# Plasma potential shaping using end-electrodes in the Large Plasma Device

R. Gueroult1<EMAIL_ADDRESS>S. K. P. Tripathi2  F.
Gaboriau 1  T. R. Look2  N. J. Fisch3  1LAPLACE, Université de Toulouse, CNRS,
INPT, UPS, 31062 Toulouse, France 2Department of Physics and Astronomy,
University of California, Los Angeles, Los Angeles, California 90095, USA
3Department of Astrophysical Sciences, Princeton University, Princeton NJ
08540, USA

###### Abstract

We perform experiments in the Large Plasma Device (LAPD) at the University of
California, Los Angeles, studying how different end-electrode biasing schemes
modify the radial potential profile in the machine. We impose biasing profiles
of different polarities and gradient signs on a set of five concentric
electrodes placed $12$ m downstream from the plasma source. We find that
imposing concave-down profiles (negative potential radial gradient) on the
electrodes create radial potential profiles halfway up the plasma column that
are comparable to those imposed on the electrodes, regardless of the biasing
polarity. On the other hand, imposing concave-up profiles (positive potential
radial gradient) leads to non-monotonic radial potential profiles. This
observation can be explained by the current drawn through the electrodes and
the parallel plasma resistivity, highlighting their important role in
controlling the rotation of plasma. Concave-down plasma potential profiles,
obtained by drawing electrons on the axis, are predicted to drive azimuthal
drift velocities that can approach significant fractions of the ion sound
speed in the central region of the plasma column.

##  1 Introduction

Producing and controlling rotation in plasmas is attractive for a number of
applications (Lehnert, 1971). For example, plasma rotation could prove
essential to advancing magnetic confinement fusion in both linear (Ellis et
al., 2001, 2005; Romero-Talamás et al., 2021; Endrizzi et al., 2023) and
toroidal (Rax et al., 2017) geometries. Plasma rotation control is also a key
element in advancing plasma mass separation technologies (Zweben et al., 2018;
Gueroult et al., 2018), both in plasma centrifuges for isotope separation
(Krishnan et al., 1981; Fetterman & Fisch, 2009; Borisevich et al., 2020) and
cross-field rotating collisionless separator concepts targeting new separation
needs (Ohkawa & Miller, 2002; Fetterman & Fisch, 2011; Gueroult et al., 2014;
Liziakin et al., 2022). Finally, controlling plasma rotation in laboratory
experiments provides unique opportunities for investigating the role of plasma
rotation in affecting the dynamics of laboratory and space plasmas — such as
the formation of accretion disks (Flanagan et al., 2020; Valenzuela-Villaseca
et al., 2023) and propagation characteristics of plasma waves (Gueroult et
al., 2019b, 2023).

In magnetized plasmas, a possible means of controlling rotation is through
cross-field drift driven by a perpendicular electric field. While new
promising schemes harnessing beams and waves have been proposed to establish
this perpendicular electric field (Putvinskii, 1981; Rax et al., 2023), a
long-studied solution is to use biased electrodes. One option is to bias
electrodes positioned at the outer edge of the plasma, referred to as
limiters, to impose a radial potential difference. Although this scheme has
been shown to effectively establish an electric field and rotation in the
plasma edge (Schaffner et al., 2012), the electric field and rotation in the
bulk plasma is due to plasma self-reorganisation (Weynants & Oost, 1993) and
cannot be directly controlled. Another option, which has been surmised to
provide greater control over the electric field in the plasma bulk, is the
end-electrodes scheme originally proposed by Lehnert (1970, 1973). The
principle is to use independently biased electrodes, on which the plasma
column terminates, to finely shape the perpendicular electric field in the
plasma bulk. Despite having been implemented in numerous experimental setups
over the years (see _e. g._ Gueroult et al. (2019c) and references therein),
the effectiveness of end-electrode biasing in controlling the perpendicular
electric field across plasma regimes remains unclear. Two outstanding issues
are determining the fraction of the imposed bias lost in the sheath and how
much the potential varies along magnetic field lines (Trotabas & Gueroult,
2022). In this study, we report the results of an experimental campaign using
end-electrodes carried out in the Large Plasma Device (LAPD) at the University
of California, Los Angeles in August 2023 that shed light on this problem.

As mentioned above, numerous experiments have used biased end-electrodes.
However, the operating conditions and size of LAPD allow for exploring
different regimes. Indeed, it has been shown (Poulos, 2019; Gueroult et al.,
2019a) that an important dimensionless parameter affecting the uniformity of
the plasma potential along a field line is

$\tau=\frac{L}{a}\sqrt{\frac{\sigma_{\perp}}{\sigma_{\parallel}}}$ (1)

with $L$ and $a$ the plasma column length and radius, respectively, and
$\sigma_{\perp}$ and $\sigma_{\parallel}$ the perpendicular and parallel
plasma conductivities. Many of the recent end-electrodes biasing experiments
used helicon plasmas in mid-size experiments where typically $\tau\geq 1$
(Gilmore et al., 2011, 2014; Zhang et al., 2015; Gueroult et al., 2016;
Liziakin et al., 2019; Désangles et al., 2021). However, this regime has been
inferred to prohibit effective electrical connection between symmetrical
electrodes positioned at opposite ends of the machine (Gueroult et al.,
2019a), contrary to the operating principle envisioned by Lehnert (1970,
1973). On the other hand the low operating pressure in LAPD typically leads to
regimes where $\tau\ll 1$, creating opportunities to test these models.

This paper is organised as follows. In Section 2 we begin by introducing the
experimental setup used in this experimental campaign, and how it both relates
to and differs from past experiments. In Sec. 3 we analyse plasma parameter
profiles obtained for the selected operating condition prior to inserting the
electrodes in the machine, which will serve as a baseline. In Section 4 we
present results obtained for a set of six complementary biasing scenarios,
which are shown to exhibit distinctly different plasma potential profiles. In
Section 5 we then propose a simple analysis to explain trends in plasma
potential modifications, considering both the voltage drop along field lines
and the dynamics of the anode-plasma system, and discuss this model in light
of previous contributions. Finally, in Sec. 6 we use the plasma potential
profiles measured in the different biasing scenarios to infer the
corresponding plasma rotation expected from cross-field drift and show how
imposing a concave-down biasing profile leads to azimuthal drift velocities
approaching a significant fraction of the ion sound speed. These results are
briefly discussed in regard to previous studies demonstrating biasing driven
rotation in LAPD. Lastly, in Section 7, we summarise the main findings of this
study.

##  2 Experimental setup

In this section, we introduce the main characteristics of the experimental
setup used in this campaign, which will be helpful for our upcoming analysis
and discussion of biasing results.

###  2.1 Biasing setup

Experiments are carried out in the Large Plasma Device (LAPD) at the
University of California, Los Angeles (Gekelman et al., 2016). LAPD uses a
large $38$ cm diameter hot LaB6 cathode (Qian et al., 2023) to produce a
highly reproducible $20$ m long, $75$ cm diameter magnetized plasma column
with a repetition rate of $0.3-1$ Hz.

