressure step is applied after (2) to restore continuity at the
edges of the cell. The experiment is repeated as many times as possible,
applying it to different reference flow fields while changing the location and
size of the perturbation cell.

Table 2 summarises the parameters of the experimental cells. They are
expressed in terms of the distance from the wall to the bottom of the cell,
$y_{\mbox{\scriptsize\it cell}}=y_{c}-l_{\mbox{\scriptsize\it cell}}/2$, which
was found to collapse some results better than the cell centre, and are
separated in two sets. In the first one, involving the 40 reference snapshots,
perturbations are applied to a $6\times 6$ grid of cells evenly spaced in $x$
and $z$, such that their centres are separated by $\pi h/6$ in each direction
(approximately 315 wall units). In the $y$-direction, perturbations are
applied at the heights detailed in the second column of table 2, ranging from
cells touching the wall to those centred at the middle of the computational
domain, $y_{c}^{+}\approx 300$. Each of them is run for 0.65 turnovers, and
consumes approximately 6 minutes in an Nvidia A100 GPU, so that the
approximately 76000 experiments in this set spent 318 GPU-days.

While these experiments test a wide range of sizes at sparsely spaced
locations across the flow, the ones in the last column of table 2 aim at
building heat maps that explore possible large-scale causality distributions
not limited to a single cubical cell. Each reference snapshot is divided into
a $30\times 30$ grid in the $x-z$ plane, approximately spaced by 75 wall
units, and perturbations are centred at each point of that grid. For cells
with $l_{\mbox{\scriptsize\it cell}}^{+}\geq 75$, this procedure uniformly
samples the whole plane but, due to its cost, it was limited to 20 initial
snapshots and five different heights, each of which only ran for 0.49
turnovers. The resulting 180000 tests spent 565 GPU-days.

In both sets of experiments, the temporal evolution of the perturbation is
measured by the energy of the perturbation velocity integrated over the whole
computational domain,

$\varepsilon_{\boldsymbol{u}}(t)=V^{-1}\,\int{|\boldsymbol{u}_{\mbox{\scriptsize\it
mod}}-\boldsymbol{u}_{\mbox{\scriptsize\it ref}}|^{2}dV},$ (3)

which evolves from some initial $\varepsilon_{\boldsymbol{u}}(0)$ at the
moment at which the perturbation is applied, to
$\varepsilon_{\boldsymbol{u}}(\infty)=2K\equiv
2V^{-1}\int|\boldsymbol{u}|^{2}{\,\rm{d}}V$ when the reference and perturbed
flow fields decorrelate after a sufficiently long time. For a chaotic system
such as turbulence,
$\varepsilon_{\boldsymbol{u}}(\infty)\gg\varepsilon_{\boldsymbol{u}}(0)$ and,
even if the evolution of the perturbation is far from linear over times of the
order of a turnover, the perturbation energy typically grows almost
exponentially for a while before levelling at
$\varepsilon_{\boldsymbol{u}}(\infty)$. These considerations lead to two
definitions of causal significance: an absolute one that disregards the
initial perturbation magnitude and vanishes as $t\to\infty$,

$\sigma_{\boldsymbol{u}}(t)=\log_{10}\varepsilon_{\boldsymbol{u}}(t)/\varepsilon_{\boldsymbol{u}}(\infty)=\log_{10}\varepsilon_{\boldsymbol{u}}(t)/2K,$
(4)

and a relative one,

$\sigma_{\boldsymbol{u}r}(t)=\log_{10}\varepsilon_{\boldsymbol{u}}(t)/\varepsilon_{\boldsymbol{u}}(0),$
(5)

which measures relative growth and vanishes at $t=0$. Both definitions
typically grow with time, but we will be interested in cases in which the
growth is particularly fast or slow, as defined by the top and bottom $\phi$
percentile of the significance distribution. For most of the paper,
experiments within the top $\phi=10\%$ of the significance distribution will
be defined as ‘causally significant’, and those in the bottom 10%, as
‘causally irrelevant’. This fraction is broadly compatible with the
percolation analysis often used to defined thresholds. For example, the
optimal percolation threshold in three-dimensional wall-bounded turbulence
fills volume fractions of order of 5%–10% (Jiménez, 2018a), while in two-
dimensional vorticity fields, which are more directly comparable with the
present application to individual planes, the covered area is closer to
20%–30% (Jiménez, 2020a). Tests using $\phi=5\%$ or 15% showed few differences
in the present results.

##  3 Temporal evolution of the significance

(a) (b)

Figure 3: Plane-averaged perturbation magnitude (6), unconditionally averaged
over all perturbations with $l_{\mbox{\scriptsize\it cell}}^{+}=75$ introduced
at a given height, normalized with its maximum at $t=0$. The light blue line
is the instantaneous position of the perturbation maximum. Contours are
${\langle\varepsilon_{\boldsymbol{u}}\rangle}(y,t)/\max_{y}\langle\varepsilon_{\boldsymbol{u}}\rangle(y,0)=10^{-4}(\times
10)10^{3}$. (a) $y_{c}^{+}=300$. (b) $y_{\mbox{\scriptsize\it cell}}=0$.

To further study the growth of the perturbations we use the $y$-dependent
averaged intensity,

${\varepsilon_{\boldsymbol{u}}}(y,t)=(L_{x}L_{z})^{-1}\,\iint|\boldsymbol{u}_{\mbox{\scriptsize\it
mod}}-\boldsymbol{u}_{\mbox{\scriptsize\it ref}}|^{2}{\,\rm{d}}x{\,\rm{d}}z,$
(6)

equivalent to (3) but integrated over wall-parallel planes instead of over the
whole domain, together with corresponding definitions for the significances.
To minimise notational clutter, we use for them the same symbols as in (3–5),
with the inclusion of $y$ as a parameter. Figure 3(a) shows the growth of
$\varepsilon_{\boldsymbol{u}}(y)$, unconditionally averaged over all the
perturbations introduced at a particular size and distance from the wall and
normalised with its maximum at $t=0$. The light blue line is the position at
which the perturbation is maximum. It initially stays at the height at which
the perturbation is introduced, but a new peak grows near the wall and becomes
dominant after $tu_{\tau}/h\approx 0.3$. In figure 3(b), where the
perturbation is initially attached to the wall, the peak is always attached.
During the very early stage of evolution $(tu_{\tau}/h\lesssim 0.05)$, a low-
intensity perturbation spanning the whole channel appears in both cases. This
is almost surely due to the pressure pulse that enforces continuity at the
edges of the perturbation cell, but it quickly dissipates and does not seem to
influence later development.

Figure 4: Attachment time as a function of $y_{\mbox{\scriptsize\it cell}}$,
for different $l_{\mbox{\scriptsize\it cell}}$. The dashed line is a least-
square fit to the curves with $y_{\mbox{\scriptsize\it cell}}>0$, with slope
$1.37u_{\tau}$. Times are computed for individual tests, and symbols and bars
are their average and standard deviation. Symbols as in table 2.

Figure 4 shows that the attachment time, $t_{att}$, defined for individual
tests as the moment when the perturbation maximum falls below $y^{+}=50$, and
later averaged over the experimental ensemble. It is approximately
proportional to $y_{\mbox{\scriptsize\it cell}}$, at least for
$y_{\mbox{\scriptsize\it cell}}^{+}\gtrsim 50$, with a propagation velocity
${\rm{d}}y_{\mbox{\scriptsize\it cell}}/{\rm{d}}t=1.37u_{\tau}$. This is
faster than the observed vertical advection velocity of coherent features in
channels, ${\rm{d}}y/{\rm{d}}t\approx\pm u_{\tau}$ (Lozano-Durán & Jiménez,
2014), and suggests that the perturbation is not simply advected by the flow,
but actively amplified by it. In fact, the production term in the evolution
equation for the perturbation energy is proportional to the mean shear (see
Appendix A), and the most likely interpretation of figure 4 is that, while all
perturbations are advected to and from the wall by the background turbulence,
those approaching the wall, where the shear is most intense, grow faster than
those moving away from it, resulting in a mean downwards migration of the
perturbation maximum. Notice, for example, the different slopes of downwards
and upwards contours in figure 3. It is also relevant that $t_{att}$ scales
better with the distance from the wall to the bottom of the cell,
$y_{\mbox{\scriptsize\it cell}}$, than with its centre, $y_{c}$ (not shown),
because it is the bottom that predominantly feels the stronger shear near the
wall.

