# Trade-off in perpendicular electric field control using negatively biased
emissive end-electrodes

B. Trotabas and R. Gueroult  LAPLACE, Université de Toulouse, CNRS, INPT, UPS,
31062 Toulouse, France<EMAIL_ADDRESS>
###### Abstract

The benefits of thermionic emission from negatively biased electrodes for
perpendicular electric field control in a magnetized plasma are examined
through its combined effects on the sheath and on the plasma potential
variation along magnetic field lines. By increasing the radial current flowing
through the plasma thermionic emission is confirmed to improve control over
the plasma potential at the sheath edge compared to the case of a cold
electrode. Conversely, thermionic emission is shown to be responsible for an
increase of the plasma potential drop along magnetic field lines in the quasi-
neutral plasma. These results suggest that there exists a trade-off between
electric field longitudinal uniformity and amplitude when using negatively
biased emissive electrodes to control the perpendicular electric field in a
magnetized plasma.

##  1  Introduction

Electric fields perpendicular to magnetic surfaces in plasmas are of
importance to a variety of applications. In magnetic confinement fusion for
instance, radial electric fields are known to play an important role on
confinement [1], as observed both in tokamaks [2, 3, 4] and stellerators [5].
Confinement enhancement is in this case believed to be enabled by transport
barriers induced by $\mathbf{E}\times\mathbf{B}$ sheared flows [6, 7, 8] in
the presence of radial electric fields. Perpendicular electric fields also
offer opportunities for the design of alternative confinement schemes in
toroidal geometry such as the magnetoelectric confinement studied by Stix [9],
or more recently the wave driven rotating torus [10, 11]. In addition and
beyond fusion, controlling perpendicular electric fields is essential for a
growing number of applications of $\mathbf{E}\times\mathbf{B}$ configurations
[12], and notably for the development of high-throughput plasma separation
technologies [13, 14, 15].

Although waves hold promise to produce such perpendicular electric fields
[16], most of the experimental effort towards perpendicular electric field
control to date has relied on electrode biasing. In magnetic confinement
fusion experiments, the high plasma temperature and density generally prohibit
inserting electrodes in the plasma core, and biasing experiments thus
typically involve edge biasing [17]. While this technique has been shown to be
effective at affecting edge properties under certain conditions [18, 19], the
use of a single polarized surface at the edge - a biasing configuration known
as a limiter - does not provide control over how the applied bias distributes
itself across magnetic surfaces. The perpendicular electric field indeed
remains an intricate function of the plasma properties [17]. Cooler and less
dense plasmas, especially in open-field line geometries, open additional
possibilities for biasing studies, and a broad array of electrode geometries
have been used for the primary purpose of instabilities and turbulence
suppression [20, 21, 22] and flow control [23, 24, 25, 26, 27, 28]. For
perpendicular electric field control, a biasing configuration of particular
interest is end-electrodes, that is electrodes intercepting magnetic field
surfaces. The basic idea here, as originally suggested by Lehnert [29, 30], is
that one could control the electric potential of individual magnetic field
surfaces through the biases imposed on a set of end-electrodes. More
specifically, the potential of a given magnetic surface is expected to be set
by the applied bias on the electrode on which this magnetic surface
terminates, allowing in principle in turn for perpendicular electric field
control. While very attractive, the practicality of this scheme remains a
question. Indeed, while control has been successfully demonstrated under
certain conditions [31, 32], other experiments reported more contrasted
results (see Ref. [13] for a more extensive discussion of end-electrodes
biasing experiments in linear geometry).

Conceptually, the ability to control the potential of a magnetic surface in a
plasma through the bias applied on an end-electrode can be split into two
problems: controlling the potential drop along magnetic field lines in the
quasi-neutral plasma and controlling the potential drop across the non-neutral
sheath formed in front of the biased electrode. So far, these two problems
have mostly been treated separately. On the former, control along magnetic
field lines in a quasi-neutral plasma is guaranteed in the limit that field
lines are isopotential, which as noted by Lehnert in his original paper [29]
can in principle be asymptotically approached using a large enough magnetic
field. Restated through conductivities, this is equivalent to the limit of a
zero perpendicular to parallel conductivity ratio
$\mu=\sigma_{\perp}/\sigma_{\parallel}$. Practically though, $\mu$ is finite,
which implies that field lines are not strictly isopotential. Examining
specifically this problem, it has recently been shown that the relative
variation in potential along field lines in a plasma column of radius $a$ and
length $L$ is about $\tau=L/a\sqrt{\mu}$ [33]. This suggests that a necessary
condition for potential control along field lines is $\tau\ll 1$. On the
latter, Liziakin _et al._ showed considering the sheath formed in front of a
negatively biased electrode that the combination of the plasma perpendicular
conductivity $\sigma_{\perp}$ and the ion saturation current sets a lower
limit on the minimal plasma potential $\phi_{p}<0$ one can expect from
applying a negative bias $\phi_{e}<\phi_{p}$ [34]. Building on this finding,
the same authors recently showed that the addition of thermionic emission from
a negatively biased electrode can lower further the plasma potential
$\phi_{p}$ at the expense of the voltage drop across the sheath
$\phi_{p}-\phi_{e}$ [35], which is consistent with earlier theoretical work
[36] and observations [37].

In this paper, we consider these two problems in a unified model with the goal
of highlighting the overall limits on perpendicular electric field control
from negatively-biased end-electrodes, and examine in particular the effect of
thermionic emission. In Section 2, we briefly introduce our model. In Section
3, the sheath models developed by Liziakin _et al._ [34, 35] are first used to
identify what the limits on potential control at the sheath edge are. In
Section 4, these insights from sheath dynamics are then coupled into models
for the potential distribution in a quasi-neutral magnetized plasma to
highlight the influence of the sheath, and theoretical predictions are
compared to numerical simulations. In Section 5, the main results are
summarized.

##  2  Model description

The configuration studied in this work is illustrated in figure 1. It consists
in a symmetrical linear machine with single full disks electrodes (sometimes
referred to as button electrodes) terminating axially a magnetized plasma
column. We note $L$ the inter-electrode distance and $r_{e}$ and $r_{g}$ the
radii of the biased electrode and grounded vacuum vessel, respectively. We
further limit ourselves in this work to negative biases imposed on the disk
electrodes, but consider both cold and hot surfaces to highlight the effect of
thermionic emission. This simpler biasing configuration is used here to
underline some of the key features expected in the more complex segmented
concentric ring electrodes configuration proposed by Lehnert [29].

Figure 1: Magnetized plasma column terminating on two single disk electrodes
(in orange). Electrodes are biased negatively (potential $\phi_{e}$) with
respect to the grounded vacuum vessel. The entire domain is permeated by a
uniform axial magnetic field $\mathbf{B}=B_{0}\bm{\hat{z}}$.

The plasma filling this volume is assumed to be produced by an external source
(_e .g._ radio-frequency or ECR) which is not modeled in this work, and we
further assume that the biased electrodes do not affect the plasma properties
other than through the plasma potential. In addition, we consider the plasma
to be uniform and quiescent, that is that density and temperature gradients,
as well as possible instabilities, are neglected. Away from the sheaths formed
in front of the biased electrodes, the plasma is therefore modeled as a
uniform anisotropic media characterized by parallel and perpendicular
conductivities $\sigma_{\parallel}$ and $\sigma_{\perp}$. Finally, since our
system is symmetrical, only the left-hand side of the domain is studied, that
is $\left[0,r_{g}\right]\times\left[-L/2,0\right]$.