In this campaign, we use a set of five stainless steel disks, electrically
separated by ceramic spacers, to form a stack of concentric electrodes as
shown in figure 1(a). Each disk, $\mathcal{E}_{i}$ with $i\in\llbracket
1,5\rrbracket$, has a radius $r_{i}=2.54i$ cm. This electrode stack, or multi-
disk electrode, is centered radially at the machine port $\\#35$ and rotated
perpendicular to the magnetic field. As illustrated in figure 2, the electrode
stack is about $11$ m from the anode. Although this multi-disk electrode setup
is similar in design and comparable in size to one previously used in LAPD by
Koepke et al. (2008, 2016), an important difference concerns the biasing
scheme on the electrodes. In this study, we bias each electrode with respect
to the machine ground, whereas electrodes were biased with respect to each
other (or a group with respect to another group) in earlier studies. As
highlighted in figure 1(b), the voltage $\phi_{i}$ of each electrode
$\mathcal{E}_{i}$ is set by positioning pins across a resistive voltage
divider ($R_{d}\sim 1.1~{}\Upomega$), which is connected to a capacitor bank
through a transistor switch. We note that this experimental setup also shares
commonalities with the work by Jin et al. (2019), though in this case, the
emissive cathode positioned in the machine was biased with respect to the
anode.

(a) Multi-disk electrode

(b) Biasing circuit

Figure 1: Picture of the multi-disk electrode used in this experimental
campaign 1(a) and electric circuit used to bias each electrode 1(b). The outer
radius of each electrode is $r_{i}=2.54i$ cm, $i\in\llbracket 1,5\rrbracket$.
$R_{w_{i}}$ is the resistance of the cable between the point of measure near
the voltage divider and the disk $\mathcal{E}_{i}$ in the machine.  Figure 2:
Experimental setup used in this campaign. The plasma is created by applying a
voltage between the cathode and a mesh anode. The five concentric disk
electrodes are installed on port #35 and biased with respect to the grounded
vacuum chamber, with bias $\phi_{i}$ for $i\in\llbracket 1,5\rrbracket$. The
magnetic field in the source $B_{s}$ is $0.2$ T whereas the field in the main
chamber $B_{0}$ is $0.1~{}$T.

We also note an important difference in this setup compared to the canonical
end-electrodes configuration proposed by Lehnert (1970). As shown in figure 2,
one end of the machine is blocked by the anode mesh. As a result, the
configuration is not symmetrical. Although it makes it impossible to test the
assumption of effective electrical connection between symmetrical end-
electrodes for $\tau\ll 1$, it is still possible to study how the potential
distributes itself along and across field lines in these conditions. In this
regard, our study builds on the study by Jin et al. (2019) by exploring
different biasing configurations using multiple electrodes $\phi_{i}$ and both
positive and negative biases.

###  2.2 Shot parameters

With the LAPD cathode upgrade (Qian et al., 2023), the discharge is fueled by
gas puffed near the new LaB6 source. In this campaign, we use helium gas puff
with a maximum chamber pressure reaching $3.45\times 10^{-5}$ Torr and a
$15~{}$ms long pulsed discharge with a repetition rate of $1$ Hz. The source
magnetic field and magnetic field in the main chamber are respectively
$B_{s}=0.2$ T and $B_{0}=0.1$ T, for a field ratio $\rho_{B}\doteq
B_{s}/B_{0}=2$. In these conditions, as can be seen in figure 3, the peak
discharge current is $I_{d}\sim 3.2$ kA with an anode-cathode voltage
difference of about $65$ V. We will verify that it corresponds to a plasma
density of about $6\times 10^{18}$ m-3, which was used as our target operating
point during the one-week-long campaign.

Figure 3: Time evolution of the anode-cathode voltage (green, right axis) and
discharge current (red, left axis) during a typical shot. The $t=0$ reference
corresponds to $I_{d}=1$ kA. The blue region $12\leq t\leq 18$ ms corresponds
to the time the electrode bias is on. The vertical dashed lines indicate $3$
different instants during the shot that we will focus on in our data analysis.
The first one ($11.1$ ms) is before bias has been turned on, the second one
($12.6$ ms) is during the main discharge with bias on, and the last one
($15.6$ ms) is in the afterglow with bias on.

The active phase during which electrodes are biased starts at $12~{}$ms and
ends at $18~{}$ms (blue region in figure 3). The active bias phase thus covers
both the end of the main discharge during which electrons are still injected
in the plasma and the afterglow regime which refers to the phase after the
discharge voltage has been switched off (Gekelman et al., 2016). Since the
electron temperature $T_{e}$ drops rapidly in the afterglow, this allows
probing the effect of biasing in different plasma conditions.

###  2.3 Diagnostics

Plasma parameters in the machine are inferred through a suite of probe
diagnostics. In this campaign, four-tip probes are mounted on probe drives on
ports $\\#20$ and $\\#34$, which correspond to distances of about $5.5~{}$m
and $30$ cm from the multi-disk electrode. The tip area is about $4$ mm2. On
each of these probes, three tips are used. Two of these three tips are used to
acquire respectively the ion saturation current $I_{sat}$ and the floating
potential $\phi_{f}$. The last tip is used as a swept Langmuir probe to
acquire $I(V)$ characteristics. Plasma potential estimates $\phi_{p}$ are
obtained from $I(V)$ probe characteristics using spline fitting and standard
inflection-point method (Godyak & Alexandrovich, 2015). The electron
temperature is determined from a linear fit of the logarithm of the electron
current as a function of bias for biases between the floating potential and
the inflection point. Practically, $200$ ms long $I(V)$ Langmuir sweeps
separated by a $100$ ms pause were used, for a total of $23$ sweeps starting
at $t=11.1~{}$ms. In addition, an emissive probe (Martin et al., 2015) is
mounted on a third probe drive on port $\\#32$, which is about $90$ cm away
from the multi-disk electrode. The heating current is first adjusted to ensure
symmetrical $I(V)$ curves, whereafter the plasma potential $\phi_{p}$ is taken
to be the floating potential of the emissive electrode. All three probe drives
are computer-controlled, making it possible to obtain 2D maps of plasma
parameters at three axial locations ($\\#20$, $32$ and $34$). In practice we
will mostly focus on results from planes at $\\#20$ and $34$, and will verify
that the plasma potential estimated from the emissive probe on port $\\#32$ is
consistent with estimates obtained from $I(V)$ data on port $\\#34$. Absolute
plasma density estimates are simply deduced from $I_{sat}$ data after
calibration using microwave interferometric data taken on port $\\#20$.
Variations in $I_{sat}$ are therefore entirely attributed to variations in
density $n$, while they could also result from variations in electron
temperature $T_{e}$. Although crude, this assumption is supported by the
weaker dependence of the ion saturation on the temperature
($\propto\sqrt{T_{e}}$) compared to the density ($\propto n_{e}$), and the
relatively limited variations in $T_{e}$ which will be observed. Lastly, the
data presented here is averaged over five shots taken at the same spatial
position, with error bars quantifying the standard deviation over these five
shots.