(a)(b)

Figure 5: Temporal development of the unconditionally averaged domain-
integrated perturbation. The cell distance from the wall increases from blue
to red, as in table 2, and the diagonal dashed lines are the exponential
Lyapunov growth rate from Nikitin (2018). (a) $l_{\mbox{\scriptsize\it
cell}}^{+}=50$. (b) $l_{\mbox{\scriptsize\it cell}}^{+}=100$.

Fig 5 shows two examples of the temporal evolution of the domain-integrated
perturbation $\varepsilon_{\boldsymbol{u}}$. The two panels in the figure are
different cell sizes, and the line colour is the cell height. The figure shows
that $\varepsilon_{\boldsymbol{u}}$ is higher for larger cells, which is to be
expected since it is an integrated quantity, and also for lower
$y_{\mbox{\scriptsize\it cell}}$, also expected for a perturbation that
removes velocity fluctuations, which are stronger near the wall. More
interesting is that cells near the wall grow faster than those away from it,
which may be understood as supporting the model in which their growth rate is
controlled by the ambient shear.

The dashed straight lines in figure 5 are the exponential growth from the
Lyapunov analysis by Nikitin (2018), who reports a Lyapunov time for
$\varepsilon_{\boldsymbol{u}}$ (the inverse of the leading exponent) of
$T_{L}^{+}\approx 19\,(u_{\tau}T_{L}/h=0.032)$ in a turbulent channel at
$Re_{\tau}=586$. The leading Lyapunov vector is concentrated in the buffer
layer, and the exponent scales in wall units, again consistent with a model in
which the growth is controlled by the near-wall shear.

It is clear that our analysis shares many characteristics with the classical
Lyapunov analysis, albeit with important differences. The most obvious is that
the classical Lyapunov exponent assumes that the perturbation behaves linearly
for an infinitely long time, while figure 5 shows that our experiments
saturate for times that, even if much longer than $T_{L}$, remain of interest
for the flow evolution. A second important difference is that our initial
perturbations, which are intended to probe the local structure of the flow
rather than its mean properties, are compact with predetermined shapes, while
those in Lyapunov analysis are allowed to spread across the flow field to
their optimal structure. It may be relevant in this respect that there is an
initial transient in which perturbations decay in most of our tests,
$u_{\tau}t/h\lesssim 0.1$, and that this period is shorter for cells near the
wall. This is reminiscent of the similar transient in Lyapunov calculations,
during which perturbations align themselves to the most unstable direction.
Our limited range of initial conditions is probably partly compensated by the
substitution of the temporal averaging of classical analysis by averaging over
tests, and it is interesting that the short-time growth rate of the smallest
perturbations in figure 5(a), which mostly sample the buffer layer,
approximately agree with Nikitin (2018). Larger or higher perturbations, which
sample weaker shears, grow more slowly. We will provide in §5.1 further
support for the relevance of local shear to perturbation growth.

(a) (b) (c)

Figure 6: (a) Lines are the absolute significance (4) of individual
experiments as a function of time; only 20% of the total are included. The red
and blue bands are envelopes that respectively contain the 10% significant and
irrelevant samples. (b) Fraction of experiments that continue to be classified
as significant or irrelevant in terms of $\sigma_{\boldsymbol{u}}$ at
different times, after being so classified at $t=0$. Red: significants; blue:
irrelevants. The black horizontal line is the probability threshold,
$\phi=10\%$. (c) As in (a), for the relative significance (5). In all cases,
$l_{\mbox{\scriptsize\it cell}}^{+}=50,y_{\mbox{\scriptsize\it
cell}}^{+}=125$.

(a)(b)(c)

Figure 7: Classification time $t_{\mbox{\scriptsize\it sig}}$ as a function of
$y_{\mbox{\scriptsize\it cell}}$. Symbols as in table 2. The dashed lines in
(a,b) are least-square linear fits, whose slope is $2.12u_{\tau}$ in (a) and
$1.28u_{\tau}$ in (b). (a) Using $\sigma_{\boldsymbol{u}}$. (b) Using
$\sigma_{\boldsymbol{u}r}$. (c) Offset between the $\sigma_{\boldsymbol{u}r}$
classification time and the attachment time.

Figure 6(a) shows a typical evolution of the absolute significance,
$\sigma_{\boldsymbol{u}}$, for perturbations with a given
$l_{\mbox{\scriptsize\it cell}}$ and $y_{\mbox{\scriptsize\it cell}}$. Each of
the grey lines is the result of a different experiment, and the red and blue
regions are the envelopes of the significant and irrelevant samples,
individually classified according to their intensity at each moment of their
evolution. The figure shows that the perturbations approximately maintain the
ordering of their initial intensity. Initially stronger perturbations tend to
remain strong for long times, although it follows from its definition that
$\sigma_{\boldsymbol{u}}$ vanishes on average as $t\to\infty$. Figure 6(b)
displays the persistence of the causality classification based of
$\sigma_{\boldsymbol{u}}$, defined as the fraction of samples identified as
significant or irrelevant at $t=0$ that remain significant or irrelevant when
classified at subsequent times. In the case illustrated in the figure, 34% of
the initially significant samples and 29% of the initially irrelevant ones
remain at the end of our experiments in the same class in which they were
classified at $t=0$. This fraction is at least 20% in all the experiments in
this paper, which is substantially higher than the 10% expected from a random
selection.

Figure 6(c) shows the evolution of the relative significance. Unlike the
absolute significance, $\sigma_{\boldsymbol{u}r}$ vanishes at $t=0$ but does
not reach the same long-time limit in all cases. In fact,
$\sigma_{\boldsymbol{u}r}(\infty)=\log_{10}(2K)-\log_{10}\varepsilon_{\boldsymbol{u}}(0)$,
and $\sigma_{\boldsymbol{u}r}(\infty)$ is essentially equivalent to the
initial perturbation magnitude.

These considerations show that both $\sigma_{\boldsymbol{u}}$ and
$\sigma_{\boldsymbol{u}r}$ characterise the evolution of the perturbations at
short and intermediate times. The former mostly reveals that the perturbation
intensity stays approximately proportional to its initial value for some time,
while the latter, which compensates for this effect, describes its intrinsic
growth. When $\varepsilon_{\boldsymbol{u}}\to 2K$ at longer times, the system
forgets its initial conditions and neither measure of significance is very
useful.

Although figure 6(b) shows that the significance classification of a given
experiment is not a completely random variable, the fact that the persistence
is not unity implies that the time at which the classification is performed is
important. Consider, for example, the mean significance of the set $\\{I\\}$
of tests classified at time $t_{c}$ as irrelevant,
$\sigma_{I}(t_{c})=N_{\\{I\\}}^{-1}\sum_{j\in\\{I\\}}\sigma_{j}$, and define a
similar $\sigma_{S}(t_{c})$ average for significant perturbations. The
difference $\sigma_{S}-\sigma_{I}$ typically increases initially and reaches a
maximum before decaying at long times. The time, $t_{\mbox{\scriptsize\it
sig}}$, at which this difference is maximum is also when the classification is
less ambiguous, and we will preferentially use it from now on to define our
significance classes.

Figure 7(a,b) shows how $t_{\mbox{\scriptsize\it sig}}$ changes as a function
of $l_{\mbox{\scriptsize\it cell}}$ and $y_{\mbox{\scriptsize\it cell}}$,
using either $\sigma_{\boldsymbol{u}}$ or $\sigma_{\boldsymbol{u}r}$ as a
causality measure. Disregarding the case $l_{\mbox{\scriptsize\it
cell}}^{+}=25$, which is well within the dissipative range of scales and tends
to behave differently from larger cells, $t_{\mbox{\scriptsize\it sig}}$ is
mainly explained by $y_{\mbox{\scriptsize\it cell}}$, and it is clear that
$\sigma_{\boldsymbol{u}r}$ is a better indicator for this purpose than
$\sigma_{\boldsymbol{u}}$. We will mostly use it from now on. It is
interesting that $t_{\mbox{\scriptsize\it sig}}$ is very close to, and
generally slightly larger than, the attachment time in figure 4, $t_{att}$, as
shown by the difference of the two values in figure 7(c), suggesting again
that the arrival of the disturbances to the wall is an important factor in
determining causality.