Having perpendicular electric field control in mind, a key element of the
plasma potential response as a function of the applied bias $\phi_{e}$ and the
plasma parameters (through the conductivities) is the evolution of potential
along magnetic field lines. For the uniform magnetic field considered here
this means the variation of $\phi(r,z)$ at constant radius $r_{0}$ as
illustrated in figure 2. To facilitate interpretation, we introduce the short-
hands $\phi_{sh}(r)=\phi(r,-L/2)$ and $\phi_{mid}(r)=\phi(r,0)$ for the plasma
potential radial profile at the sheath edge and in the mid-plane,
respectively. With this notation the voltage drop across the sheath and along
field lines in the quasi-neutral plasma are then simply

$\Delta_{sh}\phi(r)=\phi_{sh}\left(r\right)-\phi_{e}$ (1)

and

$\Delta_{\parallel}\phi(r)=\phi_{mid}\left(r\right)-\phi_{sh}\left(r\right).$
(2)

Finally, in an effort to ease comparison between experiments across a broad
range of conditions, we work in this study with dimensionless variables and
define for all electric potential quantities $\phi$ their dimensionless analog
$\psi$ through $\psi=\phi/T_{e}$ with $T_{e}$ in eV.

Figure 2: Sketch of the variation of the electric potential $\phi$ along a
magnetic field line (radius $r_{0}<r_{e}$) between the negatively biased
electrode and the midplane. The total voltage drop
$\phi_{mid}(r_{0})-\phi_{e}$ is the sum of a voltage drop across the sheath
$\Delta_{sh}\phi(r_{0})$ and a voltage drop along field line in the quasi-
neutral plasma $\Delta_{\parallel}\phi(r_{0})$, from Eqs. (1) and (2).

##  3  Effect of the sheath on the plasma potential at the sheath edge

Inserting a biased electrode in contact with a plasma leads to the formation
of a sheath, that is to a non-neutral region a few Debye length thick
connecting the bulk plasma to the electrode [38] (see figure 2). For biased
electrodes of sufficiently small surface area, the bulk plasma parameters, and
in particular the plasma potential, are to zero-th order unaffected by the
electrode bias. This is an essential hypothesis in probe theory [39]. On the
other hand, for large enough surface areas, the bulk plasma potential can be
affected by the applied bias [38]. The thin sheath region is then expected to
control to which extent the applied bias potential is passed along field lines
into the plasma bulk.

###  3.1  Sheath transfer function for the potential: saturated and non-
saturated regimes

A first picture of how the plasma potential at the sheath edge relates to the
applied bias can be obtained from the model proposed by Liziakin _et al._
[34]. In this model the potential $\phi(r,z)$ in the plasma column shown in
figure 1 is assumed to be independent of $z$, which corresponds to the limit
$\mu=\sigma_{\perp}/\sigma_{\parallel}\rightarrow 0$. In this case one simply
gets $\phi(r,z)=\phi_{sh}(r)$. This model further assumes that the potential
is constant in the shadow of the electrode (that is for $r\leq r_{e}$),
allowing to model the discharge as a constant radial current $I$ flowing from
$r_{e}$ to $r_{g}$ across magnetic surfaces. The influence of this last
simplifying hypothesis is examined in detail via a more complete model in
Section 3.3.

Under these assumptions, the plasma potential in the electrode’s shadow
$\phi_{p}=\phi(r\leq r_{e},z)$ is simply $R_{\perp}I$, where the perpendicular
resistance $R_{\perp}$ opposing the current has been obtained by integrating
between $r_{e}$ and $r_{g}$ the incremental resistance

$dR(r)=\frac{dr}{\pi L\sigma_{\perp}r}$ (3)

associated with the annular region of length $L$ located between $r$ and
$r+dr$ [34]. Note here that $R_{\perp}$ is the perpendicular resistance
associated with half the plasma column length, consistent with the fact that
we consider $I$ as the current on one axial end-electrode. This plasma
potential $\phi_{p}$ minus the voltage drop across the sheath
$\Delta_{sh}\phi$ must be equal to the applied bias $\phi_{e}$, as illustrated
in figure 3. In addition, current continuity requires for this current $I$ to
be equal to the current drawn at the biased electrode. Considering here an ion
sheath formed in front of a hot negatively biased electrode and counting
positively particle fluxes leaving the electrode, one gets
$I=I_{e}-I_{is}-I_{eth}$ where electron, ion and thermionic electron currents
have respectively been obtained from the surface integrated current densities

$\displaystyle j_{e}$ $\displaystyle=$ $\displaystyle\displaystyle
j_{is}\exp\left(\Lambda+\frac{\phi_{e}-\phi_{p}}{T_{e}}\right),$ (4a)
$\displaystyle j_{is}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{enc_{s}}{2},$ (4b) $\displaystyle j_{eth}$
$\displaystyle=$ $\displaystyle
A_{G}{T_{W}}^{2}\exp\left(-\frac{eW}{k_{B}T_{W}}\right).$ (4c)

Here $n$ is the plasma density, $c_{s}=\sqrt{eT_{e}/m_{i}}$ is the ion sound
speed, $\Lambda=\ln(\sqrt{2m_{i}/(\pi m_{e})})$ is a sheath parameter, $T_{W}$
is the temperature of the electrode, $W$ is the work function of the material
and $A_{G}$ is the Richardson’s constant taken here equal to $6.0\times
10^{5}\mathrm{A\cdot K^{-2}\cdot m^{-2}}$ [40]. Moving to dimensionless
variables, the normalized current flowing through the plasma hence writes

$\frac{I}{I_{is}}=\exp\left(\Lambda+\psi_{e}-\psi_{p}\right)-1-\Xi$ (5)

with $\Xi=j_{eth}/j_{is}$ a dimensionless parameter quantifying thermionic
emission with respect to the ion saturation current. For an applied bias
$\psi_{e}\leq\psi_{p}$ the current $I$ is negative and reaches a minimum equal
to $-\left(I_{is}+I_{eth}\right)$ for sufficiently large $\psi_{p}-\psi_{e}$.
Note that we chose here to ignore for simplicity possible dependencies of
$j_{eth}$ on $\phi_{p}$ through the Schottky effect [41], as well as possible
dependencies of $j_{is}$ on $\phi_{p}$ through a modification of the Bohm
velocity at the sheath boundary in the presence of thermionic emission [42].

Figure 3: Equivalent electric circuit for the plasma potential $\phi_{p}$ in
the shadow of the biased electrode for an axially uniform plasma proposed by
Liziakin _et al._ [34].

Plugging Eq. (5) into Ohm’s law across the plasma resistance
$\phi_{p}=R_{\perp}I$ and moving to dimensionless variables finally yields a
transcendental equation for the normalized plasma potential

$\exp\left(\Lambda+\psi_{e}-\psi_{p}\right)-1-\chi\psi_{p}-\Xi=0\text{ ,}$ (6)

where we have defined $\chi=T_{e}\big{/}I_{is}R_{\perp}$. Eq. (6) is identical
to that obtained by Liziakin _et al._ [34], other than for the choice of using
dimensionless variables and the addition in this work of thermionic emission.

