Two-stream instability with a growth rate insensitive to collisions in a dissipative
plasma jet
Yi Zhou
1,
a)
and Paul M. Bellan
1
Department of Applied Physics and Materials Science, California Institute of
Technology, Pasadena, California 91125, USA
(Dated: 3 May 2023)
The two-stream instability (Buneman instability) is traditionally derived as a colli-
sionless instability with the presumption that collisions inhibit this instability. We
show here via a combination of a collisional two-fluid model and associated experi-
mental observations made in the Caltech plasma jet experiment, that in fact, a low
frequency mode of the two-stream instability is indifferent to collisions. Despite the
collision frequency greatly exceeding the growth rate of the instability, the instability
can still cause an exponential growth of electron velocity and a rapid depletion of par-
ticle density. High collisionality nevertheless has an important effect as it enables the
development of a double layer when the cross-section of the plasma jet is constricted
by a kink-instigated Rayleigh-Taylor instability.
a)
Author to whom correspondence should be addressed: yzhou3@caltech.edu
1
arXiv:2304.06814v3 [physics.plasm-ph] 1 May 2023
I. INTRODUCTION
The two-stream instability, also known as the Buneman instability, is a fundamental
plasma behavior that can lead to rapid, unstable growth of small perturbations, resulting
in effective dissipation of currents in plasmas
1
. This instability is typically triggered when
the electron drift velocity relative to ions is faster than the electron thermal velocity
2
. The
two-stream instability is believed to be related to the formation of a double layer (DL)
3–9
which is a large, localized electric field parallel to the current flow or magnetic field inside a
plasma. This localized electric field is called a double layer because the associated charge den-
sity given by Poisson’s equation consists of two spatially-separated and oppositely-charged
layers of particles
4
. Studying the two-stream instability and double layers is of value in
understanding particle energization in astrophysical plasmas
10
, nuclear fusion
11
, plasmas
for space propulsion
12
, and laboratory plasma discharges
13
.
Traditionally, the two-stream instability is derived from two-fluid equations or Vlasov
equations by neglecting collision terms
1,2,14
, and it is commonly presumed that the inclusion
of collision terms damps the instability. In fact, many analytical and numerical studies
15–19
have argued that collisions suppress the instability. However, these studies typically do not
account for the momentum change of ions as a result of collisions, resulting in violation
of momentum conservation. In this paper, we present a collisional two-fluid model which
conserves total momentum. The momentum conservation enables this two-fluid model to
describe a very low frequency two-stream instability which, contrary to conventional pre-
sumptions, maintains its characteristic behavior even if the plasma is extremely collisional.
This low frequency two-stream instability will be referred to as an evacuation instability
because it has similarities to an evacuation mechanism proposed long ago by Alfv ́en and
Carlqvist
20
and then elaborated by Carlqvist
21
in the context of density depletion and dou-
ble layer formation with the important exceptions that here (i) collisionality is taken into
account and (ii) the driver of the evacuation instability is a naturally occurring periodic
constriction of the plasma cross-section.
2
+
-
5 kV
GND
Plasma jet
Magne
�
c
fi
eld lines
Gas nozzles
Outer electrode
Inner electrode
10 cm
FIG. 1. Setup of the Caltech plasma jet experiment with photo of one stage of plasma jet superim-
posed. A pair of coplanar, concentric electrodes in a large vacuum chamber launch the collimated
argon plasma jet shown in the photo. The green magnetic field lines are created by a coil located
behind the inner electrode. Neutral gas puffs are injected through 8 pairs of gas nozzles evenly
spaced on the electrodes, and then a capacitor bank charged to 5 kV is switched across the inner
and outer electrodes to break down the neutral gas. Initially, 8 plasma arches that follow the
magnetic field lines form. These 8 arches then merge into the collimated plasma jet shown in the
photo.
II. OBSERVATIONS IN THE CALTECH PLASMA JET EXPERIMENT
The motivation of this study is to explain previous extreme ultraviolet (EUV) and visible
light observations made in the Caltech plasma jet experiment
22
. The experiment setup,
illustrated in Figure 1, is described in detail elsewhere
22–28
, and here we describe some key
features. This experiment launches a magnetohydrodynamically-driven (MHD-driven), cold
(2 eV), dense (10
22
m
−
3
) argon plasma jet from a pair of coplanar, concentric electrodes
placed inside a large vacuum chamber. Just before the plasma jet is launched, a coil coaxial
3
(a)
(b)
(c)
kink instability
25.5
μ
s
26
μ
s
Rayleigh-Taylor
ripples
26.5
μ
s
dimming
2
c
m
FIG. 2.
