# Nonlinear interactions of ion acoustic waves explored using fast imaging
decompositions

Simon Vincent1,2 Vincent Dolique1 Nicolas Plihon1 1 Univ Lyon, ENS de Lyon,
CNRS, Laboratoire de Physique, F-69342 Lyon, France
2 École Polytechnique Fédérale de Lausanne (EPFL), Swiss Plasma Center (SPC),
CH-1015 Lausanne, Switzerland

###### Abstract

Fast camera imaging is used to study ion acoustic waves propagating
azimuthally in a magnetized plasma column. The high speed image sequences are
analyzed using Proper Orthogonal Decomposition and 2D Fourier Transform,
allowing to evaluate the assets and differences of both decomposition
techniques. The spatio-temporal features of the waves are extracted from the
high speed images, and highlight energy exchanges between modes. Growth rates
of the modes are extracted from the reconstructed temporal evolution of the
modes, revealing the influence of ion-neutral collisions as pressure
increases. Finally, the nonlinear interactions between modes are extracted
using bicoherence computations, and show the importance of interactions
between modes with azimuthal wave numbers $m$, $m-1$ and $-1$, with $m$ an
integer.

††preprint: AIP/123-QED

##  I Introduction

The propagation of ion sound waves or ion acoustic waves is ubiquitous in
plasmas and their non-linear interactions, possibly leading to ion acoustic
turbulence, is a widespread energizing process in plasma physics. The
nonlinear evolution of ion acoustic waves (IAW) generically leads to
instabilities and the development of non-linear structures. For instance, IAW
have long been observed in the solar wind, and related to the anisotropy of
the electron distribution function [1]. In this context, IAW are driven
unstable when the ratio of the electron to ion temperature is larger than
unity, as observed by the Helios spacecraft [2], and very recently for oblique
IAW by Parker Solar Probe [3]. Heating of energetic particles from ion
acoustic turbulence was also proposed in the context of polar aurorae [4]. The
non-linear evolution of ion acoustic waves into strongly non-linear structures
such as solitons [5] or double layers has been reported in electro-positive
plasmas [6] or electronegative plasmas [7], for which two branches of ion
acoustic waves exist [8]. In the context of bounded plasmas, IAW excited in
sheaths may affect particle transport at low pressure [9] or lead to strong
ion heating [10] when the ratio of the electron to ion temperature is larger
than unity. IAW may also be useful tools to probe sheath criteria in multiple
ion plasmas [11, 12, 13]. Technological plasmas may also trigger IAW, that, in
return, affect their operation, as reported for Hollow Cathodes [14, 15], Hall
thrusters [16] and diverging magnetic nozzle thrusters [17].

In this article, we report on the observation of localized ion acoustic waves
in a magnetized plasma column using high speed camera imaging. Our
observations thus shed new light on the ion acoustic activity that has been
previously reported in similar configurations [18, 19, 20, 21, 22, 23, 24]. We
do not investigate the origin of the IAW from parametric instability or waves
interactions here, as was done in these previous investigations, but we
analyse the spatio-temporal characteristics of the IAW using mode
decomposition from high-speed imaging. The IAW nonlinear interactions are
quantitatively highlighted by means of bicoherence computations.

The article is organized as follows. The experimental set-up is introduced in
Sec. II, the analysis of fast camera measurements by mode decomposition
techniques is presented in Sec. III. In particular, we highlight the
differences and complementarities of two different mode decompositions, namely
Proper Orthogonal Decomposition and 2D Fourier Transform. In section IV the
waves observed by camera imaging are identified to be IAW from the waves phase
velocities. Finally the non-linear modes interactions are characterized and
their nonlinear aspect is exhibited in section V and conclusions are drawn in
section VI.

##  II Experimental set-up and diagnostics

###  II.1 Experimental set-up

The experimental set-up [25] consists in a $20$ cm diameter, $1$ m long
stainless steel cylindrical chamber containing an argon plasma generated by a
$1$ kW, 13.56 MHz radio frequency source. The coordinates will be referenced
using a cartesian coordinate system, where $z$ denotes the axial direction
(see Fig. 1). A cylindrical coordinate system $(r,\theta,z)$ will be used for
the reconstruction of rotating modes. The plasma base pressure $p_{0}$ in the
chamber is regulated at a fixed value, between $0.8$ mTorr and $2$ mTorr by
steps of $0.1$ mTorr. The plasma is created by an inductive source around a
$11$ cm diameter borosilicate tube connected at one end of the chamber (at
$z=0$ cm). Three coils placed along the steel cylinder generate an axial
magnetic field that confines the plasma (see Fig. 1). This magnetic field is
not perfectly homogeneous along $z$. For a current in the coils set here at
$100$ A, it has an averaged amplitude along the $z$-axis of $B=170$ G.

Radial profiles of the plasma density $n$, the electron temperature $T_{e}$
and the plasma potential $\Phi_{p}$ are performed using a 5-tips probe [26,
27] and an emissive probe respectively. The probes were inserted radially at
$z=49$ cm (see blue dashed line in Fig. 1). To keep the whole apparatus in a
steady thermal state, the operation of the plasma is pulsed: the plasma is
sustained over typically 5 seconds, during which data are acquired, with a
repetition period of typically 30 s. The experiment is fully automated to
allow high repeatability and reproducibility of the plasma. The level of shot
to shot reproducibility was $\pm 0.6\%$ for the ion saturation current of a
Langmuir probe, with a standard deviation of $0.2\%$ (estimated from a series
of 40 shots at the plasma column center). Radial scans of the plasma
parameters were performed sequentially: each spatial point has been acquired
during one plasma-pulse, and the probe is translated between two pulses, from
the center of the plasma column ($r=0$ cm) to its edge ($r=10$ cm).

The results presented in this article mainly rely on high-speed imaging of the
plasma emitted light, performed through a DN 200 borosilicate window closing
the chamber at $z=80$ cm (opposite to the source tube). A Phantom v2511 camera
is placed along the $z$-axis, $3.5$ m away from this window, and the light
intensity naturally radiated by the plasma $I_{cam}$ is captured at $200$ kfps
with a resolution of $256\times 256$ px. A filter around $750\pm 5$ nm is used
in order to restrict the collected light to a single ArI spectral line.
Examples of the mean intensity $\langle I_{cam}\rangle$ and fluctuation
standard deviation $\sigma(\tilde{I}_{cam})$ images are shown in Fig. 1. Note
that the plasma column is not perfectly axisymmetric, and the fluctuations are
of the order of $10\%$ of the mean amplitude. Note also that the depth of
field of the optical set-up being of the order of the length of the chamber
(with a camera objective aperture set at f/4, and a focal length of $135$ mm),
the light intensity recorded by the camera is actually the result of an
integration along the $z$-axis. Due to the magnetic field ripple and to the
parallax, the direct comparison of the probes’ measurements (at $z=49$ cm) and
the camera images (where light is integrated along $z$), is not relevant. We
thus introduce a distorted space ($x^{*}$, $y^{*}$, $z$) in which the camera
lines of sight are parallel (see Ref. [27] for more details). The camera
images are hence observed in the plane ($x^{*}$, $y^{*}$), that may also be
referenced as $(r^{*},\theta)$ in a polar coordinate system.

Figure 1: Left: sketch of the experimental set-up. The dashed blue line
indicates where probe profiles are measured. Right: mean image of the light
intensity collected by the camera (up), and standard deviation of the
fluctuations around this average value (bottom), for $p_{0}=1$ mTorr.

