Taylor-Couette Flow of Unmagnetized Plasma C

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Taylor-Couette Flow of Unmagnetized Plasma C. Collins,1, 2 M. Clark,1 C.M. Cooper,1, 2 K. Flanagan,1, 2 I.V. Khalzov,1, 2 M.D. Nornberg,1, 2 B. Seidlitz,1 J. Wallace,1 and C.B. Forest1, 2 1)Department of Physics, University of Wisconsin, Madison WI 53706, USA 2)Center for Magnetic Self Organization, University of Wisconsin, Madison WI 53706, USA (Dated: 26 October 2018) Differentially rotating flows of unmagnetized, highly conducting plasmas have been created in the Plasma Couette Experiment. Previously, hot-cathodes have been used to control plasma rotation by a stirring technique [C. Collins et al., Phys. Rev. Lett. 108, 115001(2012)] on the outer cylindrical boundary—these plasmas were nearly rigid rotors, modified only by the presence of a neutral particle drag. Experiments have now been extended to include stirring from an inner boundary, allowing for generalized circular Couette flow and opening a path for both hydrodynamic and magnetohydrodynamic experiments, as well as fundamental studies of plasma viscosity. Plasma is confined in a cylindrical, 11 −3 axisymmetric, multicusp magnetic field, with Te < 10 eV, Ti < 1 eV, and ne < 10 cm . Azimuthal flows (up to 12 km/s, M = V/cs ∼ 0.7) are driven by edge J × B torques in helium, neon, argon, and xenon plasmas, and the experiment has already achieved Rm ∼ 65 and P m ∼ 0.2−12. We present measurements of a self-consistent, rotation-induced, species- dependent radial electric field, which acts together with pressure gradient to provide the centripetal acceleration for the ions. The maximum flow speeds scale with the Alfvéncriti- cal ionization velocity, which occurs in partially ionized plasma. A hydrodynamic stability analysis in the context of the experimental geometry and achievable parameters is also explored.

I. INTRODUCTION has been pursued as a model for understanding astrophysical systems, and theoretical investigations using both ideal magnetohydrodynam- Circular Couette flow (or Taylor-Couette ics (MHD) and extended MHD in plasma5 have flow, hereafter TCF) is one of the most fun- been examined. damental systems in fluid dynamics1–3 and was A TCF has recently been created using first investigated over 130 years ago. In TCF plasma, rather than liquids, in a laboratory geometry, a viscous liquid flows between two device named the Plasma Couette Experiment coaxial, differentially rotating cylinders. The (PCX) located at the University of Wisconsin- classic TCF profile is purely azimuthal, though Madison. The goal of PCX is to create a suffi- experiments in finite height cylinders often suf- ciently hot, steady-state, differentially rotating, fer from secondary, non-azimuthal Ekman cir- weakly magnetized plasma to study basic hy- culation due to end effects. While seemingly drodynamic and magnetic flow-driven instabil- simple, TCF exhibits a large number of distinct ities and dynamics specific to plasmas, such as flow states (which depend on the rotation rates two-fluid effects and neutral collisions. In PCX, of the inner and outer cylinders), making it toroidal plasma flow is controlled by electro- the ideal testbed for detailed experimental and static stirring assemblies at both the inner and arXiv:1403.2087v2 [physics.plasm-ph] 12 Apr 2014 theoretical studies of hydrodynamic instability outer boundaries. Flow profiles can be adjusted thresholds, nonlinear behavior, and transition so that the angular velocity decreases with ra- to turbulence4. Recently, TCF of a conduct- dius and the angular momentum increases with ing medium, such as liquid metal or plasma, radius, and thus mimic the Keplerian-like flows 2 of accretion disks (where Ω(r) = pGM/r3 ).

