Experimental Study of the Response of Sheared E × B Flow to Varying Ion–Neutral Collisions

Partially ionized plasma is a common occurrence in astrophysical and space environments. The emergence and development of plasma instabilities are significantly impacted by the inelastic collisions between the ions and neutrals in the partially ionized plasma, such as the charge exchange. In this study, the effect of the ion–neutral collisions on the sheared E × B flow was experimentally investigated. In the weak collision range, the shear-driven plasma instability, such as Kelvin–Helmholtz instability, was excited by the velocity-sheared flow. However, increasing ion–neutral collisions resulted in a decrease in the magnitude of the sheared E × B flow due to charge exchange–induced drag forces. Consequently, the Kelvin–Helmholtz instability is suppressed, and the Rayleigh–Taylor instability is triggered. The underlying mechanism was elucidated through experimental findings and numerical analysis. The result of this study proposes that a transition between the two modes occurred with increasing ion–neutral collision strength. It could be applied to the study of the solar chromosphere and prominence and planetary ionospheres, where plasma is partially ionized and the sheared E × B flow is often encountered.


Introduction
Partially ionized plasmas are characterized by an ionization degree of n e /n n < 10 −3 , which are commonly observed in a wide range of astrophysical contexts (Ballester et al. 2018).The partially ionized plasmas constitute an essential ingredient of the molecular clouds, solar atmosphere, and planetary ionospheres, giving rise to physical phenomena that are absent in fully ionized plasma.For instance, in a weakly ionized plasmas, the Hall effect arises through neutral collisions preferentially decoupling ions from the magnetic field (Martínez-Gómez et al. 2017;Pandey 2018).This typically occurs at much lower frequencies, and the Hall effect will play an important role in the dynamics of weakly ionized systems such as the Earth's ionosphere and protoplanetary disks (Pandey & Wardle 2008).Thesemultiple interactions produce numerous physical effects that can significantly impact the formation of plasma instabilities and waves (Vranjes & Poedts 2006;Shaikh & Zank 2010).In addition, the charged portion engages in collisions with the neutrals.The presence of neutral particles in the plasma can lead to the partially ionized effect, and new physical processes will be introduced, such as Cowling resistivity, isotropic thermal conduction by neutrals, heating due to ion/neutral friction, heat transfer due to collisions, charge exchange, and ionization energy (Ballester et al. 2018;Yalim 2020).With respect to the partially ionized plasmas, they are widely distributed in the early and reionization phase of the Universe, molecular clouds, interstellar, ionosphere, and comet tails (Krishan 2016).
The sheared E × B flow is widely distributed in astrophysical and planetary plasmas and has been frequently detected by spacecraft and satellites (Keskinen et al. 1981;Miura 1984;Kull 1991;Ganguli et al. 1994;Sundkvist et al. 2005;Mishra et al. 2018).Numerous electrostatic and electromagnetic instabilities can be excited by the sheared E × B flow (Amatucci 1999;Tejero et al. 2011;DuBois et al. 2013;Liu et al. 2017).For example, the transverse sheared E × B flow has been observed to trigger instabilities in a broad band frequency range, such as Kelvin-Helmholtz instability (Peñano & Ganguli 2000;Liu et al. 2018a), inhomogeneous energy-density driven instability (Ganguli & Palmadesso 1988;Koepke et al. 1994;Liu et al. 2018b), and electron-ion hybrid (EIH) instability (Amatucci 1999;DuBois et al. 2013;Liu et al. 2014).In the laboratory, the sheared E × B flow and electron density gradient were usually generated using a biased concentric ring electrode, and plasma instabilities such as KHI were excited in different collision regimes (Zhang et al. 2023).These instabilities are frequently observed in solar plasma and have been mainly investigated using numerical methods, such as analytical approaches to either single-or multiple-fluid treatment (Keskinen et al. 1988;Hysell & Kudeki 2004;Khomenko 2016).In addition, the shearedE × B flow and corresponding plasma instabilities were mainly investigated in the collisionless plasma (Pu 1989;Li et al. 2019).However, partially ionized plasma is also a common occurrence in astrophysical and space environments (Cheung & Cameron 2012;Yalim et al. 2020).In