I would like to specially thank Dr. Wendell Horton for his excellent role as my Ph.D. ... well controlled laboratory conditions should help researchers working on .... One of the most important motivations behind the design and construction of ... The VP probe was built and used in the LArge Plasma Device by the LAPD team,.
Copyright by Jean Carlos Perez 2006
The Dissertation Committee for Jean Carlos Perez certifies that this is the approved version of the following dissertation:
Theory and simulation of sheared flows and drift waves in the Large Plasma Device and the Helimak
Committee:
Wendell Horton, Supervisor
K. Gentle
C. Chiu
H. Berk
G. Hallock
Theory and simulation of sheared flows and drift waves in the Large Plasma Device and the Helimak
by
Jean Carlos Perez, B.S., M.S.
DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT AUSTIN May 2006
Dedicated to my wife Paulina and my son Juan Carlos.
Acknowledgments
Time would fail me to thank all the people that made possible the realization of this work, in one way or another. First and foremost I would like to thank my mother Zoila and my father Carlos for giving me life, family values and education. My brother and sister Carlos and Carla, not only because they helped me finance my high school and college education, but because they were always supportive in the important decisions I have taken to achieve this milestone. I would also like to thank my wife for her unconditional and constant support throughout my Ph.D. studies, and my son Juan Carlos who motivated me even more to accomplish this goal. I would like to specially thank Dr. Wendell Horton for his excellent role as my Ph.D. supervisor, financial support for graduate studies, conferences and summer schools. I would also like to thank the Institute for Fusion Studies for funding necessary to complete my Ph.D. I also thank all my other graduate student fellows, in particular Edmund Spencer, Juhyung Kim and Andrew Cole for their computing, numerical, physics and even philosophical discussions that have helped me in some way or another along the road to complete this work.
v
Theory and simulation of sheared flows and drift waves in the Large Plasma Device and the Helimak
Publication No.
Jean Carlos Perez, Ph.D. The University of Texas at Austin, 2006
Supervisor: Wendell Horton
This work develops a comprehensive understanding of the physical properties of drift waves and vortices in the presence of shear flows in strongly magnetized plasma, including comparison to experimental data. Theoretical modeling and simulation of plasma instabilities is carried out for two important university-scale basic plasma experiments, namely, the Helimak device at The University of Texas at Austin and the LArge Plasma Device (LAPD) at the University of California, Los Angeles. Both machines possess simple geometry while retaining key physical properties of major magnetic confinement devices, making theoretical study amenable to analytical and numerical treatment in a more realistic manner. A large number of probes, near steady state operation of the plasma and relative control of key external physical parameters provide unique plasma data for the study of driftwave and Kelvin-Helmholtz turbulence. In addition to thorough conventional diagnosis, a new and promising diagnostic tool is introduced to directly measure plasma vorticity. This probe, that we call the vorticity probe, was built and used at the LAPD facility to obtain vorticity data in sheared flow experiments. The data is analyzed in great detail in this work and compared to linear and nonlinear theory. The study in this work is made vi
in several stages. The first stage is to analyze probe data using standard data analysis techniques to obtain meaningful results that can be compared to theoretical models. In the second stage we perform two-fluid linear stability analysis to identify the type of fluctuations that dominate the turbulence. In the linear analysis, analytical and numerical tools are used to illustrate the plasma waves and instabilities. For the nonlinear simulations of the instabilities and vortices we develop a comprehensive fully nonlinear pseudo-spectral code that uses a Chebyshev-Fourier decomposition of the solutions, technique that is widely used in the field of computational fluid dynamics. The new modelling and interpretation of structures developed in this work under well controlled laboratory conditions should help researchers working on large fusion devices and space-astrophysical plasmas to gain a better understanding of the problem of plasma turbulent transport.
vii
Table of Contents
Acknowledgments
v
Abstract
vi
List of Tables
x
List of Figures
xi
Chapter 1.
Introduction
1
Chapter 2.