A TCF of plasma operates in a regime dis- CENTER CORE tinct from either liquid or liquid metal TCF. ISOLATION CHAMBER Specifically, the degree of ionization introduces the possibility of momentum loss through colli- CENTER CORE sions with neutral atoms, and the resistive and STIRRING ASSEMBLY viscous dissipation varies with the plasma tem- TURBOMOLECULAR & perature Te and density ne. The relevant hydro- HELMHOLTZ COILS CRYO VACUUM PUMPS dynamic Reynolds number characterizing the ratio of the momentum advection by the flow to the momentum diffusion, Re = V L/ν, is gov- OUTER STIRRING ELECTRODES erned by the flow velocity V , the characteristic PROBE DRIVES size L, and Braginski viscosity; in an unmagnetized plasma, ν = 0.96v2 τ where v is the Ti i Ti SCANNING OPTICS TABLE ion thermal speed and τi is the ion-ion collision time. Also of interest is the magnetic Reynolds number, Rm = V L/η, which describes the degree to which magnetic fields are frozen into a moving plasma with magnetic diffusivity η. In FIG. 1. Major hardware components and cut-away terms of plasma parameters, view of the vacuum chamber. 4√ −5/2 Re = 0.74 Vkm/sLmni,1011cm−3 Z µ Ti,eV (1) 3/2 −1 Rm = 0.84 Te,eV Vkm/sLmZ (2) The ratio of these quantities is the magnetic Prandtl number P m = Rm/Re = ν/η and is of order unity in PCX plasmas, a significant the plasma is achieved by extending the ro- advantage over liquid metal experiments where tation drive system to include all four cath- P m = 10−5 − 10−6. odes at the outer boundary. Plasma parame- If rotation is driven exclusively at the in- ters such as velocity, Te, ne, Vfloat are mea- ner boundary, TCF can become hydrodynam- sured with probe-based diagnostics, and a non- ically unstable above a certain critical thresh- perturbative method of line ratio spectroscopy old (set by the viscosity) and consequently ex- is also used to measure Te. A newly constructed hibits large-scale, secondary flows in the form center post assembly provides stirring on the of axisymmetric “Taylor” vortices, first studied inner boundary, resulting in a controlled, differ- in detail by G.I. Taylor in 19231. Exploring entially rotating flow which we use to explore the onset threshold for hydrodynamic instabil- the relevant dynamics of plasma TCF. Measure- ity is important in PCX, since Taylor vortices ments of a self-consistent, rotation-induced ra- govern the rate of momentum transport across dial electric field are also presented. Much ef- the sheared flow of plasma. Envisioning the use fort was made to maximize the induced flow, of this device for experiments concerning other and apparent velocity limits have been observed instabilities, such as the magnetorotational in- for various gas species (He ∼ 12 km/s, Ne ∼ 4 stability or generating a dynamo in TCF 6, re- km/s, Ar ∼ 3.2 km/s, Xe ∼ 1.4 km/s), that quires a thorough understanding of the impor- scale with a critical ionization velocity limit re- tant dynamical effects and thresholds for hydro- ported to occur in partially ionized plasmas. Fi- dynamic instability. nally, a hydrodynamic linear stability analysis This article reports on a number of recent ad- is performed to estimate the critical velocity for vancements to the experiment. Uniform, solid- the appearance of Taylor vortices at current ex- body rotation spanning the entire height of perimental geometry and parameters. 3

(a) (c) (b) 10 Swept Langmuir 9.5 OES

9 (d) 8.5 8 Te [eV] 7.5

6.5

6 0.15 0.2 0.25 0.3 0.35 (e) Radius [m]

FIG. 3. Electron temperature vs. radius measured by the swept Langmuir probe and Optical Emis- sion Spectroscopy suggest uniform temperature in the bulk, unmagnetized region (0.1 < r < 0.3 m). Measurements were taken in a ﬂowing argon plasma with 3 km/s solid body rotation (similar to Fig. 6e). FIG. 2. Newly installed center core assembly of Error bars are standard error of time-averaged mea- magnets and stirring electrodes. surements.

II. DESCRIPTION OF EXPERIMENT These emission rings are absent at locations where center post electrodes are installed due In PCX, plasma is confined by a cylindrical to increased loss area. “bucket” assembly of permanent magnets, ar- Electron temperature is routinely measured ranged in rings of alternating polarity, to form with single-tip swept or triple-tip Langmuir an axisymmetric cusp magnetic field7. The field probes and has been confirmed using Optical is localized to the boundaries, leaving a large, Emission Spectroscopy (OES), a non-invasive unmagnetized plasma in the bulk, with aspect diagnostic which uses emission line ratios to de- < termine the electron temperature. The OES ratio Γ = height/(R2 − R1) ∼ 3. Plasma is produced with 2.45 GHz microwave heating to method9 has previously been developed by col- 10 11 −3 reach Te < 10 eV and ne = 10 -10 cm . laborators on an inductively coupled argon The waveguide launches mostly O-mode, im- plasma used in plasma processing with 1 < 10 plying a cutoff density limit at ne = 7.4 × 10 Te < 6 eV , but this is the first application to cm−3. Similar to other microwave powered mul- plasma with both higher electron temperature ticusp plasmas8, the PCX plasma is observed to (up to 15 eV) and ionization fraction (30%). have two distinct states. In the overdense state, The OES electron temperature measurements the plasma is produced by a surface wave dis- in PCX (Fig. 3) agree within 15% of Langmuir charge, exhibiting bright emission from the cen- probe measurements10. tral, field-free region of the plasma, and cusp The plasma is stirred using J × B forces in losses can clearly be seen. When the plasma the magnetized edge region, where current is is underdense, bright rings of emission are ob- driven by toroidally localized, electrostatically served, especially around the center core mag- biased hot cathodes. Viscosity couples the az- nets near the ECH resonant zone (Fig. 2e), indi- imuthal flow inward to the unmagnetized re- cating resonant electron heating and trapping. gion, and the flow is optimized at low den- 4