Considering first the case without thermionic emission ($\Xi=0$), figure 4
shows the evolution of $\psi_{p}$ as a function of $\psi_{e}$ for different
values of $\chi$ as predicted by Eq. (6). For small biases applied at the
electrode $\psi_{e}$, or a sufficiently small value of $\chi$, the plasma
potential is seen to follow the applied bias, that is
$\psi_{p}=\psi_{e}+\Lambda$. In this case the plasma potential is hence
controlled by the electrode bias. The voltage drop across the sheath is then
small, $\Delta_{sh}\psi=\Lambda$, and the current drawn at the electrode is
negligible compared to $I_{is}$. For reasons that will become clear in the
next paragraph, we refer to this regime as the non-saturated regime.

Figure 4: Normalized plasma potential in the electrode shadow $\psi_{p}$ as a
function of the normalized applied bias $\psi_{e}$ for different values of
$\chi$ as predicted in Eq. (6) absent thermionic emission ($\Xi=0$).
Potentials are normalized by the electron temperature $T_{e}$ in eV,
$\psi=\phi/T_{e}$.

For larger negative biases for a given value of $\chi$, figure 4 shows that
the plasma potential progressively deviates from the floating solution
$\psi_{p}=\psi_{e}+\Lambda$ until it reaches a minimal value
$\psi_{p}=\chi^{-1}$. This behaviour is consistent with the fact that the
current $I<0$ then approaches its minimum value $-I_{is}$. At this point the
plasma potential $\psi_{p}$ is no longer controlled by the applied bias
$\psi_{e}$, and any further decrease in applied bias $\psi_{e}$ is entirely
recovered in the voltage drop across the sheath $\Delta_{sh}\psi$ which grows
as $|\psi_{e}|-\chi^{-1}$. Reflecting the property of the current in this
regime, we refer to it as saturated or current-limited regime. In contrast
with the non-saturated regime, the minimum plasma potential is indeed here set
by the maximum current drawn at the biased electrode. The dimensionless
parameter $\chi^{-1}$ can then be interpreted as a measure of the maximum
radial voltage drop the plasma column can sustain. From the definition of
$R_{\perp}$, one gets

$\chi=\frac{L}{r_{e}^{2}\ln(r_{g}/r_{e})}\frac{T_{e}\sigma_{\perp}}{j_{is}}.$
(7)

The transition from non-saturated to saturated regime, that is from a plasma
potential controlled by the electrode bias to a plasma potential controlled by
plasma parameters, takes place for $|\psi_{e}|\chi\geq 1$. The boundaries in
physical operating parameters space (densities, temperatures, bias) for this
regime transition can be obtained by developing the parametric dependencies of
$j_{is}$ and $\sigma_{\perp}$ for the plasma conditions under consideration,
as it will be done in Section 4.

###  3.2  Additional control offered by thermionic emission

Examining now specifically the effect of thermionic emission, Eq. (6) shows
that the plasma potential in the saturated regime, that is the minimal plasma
potential no matter how negative the applied bias, writes

$\psi_{p}^{sat}=-\frac{1+\Xi}{\chi}.$ (8)

Compared with the case absent thermionic emission studied above, one finds
that its amplitude is $(1+\Xi)$ times larger. This result is simply the
consequence that the discharge current is $1+\Xi$ larger when taking into
consideration thermionic emission, while the perpendicular plasma resistance
$R_{\perp}$ remains the same.

Since $\psi_{p}^{sat}$ by definition coincides with the applied bias
$\psi_{e}$ below which the transition from non-saturated to saturated regime
occurs, the above result suggests a larger range of accessible plasma
potential for a given $\chi$ (that is for a given set of plasma parameters).
Put differently, as illustrated in figure 5, increasing thermionic emission
for a given bias and a given $\chi$ allows transitioning from a saturated
regime to a non-saturated regime, and therefore regaining control over the
plasma potential. In the process the plasma potential decreases and
progressively approaches the applied bias as thermionic emission increases,
consistent with experimental observations [37].

Figure 5: Normalized plasma potential in the electrode shadow $\psi_{p}$ as a
function of the normalized applied bias $\psi_{e}$ for different values of
thermionic emission $\Xi$, as predicted in Eq. (6). Potentials are normalized
by the electron temperature $T_{e}$ in eV, $\psi=\phi/T_{e}$.

Note that while the added control offered by thermionic emission suggests
operating with large thermionic currents $I_{eth}$, there is practically a
limit to how large this current can be. Indeed, past a certain value a virtual
cathode is expected to form in front of the electrode. This will in turn
naturally limit the thermionic current reaching the bulk plasma, and thus the
thermionic contribution to the discharge current. The value of $I_{eth}$ (and
thus of $\Xi$) for which this virtual cathode is expected to form can be
derived analytically by solving Poisson’s equation in the sheath region [36].

###  3.3  Radial potential profile

Up to this point we have only examined the relation between the applied bias
$\phi_{e}$ and the plasma potential in the electrode shadow $\phi_{p}$. This
is because in the model introduced in Section 3.1 the potential is assumed
constant for $r\leq r_{e}$ and the current constant for $r>r_{e}$. Integration
of Ohm’s law with the incremental resistance given in Eq. (3) hence leads to a
simple logarithmic profile for $\phi_{sh}$ connecting $\phi_{p}$ for $r<r_{e}$
to the ground at $r=r_{g}$.

A more physical model has recently been proposed by Liziakin _et al._ [35]
noting that the radial current $I(r)$ at any radius must be equal in steady
state to the surface integrated current density for all radii $r^{\prime}<r$,
that is

$I(r)=\displaystyle\int_{0}^{r}j_{sh,\parallel}(r^{\prime})2\pi
r^{\prime}dr^{\prime}$ (9)

with

$j_{sh,\parallel}(r)=j_{e}(r)-j_{is}-j_{eth}$ (10)

the current density through the sheath. By implicitly assuming that there is
no current source past the outer radius of the electrode $r_{e}$, the
requirement for a constant potential in the electrode shadow can then be
lifted, and the plasma potential obtained by integrating from the ground
reference at $r=r_{g}$ to any radius $r$ the local Ohm’s law
$d\phi_{sh}(r)=I(r)dR(r)$. Plugging in Eq. (9) and going back to dimensionless
variables yields a linear integro-differential equation for $\psi_{sh}$,

$\displaystyle\frac{d\psi_{sh}(r)}{dr}=\int_{0}^{r}\mathcal{F}(r^{\prime},\psi_{sh}(r^{\prime}))dr^{\prime}.$
(11)

Here $\mathcal{F}$ is a function that depends on the plasma parameters through
$T_{e}$ and $\sigma_{\perp}$, on the plasma composition through the sheath
parameter $\Lambda$, on the applied bias $\psi_{e}$, on the thermionic current
parameter $\Xi$ and on the geometric parameters $r_{e}$, $r_{g}$ and $L$. In
Ref. [35], Eq. (11) was solved numerically for spatially dependent plasma
parameters (plasma density and electron temperature) measured in an
experiment, as a way to validate potential profile predictions against
experimental data. Here we similarly solve Eq. (11) using a shooting method
but instead use these results to highlight some characteristic properties of
the radial potential obtained for a uniform plasma (_i. e._ uniform
$\sigma_{\perp}$), as shown in figure 6. Another important distinction is that
Liziakin _et al._ focused specifically in Ref. [35] on the case where
$\sigma_{\perp}$ depends linearly on $d\psi_{sh}(r)/dr$ by assuming that ions
drift with velocity ${B_{0}}^{-1}d\phi_{sh}(r)/dr$ in a background static
neutral gas. This corresponds to the limit of supersonic ion rotation $\Omega
r\gg v_{thi}$. In contrast we purposely do not consider here for the moment a
particular mechanism for cross-field conductivity, but we do assume that
$\sigma_{\perp}$ does not depend on $\phi$ nor its radial derivatives. We
simply note here that this hypothesis may be representative of either ion-
neutral collisions in the slow rotation limit $\Omega r\ll v_{thi}$, or of a
different conductivity mechanism (_e. g._ electron-neutral collisions as
examined by Liziakin _et al._ in an earlier study [34]).