The LArge Plasma Device and the Vorticity Probe
6
2.1 The LArge Plasma Device (LAPD) . . . . . . . . . . . . . . . . . . . . . . .
6
2.2 The Vorticity Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3 Sheared flow regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Chapter 3.
Vortex Turbulence in the LAPD
15
3.1 E × B Vorticity and its relevance in Plasma Turbulence . . . . . . . . . . .
15
3.2 Vorticity fluctuations in Kelvin-Helmholtz turbulence . . . . . . . . . . . . .
16
3.3 Two point spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.4 KH Instability in the LAPD . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.5 Nonlinear simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Chapter 4.
The Helimak Configuration
33
4.1 The Helimak Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2 The Helimak Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.3 Helimak Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Chapter 5.
Fluctuation Measurements in the Helimak
40
5.1 Fluctuation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.2 Frequency Spectra and Wave Characteristics
. . . . . . . . . . . . . . . . .
42
5.3 Two Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.4 Quasi-neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
viii
5.5 Linear Analysis and Parallel Ohm’s Law . . . . . . . . . . . . . . . . . . . .
51
5.6 Eigenmode Equation and Dispersion Relation . . . . . . . . . . . . . . . . .
55
Chapter 6.
Conclusions
58
Appendices Appendix A.
62 Chebyshev Pseudo-spectral Techniques
63
A.1 Nonlinear Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
A.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
A.2.1 Galerkin Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
A.2.2 Tau Methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
A.2.3 Collocation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
A.3 The Hasegawa-Mima equation coupled to a passive scalar . . . . . . . . . .
69
A.3.1 Fourth Order ODE problem . . . . . . . . . . . . . . . . . . . . . . .
71
A.3.2 Second Order ODE problem . . . . . . . . . . . . . . . . . . . . . . .
73
A.3.3 Full Time Dependent Problem . . . . . . . . . . . . . . . . . . . . . .
74
Bibliography
75
Vita
80
ix
List of Tables
2.1
Sheared flow regimes arising from wall biasing experiments. These regimes are classified according to the instability that drives the turbulence. . . . .
13
3.1
Linear eigenfrequencies and growth rates . . . . . . . . . . . . . . . . . . . .
30
4.1
Helimak important typical plasma parameters . . . . . . . . . . . . . . . . .
36
x
List of Figures
2.1 2.2 2.3 2.4
3.1
3.2
3.3
3.4
3.5
3.6
3.7
4.1
Schematic of the Large Plasma Device (LAPD), including a diagram of the circuit used for plasma column biasing. . . . . . . . . . . . . . . . . . . . . .
6
Vorticity probe design. The probe is inserted into the LAPD plasma radially, so that the magnetic field is perpendicular to the surface of the probe tips.
8
Mean profiles obtained as an average from 25 experimental shots in the stationary turbulent state during the 5 ms wall-bias pulse. . . . . . . . . . . .
12
E × B flow constructed from the average electric field during the wall bias pulse and the vorticity associated with it. Top: Kelvin-Helmholtz regime, Bottom: Drift Wave+KH regime. . . . . . . . . . . . . . . . . . . . . . . .
14
Top: Typical cathode/anode bias voltage pulse with respect to the chamber wall for establishing the rotation jet. Bottom: Radial electric field measured with the triple probe in the stationary section of the bias pulse. . . . . . . .
18
Vorticity probability distribution functions for representative radius in each reported regime. PDF are constructed with vorticity fluctuation measurements from 25 experimental shots at each radius during the bias pulse, for a total of 102400 samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Radial variation on the left side of cylinder for the correlation, the kurtosis and skewness. The antisymmetry of the skewness agrees with the change if vortex rotation direction across the layer and coincides with the larger values of kurtosis that occurs in for a field of vortices. . . . . . . . . . . . . . . . .