Besides plasma parameters and general machine parameters, the voltage
$\phi_{i}$ of each electrode $\mathcal{E}_{i}$ with $i\in\llbracket
1,5\rrbracket$, the current $I_{i}$ going through each of these electrodes, as
well as the anode potential $\phi_{a}$, are recorded throughout the shot. For
practical reasons, the voltages are measured right next to the voltage divider
illustrated in figure 1(b), which is positioned some meters away from the
electrodes, outside the chamber. Due to the finite resistance of these long
connectors, the actual electrode potential $\phi_{i}$ thus has to be corrected
given the current $I_{i}$ and the resistance $R_{w_{i}}$ of each of these
connectors. These resistors were measured before the campaign with the multi-
disk electrode outside the chamber, with $R_{w_{i}}=0.160$, $0.157$, $0.167$,
$0.162$ and $0.148~{}\Upomega$ for $i\in\llbracket 1,5\rrbracket$.

##  3 Baseline parameters

###  3.1 Baseline profiles

With our goal to highlight the effect of biasing on plasma
parameters—particularly on the plasma potential—we briefly discuss the plasma
parameter profiles measured for the selected operating point, first with the
multi-disk electrode pulled out of the chamber and then with the multi-disk
electrode in the machine but disconnected. This dataset is shown in figure 4.

Figure 4: Radial profiles of the plasma potential (top panels) and the
electron temperature (middle panels) inferred from Langmuir probe sweeps, and
of the floating potential (top, cross) and of the density (bottom) deduced
from floating probe and saturation current (without $T_{e}$ corrections) at
four times during the discharge, and on two ports: port $\\#20$ (left) and
port $\\#34$ (right). The lighter curves correspond to the data taken with the
electrodes outside the machine. The vertical dashed lines represent the
electrode’s position (when inserted).

Starting with density (lower panel in figure 4), we measure a maximum of about
$6\times 10^{18}$ m-3 in the center during the main discharge, and a decrease
with time after the discharge has been turned off ($t=15~{}$ms). Both in the
main discharge and in the afterglow, the density drops rapidly for $r\geq 25$
cm. This is consistent with a plasma column radius equal to the cathode
radius. Indeed, the cathode radius is $r_{c}=19~{}$cm, which considering flux
conservation and the magnetic field ratio $\rho_{B}\doteq B_{s}/B_{0}=2$
between the source and the main chamber gives a plasma column radius
$r_{c}\sqrt{\rho_{B}}\sim 27$ cm. The density data also shows an asymmetry,
with density for positive $x$ larger than density for negative $x$. This
asymmetry does not vanish when averaging over time and is also larger than the
shot-to-shot deviation, as shown by the error bars. Although the origin of
this asymmetry remains to be determined, it might be evidence for probe
perturbation. Negative $x$ indeed corresponds to the far side of the machine,
for which the probe arm extends past the central point. This could lead to
perturbations, in particular if the plasma is rotating.

Moving on to the middle panel in figure 4, the electron temperature deduced
from $I$-$V$ characteristics is about $8$ eV in the core during the main
discharge on port $\\#20$, and drops to about $4$ eV at the same location
after the discharge has been turned off. Closer to the electrodes on the port
$\\#34$, a similar radial profile is observed but with an overall axial drop
of about $5$ eV in the main discharge and $2$ eV in the afterglow, leading to
peak temperatures of respectively $4$ and $2$ eV.

Lastly, the plasma potential $\phi_{p}$ (top panel in figure 4) is relatively
flat across the plasma column. It is respectively positive and negative by a
few volts on port $\\#20$ and on port $\\#34$. The axial voltage drop between
these two ports is about $5$ V in the main discharge and about $2$ V in the
afterglow. This axial variation is consistent with the voltage drop expected
from the axial temperature variation highlighted just above. Indeed the
electron momentum equation gives

$en_{e}\frac{\partial\phi_{p}}{\partial
z}-\frac{\partial(n_{e}T_{e})}{\partial z}+en_{e}\eta_{{\parallel}}j_{z}=0$
(2)

so that neglecting axial current

$\frac{\partial\phi_{p}}{\partial z}=\frac{1}{e}\frac{\partial T_{e}}{\partial
z}+\frac{T_{e}}{en_{e}}\frac{\partial n_{e}}{\partial z}>0.$ (3)

Over distances short compared to the axial density gradient length, which we
verify in the lower panel in figure 4 corresponds to the distance between port
$\\#20$ and port $\\#34$, the first term on the right hand side dominates. The
axial voltage drop is then equal to lowest order to the axial temperature
variation.

Finally, we note that introducing the disconnected electrodes to the plasma
produces a noticeable effect at the port $\\#34$ which is located closest to
the electrodes. A significant density depletion, consistent with enhanced
losses, occurs in the central region where the field lines connect to the
electrodes. A closer examination here also reveals a modest drop in
$T_{e}$—and consequently $\phi_{p}$.

###  3.2 Currents

From the plasma density $n_{e}$ and electron temperature $T_{e}$ measured
above, we can deduce estimates for the ion saturation current density
$j_{is}\doteq en_{e}c_{s}$ with $c_{s}\doteq\sqrt{k_{B}T_{e}/m_{i}}$ the ion
sound speed. Using $n_{e}=5\times 10^{18}$ m-3 and $T_{e}=5$ eV (see table 1)
as a baseline operating point in this campaign, we obtain a characteristic ion
saturation current density

$j_{is}^{\diamond}=8.7~{}\textrm{kA.m}^{-2}.$ (4)

Parameter | Value
---|---
Gas | helium
Discharge current $I_{d}\quad$[kA] | 3.2
Main chamber magnetic field $B_{0}\quad$ [T] | 0.1
Magnetic field ratio $\rho_{b}$ | $2$
Plasma density $n_{e}\quad[\textrm{m}^{-3}]$ | $5~{}10^{18}$
Electron temperature $T_{e}\quad$[eV] | $5$
Plasma radius [cm] | $25$
Plasma length [m] | $11$
Table 1: Typical machine and plasma parameters for the operating point
targeted in this campaign.

For reasons that will become clear later, it is informative to compare this
current to the current density deduced from the discharge current $I_{d}$.
Considering a uniform emission on the cathode, and accounting for the anode
transparency $\iota=0.66$ (Qian et al., 2023), the current density of primary
electrons injected in the plasma through the anode during the main discharge
is

$j_{inj}=-\frac{\iota I_{d}}{\pi\rho_{B}r_{c}^{2}}$ (5)

where $\rho_{B}$ is the field ratio previously defined. For the discharge
current $I_{d}\sim 3.2$ kA shown in figure 3 and $\rho_{B}=2$ one gets
$j_{inj}=-9.3~{}\textrm{kA.m}^{-2}$.

Although the amplitude of these currents will vary with the location through
the spatial dependence of $T_{e}$ and $n_{e}$ on one hand, and with the exact
discharge current on the other hand, the important point here is to note that
in the conditions investigated here these two terms are of comparable
magnitude.