A sequence of false color images of the plasma jet in the Caltech experiment. (a) The
plasma jet becomes helical due to the Kruskal-Shafranov kink instability. (b) – (c) The current
channel cross-section becomes constricted at the location of the Rayleigh-Taylor ripples and then
dims.
with and located behind electrodes generates a dipole-like magnetic field (0.03–0.06 T).
Neutral argon gas puffs are then transiently injected from 8 pairs of gas nozzles evenly
spaced on the electrodes. After the nozzles have injected a gas cloud localized near the
electrodes, a capacitor bank charged to 5 kV is switched across the electrodes to ionize
the gas cloud. The capacitor bank then drives an electric current that flows between the
electrodes along eight plasma arches that follow the dipolar magnetic field lines linking the
electrodes
24
. These eight arches then merge into a collimated plasma jet whose initial radius
is a few cm. This plasma jet then lengthens over time into regions well beyond the extent
of the initial gas cloud. Fig. 1 illustrates this collimated plasma jet after it has lengthened
to around 20 cm. The current is further sustained by a set of capacitors and inductors
connected to form a pulse forming network. Except for the initial 10
μ
s, the current is
maintained to be approximately 70 kA during the 40–50
μ
s duration of the plasma jet, and
the voltage across the electrodes varies between 2 to 3 kV. Since the plasma density in the
jet volume is many orders of magnitude larger than the density of argon gas that was present
immediately before the plasma jet was created, the jet cannot be considered as a discharge
in a pre-existing argon atmosphere.
Figure 2 shows a typical sequence of transient events that occur after the jet has length-
ened from several centimeters to tens of centimeters. At approximately 20
μ
s the jet develops
a helical instability (kink), and Fig. 2(a) shows this kink after it has grown in about 5
μ
s to a
4
finite size; the kink identification and its onset being consistent with Kruskal-Shafranov the-
ory was presented previously
23
. Since kinking is a rapid exponential growth of a corkscrew
shape, kinking produces a strong lateral acceleration of each segment of the plasma jet away
from the initial axis of the jet. This lateral acceleration provides a large effective gravity
(10
10
−
10
11
m s
−
2
) that causes a secondary instability called the Rayleigh-Taylor (RT)
instability; this rapid acceleration and the resulting RT instability ripples were reported
previously
26
. Fig. 2(a) and (c) show that in 1
μ
s these ripples significantly constrict the
cross-section of the plasma jet. Chai, Zhai and Bellan
22
observed that the region constricted
by the RT ripples became bright in EUV radiation (20–60 eV, see Fig. 3 of Chai, Zhai and
Bellan
22
) but, as can be seen in Fig.3 of Chai, Zhai and Bellan and in Fig. 2(c) here, the
constricted region also became dim in visible light. The visible light dimming suggests a
reduction of the plasma density
n
has occurred since visible light emission is proportional
to
n
2
, the EUV radiation suggests localized plasma heating. It has been unclear why the
dimming in visible light and the brightening in EUV should occur simultaneously.
III. MODEL DESCRIPTION
A. Overview of the model
We present here a collisional two-fluid model consistent with the observed behavior; this
model describes an evacuation instability that has a rapid growth rate
γ
despite
γ
ν
ei
,
where
ν
ei
is the electron-ion collision frequency. The model critically depends on the plasma
jet having a large electric current flowing along the jet axis (
z
direction). Since magnetic
forces are perpendicular to currents, there are no magnetic forces in the
z
direction, so a
1-D electrostatic model describes the dynamics along the
z
direction. The jet current
I
is
constant because the power supply driving the jet operates in a constant current mode. This
means that the current density
J
=
I/A
must increase when the RT ripples constrict the
jet cross-sectional area
A
. MHD instabilities such as the kink and RT are incompressible
29
and so imply the density
n
is constant when the plasma jet is perturbed by the kink and RT
instabilities. Because
J
=
nq
e
u
e
, where
q
e
is the electron charge and
u
e
<
0 is the electron
drift velocity relative to ions,
u
e
should become more negative when
J
increases. Hence, the
absolute value
|
u
e
|
will peak at the constricted location, as shown in Fig. 3 (a) and (b). Fig.
5
Plasma column choked by
Rayleigh-Taylor Instability
Current
(a)
n
z
(c)
I
u
e
I
z
(b)
E
z
(d)
FIG. 3. The cartoon in (a) illustrates the choking of the plasma jet current channel cross-section
A
by RT ripples as shown in Fig. 2(b). Plots (b)–(d) illustrate the electron drift velocity
|
u
e
|
,
density
n
, and electric field
E
at the constricted location respectively.