[km/s] [km/s] 0.6 3 0.6 3

2.5 2.5 0.5 0.5 2 2 0.4 0.4 1.5 1.5 z [m] z [m] 0.3 0.3 1 1

0.2 0.2 0.5 0.5

0.1 0 0.1 0 0.3 0.35 0.4 0.3 0.35 0.4 r [m] r [m] (a) (b)

FIG. 4. Argon azimuthal velocity measured with vertically scanned, 4-tipped Mach probe, with 450 V for ∼ 1 A from each biased cathode with a) two biased cathodes vs. b) four biased cathodes. Triangles mark the location of magnets, anodes (solid lines) and cathodes (stars) are located around r = 0.37 m. sity (where viscosity is largest) and a low neu- cosity of inviscid fluids such as water or liquid tral pressure (where neutral drag is minimized). metals12 is ν ∼ 10−6 − 10−7 m2/s. The plasma Mach probes measure flow using the ratio of up- in PCX is more comparable to molasses, which stream to downstream ion saturation current, has ν ∼ 3 − 7 m2/s. + − j /j = exp(KV/cs), with K = 1.34. Mea- surements of the flow profiles match the profiles predicted by a model that includes Bragin- III. ADDITION OF INNER BOUNDARY FLOW skii viscosity and momentum loss through ion- DRIVE neutral charge exchange collisions11. Rotation drive at the outer boundary has In order to control the shear flow necessary been extended to cover the entire vertical height to excite flow-driven instabilities, rotation must of the plasma. In Fig. 4, azimuthal veloc- also be induced at the inner boundary. There- ity measurements were made with a vertically fore, a center core assembly of magnets and scanned, 4-tipped Mach probe in argon. The stirring electrodes (pictured in Fig. 2) was in- plasma parameters (measured by the triple stalled, consisting of 22 radially-magnetized (Br probe) were similar for each case, with ne = = 2.3 kG), NeBFe magnet rings (OD=5.5 cm), 10 − 3 × 10 cm 3, fion = 10%, Te = 6.5 eV, and stacked in alternating polarity and separated by Ti = 0.1 eV (from fitting the velocity profile). aluminum clamps. Each inner cathode has a If all four outer cathodes are biased, the en- total surface area of 1.6 cm2, constructed from tire column of plasma spins. If only the two loops of 0.038 cm diameter thoriated tungsten central cathodes are biased, rotation is induced wire. Filaments are mounted by press-fit con- exclusively at the midplane and exponentially nections in double bore alumina tube with cop- decreases away from the midplane. Vertical per leads, and each cathode plugs in to a recep- shear is expected, since the kinematic viscosity tacle mounted on an aluminum clamp. Wiring of plasma is relatively high, ν ∼ 3 m2/s, and was gauged to ensure that the filaments are the Remax < 400. For reference, the kinematic vis- most resistive part of the circuit. Six out of the 5

5 (a) P [Torr] ECH [kW] 4 t=3.6 s (c) 4 t=0.6 s ( x 10

2.6 ) [rad/s] 3 4 2

Inner Cathode [kW] -4 kW ) Torr ( x10 2 Outer 1.6 0 1 Cathode [kW] (d) 10 0

[km/s] 6 Te [eV] (b) 14

(x 10 2 8 10 10 /s] 4 (e) 2

) cm eV

6 ) [m 4 3 n[cm ] -3 2 ( x10 2 0 0 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 Time [s] Radius [m]

