3D Two-Fluid Braginskii Simulations of the Large Plasma Device

Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

Dustin Fisher†, Barrett Rogers†, Giovanni Rossi∗, Danny Guice∗ †Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA ∗Department of Physics and Astronomy , University of California, Los Angeles, California 90095, USA

Conclusions: ? 3D global 2-ﬂuid simulations show good agreement with data from LAPD in the low-bias pa- rameter regime explored so far. ? KH turbulence at relatively large scales is the dominant driver of cross-ﬁeld transport in the low-bias simulations. ? Biased simulations are currently under study.

The work presented here builds upon an initial numerical study [Rogers and Ricci, Phys. Rev. Lett., 104, 2010] of LAPD [Gekelman et al., Rev. Sci. Instrum., 62, 1991] using the Global Braginskii Solver code (GBS) [Ricci et. al., Plasma Phys. Control. Fusion, 54, 2012].

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 1 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions LAPD Primer for a Nominal He Plasma

• Plasma 17 m in length, 30 cm in radius • Machine diameter 1m ' • n 2 1012 cm−3 ∼ × • Pulsed at 1 Hz for 10 ms ∼ • Axial magnetic ﬁeld 1kG ∼ • Te 6 eV and T 0.5 eV ∼ i ∼ • Ion sound gyroradius, ρs 1.4 cm ∼ • Plasma β 10−4 ∼ 012304-2 T. A. Carter and J. E. Maggs Phys. Plasmas 16,012304͑2009͒

Discharge Circuit generation of primary electrons with energy comparable to

− + the anode-cathode bias voltage. The mesh anode is 50% Electrical Breaks transparent, allowing half of these primaries to travel down the magnetic field into the main chamber leading to ionization and heating of the bulk plasma. The cathode is situated B at the mouth of the solenoid and therefore sits in a region of flaring magnetic field. The flared magnetic field maps the 50Ω 73 cm diameter cathode to a ϳ56 cm diameter region in the Cathode Anode main chamber. Primary electrons from the source are isolated 0.2Ω to this region in the chamber and lead to a fast electron Wall Bias Circuit − + tail for rՇ28 cm. Typical plasma parameters in LAPD 12 −3 discharges are ne Շ5ϫ10 cm , Te ϳ7 eV, Ti ϳ1 eV, and BϽ2 kG. In the experiments reported here, the primary T.A. Carter and J.E. Maggs, Physics of Plasmas, 16 (2009) FIG. 1. Schematic of LAPD including wall biasing circuit. plasma species was singly ionized helium and the magnetic http://plasma.physics.ucla.edu/pages/gallery.html (BaPSF) field strength used was 400 G. Modeling18 and spectroscopic measurements show that the ionization fraction of the LAPD Dustin M. Fisher Sherwood 2015 perform a detailedModeling study the of LAPD modifications of turbulence and2 / 21 plasma is տ50% and therefore Coulomb collisions are the turbulent transport in a system free from complications asso- most important collisional process. However, neutral colli- ciated with toroidal systems ͑e.g., field line curvature, poloi- sions ͑charge exchange in particular͒ are important for estab- dal asymmetry, trapped particles͒. Good diagnostic access to lishing the radial current during biasing. LAPD provides for detailed measurements of the spatial and Measurements of density, temperature, floating potential, temporal characteristics of the turbulence. and their fluctuations are made using Langmuir probes. A Here we summarize the primary results reported in this four-tip probe with 0.76 mm diameter tantalum tips arranged paper. Measured turbulent transport flux is reduced and then in a diamond pattern is used as a triple Langmuir probe and suppressed, leading to a confinement transition, as the ap- particle flux probe. Two tips are separated 3 mm along the plied bias is increased. The threshold in the applied bias is field and are used as a double probe to measure ion satura- linked to radial penetration of the driven azimuthal flow. tion current I ϰn ͱT . The remaining two tips are sepa- Two-dimensional measurements of the turbulent correlation ͑ sat e e͒ rated 3 mm perpendicular to the field and measure floating function show that the azimuthal correlation increases dra- potential for deriving azimuthal electric field fluctuations and matically during biasing, with the high-m-number modes in- temperature using the triple Langmuir probe method.24 Ra- volved in turbulent transport becoming spatially coherent. dial particle transport can be evaluated directly using mea- However, no significant change in the radial correlation sured density and electric field fluctuations through Eq. 1 . length is observed associated with the confinement transi- ͑ ͒ Flows are measured using a Gundestrup Mach probe with tion. As the bias is increased above threshold, there is an ͑ ͒ six faces.25 The flow measurements are corrected for the fi- apparent reversal in the particle flux ͑indicating inward trans- nite acceptance angle of the probe faces, which are smaller port͒. The peak amplitude of density and electric field fluc- than the ion gyroradius.26 tuations do decrease, but the reduction is only slight and does Using the difference in floating potential to determine not fully explain the transport flux reduction. The cross- the azimuthal electric field is problematic in the presence of phase between density and electric field fluctuations changes fast electron tails, such as exist on field lines connected to significantly as the threshold for confinement transition is the cathode source. In the presence of fast electrons, differ- reached, and explains the reduction and reversal of the mea- ences in floating potential may no longer be proportional to sured transport flux. The dynamics of the transition have also differences in plasma potential and thus may not accurately been studied and show a correlation between transport sup- measure the electric field. In addition, the fast electron tail, pression and the overlap of the flow ͑shear͒ profile and the which sets floating potential, is much less collisional than the density ͑gradient͒ profile. bulk plasma so that the floating potential is no longer a lo- calized quantity. This delocalization skews the correlation II. EXPERIMENTAL SETUP with locally measured density fluctuations and thereby af- The experiments were performed in the upgraded Large fects the flux measurement. Measurements of Isat are made Plasma Device ͑LAPD͒,21 which is part of the Basic Plasma with a double-Langmuir probe biased to 70 V to reject pri- Science Facility ͑BaPSF͒ at UCLA. The vacuum chamber of mary electrons, and therefore should not be affected by the LAPD is 18 m long and 1 m in diameter and is surrounded fast electron tail. Thus, in this study, we report on the prop- by solenoidal magnetic field coils. The plasma is generated erties of ion saturation fluctuations everywhere in the plasma by a cathode discharge.22 The cathode is 73 cm in diameter column. Properties of electric field fluctuations and cross- and is located at one end of the vacuum chamber, as shown correlation flux measurements are presented everywhere in schematically in Fig. 1. A molybdenum mesh anode is situ- the plasma column, but a caution is issued in regard to mea- ated 50 cm away from the cathode. A bias of 40–60 V is surements made on field lines where primary electrons are applied between the cathode and anode using a solid-state present ͑rՇ28 cm͒. switch,23 resulting in 3–6 kA of discharge current. An im- The edge plasma in LAPD is rotated through biasing the portant aspect of the operation of the plasma source is the vacuum vessel wall positively with respect to the source