2(a) and (c) show that the radius of the plasma jet current channel is reduced from initially
greater than 1 cm to eventually less than 0.3 cm. As a result,
A
=
πr
2
is constricted
approximately by a factor of 10 to 20 (from
>
3 cm
2
to
<
0
.
2 cm
2
). With
I
= 70 kA,
n
= 10
22
m
−
3
,
|
u
e
|
=
I/Anq
e
can increase from below 1
×
10
5
m/s to above 2
×
10
6
m/s. The
electron thermal velocity of the 2 eV plasma jet is approximately
v
T e
=
√
κT
e
/m
e
≈
6
×
10
5
m/s. Therefore, the initially slower electron drift velocity will greatly exceed
v
T e
due to the
constriction of
A
. It will be shown that this suprathermal electron drift velocity triggers the
evacuation instability that can create a density cavity at the constricted location, as shown
in Fig. 3(c). This density cavity would then cause the dimming of visible light.
Due to the combination of low temperature (
T
e
≈
T
i
≈
2 eV) and high density (
n
e
≈
n
i
≈
10
22
m
−
3
), the characteristic electron-ion collision frequency for the plasma jet
30
is
ν
ei
= 2
.
9
×
10
−
12
n
i
ln Λ
T
3
/
2
e
≈
10
11
s
−
1
,
(1)
where ln Λ is the Coulomb logarithm whose value is assumed to be around 10, and
T
e
is
in units of electron volts. This fast collision frequency corresponds to a 10
−
10
s – 10
−
11
s
characteristic collision time scale. Since this collision time is five to six orders of magnitude
smaller than the 10
−
6
s – 10
−
5
s jet dynamic time scale, the jet dynamics is highly collisional.
6
Thus, a model explaining how the RT ripples lead to simultaneous visible light dimming
(density evacuation) and EUV brightening must take into account that there are 10
5
- 10
6
electron-ion collisions in the 1
μ
s observed time scale of the visible light dimming and EUV
brightening. Clearly whatever is happening cannot be described by a collisionless instability.
One may argue that the subset of suprathermal electrons at the constricted region could
be collisionless because the electron-ion collision frequency for these fast electrons should
be significantly smaller
31
. A quick estimate of this smaller collision frequency shows that
this argument is not correct: we can roughly estimate the collision frequency by replacing
the
T
e
in Eq. (1) with the kinetic energy of electrons traveling at
u
e
≈
2
×
10
6
m/s, i.e.,
m
e
u
2
e
/
2
≈
11 eV. This kinetic energy reduces the collision frequency to be around 8
×
10
9
s
−
1
,
which corresponds to a 1
×
10
−
10
s characteristic collision time.
Because the plasma jet is extremely collisional, we can consider the plasma jet as a resistor
whose resistance is inversely proportional to
A
. When
A
is constricted by RT ripples, the
resistance at the constricted location should increase. As the current carried by the plasma
jet flows through this location with larger resistance, a DL, as shown in Fig. 3(d), would
develop to heat the plasma locally. This heating can potentially explain the observed EUV
radiation. A collisional two-fluid model that contains the presumptions discussed above will
be derived in the next section.
B. Derivation of the collisional two-fluid model
We can derive our collisional two-fluid model by neglecting various small terms from
the familiar 1-dimensional, unmagnetized, two-fluid equations for a fully ionized, collisional
plasma, namely
∂n
e
∂t
+
∂
∂z
(
n
e
u
e
) = 0
,
(2)
∂n
i
∂t
+
∂
∂z
(
n
i
u
i
) = 0
,
(3)
m
e
(
∂u
e
∂t
+
u
e
∂u
e
∂z
)
=
q
e
E
−
1
n
e
∂P
e
∂z
−
R
ei
n
e
,
(4)
m
i
(
∂u
i
∂t
+
u
i
∂u
i
∂z
)
=
q
i
E
−
1
n
i
∂P
i
∂z
−
R
ie
n
i
,
(5)
ε
0
∂E
∂z
=
n
i
q
i
+
n
e
q
e
.
(6)
7
Variables in the above equations are defined in the usual way. The electron energy equation
is assumed to be the isothermal equation of state, i.e.,
P
e
=
n
e
κT
e
with
T
e
being a constant,
and the ion energy equation is assumed to be the adiabatic equation of state
P
i
∝
n
Γ
i
where Γ
is an adiabatic constant of order unity. Justification of these energy equations can be found
below. The collision terms satisfy
R
ei
+
R
ie
= 0 because the collisions between electrons
and ions must conserve the overall momentum. The derivation of Eqs. (2–6) can be found
in several plasma physics textbooks
32–34
.