FIG. 5. a) Plasma is created with a neutral gas puff and ECH heating. Rotation is induced with four outer cathodes (each 350 V, ∼ 1.1 A) and five inner cathodes (each 575 V, ∼ 0.75 A ). b) Plasma density and electron temperature vs. time, measured with the triple probe. c) Angular velocity profile at two different times during plasma discharge (times marked by dashed line in (a) and (b)), along with corresponding azimuthal velocity (d) and angular momentum (e) profiles. Error bars are standard deviation of fluctuation in time. possible ten cathodes are installed; additional ∼ 120◦C due to the radiant heat flux from fila- cathodes would require modification to both the ments. Therefore, the inside of the clear quartz center post and electrical vacuum feedthroughs. tube was coated with silver to act as a heat The inner anodes are 2 cm long cylinders of 0.64 shield, resulting in acceptable magnet temper- cm diameter molybdenum, mounted on molyb- atures of < 60◦C during operation. Over time, denum threaded rod insulated by quartz tubes. the quartz tube becomes coated with a conduc- All inner anodes are mounted to a copper bus tor, e.g. tungsten evaporating from hot fila- bar and biased together. Cathodes are ohmi- ments. In order to facilitate efficient cleaning cally heated to temperatures around 1700◦C and filament replacement, a gate valve system using variable-voltage AC power supplies and has been installed to isolate and remove the cen- isolation transformers. The cathodes are neg- ter core while preserving vacuum in the main atively biased up to 600 V with respect to the chamber (see Fig. 1). cold anodes, and the entire electrode assembly is floating relative to ground (the chamber wall). In general, filaments most often fail in the The inner and outer electrode assemblies are overdense state of the plasma discharge (ne > biased independently with separate DC power 7 × 1010 cm−3), during which large spikes in supplies. currents between biased cathodes and adjacent anodes are often observed (see Fig. 5a , t=0.8 The center post magnets were initially cov- s). The inner cathodes were particularly vul- ered by a clear quartz tube (OD=6.02 cm). nerable and tended to be destroyed by a single The aluminum center rod is water cooled to “arc” event. It is not clear if this was caused prevent magnets from overheating, but mag- by shorting across the face of the magnets due net surfaces still reached high temperatures of to contamination by deposition of conducting 6 material, or if it was a property of the cath- optimized. The radial location for the center odes themselves. One possibility is that higher post stirring electrodes (r ∼ 7 cm) was found plasma density could cause increased and possi- through a series of trial-and-error experiments bly non-uniform heating through ion bombard- to test cathodes of various lengths. ment on the tungsten filaments, leading to localized hot spots and sudden bursts of emission current. Similar effects might occur because the IV. MEASUREMENTS OF coil shaped filaments could pick-up strong E- ROTATION-INDUCED RADIAL ELECTRIC fields from microwaves in the edge region. FIELD Cathode behavior might be better under- stood through further studies of the edge region In a strongly rotating plasma, where the mag- in overdense plasmas. Perhaps this problem can nitude of the toroidal velocity of the plasma be solved by using a different cathode mate- is comparable to the thermal velocity, the cen- rial and design, such as the LaB6 cathode de- trifugal force causes ions to move to larger radii. sign currently being tested on the recently con- As with any gas centrifuge, the outward dis- structed, much larger Madison Plasma Dynamo placement of ions creates a pressure gradient, Experiment13. resulting in an inward force to balance the out- Basic characteristics of the microwave- ward centrifugal acceleration. In a rotating heated, differentially rotating plasma are pre- plasma, however, a radial electric field arises to sented in Fig. 5. Plasma is created using si- prevent charge separation from the difference in 14 multaneous gas puff and microwaves, and cath- centrifugal forces . The ions are thus held in ode bias is applied to both inner and outer orbit by both the ion pressure gradient force and cathodes. Density and electron temperature the electric field as determined by the electron are measured with a triple probe. Azimuthal pressure. rotation profiles are measured with a radially In the case of an unmagnetized flow with scanned Mach probe at the midplane, and pro- uniform temperature, the radial force balance files are fitted with the Bessel function solution equation for ions can be written: 11 based on measured parameters . In the begin- 2 Vφ(r) dΦ dni ning of the shot, the plasma is overdense with − niMi = −Zeni − Ti (3) −4 11 −3 r dr dr P = 2.8 × 10 Torr, ne = 1.2 × 10 cm , Te = 6.8 eV, and Ti = 1.1 eV (found from the where Φ is the electrostatic potential. Neglect- fitted velocity profile), resulting in P m = 12 ing the small centrifugal force on the electrons, and Rm = 30 with peak flow speeds of V1 = 2.3 the electrons simply obey the Boltzmann re- km/s and V = 5.7 km/s at the inner and outer 2 lation, ne = n0exp(eΦ/Te). Since the elec- boundaries, respectively. Later in the shot, the tron density must match the ion density to −4 plasma is underdense with P = 1.1 × 10 maintain quasineutrality, a radial electric field 10 −3 Torr, ne = 2.3 × 10 cm , Te = 7.5 eV, and arises. The resulting rotation-induced, mass- Ti = 0.3 eV. At these parameters, peak flow dependent plasma potential profile is speeds reached V1 = 3.4 km/s and V2 = 12.3 Z R2 2 km/s, with P m = 2.5 and Rm = 65. 1 Vφ(r) Φ(r) = Mi dr (4) Maximum flow in the bulk was obtained when Ti e(Z + ) R r the neutral drag is minimized (at low neutral Te 1 pressure and high ionization fraction) and ion where the inner and outer radii are denoted by viscosity ν is maximized (at low density, since R1 and R2. 5/2 −1 −1/2 ν ∝ Ti ni µ for unmagnetized plasma). The PCX rotation drive scheme is unique in For the J × B flow drive to be effective, the that it does not require external radial electric current has to be large enough and the elec- fields applied to the bulk, in contrast to most trode position in the magnetized edge must be homopolar-type devices which create E × B 7