Downloaded 14 Jun 2010 to 129.170.241.133. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Standard Bohm Sheath B.C.’s at the End Walls

Vki = ±cs (1) 0 ¨¨* Vke = ±cs exp([Λ − e{φplasma − ¨φwall}/Te ]) (2) with p p cs = Te0/mi ; Λ = ln mi / (2πme ) ∼ 3 (3)

The B.C.’s on the outﬂows of ions and electrons to the end walls lead to an approximate global balance Vki ∼ Vke , or

φplasma ∼ Λ Te (4)

Relaxation due to turbulence

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 3 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

The code evolves a set of electrostatic two-ﬂuid drift-reduced Braginskii

equations assuming Ti Te :

dn ∂ nV e = − k + Dn (n) + Sn (5) dt ∂z dTe ∂Te 2 Te ∂j = −V e + 0.71 k dt k ∂z 3 en ∂z 2 ∂V e − Te k + Dk (Te ) + DT (Te ) + ST (6) 3 ∂z Te e 2 dV e ∂V e 4 η0e ∂ V e ej me k = −me V e k + k + k dt k ∂z 3 n ∂z2 σ k ∂φ T ∂n ∂T + e − e − 1.71 e (7) ∂z n ∂z ∂z 2 dV i ∂V i 4 η0i ∂ V i 1 ∂pe mi k = −mi V i k + k − (8) dt k ∂z 3 n ∂z2 n ∂z 2 dω ∂ω mi Ωci ∂j = −V i + k − νinω + Sω (9) dt k ∂z e2n ∂z

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 4 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Evolution of Turbulence and Transport

t = 0.25 ms t = 0.40 ms • Starting top-hat 0.5 0.54

shaped density 0.48

source. 0.0 0.42 y (m) • Onset of drift waves 0.36 a) b) 0.5 with kθρs ∼ 0.5, − 0.30 t = 1.13 ms t = 3.33 ms kkLeq ∼ 1. 0.5 0.24

