The slow collisional ExB ion drift characterized as the major instability mechanism of a poorly magnetized plasma column with an inward-directed radial electric field
TThe slow collisional ExB ion drift characterized as the major instability mechanism ofa poorly magnetized plasma column with an inward-directed radial electric field
Thi´ery PIERRE Centre National de la Recherche Scientifique, UMR 7345 Laboratoire PIIM,Aix*Marseille University, Marseille, France.
The low-frequency instability of a cylindrical poorly magnetized plasma with aninward-directed radial electric field is studied changing the gas pressure and the ioncyclotron frequency. The unstable frequency always decreases when the gas pressureis increased indicating collisional effects. At a fixed pressure, the unstable frequencyincreases with the magnetic field when the B-field is low and decreases at largermagnetic field strength. We find that the transition between these two regimes isobtained when the ion cyclotron frequency equals the ion-neutrals collision frequency.This is in agreement with the theory of the slow-ion drift instability induced by thecollisional slowing of the electric ion drift (A. Simon, Phys. Fluids 6, , 1963).PACS numbers: 52.25.Xz,52.55.Dy,52.55.DyKeywords: Magnetized plasmas, Anomalous transport1 a r X i v : . [ phy s i c s . p l a s m - ph ] F e b . INTRODUCTION Understanding the physical mechanisms leading to instabilities and anomalous transportin a magnetized plasma has been a major goal in plasma physics during the past sixtyyears. In general, magnetized plasmas exhibit macroscopic non-MHD instability due to theradial decrease of the density and to the existence of a radial electric field. In the case of amagnetized plasma column with cylindrical geometry, if the physics of the sheath at the endof the plasma is not included, the situation is highly simplified. However, the radial densitygradient, the radial profile of the plasma potential and the radial profile of the electrontemperature can still lead to many instabilities. Among them, the electrostatic instabilitiesare the most violent and they dominate the other types, e.g. density gradient instability,temperature gradient instability, and micro-instabilities.In a previous paper , we have shown experimentally that the rotation of the magnetizedplasma column obtained in laboratory plasma devices is due to the large radial inward-directed electric field that can trigger flute-like modes, especially at low B-field. The aim ofthe present work is to carry out the analysis of the evolution of that dominant low-frequencyinstability when the pressure and the magnetic field strength are varied. The role of the ion-neutral collisions is investigated. We conclude the paper about the physical destabilizationmechanism. To the best of our knowledge, the role of the ion-neutral collisions has not beencarefully examined yet in the case of the physical system described here. II. THE MAGNETIZED PLASMA COLUMN
The plasma is produced in the MISTRAL Machine that has been depicted in previouspapers.
The device is similar to the linear plasma device MIRABELLE that was usedin previous investigations. A weakly ionized magnetized plasma column is produced at lowmagnetization. The device consists in a stainless steel cylindrical vessel (internal diameterD = 40 cm) evacuated to a base pressure of about 10 − Pa. The formation of a sharpboundary between the magnetized plasma and the vacuum is obtained inserting a stainlesssteel limiter made of a diaphragm with a free diameter of 8 cm inserted at the entrance of thetube covered by the solenoidal coils. The solenoid is made of 20 water-cooled coils equallyspaced along the cylindrical vacuum chamber. The maximum magnetic field strength used2n this investigation is 25 mT with a low ripple of the field lines along the plasma column.The plasma column is terminated by a glass end-plate. A weakly-ionized plasma is createdinside the large source chamber (80 cm diameter) by thermionic discharge using 32 tungstenfilaments (0.2 mm in diameter, 14 cm in length) located in front of a large multipolarmagnetic anode. The filaments are Joule-heated at 2000 K and they emit the energeticionizing electrons that are injected inside the magnetized column. A very fine tungstenmesh grid (78% optical transparency) inserted at the entrance section of the column allowsan electric insulation between the source plasma and the target plasma. The grid is kept atfloating potential in the experiments reported here. Only energetic electrons overcome thepotential of the grid (typically -25 volts) and produce the ionization of the column enteringthe solenoid through the circular aperture. A linear magnetized plasma column is produced(14 cm diameter, 90 cm length) inside the target chamber.With a floating injection grid and a biasing of the anode inside the source chamberbelow ground are established, a saturated low-frequency instability is present around themagnetized plasma column. Decreasing the potential of the anode inside the source chamber,the radial electric field across the magnetized plasma column is increased and the frequencyof the unstable mode is higher. This phenomenon has been studied and reported in arecent paper. It is easy to explain the role of the biasing of the anode by the fact thatthe decrease of the potential of the anode inside the source plasma leads to a larger fluxof ionizing electrons injected in the target plasma. This in return gives a lower plasmapotential inside the magnetized plasma column and as a consequence, the inward-directedradial electric field increases. More precisely, when the anode inside the source chamber ispositively biased, no instability is present. Decreasing the potential of the anode to -5 voltsleads to the destabilization of the plasma column and a strongly nonlinear regime of thedensity fluctuations is obtained. As a consequence of the increased electric field, the