Helium Neon Argon 8 3.5 Shot=656 t=4 s t=3.9 s 4 3 Shot=650 t=0.8 s t=1.9 s 6 2.5 3 2 4 [km/s] [km/s] [km/s] φ φ 2 φ 1.5 V V V 1 2 1 0.5 (a) (c) (e) 0 0 0

1 0.6 0.6 0.8 0.4 0.6

[ V ] 0.4 Φ [ V ] 0.2 [ V ] 0.4 Φ Φ 0.2 0 0.2

0 0 −0.2 (b) −0.2 (d) (f) −0.4 −0.2 0.2 0.25 0.3 0.2 0.25 0.3 0.2 0.25 0.3 Radius [m] Radius [m] Radius [m]

FIG. 6. Floating potential profiles (b,d,f) are measured for various flow profiles (a,c,e) generated in helium, neon, and argon. As the flow velocity increases, the radial floating potential increases as predicted by Eq. 4.

flow. It would therefore be expected that a ra- however, the measurements tended to be very dial electric field appearing in the PCX plasma noisy (|Φfl| < 10 V, with ±2 V fluctuations) would be self-generated. The thermal diffusion and better results were obtained by simply mea- gradient scale-length is much larger than the suring the potential profile and comparing to machine size and much smaller than the particle Eq. 4. Over the course of many plasma pulses, diffusion gradient scale-length in PCX, justify- a number of measurements were recorded at ing the isothermal model assumed in Eq. 4. In each radial location, and the floating poten- the bulk, unmagnetized region away from any tial was then taken as the mean value between possible sources of electron heating (ECH res- multiple shots. Efforts were made to main- onance, joule heating due to biased cathodes), tain consistent plasma conditions, but the bulk the electrons undergo multiple bounces, or re- plasma potential still varied slightly from shot- flections by the mirror force at the cusps, re- to-shot. To adjust for these shot-to-shot differ- sulting in uniform distribution in the bulk. Uni- ences, the floating potential was simultaneously form electron temperatures in the bulk, unmag- measured by the fixed triple probe and sub- netized region (0.1 m < r < 0.3 m) in both tracted from the rake probe floating potential rotating and non-rotating plasmas have been measurements. In order to extract the mean- confirmed with both a radially scanned, swept ingful potential data (which is DC in character), Langmuir probe and OES, as shown in Fig. 3. a first order Savitzky-Golay smoothing routine (sgolayfilt in Matlab) has been applied to reduce Potential profile measurements were obtained the magnitude of the time-varying fluctuations with a rake probe consisting of a single Mach and remove large voltage spikes from the data, probe along with multiple radially-spaced tips which are often encountered in discharges with to measure floating potential. Assuming con- strongly biased cathodes. stant Te,Φfl and Φp vary by a constant sheath potential Φsh, and Φfl can be substituted into Measurements of the radial floating potential Eq. 4 by absorbing Φsh into the integra- show the presence of an outward rising potential tion constant15. The rake probe measurements which only arises when the plasma is flowing. could be used to derive the radial electric field, The velocity and corresponding floating poten- 8

tial measurements in helium, neon, and argon He (a) Ne are shown in Fig. 6. The Mach probe velocity 6 Ar Xe measurements are ﬁtted with the Bessel func- 4 [km/s] φ

tion model based on the the measured plasma V parameters for each dataset. The floating po- 2 tential measurements are fitted with the pre- (b) dicted potential according to Eq. 4, where the 0.6 velocity profile is the Bessel function fit of the 0.4 [ V ]