For simplicity, we consider the two-fluid equations in a reference frame that moves with
the plasma jet center of mass velocity. In this frame, ions are nearly stationary because the
mean ion velocity approximately equals the plasma jet center of mass velocity. As a result,
u
e
in this frame not only describes the mean electron drift velocity but also the electron drift
velocity relative to ions. This use of
u
e
is consistent with the definition of
u
e
in section III A.
We will analyze equations near an unstable equilibrium with an initial density
n
0
and an
initial suprathermal drift velocity
u
e
0
that satisfies 1
u
e
0
/v
T e
√
m
i
/m
e
. This fast
u
e
0
is achieved when the plasma jet cross-section is significantly constricted. This equilibrium
is perturbed by an evacuation instability that has a space-time dependence proportional
to exp(
ik
z
z
+
γt
) where the wavenumber
k
z
= 2
π/λ
is defined by the wavelength
λ
of the
RT ripples (
∼
1 cm) and the growth rate
γ
is assumed to be faster than that of the RT
instability (
γ
RT
is experimentally measured to be on the order of 1
×
10
6
s
−
1
) but slower
than
k
z
v
T e
.
Fig. 4 shows the assumed
γ
is either much smaller or much bigger than several charac-
teristic frequencies relevant to the Caltech plasma jet experiment. After an expression for
γ
is derived, it can be directly verified that
γ
indeed falls in the range indicated by this plot,
i.e.,
γ
a
k
z
v
T i
γ
RT
γ
k
z
v
T e
k
z
u
e
0
ω
pi
ν
ei
,
(7)
where
γ
a
is the growth rate of the ion-acoustic instability (see section V B), and
ω
pi
is the
ion-plasma frequency. The physically relevant inequality (7) establishes that a number of
terms in Eqs. (2)–(6) are extremely small and so may be neglected; the detailed arguments
for dropping these terms are as follows:
1. The plasma is quasi-neutral, so
n
e
=
n
i
=
n
when ions are singly charged. From now
on,
n
will be used to denote either the electron or ion density. This simplification is
8
FIG. 4. A plot of various characteristic frequencies in the Caltech plasma jet as a function of
k
z
.
The dashed lines represent characteristic frequencies estimated with experimental measurements.
The growth rate
γ
RT
of the RT instability is approximated by the formula
γ
RT
≈
√
g
eff
k
z
, where the
effective gravity
g
eff
is estimated from image data to be
∼
10
10
m/s
2
. The ion acoustic instability
growth rate
γ
a
is approximated to be an order of magnitude smaller than
k
z
c
s
where
c
s
is the
ion acoustic velocity
√
κT
e
/m
i
(see section V B). The growth rate
γ
of the evacuation instability,
indicated by the shaded ellipse, is assumed to be at a intermediate location that is either much
bigger or smaller than a characteristic frequency. This separation in frequencies enables several
simplifications of the original two-fluid equations.
equivalent to assuming the left-hand side (LHS) of Eq. (6) is negligible compared to
either one of the two terms on the right-hand side (RHS), i.e.,
∣
∣
∣
∣
ε
0
∂E
∂z
∣
∣
∣
∣
|
n
i
q
i
|
,
|
n
e
q
i
|
.
(8)
so the two terms on the RHS approximately balance each other.
Justifying this quasi-neutral simplification requires the wavelength of the perturbation
to be much longer than the electron Debye length
λ
De
=
√
εκT
e
/nq
2
e
. This requirement
can be written as
k
2
z
λ
2
De
1
,
(9)
9
z
1
z
2
nu
e
A(z
1
)
A(z
2
)
nu
e
FIG. 5. A current channel with varying cross-sectional area extending from
z
1
to
z
2
. The product
nu
e
A
is independent of
z
.
which is satisfied when
k
z
v
T i
γ
ω
pi
because
k
2
z
λ
2
De
=
k
2
z
v
2
T e
ω
2
pe
=
k
2
z
v
2
T i
ω
2
pi
γ
2
ω
2
pi
1
.
(10)
2. The partial time derivative in Eq. (2) is negligible compared to the spatial derivative
of
n
, i.e.,
∣
∣
∣
∣
∂n
∂t
∣
∣
∣
∣
∣
∣
∣
∣
u
e
∂n
∂z
∣
∣
∣
∣
.
(11)
Ignoring the time derivative simplifies Eq. (2) to be
∇ ·
(
nu
e
ˆ
z
) = 0 which when
integrated over the volume of a current channel extending from
z
1
to
z
2
, as shown in
Fig. 5, gives
n
(
z
2
)
u
e
(
z
2
)
A
(
z
2
)
−
n
(
z
1
)
u
e
(
z
1
)
A
(
z
1
) = 0. Since
z
1
and
z
2
are arbitrary,
the equation above is equivalent to
n
(
z
)
u
e
(
z
)
A
(
z
) = constant for any
z
(specifically,
nu
e
A
is constant at the constriction location). This corresponds to stating that the
electric current flowing along the channel is independent of
z
.