##  4 Effect of biasing on the radial profile of plasma parameters

###  4.1 Biasing profiles

We choose six biasing profiles to impose on the multi-disk electrode to study
the effect of differential electrode biasing on plasma parameters. Our
motivation for choosing these six specific biasing scenarios, which are listed
in table 2 and illustrated in figure 5, is that they cover different
polarities with respect to machine ground as well as different directions of
the radial gradient of the potential. In other words, we examine the effect of
both positive and negative biases, as well as concave-up and -down potential
profiles. The amplitude in all cases is $30$ V, though as mentioned above the
resistance of the connectors is such that the actual bias on the electrode can
be slightly larger or smaller depending on the current drawn by this
particular electrode.

Biasing profile | Shorthand | Color code
---|---|---
Positive, concave-down (negative gradient) | $\frown^{+}$ |
Positive, flat |  + |
Positive, concave-up (positive gradient) | $\smile^{+}$ |
Negative, concave-up (positive gradient) | $\smile^{-}$ |
Negative, flat |  - |
Negative, concave-down (negative gradient) | $\frown^{-}$ |
Table 2: Six biasing scenarios studied in this campaign. The amplitude is $30$
V in all cases.  Figure 5: Potential profile imposed on the electrodes for the
six biasing scenarios tested in this campaign, as listed in table 2. The thin
solid coloured lines represent the continuous targeted profile, while the
thicker coloured segments represent the actual steplike profile imposed on the
electrodes. The grey hatched regions represent the position of the different
electrodes $\mathcal{E}_{i}$.

A consequence of this resistance, and of the biasing circuit shown in figure
1(b), is that the plasma potential profiles already show a weak dependence on
the particular biasing scenario implemented before the active biasing phase.
This is illustrated in figure 6, which corresponds to $t=11.1$ ms, that is
$0.9$ ms before the biasing power supply is turned on. Indeed, even if the
bias is off, electrodes are not floating. They are connected through the
connectors and the voltage divider. Because the position of a given pin on the
voltage divider differs with the different biasing scenarios, the resistance
between a given electrode and ground also differs. This leads to a different
electrode bias, and from there to a different plasma potential in front of
this electrode. Note that the larger error bars seen in figure 6 for $r\gtrsim
25$ cm corresponds to the region beyond the projected cathode radius
$r_{c}\sqrt{\rho_{B}}\sim 27$ cm where the density drops rapidly.

Figure 6: Plasma (solid lines) and floating (cross markers) potential profiles
on two ports ($\\#20$ top, $\\#34$ bottom) at $t=11.1$ ms (before biasing) for
the six biasing scenarios listed in table 2. The vertical dashed lines
represent the electrodes position. The thicker and lighter colored horizontal
lines represent the electrodes potential $\phi_{i}$ at that time. The
horizontal grey dashed lines with coloured circles symbols represent the anode
potential $\phi_{a}$. The triangle symbol in the bottom panel corresponds to
plasma potential measurement from emissive probe on port $\\#33$ (when
available).

We also verify in figure 6 that, while there is a small offset of about $2$ V
between the two datasets, the plasma potential estimated from $I(V)$ sweeps on
port $\\#34$ (solid lines) is consistent with that obtained about $60$ cm away
from the emissive probe on port $\\#32$ (triangle symbols). The radial
dependence predicted by both methods is notably remarkably similar. This is
particularly noteworthy in that these two datasets were obtained in different
runs (days apart), so that small variations in plasma parameters cannot be
excluded.

###  4.2 Potential profiles under active biasing

As shown in figure 7, biasing has a very noticeable effect on the radial
profile of plasma potential during the main discharge ($t=12.6$ ms). We can
make a number of general observations before attempting to explain this plasma
response in the next section.

First, the amplitude of the plasma potential variations in response to biasing
is significant. For the conditions used here, one finds $-15\leq\phi_{p}\leq
35$ V with active biasing, compared to $5\leq\phi_{p}\leq 10$ V with biasing
off. The amplitude of this shift is thus comparable to the maximum bias of
$\pm 30$ V applied on the electrodes. This suggests that only a modest
fraction of the applied bias is lost through the sheaths (Poulos, 2019;
Trotabas & Gueroult, 2022). Second, the radial profile of the plasma potential
is found to track—if not to be driven by—the applied bias profile. More
precisely, figure 7 shows that it is the floating potential
$\phi_{f}(r,z_{34})$ rather than the plasma potential $\phi_{p}(r,z_{34})$
which tracks the applied bias $\phi_{b}(r)$, and that concave-down profiles
which have negative radial gradients (blue and purple) appear to be more
easily passed into the plasma column than those with positive gradients (red
and green). Lastly, we note that these variations in plasma potential are
accompanied by smaller yet significant variations in anode potential
$\phi_{a}$. The anode potential is indeed observed to vary in these conditions
from $-11$ to $11$ V depending on the biasing scenario.

Figure 7: Plasma (solid lines) and floating (cross markers) potential profiles
on two ports ($\\#20$ top, $\\#34$ bottom) at $t=12.6$ ms (_i. e._ in the main
discharge during biasing) for the six biasing scenarios listed in table 2. The
vertical dashed lines represent the electrodes position. The thicker and
lighter colored horizontal lines represent the electrodes potential $\phi_{i}$
at that time. The horizontal grey dashed lines with coloured circles symbols
represent the anode potential $\phi_{a}$. The triangle symbol in the bottom
panel correspond to plasma potential measurement from emissive probe on port
$\\#33$ (when available).

Looking more closely at the ordering between the plasma potential and the
electrode bias, we find that for constant applied bias across the electrodes
(orange and brown) the plasma potential is larger than the bias on any
electrode. The difference between the plasma potential on the port $\\#34$ and
the applied bias on the electrodes is then relatively uniform radially. For
non-uniform biasing profiles, the overall profile appears to be set by the
condition that the plasma potential is to be slightly more positive than the
most positive electrode in the region in front of this electrode. Because the
plasma potential does not precisely follow the applied bias, the voltage drop
between a given electrode $\mathcal{E}_{i}$ and the plasma potential in front
of it varies with each electrode. Anticipating our discussion in the next
section, we see that this will lead to non-uniform currents to the electrodes.

Irrespective of the biasing case, the plasma potential profiles measured on
ports $\\#20$ and $\\#34$ have almost the same shape, at least in the region
in front of the electrodes. In fact one verifies that they can be simply
deduced from one another via a uniform shift up (going from $\\#34$ to
$\\#20$) or down (going from $\\#20$ to $\\#34$). As mentioned above, this
shift is consistent with a radially uniform temperature drop between the two
axial locations. Other than for this temperature effect, the data in figure 7
shows that the imprint of the bias on the plasma potential profile is
recovered at least $6$ m away from the multi-disk electrodes.

Finally, we verify in figure 8 that the strong influence of the biased
electrodes on the plasma potential, as well as the general trends identified
above, persist in the afterglow after the main discharge is shut off. In the
afterglow, however, we find that the plasma potential profile is much closer
to the floating potential profile. This behaviour is consistent with the rapid
drop in $T_{e}$ observed in the afterglow.