the partially ionized plasma, the inelastic collisions between the neutrals and Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.charged particles are frequent, which have a significant impact on the development of the sheared E × B flows and plasma instabilities (Kumar & Roberts 2003).These inelastic collisions include electron (ion) collision ionization, recombination, charge exchange, and ion collision ionization.Charge exchange is a process in which an ion and an atom exchange electron(s) in the process of collisions, which is considered to significantly influence the dynamics of ions.When the chargeexchange interaction is dominant, the directed flow of neutrals may become a source of inward momentum, i.e., during unit time interval, a momentum P ni = ν ni Mn n v n (ν ni : chargeexchange collision frequency of neutrals with ions) is brought into the ion fluids, while in the same interaction, the ions lose momentum (Okamoto et al. 2003).Therefore, the charge exchange between the ions and neutrals will generate a drag force that decreases the strength of the sheared E × B flow.Previous studies (Keskinen et al. 1988;Khomenko 2016;Rathod et al. 2021) have reported that the electrostatic KHI can be damped by collisions and that ion-neutral interactions can lead to a modification of the onset criteria and the growth rates of KHI and Rayleigh-Taylor instability (RTI) encountered in chromospheric heating solar plasmas.Thus, the effect of ionneutral collisions on the sheared E × B flow and the accompanying plasma instabilities in the ion cyclotron frequency range has been investigated in order to interpret the dynamical processes in partially ionized plasma such as the solar chromosphere and planetary ionosphere in this work.
In this study, the sheared E × B flow was artificially produced in partially ionized plasma in the laboratory, and the ion-neutral collisions were successively adjusted by controlling the neutral density.It was found that the enhanced ion-neutral collisions decrease the radial electric field, resulting in the velocity of sheared flow being dissipated.Meanwhile, the electron density gradient was increased with the collision damping, accompanying the residence electric field to produce a centrifugal force in the partially ionized plasma.Consequently, the shear-driven KHI was suppressed, and the RTI was simultaneously excited.This study interpreted the characteristics and evolution of the sheared E × B flow varying with the increased ion-neutral collisions, investigated by experimental and numerical simulation.The findings can aid in an improved understanding of the generation mechanisms of KHI and RTI observed in interstellar and solar-terrestrial space and also provide evidence to interpret the formation mechanisms of ionospheric irregularities in different altitudes of the ionosphere.

Experimental Apparatus and Setup
The experiment was conducted using the ground-based space plasma simulation platform: Keda Space Plasma EXperiment (KSPEX; Liu et al. 2016), which is a linear magnetized plasma device, with flexible control of the plasma parameters (Figure 1).An oxide-coated emissive cathode source (left of Figure 1) with a circular mesh as the anode electrode was utilized to produce the large-area uniform plasma.A metal mesh made of nickel (16 cm in diameter) was used as the anode.In this work, a partially ionized plasma environment was achieved by injecting neutral gas into the experimental chamber.The collision frequency was adjusted by controlling the neutral gas pressure.The sheared E × B flow was achieved by an inhomogeneous electric field, which was generated using a biased concentric ring electrode.The ring electrode was installed in front of the coated-oxide plasma source 1 m away from the anode mesh, as illustrated in Figure 1.The schemes of the concentric ring electrode design and alignment to adjust the distribution of plasma potential are illustrated in the blue-gray area.Four rings and three pole supports are required to maintain the ring electrode concentric.These rings were electrically isolated from each other with two machinable ceramics.The largest ring with a radius of 24.5 cm was attached to the vacuum chamber to keep the ring electrode more stable, and the radii of the other three electrode rings were 8, 5, and 2.5 cm.These three rings were biased by different voltages to flexibly adjust the radial electric field distributions.A macroscopic sheared E × B flow can be subsequently achieved in combination with the ambient magnetic field.