ExB drift velocity of both ions and electrons is higher. The flute-like character of the instabilityhas been analyzed in previous papers with no phase detection along the plasma column. III. EXPERIMENTAL RESULTS
During the measurements described in this paper, the typical B-field strength on theaxis is varied between 5 mT and 20 mT. As shown in Fig.1, the radial density profile is3
IG. 1. Mean radial profile of the electron density. almost Gaussian near the axis of the column with a central density in the range 10 to10 m − . The electron temperature ranges from 3 to 4 eV depending on the working gaspressure , ranging from 5.10 − Pa to 5.10 − Pa in argon. Several Langmuir probes are usedfor measuring the radial profile of the plasma density, the plasma potential and the electrontemperature. The time-averaged plasma potential profile is roughly parabolic in the centralpart of the plasma column corresponding to an inward-directed electric field with linearradial profile. This corresponds to a rigid-body rotation of the core plasma column.The electron density fluctuations are recorded using cylindrical probes biased at plasmapotential and recorded by digital storage oscilloscopes. The power spectrum of the densityfluctuations is recorded using an analog spectrum analyzer (HP8560A).We have chosen to set the experiment in the most unstable regime which is obtained usinga floating collector. The potential of the anode in the source plasma is 5 volts below theground. Time series of the density fluctuations are detected by Langmuir probes locatedat the edge of the central plasma column. The study of the correlation between time seriesrecorded by two probes located on opposite positions at the edge of the plasma columnexhibits opposite phase in the signals and allows to determine the m = 1 structure of themode. The relative fluctuation level is maximum at the edge of the magnetized plasmacolumn and the absolute fluctuation level is maximum at the radial position where thedensity gradient is maximum. The time series of the density fluctuations detected by theLangmuir probe at the edge of the plasma column is shown in Fig. 2.The shape of the signal is similar to a cnoidal wave that can be considered as an infinitesum of periodically repeated solitary waves. This indicates the strongly nonlinear saturated4
IG. 2. Time series of the electron density recorded at the edge of the magnetized plasma columnin typical conditions exhibiting a strong nonlinear modulation . state of the instability. It is confirmed by the spectral analysis exhibiting multiple harmonicsof the unstable frequency. In this situation, the instability is established at a resonantazimuthal wave number. Changing the plasma parameters, the evolution of the unstablefrequency is investigated in order to accurately determine the mechanism of the instability.As mentioned before, the relative fluctuation level is maximum at the edge of the plasmacolumn where the density sharply decreases. The frequency of the unstable mode is close tothe ion cyclotron frequency and the frequency increases when the potential of the anode ofthe source chamber is decreased. As explained hereupon, the flux of the ionizing electronsentering the cylindrical magnetized target plasma is larger when the anode potential insidethe source chamber is decreased. The subsequent change in the plasma potential on theaxis of the magnetized plasma column leads to the enhancement of the radial electric field.It is important to note that the radial electric field is inward-directed. This is of majorimportance in the investigation of the destabilization mechanism. During the experiments,the radial scan of a swept Langmuir probe gives the mean plasma potential profile and thisin return gives an estimation of the profile of the mean radial electric field. A parabolicradial profile of the plasma potential is most often recorded near the center of the plasmacolumn giving rise to a static inward-directed radial electric field with a linear radial profile.This leads to a rigid-body rotation of the magnetized plasma column, at least in the centralpart. It is important to note that this experiment is a complex multi-parameters experiment.For instance, when investigating the evolution of the unstable frequency with changing thepressure, it is of major importance to monitor the evolution of the other parameters e.g. thedensity gradient length and the radial electric field during the variation of the pressure.5
IG. 3. Unstable frequency in the edge of the magnetized plasma column (squares) at B = 8 mTwhen the pressure is increased from 0.01 Pa to 0.03 Pa. The line displays the theoretical evolutionof the rotation frequency.