###  II.2 Radial profiles of the plasma parameters

The top row of Fig. 2 displays the radial profiles of the plasma density,
electron temperature and plasma potential as a function of $r$ for a pressure
$p_{0}=1$ mTorr. As previously introduced, these profiles cannot be directly
compared to the images in the $(r^{*},\theta)$ plane. However, assuming
axisymmetry and invariance of the plasma parameter along magnetic field lines,
a synthetic integration process detailed in Ref [27] allows to map the probe
measurements along r (at $z=49$ cm) to the images expressed along $r^{*}$,
enabling quantitative comparison. The density is approximately constant in a
core region of the plasma for $r\leq 4$ cm ($r^{*}\leq 3$ cm) and then
decreases towards the edge. A clear peak can be seen around $r=4.5$ cm
($r^{*}\sim 3$ cm) for the electron temperature, produced by the higher
ionization rate of the RF inductive source close to the wall at $r=5.5$ cm,
$z\sim-10$ cm. This higher temperature is also responsible for the higher
light emission observed on the images at $r^{*}\sim 3$ cm in Fig.1. Finally,
the plasma potential decreases from the center to $r\sim 4$ cm (i.e.
$r^{*}\sim 3$ cm), and presents a strong positive gradient at the edge of the
plasma column. This is responsible for an $\vec{E}\times\vec{B}$ drift that
drives plasma rotation in the $-\vec{e}_{\theta}$ direction, discussed later
in this work.

Figure 2: Radial profiles of the density, electron temperature, and plasma
potential mean values (top row) and fluctuations (middle row) at $p_{0}=1$
mTorr. Plasma parameters measurements were made with 5-tips ($n$, $T_{e}$) and
emissive ($\Phi_{p}$) probes. Bottom: Power Spectral density of the plasma
density and light intensity fluctuations (computed from the average value of a
$5\times 5$ pixel box on the images).

The radial profiles of the fluctuations of the plasma parameters are shown in
the middle panel of Fig. 2. The fluctuations are peaked at the edge of the
plasma column at $r\sim 5.5$ cm (i.e. $r^{*}\sim 4$ cm).

Filtered light fluctuations recorded by fast camera are usually considered to
be a proxy for density fluctuations [28, 29, 30]. For the magnetic field value
of $B=170$ G reported in this article, simultaneous probe and camera
measurements showed the fluctuations of density $\tilde{n}$ and light
intensity $\tilde{I}_{cam}$ at the probe location to have very similar
spectra, as shown in the bottom panel of Fig. 2. However, as it was shown in
our previous work [27], for magnetic field values in the range 100 to 700 G,
light intensity naturally radiated by low temperature plasmas are also highly
correlated to the electron temperature. In Sec. IV, the comparison of
experimental phase velocities with the theoretical ion acoustic speed
nonetheless requires to assume $\tilde{I}_{cam}$ to be a reasonable proxy for
$\tilde{n}$. We therefore underline that this is a rather strong assumption,
and that for further quantitative comparison $\tilde{I}_{cam}$ should be
interpreted as a combination of both $\tilde{n}$ and $\tilde{T}_{e}$ \- a task
beyond the scope of this article. The strong spectral component observed at
$5.6$ kHz is identified as a Kelvin-Helmholtz mode [31]. In the present work,
we will focus on fluctuations observed between $50$ and $70$ kHz, which are
unambiguously identified as ion acoustic waves.

##  III Image analysis

The first and most natural tool that comes to mind for the images analysis is
the Fourier decomposition. This is used later on; we prefer here to start the
analysis by using an alternative method, the Proper Orthogonal Decomposition
(POD).

Note that before performing these decomposition, we choose to normalize each
pixel by its fluctuations mean, as was done in similar conditions [32]. This
choice greatly enhances the contrast, allowing to nicely extract the relative
amplitudes of the modes (especially in the regions of low light intensity),
but at the cost of losing information on the absolute amplitude of the modes.

Note finally that in the following the light fluctuations recorded by the
camera $\tilde{I}_{cam}$ will simply be denoted $I$ for easier readability.

###  III.1 Proper Orthogonal Decomposition

The POD consists in extracting the spatial structures that are dominant
throughout time in a given data set $I(\mathbf{r^{*}},t)$, where
$\mathbf{r^{*}}$ denotes space. This is done by computing the eigenmodes
$\Psi_{i}$ of the spatial autocorrelation of the time-averaged field $\langle
I\rangle(\mathbf{r^{*}})$. These so-called spatial modes $\Psi_{i}$ then
define an orthonormal basis onto which the original data can be projected.
This can be written:

$I(\mathbf{r^{*}},t)=\sum_{i}\sigma_{i}\,a_{i}(t)\psi_{i}(\mathbf{r^{*}})$

with $\sigma_{i}\,a_{i}(t)$ being the time evolution of the data projected on
the spatial modes $\Psi_{i}$. Here $a_{i}$ and $\Psi_{i}$ are of norm unity;
the amplitude of the various components of the decomposition are thus given by
the values of $\sigma_{i}$.

One of the most interesting aspects of this decomposition is that it is done
without any a priori on the shape of the $\Psi_{i}$ structures: they simply
come out from the computation process, as natural modes, contrary to Fourier
analysis, which projects the data onto predefined spatial and temporal
structures. Hence POD might allow the emergence of structures with physical
significance that are not well described by mere Fourier modes. Thanks
moreover to its simplicity of implementation and computational speed when
performed onto a discrete set of data, as is explained later, POD becomes a
very attractive and efficient analysis tool for experimentalists, and has
grown very popular in the last decades for the analysis of data from
experiments or from numerical simulations. Note that depending on the field,
this technique is also referred to as Karhunen-Loève decomposition (as a
reference to the original mathematical theorem) or principal component
analysis.

The set of spatial modes $(\Psi_{i})$ has the property of being the optimal
basis for approximating the data $I$ [33]: for any $N$, the norm of the
projection of $I$ onto $(\Psi_{i})_{1,N}$, which reads
$\sum_{1}^{N}\sigma_{i}^{2}$, is higher than the projection onto any other
basis than one might choose. For a given value of $N$, the spatial modes
$(\Psi_{i})_{1,N}$ can then be interpreted as the vectors that are the best
suited to reproduce the information carried by I, in the most efficient way.
Applied to physical data, this property is even more interesting if the
$\sigma_{i}^{2}$ have a clear physical meaning. The use of POD on experimental
data has been initiated in fluid dynamics, for the analysis of turbulent
velocity fields [34]. In this context the norm $\sum_{1}^{N}\sigma_{i}^{2}$ of
the projected data represents a kinetic energy, and the modes $\Psi_{i}$ may
then be interpreted as the most important flow structures in terms of kinetic
energy. POD has then been applied to spatio-temporal measurements of plasma
fluctuations in tokamaks, either measured by sets of Langmuir probes [35, 36,
37] or by means of soft x-ray emission [38]. It has also been applied to
camera imaging data of plasma naturally radiated light to exhibit spiral
shaped structure generated by a $m=2$ instability in a linear device [39] and
in a tokamak to highlight plasma response to resonant magnetic perturbations
[40]. More recently, the technique was used to characterize instabilities in
the plume of a Hall thruster from fast imaging data [41]. Following this
study, POD has been applied to decompose the camera imaging data of a plasma
plume produced by a high-current hollow cathode [42]. Unfortunately, in the
latter cases, the physical interpretation of the $\Psi_{i}$ vectors is not as
straightforward as in fluid mechanics, since the extracted modes result from
the decomposition of light intensity fields, which depends in a non trivial
way on the plasma parameters. And even by considering as a crude approximation
$\tilde{I}_{cam}\propto\tilde{n}$, not much can be said on the norm
$\sum_{1}^{N}\sigma_{i}^{2}$ in terms of physical significance. This does not
mean the amplitude of the modes extracted from plasma emitted light is void of
meaning, but simply that one has to be careful before thinking of it as a
precise energy estimation. In this article POD decomposition is thus discussed
in a purely qualitative way. Finally, POD does not require any symmetry, which
is a significant advantage over Fourier decomposition for instance. In the
case of complex geometries, POD can be an efficient alternative for capturing
the physical structures in the data.