Figure 8: Plasma (solid lines) and floating (cross markers) potential profiles
on two ports ($\\#20$ top, $\\#34$ bottom) at $t=15.6$ ms (_i. e._ in the
afterglow during biasing) for the six biasing scenarios listed in table 2. The
vertical dashed lines represent the electrodes position. The thicker and
lighter colored horizontal lines represent the electrodes potential $\phi_{i}$
at that time. The horizontal grey dashed lines with coloured circles symbols
represent the anode potential $\phi_{a}$. The triangle symbol in the bottom
panel correspond to plasma potential measurement from emissive probe on port
$\\#33$ (when available).

###  4.3 Density and temperature profiles under active biasing

Looking now at the effect of biasing on other plasma parameters, figure 9
plots the radial profiles of plasma density $n_{e}$ and of electron
temperature $T_{e}$ during active biasing in the main discharge ($t=12.6$ ms),
just in front of the electrodes (port $\\#34$).

Starting with the density, one observes for non-uniform biasing scenarios a
decrease in density in the region in front of the most positive electrode.
This behaviour is most noticeable for the negative gradient biasing scenarios
($\frown^{-}$, bottom-right and $\frown^{+}$, top-left panels in figure 9),
for which a clear density drop of $30-50\%$ is observed in front of the
central electrode $\mathcal{E}_{1}$. This depletion is associated, as we will
show, to an electron current collected on the electrode with the highest
potential since, as seen in figure 7, the sheath in front of this electrode is
the smallest. Looking more closely, this localised decrease in density appears
to be accompanied by a density increase in the adjacent region. This is
notably visible for $x>0$ and the positive gradient scenarios ($\smile^{-}$,
top-right and $\smile^{+}$, bottom-left panels in figure 9), for which the
density appears to drop when actively biasing for $r\leq 12.5$ cm, but to rise
for $r>12.5$ cm (_i. e._ past the outer edge of the outermost electrode
$\mathcal{E}_{5}$).

In this same figure we see that this local depletion is accompanied by an
increase in the local electron temperature $T_{e}$. This is particularly clear
for the negative gradient biasing scenarios ($\frown^{-}$, bottom-right and
$\frown^{+}$, top-left panels in figure 9), for which a significant increase
in $T_{e}$ is seen in front of the central electrode $\mathcal{E}_{1}$. We
note that since the density is here deduced directly from $I_{sat}$, and thus
does not account for the effect of a change in $T_{e}$ through the ion sound
speed, this local heating will lead to an even stronger density drop.

Exploring how these effects vary axially, that is with the distance from the
multi-disk electrode, we found (not shown here) that both the depletion and
heating were clearly recovered in the density and temperature profiles
measured on port $\\#20$, that is about $6$ m away from the electrodes.
Finally, these features are observed to persist in the afterglow. In fact the
depletion is even more pronounced in the afterglow, with the plasma density
then approaching zero. This is because there is in this case no longer an
electron beam to act as a plasma source to replenish the depleted flux tube.

Figure 9: Density (blue) and temperature (red) radial profiles on port $\\#34$
before active biasing ($t=11.1$ ms, thicker lighter color lines) and during
active biasing in the main discharge ($t=12.6$ ms, thinner solid lines) for
the six biasing scenarios listed in table 2 (six different panels). The
vertical dashed lines represent the electrodes position. Density and
temperature are normalised to $5~{}10^{18}$ m-3 and $8$ eV respectively.

##  5 Model for the plasma potential response to biasing

Having highlighted in the previous section how different biasing profiles
imposed on the multi-disk electrode lead to different plasma potential
profiles and anode potential, we now try to shed light on the dynamics at
play. We begin by discussing how the different plasma potential profiles
observed for the different biasing scenarios can be understood in terms of the
axial current drawn at the electrodes. We then show how the current balance at
the anode forces the anode potential to respond to the plasma potential
created by the multi-disk electrodes. Throughout this section, we assume that
the plasma column is axisymmetric with radial profiles consistent with the
profiles along $x$ presented in the previous section.

###  5.1 Voltage drop along field lines

The first element to analyse the evolution of the plasma potential with
biasing is to appreciate that although it is small, the non-zero axial plasma
resistivity can be the source of significant (with respect to the applied
bias) axial voltage drops.

To see this recall that, for the low operating pressure used on LAPD, the
dominant contribution to parallel resistivity is electron-ion collisions
(Poulos, 2019). The parallel resistivity is thus the parallel Spitzer
resistivity,

$\eta_{ei}=1/\sigma_{\parallel}^{ei}=\frac{1}{1.96}\frac{m_{e}\nu_{ei}}{n_{e}e^{2}}$
(6)

with

$\nu_{ei}=\frac{n_{e}Z^{2}e^{4}\Lambda_{c}}{6\sqrt{2}\pi^{3/2}\epsilon_{0}^{2}\sqrt{m_{e}}(k_{B}T_{e})^{3/2}}.$
(7)

the Coulomb collision frequency and $\Lambda_{c}$ the Coulomb logarithm. For
the typical plasma parameters given in table 1 one gets $\eta_{ei}\sim
7~{}10^{-5}~{}\Upomega.$m. From there one finds that the voltage drop along
field lines over a distance $s$ for a given parallel current $j_{\parallel}$
is

$\Delta_{\parallel}\phi_{p}(r)=\int_{0}^{s}\eta_{ei}j_{\parallel}(r,z)dz.$ (8)

Using as an estimate the characteristic ion saturation current density
$j_{is}^{\diamond}=8.7$ kA.m-2 defined in Eq. (4) this leads to
$\eta_{ei}j_{is}\sim 0.6~{}$V.m-1. Integrating over the $11$ m separating the
anode from the multi-disk electrode, this would lead to
$\Delta_{\parallel}\phi_{p}\sim 7$ V, which is indeed significant compared to
the $\pm 30$ V of applied bias.

###  5.2 Current driven plasma potential radial profile

To confirm this finding and the importance of a voltage drop along field lines
in the quasi-neutral plasma, one still needs to know how large the actual
parallel current $j_{\parallel}$ is compared to the characteristic ion
saturation current density $j_{is}^{\diamond}$. To do so, we assume current
conservation along magnetic field lines to use here as a proxy the current
density, $j_{\mathcal{E}_{i}}$ which is deduced from the current
$I_{i}=j_{\mathcal{E}_{i}}\mathcal{A}_{i}$ collected on each electrode
$\mathcal{E}_{i}$ of surface $\mathcal{A}_{i}$. This dataset is shown in
figure 10. Beyond a substantial overall increase in current drawn when bias is
turned on, one verifies that current densities $|j_{\mathcal{E}_{i}}|$
exceeding $j_{is}$ are collected on the innermost electrode $\mathcal{E}_{1}$
when this electrode is the most positive (negative potential gradients on the
electrodes, blue and purple). This confirms that the axial currents that
result from active biasing are indeed the source of voltage drops along field
lines of more than $10$ volts.

Figure 10: Current density $j_{{\mathcal{E}}}$ on each electrode
$\mathcal{E}_{i}$ for $i\in\llbracket 1,5\rrbracket$ ($5$ leftmost columns)
and total current $\Sigma I_{i}$ drawn by the multi-disk electrode (rightmost
column) at the three different instants of interest (rows) and for all biasing
cases (color code) shown in table 2. The current density is here normalized by
the characteristic current density ${j_{is}}^{\diamond}$ (see Eq. (4)), and
obtained by assuming a uniform current distribution on each electrode. The
horizontal dashed gray lines highlights
$j_{{\mathcal{E}}}=\pm{j_{is}}^{\diamond}$.