• Onset of KH from 0.18

sheared ﬂow with 0.0 0.12 y (m) kk ∼ 0. 0.06 c) d) • Steady-state region 0.5 0.00 − 0.5 0.0 0.5 0.5 0.0 0.5 − − where turbulence has x (m) x (m) reached saturation. Plots show mid-plane cuts perpendicular to B.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 5 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Evolution of Turbulence and Transport

t = 0.25 ms t = 0.40 ms • Starting top-hat 0.5 0.54

shaped density 0.48

source. 0.0 0.42 y (m) • Onset of drift waves 0.36 a) b) 0.5 with kθρs ∼ 0.5, − 0.30 t = 1.13 ms t = 3.33 ms kkLeq ∼ 1. 0.5 0.24

• Onset of KH from 0.18

t = 0.25 ms t = 0.40 ms • Starting top-hat 0.5 0.54

shaped density 0.48

source. 0.0 0.42 y (m) • Onset of drift waves 0.36 a) b) 0.5 with kθρs ∼ 0.5, − 0.30 t = 1.13 ms t = 3.33 ms kkLeq ∼ 1. 0.5 0.24

• Onset of KH from 0.18

t = 0.25 ms t = 0.40 ms • Starting top-hat 0.5 0.54

shaped density 0.48

source. 0.0 0.42 y (m) • Onset of drift waves 0.36 a) b) 0.5 with kθρs ∼ 0.5, − 0.30 t = 1.13 ms t = 3.33 ms kkLeq ∼ 1. 0.5 0.24

• Onset of KH from 0.18

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 5 / 21 t = 0.25 ms t = 0.40 ms 0.5 0.54

0.48

0.0 0.42 y (m) 0.36 a) b) 0.5 − 0.30 t = 1.13 ms t = 3.33 ms 0.5 0.24

0.18

0.0 0.12 y (m) 0.06 c) d) 0.5 0.00 − 0.5 0.0 0.5 0.5 0.0 0.5 − − x (m) x (m)

Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

Fluctuations Predominantly kk = 0

Top Starting Density at t = 0.04 ms 0.50 0.072 0.068 exponential 0.25 0.064 0.00 0.060 0.25 0.056 source y (m) −0.50 0.052 − 0.048 8 6 4 2 0 2 4 6 8 dependence. − − − − z (m)

Density t = 3.44 ms 0.48 0.50 0.42 Bottom Proﬁle modiﬁed 0.25 0.36 0.30 0.00 0.24 by turbulence. 0.25 0.18 y (m) −0.50 0.12 − 0.06 8 6 4 2 0 2 4 6 8 − − − − z (m) Plots show mid-plane cuts parallel to B.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 6 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

Fluctuations Predominantly kk = 0

Top Starting Density at t = 0.04 ms 0.50 0.072 0.068 exponential 0.25 0.064 t = 0.25 ms t = 0.40 ms 0.00 0.060 0.5 0.25 0.056 source 0.54 y (m) −0.50 0.052 − 0.048 8 6 4 2 0 2 4 6 8 dependence. 0.48 − − − − z (m) 0.0 0.42 y (m) Density t = 3.44 ms 0.36 0.48 0.50 0.42 a) Bottom b)Proﬁle modiﬁed 0.25 0.36 0.5 0.30 0.30 0.00 0.24 − by turbulence. 0.25 0.18 y (m) 0.12 t = 1.13 ms t = 3.33 ms −0.50 0.06 0.5 0.24 − 8 6 4 2 0 2 4 6 8 − − − − 0.18 z (m)

0.0 0.12 Plots show mid-plane cuts parallel to B. y (m) 0.06 c) d) 0.5 0.00 − 0.5 0.0 0.5 0.5 0.0 0.5 − Dustin M.− Fisher Sherwood 2015 Modeling the LAPD 6 / 21 x (m) x (m) Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Low Bias with Intrinsic Rotation due to Sheath B.C.’s

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 7 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

12 φ 10 ΛT 8

4 Amplitude (V) 2

0 0.6 0.4 0.2 0.0 0.2 0.4 0.6 − − − r (m)

Potential proﬁle that forms self-consistently from the temperature proﬁle and the boundary sheath conditions. Here Λ = 3.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 8 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

Time-averaged density proﬁles 1.0 t=0.02–0.43 ms t=0.85–1.27 ms 0.8 t=1.93–2.34 ms LAPD 0.6