This simplification is justified when
γ
is small compared to
k
z
u
e
0
. This is because
the linearized form of inequality (11) can be reduced to
γ
k
z
u
e
0
due to the
exp (
ik
z
z
+
γt
) dependence of perturbations.
3. Similar to simplification 2, the partial time derivative in Eq. (4) can be ignored com-
pared to the convective term, i.e.,
∣
∣
∣
∣
∂u
e
∂t
∣
∣
∣
∣
∣
∣
∣
∣
u
e
∂u
e
∂z
∣
∣
∣
∣
.
(12)
4. Electrons are isothermal, so the electron pressure is
P
e
=
nκT
e
where
T
e
is a constant.
The validity of this assumption relies on
γ
being much smaller than
k
z
v
T e
. This is be-
cause electrons can be considered as isothermal for a perturbation with a characteristic
velocity
γ/k
z
that is slow compared to the electron thermal velocity
v
T e
.
10
5. The ion pressure term can be ignored compared to the partial time derivative in Eq. (5),
i.e.,
∣
∣
∣
∣
1
n
∂P
i
∂z
∣
∣
∣
∣
∣
∣
∣
∣
m
i
∂u
i
∂t
∣
∣
∣
∣
.
(13)
This simplification is equivalent to assuming
k
z
v
T i
γ
because the the linearized form
of inequality (13) can be rewritten as Γ
k
2
z
v
2
T i
γ
2
and the adiabatic constant Γ is of
order unity. To see how this can be done, we first multiply both sides of inequality (13)
by
n
0
/m
i
and express
P
i
1
in terms of
n
1
using the linearized adiabatic equation of state
P
i
1
= Γ
κT
i
n
1
:
∣
∣
∣
∣
Γ
v
T i
∂n
1
∂z
∣
∣
∣
∣
∣
∣
∣
∣
n
0
∂u
i
1
∂t
∣
∣
∣
∣
.
(14)
We can then differentiate both sides with respect to
z
and eliminate
u
i
1
using the
linearized form of Eq. (3) (see Eq. [25]) to get
∣
∣
∣
∣
Γ
v
T i
∂
2
n
1
∂z
2
∣
∣
∣
∣
∣
∣
∣
∣
∂
2
n
1
∂t
2
∣
∣
∣
∣
.
(15)
Assuming
n
1
is proportional to exp (
ik
z
z
+
γt
) gives us desired inequality Γ
k
2
z
v
2
T i
γ
2
.
The 5 simplifications listed above reduce the original system of two-fluid equations to be
nq
e
u
e
A
=
I
= constant
,
(16)
∂n
∂t
+
∂
∂z
(
nu
i
) = 0
,
(17)
m
e
u
e
∂u
e
∂z
=
q
e
E
−
κT
e
n
∂n
∂z
−
R
ei
n
,
(18)
m
i
(
∂u
i
∂t
+
u
i
∂u
i
∂z
)
=
q
i
E
−
R
ie
n
.
(19)
Adding Eqs. (18) and (19) gives
m
e
u
e
∂u
e
∂z
+
m
i
(
∂u
i
∂t
+
u
i
∂u
i
∂z
)
+
κT
e
n
∂n
∂z
= 0
.
(20)
which has the interesting feature of being a two-fluid equation that, while taking collisions
into account, does not explicitly depend on collisions. By two-fluid, it is meant that the
equation is beyond the scope of MHD as there is an explicit dependence on electron mass.
The collision terms in Eqs. (18) and (19) can be arbitrarily large without affecting Eq. (20).
If the collision effects are neglected and the cross-sectional area
A
is assumed to be
uniform, the system of Eqs. (16)–(20) reduces to Carlqvist’s collisionless evacuation mech-
anism
21
and to the collisionless equations studied by Galeev et al.
35
and by Bulanov and
11
Sarosov
36
. Because collision effects and a constricted
A
are not considered in Refs. 21, 35, and
36, these studies failed to describe the unidirectional DL electric field shown in Fig. 3(d)
(note that the electric field in Fig. 2d of Carlqvist
21
is bidirectional). It will be shown in
section III D that collision effects and a constricted
A
are critical for producing the unidi-
rectonal DL electric field.
C. Growth rate of the evacuation instability
We now consider the cross-section
A
to vary on the time scale of the RT instability
which is assumed to be much slower than the evacuation instability being derived. Thus,
A
is a slowly varying parameter from the point of view of the evacuation instability, and
a decrease of
A
with a consequent increase of
u
e
is effectively the “knob” that triggers the
fast evacuation instability. Because the decrease of
A
is caused by the RT instability, the
evacuation instability can be considered as a tertiary instability triggered by the secondary
RT instability.