20
Left: Auto power spectrum for plasma potential at the center of the shear layer in the KH regime. Right: Cross phase between two angularly separated probes for the KH regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Frequency and growth rate as a function of kθ a for the model of the equilibrium radial electric field measured in the plasma. There are two modes: the unstable Kelvin Helmholtz mode and the neutrally stable modes with odd symmetry corresponding to a wavy motion of the jet. . . . . . . . . . . . . .
28
Stream function ψ = φ/B from the iso-potentials of the unstable eigenmode of Eq. (3.12). The last frame in the saturated state shows the alternation of the vortex directions across the jet. . . . . . . . . . . . . . . . . . . . . . . .
29
Proceeding clockwise from the upper left, the figures show four stages in the growth and evolution of the vortices from the jet flow modeled for the LAPD. (Click on figure for animation) . . . . . . . . . . . . . . . . . . . . . . . . .
32
Picture of the Helimak device located in the Robert Lee Moore Hall at the University of Texas at Austin. . . . . . . . . . . . . . . . . . . . . . . . . . .
34
xi
4.2
4.3
5.1
5.2
5.3
5.4
5.5 5.6
Left: Helimak cross section showing the magnetic field lines and probe geometry. Four sets of four conducting plates contain over 700 Langmuir probes as shown. Plates are isolated from each other and can be independently biased, however, in this experiment all of them are connected to the vessel. Right: Vacuum magnetic field lines spiraling from bottom to top with radially varying pitch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
The Helimak operates in a steady state with a duration of 30 s. Radial equilibrium density, electron temperature, and floating potential profiles are shown. All profiles are measured as a function of radius by Langmuir probes spanning the vessel. Also shown is a radial profile of the density fluctuation level, δn/n, measured by probes collecting ion saturation current. Distance is measured from the central vertical axis of the torus. . . . . . . . . . . . .
39
(a) Sample of time series lasting 5 ms from a probe collecting ion saturation current during a discharge. (b) Measured density fluctuation level amplitude normalized by the mean as a function of the connection length along helical field lines for both the LFS and the HFS. . . . . . . . . . . . . . . . . . . .
41
(a) Frequency spectrum of 3 probes collecting ion saturation data. The density maximum is located at r = 114 cm from the vertical axis. The two LFS probes are at r = 122 cm(solid black) and r = 130 cm(solid red). The HFS probe is located at r = 101 cm(dotted blue). (b) Coherency, γnn , and cross phase, αnn , between two probes separated by a vertical distance of 10cm, located on the LFS. The slope of αnn gives a phase velocity of 1000 m/s, and its positive sign indicates propagation in the +z direction. . . . . . . . . . .
42
Poloidal cross correlation functions between two sets of probes, ∆z = 9 cm, located at radial positions on the LFS r = 122 cm(dashed red) and r = 130 cm(solid black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Cross coherency, γnφ (solid red), and cross phase,αnφ (dotted black), between adjacent probes measuring ion saturation and floating potential respectively. The upper plot corresponds to a probe pair 8 cm radially outward from the density peak on the LFS. The lower plot corresponds to a probe pair 26 cm radially outward from the density peak on the LFS. . . . . . . . . . . . . .
45
Radial cross correlation function between sets of probes with ∆z = 0 cm and ∆r = 0, 4, 8, 12 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
(a) Frequency spectrum from the quadratic linear dispersion relation. (b) Unstable growth rates as function of k⊥ ρs . The higher frequency root in the left panel is heavily damped and not shown. Each diamond corresponds to integer mode numbers kz = mπ/2H. . . . . . . . . . . . . . . . . . . . . . .