We have seen above that the initial intensity of the perturbations has an
effect on their subsequent evolution. This is also true of our significance
classification, and can be quantified by the correlation of
$\sigma_{\boldsymbol{u}r}(t_{c})$ with $\varepsilon_{\boldsymbol{u}}(0)$ (not
shown). This correlation tends to $-1$ at long times, as explained above, but
remains moderately positive for $t_{c}\lesssim t_{\mbox{\scriptsize\it sig}}$,
confirming that the initial relative growth rate for strong perturbations is
faster than for weak ones. In most cases, $t_{\mbox{\scriptsize\it sig}}$
approximately coincides with the moment at which the correlation changes sign
and is close to zero, making the classification relatively independent of the
initial perturbation intensity. At this moment, the energy of the perturbation
is still a small fraction of the total energy of the flow. Taking as example
the topmost curve in figure 5(b) $(l_{\mbox{\scriptsize\it
cell}}^{+}=100,\,y_{\mbox{\scriptsize\it cell}}=0)$, the average energy of the
initial perturbations is approximately $3\times 10^{-4}K$, and grows to $0.6K$
at the end of the experimental runs, but it is still $8\times 10^{-3}K$ at the
optimal classification time $t_{\mbox{\scriptsize\it sig}}\approx
0.17h/u_{\tau}$. This does not mean that the perturbation can be linearised up
to that time. The intensity of the perturbation is always $O(K)$, and the
growth of its integrated energy is mostly due to its geometric spreading(see
figure 2).

##  4 Diagnostic properties for causal significance

Having described how the significance of an initial condition can be
characterised, we recover our original task of determining which properties of
the perturbed cells are responsible for their causality. The basic assumption
is that the characterisation of causality can be reduced to a single
observable of the cell at the perturbation time, $t=0$, such as its average
vorticity, rather than requiring several conditions to be simultaneously
satisfied, or even some property of the extended environment of the cell, or
of its history. As mentioned in the introduction, the strategy is to perform
many experiments modifying individual cells, to label them according to their
significance at some later time, and to test which cell observables at $t=0$
can be used to separate the classes thus labelled.

$\langle{u_{i}}\rangle_{c}$, $\langle{\omega_{i}}\rangle_{c}$ | Mean velocities and vorticities.
---|---
$\langle{u_{i}^{2}}\rangle_{c}$ | Kinetic energy.
$\langle{uv}\rangle_{c}$, $\langle{(uv)^{2}}\rangle_{c}$ | Mean and mean-squared Reynolds product.
$\langle{u_{i}^{2}}\rangle_{c}$, $\langle{\omega_{i}^{2}}\rangle_{c}$ (no sum over $i$) | Mean-squared components.
$\langle{\mbox{\it Prod}}\rangle_{c}=-\langle{uv\partial_{y}U}\rangle_{c}$ | Mean energy production.
$\langle{\mbox{\it Tdif}}\rangle_{c}=-\langle{\partial_{y}vu_{i}^{2}}\rangle_{c}/2$ | Mean energy transport.
$\langle{\partial_{y}u}\rangle_{c}$ | Mean shear.
$\langle\partial_{y}u\rangle_{\mbox{\scriptsize\it fw}}$ | Mean shear using only $y^{+}>5$.
$\langle{\partial_{i}u_{i}}\rangle_{c}$ (no summation) | Mass conservation.
Table 3: Cell observables. All averages are taken over cells. $\varepsilon_{\boldsymbol{u}}$, $\varepsilon_{\boldsymbol{\omega}}$ | Mean squared velocity and vorticity fluctuations.
---|---
$\langle{\omega_{i}^{2}}\rangle_{c}$, $\langle{s_{ij}s_{ij}}\rangle_{c}$ | Enstrophy, strain.
${u_{i}}^{\prime c}$, ${\omega_{i}}^{\prime c}$ (no sum over $i$) | In-cell standard deviations.
$\langle{P_{\varepsilon u}}\rangle_{c}$, $\langle{C_{\varepsilon u}}\rangle_{c}$, $\langle{D_{\varepsilon u}}\rangle_{c}$ | Production, transport and dissipation of $\varepsilon_{\boldsymbol{u}}$ (AppendixA).
Table 4: Perturbation and small-scale observables

Following Jiménez (2020b) and Encinar & Jiménez (2023), the ranking of
observables uses a linear kernel Support Vector Machine (SVM, Cristianini &
Shawe–Taylor, 2000), implemented in the scikit-learn Python library (Pedregosa
et al., 2011), which determines an optimal separating hyperplane between two
pre-labelled data classes. In our case, we look for the optimal separation of
significant or irrelevant experiments in terms of a single quantity, and the
SVM hyperplane reduces to a threshold. For each combination of
$l_{\mbox{\scriptsize\it cell}}$ and $y_{\mbox{\scriptsize\it cell}}$ in the
second column of table 2, and for each classification time $t_{c}$, two-thirds
of the initial conditions are collected into a training set, with the
remaining third reserved for testing. The 10% most significant experiments of
the training set are labelled as significant, and the bottom 10%, as
irrelevant. The remaining 80% are not used for classification purposes. An
optimum partition threshold is computed for each of the observables detailed
below, and a SVM classification score is assigned to each observable using the
test set. The score measures the fraction of data allocated to their correct
class by the SVM threshold, and ranges from unity for perfect separability to
0.5 for cases in which the two classes are fully mixed. The procedure is
repeated three times after randomly separating the data into training and test
sets, and the diagnostic score for the observable is defined as the average of
the three results.

The whole process can be automated and is reasonably fast. The experimental
description in §2.1 shows that each SVM run is only requested to classify two
sets of 96 points each, and to test the classification on two sets of 48
points. This allows us to minimise preexisting biases by testing many possible
observables.

The observables can be physically classified into average cell properties,
such as its kinetic energy, and perturbation or small-scale properties, such
as the kinetic energy of the velocity fluctuations with respect to the cell
mean. The former are summarised in table 3, and the latter in table 4. In both
cases, properties that are statistically symmetric with respect to reflections
on $z$ are used as absolute values, and positive definite quantities, such as
mean squares, are used as logarithms. Otherwise, all observables are processed
in the same way.

Figure 8: Classification score of the three best observables for a
classification based on absolute significance, $\sigma_{\boldsymbol{u}}$. Row:
rank; Column: evaluation time in turnovers. Colour: classification score.
$l_{\mbox{\scriptsize\it cell}}^{+}=50$, $y_{\mbox{\scriptsize\it
cell}}^{+}=125$.

The diagnostic score of an initial condition depends on the cell height, on
its size, and on the moment at which it is classified. Figure 8 shows a
typical table of the three best observables identified by the absolute
significance $\sigma_{\boldsymbol{u}}$, as functions of the classification
time. Cells are coloured by the classification score. In all cases, the best
observable is the initial perturbation amplitude
$\varepsilon_{\boldsymbol{u}}$, a related quantity such as
$\varepsilon_{\boldsymbol{\omega}}$, or the viscous dissipation of the
perturbation intensity $\langle{D_{\varepsilon u}}\rangle_{c}$. Although not
included in the table, the next best observable is usually also a small-scale
quantity closely correlated with the intra-cell velocity fluctuations, such as
$s_{ij}s_{ij}$, $\omega_{i}^{2}$ or the in-cell standard deviation
${v}^{\prime c}$. Table 8 is thus equivalent to the fluctuation persistence in
figure 6(a,b).