Parameters, such as the electron density, temperature, and plasma potential, were diagnosed by electrostatic probes.These probes were placed 0.3 m away from the ring electrode.The radial distributions of electron density and temperature can be diagnosed by the radially movable Langmuir single probe biased with a triangular wave.A radially movable electrically floating emissive probe was utilized to diagnose the plasma potential.From the curves of the sampled current and voltage, the plasma potential can be acquired by the singular point of the I-V curve.The inhomogeneous electric field can be calculated by using the one-order differentiation of the plasma potential distribution.All the characteristics of these probes have been described in our previous articles (Liu et al. 2016;Zhang et al. 2018).These probes were integrated into the National Instruments ( NI )data acquisition (DAQ) integrated system for motion control and DAQ controlled with 2 MHz sampling rate.Here, both the working and the injected gas were argon.The electron density varied from ∼2 × 10 9 cm −3 to 2.5 × 10 10 cm −3 , and the electron temperature is in the range of 2-5 eV, with the ion temperature at ∼0.1 eV.The magnetic field was kept at 130 G, which yields an ion cyclotron frequency f ci of 4.9 kHz and an ion gyroradius of ρ i ≈ 1.1 cm.

Results and Discussion
In this study, the working pressure in the vacuum chamber was controlled by a mass-flow meter, which was increased from 0.04 to 0.28 Pa as the mass flow increased from 12 to 80 standard cubic centimeter per minute (sccm).The ion-neutral collision frequency increased from 1.7 to 11.7 kHz, which was calculated by n s = n kT m in n i i (k is the Boltzmann constant).Therefore, the normalized collision frequency, which is defined as the ratio of the ion-neutral collision frequency to the ion gyrofrequency n f in ci lab

(
) , increased from 0.34 to 2.39 in this study, implying the ion-neutral collision frequency becomes larger than the ion cyclotron frequency.n » f in ci ionosphere in ci solaratmosphere ( ) - (Vernazza et al. 1981; Dong  & Paty 2011) in the solar atmosphere, respectively.It is suggested that both weak collision plasma and strong collision plasma were achieved in this study, which can be used to simulate the collision effects on the sheared E × B flow and the accompanied plasma instabilities in partially ionized plasmas, such as the ionosphere and solar atmosphere.
Figure 2(a) shows the radial profiles of the plasma potential (left axis) and electric field (right axis), respectively.Plasma potential can be modulated by an extra biased electrode, when the outer two rings are biased at 10 V relative to the ground.The radial profiles of the modulated plasma potential have been diagnosed across the plasma column by a movable emissive probe.In the sheared-layer region, the plasma potential is larger than that in the inner region.There is a clear potential drop in the region between the inner and the second ring electrodes, regardless of whether the inner ring was biased at a negative DC voltage or floating.Simultaneously, a larger potential gradient can be generated, which can subsequently drive the nonlinear radial electric field (∼2 V cm −1 ).Thus, the sheared E × B flow can be produced using the biased concentric ring electrode.Previous studies have suggested that the peak value and distribution of the electric field mainly depended on the biased potential of the ring electrode (Amatucci 2006;Desjardins & Gilmore 2016).In combination with the background magnetic field, a sheared E × Bflow was subsequently achieved.