Figure 6: Normalized plasma potential $\psi_{sh}(r)\big{/}|\psi_{sh}(0)|$
radial profile for different values of $\chi$, and normalized current density
radial profile (inset). Green and red curves highlight the potential profiles
obtained for the non-saturated and saturated regimes, respectively. The black
dots for $r\leq r_{e}$ illustrate a parabolic radial dependence, whereas the
dotted black curve near $r_{g}$ represents a logarithmic radial profile.

Looking first at the normalized current density, one recovers the non-
saturated and saturated regimes for respectively small and large values of
$\chi$. Indeed, consistent with the simpler model used in Sections 3.1 and
3.2, the current density on the electrode $j_{\parallel}(r)$ is observed to be
uniform and equal to its maximum value (in absolute value) in the saturated
regime, whereas it is negligible in the non-saturated regime. However, we also
observe that these two regimes are now separated by a group of curves obtained
for intermediate values of $\chi$ for which the current goes from near zero
on-axis to its saturated value at the outer edge of the biased electrode. This
shows that saturation is actually a local phenomenon. By analogy with the
nomenclature introduced earlier, we refer to this regime as partially
saturated. A closer examination reveals that saturation first appears at the
outer edge of the biased electrode and progressively moves radially inward
with increasing $\chi$ until full saturation has been reached, which is to be
expected for a monotonically increasing radial potential profile.

Moving on now to the normalized plasma radial profile and starting from small
value of $\chi$, that is in the non-saturated regime, one observes a nearly
constant profile in the shadow of the electrode ($r<r_{e}$). This result can
be explained from Ohm’s law and the fact that the current drawn at the
electrode is in this case very small. In this regime in the electrode shadow
one simply finds $\psi_{p}=\psi_{e}+\Lambda$. This same profile holds as
$\chi$ increases up until partial saturation begins at the outer edge of the
electrode. As the current then becomes larger (in amplitude), Ohm’s law
predicts an increase of $\psi_{sh}$. Since, as showed above, partial
saturation move radially inward with $\chi$, the position where $\psi_{sh}(r)$
begins to deviate from its constant on-axis value is also observed to move
radially inward. Finally, once the fully saturated regime has been reached, a
new profile sets in and holds for arbitrary large value of $\chi$.

Because the saturated regime is characterized by a saturated current at the
electrode, the flux of electrons reaching the electrode is then by definition
null. This property allows for further analysis. Indeed, Eq. (11) then becomes
a simple ordinary differential equation whose solution for $r\leq r_{e}$
writes

$\psi_{sh}^{sat}(r)=-\frac{1+\Xi}{\chi}\left[1+\frac{1}{2\chi\ln\left(r_{g}/r_{e}\right)}\left(1-\frac{r^{2}}{r_{e}^{2}}\right)\right].$
(12)

Here one recognizes that the first term in brackets on the right hand side is
$\psi_{sh}^{sat}(r_{e})$ given in Eq. (8), which was assumed to be the
constant potential found in the electrode shadow in Sections 3.1 and 3.2. In
contrast it is found here that the potential in the electrode shadow actually
exhibits a parabolic radial dependence. This is noteworthy insofar that this
profile leads to a constant angular $\mathbf{E}\times\mathbf{B}$ drift
frequency, and thus to solid body rotation in the shadow of the electrode
assuming crossed-field drift is the dominant contribution to rotation. It
should also be noted here that the fact that a parabolic profile is obtained
in Eq. (12) instead of the $r^{3/2}$ profile derived by Liziakin _et al._ [35]
is the consequence that, as mentioned above, $\sigma_{\perp}$ is assumed in
this work independent of $d\phi_{sh}(r)/dr$.

Finally, one notes that all profiles feature a logarithmic radial dependence
past the outer radius of the electrode $r_{e}$, no matter the regime. This is
the direct consequence of the fact that we assumed zero axial current past
$r_{e}$, so that the radial current $I(r)$ in Eq. (9) is constant for
$r>r_{e}$. In this case the potential simply writes

$\psi_{sh}(r>r_{e})=I(r_{e})\displaystyle\int_{r_{g}}^{r}\frac{dr}{\pi
L\sigma_{\perp}r}$ (13)

which indeed yields the observed logarithmic radial dependence, consistent
with Ref. [34]. Profiles then only differ through their value at $r=r_{e}$,
which is a function of the total current emitted by the electrode $I(r_{e})$.

To summarize our findings, we have shown in this section that the ability to
pass the negative applied bias $\psi_{e}$ unchanged (up to the sheath
parameter $\Lambda$) along field lines into the plasma bulk is conditioned
upon the smallness of the quantity $|\psi_{e}|\chi(1+\Xi)^{-1}$. The value of
this quantity can be obtained for actual plasma parameters through Eq.(7) and
the definition of $\Xi$. For values of $\psi_{e}$ such that this quantity is
greater than $1$, the regime is said to be saturated. The plasma potential
then no longer varies with $\psi_{e}$, $\psi_{p}=\psi_{p}^{sat}$, whereas the
sheath drop $\Delta_{sh}=\psi_{p}-\psi_{e}$ grows as
$|\psi_{e}|-(1+\Xi)\chi^{-1}$. These results point to the value of thermionic
emission to increase potential control. Finally, a more refined model of
currents in the system shows that the saturation phenomenon is local, starting
at the outer edge of the biased electrode and moving radially inward until the
entire surface of the biased electrode is collecting maximum current density.

##  4  Effect of the sheath on the voltage drop along field lines

Controlling the potential drop across the sheath is essential for
perpendicular electric field control through end-electrodes biasing but it is
not enough. One indeed must also ensure that the potential does not vary
significantly along field lines in the quasi-neutral plasma. Assuming a
monotonic variation of $\phi(r,z)$ along $z$, this condition translates into
the smallness of $\Delta_{\parallel}\phi(r)=\phi_{mid}(r)-\phi_{sh}(r)$.
Considering only the quasi-neutral plasma (_i. e._ neglecting sheath effects),
Gueroult _et al._ [13] showed that the normalized voltage drop along field
lines, that is $\Delta_{\parallel}\phi(r)/\phi_{sh}(r)$, is small under the
condition $\tau=L/r_{g}\sqrt{\sigma_{\perp}/\sigma_{\parallel}}\ll 1$. This
condition, however, does not provide a measure of $\Delta_{\parallel}\phi(r)$
since it depends on $\phi_{sh}(r)$ which is governed by the sheath. In this
section we revisit this question in light of the insights in sheath physics
obtained in Section .3.

###  4.1  Insights from theory

Ohm’s law at the sheath writes

$\displaystyle\left.\begin{matrix}\displaystyle
j_{\parallel}\left(r,-\frac{L}{2}\right)=\displaystyle-\sigma_{\parallel}\frac{\partial\phi}{\partial
z}\end{matrix}\right|_{z=-L/2}.$ (14)

Although local, this results suggests that the voltage drop along field lines
in the quasi-neutral plasma will depend on the current collected at the
electrode. In fact, one shows as done in A that in the particular case that
the current density through the sheath does not depend on the radius, which is
precisely verified in the saturated regime, this result can be generalized to
obtain to lowest order in $\tau$ the global result

$\Delta_{\parallel}\psi^{sat}=\frac{L}{4T_{e}\sigma_{\parallel}}j_{is}\left(1+\Xi\right).$
(15)

This result shows that, at least in the saturated regime, the voltage drop
along field lines in the quasi-neutral plasma $\Delta_{\parallel}\psi$ is
expected to grow with thermionic emission.