Exploring this idea further, we can examine how the radial dependence of
$j_{{\mathcal{E}}}$ can possibly explain the observed radial profile of the
plasma potential $\phi_{p}(r)$. To do so, we simply construct the radially
dependent axial voltage drop $\Delta_{\parallel}\phi_{p}(r)$ computed from a
constant and uniform resistivity $\eta_{ei}$ and the time-dependent and
electrode dependent $j_{{\mathcal{E}_{i}}}$. Comparing these simple results
with the actual plasma potential profiles $\phi_{p}(r)$ as done in figure 11,
we see that this simple model reproduces very well the radial variations of
the plasma potential. Specifically, figure 11 shows that

$\phi_{p}(r,z_{20})\approx\phi_{a}+\varphi+\eta_{ei}j_{\mathcal{E}_{i}}(z_{a}-z_{20}),$
(9)

where $\varphi$ is an offset and $\phi_{a}$ is the anode potential. Although,
as summarized in table 3 $\phi_{a}$ and $\varphi$ are both biasing scenario
dependent, they are importantly radially uniform. The remarkable fit observed
in figure 11 thus supports further the idea that the plasma potential profile
is driven by the axial currents $j_{\mathcal{E}_{i}}$ collected on biased
electrodes. Radial potential shaping in the plasma is thus achieved by drawing
different axial currents on the different electrodes. On the other hand the
reason for this offset remains at this point unclear. To answer this question,
we need to look at the anode-cathode system and how its dynamics is coupled to
that of the plasma, as we will do next.

(a) $t=12.6$ ms (b) $t=15.6$ ms

Figure 11: Radial profile of the plasma potential $\phi_{p}(r,z_{20})$ minus the anode potential $\phi_{a}$ measured at $t=12.6$ (main discharge, left) and $t=15.6$ ms (afterglow, right) on port $\\#20$ (solid lines with error bars) compared with the profiles constructed from an offset $\varphi$ and the voltage $\eta_{ei}j_{\mathcal{E}_{i}}l$ (thick lines, see Eq. (9)), with $j_{\mathcal{E}_{i}}$ the current density measured on electrode $\mathcal{E}_{i}$ at this instant, $l=z_{a}-z_{20}=6$ m for the distance between the anode and port $\\#20$, and $\eta_{ei}=7~{}10^{-5}~{}\Upomega.$m which corresponds to the baseline parameters given in table 1.  Biasing scenario | Offset $\varphi$ [V] | Anode potential $\phi_{a}$ [V]
---|---|---
|  $12.5$ ms |  $15.5$ ms |  $12.5$ ms |  $15.5$ ms
$\frown^{+}$ | $18$ | $12$ | $5$ | $4$
+ | $25$ | $20$ | $11$ | $12$
$\smile^{+}$ | $22$ | $17$ | $8$ | $8$
$\smile^{-}$ | $8$ | $2$ | $-5$ | $-6$
- | $0$ | $-7$ | $-11$ | $-14$
$\frown^{-}$ | $4$ | $-3$ | $-8$ | $-9$
Table 3: Offset $\varphi$ to be added to the anode potential $\phi_{a}$ and
the axial voltage drop $\eta_{ei}j_{\parallel}l$ to reproduce the observed
radial plasma potential profile $\phi_{p}(r,-l)$ on port $\\#20$. For
comparison $\varphi=11\pm 1$ V for all scenarios at $t=11.6$ ms, that is prior
to active biasing.

Note finally that we could have captured the radial and biasing scenario
dependencies of $\eta_{ei}$ via the measured profiles of $T_{e}$. Our choice
to simply use here a constant value $\eta_{ei}=7~{}10^{-5}~{}\Upomega.$m is
motivated by our goal in this study to highlight global trends rather than to
do a quantitative analysis.

###  5.3 Current balance at the anode

We now would like to explain the offset $\varphi$ needed to reproduce the
plasma potential in Eq. (9), and because this offset is added to the anode
potential $\phi_{a}$ we must also consider the anode dynamics.

A key element to understanding the response of the plasma-anode system to
biasing is the overall current balance in the machine. During the main
discharge, an electron current $\iota I_{d}$, with $\iota$ the anode
transparency, is injected through the mesh anode into the main chamber. This
is equivalent to the current density $j_{inj}$ defined in Eq. (5). Although
the anode-cathode system is floating with respect to the machine ground (Qian
et al., 2023), this injected current $-\iota I_{d}$ must be balanced by an
equal excess of electron current returning to the anode from the plasma in the
main chamber. This in turn sets constraints on the voltage drop one can have
between the anode potential $\phi_{a}$ and the plasma potential in front of
the anode $\phi_{p}(r,z_{a})$. Indeed, assuming an ion sheath, the current
density collected on the anode is (Poulos, 2019; Trotabas & Gueroult, 2022)

$j_{a}(r)=j_{is}\left(1-\exp\left[\Lambda+\frac{\phi_{a}-\phi_{p}(r,z_{a})}{T_{e}}\right]\right)$
(10)

with

$\Lambda=\ln\left[\sqrt{\frac{m_{i}}{2\pi m_{e}}}\right]$ (11)

a parameter that is about $3.5$ in helium. The anode potential $\phi_{a}$ and
the plasma potential in front of the anode $\phi_{p}(r,z_{a})$ must thus
adjust so that overall

$j_{inj}\pi\rho_{B}r_{c}^{2}=\int_{0}^{{\sqrt{\rho_{b}}r_{c}}}2\pi
rdrj_{a}(r).$ (12)

Note importantly that this condition is global as opposed to local, so that
there can still be current along magnetic field lines, _i. e_ $j_{a}(r)\neq
j_{inj}$.

We now see the origin of the offset $\varphi$ in Eq. (9). It is the height of
the anode sheath required to ensure the needed electron return current $-\iota
I_{d}$ to the anode.

###  5.4 Effect of biasing on the anode sheath height

To see why the anode sheath and thus the offset $\varphi$ depend on the
biasing scenario, we need to consider the total current balance.

In the absence of parallel current, we can safely disregard the dynamics at
the multi-disk electrode and focus exclusively on the anode. Assuming further
a uniform plasma potential in front of the anode, $j_{a}=j_{inj}$ and Eq. (10)
then gives

$\varphi\approx\left[\Lambda-\log\left(1-\frac{j_{inj}}{j_{is}}\right)\right]T_{e},$
(13)

where we recall from Eq. (5) that $j_{inj}<0$. For comparable current
densities $j_{inj}$ and $j_{is}^{\diamond}$ this would lead to a voltage drop
of $2.5-3~{}T_{e}$.