57
xii
Chapter 1 Introduction
The main objective of this work is to provide theoretical understanding of important basic plasma processes that take place in two plasma configurations: The Helimak and the LArge Plasma Device (LAPD). One of the most important motivations behind the design and construction of these devices is that of creating a steady state plasma in a rather simple configuration with good diagnostics, but still maintaining many important features present in most magnetic confinement devices. For instance, in fusion devices, the intricacies of complex magnetic field geometries together with the difficult access of diagnostics to probe the plasma, make difficult the theoretical analysis and comparison with experimental data. The pursuit of magnetic plasma confinement for the generation of fusion power has been one of the main drivers of plasma theory research. In particular, understanding the underlying mechanisms of cross-field plasma transport is still an open problem of crucial importance to experimentally achieve the necessary temperatures, density and confinement time, determined by the Lawson’s criterion (see for instance [1]), for a deuterium-tritium plasma to generate net fusion power. Classical and neoclassical (see [2, 3]) theories fail to predict the measured transport coefficients, so it is strongly believed that most of the cross-field transport is due to the turbulence driven by unstable plasma waves, which is commonly called anomalous transport. A plasma can support a number of waves that can go unstable, drive the plasma 1
into a turbulent state and produce anomalous transport. In magnetic confinement devices, like tokamaks, it is believed that most of the transport is due to low frequency modes associated to plasma inhomogeneities like pressure, density, temperature and flow. These plasma inhomogeneities excite drift-waves, in the case of pressure, density and temperature gradients, and Kelvin-Helmholtz waves, for flow gradient or flow shear. An extensive review of the role of drift wave instability in the turbulent transport of magnetized fusion plasmas can be found in [4] and references therein. This instability is universally present in fusion plasmas due to the unavoidable presence of density and temperature gradients in confined plasmas. As mentioned before, the experimental study of drift wave instabilities, the associated turbulence and turbulent transport are not simple problems. This difficulty comes from the fact that in order to obtain good magnetic plasma confinement a complex toroidal magnetic geometry is necessary. In addition, a complete diagnostic of the fluctuations is not possible either because of the very high temperatures of fusion grade plasmas or access limitations due to configurational constraints in the confinement devices. This work investigates, with a combination of laboratory experiments and plasma theory, the properties of high temperature ionized gasses in confining magnetic fields. Once the ionized gas reaches the temperature of about one hundred thousand degrees centigrade, which is about ten times higher than the ionization temperature at the low densities used in the experiments, the plasma has the same characteristics as fusion grade plasmas that have attempted to reach the breakeven conditions for fusion power. These experiments have fallen short of the breakeven condition, called the Lawson condition, due to the anomalous transport of plasma from the drift wave instabilities. By operating in an MHD stable equilibrium [5], the Helimak experiment is suitable
2
for the study of drift wave instabilities and the associated turbulent transport. In this work we report density fluctuations that are identified as drift waves traveling along the axis of the Helimak machine with high levels of fluctuation in the bad curvature side of the cylindrical shell. Another important instability that can affect the turbulence, associated with flow shear, is the Kelvin-Helmholtz instability. A key quantity in the understanding of this type of plasma fluctuations is the vorticity associated with E × B flows. The dynamical equation that describes it is a statement of vorticity conservation, usually the case in plasmas with low viscosity. The LArge Plasma Device provides an excellent setting to experimentally study the basic properties of the Kelvin-Helmholtz (KH) instability. Cross-field sheared flow experiments in a strong and fairly uniform magnetic field can be performed in the LAPD with very good reproducibility for a wealth of turbulence data. In addition to this, we introduce a new diagnostic tool, named the Vorticity Probe (VP) that allows the direct measurement of E×B plasma vorticity. This is an invaluable tool that does not have a fluid counterpart and that was not previously available in plasma diagnostic. The advantage of the KH instability in plasmas over fluids lies in the fact that for plasmas, the KH instability manifest itself in the statement of local charge neutrality, leading to an equation for the electrostatic potential that is isomorphic to the fluid version in the vorticity-stream function formulation. In the plasma, the electrostatic potential plays the role of the stream function, whose Laplacian gives the fluid vorticity. Therefore, measuring plasma vorticity is equivalent to measuring the Laplacian of the electrostatic potential, precisely what the VP probe proposed and used in this work does. The VP probe was built and used in the LArge Plasma Device by the LAPD team,