Figure 3: Proper Orthogonal Decomposition modes (a) and singular values (b) of
light intensity normalized fluctuations, for $p_{0}=1.3$ mTorr. c) Zoom on the
evolution of $a_{1}$ and $a_{2}$. See text for details.

In practice, it can be shown that a direct extraction of the spatial modes
$\Psi_{i}$ associated to their temporal evolution $a_{i}$, is in fact achieved
by applying a mere singular value decomposition (SVD) to the matrix containing
the data $I$, rearranged in such a way that one dimension of the matrix
represents space, and the other time. This way of computing a POD, also
referred to as bi-orthogonal decomposition [43] is the one implemented here.
Each image (of $p$ pixels) of a video containing $q$ frames is rearranged to
form a matrix $I$ of size p$\times$q to which a singular value decomposition
is applied $I=\Psi\Sigma A^{t}$. $\Psi$ and $A$ are orthonormal matrices of
respective sizes $p\times p$ and $q\times q$, and $\Sigma$ is a matrix of the
same size as $I$. The matrix $\Sigma$ only contains diagonal elements, which
are the decomposition’s singular values $\sigma_{i}$:

$\stackMath\stackon{\mathrm{\textit{space}}\left\\{\begin{bmatrix}&\ &\\\ &\
&\\\ &\ &\\\ &\ (I)_{ij}\\\ &\ &\\\ &\ &\\\ &\ &\\\
\end{bmatrix}\right.}{\overbrace{\phantom{\smash{\begin{bmatrix}&\ &\\\ &\
&\\\ &\ &\\\ &\ (I)_{ij}\\\ &\ &\\\ &\ &\\\ &\ &\\\
\end{bmatrix}\mkern-35.0mu}}}^{\mathrm{\textstyle\textit{time}}}\mkern
20.0mu}=\begin{bmatrix}\ &\ &\ &\ &\ \\\ \vdots&\vdots&\ &\ &\vdots\\\
\Psi_{1}&\Psi_{2}&\ &\ &\Psi_{q}\\\ \vdots&\vdots&\ &\ &\vdots\\\ \ &\ &\ &\
&\ \\\ \end{bmatrix}\begin{bmatrix}\sigma_{1}&\ &\\\ \ &\ddots&\\\ \ &\
&\sigma_{q}\\\ \ &\ &\ \\\ \ &\ &\ \\\ \ &\ &\ \\\
\end{bmatrix}\begin{bmatrix}&\cdots\ a_{1}\ \cdots&\\\ &\cdots\ a_{2}\
\cdots&\\\ &\ &\ \\\ &\cdots\ a_{p}\ \cdots&\\\ \end{bmatrix}$

Figure 3 a) shows the result of a POD applied to a 100 ms time series of
intensity fluctuations (i.e. 20000 images) recorded at a pressure $p_{0}=1.3$
mTorr. The spatial modes $\Psi_{i}(\mathbf{r^{*}})$ are displayed in the top
row, the time series of the amplitudes $a_{i}(t)$ in the the middle row, and
the corresponding power spectral density $S(f)$ in the bottom row. The
interpretation of the decomposition requires to consider pairs of modes, such
as
$I^{POD}_{1,2}(\mathbf{r^{*}},t)=\sigma_{1}a_{1}(t)\Psi_{1}(\mathbf{r^{*}})+\sigma_{2}a_{2}(t)\Psi_{2}(\mathbf{r^{*}})$,
which yields rotating azimuthal waves of the type $e^{-i\omega t-im\theta}$
(shown later in subsection III.3), since the spatial modes $\Psi_{1}$ and
$\Psi_{2}$ are shifted by a quarter wavelength and the temporal modes $a_{1}$
and $a_{2}$ are in quadrature (see Fig. 3 c))111Due to the very high frequency
of the waves (75 kHz) relative to the sampling frequency (200 kHz), the
signals $a_{1}$ and $a_{2}$ are closer to triangular than sinusoidal shapes;
but other measurements of lower frequency waves clearly show sinusoidal
evolutions for the $a_{i}$ signals. In the example shown in Fig. 3, the modes
($\Psi_{1}$, $a_{1}$) and ($\Psi_{2}$, $a_{2}$) correspond to a $m=-5$
rotating azimuthal wave, the modes ($\Psi_{3}$, $a_{3}$) and ($\Psi_{4}$,
$a_{4}$) correspond to a $m=-6$ rotating azimuthal wave, and the modes
($\Psi_{6}$, $a_{6}$) and ($\Psi_{7}$, $a_{7}$) correspond to a $m=-4$
rotating azimuthal wave.

The rotation frequencies of the $m$-modes can be deduced from the spectra of
the temporal signals $a_{i}$, shown in the bottom row of Fig. 3 a). A very
clear peak at frequency $f=55.6$ kHz is observed for the modes $1\&2$,
capturing the $m=-5$ azimuthal wave. The $m=-6$ wave (POD modes $3\&4$) has a
$f=65.1$ kHz frequency, and the $m=-4$ wave (POD modes $6\&7$) has a $f=45.2$
kHz frequency. Section IV shows that these modes are ion acoustic waves.

The singular values $\sigma_{i}$ of the modes are plotted in Fig. 3 b). The
amplitudes of $\sigma_{1}$ and $\sigma_{2}$ are nearly identical (within
$0.3\%$), and more than twice larger than the other singular values, showing
that the dynamics is dominated by a $m=-5$ rotating mode. The time series
$a_{i}(t)$ show sudden changes in amplitude, for instance at times $t\sim 3.8$
ms, $29.5$ ms and $65.2$ ms, corresponding to an energy exchange between modes
$m=-6$ and $m=-5$, that will be investigated in Section V. The relatively
intense POD mode ($\Psi_{5}$, $a_{5}$), with a strong spectral component at
$f=5.6$ kHz, was identified as a $m=3$ Kelvin-Helmoltz mode [31], not
discussed here. The POD analysis presented here was straightforward to
implement, and provide a very efficient way of extracting global features
captured in a video sample. Now we present the results of a Fourier analysis
performed on the same data that complements the POD analysis.