Taking a step back, we have seen in Section 3 that thermionic emission can
help achieve control over the plasma potential over a larger range of
operating conditions compared to the case of a cold biased electrode. In has
notably been shown to help limit the voltage drop which exists across the
sheath in the case of a strongly negative bias such that $|\psi_{e}|\chi>1$.
On the other hand, Eq. (15) now suggests that in these same saturated
conditions thermionic emission would lead to a greater voltage drop along
field lines. To better understand this apparent trade-off and to which extent
thermionic emission is desirable for perpendicular electric field control, we
now turn to numerical simulations.

###  4.2  Numerical simulations

While the anisotropic Laplace equation

$\frac{1}{r}\frac{\partial}{\partial r}\left[r\frac{\partial\phi}{\partial
r}(r,z)\right]+\frac{\sigma_{\parallel}}{\sigma_{\perp}}\frac{\partial^{2}\phi}{\partial
z^{2}}(r,z)=0$ (16)

obtained from the combination of Ohm’s law for a static background neutral
$\bm{j}=\bm{\sigma}\mathbf{E}$ and charged conservation
$\bm{\nabla}\cdot\bm{j}=0$ allowed for analytical solutions when imposing
Dirichlet conditions [33], the use of more physical flux conditions requires
numerical modeling. Eq. (16) is thus solved here using finite differences in
the interior of the domain shown in figure 7 and implementing flux conditions
at the electrodes in a way very similar to that employed by Von Compernolle
_et al._ [43]. Specifically, noting $z_{0}=-L/2$ the axial position of the
left boundary of the domain, the ion-sheath in front of the electrode is
modeled via the non-linear Neumann condition

$\left.\frac{\partial\psi(r,z)}{\partial
z}\right|_{z=z_{0}}=\frac{j_{is}}{T_{e}\sigma_{\parallel}}\left[1+\Xi-\exp\left(\Lambda+\psi_{e}(r)-\psi(r,z_{0})\right)\right].$
(17)

In addition, we model the axial boundary in the electrode plane $z=z_{0}$
between $r_{e}$ and $r_{g}$ by enforcing $\nabla_{r}^{2}\psi(r,z_{0})=0$,
ground potential in $r_{g}$ and a potential in $r_{e}$ computed from the flux
condition Eq. (17). This leads to the Dirichlet condition

$\psi(r,z_{0})=\psi(r_{e},z_{0})\frac{\ln(r\big{/}r_{g})}{\ln(r_{e}\big{/}r_{g})}$
(18)

which is consistent with the radial profile derived in Sec. 3.3. The rest of
the domain’s boundary conditions, as shown in figure 7, are straightforward
and imposed from symmetry or zero potential.

Figure 7: Computational domain used for numerical simulations. The green dash-
dot boundary condition (BC) corresponds to axisymmetry. The pink dash-dot BC
corresponds to symmetry with respect to the plane $z=0$. The blue BC
represents the grounded vacuum vessel. The orange BC corresponds to the flux
condition Eq. (17) modeling the ion-sheath in front of the electrode. The
purple BC is the Dirichlet condition given in Eq. (18).

To offer a more physical illustration of the effect of thermionic emission, we
discuss here the results obtained for a set of dimensional geometric and
plasma parameters (see Table 1) which correspond loosely to the conditions
expected in a RF or Helicon laboratory plasma.

Table 1: Set of plasma and geometrical parameters used in numerical simulations. Neutral pressure $P$ [Pa] | $0.15$
---|---
Plasma density $n$ [m-3] | $10^{18}$
Electron temperature $T_{e}$ [eV] | $3$
Ion temperature $T_{i}$ [eV] | $0.2$
Magnetic field $B$ [mT] | $140$
Cross section $\sigma_{0}$ [$10^{-19}$ m2] | $5$
Plasma column length $L$ [m] | $2$
Plasma column radius $r_{g}$ [cm] | $20$
Electrode radius $r_{e}$ [cm] | $2.5$
Electrode bias $\phi_{e}$ [V] | $-250$
Ion mass $m_{i}$ [amu] | $40$
Normalized electrode potential $\psi_{e}$ | $-83.3$
Admissible plasma potential $\chi$ | $1.5$
Cold control parameter $|\psi_{e}|\chi$ | $123$

Focusing first on the on-axis behavior, figure 8 confirms as expected from the
value of $|\psi_{e}|\chi\gg 1$ in Table 1 that the regime is saturated for
zero thermionic emission ($\Xi=0$). The current is indeed equal to its maximum
value (in absolute value) while the voltage drop across the sheath
$\Delta_{sh}\psi$ is significant. In these conditions the voltage drop along
field lines $\Delta_{\parallel}\psi$ is very small. As the thermionic current
is increased, that is as $\Xi$ increases, the voltage drop across the sheath
$\Delta_{sh}\psi(0)$ decreases, as expected from the
$|\psi_{e}|-(1+\Xi)\chi^{-1}$ scaling in the saturated regime. Meanwhile, the
voltage drop along field lines increases, and numerical simulation results are
observed to be in very good agreement with the theoretical prediction obtained
in Eq. (15). This trend continues up until $\Xi$ is large enough to lead to
the transition from a saturated to a non-saturated regime on-axis. This
transition is indicated by the sudden drop in normalized current in figure 8.
Depending on plasma and geometrical parameters, the voltage drop along field
line can already be larger than the voltage drop through the sheath when the
regime transition takes place. In the non-saturated regime, both the voltage
drop across the sheath and the voltage drop along field lines are observed to
decrease with $\Xi$.

Figure 8: On-axis voltage drop across the sheath $\Delta_{sh}\psi(0)$ (black
dashdot, left axis) and along field lines $\Delta_{\parallel}\psi(0)$ (black
solid, left axis) and normalized current density (red solid, right axis) as a
function of thermionic emission $\Xi$. The green diamond represents the
analytical solution $\Delta_{\parallel}\psi^{sat}$ given in Eq. (21).
Potentials are normalized by the electron temperature $T_{e}$ in eV,
$\psi=\phi/T_{e}$.

Additional insights can be gained by examining how this behavior depends on
radius, as shown in figure 9. Starting again from zero emission ($\Xi=0$), one
finds a uniform behavior across the radius, consistent with the fact that the
regime is fully saturated. This is confirmed by the result that
$\tilde{j}=j_{\parallel}(r)/j_{max}=1$ over the entire electrode. As
thermionic emission is turned on and $\Xi$ increases, the voltage drop along
field lines $\Delta_{\parallel}\psi$ is first observed to grow nearly
uniformly in the electrode shadow ($r<r_{e}$). As $\Xi$ is increased further,
the current iso-contours confirm that the transition from saturated to non-
saturated regime occurs first on-axis, and progressively moves radially
outward, as predicted in Section 3.1. This radial current density non-
uniformity is accompanied by a similar variation in the voltage drop along
field lines, consistent with Eq. (15). For a given $\Xi$ the voltage drop
along field lines $\Delta_{\parallel}\psi(r)$ hence grows with $r$ in this
regime.