0.4

0.2 Normalized Amplitude 0.0 0.0 0.1 0.2 0.3 0.4 0.5 r (m)

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 9 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions CCD Camera Comparisons

Luminosity data from a Phantom camera looking down the length of the LAPD. LAPD Simulation

Luminosity Luminosity Fluctuations n δn 120 0.5 0.50 0.20 100 0.40 0.10 80 0.30 60 0.0 0.00 y-pixel

y (m) 0.20 40 -0.10 20 0.10 0.5 -0.20 0 − 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0.5 0.0 0.5 0.5 0.0 0.5 x-pixel x-pixel − − x (m) x (m) • Presence of density holes inside the cathode edge and blobs outside. • Scale size of visible density ﬂuctuations comparable to simulations.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 10 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

20 − 40 − 60 − 60 40 20 0 20 40 60 − − −

28 28

26 26

24 24

22 22 y (cm) 20 20

18 LAPD 18 GBS

6 4 2 0 2 4 6 6 4 2 0 2 4 6 − − − − − − x (cm) x (cm)

LAPD Filtered, mean-subtracted luminosity Simulation Line-averaged density ﬂuctuations from data from LAPD using a periscopic GBS zoomed to the same window size mirror arrangement. as the CCD data.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 11 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

Max shear ﬂow 0.14 0.12 0.10 | 0.08 max 0.06 δn/n | 0.04 LAPD 0.02 GBS GBS with νin 0.00 0.0 0.1 0.2 0.3 0.4 0.5 r (m)

• Peak ﬂuctuations occur where the shear ﬂow is greatest.

• Theorized νin leads to modest stabilizing eﬀect on KH modes.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 12 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

100

1 10−

2 10− LAPD GBS GBS with ν Power Spectrum (a.u.) in 3 10− 4 6 8 10 12 14 16 18 20 f (kHz)

• Exponential spectra consistent with intermittent turbulent structures (Pace, Shi, Maggs, Morales, and Carter, Phys. Rev. Lett., 101, 2008).

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 13 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

100

1 10− PDF 2 10− LAPD GBS GBS with νin 3 10− 3 2 1 0 1 2 3 − − − δn/RMS(δn)

Probability distribution function of density ﬂuctuations averaged over the interval 22 cm r 28 cm. ≤ ≤

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 14 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

Intermittent turbulence can spotted by its eﬀect on the third standardized moment of the distribution function know as the skewness:

1.5 1 P δn3 Skewness = N N LAPD 1 P 23/2 1.0 GBS N N δn δn GBS with νin 0.5 Blobs 0.0

Skewness of 0.5 Holes − 1.0 − 0.15 0.20 0.25 0.30 0.35 0.40 0.45 r (m)

Negative skewness indicates a left tail in the PDF which is linked to the presence of density holes. Positive skewness indicates the PDF is skewed to the right by a density tail which may signal the presence of density blobs.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 15 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

2D cross-ﬁeld correlation function of the density ﬂuctuations referenced to a point near the cathode. edge

A solid line marks the correlation at 0.5 below the maximum value to give a correlation length of 5.5cm consistent with KH scale lengths. ' Dustin M. Fisher Sherwood 2015 Modeling the LAPD 16 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions

1020 2.0 × LAPD Primary electrons present GBS 1.5 GBS with νin ) /s 2

− 1.0 ? m

Γ( 0.5

0.0 0.15 0.20 0.25 0.30 0.35 0.40 0.45 r (m)

• Data inside cathode edge unreliable due to fast electrons.

• Theorized νin value drops transport by a factor of 2.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 17 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Biasing in LAPD

055907-3 Schaffner et al. Phys. Plasmas 20, 055907 (2013)

steady-state azimuthal flow, Vh, is determined through the radial derivative of plasma potential profiles measured using a swept-Langmuir probe technique again assuming only E B flow. The shearing rate in a cylinder has been defined by many authors16,27 as