In the experiments reported here, two crucial parameters for the frequency selection areidentified : 1- when the gas pressure is increased, the frequency of the instability alwaysdecreases; 2- when the B-field is changed, the unstable frequency varies non-monotonically.Investigating first the effect of the pressure parameter, the unstable frequency is recordedwhen the gas pressure is changed over the range 0.01 Pa to 0.03 Pa at a magnetic field B= 8 mT. The recorded evolution is depicted in Fig. 3 (squares). We note that the unstablefrequency is roughly inversely proportional to the pressure. This will be compared to thetheory in the next section. These measurements show that the collisionality (ions-neutralcollisions) is clearly a major parameter in this experiment. Note that the evolution ofthe electron temperature changing the pressure cannot be invoked for the decrease of theunstable frequency. In fact, in the range of pressure investigated in this work, the electrontemperature decreases from 4 eV to 3 eV, inducing a relatively small change of the ion-acoustic velocity (about 15%) that cannot explain the recorded fast decrease of the unstablefrequency when the pressure is increased.In the second set of experiments, we investigate the influence of the second crucial param-eter, namely the magnetic field strength. We observe that the increase of the B-field leads6
IG. 4. Evolution of the unstable frequency (triangles) and the fluctuation level (dots) at r = 6cm with increasing the magnetic field. first to an increase of the unstable frequency when the B-field is low and then to a decreaseof the frequency at higher B-field. The evolution of the frequency has been recorded ata fixed pressure 0.02 Pa in argon changing the B-field from 5 mT to 20 mT. The resultsare depicted in Fig. 4 that displays the unstable frequency (triangles) versus the magneticfield strength. The unstable frequency is maximum at B = 9 mT. At higher magneticfield strength, the unstable frequency progressively change for a 1/B decrease. In Fig. 4,the fluctuation level is displayed during the B-field evolution (dots). The maximum of thefluctuation level is obtained on a B-field range corresponding to the maximum values of theunstable frequency. As will be explained hereafter when detailing the theory, we argue thatthe maximum of the unstable frequency and its highest level are obtained when the ionmean free path is equal to the ion cyclotron Larmor radius. In other words, it seems thatthe parameter ν in /Ω ci , where ν in is the global ion-neutral collision frequency and Ω ci theion cyclotron angular frequency, is an important parameter in the selection of the unstablefrequency. 7 V. THEORETICAL ANALYSIS
Considering the theoretical analysis, it is well known that the transverse diffusion coef-ficient and the transverse mobility in a weakly-ionized magnetized plasma are dependenton the collision time between ions and neutrals. Analyzing the mobility and the diffusionof ions in a weakly ionized magnetized plasma with ion cyclotron angular frequency Ω ci and ion-neutral collision time τ in , with the parallel mobility µ and the parallel diffusioncoefficient D , the perpendicular mobility and perpendicular diffusion coefficient for ions areexpressed as : µ ⊥ = µ (1 / ci .τ in ) and D ⊥ = D (1 / ci .τ in ) (1)As a consequence, the ExB velocity of the ions taking into account the collisions is expressedas : V E = ( E/B ) / (1 + ν in / Ω ci ) (2)In this analysis, the collision time include the global ion-neutral collision time (elastic colli-sion time and charge exchange collision time). As a consequence, the drifting ions experiencea collisional drag and the ExB ion drift is slow compared to the electron drift. For someplasma parameters, the reduced collisionality factor ν in /Ω ci is dramatically changing theelectric convection of the particles. Considering the instability mechanism in the case of aslab model, if a positive density disturbance is created