3
led by Walter Gekelman et al, and later more machine time was requested by R. D. Bengtson, W. Horton and J.C. Perez to obtain further data. The plasma conditions obtained in all experiments were such that either the Kelvin-Helmholtz or a hybrid of Drift-Waves and KH instability were responsible for the turbulence. Fluctuating vorticity data is analyzed in both regimes to obtain vortex characteristics, such as vortex intensity and size, and contrasted to simulations. The use of the vorticity probe is not only limited to measuring vorticity, but it is also able to measure other electrostatic potential gradients that can be used to construct anomalous fluxes, such as turbulent vorticity flux, Reynolds stress, etc. Another use of the vorticity probe we present in this work is the use of standard two point correlation techniques to detect the existence of waves, determine wave characteristics and possibly wave-induced transport analysis. Nonlinear simulations that reproduce the main important physical properties of the vortices created by the KH turbulence are presented. Nonlinear simulations are carried out with a Chebyshev-Fourier-tau pseudo-spectral method. Chebyshev methods have been widely used in the description of fluid turbulence, and only recently in plasma for Magnetohydrodynamic instabilities and turbulence. To the best of our knowledge, this numerical technique has not been used in more general two-fluid models. In this work we present numerical solutions to several reduced plasma equations based on approximations to the more general two-fluid Braginskii’s equations. This work is organized in six chapters, including this introduction, as follows. In chapter 2 we present a general description of the LArge Plasma Device, the vorticity probe design and the experiments of sheared plasma rotation. Data from VP in the sheared rotation experiments is analyzed in chapter 3 to demonstrate the existence of vortex structures generated by the KH instability, showing estimated values of vortex intensity and size, and evidence of intermittent transport due to the coherent vortices. Vorticity and potential
4
power spectrum are shown, as well as evidence of coherent modes. Linear and nonlinear simulation of KH instability for similar condition to the experiment are also presented in this chapter. In chapter 4 we move to the description of the Helimak device and discuss its MHD equilibrium. In chapter 5 we show experimental data, courtesy of the Helimak team, of fluctuating density, electrostatic potential and temperature. Frequency spectra and wave characteristics from fluctuating data are shown. In order to model the experimental data, we introduce the two-fluid model equations tailored to the Helimak geometry and plasma conditions. Linear dispersion relation, radial mode structure and growth rates are obtained for the Helimak operating conditions. In chapter 6 we conclude.
5
Chapter 2 The LArge Plasma Device and the Vorticity Probe
2.1
The LArge Plasma Device (LAPD) The LArge Plasma Device [6], or LAPD, is part of the Basic Plasma Science Facility
(BaPSF) at UCLA. This device is an 18 m long , 1 m diameter cylindrical vacuum chamber surrounded by around 90 magnetic field coils properly positioned to give rise to an fairly uniform axial magnetic field with δB/B < 0.5%. A schematic of the experiment is provided in Fig. 2.1.
Figure 2.1: Schematic of the Large Plasma Device (LAPD), including a diagram of the circuit used for plasma column biasing. The plasma is created in pulsed discharges from a heated Barium Oxide coated emissive cathode. Electrons emitted from this cathode, called primary electrons, are then 6
accelerated into the main chamber by an electrostatic field resulting in the ionization of the fill gas. Usually a noble gas is used in order to avoid chemical reactions within the device. The acceleration of the primary electrons is attained by placing the anode, a stainless steel mesh, around 30 cm away and biased to a higher potential than the cathode. This cathodeanode configuration is immersed in a strong magnetic field so that the primary electrons will have a small gyro-radii which leads to an ionization front with the same spatial distribution as the primaries. Given that the primary distribution strongly depends on the emitter temperature, the cathode needs to be uniformly heated to attain a uniform plasma density across the magnetic field. Typical plasma parameters for the experiments reported in this work are n e ∼ 1.2 × 1018 m−3 , Te ∼ 10 eV and B ∼ 0.08 T with helium as the working gas. Background neutral density at the position of the probe measurements is less than 1012 m−3 as determined by spectroscopic measurements which is consistent with measured pressures in LAPD. This level of neutral background gives a small ion-neutral collision frequency of order 100 /s, providing a small background viscosity that is negligible for the wave-numbers considered here.