@ V @V V c ¼ r h ¼ h h : (7) s @r r @r r

This definition results in zero shearing rate for rigid [Schaffner et. al., Phys. Plasmas, 20, 2013] body rotation in a cylinder. In these experiments, the flow profiles are far from rigid body and measurements are performed at large radius (r 30 cm). Therefore the Vh=r term • Original biasing of LAPD was done is very small and a local definition of shearing rate is reason- by biasing the chamber walls relative 4,6,14,15 able and has been used in other experimental work. to the cathode [Maggs, Carter, and Therefore, the shearing rate definition adopted for the data Taylor, Phys. Plasmas, 14, 2007]. presented here is computed as cs ¼ @Vh=@r. The shearing rate is normalized using the autocorrela- • Recently, a biasable limiter was used tion time of the turbulence sac. This is determined by finding for continuous variation of the shear the width of the autocorrelation function of the density time flow by David Schaffner and series at zero shearing rate. Spectral calculations (FFT) are performed using a time window of 3.2 ms (the flat-top of the colleagues. bias-driven flow phase). The plasma potential is determined from Langmuir probe sweeps 2.5 ms long during the flat-top period.Dustin M. Fisher Sherwood 2015 Modeling the LAPD 18 / 21 A large annular aluminum limiter was installed in LAPD to provide a parallel boundary condition for the edge plasma and is biased relative to the cathode of the plasma source to control plasma potential and cross-field flow. A FIG. 1. (a) Diagram of the LAPD device showing relative location of the an- diagram of the limiter arrangement and biasing circuit is nular limiter and basic biasing setup. (b) Velocity profiles using plasma potential from swept measurements. (c) Flow at the limiter edge (black, triangles) shown in Fig. 1(a). and mean shearing rate, averaged over 27 < r < 31 cm (red, circles). A recent experiment on the LAPD demonstrated the ability to achieve continuous control of steady-state azimuthal flow and flow shear through the use of these biasable limiters.8 Spontaneous rotation is observed in LAPD in the ion diamagnetic drift direction (IDD). This spontaneous flow can be reduced and reversed into the electron diamagnetic drift direction (EDD) as the limiter bias is increased. This results in a continuous variation of edge flow and flow shear including zero flow and flow shear states. Shearing rates are achieved up to about five times the turbulent inverse autocor- 1 relation time or decorrelation time (sac ) as measured in the state with minimum shearing rate. Radial particle flux and fluctuation amplitude are reduced as shearing rate is increased and the resulting transport changes cause observ- able steepening of the density gradient. Examples of time series of density and radial velocity measurements for the minimum flow shear state and a high flow shear state are shown in Figure 2 specifically for r ¼ 29 cm, in the middle of the shear layer. Comparison of the curves shows a decrease in fluctuations throughout the entire time averaging region. Profiles made from these time- averaged quantities are shown in Figure 3 for density fluctuations, radial velocity fluctuations, particle flux, and shearing rate. A minimum shearing rate, intermediate shearing rate, FIG. 2. (a) Single shot (not averaged) time series of density fluctuations and high shearing rate state are shown and with suppression from a Langmuir probe at 29 cm. The black curve shows the trace for the of these measurements clearly visible especially within the minimum shearing rate state while the red is for a high flow shear state. The 3.2 ms shown is the temporal averaging region. (b) Time series for radial ve- spatial averaging region indicated by the grey dashed lines locity fluctuations (mean subtracted) taken using two vertically spaced float- (27 cm to 31 cm). The averaging region was chosen to avoid ing potential probe tips also at 29 cm.

This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.170.26.119 On: Mon, 25 Nov 2013 18:43:47 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Increasing the Shear Flow

LAPD Biased Run m 6 mode ∼

5 φ φw φw + (Λ ln(G)) Te −ˆ 4 G = 1 + Sω ΛTe

2 Normalized Units 1

0 0 10 20 30 40 50

x (ρs0) Dustin M. Fisher Sherwood 2015 Modeling the LAPD 19 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Reducing/Nulling/Reversing the Shear Flow

Density with Decreased Shear Flow

Density with 2.0 φ Increased Shear Flow 1.5 φw φw + (Λ ln(G)) Te Current work focuses on −ˆ 1.0 G = 1 + Sω ΛTe completely nulling the 0.5 shear flow to explore the 0.0 0.5 effects drift wave modes − 1.0 have on turbulence and Normalized Units − 1.5 − transport without KH 2.0 − from shear flow. 2.5 − 0 10 20 30 40 50

x (ρs0) Dustin M. Fisher Sherwood 2015 Modeling the LAPD 20 / 21 Large Plasma Device (LAPD) Model Equations and Numerical Setup Evolution of Turbulence and Transport LAPD Comparisons Conclusions Conclusions

• Overall good agreement between the simulations and LAPD data.