(a)(b)(c)

Figure 9:  Classification score of the two best observables for a
classification based on relative significance, $\sigma_{\boldsymbol{u}r}$. (a)
$l_{\mbox{\scriptsize\it cell}}^{+}=150,\,y_{\mbox{\scriptsize\it cell}}=150$,
(b) $l_{\mbox{\scriptsize\it cell}}^{+}=75,\,y_{\mbox{\scriptsize\it
cell}}=138$, (c) $l_{\mbox{\scriptsize\it
cell}}^{+}=25,\,y_{\mbox{\scriptsize\it cell}}=138$. Row: rank; Column:
evaluation time in turnovers. The highlighted columns are
$t_{\mbox{\scriptsize\it sig}}$. Colour: classification score.
$y_{\mbox{\scriptsize\it cell}}^{+}\approx 150$.

(a)(b)(c)

Figure 10: Classification score of the two best observables for a
classification based on relative significance, $\sigma_{\boldsymbol{u}r}$. (a)
$y_{\mbox{\scriptsize\it cell}}^{+}=275$, (b) $y_{\mbox{\scriptsize\it
cell}}^{+}=125$, (c) $y_{\mbox{\scriptsize\it cell}}=0$. Row: rank; Column:
evaluation time in $h/u_{\tau}$. The highlighted columns are
$t_{\mbox{\scriptsize\it sig}}$. Colour: classification score.
$l_{\mbox{\scriptsize\it cell}}^{+}=50$.

(a) (b)

(c) (d)

(e) (f)

Figure 11: (a) Time at which the sum of the scores of the four best four
observables is lowest. Dashed: $t_{\mbox{\scriptsize\it sig}}$ from figure
7(b). Other symbols as in table 2. (b-f) Classification score of selected
observables as a function of time, computed from $\sigma_{\boldsymbol{u}r}$. ,
Small-sale quantities, average of $\varepsilon_{\boldsymbol{u}}$ and
$\langle{s_{ij}s_{ij}}\rangle_{c}$; $\times$, cell-scale quantities, average
of $\langle{u}\rangle_{c}$ and $\langle{v}\rangle_{c}$. Abscissae are offset
by the attachment time $t_{att}$. Colour intensity increases with distance
from the wall, from $y_{\mbox{\scriptsize\it cell}}^{+}=12.5$ to
$y_{\mbox{\scriptsize\it cell}}^{+}=275$, excluding $y_{\mbox{\scriptsize\it
cell}}=0$. (b) $l_{\mbox{\scriptsize\it cell}}^{+}=25$. (c)
$l_{\mbox{\scriptsize\it cell}}^{+}=50$. (d) $l_{\mbox{\scriptsize\it
cell}}^{+}=75$. (e) $l_{\mbox{\scriptsize\it cell}}^{+}=100$. (f)
$l_{\mbox{\scriptsize\it cell}}^{+}=150$.

Much of this persistence can be compensated by using the relative significance
$\sigma_{\boldsymbol{u}r}$. Figure 10 displays the two best observables for
different sizes at a fixed cell height, and figure 10 displays results for a
given size and different heights. In both cases the best score starts being
relatively high at short classification times, decreases for intermediate
ones, and increases again towards the end of the experimental run. The
evolution of the optimum diagnostic variables with the classification time can
be divided in three phases.

During the initial phase, up to the time when the scores are lowest, the best
observables include $s_{ij}s_{ij}$, the magnitude of the initial disturbance,
and the disturbance production and dissipation, all of which are either highly
correlated with the initial value of $\varepsilon_{\boldsymbol{u}}$, or are
terms in its evolution equation. This part of the table is equivalent to the
observation in §2.1 that initially stronger perturbations not only remain
strong, but also grow faster than weaker ones.

The second phase is the broad minimum of the score around
$t_{\mbox{\scriptsize\it sig}}$. While the scores in this phase are not high,
the best observables change from the small-scale perturbation properties of
the initial phase to properties of the cell that do not include fluctuations,
such as the cell average of some velocity component or, equivalently, the mean
shear when the cells are very close to the wall. It is interesting that the
best measure of shear near the wall is
$\langle\partial_{y}u\rangle_{\mbox{\scriptsize\it fw}}$, which excludes the
viscous sublayer (see figure 10c). This is consistent with the idea that the
growth of the perturbation is due to the energy production by the local shear,
because the fluctuation production term in Appendix A is proportional to the
shear, but also to the Reynolds stresses, $\widehat{u}_{i}\widehat{u}_{j}$,
which are inactive in the sublayer.

It is interesting that the longitudinal velocity derivatives,
$\langle{\partial_{x}u}\rangle_{c}$ and $\langle{\partial_{z}w}\rangle_{c}$,
appear among the most diagnostic cell properties for wall-attached
perturbations in figure 10(c). These derivatives are involved in mass
conservation, and follow naturally from the meandering of near-wall streaks,
which has been associated with streak breakdown (Jiménez & Moin, 1991;
Hamilton et al., 1995; Waleffe, 1997) and with the generation of Orr bursts
(Orr, 1907; Jiménez, 2013). Although not apparent from figure 10(c), it can be
shown that significant cells are associated with
$\partial_{x}u<0,\partial_{z}w>0$, with the opposite association for
irrelevant ones.

Finally, at longer times of the order of $t-t_{\mbox{\scriptsize\it
sig}}\approx 0.25h/u_{\tau}$ the score recovers, and the most diagnostic
observable reverts to the small-scale quantities that dominate short times.
Since we saw in §2.1 that $\sigma_{\boldsymbol{u}r}$ at long times is
essentially equivalent to $\varepsilon_{\boldsymbol{u}}(0)$, this final phase
is a reflection of the behaviour at short times, and does not represent new
physics.

Figure 11(a) shows the dependence on $y_{\mbox{\scriptsize\it cell}}$ of the
time at which the accumulated score of the top four observables reaches its
minimum. After an initial transient that gets shorter as the cell size
increases, $t_{\mbox{\scriptsize\it min}}$ grows with the distance from the
wall, and approximately tracks the optimum classification time,
$t_{\mbox{\scriptsize\it sig}}$.

Figure 11(b–f) summarises the evolution of the scores as functions of time.
The blue lines are averages of the scores of several fluctuation quantities,
and the red ones are averages of cell-scale properties. The figures are offset
by their attachment time, $t_{att}$, which improves their collapse
significantly, and reflect the decreasing influence of the small-scale
quantities as the perturbations approach the attachment time, as well as the
increasing importance of the cell-scale properties as the perturbations
intensify. It should be mentioned that offsetting $t$ with the optimum
classification time, $t_{\mbox{\scriptsize\it sig}}$, instead of with
$t_{att}$, also collapses most scores, as could be expected from the
similarity of both times in figure 7. It also collapses better the case
$y_{\mbox{\scriptsize\it cell}}=0$, which is not included in figure 11, and
for which $t_{att}$, defined at the arbitrary distance $y^{+}=50$, does a poor
job. In spite of this, $t_{att}$ is used in figure 11 because it improves the
case $l_{\mbox{\scriptsize\it cell}}^{+}=25$, and underscores the already
mentioned connection between significance and the energy production from the
near-wall shear. It is also interesting that the score of the velocities has a
secondary maximum at $t=0$, probably due to the known correlation of small-
scale vorticity with the large-scale streamwise-velocity streaks (Tanahashi et
al., 2004).

Figure 12: Best observable at $t_{\mbox{\scriptsize\it sig}}$. Color: score.
The bottom of each tile in the figure aligns with the left-hand
$y_{\mbox{\scriptsize\it cell}}$ axis.

From now on, we will mostly focus on results classified at
$t_{\mbox{\scriptsize\it sig}}$, for which the optimum observables are
collected in figure 12. As seen above, they are mostly the average cell
velocities, $\langle{u}\rangle_{c}$ or $\langle{v}\rangle_{c}$. The exceptions
are cells in the buffer layer $y_{\mbox{\scriptsize\it cell}}^{+}\lesssim 50$,
in which shear can probably be taken as a proxy for the streamwise velocity,
and cells with $l_{\mbox{\scriptsize\it cell}}^{+}=25$ very far from the wall.
We have already mentioned that these perturbations are probably too small to
survive for the relatively long times required to reach the wall, in agreement
with the assimilation results from Wang & Zaki (2022) mentioned in the
introduction.