2.2
The Vorticity Probe It is well known that the most natural quantity for describing eddies or vortices in
neutral fluids or plasmas is the fluid vorticity, defined as ω = ∇ × v.
(2.1)
This gives a local measure of the circulation of the velocity field at every point in the fluid plasma.
7
In the limit of a uniform magnetic field B = Bˆ z and with an electrostatic field E = −∇ϕ(x, t), the ions and electrons move across the B-field with velocities 1 ∂ϕ (x, y, t) B ∂y 1 ∂ϕ (x, y, z, t), B ∂x
vx = −
(2.2)
vy =
(2.3)
so that the parallel component of the vorticity vector ω = ∇ × v is given by ω=
∂vy ∂vx 1 − = ∇2⊥ ϕ(x, y, z, t). ∂x ∂y B
(2.4)
Figure 2.2: Vorticity probe design. The probe is inserted into the LAPD plasma radially, so that the magnetic field is perpendicular to the surface of the probe tips.
To measure the vorticity we use the vorticity probe shown in Fig. 2.2. Then with the spacing ∆x = ∆y = h between the four corners of the probe with respect to the center of the probe, we use the sum of the floating potentials of the corner, minus four times the floating potential at the center, ϕ1 + ϕ2 + ϕ3 + ϕ4 − 4ϕ0 to obtain the vorticity ω. It is worth mentioning that the vorticity calculated in this way from the floating potential is valid as long as one assumes that the number of primary electrons at probe positions is small and that the temperature is the same at all probe tips. For the measurements we report here these assumptions are justified given that the probe is introduced in the LAPD more 8
than 10m from the plasma source where the primary electrons are substantially reduced. Furthermore, measurements of temperature profiles show typical temperature scale length of the order of LTe ∼ 10 − 20 cm which is sufficiently greater than the probe tip distance h = 5 mm. If this were not the case we would need to measure the temperature profile at each probe tip in order to get the plasma potential through the equation [7] µ ¶ Te me ϕp (eV ) = ϕf (eV ) − ln 2π 2e mi
(2.5)
= ϕf (eV ) + 3.5Te
(2.6)
for Helium
The origin of the electron temperature peak in the shear flow layer is not well understood and suggests further research studies on the anomalous electron viscous heating. This stencil is based on the finite difference approximation to ∇2 ϕ given by ∇2 ϕ =
(ϕ1,0 + ϕ0,1 + ϕ−1,0 + ϕ0,−1 − 4ϕ0,0 ) . h2
(2.7)
The vorticity is then given by dividing by the magnetic field B as in Eq. (2.4). Clearly, the vorticity in Eq. (2.7) vanishes for ϕ = C + Ax + By. Only quadratic variations over the probe yield nonzero vorticity. Just as in numerical simulations, the 5-point sample of ϕ(x, y, z, t) yields the true vorticity only for that part of the spectrum ϕ(kx , ky , z, t) that has kx h < 1 and ky h < 1. The velocities into and out of the four sides of the square are also of interest. For the left side, for example, the velocity is given by vx (left) =
1 (ϕ1 − ϕ2 ) . B 2h
(2.8)
The rate of convective transport of ω is given by ∂ω + v · ∇ω = Sω (x, t), ∂t 9
(2.9)
where the source/sink function Sω includes various mechanisms that produce and absorb vorticity. Since the E × B drift is incompressible for a uniform B-field, Eq. (2.9) also has the conservation form ∂ω + ∇ · (ωv) = Sω . ∂t
(2.10)
Averaging using Eq. (2.10) over the symmetry direction y (or θ) yields the transport equation, ∂ ∂ 0 vy (x, t) + hωvx i = S ω (x, t) ∂t ∂x
(2.11)
for the generation of sheared zonal flows vy0 = d hvy i /dx. It is worth noticing that the average flux of vorticity hvx ωi equals the divergence of the Reynolds stress hvx vy i per unit mass. Both the vorticity flux and the Reynolds stress can be measured by the vorticity probe. In this experiment, the differential axial confinement of the electrons at the radius of the cathode controls the source of vorticity Sω through the biasing of the cathode/anode with respect to the chamber walls. For tokamaks a similar Er structure, usually an Er well, is formed by the differential confinement of the ions at the edge of the plasma. This becomes a strong feature after the transition to the H mode [8]. The vorticity probe was constructed from seven Tantalum tips, as