According to early theoretical analyses (Jassby 1970(Jassby , 1972)), the first-order collision viscosity tends to reduce the growth rate of KHI.Moreover, the damping mechanism of viscosity determines the threshold of an additional inhomogeneous electric field to excite KHI due to the ion-neutral collision effect, particularly for the smaller azimuthal mode number m, and requires a larger electric field for instability excitation.In this study, the ion-neutral collision was produced by injecting neutral gas into the vacuum chamber, and the collision frequency was adjusted by controlling the neutral gas pressure.Figure 3 shows the profiles of the electric field, electron density, and its gradient as the normalized collision frequencies, ν in /f ci increasing until they are larger than unity.The profiles of the electric fields are plotted in Figure 3(a).The electric field is nonuniform in the radial direction.For example, the electric field peaks at around −4.5 cm (0 cm refers to the geometric center of the vacuum chamber, and the negative sign indicates that the direction is from the top to the central) with a magnitude of approximately −2 V cm −1 at a normalized collision frequency of 0.56, and the scale length can be estimated from the full width at half maximum of the profile, L E ≈ 1 cm (Tejero et al. 2011).Similar phenomena can also be seen from other lines, and the electric field decreases from ∼−2 V cm −1 to ∼0 V cm −1 as the ratio of ν in /f ci increases.A net momentum transfer during the charge-exchange interaction is suggested to produce an effective force acting on the ions.In noble gases such as argon, the charge exchange is more dominant than the usual elastic scattering, which plays a dominant role in the evolution of the sheared E × B plasma flow.Considering the ion-neutral collision effect, the equilibrium perpendicular velocity can be expressed as (Okamoto et al. 2003) Here, m i is the mass of the ion, n i (n n ) is the density of the ions (neutrals), T i (T e ) is the temperatures of the ions (electrons), respectively.ω ci and ν in are the ion gyrofrequency and ionneutral collision frequency, respectively.D eff is the diffusion constant of the neutrals.The first, second, and third terms in the bracket of Equation ( 1) are E × B drift, diamagnetic drift, and F × B drift (due to the "effective pressure" of neutrals, F is the force attributed to the charge-exchange collisions), respectively.As T i = T e , the second term could be neglected.From Equation (1), one can find that the ion drift can be directly damped by ion-neutral collision when ν in is large enough.On the contrary, with the increases of neutrals, the F × B term can also reduce the ion drift, as is found by Okamoto et al. (2003).
Therefore, the sheared E × Bflow is expected to become weaker and unstable in the observed regimes like Okamoto's numerical analysis, which is consistent to the experimental results in our following discussion.Figure 3(b) shows the profiles of the electron density, which range from ∼2 × 10 9 cm −3 to 2.5 × 10 10 cm −3 .The electron density gradually decreases with the increased mass flow.Moreover, the radial distributions of electron density become much more nonuniform as the diffusion is reduced by the neutral particles, and the electron density in the sheared region decreases versus the ratio of ion-neutral collision frequency to ion gyrofrequency (illustrated in Figures 2(b) and 3(b)).Figure 2(b) shows that the electron density decreases from 2.15 ×10 10 cm −3 to 6 × 10 9 cm −3 with the injection of neutral gas flow.Thus, this is an import factor on the electron density distribution and leads to a much more nonuniform profile of electron density.The electron density gradient can be calculated from the first-order differential of the density profile (shown in Figure 3(c)).It reaches the peak value in the radial region r = − 4.5 cm, and the scale length is estimated as L ne ≈ 2 cm.The ion-neutral collision is an important factor in the electron density distribution, and the nonuniform density was enhanced in the radial direction, resulting in a steeper electron density gradient.It can be therefore observed that the ion-neutral collision significantly reduces the electric field and increases the electron density gradient at a certain collision range.As ion-neutral collisions further increases (>1.98), the gradient of electron density decreases, and the accurate electric field can be difficult to obtain due to the fact that the decreasing electron density and the emissive probe are unstable with emitted hot electrons.