Mach probe velocity data. Φ 0.2 According to Fig. 6, the measured radial in- 0 −0.2 crease in potential is not much more than 0.8 (c) 2 V, and the corresponding density displacement 10 is 10 − 20% of the core density, according to 1 ne/no = exp(eΦ/Te). A density gradient has 10 not yet been observed, because the interpreta- Derived a.m.u. 0 tion of probe measurements based on collection 10 0.15 0.2 0.25 0.3 0.35 of ion saturation current in a rotating plasma Radius [m] were ambiguous, and shot-to-shot variation in the measurement also tends to be on the order of 10 − 20%. FIG. 7. The measured values of velocity and potentials in four diﬀerent gases (helium, neon, argon, and xenon) result in derived ion masses (based on Eq. 5 with Z=1) which are close to the expected values (dashed lines).

A. Mass Dependence V. VELOCITY LIMITS The best evidence for mass-dependent, outward displacement of spinning plasma in PCX The velocities measured in the PCX exper- is found by comparing the profiles for flow and iment have increased through the optimiza- potential of various gases. One can solve for tion of electrode locations and realization that the ion mass using the measured potential and plasma conditions are optimized at low density fitted velocity profiles: and high ionization fraction. It seems, however, that as long as the plasma remains partially ionized, velocity limits do exist and are consis- eΦ(r)(Z + Ti ) Te tent with the species-dependent critical ioniza- Mi = 2 (5) R R2 Vφ(r) tion velocity first proposed by Alfvénin 1954. R r dr 1 Alfvénhypothesized that if the ion flow kinetic energy exceeds the ionization potential of the In Fig. 7, experiments with four different gases neutrals, the ionization rate of a neutral gas (helium, neon, argon, and xenon) were per- should strongly increase. The critical ionization formed. Since the centrifugal force is stronger velocity (CIV) is reached when the relative ve- on heavier ions, lower rotation speeds are re- locity between the plasma ions and neutral gas quired to obtain radial potentials which are reaches the value equivalent to the faster rotating, lighter species. The measured values of the potential and veloc- r ity are used to calculate the ion mass to within 2eUi Vcrit = (6) a factor of two for the distinct species. mn 9 where Ui is the ionization potential and mn is 10 the mass of the neutral atom. Rather remark- Gas Ui Mi Vcrit Vmax ne × 10 fion Te Tifit (eV) (a.m.u) (km/s) (km/s) (cm−3) (%) (eV) (eV) ably, the simple velocity limit set by Eq. 6 has He 24.6 4 34 12 2.3 0.7 6.9 0.21 been confirmed in numerous experiments with Ne 21.6 20.2 14 4 3.4 1.2 7.7 0.3 a variety of geometries, spanning orders of mag- Ar 15.8 40 8.7 3.2 3 9.5 5.9 0.135 nitude in various neutral gas, plasma, and mag- Xe 12.1 131.3 4.2 1.35 6 32 3.6 0.15 netic field conditions, as summarized in several 16,17 TABLE I. Summary of ionization energy, mass, crit- extensive review articles . Laboratory ex- ical ionization velocity, maximum induced velocities periments that have been designed especially to for different gas species in PCX as measured by study CIV generally create flowing plasma with the Mach probe. Also listed for reference are the either homopolar-type devices that implement corresponding ne, Te, and the ionization fraction cross-field discharges, or with coaxial plasma fion ≡ n/no. Note that Vmax is always less than guns to create a plasma beam colliding with Vcrit, with Vmax/Vcrit ∼ 0.3 in all cases. neutral gas. Investigations of many gas species found that the maximum E/B velocity was typ- VI. HYDRODYNAMIC STABILITY OF ically within 50% of Vcrit (as long as the plasma remained partially ionized), where the magni- TAYLOR-COUETTE FLOW OF PLASMA tude of the velocity was inferred by |E/B| or by the Doppler shift of spectral lines. In PCX, flow which is driven exclusively at the outer boundary results in nearly solid-body rotation, resembling a centrifuge. Such profiles are hydrodynamically stable because the angu- The peak velocities measured in PCX for lar momentum increases with radius , and the four different gas species (helium, neon, argon, outward centrifugal force is balanced by the in- xenon) are listed in Table I. The velocities were ward pressure gradient force in accordance with measured with a Mach probe at the midplane the Rayleigh stability criterion. As sheared flow near the outer edge of the unmagnetized re- is driven by inner boundary flow drive, the flow gion, around r = 0.34 m. Density and elec- can become hydrodynamically unstable above tron temperature were measured with both the a certain critical threshold. The fluid stability single swept Langmuir probe and triple probe. problem of TCF can be formulated using tra- In all cases, the peak speeds were always less ditional linear stability methods2, except with than the calculated Vcrit for each species. While the inclusion of the plasma-specific ion-neutral Vmax < Vcrit, it is conceivable that peak ve- drag term. In the unmagnetized bulk region, locities reached higher speeds in regions of the the toroidal force balance is given by plasma that were not measured by the Mach ∂V ∇P V probe, for example directly in front of the per- + (V · ∇)V = − − + ν∇2V (7) manent magnets where J × B forces are large. ∂t ρ τi0 Attempts were made to further increase the ve- where ν is viscosity due to ion-ion collisions and locity by increasing J (by increasing the fila- −1 τi0 ≈ (n0hσcxvi) is the ion-neutral collision ment temperature and bias). This resulted in time; σcx is the charge-exchange cross-section. increased density and often caused the plasma Assuming axisymmetry with a perturbed flow δP to switch from the underdense to overdense V + u = [ur,V0 + uθ, uz] andω ¯ = ρ , the state in microwave discharges, leading to unde- linearized momentum equation is: sirable filament arcing. Interestingly, the ratio ∂u V u ∂ω¯ u u r − 2 0 θ = − − r + ν ∇2u − r (8) of V /V ∼ 0.3 for all gases tested. Thus, r 2 max crit ∂t r ∂r τi0 r it seems that a velocity limit which depends on ∂u V ∂V u u θ + 0 + 0 u = − θ + ν ∇2u − θ (9) the species mass and ionization potential, con- r θ 2 ∂t r ∂r τin r sistent with CIV phenomenon, is observed in ∂uz ∂ω¯ uz 2 PCX. = − − + ν∇ uz (10) ∂t ∂z τin 10