Figure 9: Colormap of the voltage drop along field lines
$\Delta_{\parallel}\psi(r)$ as a function of the radius $r$ and the thermionic
emission $\Xi$. Red solid lines indicate iso-contours of
$\Delta_{\parallel}\psi(r)$. Black dashdot lines indicate iso-contours of
normalized current density $\tilde{j}=j_{\parallel}(r)\big{/}j_{max}$. The
thick dashdot line represent the local regime transition, that is
$\tilde{j}=1$. Black dotted lines indicate values of $\Xi$ for which
transition for non-saturated to partially saturated and finally saturated
regimes occur. Potentials are normalized by the electron temperature $T_{e}$
in eV, $\psi=\phi/T_{e}$.

An interesting result revealed by figure 9, contrasting with on-axis results
observed in figure 8, lies in the behavior in the non-saturated regime.
Indeed, while both the voltage drop along field lines and the current density
through the sheath on-axis were seen to drop rapidly with $\Xi$ once the non-
saturated regime has been reached, one observes here that the voltage drop can
actually grow with $\Xi$ in the outer region of the biased electrode, and that
even in the non-saturated regime. The origin for this apparently counter-
intuitive result is to be found in the fact the while the normalized current
density $\tilde{j}$ falls off rapidly with $\Xi$ on-axis, the decrease is much
slower near $r=r_{e}$. As a result the absolute parallel current density
through the sheath can still locally grow with $\Xi$ near the outer radius of
the electrode. As shown in A this growth in parallel current can explain the
observed growth of the voltage drop along field lines even in the non-
saturated regime on the condition that the radial gradient length for the
current is sufficiently large compared to the electrode radius, which is
indeed what is observed in figure 9.

Returning to our motivation of assessing the potential drop along field lines,
in particular in the case of thermionic emission, these results show that
strong emission can lead to potential variations along field lines in the
quasi-neutral plasma of the order of a few electron temperature. While
moderate, these variations could be sufficient to affect perpendicular field
homogeneity along field lines. This is particularly true when considering
that, as discussed next, these variations may be even larger for a different
set of operating conditions and geometric parameters. In addition, it is found
here that the axial voltage drop can vary by a comparable amount perpendicular
to field line across the radius of the biased electrode. This, combined with
the possibility of a radially dependent voltage drop across the sheath, could
lead to potential differences between field lines that intersect the same
electrode, hindering the ability to control perpendicular electric fields.

###  4.3  Parametric dependencies

While the results presented in figure 8 and figure 9 were obtained for
specific plasma and geometric parameters, they can be used in combination with
the analytical results obtained in Sec. 4.1 to offer a more general analysis.
Indeed, although Eq. (15) already provided insights into the effect of
thermionic emission, a deeper understanding can be gained by developing the
parametric dependencies of conductivities.

Expressions for parallel and perpendicular conductivity in a magnetized plasma
are in general non-trivial and require solving a full set of fluid
equations[44, 45, 46] but one can often reasonably use simplified formulas
within an appropriate operating parameter space. For instance, electron-
neutral and ion-neutral collisions are found to be main contribution to
respectively parallel and perpendicular conductivity across a range of low-
temperature partly ionized and magnetized plasma laboratory experiments [35].
In this case the perpendicular conductivity reduces to the Pedersen
conductivity

$\displaystyle\sigma_{\perp}=\frac{nm_{i}\nu_{in}}{{B_{0}}^{2}}$ (19)

while the parallel conductivity is

$\sigma_{\parallel}=\frac{e^{2}n}{m_{e}\nu_{en}}$ (20)

with $\nu_{en}$ and $\nu_{in}$ the electron-neutral and ion-neutral collision
frequency, respectively. Here we simply take $\nu_{\alpha
n}=\sigma_{0}N_{n}\sqrt{T_{\alpha}/m_{\alpha}}$ with $\alpha=e,i$,
$m_{\alpha}$ the particle mass, $\sigma_{0}$ a fiducial cross section and
$N_{n}$ the neutral density, and ignore for simplicity the temperature
dependence of $\sigma_{0}$. Plugging Eq. (20) into Eq.(15) yields

$\Delta_{\parallel}\psi^{sat}_{en}\propto\frac{\left(1+\Xi\right)LN_{n}}{\sqrt{m_{i}}}.$
(21)

Meanwhile, plugging Eq. (19) into Eq. (7) leads to

$\displaystyle\chi_{in}\propto\frac{L\sqrt{T_{e}T_{i}}m_{i}N_{n}}{{r_{e}}^{2}\ln(r_{g}/r_{e}){B_{0}}^{2}}.$
(22)

Comparing Eq. (21) and Eq. (22), one finds that in regimes where the
simplified conductivities Eqs. (19) and (20) hold a decrease in the magnetic
field intensity $B_{0}$ will increase the range of operating conditions
leading to saturation. In particular, it will require a larger $\Xi$ to
transition from saturated to non-saturated operation. Looking at figure 9,
this means moving the transition region higher up. Meanwhile, since the
voltage drop along field lines does not depend on $B_{0}$ but does depend on
$\Xi$, the expansion of the saturated region means that higher voltage drop
along field lines will be observed in this regime. From Eqs. (19) and (20), a
similar growth is expected for heavier ion species since $\chi_{in}\propto
m_{i}$ while $\Delta_{\parallel}\psi^{sat}_{en}\propto 1/\sqrt{m_{i}}$, or for
a smaller electrode radius $r_{e}$.

The generic results Eq. (7) and Eq.(15) can be similarly used to explore other
regimes. For instance, if ones considers a denser plasma where parallel
conductivity is now governed by Spitzer conductivity,

$\sigma_{\parallel}=\frac{e^{2}n}{m_{e}\nu_{ei}}$ (23)

with $\nu_{ei}$ the electron-ion collision frequency then

$\Delta_{\parallel}\psi_{ei}^{sat}\propto\frac{\left(1+\Xi\right)Ln}{\sqrt{m_{i}}{T_{e}}^{2}}.$
(24)

This last result shows that in these conditions an increase of the plasma
density $n$ will lead to an increase in the voltage drop along field lines in
the saturated regime. On the other hand the value of $\Xi$ for which
transition occurs will decrease with $n$ since
$\Xi\propto\sqrt{m_{i}/T_{e}}/n$ for $j_{eth}\gg j_{is}$. As a result the bias
for which transition occurs will also decrease with $n$.

##  5  Summary

In this study the problem of plasma potential control from negatively biased
emissive electrodes positioned at the axial ends of a magnetized plasma column
is investigated through a combination of theory and numerical simulations.

By limiting the radial current flowing from the grounded vacuum vessel to the
negatively biased electrode through the plasma, the ion sheath formed in front
of the electrode is shown to control how much of the applied bias is passed on
along field lines to the plasma potential. Besides geometrical parameters the
minimum achievable plasma potential is controlled by the ion saturation
current and the perpendicular plasma conductivity. When the applied bias
amplitude is smaller than the voltage constructed from the plasma
perpendicular resistance and the ion saturation current at the biased
electrode, the plasma potential is found to be about the applied bias. In
contrast, when the applied bias amplitude is larger than this voltage the
plasma potential is no longer controlled by the applied bias, and the voltage
difference is found across the sheath. These two regimes are referred to as
non-saturated and saturated, respectively.