###  III.2 2D Fourier Transform

The 2D Fourier Transform (2D-FT) of a two variables function $f(\mathbf{x},t)$
reads: $\hat{f}(\mathbf{k},\omega)=\iint
f(\mathbf{x},t)e^{-i(\mathbf{k}\cdot\mathbf{x}+\omega t)}d\mathbf{x}dt$. 2D-FT
is classically used to decompose the spatio-temporal signals collected by
azimuthally distributed probe arrays into azimuthal modes [45, 46]. Following
many studies using camera imaging and performed in the context of linear
plasma devices [47, 29, 48, 49] and plasma thrusters [50, 51], the 2D-FT is
here performed on virtual rings at various radii $r^{*}$. For a given value of
the radius $r^{*}$, a time series $I(\theta,t)|_{r^{*}}$ is extracted from the
camera images. For each angle $\theta_{n}=\frac{2\pi n}{N_{\theta}}$ with
$n\in[0:N_{\theta}-1]$, the value of the pixel at position ($r^{*}$,
$\theta_{n}$) is extracted. The angle resolution of $N_{\theta}=700$ is chosen
here, such that no interpolation is needed in the processes of converting
either a ring of pixels in the image space ($x^{*}$, $y^{*}$) into a vector
along the $\theta$ direction, nor in the inverse process, when reconstructing
images in the ($x^{*}$, $y^{*}$) plane from the mode decomposition results.
Since the images are $2\pi$ periodic in the $\theta$ direction, the wave-
vectors read $m/r^{*}$, with $m$ an integer, and the 2D-FT of
$I(\theta,t)|_{r^{*}}$ is computed as

$\hat{I_{r^{*}}}(m,f)=\iint I(\theta,t)|_{r^{*}}e^{-i(m\theta+2\pi ft)}d\theta
dt$

with $f$ the frequency. The resulting 2D power spectrum
$S_{r^{*}}(m,f)=|\hat{I_{r^{*}}}(m,f)|^{2}$ displays the amplitudes of light
intensity fluctuations as a function of the spatial mode $m$ and the frequency
$f$, at a given radius $r^{*}$. Note that at radius $r^{*}\sim$ 3.5 cm on the
images, the intensity $I(\theta,t)|_{r^{*}}$ is reconstructed from a corona of
$\sim 400$ pixels, which ensures a very good precision in the extraction of
the first modes $m$ up to $m\sim 20$. The power spectrum at $r^{*}=3.3$ cm and
$p_{0}=1.3$ mTorr is shown in Fig. 4. The observations are similar to those
drawn from the POD analysis : the dominant mode is an $m=-5$ mode whose
frequency is peaked at 55.6 kHz, and the other important modes are an $m=-6$
mode peaked at 65.2 kHz and an $m=-4$ mode peaked at 44.9 kHz. Note that this
2D power spectrum provides a dependence of the dominant frequency on the mode
number, and can therefore be seen as an experimental dispersion relation. This
is used in section IV for the identification of the modes.

Figure 4: Power spectrum of the raw light intensity fluctuations from camera
imaging, taken on a corona of radius $r^{*}=3.3$ cm for $p_{0}=1.3$ mTorr.
Dispersion relations are plotted from experimental fits of the power spectrum
maxima (red) and from the ion acoustic speed (black) (see section IV).

The full spatio-temporal evolution of any given $m$ mode can also be extracted
by 2D-FT. To this end, the 2D Fourier Transform is computed for radii $r^{*}$
covering the full image (here 2D-FT are computed for $r^{*}=2i\ \mathrm{px},\
i\in[1:64]$, instead of $r^{*}=[1:128]$ px, to limit memory storage and
increase computational speed). For given $m$ and $r^{*}$ values, the inverse
Fourier Transform of $\hat{I}_{r^{*}}(m,f)$ is computed, resulting in the
spatio-temporal signal of the $m$ mode, at radius $r^{*}$. Performing this
inverse computation for all radii previously mentioned leads to the full
spatio-temporal reconstruction of the $m$ mode. Examples of snapshots of such
reconstructed 2D-FT modes are shown and commented in subsection III.3.

The average amplitude of the reconstructed modes is then computed along time,
providing a global picture of the modes dynamics. The global $m$-modes time
evolution for $p_{0}=1,3$ mTorr is plotted in Fig. 5 (top). As already
observed on the time signals of the POD in Fig. 3, clear exchange events can
be observed involving modes $m=-5$ and $m=-6$. The $m=-4$ mode is seen to
follow the dynamics of $m=-5$ while the $m=-7$ mode follows the dynamics of
$m=-6$ mode, a feature that was not detected by the POD analysis. From the
reconstructed signals of the individual $m$-modes, the instantaneous mean
radial profile is computed by an integration over $\theta$, allowing for the
computation of the radial location $r^{*}_{max}$ where the wave amplitude is
maximal. Figure 5 (bottom) shows $r^{*}_{max}$ for the $m=-5$ and $m=-6$
modes, that are highly correlated to the global dynamics of the modes. Again
this could not be deduced from POD, since spatial modes structure are deduced
from a time-averaged analysis. Figure 5 is further discussed in section V. Now
let us compare the results obtained from both POD and 2D-FT analysis.

Figure 5: Time evolution of $m$-mode average amplitudes extracted by 2D-FT,
for $p_{0}=1.3$ mTorr.

###  III.3 Comparison between POD and 2D Fourier Transform

Figure 6 shows snapshots of the $m=-5$ and $m=-6$ 2D-FT reconstructed modes,
as well as the corresponding modes reconstructed from the POD analysis
$I^{POD}_{1,2}$ and $I^{POD}_{3,4}$, for the experiment achieved at
$p_{0}=1.3$ mTorr. The snapshots are shown every 0.06 ms following
$t_{0}=77.27$ ms, marking the beginning of an energy exchange between modes
$m=-5$ and $m=-6$ (see Fig. 5 (top)). The time interval 0.06 ms represents
slightly more than 3 wave periods for the $m=-5$ mode, and nearly 4 wave
periods for the $m=-6$ mode. The spatial shape of the 2D-FT modes varies
significantly. On the contrary the shapes of the POD reconstructed modes
remains almost unchanged. This is actually expected since each of these
signals is merely composed of the linear combination of two spatial fields.
Note also that the spatial structures $\Psi_{i}$ were extracted from the time-
averaged data field (see subsection III.1): the reconstructed mode are unable
to account for spatially localized variations.

Figure 6: Evolution of 2D-FT modes $m=-5$, $m=-6$ and POD reconstructed modes
$I^{POD}_{1,2}$ and $I^{POD}_{3,4}$, during the exchange event highlighted by
the dashed-black box in Fig. 5, for $p_{0}=1.3$ mTorr.

Let us now compare the spatial structures provided by POD and 2D-FT. Figure 7
(top) shows time-average radial profiles of the modes for both decompositions.
The profiles are computed by an integration along $\theta$, averaged over the
20000 images. Fig. 7 (bottom) shows the azimuthal profiles taken at a given
time and for $r^{*}=4$ cm. The comparison between POD modes ($1\&2$) and the
$m=-5$ 2D-FT mode shows an almost perfect match (note that the match slightly
decreases when the amplitude of the $m=-5$ mode strongly decreases). The
comparison between POD modes $I^{POD}_{3,4}$ and the $m=-6$ 2D-FT mode give
similar results, although with a lower agreement on the outward part
($r^{*}\geq 4$ cm) of the radial profiles. An overall good match is observed
between the lower amplitude POD modes ($6\&7$) and the $m=-4$ 2D-FT mode
radial profiles. The instantaneous azimuthal profile are not identical, with a
phase shift up to $\sim\pi/8$ depending on the frame. These results show that,
in the context of data having 2-$\pi$ periodicity, POD and 2D-FT
decompositions share several common features, while they do provide exactly
the same knowledge. Note finally that for the computations performed here with
a number of images $N=20000$, the POD is twice faster than the 2D-FT (even
though it was taken into account for the 2D-FT only 20 mode reconstructions,
and half of the images pixels as mentioned in subsection III.2). Then when
only taking $N=2000$, the POD is more than 12 times faster than the 2D-FT.