The fluctuation of the plasma potential induced by the sheared plasma flow and electron density gradient were sampled by a multiple-tip probe and analyzed by a fast Fourier transform algorithm to acquire power spectral density (PSD). Figure 4 illustrates the PSDs and wavenumbers versus the normalized collision frequencies for the lower collision frequency ν in /f ci 1.The instability is the KHI excited by the sheared E × B flow.The peak values of PSD are ∼1.6 kHz when the corresponding normalized collision frequency is ∼0.56-0.72.As the neutral gas pressure increases, the peak value shifts toward a larger frequency, and the instability is damped when the normalized collision frequency ν in /f ci ∼ 1, i.e., the ion collision frequency is equal to the ion cyclotron frequency.In this case, ions will be demagnetized, and the characteristics of the wave mode should be subsequently influenced.The corresponding azimuthal k θ and axial numbers k z with varying ratios of ν in /f ci are shown in Figure 4(b).k θ and k z range from 0.38 to 0.78 cm −1 and from 0.11 to 0.33 cm −1 , respectively.The ratio of the azimuthal number and axial number k θ /k z is larger than 1.5, and the mode propagates transversely.At a radius of 4.5 cm (the maximum electric field), the azimuthal mode number can be estimated by k θ • r, (c) electron density gradient for weak and strong collisions (ν in /f ci < 1 and ν in /f ci > 1).The negative sign indicates that electric fieldis from the top to the central direction, and the black line is the peak value position of the parameters (note that in (a), the electric field for ν in /f ci > 1 was offset by 3 V cm −1 for better visualization).and m = 4.5 × (0.4-0.8) ≈ 3 (Perez et al. 2006;Liu et al. 2017).The characteristics of KHI are similar with those reported in an early paper (Liu et al. 2018a), and the KHI is suppressed as the ion-neutral collision frequency increases.As discussed previously, the collisional viscosity induced by the ion-neutral collisions can suppress the KHI by decreasing the sheared electric field (Khomenko 2016).Moreover, early simulation studies have reported that the influence of the density gradient can suppress the KHI evolution (Wang et al. 2009(Wang et al. , 2010)).As shown in Figure 2(b), the ion-neutral particle collision is an important factor in electron density distribution.With an increasing collision frequency, the electron density is lower, and the inhomogeneity can be enhanced due to the reduced diffusion of neutral particles, which can result in an increased gradient of electron density in the radial direction.Hence, the KHI evolution in the regime of ν in /f ci < 1 can be due to ion-neutral collisional viscosity, resulting in the gradient of electron density increment and decreasing the electric field in the partially ionized plasma.This can also be derived from the numerical analysis shown in the Appendix.In the weak collision regime, . We solve the dispersion relation of KHI, i.e., Equation (A10), and show the normalized real frequency and growth rate of KHI in Figure 5. Here, From Figure 5, one can find that the growth rate is significantly damped by the ionneutral collisions.However, the real frequency also decreases with ν in .This may be due to the fact that some key physical effects, e.g., the electromagnetic effect, are not included here.In the future, we will include these and make a more systematical comparison between experiments and theories.
Figure 6 illustrates the PSDs (a) and wavenumbers (b) of the wave mode versus the normalized collision frequencies for strong collision (ν in /f ci > 1).Compared to Figure 4(a), a new instability appears as the neutral gas pressure further increases as ν in /f ci > 1.The fundamental frequency is ∼2 kHz and is larger than that of the lower collision frequency.Related to the varied characteristic frequency of the lower neutral flow, it does not shift with the increasing neutral gas and is almost fixed above 2 kHz.The wavenumbers k θ and k z are 0.38-0.78cm −1 and 0.11-0.33cm −1 , respectively.The azimuthal mode number of m = 1 is smaller than that of the KHI in the lower ν in /f ci case.In this case, the electric field is reduced by the increased  ion-neutral collisions.Therefore, the effect of the radial electric field-driven energy source becomes weaker, and the electron density gradient can be dominant.Previous studies have shown that when the driven source of electron density gradient and radial electric field can generate a centrifugal force, which can be equivalent to gravity, RTI is subsequently excited.The centrifugal force only is dominant for the azimuthal mode numbers of m = 1 and 2 (Jassby & Perkins 1970).This can be the reason why the centrifugal force driven source can be ignored for the instability of KHI (ν in /f ci < 1).The instability evolution that changes with the increasing neutral gas flow can be due to the gradient of the electron density increment and decreasing electric field with the varied collision frequencies in the partially ionized plasma (Spicher et al. 2020).The numerical analysis theory also provides similar results; because the effect of ion-neutral collisions is introduced (Zaqarashvili et al. 2015), one can find that the stable m = 1 mode is driven, just as is shown in Figure 7.