The growth rate
γ
of the evacuation instability can be calculated in the jet frame (ion
velocity is nearly zero in this frame) by first linearizing Eqs. (16), (17), and (20) about an
initial equilibrium with a uniform density
n
0
and an electron drift velocity
u
e
0
to obtain
n
0
u
e
1
+
n
1
u
e
0
= 0
,
(21)
∂n
1
∂t
+
n
0
∂u
i
1
∂z
= 0
,
(22)
m
e
u
e
0
∂u
e
1
∂z
+
m
i
∂u
i
1
∂t
+
κT
e
n
0
∂n
1
∂z
= 0
,
(23)
where the subscript 1 denotes a small perturbation to the initial equilibrium. The cross-
sectional area
A
is treated as a constant when linearizing Eq. (16). We can express
u
e
1
and
u
i
1
in terms of
n
1
using Eqs. (21) and (22):
u
e
1
=
−
n
1
u
e
0
n
0
,
(24)
∂u
i
1
∂z
=
−
1
n
0
∂n
1
∂t
.
(25)
Multiplying Eq. (23) by
−
n
0
and differentiating the resulting equation with respect to
z
give
−
n
0
m
e
u
e
0
∂
2
u
e
1
∂z
2
−
n
0
m
i
∂
2
u
i
1
∂t∂z
−
κT
e
∂
2
n
1
∂z
2
= 0
.
(26)
12
We can get an equation that only involves
n
1
by eliminating
u
i
1
and
u
e
1
using Eqs. (24)
and (25):
m
e
u
2
e
0
∂
2
n
1
∂z
2
+
m
i
∂
2
n
1
∂t
2
−
κT
e
∂
2
n
1
∂z
2
= 0
.
(27)
We can then derive a dispersion relation by assuming the density perturbation
n
1
has an
exp (
ik
z
z
+
γt
) dependence:
−
k
2
z
m
e
u
2
e
0
+
γ
2
m
i
+
k
2
z
κT
e
= 0
,
(28)
which can be solved for
γ
to obtain
γ
=
k
z
√
m
e
m
i
(
u
2
e
0
−
v
2
T e
)
.
(29)
Equation (29) shows
γ
is real and positive when
u
e
0
is greater than
v
T e
, so this instability
is triggered when reduction of
A
causes the electron drift velocity to exceed the electron
thermal velocity. It is important to note that, while collisions have been taken into account,
this instability does not depend on whether or not the plasma is collisional. Thus, the
instability should occur in a highly collisional plasma such as the Caltech plasma jet. Using
k
z
= 700 m
−
1
,
m
e
= 9
.
1
×
10
−
31
kg,
m
i
= 6
.
7
×
10
−
26
kg,
u
e
0
= 2
×
10
6
m/s, and
v
T e
= 6
×
10
5
m/s, we can estimate the instability growth rate
γ
for the Caltech argon plasma jet to be
γ
≈
5
×
10
6
s
−
1
, which is indeed consistent with Fig. 4.
D. Electric field
The electric field in the plasma jet can be found from Eq. (4):
E
=
m
e
q
e
(
∂u
e
∂t
+
u
e
∂u
e
∂z
)
+
1
nq
e
∂P
e
∂z
+
R
ei
nq
e
,
(30)
As discussed in section III A and shown in Figure 4, the electron-ion collision frequency
ν
ei
is much larger than any other characteristics frequencies, such as
k
z
u
e
0
and
k
z
v
T e
. The
collisional term
R
ei
/nq
e
is proportional to
ν
ei
:
R
ei
nq
e
=
ν
ei
m
e
(
u
e
−
u
i
)
q
e
≈
ν
ei
m
e
u
e
q
e
.
(31)
Thus, we can assume the collision term in Eq. (30) is the dominant term that balances the
electric field, i.e.,
E
≈
ν
ei
m
e
u
e
q
e
.
(32)
13
This electric field is in fact a DL that is inversely proportional to the cross-sectional area
A
.
Before the plasma jet is constricted significantly, the electron drift velocity is slower than the
electron thermal velocity
v
T e
. To first order, we can assume
ν
ei
is initially proportional to
nT
−
3
/
2
e
ln Λ. Since
T
e
is a constant for isothermal electrons and ln Λ does not vary much,
ν
ei
is approximately proportional to
n
. Thus, the electric field in Eq. (32) is approximately pro-
portional to the electron flux
nu
e
, which is inversely proportional to the cross-sectional area
A
because
nu
e
=
I/q
e
A
. Therefore, this electric field is enhanced at the region constricted
by RT ripples and appears as a DL as shown in Fig. 3(d).