Figure 7: Comparison of (top) radial and (bottom) azimuthal profiles of the
POD and 2D-FT modes, at $p_{0}=1.3$ mTorr, for the (left) $m=-5$, (center)
$m=-6$ and (right) $m=-4$ modes.

The main strengths of both POD and 2D-FT techniques are summarized:

  * •

POD is fast and easy. It is extremely simple to implement, and it provides
quick and direct results on the spatio-temporal dynamics of a dataset.

  * •

POD is flexible. It does not rely on any particular shape of the physical
structure at play, nor on a specific location in the images analysed. It will
therefore be particularly well suited to study for instance non-linearly
saturated modes exhibiting a complex spatial or temporal pattern. Note however
that if the results can be particularly insightful, they might also be
difficult to interpret (and in some cases even unusable).

  * •

2D-FT is explicit, hence robust. Projecting the data onto a predefined set of
wave modes (here for instance of the form $e^{-i\omega t-im\theta}$) prevents
the emergence of unexpected structures, but it provides the results with a
well identified physical meaning.

  * •

2D-FT is exhaustive for linear mode analysis. Since it provides the full
spatio-temporal evolution of linear wave, 2D-FT is particularly attractive to
study their dynamics, exhibit the corresponding dispersion relations, or use
for instance the phase correlations between modes to study weakly non-linear
interactions (see the use of bicoherence in section V).

Both techniques can provide insightful and complementary results. A recent
preprint, reporting on the specific comparison between POD and 2D-FT applied
to Hall thruster camera imaging [52], concludes similarly. Applied to the
present datasets, POD shows that the dominant physical structures are
$m$-modes of the form $e^{-i\omega t-im\theta}$. This indicates that the 2D-FT
as implemented here, is an appropriate numerical tool for the mode
decomposition. Hence POD does not constitute a strong gain for further
analysis here. In the following, for the identification of the waves and the
in-depth study of their weakly non-linear interactions, we will use the
results from the 2D-FT decomposition.

##  IV Waves identification

The azimuthal waves detected by both POD and 2D-FT are now unambiguously
identified as ion acoustic waves. A series of high-speed imaging acquisitions
was performed for pressures $p_{0}$ in the range $[0.8;2]$ mTorr by steps of
0.1 mTorr. For each value of the pressure, the radius $r^{*}_{max}$ at which
the wave amplitude is maximal is deduced from the time-average of the raw
images. The experimental phase velocity $v_{\phi}$ is determined by a linear
fit of the most energetic modes observed on the spectrum
$S_{r^{*}_{max}}(f,m)$ as $f(m)=v_{\phi}m/(2\pi r^{*}_{max})$. A typical
linear fit is shown in Fig. 4 for $p_{0}=1.3$ mTorr. The experimental phase
velocities $v_{\phi}^{exp}$ are displayed in Fig. 8 as red dots. The errorbars
are estimated by the combination of the uncertainties on the fit on $S(m,f)$,
and on the evaluation of $r^{*}_{max}$ ($r^{*}_{max}(t)$ fluctuates around its
mean value with a standard deviation of $\sim 3\%$, see Fig. 5 (bottom)).

These experimental phase velocities are compared to the theoretical ion
acoustic speed $c_{s}=\sqrt{eT_{e}/m_{i}}$, with $e$ the elementary charge and
$m_{i}$ the ion mass. The computation of the latter requires careful estimates
of $T_{e}$ where the phase velocity is measured on the high-speed images. Note
that at $z=49$ cm, where the probe measurement is performed, the radial
position that is best representative of what is seen at $r^{*}=3.3$ cm on the
images is in fact at $r=5$ cm (see Appendix A and for a detailed explanation
see [27]). A detailed pressure scan of the electron temperature $T_{e}$ was
performed with the 5-tips probe at a radius $r=4$ cm, and from a finely
resolved radial scan at $p_{0}=1$ mTorr [27, 31], we have
$T_{e}(5~{}$cm$)\approx T_{e}(4~{}$cm$)+0.2$ eV. Therefore, from the measured
values $T_{e}(4~{}$cm$)$, $T_{e}(5~{}$cm$)$ is evaluated to lie in the range
$[T_{e}(4~{}$cm$);T_{e}(4~{}$cm$)+0.5]$ eV. The resulting theoretical ion
acoustic speeds $c_{s}(p_{0})$ are shown in Fig. 8 (gray area).

The experimental phase velocities follow the trend of $c_{s}(p_{0})$, with
values shifted down by approximately $700$ m/s. This is well explained by a
Doppler shift due to the plasma column rotation. The plasma column indeed
rotates, as was reported previously [53], where the electric drift
$\displaystyle\frac{\vec{E}\times\vec{B}}{B^{2}}=-\nabla_{r}\phi_{p}/B\,\vec{e}_{\theta}$
was shown to overcome the diamagnetic drift
$-\displaystyle\frac{T_{i}}{n}\frac{\vec{\nabla}n\times\vec{B}}{B^{2}}$. Two
damping mechanisms also need to be accounted for: ion-neutral friction and
effective friction due to ionization. The ion-neutral collision frequency
reads $\nu_{in}=n_{n}\sigma_{in}v_{th,i}$, with
$v_{th,i}=\sqrt{eT_{i}(eV)/m_{i}}$, $n_{n}$ being the neutral density and
$T_{i}$ the ion temperature. We consider $n_{n}\approx p_{0}/k_{B}T_{n}$ with
$T_{n}(p_{0})=350$ K, $\sigma_{in}=1.6\times 10^{18}$ m-2 from experimental
cross sections [54], and $T_{i}\approx 0.2$ eV using previous LIF
measurements. The effective friction due to the ionization originates from
ions created with a temperature much lower than the surrounding $T_{i}$ and
depends upon the ionization frequency $\nu_{iz}$, computed as
$\nu_{iz}=n_{n}K_{iz,0}T_{e}^{0.59}\exp(-\varepsilon_{iz}/T_{e})$, with
$K_{iz,0}=2.34\times 10^{-14}$ m3/s and $\varepsilon_{iz}=17.44$ eV, and
$T_{e}$ in eV [55]. A global damping factor $K$ is then given [53, 31] as
$K=1+\left(\displaystyle\frac{\nu_{in}+\nu_{iz}}{\omega_{ci}}\right)^{2}$,
with $\omega_{ci}$ the ion cyclotron frequency. This finally gives a
background azimuthal rotation of the ions as
$v_{i0,\theta}\approx-(1/K)\partial_{r}\phi(r)/B$.

The rotation velocity $v_{i0,\theta}$ is estimated from the experimental
profiles $\phi_{p}(r)$ shown in Fig. 2 (measured at $p_{0}=1$ mTorr, and
assuming variations with pressure within $\pm 20\%$). The estimated values of
$n_{n}$ and $T_{i}$ are considered to be bounded within $\pm 10\%$, and
$T_{e}$ is estimated from $T_{e}^{r=4cm}(p_{0})$ as explained above. The
results for the estimate of $c_{s}+v_{i0,\theta}$ are shown in Fig. 8 (green
curve). In spite of all the approximations made, the comparison between
experimental phase velocities and the Doppler shifted values of $c_{s}$
provides a very satisfactory agreement. This allows us to identify with great
confidence the azimuthal waves observed at $B=170$ G as ion acoustic waves. An
interesting feature is that the ion acoustic waves travel in the positive
$\theta$ direction, i.e. opposite to the $E\times B$ drive.