This regime of negligible parallel current happens to be relatively
representative of the regime before active biasing ($t\leq 12$ ms). As shown
in the first panel in figure 10, the current density on the electrodes is
indeed much smaller than to the ion saturation current density
$j_{is}^{\diamond}$. From Eq. (8) this implies a negligible voltage drop along
field lines, so that we can in first approximation consider
$\phi_{p}(r,z_{a})\approx\phi_{p}(r,z_{20})$. Now, looking back at the left
panel in figure 6, we find that the plasma potential on port $\\#20$ is about
$10$ to $15$ V higher than the anode potential $\phi_{a}$. This voltage drop
is relatively consistent with the $2.5-3~{}T_{e}$ predicted above, given the
temperature $6\leq T_{e}\leq 8$ eV measured on the port $\\#20$. It is also
very consistent with the offset $\varphi=11\pm 1$ V computed at this instant
for all biasing scenarios, supporting the interpretation of $\varphi$ in terms
of the anode sheath height.

In contrast, we have seen in the middle panel of figure 10 that current
densities comparable or even larger in amplitude than $j_{is}^{\diamond}$ are
collected on the multi-disk electrode during active biasing. Beyond leading,
as discussed earlier, to a significant axial voltage drop along the field line
in the plasma, this collected current also modifies the total current balance.
A negative current drawn on the multi-disk electrode must be compensated by a
decrease in the electron current collected at the anode, which is achieved by
increasing the sheath height. Going back to table 3 and the last column of
figure 10, we verify that the offset indeed is larger for positive biases
(blue, orange and green), and that the larger the amplitude of the total
current drawn $\Sigma I_{i}<0$, the larger the offset. Conversely, a positive
current drawn on the multi-disk electrode, which for an ion sheath implies a
reduced electron current, must be compensated by an increase of the electron
current collected at the anode, which is achieved by decreasing the sheath
height. Going once again back to table 3 and the last column of figure 10, we
verify this time that the offset is indeed smaller for negative biases (red,
brown and purple), and that the larger the amplitude of the total current
drawn $\Sigma I_{i}>0$, the smaller the offset. Finally, the offset $\varphi$
pre-biasing, for which the potential $\phi_{i}$ on the electrodes is as shown
in figure 6 only a few volts below ground, seats as expected in between those
of positive and those of negative biases.

Finally, we note in table 3 that the offset changes, and more precisely
decreases slightly, when moving into the afterglow. This behaviour was to be
expected as there is in this case no longer current injection at the anode,
and thus no longer the need to collect this current back to the anode from the
plasma.

###  5.5 Discussion

In summary, the analysis proposed here of the experimental results obtained in
this campaign indicate that the plasma potential profile is primarily
controlled via the boundary condition imposed on a given field line by the
multi-disk electrode. This suggests that the dynamics is essentially axial
(_i. e_ parallel to the background magnetic field), which is consistent with
earlier results on LAPD by Jin et al. (2019), and with the analysis of these
results proposed by Poulos (2019) in the limit that $\tau\ll 1$ (see Eq. (1)).
This also suggests that more pronounced profiles, _e. g._ stronger radial
electric field, could be produced by drawing appropriate currents on the
biased end-electrodes.

A few comments are however called for at this point. Jin et al. (2019) used
hot electrodes producing parallel current densities a few times the ion
saturation current density (in amplitude), which is significantly larger than
those measured for cold electrodes in this study. This weaker current could
have suggested a larger voltage drop across the sheath (Poulos, 2019; Trotabas
& Gueroult, 2022), contrary to what was observed in the results reported here.
The reason for this finding is twofold. First, in biasing with respect to the
chamber wall as opposed to the anode, the anode was here free to self-adjust,
limiting in turn the amplitude of the voltage drop across the sheath formed in
front of the multi-disk electrode for negative biases. Second, although they
remain smaller than those produced by hot electrodes, the parallel currents
generated here for positive biases were large enough to support axial voltage
drops along field lines that are comparable to the applied bias, limiting
again the voltage drop across the sheath.

##  6 Biasing driven plasma rotation

Having established in the previous sections that differentially biasing the
multi-disk electrodes with respect to the machine ground has a clear effect on
the radial profile of plasma potential $\phi_{p}(r)$ in the machine, we can
now infer how these different equilibria lead to different rotation profiles.

###  6.1 Cross-field rotation from end-electrodes biasing

In a linear machine with $\mathbf{B}=B_{0}\mathbf{\hat{e}}_{z}$ radial
gradients of pressure $p(r)$ and plasma potential $\phi_{p}(r)$ are the source
of different azimuthal drifts that can both contribute to plasma rotation. One
is the diamagnetic drift $\bm{\Omega}_{D}=\Omega_{D}\mathbf{\hat{e}}_{z}$
where

$\Omega_{D}=-\frac{|\bm{\nabla}p\times\mathbf{B_{0}}|}{rqnB_{0}^{2}}=\frac{1}{rqnB_{0}}\frac{\partial
p(r)}{\partial r}$ (14)

with $q$ the signed particle charge. The other is the
$\mathbf{E}\times\mathbf{B}$ drift, which leads to a cross-field angular
plasma frequency $\bm{\Omega}_{E\times B}=\Omega_{E\times
B}\mathbf{\hat{e}}_{z}$ with

$\Omega_{E\times B}=\frac{1}{rB_{0}}\frac{\partial\phi_{p}(r)}{\partial r}.$
(15)

A positive potential radial gradient thus leads to $\Omega_{E\times B}>0$ and
a cross-field drift in the direction of the electron diamagnetic drift
(assuming $\partial p/\partial r<0$) while a negative potential radial
gradient leads to $\Omega_{E\times B}<0$ and a cross-field drift in the ion
diamagnetic drift.

Comparing these two contributions, one finds

$\frac{\Omega_{E\times
B}}{\Omega_{D}}=\frac{e}{kT}\frac{\partial\phi_{p}}{\partial
r}\bigg{/}\left[\frac{1}{T}\frac{\partial T}{\partial
r}+\frac{1}{n}\frac{\partial n}{\partial r}\right]$ (16)

where $\phi_{p},n$ and $T$ are all functions of $r$. This shows that the
$\mathbf{E}\times\mathbf{B}$ contribution dominates over the diamagnetic
contribution if the drop in plasma potential over the characteristic gradient
lengths for the density and the temperature is equal or larger than a few
$T_{e}$. Because we observe radial voltage drops as large as $2$ to
$3~{}T_{e}$ across the radius of the multi-disk electrode while the density
and temperature vary by at most $30\%$ in this same region, this assumption
appears justified here, at least in the central region.

Under this simplifying assumption, it is possible to infer rotation from the
plasma potential profiles measured for the different biasing scenarios. To do
so we fit the plasma potential profiles using cubic smoothing splines, then
fold back the negative $x$ over the positive $x$ to obtain an average radial
profile $\phi_{p}(r)=[\phi_{p}(x=-r<0)+\phi_{p}(x=r>0)]/2$, which is finally
differentiated with respect to $r$ to get minus the radial electric field. The
results obtained for active biasing in the main discharge ($t=12.6~{}$ms) and
in the early afterglow ($t=15.6$ ms) are shown in figure 12.