By increasing the current flowing through the plasma, thermionic emission from
the biased electrode makes it possible to lower the minimum achievable plasma
potential for a given set of plasma conditions and geometric parameters. This
in turn allows accessing a broader range of plasma potential, or in other
words improves plasma potential control through electrode biasing. On the
other hand, thermionic emission from the biased electrode is shown to lead to
a larger plasma potential variation along magnetic field lines in the quasi-
neutral plasma. Scaling laws for this potential variation along field lines
are derived analytically through the parametric dependencies of the plasma
parallel and perpendicular conductivities, and verified against numerical
simulations. Quantitatively, one finds plasma potential variations along field
lines of a few electron temperature for typical RF or Helicon laboratory
experiments under strong thermionic emission.

Although such potential variations along field lines may not in themselves be
a showstopper, they suggest that the possibility of using thermionic emission
to minimize the voltage drop across the sheath and hence to maximize control
over the plasma potential at the sheath edge should be consider carefully, and
that beyond the practical limit resulting from the formation of a virtual
cathode. This is particularly true in the non-saturated regime where a
radially dependent axial voltage drop could lead to a rotation of the electric
field vector.

##  Appendix A  Analytical model for the voltage drop along field lines

In the following we ignore thermionic emission for simplicity. Results
accounting for thermionic emission can simply be obtained by substituting
$j_{is}+j_{eth}$ in lieu of $j_{is}$ in the following results.

Taylor expanding the first partial derivative of $\phi$ with respect to $z$ in
$\left(r,z_{0}=-L/2\right)$ yields

$\displaystyle\frac{\partial\phi}{\partial z}(r,z)$
$\displaystyle=\frac{\partial\phi}{\partial
z}\left(r,z_{0}\right)+\frac{\partial^{2}\phi}{\partial
z^{2}}\left(r,z_{0}\right)\left(z-z_{0}\right)$
$\displaystyle\qquad+\frac{1}{2}\frac{\partial^{3}\phi}{\partial
z^{3}}\left(r,z_{0}\right)\left(z-z_{0}\right)^{2}+\mathcal{O}\left(z-z_{0}\right)^{3}.$
(25)

Additional terms in $\partial/\partial r$ expected for the expansion of a bi-
variate function do not appear here since we only consider a change in
position along $\mathbf{\hat{z}}$. From Ohm’s law at the sheath edge the first
term on the RHS of Eq. (25) is

$\frac{\partial\phi}{\partial
z}\left(r,z_{0}\right)=-\frac{j_{sh,\parallel}}{\sigma_{\parallel}}.$ (26)

Then, from the anisotropic Laplace equation Eq. (16), the second and third
partial derivative of $\phi$ with respect to $z$ write

$\frac{\partial^{2}\phi}{\partial
z^{2}}(r,z)=-\frac{\sigma_{\perp}}{\sigma_{\parallel}}\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial\phi}{\partial r}(r,z)\right)$ (27)

and

$\displaystyle\frac{\partial^{3}\phi}{\partial z^{3}}(r,z)$
$\displaystyle=-\frac{\partial}{\partial
z}\left[\frac{\sigma_{\perp}}{\sigma_{\parallel}}\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial\phi}{\partial r}(r,z)\right)\right]$
$\displaystyle=-\frac{\sigma_{\perp}}{\sigma_{\parallel}}\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial}{\partial r}\frac{\partial\phi}{\partial
z}(r,z)\right)$
$\displaystyle=\frac{\sigma_{\perp}}{\sigma_{\parallel}}\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial}{\partial
r}\frac{j_{\parallel}}{\sigma_{\parallel}}(r,z)\right).$ (28)

Writing $\bar{r}=r/r_{e}$ and $\bar{z}=z/L$, Eq. (28) rewrites

$\frac{\partial^{3}\phi}{\partial\bar{z}^{3}}(\bar{r},\bar{z})=\tau^{2}L\left(\frac{r_{g}}{r_{e}}\right)^{2}\frac{1}{\bar{r}}\frac{\partial}{\partial\bar{r}}\left(\bar{r}\frac{\partial}{\partial\bar{r}}\frac{j_{\parallel}}{\sigma_{\parallel}}(\bar{r},\bar{z})\right)$
(29)

with

$\tau=\frac{L}{r_{g}}\sqrt{\frac{\sigma_{\perp}}{\sigma_{\parallel}}}$ (30)

as previously defined in Ref. [33]. This qualitatively shows that the
contribution of the third partial derivative in Eq. (25) scales as
$(\tau/L_{\nabla_{r}}j_{\parallel})^{2}$ where ${L_{\nabla_{r}}j_{\parallel}}$
characterizes the radial gradient length of the parallel current density in
units of $r_{e}$.

Similarly, higher order derivatives can be written as

$\frac{\partial^{n}\phi}{\partial\bar{z}^{n}}(\bar{r},\bar{z})=\tau^{2}L\left(\frac{r_{g}}{r_{e}}\right)^{2}\frac{1}{\bar{r}}\frac{\partial}{\partial\bar{r}}\left(\bar{r}\frac{\partial}{\partial\bar{r}}\left[\frac{\partial^{n-3}}{\partial\bar{z}^{n-3}}\frac{j_{\parallel}}{\sigma_{\parallel}}(\bar{r},\bar{z})\right]\right).$
(31)

A similar analysis as that done for the third derivative can thus be made
here, with the difference that the gradient length is now taken on the
$(n-3)^{th}$ partial derivative of $j_{\parallel}$ with respect to $\bar{z}$.
Because $j_{\parallel}$ is expected to vary smoothly along field lines from
its value at the electrode to $0$ for $z=0$ (by symmetry), the contribution of
these extra terms is expected to decrease rapidly with $n$, and in any case to
be dominated by third order contribution.

In the saturated regime the parallel current density is constant along the
entire electrode, $j_{sh,\parallel}=-j_{is}$, and thus $j_{\parallel}(r\leq
r_{e},z_{0})$ does not depend on $r$ (_i. e._
$L_{\nabla_{r}}j_{\parallel}\rightarrow\infty$). Eq. (28) then implies that
the last term in Eq. (25) simply cancels out in this case. In addition,

$\frac{\partial\phi}{\partial
r}\left(r,z_{0}\right)=\frac{j_{is}r}{L\sigma_{\perp}}$ (32)

so that

$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial\phi}{\partial
r}\right)\left(r,z_{0}\right)=2\frac{j_{is}}{L\sigma_{\perp}}.$ (33)

Putting these pieces together Eq. (25) finally leads to

$\frac{\partial\phi}{\partial
z}(r,z)=\frac{j_{is}}{\sigma_{\parallel}}\frac{2z}{L}\left[1+\mathcal{O}\left(\tau^{2}\right)\right]$
(34)

where the $\mathcal{O}\left(\tau^{2}\right)$ term comes from higher order
terms involving $\partial j_{\parallel}/\partial z$. Integration between
$z_{0}$ and $0$ with the boundary conditions $\phi(r,z_{0})=\phi_{sh}(r)$ and
$\phi(r,0)=\phi_{mid}(r)$ yields to lowest order in $\tau$

$\Delta_{\parallel}\phi^{sat}(r)=\frac{L}{4\sigma_{\parallel}}j_{is}.$ (35)

In the non-saturated regime, the contribution of the third derivative is no
longer zero but instead scales as $(\tau/L_{\nabla_{r}}j_{\parallel})^{2}$.
This suggests that an analog to Eq. (35)