• Sheath boundary conditions lead to φplasma ΛTe + φwall. ∼ • Large shear flow destabilizes KH which appears to be the dominant driver of turbulence and transport in the unbiased case. Pressure gradients also destabilize small scale drift waves. • Ion-neutral collisions have a modest stabilizing effect on the KH modes, and reduce the radial transport by approximately a factor of two for the theoretically predicated ion-neutral collision frequency. • Biasing can increase, null, or reverse shear flow in the plasma. The physics of these biased runs are currently being studied.

Our next step is to null the shear ﬂow to study driftwaves in the absence of KH driven shear ﬂows.

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 21 / 21 Vorticity source, Sω Cross-ﬁeld ion-neutral collisions via a Petersen conductivity term Modiﬁcation of Bohm sheath factor due to vorticity source

To add a source to the vorticity equation, one assumes a small electron source term representing the primary electronics coming from the hot cathode. ∂n i + ∇ · n v + v + v + v b = S (10) ∂t E×B di pol ki i ∂n e + ∇ · n v + v + v b = S + S (11) ∂t E×B de ke e e,f Subtracting (11) from (10) and assuming quasi-neutrality gives a current continuity equation with a small source term that can be physically thought of as relating to the discharge current each time the plasma is pulsed.

∂ j ∇ · [n (v − v )] + ∇ · (nv ) + k = −S (12) di de pol ∂z e e,f

Using the Boussinesq approximation and neglecting magnetic curvature terms

nc d 2 ∇ · (nvpol) ' − ∇⊥φ (13) Bωci dt

2 the vorticity equation with ω = ∇⊥φ can be written

dω m ω2 ∂ j = i ci ∇ · (n (v − v )) + k + S (14) dt en di de ∂z e ω

The Pedersen conductivity can be written as

2 (1 + κ) νe σ1 = σ0 (15) 2 2 2 (1 + κ) νe + ωce where ne2 σ0 = (16) me νe ω ω κ = ce ci (17) νe νin νe = νen + νei (18)

and the collision frequencies for the electrons with neutrals, νen, the electrons with ions, νei , and the ions with neutrals, νin are all known from theory or experiment.

In the limit where νin/ωci 1 (valid for LAPD estimates of −3 νin/ωci ∼ 2 × 10 ) the Pedersen conductivity term can be written as 2 ne νin σ1 = 2 (19) mi ωci

Ohm’s law dictates that J⊥ = −σ1∇φ so that the perpendicular component of current in the current continuity equation becomes:

∇ · J⊥ = ∇ · (−σ1∇φ) (20) 2 ne νin = − 2 ω (21) mi ωci

where it’s assumed that σ1 is not spatially dependent as estimated for LAPD [Maggs, Carter, and Taylor, Phys. Plasmas, 14, 2007].

To prevent a buildup of charge in the plasma and maintain quasi-neutrality Jkcathode = Jkw1 + Jkw2 (22)

where Jkcathode is the discharge current into the source and Jkw1,w2 are the currents out of the near and far walls. Balancing these terms, Eq. (1) and Eq. (2) can be written Jkcathode e Jˆ≡ = 2− exp Λ − (φse − φw1) (23) qnse cse Te e × 1 + exp (φw2 − φw1) Te

where nse is the plasma density at the sheath edge, cse is the sound speed at the sheath edge at which ions are assumed to enter, φse is the plasma potential at the sheath edge, and φw1 and φw2 are the near and far wall potentials.

When φw1=φw2 the exponential factor on the far RHS goes to unity. When φw1 > φw2 the exponential becomes negligible and vanishes. Solving for the plasma potential at the sheath edge one can show T 1 φ − φ = e Λ − ln 2 − Jˆ (24) se w1 e f

where f = 1 when φw1 > φw2 and f = 2 when φw1 = φw2 and Jˆ< 0 since it’s modeling an electron beam. Thus a vorticity source, Sω which acts as a source of current in the current continuity equation, also eﬀectively shifts the Bohm sheath factor to a lower value when setting the plasma potential.

A modest correction to this calculated potential can likewise be made with the inclusion of ion-neutral collisions. Thus in solving the vorticity equation for a perturbed potential φ = φ0 + φ1, with φ1 φ0,

0 eφ = Λ Te (25)

where

Λ0 = Λ − ln (G) (26)

and Sω νin 2 G ∼ 1 + − ∇⊥φ0 . (27) cse cse

Dustin M. Fisher Sherwood 2015 Modeling the LAPD 21 / 21