at the edge of the plasma, a spacecharge separation is built inside the disturbance by the difference in the azimuthal electricdrift velocity of positive and negative charges. The induced transverse electric field inducedby the charge separation is perpendicular to the inward-directed electric field and it can am-plify the initial perturbation because the plasma will be locally displaced at a larger radialposition. The condition for amplification is that the density gradient and the electric fieldmust be oriented in the same direction, as explained in the seminal paper by Simon (1963) and in a following paper by Hoh (1963). The mechanism of the slow ion drift instability has been analytically studied in thecase of a high temperature plasma including finite Larmor radius effects. The nonlineartheoretical analysis has been detailed later. This instability has also been identified as animportant mechanism in a toroidal laboratory plasma. The difference in velocity drift ofpositive and negative species has been shown responsible for instabilities in non-collisional8lasmas exhibiting large ion radius effects inside a cylindrical magnetized plasma , thatis the so-called Modified Simon-Hoh high frequency instability. It is important to have inmind that the ion diamagnetic drift that experience also a collisional drag is negligible inour experiment. The low ion temperature (about 0.1 eV) and the gradient length (2 to 4cm) leads to an ion diamagnetic drift (opposite to the ExB drift) that is least ten timeslower that the electric drift.
V. COMPARISON WITH EXPERIMENT
We now compare the measurements to the theoretical analysis of the slow ion drift insta-bility. We investigate first the theoretical variation of the unstable frequency when the gaspressure is changed. The B-field value in the calculation is 8 mT. The main parameter inthe physical mechanism described here is the global ion-neutral collision frequency. In fact,the choice of the numerical value for the collision frequency is rather complex. We refer tothe literature detailing the collisions between ions and neutral, including elastic collisions and charge exchange collision , giving the cross-section for the interaction in the range 0.810 − m and 1.2 10 − m . At the pressure P = 0.02 Pa in argon with ion temperature T i =0.1 eV, we have chosen ν in = 10 Hz for the calculation. This corresponds to a mean freepath of 5 cm at the ion thermal velocity 490 m/s. This value for the collision frequencyis similar to the value used in previous numerical simulation of the unstable magnetizedplasma column where the normalized collision frequency was ν in /Ω ci = 0.06 at B = 60mT though the charge exchange collisions were not taken into account. In the theoreticalevaluation of the E x B ion drift velocity when the pressure is varied, we use that referencevalue of the global ion-neutral collision time τ in = 10 − s at P = 0.02 Pa with Ω ci = 1.9 10 rad/s at B = 8 mT.In order to compare the experimental results to the theory, we investigate first the vari-ation of the unstable frequency when the gas pressure is changed. The B-field value in thecalculation is 8 mT. Assuming that the angular frequency of the drifting ions determinesthe detected unstable frequency at the radial position r= 6.5 cm (assuming m=1 mode), thetheoretical results are superimposed in Fig. 3. The solid line gives the frequency obtainedwith an azimuthal velocity equal to the slow ion drift velocity with a collision time changingfrom 0.2 10 − s to 1.8 10 − s when the pressure is changed from 0.012 Pa to 0.03 Pa. A9 IG. 5. Theoretical evolution of the angular rotation frequency of the drifting ions with param-eters: radial position r = 6.5 cm, radial electric field 20 V/m, ion-neutral collision frequency 18kHz. A lower collision frequency, for instance 11 kHz, would give B = 6 mT for the value of theB-field corresponding to the most unstable frequency. good correlation with the experimental points is obtained. We