The combination of this DL and high collisionality of the plasma jet can possibly cause
Ohmic heating which can potentially explain the 20–60 eV EUV radiation observed in the
plasma jet experiment. Because the Caltech plasma jet is highly collisional, this DL electric
field can be quite large. For example, this DL can be as large as
∼
6
×
10
4
V/m, when
the relative drift velocity is
u
e
≈
1
×
10
5
m/s and the electron-ion collision frequency is
ν
ei
≈
1
×
10
11
s
−
1
. A complication associated with Ohmic heating is if the local electron
temperature is increased to become sufficiently large, then the condition
u
e
v
T e
would
cease and the evacuation instability would be quenched; consideration of this higher order
issue will be left for future consideration and so will not be addressed here.
The 1
/A
dependence derived above depends on
ν
ei
being proportional to
nT
−
3
/
2
e
ln Λ.
However, assuming
ν
ei
is proportional to
nT
−
3
/
2
e
ln Λ is not valid when the electron drift
velocity becomes significantly faster than the electron thermal velocity
31
. As a result, the
DL due to collisions will probably not scale as 1
/A
when the plasma jet is constricted
significantly. In fact, the strength of the DL may decrease because the collisionality of
suprathermal electrons decreases as the relative drift velocity increases
31
. When the strength
of DL has decreased to a point that it no longer dominates the RHS of Eq. (30), we can
retain an extra term from Eq. (30):
E
=
R
ei
nq
e
−
m
e
2
|
q
e
|
∂u
2
e
∂z
.
(33)
The newly included term on the RHS of Eq. (33) is the leading order correction because
the linearized form of this term is proportional to
k
z
u
e
0
, which is smaller than
ν
ei
but larger
than other relevant characteristic frequencies according to Fig. 4.
The new correction term is proportional to the partial
z
derivative of
u
2
e
. Across the
constricted region, the velocity profile in Fig. 3(b) has a maximum, so the partial
z
derivative
14
of
u
2
e
will be positive on the LHS of the maximum and negative on the RHS of the maximum.
Because the constant in front of the partial
z
derivative is negative, the second term should
point away from constricted region. Since
u
e
can grow exponentially once the evacuation
instability is triggered, this initially insignificant bidirectional electric field will also grow
exponentially and possibly become stronger than the unidirectional DL eventually. Unlike a
unidirectional DL, this bidirectional electric field cannot accelerate the bulk of the streaming
electrons since the associated electric potential does not have a net jump (the integral of the
bidirectional electric field across the constricted region equals 0). However, a small fraction
of electrons can possibly be accelerated by this bidirectional electric field in two opposite
directions. These fast electrons could possibly explain why localized X-ray sources near the
electrode and far from the electrode have been observed simultaneously in the Caltech jet
experiment (see Fig. 9 of Zhou, Pree, and Bellan
37
).
IV. NUMERICAL SOLUTION
A numerical solution of the problem is now presented to demonstrate the evacuation
instability when the cross-sectional area of Caltech’s plasma jet is significantly constricted
by the RT instability. The evacuation instability manifests itself as a fast growth of electron
drift velocity and a rapid depletion of density in the numerical solution. This numerical
solution replicates and extends the linear analysis.
A. Dimensionless equations and Discretization
For the numerical treatment, we use Caltech-experiment-relevant reference quantities
defined in Table I to normalize the equations of our collisional two-fluid model. For the
reference quantities in Table I, the reference length
z
ref
, density
n
ref
, and electric current
I
ref
are experiment-specific independent quantities that need to be prescribed. Other reference
quantities can be derived from these three prescribed quantities, the elementary charge
e
, and the electron mass
m
e
. With these reference quantities, the dimensionless form of
15
TABLE I. Definitions of reference quantities relevant to the Caltech plasma jet experiment.
Symbol
Reference quantity
Value
z
ref
length
1 cm
n
ref
density
10
22
m
−
3
I
ref
current
100 kA
q
ref
charge
e
= 1
.
6021
×
10
−
19
C
m
ref
mass
m
e
= 9
.
1094
×
10
−
31
kg
A
ref
=
z
2
ref
cross-sectional area
1 cm
2
u
ref
=
I
ref
/A
ref
n
ref
q
ref
velocity
6
.