Figure 8: Comparison between the experimental phase velocity (red dots), the
ion sound velocity $c_{s}$ (black curve), and its Doppler shifted values
(green curve).

We stress here that adding the ion background velocity to the classical ion
acoustic wave speed is a crude approximation, deemed sufficient here for the
purpose of wave identification. However, a careful calculation would require
to compute a complete dispersion relation from the governing equations, which
couple in a complex way and prescribe direct analytical computation. Indeed
the effect of an ion background velocity on the ion acoustic phase velocity is
likely to be coupled with other effects such as electron magnetization or
friction with the neutrals, leading to computations well beyond the scope of
this article.

Interestingly, we observed that the ion acoustic waves are only observed over
a narrow range of magnetic field values. For $B=80$ G no clear wave emerges
from the fluctuations of the plasma density or emitted light intensity; on the
other hand for $B\geq 300$ G low frequency waves develop [31].

##  V Modes dynamics and interactions

The spatio-temporal dynamics and the non-linear nature of the energy exchanges
between the ion acoustic modes, as clearly shown in Fig. 5, are now described.

###  V.1 Growth rates of ion acoustic modes

The time series shown in Fig. 6 is taken around the exchange event highlighted
at $t_{0}$ in Fig. 5. At time $t_{0}$ the amplitude of the $m=-6$ mode is
close to its maximum, while the amplitude of the $m=-5$ mode is close to its
minimum. At time $t_{0}+18$ ms, the amplitude of the $m=-6$ mode has decreased
close to its minimum value, and the $m=-5$ mode dominates. Figure 5 (bottom)
shows that the radial position of the dominant mode (either the $m=-5$ or the
$m=-6$ mode) is indeed very stable. On the other hand, the radial position of
the low amplitude mode strongly fluctuates around its equilibrium value (with
standard deviations around $~{}0.6$ cm for the $m=-5$ mode and $\sim 0.5$ cm
for the $m=-6$ mode).

The exchange events observed for $p_{0}=1.3$ mTorr between modes $m=-5$ and
$m=-6$ (Figures 3 and 5) are similarly observed at $p_{0}=1.1$ mTorr and
$p_{0}=0.9$ mTorr. The timescales of the exchange events are now determined at
these three values of the pressure. This is done by fitting the mode amplitude
$A_{m}$ as exponentially growing: $A_{m}\propto\exp(t/\tau)$. Figure 9 (left)
shows a typical fit around $t_{0}$: the green part shows the interval over
which the raw signal (blue) is fitted; a low-pass filtered signal is shown for
clarity (red). Figure 9 (right) shows the resulting values of $\tau$ found for
the $m=-5$ mode. The growth-time $\tau$ significantly increases with the
pressure, its value doubles from $p_{0}=0.9$ mTorr to $1.3$ mTorr. This is
interpreted as being the result of an increased friction from the neutrals at
higher pressure. Note that this observation of a decrease of the ion acoustic
wave growth rate with increasing pressure is consistent with theoretical
predictions [9].

Figure 9: Left: time evolution of $m$-modes average amplitudes, extracted by
2D-FT. Left: fit of the 2D-FT $m=-5$ mode growth rate at an exchange event
with $m=-6$ mode, for $p_{0}=1.3$ mTorr. The fit is done on a selected
interval of the raw data (light blue). The red curve is the result of a
filter. Right: Evolution of the growth time scale of $m=-5$ mode, evaluated
during exchange events, as a function of the pressure $p_{0}$.

###  V.2 Residence time distribution

The statistics of the transitions between the $m=-5$ and $m=-6$ modes were
obtained in a new set of experiments, performed at a lower sampling frequency
(20 kfps) 222Note that with the lower sampling frequency of 20 kfps, the
extracted IAW modes with frequencies $\sim 70$ kHz can not be resolved
temporally, which prevent the distinction between modes $+m$ and $-m$ for a
given integer $m$. However it is observed that in the same conditions, with
higher sampling frequency acquisition, the (m=+p) mode amplitude is negligible
in front of the (m=-p) amplitude for $p=[4,7]$. We therefore use at $F_{s}=20$
kfps the sum of the amplitudes of extracted modes +p and -p, to estimate the
amplitude of mode (m = -p), for $p=[4,7]$. over longer times (2 seconds). This
allows to extract the time evolution of the modes average amplitude, extracted
by 2D-FT. The probability distribution function of the residence time of the
$m=-4$, $m=-5$ and $m=-6$ modes are shown in Fig. 10, for a total duration of
4 seconds (i.e. more than one thousand transitions between modes). The
distributions are compatible with an exponential distribution, which implies
that the transition events are not correlated. Such distributions of residence
times or waiting times are ubiquitous to transitions observed in aerodynamics
[57], turbulent flows [58, 59] or convection [60], to the waiting time between
reversals in dynamo experiments [61], or the turbulent dynamics of the scrape-
off layer in tokamaks [62, 63].

For all modes, the probability distribution function is compatible with a
functional fit of the form $e^{-t/\tau}$, with $\tau=5.4$ ms for $m=-4$, and
$\tau=6.0$ ms for $m=-5$ and $\tau=3.2$ ms for $m=-6$. As already observed in
Fig. 5, the $m=-4$ mode is tied to the $m=-5$ mode, resulting in similar pdf.
Figure 5 also shows that the system is more often dominated by a $m=-5$ mode,
which results in an exponential pdf with a larger characteristic time for the
$m=-5$ mode as compared to the $m=-6$ mode. High speed imaging of the dynamics
of the plasma allows to probe long-time statistics of the waves dynamics. It
opens the possibility to probe the evolution of the characteristic residence
time as a function of the control parameters (for instance pressure), possibly
shedding light to the physical processes leading to exchange events. Note that
the dominant mode (and the associated characteristic time) was observed to
strongly evolve with pressure (data not shown and beyond the scope of this
article).

Figure 10: Residence time PDF, obtained for a neutral pressure $p_{0}=1.30$
mTorr. Figure 11: Maps of the threshold $b_{0}^{2}$ (a) and bicoherence
$b^{2}$ (b), corresponding to the three-wave interaction
$(m=-5)+(m=-1)\leftrightarrow(m=-6)$. c) Frequency power spectra of modes
$m=-1$, $m=-5$ and $m=-6$ from 2D-FT computed at $r^{*}=3.3$ cm. $B=170$ G and
$p_{0}=1.3$ mTorr.