The results show that f r ∼ 0.5 and f ci ≈ 2 kHz when f ci < ν in < 2f ci .In fact, the real frequency almost increases with ν in in this regime.However, dependence of the growth rate on ν in is non-monotonic.We also note that the m = 1 mode is stable when ν in = 0. Therefore, for the m = 1 mode, the stabilization terms are partially canceled by ν in .Our calculation results show that the m = 1 mode is stable (1) when ion-neutral collision effects are neglected and (2) the number density of ion and the azimuthal flow are constants across the boundary layer.Therefore, the m = 1 mode is the collision-induced instability.For m = 1 mode, the stabilization terms are partially canceled by the finite ion-neutral collisions.The m = 1 mode could be driven when the flow shear is negligible (Ω 0+ = Ω 0− ≠ 0) while the density gradient still holds.Similarly, the growth rate of the m = 1 mode could be positive in the regime where the density gradient disappears while the flow shear is kept constant.Then, in the strong collision regime, the m = 1 mode may be a hybrid of the KHI and RTI, which is decided by the distribution of the ion density gradient and azimuthal E × B flow.However, one can find that the density gradient is enhanced by the ion-neutral collisions, as is shown in Figure 3(b).Therefore, the m = 1 mode is assumed to be RTI.All the conclusions are based on a local boundary layer theory and neglect the thickness of the boundary layer, which may deviate from the experimental settings.In the future, we will try to develop a more accurate theory and explore the dominant driven source of the m = 1 mode.With the increasing frequency of collisions, the electron density decreases, and the inhomogeneity can be enhanced because of the reduced diffusion of neutral particles, which can result in a relatively larger gradient of electron density in the radial direction and a decrease in the radial electric field.According to an early report (Jassby & Perkins 1970), the firstorder collisional viscosity tends to reduce the growth rate of KHI by the term n r q k 1 8 i i 2 2 due to the decreased electric field, so a weakening PSD with increasing collisional frequency can be attributed to the collisional viscosity, while the instability-ignited threshold is limited by the collisional viscosity.Moreover, the KHI is usually destabilized by the electron density gradient-driven flow, and the sheared E × B flow also becomes weaker because of the decreased electric field.Hence KHI can be suppressed by the ion-neutral collision damping.Finally, the increased gradient of electron density can be an important source for instability excitation.The electron density gradient-driven centrifugal force can be formulated (Jassby 1972): where, ω E is the sheared frequency, n e is the electron density, r is the radial location function, and k θ is the wavenumber, respectively.The above equation shows that the centrifugal force can be directly proportional to the gradient of electron density and be inversely proportional to electron density.Therefore, in this case the centrifugal force is significant in driving the instability (ν in /f ci > 1) as the gradient of electron density can increase and density decrease with the collisional frequency.Additionally, we observed that as the ion-neutral collision further increases, the gradient of electron density decreases, which can be the reason why the magnitude of RTI is weak as ν in /f ci > 2. In this case, the plasma density is much lower than that of the relatively weak collisions, to the degree where it is challenging to keep the RTI evolving.Furthermore, the electron density gradient-driven flow can usually suppress KHI evolution and enhance the RTI growth.From the above discussion, we can find that the KHI can easily exist in the relatively lower collision frequency condition (ν in /f ci < 1).
With the collision frequency increase, however, where the electron density gradient can be a significant free energydriven source, RTI is more easily excited.In our experimental regimes, the length scale of the electric field is L E = 1 cm and is close to the ion gyroradius ρ i =1 while the electron Larmor radius is about 0.05 cm and the electron inertial length is 0.29-0.82m and greater than the electric scale length.Thus, the electron-scale dynamical effect can be ignored in our work, and the ion-scale dynamical movement is a significant factor in the wave evolution.With the neutral gas flow injects, the ratio of ion-neutral collision frequency is more significant, and results in the gradient and plasma density are changing.Therefore, the ion-scale dynamical movement is dominant in the sheared E × B flow to drive the KHI in weak collision regimes, and the ion-neutral collisions become significant for the RTI growth in strong collision regimes.