shown in Fig. 2.2. The tips are cylindrical, 0.02 inch in diameter and 1 mm long. The tips are oriented along the direction of the magnetic field (into the page in Fig. 2.2). The Tantalum tips are held in alumina ceramics, which in turn are held by a stainless steel structure that is slotted in order to minimize perturbation to the plasma. The central five tips are arranged in a diamond pattern, with the outer four tips separated from the central tip by 5 mm. The five inner tips are used to measure the floating potential, which is then used to compute a finite difference estimate of the vorticity, while the two outer tips allow measurement of the ion 10
saturation current, in order to obtain the plasma density and radial particle flux hnE θ i /B, ¡ ¢ as well as the Reynolds stress nmi hvr vθ i = ρ/B 2 hEr Eθ i. In order to compute the finite-difference estimate of the vorticity, four times the
floating potential on the central tip must be subtracted from the floating potential on the surrounding four tips. During biased rotation experiments in LAPD, the DC floating potential can reach values of order 200V, substantially larger than the observed fluctuation amplitude (∼ 1V). The floating potential measurements are therefore performed using AC coupled amplifiers in order to reject the large low-frequency floating potential signal and to maximize the use of the dynamic range of the available digitizers. The amplifiers are constructed using wide-band operational amplifiers (National Semiconductor LM7171) and flat response from 1 KHz to 10 MHz. A schematic of the measurement circuit is shown in Figure 2.1. Data is acquired using 100 MS/s, 14 bit VME-based digitizers (eight channels per board, four available boards).
2.3
Sheared flow regimes A sheared poloidal velocity profile is established by biasing the chamber wall with
respect to the cathode (see Fig. 2.1) for 5 ms during the discharge. This biasing results in different sheared E × B flow turbulent regimes at the edge of the plasma column. We classify these regimes according to the dominant instability driving the turbulence as shown in table 2.1. Fig. 2.3a shows the background mean profiles for the first of these regimes, the MHD Kelvin-Helmholtz in which the E × B flow dominates the perpendicular dynamics with weak coupling to the parallel direction. In this case potential fluctuation levels are significantly higher than density fluctuations, eϕ/Te >> δn/n. In table 2.1 we see that
11
eϕ/Te = (5.0 ± 0.5)˜ n/n for the KH regime.
Kelvin Helmholtz unstable (2004)
Drift wave unstable (2005)
15
n (1011 cm -3)
10 10
Te(eV)
5
n (1011 cm -3)
0
"$ #&%
-5 -10 10
0
Er(100 V/m)
-5
15
20 25 r(cm)
30
35
Te(eV)
5
-10 10
Er (100 V/m)
"( ') !% 15
20
25 r (cm)
30
35
40
Figure 2.3: Mean profiles obtained as an average from 25 experimental shots in the stationary turbulent state during the 5 ms wall-bias pulse. From table 2.1 we can also see that in the KH regime, the density scale length is about 7 times larger than the velocity scale length which allows us to ignore effects related to density gradient driven drift waves, supporting the hypothesis that KH fluctuations are dominating the turbulence. Fig. 2.3b shows the second regime, a hybrid Drift-Wave KelvinHelmholtz in which both mechanisms are believed to drive the turbulence. Fluctuation levels for the dimensionless quantities eϕ/T ˜ e and n ˜ /n are similar up to a 10 % error, characteristic of drift waves. Furthermore, the gradient scale length of density Ln and velocity Lv are comparable, giving an indication that the drift wave component of the fluctuation can no longer be neglected. This regime, with a significant drift wave component [9] shows the clearest formation of the internal transport barrier [10] in the density profile consistent with the idea of suppression of turbulence and anomalous transport by sheared flows [11, and references therein]. The mode spectrum and vorticity statistics for the KH regime are reported in chapter 3. A third regime for which there was no available data at this time is one of very weak shear with steep density gradient driven drift wave turbulence.