(a) $t=12.6$ ms

(b) $t=15.6$ ms

Figure 12: Angular plasma frequency on port $\\#34$, that is about $30$ cm
away from the multi-disk electrode, at $t=12.6$ ms 12(a) and $t=15.6$ ms 12(b)
for the six biasing scenarios (colour code) listed in table 2. Rotation is
computed from the plasma potential profiles assuming pure
$\mathbf{E}\times\mathbf{B}$ rotation. The dashed and dash-dotted black curves
represent sonic rotation for $T_{i}=1$ and $0.1$ eV, respectively. The
vertical grey dashed lines show the electrodes position.

As anticipated from the plasma potential profiles, we find the largest
rotation in the core for the two negative gradient biasing scenarios (blue and
purple) for which electrons are drawn by the electrodes on the axis. Rotation
is counterclockwise, that is in the ion diamagnetic direction. Although
rotation likely remains subsonic, we see on this same figure that the
azimuthal drift locally approaches what would be the sonic speed for an ion
temperature of a fraction of an eV. We also verify that the non-monotonic
plasma potential profiles measured for the two positive gradient biasing
scenarios (green and red) lead to a rotation reversal near the edge of the
outermost electrode. Rotation is in this case clockwise in the core, and
counterclockwise in the outer region. Finally, uniform biasing profiles
(orange and brown), which we have seen lead to comparatively flat plasma
potential profiles, consistently yield weak rotation.

Note however again that this analysis assumed $|\Omega_{E\times
B}|\gg|\Omega_{D}|$. Clearly this is not valid for non-monotonic profiles near
the point of reversal, where by definition $\Omega_{E\times B}\sim 0$. This
ordering is also questionable for flat potential profiles. In short, the
examination of the potential profiles points to a strong plasma rotation
driven by biasing for the negative gradient biasing scenarios, but a more in-
depth analysis of the rotating equilibrium is called for other biasing
scenarios.

###  6.2 Discussion

Having underlined the possibility to produce rapidly rotating plasmas via end-
electrodes biasing, we now briefly discuss this result in light of related
experiments previously conducted in LAPD. Of particular interest here is the
work by Maggs et al. (2007) and Carter & Maggs (2009) in which a section of
the chamber wall was biased positively with respect to the cathode to drive
rotation.

An interesting observation is that although both biasing schemes may lead (did
lead in the case of Maggs et al. (2007)) to rotation, they do so in very
different ways. In the experiment discussed here, we surmised that rotation
results from axial currents drawn by end electrodes, which then support a
radial electric field. In this case, we have seen that rapid rotation appears
more easily achievable in the counterclockwise direction ($E_{r}>0$). On the
other hand, in Maggs et al. (2007) it is an inward ion radial current driven
by a radial voltage difference that led to a clockwise ($E_{r}<0$) plasma
rotation. Biasing the chamber negatively compared to the cathode, which would
have led to counterclockwise rotation, was tried but shown to result in
arcing. These basic differences are then expected to lead to distinct
behaviours, which future studies may be able to examine.

First, the axial voltage drop model proposed here to explain end-electrodes
biasing results would lead to a radial electric field that varies along $z$,
decreasing in amplitude from the multi-disk electrode to the anode. The
corresponding axially dependent crossed-field rotation would in turn translate
into a twist of the plasma. On the contrary, the edge biasing, if done over
the entire length of the plasma column, would in principle lead to an axially
uniform rotation. Second, with end-biasing, the axial voltage drop and thus
the radial electric field amplitude scale with the parallel Spitzer
resistivity Eq. (6). Meanwhile, in edge biasing experiments ions are
transported radially due to ion-neutral collisions, and the radial electric
field is hence inversely proportional to the ion-driven Pedersen conductivity

$\sigma_{p}=\frac{n_{e}m_{i}\nu_{in}}{{B_{0}}^{2}}$ (17)

with $\nu_{in}$ the ion-neutral collision frequency. Comparing these two
scaling, one expects completely different dependencies on plasma parameters.

Lastly, an important finding from Maggs et al. (2007) was that rotation driven
by biasing led to a modification of transport properties, and more
specifically to a reduction of cross-field transport from Bohm to classical
rates, which translated into a steepening of the plasma density radial
profiles. Although this is not the focus of our study, we note here in
contrast looking back at figure 9 that there does not appear to be a
noticeable change in density profile in the end-electrodes biasing results
reported here.

##  7 Conclusion

An experimental campaign was carried out on the Large Plasma Device (LAPD) at
the University of California, Los Angeles to study how biased end-electrodes
can be used to control the radial potential profiles in the plasma bulk in the
machine. It uses a set of five independently biased disk electrodes to control
and impose boundary conditions for the potential about $11$ m from the anode.
Typical operating conditions in the main discharge were $n_{e}=5~{}10^{18}$
m-3 and $T_{e}=6-8$ eV.

Under these conditions, we observed that biasing has a very noticeable effect
on the plasma potential in the machine, at least up to $6$ m from the
electrodes, with shifts in plasma potential that are comparable to the $\pm
30$ V amplitude bias imposed on the electrodes. Examining the plasma response
to non-uniform (though monotonic) biasing profiles, it was found that concave-
down biasing profiles lead to the largest radial drops in plasma potential, no
matter whether the applied bias is positive or negative. On the other hand,
concave-up biasing profiles were shown to lead to non-monotonic profiles and
less pronounced plasma potential radial gradients.

Analysing these results, it was shown that the radial profile of plasma
potential in the machine is governed by the current drawn by each of the
concentric electrodes and the axial resistance associated with Spitzer
conductivity. The largest radial variations observed for concave-down biasing
profiles are notably driven by the larger electron current drawn by the on-
axis electrode in this case, suggesting that steeper profiles may be
achievable if drawing suitable parallel currents. It was also found that the
nature of the plasma source on LAPD, namely the injection of energetic
electrons through the anode, further imposes constraints between the plasma
potential and the anode potential, which in turn forces the anode potential to
track variations in plasma potential.

Finally, we computed the cross-field rotation expected from the different
plasma potential profiles resulting from the various biasing scenarios. This
analysis suggests that concave-down biasing profiles on end-electrodes may be
used to produce rotating plasmas that locally approach sonic rotation,
complementing other biasing-driven rotation techniques that had been
previously demonstrated on LAPD. On the other hand, in producing non-monotonic
potential profiles, concave-up biasing profiles would lead to flow reversal.
In this latter case, though, one expects diamagnetic contributions to modify
this picture.

## Acknowledgments

The authors would like to thank Patrick Pribyl for his guidance on pulser
biasing and emissive probes, Vadim Murinov for his help developing a script to
extract electron temperature data from Langmuir scans, as well as Zoltan
Lucky, Marvin Drandell, and Tai Ly for their technical support and assistance.
The authors would also like to thank Troy Carter and Stephen Vincena for
constructive discussions of the experimental results, and Stewart J. Zweben
for his help designing this experiment.

## Funding

This research was performed at the Basic Plasma Science Facility at UCLA,
which is funded by the United States Department of Energy and the National
Science Foundation. RG gratefully acknowledges financial support for travel
expenses from the International Action Program from CNRS INSIS.

## Declaration of interests

The authors report no conflict of interest.

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