$\Delta_{\parallel}\phi(r)=-\frac{L}{4\sigma_{\parallel}}j_{sh,\parallel}(r).$
(36)

roughly holds as long as

$j_{sh,\parallel}(r,z_{0})\Big{/}\left[r_{e}\frac{\partial
j_{sh,\parallel}(r,z_{0})}{\partial r}\right]\gg\tau.$ (37)

## References

## References

  * [1] Itoh K and Itoh S I 1996 Plasma Phys. Control. Fusion 38 1–49
  * [2] Taylor R, Brown M, Fried B, Grote H, Liberati J, Morales G, Pribyl P, Darrow D and Ono M 1989 Phys. Rev. Lett. 63 2365–2368
  * [3] Groebner R J, Burrell K H and Seraydarian R P 1990 Phys. Rev. Lett. 64 3015–3018
  * [4] Oost G V, Bulanin V V, Donné A J H, Gusakov E Z, Kraemer-Flecken A, Krupnik L I, Melnikov A, Nanobashvili S, Peleman P, Razumova K A, Stöckel J, Vershkov V, Adamek J, Altukov A B, Andreev V F, Askinazi L G, Bondarenko I S, Brotankova J, Dnestrovskij A Y, Duran I, Eliseev L G, Esipov L A, Grashin S A, Gurchenko A D, Hogeweij G M D, Hron M, Ionita C, Jachmich S, Khrebtov S M, Kouprienko D V, Lysenko S E, Martines E, Perfilov S V, Petrov A V, Popov A Y, Reiser D, Schrittwieser R, Soldatov S, Spolaore M, Stepanov A Y, Telesca G, Urazbaev A O, Verdoolaege G, Zacek F and Zimmermann O 2007 Plasma Phys. Controlled Fusion 49 A29–A44
  * [5] Stroth U, Itoh K, Itoh S I, Hartfuss H, Laqua H, the ECRH team and the W7-AS team 2001 Phys. Rev. Lett. 86 5910–5913
  * [6] Burrell K H 1997 Phys. Plasmas 4 1499–1518
  * [7] Terry P W 2000 Rev. Modern Phys. 72 109–165
  * [8] Burrell K H 2020 Phys. Plasmas 27 060501
  * [9] Stix T H 1971 Phys. Fluids 14 692
  * [10] Rax J M, Gueroult R and Fisch N J 2017 Phys. Plasmas 24 032504
  * [11] Ochs I E and Fisch N J 2017 Phys. Plasmas 24 092513
  * [12] Kaganovich I D, Smolyakov A, Raitses Y, Ahedo E, Mikellides I G, Jorns B, Taccogna F, Gueroult R, Tsikata S, Bourdon A, Boeuf J P, Keidar M, Powis A T, Merino M, Cappelli M, Hara K, Carlsson J A, Fisch N J, Chabert P, Schweigert I, Lafleur T, Matyash K, Khrabrov A V, Boswell R W and Fruchtman A 2020 Phys. Plasmas 27 120601
  * [13] Gueroult R, Zweben S J, Fisch N J and Rax J M 2019 Phys. Plasmas 26 043511
  * [14] Zweben S J, Gueroult R and Fisch N J 2018 Phys. Plasmas 25 090901
  * [15] Dolgolenko D A and Muromkin Y A 2017 Phys.-Uspekhi 60 994–1017
  * [16] Fisch N J and Rax J M 1992 Phys. Rev. Lett. 69 612–615
  * [17] Weynants R R and Oost G V 1993 Plasma Phys. Control. Fusion 35 B177–B189
  * [18] Bagryansky P, Lizunov A, Zuev A, Kolesnikov E Y and Solomachin A 2003 Fusion Sci. Technol. 43 152–156
  * [19] Schaffner D A, Carter T A, Rossi G D, Guice D S, Maggs J E, Vincena S and Friedman B 2012 Phys. Rev. Lett. 109 135002
  * [20] Mase A, Itakura A, Inutake M, Ishii K, Jeong J, Hattori K and Miyoshi S 1991 Nucl. Fusion 31 1725–1733
  * [21] Yamada T, Inagaki S, Kobayashi T, Nagashima Y, Mitsuzono T, Miwa Y, Nakanishi K, Fujino H, Sasaki M, Kasuya N, Lesur M, Kosuga Y, Fujisawa A, Itoh S I and Itoh K 2014 Nucl. Fusion 54 114010
  * [22] Desjardins T R and Gilmore M 2016 Phys. Plasmas 23 055710
  * [23] Amatucci W E, Walker D N, Ganguli G, Antoniades J A, Duncan D, Bowles J H, Gavrishchaka V and Koepke M E 1996 Phys. Rev. Lett. 77 1978–1981
  * [24] Shinohara S and Horii S 2007 Jpn. J. Appl. Phys. 46 4276–4281
  * [25] Collins C, Katz N, Wallace J, Jara-Almonte J, Reese I, Zweibel E and Forest C B 2012 Phys. Rev. Lett. 108(11) 115001
  * [26] Plihon N, Bousselin G, Palermo F, Morales J, Bos W J T, Godeferd F, Bourgoin M, Pinton J F, Moulin M, Aanesland A and et al 2015 J. Plasma Phys. 81 345810102
  * [27] Gueroult R, Evans E, Zweben S, Fisch N J and Levinton F 2016 Plasma Sources Sci. Technol. 25 035024
  * [28] Désangles V, Bousselin G, Poyé A and Plihon N 2021 J. Plasma Phys. 87 905870308
  * [29] Lehnert B 1970 Phys. Scripta 2 106–107
  * [30] Lehnert B 1973 Phys. Scripta 7 102–106
  * [31] Tsushima A, Mieno T, Oertl M, Hatakeyama R and Sato N 1986 Phys. Rev. Lett. 56 1815–1818
  * [32] Bardakov V M, Ivanov S D and Strokin N A 2014 Phys. Plasmas 21 033505
  * [33] Gueroult R, Rax J M and Fisch N J 2019 Phys. Plasmas 26 122106
  * [34] Liziakin G, Gavrikov A and Smirnov V 2020 Plasma Sources Sci. Technol. 29 015008
  * [35] Liziakin G, Oiler A, Gavrikov A, Antonov N and Smirnov V 2021 J. Plasma Phys. 87 905870414
  * [36] Poulos M J 2019 Phys. Plasmas 26 022104
  * [37] Jin S, Poulos M J, Compernolle B V and Morales G J 2019 Phys. Plasmas 26 022105
  * [38] Baalrud S D, Scheiner B, Yee B T, Hopkins M M and Barnat E 2020 Plasma Sources Science and Technology 29 053001
  * [39] Hershkowitz N 1989 How langmuir probes work Plasma Diagnostics (Elsevier) pp 113–183
  * [40] Richardson O 1912 Phil. Mag. 23 594–627
  * [41] Nottingham W B 1956 Thermionic emission Encyclopedia of Physics / Handbuch der Physik (Springer Berlin Heidelberg) pp 1–175
  * [42] Palacio Mizrahi J H and Krasik Y E 2013 Physics of Plasmas 20 083510
  * [43] Van Compernolle B, Poulos M and Morales G 2019 Physics of Plasmas 26 122102
  * [44] Song P, Gombosi T and Ridley A 2001 Journal of Geophysical Research: Space Physics 106 8149–8156
  * [45] Rax J M, Kolmes E J, Ochs I E, Fisch N J and Gueroult R 2019 Phys. Plasmas 26 012303
  * [46] Kolmes E J, Ochs I E, Mlodik M E, Rax J M, Gueroult R and Fisch N J 2019 Phys. Plasmas 26 082309