As shown previously, TCF of partially ion- 1500 ized plasmas is modified by drag induced from α =0 plasma ion-neutral charge exchange collisions. α =2 α =5 Hence, the base flow V0 is the equilibrium ion- α Unstable 1000 =10 neutral damped plasma TCF profile given by ν L/ modified Bessel functions11 1 =V 1

V0(r) = AI1(r/Lv) + BK1(r/Lv), (11) Re 500 Stable where the constants A and B are determined by the boundary values of the velocity and L = Rayleigh Line √ v τi0ν. Suppose perturbations of the form 0 −200 −100 0 100 200 300 400 500 γt Re =V L/ν ur = e u(r)cos(kz) (12) 2 2 γt uθ = e v(r)cos(kz) (13) γt uz = e w(r)sin(kz) (14) FIG. 8. Marginal stability diagram for Taylor- γt Couette flow. As the momentum diffusion length ω¯ = e ω¯(r)cos(kz) (15) decreases, flow becomes unstable in regions which where the growth rate is γ, the system is pe- would otherwise be Rayleigh stable. riodic in the z-direction (described by the axial wavenumber k), and the perturbation am- The results of the marginal stability analy- plitudes depend on the radial position. Assum- sis are shown in Fig. 8. Stability curves for ing incompressible flow and using the notation several values of α are plotted in the parame- d → D and d + 1 → D , we find dr dr r ∗ ter space of dimensionless fluid Reynold’s num- ν 1 γ bers of the inner and outer cylinders (Re1, Re2). DD − k2 + + × k2 ∗ L2 ν At small values of α, ion-neutral collisions are v unimportant, and the azimuthal velocity pro- 2 2V0 DD∗ − k u = v (16) file takes on the classical form for TCF of a r viscous fluid, where α = 0. In this limit, the 1 γ 2 minimum critical Re1 ∼ 330 is determined by ν DD∗ − k + 2 + v =(D∗V0)u Lv ν viscosity, and the stability curve asymptotes (17) to Rayleigh’s prediction for stability of invis- 2 2 cid fluid (Ω1R1 = Ω2R2), which is plotted as a These equations can then be normalized, dis- dashed line in Fig. 8. cretized, transformed into an eigenvalue prob- As α increases, ion-neutral collisions become lem at marginal stability (where γ = 0), and more important, and the plasma flow becomes solved numerically. The numerical calculation unstable in regions which would otherwise be is in terms of the normalized momentum diffu- Rayleigh stable in classical fluid flows. This is sion length parameter, α = L/Lv, or due to the increased shear in the velocity profile 2√ −1 created by momentum loss in ion-neutral colli- α = 153 LmZ σcx,10−19m2 ni,1011cm−3 T (18) i,eV sions. In Fig. 9, profiles of flow and d(r2Ω)/dr where σcx,10−19m2 is the charge exchange cross- are plotted for various neutral gas pressures section in units of 10−19m2. The assumed (8 × 10−6 − 2 × 10−4 Torr), which correspond axial wavenumber is k = π/(R2 − R1) with to different values of α. In this example, Re1 = PCX dimensions R1 = 0.1 m and R2 = 0.3 800 and Re2 = 300, and azimuthal velocity pro- m. The eigenvalue solver then finds the crit- files are calculated based on fixed helium pa- 10 −3 ical Reynold’s number at the inner boundary rameters L = 0.3 m, n = 4 × 10 cm , Te = 8 (Re1 = V1L/ν) for each prescribed Re2. eV, and Ti = 0.1 eV. Centrifugal instability can 11