###  V.3 Non-linear behaviour

In order to further assess the non-linear nature of the dynamics between the
dominant ion acoustic modes, the bicoherence $b^{2}(f_{m=-5},f_{m=-1})$ is
computed for the three wave interaction $(m=-5)+(m=-1)\leftrightarrow(m=-6)$,
and shown in Fig 11. Note that to increase statistics, the 2D-FT at all radii
$1\leq r^{*}\leq 5$ around the wave maximal amplitude are used. More details
on the bicoherence computations are provided in appendix B. The threshold map
shown in Fig 11 a) was computed using a basic surrogate technique where the
phases of the 2D-FT signal are randomly mixed. This yields bicoherence values
for signals without any preferential phase relations, from which a threshold
value of $\max(b_{0}^{2})=0.12$ is estimated. Figure 11 b) shows the map of
$b^{2}(f_{m=-5},f_{m=-1})$ with $f_{m=-5}$ and $f_{m=-1}$ the frequencies of
modes $m=-5$ and $m=-1$ respectively. For the sake of visibility, the areas of
bicoherence high values are highlighted by gray contours (defined at $40\%$ of
the maximum value of a Gaussian filtered $b^{2}$ map). Most of the bicoherence
highest values lie around the diagonal $f_{m=-5}+f_{m=-1}=65$ kHz, that is the
dominant frequency of the $m=-6$ mode. This reveals the strong non-linear
behaviour of the ($m=-6$, $f\sim 65$ kHz) mode component, which interacts with
$m=-5$ and $m=-1$ modes via continuous sets of frequencies. The points
displayed as red dots in Fig. 11 b) are also enlarged for clarity: they
correspond to $b^{2}\gtrsim 0.36$, i.e. more than three times the threshold
value. The points for which $f_{m=-1}=0$ kHz, and $f_{m=-5}\in[64.6;65.4]$ kHz
correspond to frequency components of the $m=-5$ mode being fed by the high
amplitude of the ($m=-6$, $f\sim 65$ kHz) component. Note that these
interactions are not the dominant process characterizing the energy exchanges
detailed in subsection V.2, since they only involves frequency components of
the $m=-5$ mode around 65 kHz, with a low energy. The point at $f_{m=-5}=55.4$
kHz and $f_{m=-1}=11.2$ kHz however corresponds to the interaction:

$(m=-5,f=55.4)\ +\ (m=-1,f=11.2)\ \leftrightarrow\ (m=-6,f=66.7)$

which involves the dominant frequency components of the $m=-5$ and $m=-6$
modes. The very high bicoherence value at this location ($b^{2}=0.39$)
definitively establishes the non-linearity of the interactions between the ion
acoustic modes $(m=-5,f=55.4\mathrm{~{}kHz})$ and $(m=-6,f=66.7\
\mathrm{kHz})$, at the origin of the transitions observed in Fig. 5.

Finally, Fig. 11 c) shows the frequency spectra of the $m=-6$, $m=-5$ and
$m=-1$ modes involved in the three-wave interactions described above. These
spectra correspond to 1D cuts along the frequency axis of the 2D-FT spectrum
shown in Fig. 4. These spectra clearly display the non-linear feeding of the
$m=-5$ mode by the high amplitude $m=-6$ mode around 65 kHz. The non-linear
feeding of modes $m=-1$ and $m=-6$ by the high amplitude $m=-5$ mode around 55
kHz is also visible. The component $(m=-6,f=55\mathrm{~{}kHz})$ is then non-
linearly interacting with $(m=-5,f=44)\mathrm{~{}kHz}$ and
$(m=-1,f=11\mathrm{~{}kHz})$, as can be deduced by the high values of
$b^{2}(f_{m=-5}\sim 44,f_{m=-1}\sim 11)$ from Fig. 11 b).

The computation of other bicoherence maps (not shown here) reveals additional
non-linear behaviours. The bicoherence computation of the
$(m=-6)+(m=-1)\leftrightarrow(m=-7)$ coupling unambiguously shows that
$(m=-6,f=65.2\ \mathrm{kHz})$ non-linearly interacts with $(m=-7,f=74.0\
\mathrm{kHz})$ via an $(m=-1,f=8.8\ \mathrm{kHz})$ mode component (with
$b^{2}=0.36>3\max(b_{0}^{2})$). Similarly, bicoherence computation of the
$(m=-4)+(m=-1)\leftrightarrow(m=-5)$ coupling highlights that $(m=-4,f=44.2\
\mathrm{kHz})$ and $(m=-5,f=55.4\ \mathrm{kHz})$ modes non-linearly interact
via $(m=-1,f=11.2\ \mathrm{kHz})$ (with $b^{2}=0.38>3\max(b_{0}^{2})$). As a
last example, the bicoherence map for the interaction
$(m=-4)+(m=-2)\leftrightarrow(m=-6)$ does not exhibit high values indicating
the absence of non-linear interaction between the corresponding ion acoustic
modes. It however reveals that the frequency components $(f=55\ \mathrm{kHz})$
of $(m=-4)$ and $(m=-6)$ modes (resulting from the spread of the $m=-5$ mode,
visible in Fig. 4) are non-linearly linked via the $(m=-2,f=0\ \mathrm{kHz})$
mode component.

Thanks to the rich spatio-temporal information provided by camera imaging and
to the use of bicoherence, the weakly non-linear interactions are clearly
highlighted. In particular the existence of three-wave interactions between
ion acoustic modes $m=-p$, $m=-p-1$ and $m=-1$, for $p\in[4,5,6]$, is
demonstrated.

##  VI Conclusion

We have presented the first report of temporally and spatially entirely
resolved ion acoustic waves in a magnetized plasma column. The ion acoustic
waves were observed by means of fast camera imaging in a low temperature argon
plasma column, with dominant azimuthal mode numbers $m=-4$, $m=-5$ and $m=-6$
depending on the neutral pressure that was varied from 0.8 mTorr to 2 mTorr.

Two image analysis techniques, namely proper orthogonal decomposition (POD)
and 2D Fourier transform (2D-FT), were presented and thoroughly compared.
These tools are found to be complementary. POD is easy to implement and
adaptable to any type of data, and useful to provide a fast overview of the
underlying dynamics of a given dataset. This helps focusing in a second time
on a more precise and targeted analysis, that 2D-FT can then provide, yielding
detailed and unambiguous information.

Using 2D-FT analysis of high speed images, the ion acoustic waves were found
to rotate in opposite direction to the global $E\times B$ drift of the plasma
column, with a phase velocity Doppler shifted by this actual electric drift
velocity.

The dynamics of the dominant ion acoustic modes was then explored using the
2D-FT decomposition. Growth rates, which extraction was made possible by the
camera high temporal resolution, were found to decrease as pressure increases,
following previous numerical predictions. A detailed analysis was then carried
out in the particular case of $p_{0}=1.3$ mTorr. At this pressure the exchange
dynamics between dominant modes $m=-5$ and $m=-6$ was shown to be of a
bistable nature. More generally the weakly non-linear nature of the $m=-p$ and
$m=-p-1$ mode interaction ($p\in[4,5,6]$), involved in a three-wave
interaction with a $m=-1$ mode, was demonstrated by means of bicoherence
computation.

Finally we emphasize that, except from probe measurements that were needed for
the wave identification, all the results that were presented exclusively rely
on fast camera imaging measurements. This work can therefore be considered as
a case study demonstrating the very powerful capabilities of fast camera
imaging as a plasma diagnostics, notably for the exploration of complex waves
dynamics.

## Acknowledgements

This work was partly supported by the French National Research Agency under
Contract No. ANR-13-JS04–0003-01. We acknowledge support from the CNRS for the
acquisition of the high-speed camera and useful discussions with V. Désangles
and G. Bousselin and warmly thank P. Borgnat for advises on surrogate
techniques.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Radial scale: camera imaging v.s. probe

The magnetic field ripple and parallax in our experimental set-up leads the
camera lines of sight to cross regions of different plasma parameters. The
light recorded by camera, resulting of an integration process along these
lines of sight, cannot be directly compared to probe measurements that are
performed at $z=L_{2}$.

A transformation is implemented, modeling the integration along the camera
lines of sight of any plasma parameter that is measured at $z=L_{2}$. The
details of this transformation are provided in Ref. [27].