Conclusions
The sheared E × B flow is one of the dominant sources of free energy for the excitement of plasma instabilities, while the partially ionized plasma is also universal in astrophysical and space enviroments.Studying the response of the sheared E × B flow to varying ion-neutral collisions is therefore important.In this study, the effects of the ion-neutral collision on the sheared E × B flow were experimentally investigated.In a weak collision plasma, the plasma instabilities such as the KHI were observed to be excited by the sheared E × B flow.However, as the neutral-ion collision frequency increases, it was found that the enhanced ion-neutral collision can decrease the magnitude of the sheared plasma flow and increase the density gradient.In combination with the density gradient and the residence electric field, a centrifugal force was subsequently generated, which leads to the suppression of the KHI and the emergence of the RTI in the partially ionized plasma.This study indicates the response of the sheared plasma flow to a varying ion-neutral collision and how sources of free energy excite plasma instabilities.Specifically, the sheared E × B flow-driven KHI and density gradient flow-driven RTI can be excited successively as the change in the sheared plasma flow is significantly modified by the collision viscosity.This basic physics process is significant in understanding the physical processes that appear in the solar chromosphere and ionosphere environments.Interpreting the different excited mechanisms of ionospheric irregularities at different altitudes is also useful.
where q i = e and n i are the ion charge and number density, respectively.ρ i , p i , and  v i (ρ n , p n , and  v n ) are the ion (neutral) density, pressure, and velocity, respectively.ν in and ν ni are the ion-neutral and neutral-ion collision frequencies, respectively.Usually ν in ?ν ni in the weakly ionized plasma.
To take a further step, we linearize Equations (A1)-(A4) and hypothesize that any perturbed quantity can be expressed as f = f 0 + δf.Here, f 0 and d µ q w -f e i m k z t z ( ) (m, k z , and ω are the azimuthal mode number, the axial wavenumber, and the complex wave frequency, respectively) are the equilibrium part and the perturbed term.Then, the governing equations can be reduced to Here, m i is the ion mass, and f = -  E 1 1 and f 1 are the perturbed electric field and electrostatic potential, respectively.

Figure 1 .
Figure 1.Schematic diagram of the KSPEX and biased concentric ring electrode.

Figure 2 .
Figure 2. (a) Typical radial profiles of plasma potential and electric field vary with biased voltages; the magnetic field is ∼130 G, and the ratio of ion-neutral collision frequency to ion gyrofrequency is ∼0.56.(b) The electron density in the sheared region vs. the ratio of ion-neutral collision frequency to ion gyrofrequency.

Figure 3 .
Figure 3.The radial profiles of the (a) electric field, (b) electron density, and(c) electron density gradient for weak and strong collisions (ν in /f ci < 1 and ν in /f ci > 1).The negative sign indicates that electric fieldis from the top to the central direction, and the black line is the peak value position of the parameters (note that in (a), the electric field for ν in /f ci > 1 was offset by 3 V cm −1 for better visualization).

Figure 4 .
Figure 4.The PSD of the potential fluctuation and wavenumbers vs. the normalized collision frequencies ν in /f ci 1.(a) is the PSD of the potential fluctuation; (b) is the variation of azimuthal wavenumber k θ and axial wavenumber k z .

Figure 5 .
Figure 5.The normalized real frequency (a) and growth rate (b) of the KHI with n f in ci .The plasma parameters are m = 3, = w W + 1 ci 0

Figure 6 .
Figure 6.PSDs of the potential fluctuation and wavenumbers vs. the ratio of normalized collision frequencies ν in /f ci > 1.(a) PSDs of the potential fluctuation.(b) The variation of azimuthal k θ and axial number k z .