##  5 Conditional flow fields

(a) (b)

(c) (d)

(e) (f)

Figure 13: Streamwise section at $z=z_{c}$ of the conditional velocity field
of the reference flow at $t=0$ around the perturbation cell. The colour
background is the conditional streamwise velocity. Arrows are velocity
fluctuation vectors parallel to the plane of the figure, and the light-
coloured box is the perturbation cell. $l_{\mbox{\scriptsize\it
cell}}^{+}=75$. (a,b) $y_{\mbox{\scriptsize\it cell}}^{+}=113$. (c,d)
$y_{\mbox{\scriptsize\it cell}}^{+}=62.6$ (e,f) $y_{\mbox{\scriptsize\it
cell}}=0$. (a,c,e) Significants. (b,d,f) Irrelevants.

(a) (b)

(c) (d)

(e) (f)

Figure 14: As in figure 14, for a wall-parallel section at $y=y_{c}$.

(a)(b)

Figure 15: As in figure 14(a,b), for the $(z-y)$ cross-flow section at
$x=x_{c}$. (a) Significants. (b) Irrelevants. $y_{\mbox{\scriptsize\it
cell}}^{+}=113,\,l_{\mbox{\scriptsize\it cell}}^{+}=75$.

While we have seen that the cell-averaged velocity fluctuations are diagnostic
quantities for causality, our analysis did not include their sign. Figures 14
to 15, which display averaged velocity fields conditioned to the position of
significant or irrelevant perturbation cells, shows that the sign is
important.

Figure 14 displays longitudinal $(x-y)$ sections of the streamwise and wall-
normal velocities, conditioned to either significant (left column) or
irrelevant cells (right column). It is evident that the former are biased
towards fourth-quadrant regions $(u>0,\,v<0)$, while the latter are in second-
quadrant regions $(u<0,\,v>0)$.

The three rows in figure 14 corresponds to perturbations introduced at
decreasing distances from the wall. The frames are centred at the streamwise
position of the perturbation cell, and the figure shows a fluid wedge entering
the frame from its downstream right-hand edge, becoming more prominent as the
cell approaches the wall. At the same time, there is an upstream drift of the
darker core of the velocity distribution. This is clearest in the significant
cases in the left-hand column, where the incoming wedge is low-speed fluid,
but it can also be traced in the irrelevant cases in the right-hand column.
The result is that the position of the causally significant cells moves
downstream towards an interface at which high-speed fluid overtakes a low-
speed one. Irrelevants are associated to an interface in which low-speed fluid
is left behind by higher speed ahead of it.

Figure 14 displays wall-parallel $(x-z)$ sections of the same cases as figure
14, and figure 15 shows cross-flow $(z-y)$ sections of figure 14(a,b). There
is no orientation ambiguity in the longitudinal sections in figure 14, but
figures 14 and 15 would be statistically symmetric even if individual flow
fields were not. To preserve possible systematic asymmetries, the $z$
coordinate of all the flow fields is reflected so that the spanwise velocity
averaged over a cube of side $3l_{\mbox{\scriptsize\it cell}}$, centred at the
perturbation cell and possibly truncated by the wall, is $\langle
w\rangle_{3c}<0$. The orientation of the sections in figures 14 and 15 is
therefore not physically meaningful, but the asymmetry of the different frames
is consistent and complements the information in figure 14.

Figure 14 shows that the velocity interfaces in figure 14 correspond to kinks
in the large-scale streaks that dominate the flow.

Although the frames in figures 14 or 14 represent unrelated experiments, it is
tempting to interpret them as a temporal evolution in which perturbations
introduced farther from the wall correspond to earlier times, and travel
downstream and towards the wall until they reach it at $t_{att}$ (or
$t_{\mbox{\scriptsize\it sig}}$). The velocity maximum in figures 14(a,c,e)
shifts by $\Delta x^{+}\approx 200$ from the top to the bottom row of frames.
Assuming, from figure 4, that $u_{\tau}t_{att}/h\approx 0.2$, this corresponds
to a velocity difference $\Delta u^{+}\approx 1.5$, which is a reasonable
estimate for the difference in the advection velocity of features in a high-
velocity streak with respect to the average flow velocity (Krogstad et al.,
1998; Lozano-Durán & Jiménez, 2014).

Interestingly, the streamwise drift of the irrelevant velocity features in the
right-hand columns of figures 14 and 14 is less clear than in the significant
ones in the left-hand columns. The environment of the irrelevants can rather
be described as an interface that gets wider as the cell approaches the wall.
The flow cross-sections in figure 15 support this description, and the
combined evidence from the three sets of sections is consistent with a model
in which causally significant cells are associated with the front of a high-
speed sweep that steepens as it approaches the wall and overtakes a lower-
speed region. Conversely, irrelevant cells are located at the trailing end of
a high-speed region that leaves behind a lower-speed flow. We may recall at
this point that the table in figure 10(c) showed that the wall-parallel mass-
conservation derivatives are diagnostic of causality near the wall and that,
while $\partial_{x}u<0$ signals significance, $\partial_{x}u>0$ signals
irrelevance. This asymmetry suggests that the generation of structures strong
enough to have a global effect on the flow depends on mass conservation
failures when streaks of different velocity run into each other. This is most
probably due to meandering, as in figure 14(a,c,e), and the effect is
strongest when this happens within the strong shear near the wall. The
trailing edge of the meander, as in figure 14(b,d,f), or the tails at which
high-speed streaks pull away from low speed ones, are passive.

The association of strong near-wall $v$ structures with the downstream end of
high-speed streaks and with the upstream end of low-speed ones was already
noted by Jiménez et al. (2004) and Jiménez & Kawahara (2013). Their
interpretation was that $v$ creates the streaks, but the arguments above,
together with the fact that causality can be traced to flow locations far from
the wall, suggest that the sequence of events is the other way around, and
that the formation of near-wall bursts depends on continuity failures of
nonuniform $u$-streaks.

###  5.1 The shear time

(a) (b)

Figure 16: Plane-averaged perturbation magnitude (6), conditionally averaged
over significant or irrelevant perturbations and normalized with the maximum
of the conditioned initial value. The bold cyan lines are the instantaneous
position of the perturbation maximum. Filled contours and solid line are
significants; line contours and dashed line are irrelevants. Contour levels
are
${\langle\varepsilon_{\boldsymbol{u}}\rangle}(y,t)/\max_{y}\langle\varepsilon_{\boldsymbol{u}}\rangle(y,0)=10^{-4}(\times
10)10^{3}$. Classification is done at $t_{\mbox{\scriptsize\it sig}}$ using
$\sigma_{\boldsymbol{u}r}$. $l_{\mbox{\scriptsize\it cell}}^{+}=75$. (a)
$y_{\mbox{\scriptsize\it cell}}^{+}=113$. (b) $y_{\mbox{\scriptsize\it
cell}}^{+}=263$

Figure 16 is similar to the evolution of the plane-averaged fluctuation
magnitude in figure 3, but is here separated into conditionally significant
cases (filled contours) and irrelevant ones (lines). In both cases, as well as
in the unconditional evolution in figure 3, the perturbation initially remains
at the height at which it is introduced, before spreading vertically. All
perturbations eventually fill the channel, as expected for a chaotic system,
but the growth is faster in the significant cases. The bold cyan lines in
figure 16 are the wall-normal position of the perturbation maximum. It is
clear that the solid lines representing significants initially trend downwards
and attach to the wall faster than the dashed lines representing irrelevants,
which initially trend away from the wall or drift little. This supports the
interpretation that significants are sweep-like, and the conjecture in §3 and
§4 that causal significance depends on the amplification of perturbations by
the strong shear near the wall.

(a) (b)

Figure 17: Relative perturbation growth, $\sigma_{\boldsymbol{u}r}(t)$,
conditioned to significant (red) and irrelevant (blue) samples, classified at
$t_{\mbox{\scriptsize\it sig}}$. Solid line, mean; shading: standard
deviation; , $t_{\mbox{\scriptsize\it sig}}$. (a) In eddy turnovers. (b) In
local shear time. The dashed part of the irrelevant line in (b) corresponds to
times for which not all experiments are available, because some of them end
within the plot. $l_{\mbox{\scriptsize\it cell}}^{+}=75$,
$y_{\mbox{\scriptsize\it cell}}^{+}=133$.