15 is in agreement with the predicted proﬁle (a) α = 0 α = 2 calculated from ﬁtted velocity measure- 10 α = 5 ments. α =10 [km/s] φ

V 5 • The Te measurements of Langmuir probes have been independently conﬁrmed by a newly developed Optical Emission Spec- 0 (b) troscopy technique, leading to further 4 conﬁdence in the magnitude of the Mach )/dr

Ω 2 2 probe velocity measurements. Further-

d(r 0 more, the T proﬁles are ﬂat, so the

4 e −2 changes in potential profiles are an impor- x 10 −4 tant confirmation of Mach probe measure- 0.1 0.15 0.2 0.25 0.3 ments. Radius [m] • Maximum velocity limits have been observed for various gas species ( He ∼ 12 FIG. 9. Top) Representative flow profiles for var- km/s, Ne ∼ 4 km/s, Ar ∼ 3.2 km/s, Xe ∼ ious momentum diffusion lengths with Re1 = 800 1.4 km/s ), consistent with the well doc- and Re2 = 300. Bottom) Corresponding calcula- umented critical ionization velocity limit tion of d(r2Ω)/dr. Flows can be unstable in regions phenomenon, which exists in partially ion- where perturbations are not damped by viscosity and d(r2Ω)/dr < 0. ized plasma. • Flow can also be driven at the in- occur in regions where d(r2Ω)/dr < 0. Note ner boundary, where the geometry is that high neutral pressure can also increase the slightly different with smaller cathodes instability threshold for rotation driven exclu- and stronger center stack magnets. sively at the inner boundary. Thus far, shear The next step for the experiment is to expand flows in PCX have been generated at high he- 4 the parameter space by further maximizing flow lium neutral pressures (> 10 ) and lower inner speeds at the inner boundary and driving flow boundary velocity, where Re1 < 10. According at higher densities. Rotation would improve at to the marginal stability analysis, the achieved increased ionization fractions, which could be flows would be expected to be hydrodynami- accessed through additional vacuum pumping cally stable. and improving confinement, for example by re- placing the first generation of ceramic magnets on the endcaps and outer boundaries with a VII. SUMMARY second generation of the high-field, neodymium type. Spinning at higher density may also re- The demonstration of sheared flow of plasma quire improvements to cathode design. Addi- in the unmagnetized core region, driven by bi- tional microwave power and launching of X- ased electrodes in the magnetized plasma edge, mode waves (E ⊥ B), which are not restricted is a major advance for carrying out future flow- by density cutoffs, could also be explored. driven MHD experiments. A number of experimental observations have been reported here: • A self-consistent, rotation-induced radial VIII. ACKNOWLEDGMENTS electric field is measured. It only appears when there is flow, it scales with the mag- This work was funded in part by NSF award nitude of the flow and the ion mass, and it AST-1211937 and the Center for Magnetic Self 12

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