Figure 12 shows the result of this artificial integration process, applied to
a test profile peaked at $r=5$ cm (blue curve). The resulting profile (red
curve), expressed along the camera imaging coordinate $r^{*}$, shows that what
is seen on the camera images at $r^{*}=3.3$ cm mainly corresponds to the
plasma parameter evolution that is located at $r=5$ cm on the axis $z=L_{2}$.

Figure 12: Camera lines of sight integration process, applied to a test
profile measured at $z=L_{2}$.

##  Appendix B Bicoherence and confidence level

Bicoherence is a spectral analysis tool that is commonly used in physics, for
the detection of non-linear three waves interactions. Bicoherence computation
essentially consists in extracting the frequency components of one of several
signals, and comparing their phases. The signal decomposition at the basis of
a bicoherence analysis can be done by Fourier transform [64, 65] as it is the
case in this work, or based on a wavelet approach [66, 67]. In this appendix,
we first remind the basic principle of bicoherence, and then explained how
bicoherence is computed in the particular case of camera images. Then we
provide the definition of a clear and mathematically meaningful threshold,
that is often lacking when bicoherence is used onto experimental data in
plasma physics.

#### Evaluate three wave interactions by bicoherence

Let us consider three signals $x$, $y$ and $z$ with corresponding Fourier
transforms $\hat{x}$, $\hat{y}$ and $\hat{z}$. The cross bispectrum of $x$,
$y$ and $z$ is defined as a function of frequencies $(f_{1},f_{2})$ as :
$B(f_{1},f_{2})=\hat{x}(f_{1})\hat{y}(f_{2})\hat{z}^{*}(f_{1}+f_{2})$. If the
frequency components $f_{1}$ and $f_{2}$ of $x$ and $y$ respectively (with
phases $\phi^{x}_{1}$ and $\phi^{y}_{2}$) are involved in a three-wave
interaction with the frequency component $f_{1}+f_{2}$ of $z$ (with a phase
$\phi^{z}_{1+2}$) the phase difference between these signals is a constant.
Computing the bispectrum onto successive reduced parts $\delta_{t}$ of the
signals is therefore a way of measuring this phase locking, since $B_{\delta
t}(f_{1},f_{2})\propto\exp^{-i(\phi^{x}_{1}+\phi^{y}_{2}-\phi^{z}_{1+2})}$.
The bicoherence is defined by the normalized average over a statistically
significant number of such bispectrum computations:

$b^{2}(f_{1},f_{2})=\frac{|\langle\hat{x}(f_{1}).\hat{y}(f_{2}).\hat{z}^{*}(f_{1}+f_{2})\rangle_{\delta_{t}}|^{2}}{\langle|\hat{x}(f_{1}).\hat{y}(f_{2})|^{2}\rangle_{\delta_{t}}\langle|\hat{z}(f_{1}+f_{2})|^{2}\rangle_{\delta_{t}}}$

If the signal frequency components previously mentionned are perfectly
uncorrelated, $b^{2}$ corresponds to the average of random complex numbers,
and tends to cancel out. If those frequency components are on the contrary
perfectly phase locked, the computations of $B_{\delta t}(f_{1},f_{2})$ have a
constant value and $b^{2}=1$. In the case of experimental data, neither case
is realistic, and a threshold value $b_{0}^{2}$ above which the bicoherence
can be considered significant needs to be defined (see last paragraph of this
appendix).

#### Bicoherence on camera images

With camera images that provide 2D spatio-temporal signals, bicoherence can be
computed between the frequency components of distinct modes $m$. Bicoherence
allows to probe the phases of signal components for given set of wave vector
and frequency ($m,f$). This analysis is applied on the present camera images,
following the work of Ref. [46]. For a given radius $r^{*}$, let us denote the
2D Fourier decomposition of the light intensity:

$A(t,\theta)=\sum_{n,p}a(f_{n},m_{p})e^{i(2\pi
f_{n}t-m_{p}\theta+\phi_{n,p})}$

The spectrum associated with a single $m_{p}$ mode is a part of this
decomposition:

$\hat{A}_{m_{p}}(f_{n})=a(f_{n},m_{p})e^{i\phi_{n,p}}$

Similarly to computations achieved for 1D signals, the bispectrum is defined
as a statistical averaging, over parts of lengths $\delta t$ of the signal. In
order to improve the statistical averaging here, the sum is also done over the
signals from various radii $r^{*}$. This double averaging process is denoted
$\langle.\rangle_{r^{*},\delta_{t}}$. The bispectrum between components
($m_{1}$,$f_{1}$) and ($m_{2}$,$f_{2}$) is then defined as
$B_{m_{1},m_{2}}(f_{1},f_{2})=\hat{A}_{m_{1}}(f_{1}).\hat{A}_{m_{2}}(f_{2}).\hat{A}_{m_{1}+m_{2}}(f_{1}+f_{2})^{*}$,
and the bicoherence is computed as:

$b_{m_{1},m_{2}}^{2}(f_{1},f_{2})=\frac{|\langle\hat{A}_{m_{1}}(f_{1}).\hat{A}_{m_{2}}(f_{2}).\hat{A}_{m_{1}+m_{2}}^{*}(f_{1}+f_{2})\rangle_{r^{*},\delta_{t}}|^{2}}{\langle|\hat{A}_{m_{1}}(f_{1}).\hat{A}_{m_{2}}(f_{2})|^{2}\rangle_{r^{*},\delta_{t}}\langle|\hat{A}_{m_{1}+m_{2}}(f_{1}+f_{2})|^{2}\rangle_{r^{*},\delta_{t}}}$

The bicoherence as it is implemented in our code takes mode numbers $m_{1}$
and $m_{2}$ as an entry and explores all possible three-wave interactions
$(m_{1},f_{1})+(m_{2},f_{2})\leftrightarrow(m_{1}+m_{2},f_{1}+f_{2})$ in terms
of frequencies $f_{1}$ and $f_{2}$. The operation is fixed as an addition, and
the result is in a form of a 2D map of $b_{m_{1},m_{2}}^{2}(f_{1},f_{2})$,
with $[f_{1},f_{2}]\in[0,F_{s}/2]^{2}$, $F_{s}$ being the data sampling
frequency. Here for simplicity, the bicoherence applied to camera images is
simply denoted $b^{2}$.

#### Definition of a threshold

The phase correlation between any set of experimental signals is likely to be
imperfect or partial, leading to $0<b^{2}<1$. Moreover the absolute values of
the bicoherence are relative to each set of signals investigated: a general
threshold value is not relevant. A method to systematically determine the
level above which the value of $b^{2}$ becomes physically meaningful, that
depends on each bicoherence computation, is therefore needed.

A possible method consists in the creation of an artificial set of signals,
sharing the same characteristics than the original signals, but without any
preferential relation between its frequency components. The bicoherence of
this artificial set of signals is then computed, providing a lower limit for
the values of $b^{2}$. This type of method is called surrogate technique [68],
and can be very sophisticated. Here we use a very basic version of the
surrogate techniques: the phases of each 2D-FT spectra are randomly mixed. The
bicoherence computation applied to this modified data defines a threshold map
$b_{0}^{2}(f_{1},f_{2})$. Then for simplicity we take the maximal value
$\max(b_{0}^{2})$ and define it as a global threshold value for the real
bicoherence computation $b^{2}(f_{1},f_{2})$ of this dataset.

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