This is tested directly in figure 17. Figure 2 shows that perturbations spread
with time along the three directions, and that it is difficult to define an
instantaneous location to measure the shear that they encounter, but figures 3
and 16 suggests that it is possible to define an effective shear by weighting
the mean velocity profile, which depends only on $y$, with the perturbation
magnitude,

$S_{\varepsilon}(t)=\frac{\int\varepsilon_{\boldsymbol{u}}(y,t)\,\partial_{y}U(y){\,\rm{d}}y}{\int\varepsilon_{\boldsymbol{u}}(y,t){\,\rm{d}}y},$
(7)

and an effective shear time

$T_{S}(t)=\int{S_{\varepsilon}{\,\rm{d}}t}.$ (8)

Figure 17(a) illustrates the development of the relative growth of causally
significant and irrelevant perturbations in terms of the global eddy-turnover-
time. The significant perturbations grow from the start, while the irrelevant
ones initially decay and only later grow to match the causal case. Most of the
initial decay and of the slow growth of irrelevants can be attributed to their
failure to initially approach the wall. Figure 17(b) plots the same data using
the local shear time computed for individual experiments. The two evolutions
now approximately coincide, supporting the importance of the local shear, and
providing an explanation for the association of sweep-like flows with
causality. The effect of the negative wall-normal velocity is to bring
perturbations close to the wall. Ejection-like regions move perturbations away
from the wall to layers where the shear is low, and they become causally
relevant only after they eventually diffuse into the near-wall layer.

The collapse with the shear time only applies to significant and irrelevant
perturbations at the same distance from the wall. Perturbations introduced at
different distances behave differently, at least at the relatively low
Reynolds number of our experiments for which self-similar behaviours with
respect to $y$ are necessarily limited.

###  5.2 Geometry of causal events

While we have seen that significant and irrelevant cells are associated with
sweep- and ejection-like regions of the flow, it remains unclear whether the
quadrants discussed in the previous section are the same as the intense events
traditionally associated with sweeps and ejections in wall turbulence (Lu &
Willmarth, 1973). This is the purpose of the experiments in the second column
of table 2, which cover selected wall-parallel planes with a dense grid of
perturbation experiments.

(a) (b)

Figure 18: Joint probability density function of the cell-averaged velocities. Line contours are unconditional. Filled ones are conditioned to: (a) significant cells; (b) irrelevants. Both contain 60% and 99% of the data. $y_{\mbox{\scriptsize\it cell}}^{+}=113$, $l_{\mbox{\scriptsize\it cell}}^{+}=75$. The solid hyperbolae are the $H_{-}$ threshold that isolates the 10% most intense velocity quadrants with $\langle{v}\rangle_{c}<0$, as in (9), and the dashed ones are $H_{+}$ for $\langle{v}\rangle_{c}>0$.  $y_{\mbox{\scriptsize\it cell}}^{+}$ |  $H_{-}$ |  $H_{+}$ |  $S_{1}(\%)$ |  $S_{2}(\%)$ |  $S_{3}(\%)$ |  $S_{4}(\%)$
---|---|---|---|---|---|---
1 |  0.74 |  0.75 |  4.17 |  45.94 |  3.14 |  46.75
38 |  0.77 |  0.87 |  2.58 |  47.17 |  1.81 |  48.44
113 |  0.80 |  1.03 |  1.44 |  48.08 |  2.00 |  48.47
188 |  0.76 |  1.04 |  1.39 |  48.17 |  2.72 |  47.72
263 |  0.75 |  0.99 |  0.92 |  48.25 |  4.14 |  46.69
– |  1.75 |  1.75 |  4.40 |  61.54 |  6.59 |  27.47
Table 5: Parameters of intense quadrant structures for cell-averaged
velocities, compiled over wall-parallel planes for $l_{\mbox{\scriptsize\it
cell}}^{+}=75$. The thresholds $H_{-}$ and $H_{+}$ are as in figure 18, and
the $S_{j}$ are the fraction of the intense area associated to each quadrant.
The bottom row are volume fractions for point-wise quadrant structures in the
$Re_{\tau}=935$ channel of Lozano-Durán et al. (2012), in which the combined
intense quadrants fill 9% of the channel volume.

Figure 18 displays the resulting quadrant plot, drawn for cell-average
quantities. The line contours are the unconditional joint probability density
function of $\langle{u}\rangle_{c}$ and $\langle{v}\rangle_{c}$, while the
coloured ones are conditioned to either significant cells in figure 18(a), or
irrelevants in figure 18(b). The hyperbolic lines in the figure are intensity
limits for sweeps and ejections, defined as

$|\langle{u}\rangle_{c}(\boldsymbol{x})\langle{v}\rangle_{c}(\boldsymbol{x})|\geq
H_{\pm}\langle{u}\rangle_{c}^{\prime}(y)\langle{v}\rangle_{c}^{\prime}(y).$
(9)

In the classical quadrant plot for point velocities, the threshold $H$ only
depends on $y$ (Lozano-Durán et al., 2012), and results in different volume
fractions for sweeps and for ejections (see the last line in table 5). To
facilitate comparison with the our choice of a common area fraction for
significant and irrelevant cells, figure 18 uses two different thresholds:
$H_{-}$ for sweep-like structures with $\langle{v}\rangle_{c}<0$, and $H_{+}$
for ejection-like ones with $\langle{v}\rangle_{c}>0$. They are adjusted so
that the total areas for sweeps and for ejections are the ones used for the
significance analysis, $\phi=10\%$. They are given in table 2 for the five
experimental wall distances, and are lower than the $H\approx 1.75$ used in
Lozano-Durán et al. (2012) and in other studies. Correspondingly, they select
a larger area fraction, 20% in total, rather than the approximately 9% volume
fraction in Lozano-Durán et al. (2012). However, table 5 shows that the
distribution of quadrants in these intense regions is not very different from
the classical values, once the relative fractions of sweep- and ejection- like
structures are taken into account. As in the case of point velocities, most
strong structures are either pure sweeps,
$\langle{u}\rangle_{c}>0,\,\langle{v}\rangle_{c}<0$, or pure ejections,
$\langle{u}\rangle_{c}<0,\,\langle{v}\rangle_{c}>0$, and there are
comparatively few intense $Q_{1}$ or $Q_{3}$. The filled contours in figure
18(a) are cell-averaged velocities of the significant cells, and those in
figure 18(b) are irrelevant ones. It is evident that significants tend to be
in strong $Q_{2}$ sweeps, while irrelevants are in strong $Q_{4}$ ejections.

(a) (b)

Figure 19: Area fraction of the significance structures intersected by intense
quadrants. $l_{\mbox{\scriptsize\it cell}}^{+}=75$. Red, $Q_{4}$; blue,
$Q_{2}$; black, $Q_{1}\cup Q_{3}$. Solid lines are conditioned to: (a)
significants, (b) irrelevants. Dashed ones are unconditional.

This association is quantified in figure 19. The area fraction of the
intersection between two classes, $A$ and $B$, is defined as
$\Gamma(A,B)=2S(A\cap B)/(S(A)+S(B))$, where $S$ denote the area covered by
each class. Figure 19(a) shows the intersection of significant structures with
intense quadrants, $Q_{j}$. The dashed lines are area fractions of the
intersection of randomly $Q_{j}$ structures with the same area as the
significance structures. Figure 19(b) repeats the analysis for irrelevants. It
is again clear that significants predominantly overlap $Q_{4}$, and
irrelevants overlap $Q_{2}$, with a maximum at $y^{+}\approx 100-150$.

(a) (b) (c)

Figure 20: Heat maps of the relative significance $\sigma_{\boldsymbol{u}r}$
at various evaluation times. Note that the grid shows the position of the
initial cells, whereas the colour indicates their relative significance
evaluated some time in the future. $l_{\mbox{\scriptsize\it cell}}^{+}=75$,
$y_{\mbox{\scriptsize\it cell}}^{+}=113$. (a) Evaluation time,
$0.01h/u_{\tau}$. (b) $0.14h/u_{\tau}$. (c) $0.28h/u_{\tau}\approx
t_{\mbox{\scriptsize\it sig}}$.