2
×
10
5
m/s
t
ref
=
z
ref
/u
ref
time
16 ns
T
ref
=
m
ref
u
2
ref
Temperature
2.2 eV
Eqs. (16), (17), and (20) are
̄
n
̄
u
e
̄
A
=
−
̄
I
= constant
,
(34)
∂
̄
n
∂
̄
t
+
∂
∂
̄
z
( ̄
n
̄
u
i
) = 0
,
(35)
1
2
∂
̄
u
2
e
∂
̄
z
+ ̄
m
i
(
∂
̄
u
i
∂
̄
t
+
1
2
∂
̄
u
2
i
∂
̄
z
)
+
̄
T
e
∂
∂
̄
z
ln ̄
n
= 0
,
(36)
where the barred (dimensionless) variables are the original (dimensioned) variables divided
by their associated reference quantities.
The 3 dimensionless equations above are discretized with the Forward Time Centered
Space (FTCS) method. The spatial dimension ̄
z
is discretized over the interval [
−
π,π
]
with a mesh width of
h
=
π/
128. The dimensionless time
̄
t
is discretized with a discrete
time step of
k
= 0
.
5
h
. For any variable
f
defined on the ̄
z
−
̄
t
space-time plane,
f
m
j
denotes the pointwise value
f
( ̄
z
j
,
̄
t
m
). All quantities are assumed to be periodic so that
16
FIG. 6. The area function used in the numerical calculation. The area near ̄
z
= 0 has been reduced
significantly to trigger the evacuation instability.
f
(
−
π,
̄
t
m
) =
f
(
π,
̄
t
m
). The discretized equations read
̄
u
m
e,j
=
−
̄
I
̄
n
m
j
̄
A
m
j
,
(37)
̄
n
m
+1
j
= ̄
n
m
j
−
k
2
h
(
̄
n
m
j
+1
̄
u
m
i,j
+1
−
̄
n
m
j
−
1
̄
u
m
i,j
−
1
)
,
(38)
̄
u
m
+1
i,j
= ̄
u
m
i,j
−
k
2
h
̄
m
i
[
(
̄
u
m
e,j
+1
)
2
−
(
̄
u
m
e,j
−
1
)
2
2
+
̄
T
e
(
ln ̄
n
m
j
+1
−
ln ̄
n
m
j
−
1
)
]
−
k
4
h
[
(
̄
u
m
i,j
+1
)
2
−
(
̄
u
m
i,j
−
1
)
2
]
.
(39)
B. Numerical results
Given an area function
̄
A
and parameters
̄
I
,
̄
T
e
, and ̄
m
i
, Eqs. (37)–(39) can be solved
recursively for unknowns ̄
u
e
, ̄
n
, and ̄
u
i
from an initial ion velocity ̄
u
0
i
and an initial density
̄
n
0
. For the numerical solution to be presented in this section, we choose the area function to
be
̄
A
=
π
[
1
−
15
/
16
e
̄
t/
1000
sech(2
π
̄
z
)
]
, so that the cross-sectional area
̄
A
at ̄
z
= 0 has been
exponentially reduced by RT instability to
π/
16 at
̄
t
= 0. A plot of this area function at
̄
t
= 0
is shown in Fig. 6. The
e
-folding time is chosen to be 1000, so that the area is essentially
constant on the time scale of the evacuation instability. The dimensionless parameters
̄
I
,
̄
T
e
, and ̄
m
i
are chosen to be 0.7, 1, and 70000 respectively to match the conditions in the
plasma jet experiment. For the initial condition, we use an initially uniform density profile
̄
n
0
j
= 1 and an initially stationary ion velocity profile ̄
u
0
i,j
= 0.
17
(a)
(b)
FIG. 7. The numerically calculated (a) electron drift velocity and (b) density at different times
around ̄
z
= 0. The electron drift velocity becomes faster and the density dip becomes deeper due
to the evacuation instability triggered by the constriction of the plasma jet cross-section. ∆
|
̄
u
e
|
in
(a) denotes the change in electron drift velocity. Fig. 8 shows how ∆
|
̄
u
e
|
grows exponentially with
time due to the evacuation instability.
Fig. 7 shows the numerically calculated
|
̄
u
e
|
and ̄
n
around ̄
z
= 0 at different times.
These numerical results agree with the apparent profiles illustrated in Fig. 3(b) and (c)
qualitatively. In Fig. 7(a), the absolute value of ̄
u
e
peaks at ̄
z
= 0 due to the constriction
of the plasma jet.
|
̄
u
e
|
then grows rapidly due to the evacuation instability. In order to
conserve the current
̄
I
, a density dip starts developing as shown in Fig. 7(b). As time
increases, the density dip becomes deeper and deeper. This density dip is consistent with
the dimming of visible light observed in the plasma jet experiment. In Fig. 8, we compare
the numerical growth of
|
̄
u
e
|
to the exponential growth obtained from linear theory. The
blue line represents the numerically calculated ∆
|
̄
u
e
|
indicated in Fig. 7, and the red line is
18