conclude that the measuredunstable frequency and its evolution when the pressure is increased are in accordance withthe theory of the slow ion drift instability when the most unstable frequency is selected bythe m=1 mode and by the slow collisional E x B ion drift velocity.The theoretical investigation of this physical situation was first detailed in the paper bySimon describing a flute-type Rayleigh-Taylor instability predicted in the case of a weaklyionized magnetized plasma column with an inward radial electric field and a Gaussian profileof the density. In this physical situation, the calculation indicates that the most unstablefrequency is maximum when the parameter ν in /Ω ci is close to unity. The details of thecalculation and theoretical arguments are been revisited recently. The second important parameter determining the value of the unstable frequency is themagnetic field strength. In order to compare the theoretical evolution of the unstable fre-quency to the measurements when the B-field is increased, the evolution of the slow ion driftvelocity given by Eq. (2) is computed between B = 5 mT and B = 20 mT at a pressure 0.02Pa.The evolution of the angular frequency of the magnetized ions drifting at the slow col-lisional E x B drift velocity is displayed in Fig. 5. The global collision frequency has beenadjusted in order to get a maximum of the unstable frequency at 9 mT, that leads to ν in =10.8 10 Hz. This value leads to a mean free path of 2.7 cm (ion temperature T i = 0.1 eV). Arigid-body rotation induced by a radial electric field of 20 V/m at r = 6.5 cm is assumed inthe calculation. The non-monotonous variation of the angular frequency displayed in Fig. 5is similar to the experimental results depicted in Fig. 4. At the frequency maximum (B =9 mT), the ion Larmor radius is 2.27 cm i.e. a value very close to the mean free path. It isimportant to note that this evolution of the unstable frequency when the B-field is increasedis very sensitive to the ion-neutral collision time. For instance, choosing ν in = 1.1 10 Hzwould give a maximum of the unstable frequency at B = 6 mT.
VI. CONCLUSION
The low-frequency flute-type instability observed in the magnetized plasma column pro-duced in our laboratory device has been investigated taking into account the low magnetiza-tion of the ions leading to a relatively large ion Larmor radius of argon ions. The cyclotronicmovement of the ions moving at thermal velocity leads to a radial excursion comparable tothe radius of the plasma column. Moreover the gas pressure used in the experiment leadsto a high probability for created ions to be deflected or neutralized after a few centimeters.The existence of a radial electric field directed toward the axis of the plasma column in-duces a global rotation of the plasma ions. The rotation is slowed down by the collisionswith neutrals and in return induces the low-frequency instability of the magnetized plasmacolumn. This experimental situation has been compared successfully to the description ofthe slow electric drift instability described first by A. Simon in the early 60’s. The mea-surements indicate a good agreement with the theory, especially about the selection of themost unstable frequency when the mean free-path of the ions is close to ion Larmor radius.In this situation, an enhancement of the transport across the magnetic field is predicted.In conclusion, the parameter ν in /Ω ci , where ν in is the global ion-neutral collision frequencyand Ω ci the ion cyclotron angular frequency, is an important parameter for the stability ofa poorly magnetized collisional plasma. 11 CKNOWLEDGMENTS
We are grateful to Prof. F. F. Chen for suggestions about the Simon-Hoh instability andto Prof. G. Tynan for proposing this mechanism when visiting our experimental set-up.This paper is dedicated to the memory of Dr. Steve Jaeger (1976-2014), a dear friend andan inspiring collaborator.
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