The question of whether significant cells are organised into structures
similar to those of intense quadrants is addressed in figure 20. Each panel
displays the same plane away from the wall. Cells are shown at their position
at $t=0$, but labelled by their relative significance evaluated some time
after they are perturbed. The evaluation time increases from left to right.
The heat map in figure 20(a) is featureless, reflecting the difficulty
discussed in §3 of predicting the future significance of a cell from its
growth at short times. The organisation increases in figure 20(b), and is best
developed in figure 20(c), where significance is evaluated at the optimum
classification time, $t_{\mbox{\scriptsize\it sig}}$. The size and
organisation of the significance in the last figure are very similar to those
in the velocity maps in figure 14(a,b), with structures of $O(h)$ organised
into longer streaky structures.

This suggests the possibility that irrelevants and significants are
essentially the same as sweeps and ejections. To test this hypothesis, both
sets of cells are collected into individual connected objects for which the
product $\langle{u}\rangle_{c}\langle{v}\rangle_{c}$, or
$\sigma_{\boldsymbol{u}r}$, are above or below the threshold required to
isolate the $\phi\%$ area fraction of their wall-parallel plane, as in Lozano-
Durán et al. (2012). It is known that the wall-attached sweeps and ejections
of the point-wise velocity form spanwise pairs (Lozano-Durán et al., 2012). A
similar analysis is done here for the cell-averaged quadrants and for the
significance structures. The centroid of all the structures and their pairwise
distance is computed first, and two structures, such as $A_{i}$ and $B_{j}$
are defined as a pair if $B_{j}$ is the closest object to $A_{i}$ and $A_{i}$
is the closest object to $B_{j}$. It can be shown that the probability that
significant and irrelevant structures are part of a close pair is similar to
that of the $Q_{2}$ and $Q_{4}$, and much larger than for randomly located
objects.

(a) (b) (c)

Figure 21: Joint probability density function of the relative position of
structures in close pairs. Filled contours are irrelevant–significant pairs,
and lines are $Q_{2}$–$Q_{4}$. Contours contain 60% and 99% of the data, and
$z$-symmetry is enforced. $l_{\mbox{\scriptsize\it cell}}^{+}=75$. (a)
$y_{\mbox{\scriptsize\it cell}}^{+}=0$. (b) $y_{\mbox{\scriptsize\it
cell}}^{+}=113$. (c) $y_{\mbox{\scriptsize\it cell}}^{+}=263$.

Figure 22: Mean spanwise distance among nearest significance or quadrant
structures. Lines with symbols are significant–irrelevant pairs, with symbols
denoting $l_{\mbox{\scriptsize\it cell}}$, as in table 2. The dashed line is
$Q_{2}$–$Q_{4}$.

Figure 21 presents joint probability density functions of the position of the
nearest structures of similar kinds. Filled contours depict irrelevants around
significants, and lines indicate ejections around sweeps. Both use spanwise
symmetry to enhance statistical convergence, and it is clear that the pairs of
the two types of structures have a similar organisation. Figure 22 shows the
average spanwise width of the pairs as a function of wall distance and cell
size. It is known from Lozano-Durán et al. (2012) that this width scales with
$y$ for attached $Q_{2}$–$Q_{4}$ pairs far from the wall. Our Reynolds number
is too low for this self-similarity to hold, but figure 22 shows that, at
least for the two smallest cell sizes, the width of the significance pairs
grows with the distance from the wall, and approximately follows that of the
quadrants. It is difficult to make an exact correspondence between single
planes and slabs of relatively large cells, and the green line in the figure
is probably a reflection of this difficulty. Its cell size,
$l_{\mbox{\scriptsize\it cell}}^{+}=150$, is of the same order as the distance
from the wall. Similarly, the apparently large discrepancy of the
$l_{\mbox{\scriptsize\it cell}}^{+}=75$ red line with the quadrant pairs at
$y_{\mbox{\scriptsize\it cell}}=0$ is put in perspective by figure 21(a),
which represents the same data.

(a) (b)

Figure 23: Joint probability density function of the relative position of
closest structures of different types. (a) Quadrants around significants. (b)
Quadrants around irrelevants. Red, nearest $Q_{4}$; blue, nearest $Q_{2}$.
Contours contain 60% of data. $y_{\mbox{\scriptsize\it cell}}^{+}=113$.
$l_{\mbox{\scriptsize\it cell}}^{+}=75$.

Finally, figure 23 shows the relative position of intense quadrants with
respect to significance structures. Figure 23(a) is centred on significants,
and shows that the closest sweep coincides with the significant structure,
while the closest ejection avoids it. Figure 23(b), which is centred on
irrelevants, shows that the opposite is true for them.

In summary, the results in this section show that highly significant and
irrelevant structures respectively coincide with intense ejections and sweeps,
at least statistically. They are organised in a similar way, and they most
probably refer to the same structures, although it should be emphasised that
this does not imply that all significance structures are intense quadrants, or
vice versa.

##  6 Discussion and conclusions

We have analysed the causal relevance of flow conditions in wall-bounded
turbulence, using ensembles of interventional experiments in which the effect
of locally perturbing the flow in a small cell is monitored at some future
time. We have shown that the evolution of the kinetic energy of the
perturbation velocity is mostly determined by its initial intensity, but that,
when the effect is characterised by the relative amplification of the
perturbation energy, causality depends on the cell size, on the flow condition
within the cell, and on its distance from the wall. It is then possible to
enquire which properties of the flow at the time of the initial perturbation
determine causality and, in this way, use the causality experiments as probes
of the flow dynamics, rather than simply as a reflection of the dynamics of
the perturbations (Jiménez, 2020b; Encinar & Jiménez, 2023).

We have shown that there is an optimum time at which causality can be measured
most effectively, because the influence of different cell conditions is most
pronounced. This time is proportional to the distance from the wall of the
original intervention, and we have related it to the mean shear that the
perturbation experiences as it evolves. When time is normalised with this
shear, the evolution of perturbation applied to causally significant and to
causally irrelevant cells collapses reasonably well (figure 17).

For perturbations away from the wall, the variables that predominantly
determine causality are the streamwise and wall-normal velocities within the
cell, $\langle{u}\rangle_{c}$ and $\langle{v}\rangle_{c}$, with the latter
becoming more influential farther from the wall. For wall-attached
perturbations, the dominant variable is the local cell-averaged wall shear.
Positive $\langle{u}\rangle_{c}$, negative $\langle{v}\rangle_{c}$ and high
wall shear are associated with high causality, and negative
$\langle{u}\rangle_{c}$, positive $\langle{v}\rangle_{c}$ and low shear are
associated with irrelevance. This, together with the shear scaling mentioned
above, suggests that wall-detached significant cells are predominantly
associated with sweeps that carry the perturbation towards the stronger shear
near the wall, whereas irrelevant ones are associated with ejections that
carry it towards the weaker shear in the outer layers. This is confirmed by
the conditional flow fields in figures 14–15, and by the quadrant analysis in
figures 18–19.

We have also shown that causally significant and irrelevant cells are
themselves organised into structures that share many characteristics of
classical sweeps and ejections. For example, the latter are known to be
organised in spanwise pairs, and we have shown that the same is true of
causally significant and irrelevant structures. The dimensions of the two
types of pairs are similar, and their relative positions are consistent with
the identification of sweeps with significants and of ejections with
irrelevants (figure 23).

However, as already noted at the end of §5.2, not all sweeps and ejections are
causally significant or irrelevant. Figures 14–14 show that, as the
perturbation experiments are performed closer to the wall, significant cells
move towards the downstream end of the sweep, while irrelevant ones drift
towards an interface between the ejection and a sweep downstream and
underneath it. At the wall, this is consistent with a causally significant
configuration in which a high-speed streak overtakes a low-speed one, and with
a causally irrelevant situation in which the two streaks pull apart. In fact,
$\langle{\partial_{z}w}\rangle_{c}$ and $\langle{\partial_{x}u}\rangle_{c}$
are among the leading indicators of causality for some wall-attached
perturbations (figure 10).

This raises the question of how two structures that form a close pair can lead
to different outcomes. A similar question was raised by Lozano-Durán & Jiménez
(2014) when they found that the vertical advection velocity of the sweep and
ejection components of attached pairs are $-u_{\tau}$ and $+u_{\tau}$,
respectively. During the lifetime of the pair, this leads to relative vertical
displacements of the order of the height of the pair, and to its dissolution.
The answer offered at the time was that this was the mechanism that limits the
lifetime of the pair, and a similar answer may apply here, since one of the
results of this paper is that causality can only be traced to the moment that
significant perturbations reach the wall. In fact, sweep–ejection pairs are
known to be located at the interface between high- and low-velocity streaks
Lozano-Durán et al. (2012), and it is easy to see that their effect on the
streamwise velocity would be to deform the streaks into meandering. Although
the statistical confirmation of this model is beyond the scope of the data
used in the present paper, figure 14 strongly suggests that the association of
causality at the wall with the streamwise variation of $\langle{u}\rangle_{c}$
refers to the leading and trailing edges of a streak meander. The causal
significance of the off-wall sweeps would then reduce to their role in
modifying the active downstream edge of the meander.

From an application perspective, the results of this study suggest new
possibilities for turbulence control. Currently, most active control research
centres on wall-attached devices, but some of the results above suggest that,
if the focus is on altering the overall state of the flow, it may be more
efficient to act on motions detached from the wall and moving towards it.
While the technical challenges associated with manipulating the flow in the
far field are clear, this approach is intriguing because it is not currently
receiving much attention, although one cannot avoid to be reminded of LEBUs
(Alfredsson & Örlu, 2018).

As for further work, several avenues should be explored, although most would
require substantially more data and processing than the ones used here. For
example, statistical confirmation of the continuity model outlined above
should probably be done on larger computational boxes than the present one, to
avoid artefacts on the behaviour of streaks. The same can be said about
different Reynolds numbers. Similarly, the characterisation of causality by
the integrated perturbation energy over the whole domain could probably be
gainfully substituted by more specific measures, such as the energy of a
particular layer, near or far from the wall, or by more practical ones, such
as the skin friction. This would make the results more relevant to control,
and probably illuminate flow interactions that are relevant to the physical
understanding of the flow (e.g. small near-wall scales with large far-wall
ones). Unfortunately, studies like the present one involve large amounts of
data that cannot be fully stored for re-processing. Much of the analysis is
done ‘on the fly’ while the experiments are being carried out, and applying a
new processing strategy involves a new set of simulations. On the other hand,
prospective studies like the present one are crucial to the design of any such
future extension.

[Funding]This work was supported by the European Research Council under the
Caust grant ERC-AdG-101018287.

[Declaration of interests]The authors report no conflict of interest.

[Author ORCIDs]K. Osawa, https://orcid.org/0000-0001-6535-6241; J. Jiménez,
https://orcid.org/0000-0003-0755-843X

##  Appendix A The perturbation evolution equation

Consider a generic quantity $T$ with source $S$, advected by a velocity field
$u_{i}$,

$\partial_{t}T=-u_{j}\partial_{j}T+S,$ (10)

where repeated indices imply summation, and two independent experiments ‘$a$’
and ‘$b$’, denoted by superscripts of the respective fields. Consider now the
evolution equation for the difference between the experiments. Define

$\displaystyle\widehat{g}$ $\displaystyle=g^{a}-g^{b},$ (11)
$\displaystyle\overline{g}$ $\displaystyle=(g^{a}+g^{b})/2.$ (12)

for any $g$. Particularising (10) for $T^{a}$ and $T^{b}$, and subtracting one
from the other,

$\partial_{t}\widehat{T}=-u_{j}^{a}\partial_{j}T^{a}+u_{j}^{b}\partial_{j}T^{b}+\widehat{S},$
(13)

which, since

$\displaystyle\widehat{u}_{j}\partial_{j}\overline{T}+\overline{u}_{j}\partial_{j}\widehat{T}=$
$\displaystyle\,(u_{j}^{a}-u_{j}^{b})\partial_{j}(T^{a}+T^{b})/2(u_{j}^{a}+u_{j}^{b})\partial_{j}(T^{a}-T^{b})/2$
$\displaystyle=$
$\displaystyle\,u_{j}^{a}\partial_{j}T^{a}/2-u_{j}^{b}\partial_{j}T^{b}/2+u_{j}^{a}\partial_{j}T^{b}/2-u_{j}^{b}\partial_{j}T^{a}/2$
$\displaystyle+u_{j}^{a}\partial_{j}T^{a}/2-u_{j}^{b}\partial_{j}T^{b}/2-u_{j}^{a}\partial_{j}T^{b}/2+u_{j}^{b}\partial_{j}T^{a}/2$
$\displaystyle=$
$\displaystyle\,u_{j}^{a}\partial_{j}T^{a}-u_{j}^{b}\partial_{j}T^{b},$ (14)

can be written as

$\partial_{t}\widehat{T}=-\widehat{u}_{j}\partial_{j}\overline{T}-\overline{u}_{j}\partial_{j}\widehat{T}+\widehat{S}.$
(15)

Multiplying (15) by $\widehat{T}$,

$\displaystyle\partial_{t}\widehat{T}^{2}$
$\displaystyle=-2\widehat{u}_{j}\widehat{T}\partial_{j}\overline{T}-2\overline{u}_{j}\widehat{T}\partial_{j}\widehat{T}+2\widehat{T}\widehat{S}$
$\displaystyle=-2\widehat{u}_{j}\widehat{T}\partial_{j}\overline{T}-\overline{u}_{j}\partial_{j}\widehat{T}^{2}+2\widehat{T}\widehat{S}.$
(16)

The evolution of the velocity perturbation magnitude follows from substituting
$u_{i}$ for $T$ in (16). The source of the evolution equation for $u_{i}$ is
$S_{i}=-\partial_{i}p+\nu\partial_{j}^{2}u_{i}$, so that
$\widehat{S}_{i}=-\partial_{i}\widehat{p}+\nu\partial_{j}^{2}\widehat{u}_{i}$,
and

$\displaystyle\partial_{t}\widehat{u}_{i}^{2}$
$\displaystyle=-2\widehat{u}_{i}\widehat{u}_{j}\partial_{j}\overline{u}_{i}-\overline{u}_{j}\partial_{j}\widehat{u}_{i}^{2}-2\widehat{u}_{i}\partial_{i}\widehat{p}+2\nu\widehat{u}_{i}\partial_{j}^{2}\widehat{u}_{i}$
$\displaystyle=-2\widehat{u}_{i}\widehat{u}_{j}\partial_{j}\overline{u}_{i}-\overline{u}_{j}\partial_{j}\widehat{u}_{i}^{2}-2\partial_{i}(\widehat{u}_{i}\widehat{p})+2\nu\partial_{j}(\widehat{u}_{i}\partial_{j}\widehat{u}_{i})-2\nu(\partial_{j}\widehat{u}_{i})^{2}$
$\displaystyle=-2\widehat{u}_{i}\widehat{u}_{j}\partial_{j}\overline{u}_{i}-\partial_{j}\\{\overline{u}_{j}\widehat{u}_{i}^{2}+2\widehat{u}_{i}\widehat{p}\delta_{ij}-\nu\partial_{j}\widehat{u}_{i}^{2}\\}-2\nu(\partial_{j}\widehat{u}_{i})^{2}$
$\displaystyle=P_{\varepsilon u}+C_{\varepsilon u}+D_{\varepsilon u},$ (17)

where $\delta_{ij}$ is Kronecker’s delta. The first and last term in (17)
represent the production and dissipation of the perturbation energy,
respectively. Note that velocity gradient in the production term is the
average of two independent fields, and does not necessarily agree with the
usual ensemble averaged gradient. The terms in curly brackets are fluxes that
do not contribute to $\varepsilon_{\boldsymbol{u}}$ when (17) is integrated
over the whole computational box. From left to right, they represents
convection, pressure-strain and viscous diffusion, respectively. It is
noteworthy that swapping $a$ and $b$ do not change the (17), in agreement with
the symmetric way in which they are defined. In fact, (17) is similar to the
evolution equation for the structure function between the velocities at two
neighbouring points, with the difference that there are no interactions here
between the fields $a$ and $b$.

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