12
Table 2.1: Sheared flow regimes arising from wall biasing experiments. These regimes are classified according to the instability that drives the turbulence.
eϕ ˜ Te n ˜ n Lv
Ln vE vde
Kelvin-Helmholtz (KH)
Drift-Wave+KH
∼1 ∼ 0.2 ∼ 2 cm ∼ 15 cm ∼ 10 km/s ∼ 0.6 km/s
∼ 0.4 ∼ 0.4 ∼ 5 cm ∼ 5 cm ∼ 5 km/s ∼ 1.5 km/s
An early study of the KH mode in small laboratory plasma devices include Kent et. al. [12] with a Q-machine with L/a = 100 cm/1 cm= 100, B = 0.1 to 0.4 T at T e = 0.2 eV, and n = 5 × 108 to 5 × 1010 cm−3 . Kent et. al. conclude that the edge oscillations over 5 kHz in frequency where Ωmax = 2 × 104 rad/s at the edge are KH modes and caution neglecting these modes in the stability analysis of lower frequency drift waves. In the larger Columbia Linear Machine (CLM) Sen et. al. 2001 [13] report the identification of KH modes at the frequency of 65 kHz. In the CLM with L/a = 100 cm/3 cm= 50, B = 0.1 to 0.15 T and Te ∼ Ti ∼ 5 eV, n ∼ 5 × 108 to 5 × 109 cm−3 , the KH instability is a permanent feature in the region of maximum Er and has a laboratory frequency of 55 to 65 kHz. Fig. 2.4a shows the background E × B flow as calculated from the floating potential and temperature measurements using the triple probe for the strong shear conditions. Fig. 2.4b shows the E × B flow from the triple probe measurements compared to flow measurements using a Mach probe. The Mach probe used has 6 tantalum faces (three pairs), flush-mounted on the surface of a cylindrical probe tip. The probe is inserted so that the magnetic field is perpendicular to the axis of the cylindrical tip. Defining the angle from the direction of the magnetic field, the 3 face pairs are located at (0◦ , 180◦ ), (45◦ , 225◦ ), and
13
Background flow and vorticity (DWKH)
Background flow and vorticity (KH)
20
6
15
Vorticity (105 /s) ExB flow (km/s)
10
4
Vorticity (105 /s) ExB flow (km/s) Mach probe (km/s)
2
5 0
0
-2
-5 10
15
20 25 r(cm)
30
35
10
15
20
25 r(cm)
30
35
Figure 2.4: E × B flow constructed from the average electric field during the wall bias pulse and the vorticity associated with it. Top: Kelvin-Helmholtz regime, Bottom: Drift Wave+KH regime. (315◦ , 135◦ ). The first tip pair is used to measure parallel flows, while the other two pairs are used to measure the perpendicular Mach number, as outlined in Ref. [14]. The probe tips used are comparable in size to the ion gyro-radius, and for this reason the computed Mach numbers are corrected for an increased effective ion collection area in unmagnetized plasmas, following Shikama, et. al. [15]. A constant offset, associated with the imbalance in the area of the face pairs, has been removed from the Mach probe data by forcing the flow velocity at r = 10 cm to be zero. The Mach probe measured flow profile is consistent with the E × B speed profile computed from the potential profile, although the Mach probe reports a slightly higher flow velocity. The discrepancy between the E × B speed and the Mach probe measurement appears largest in the density gradient region, where the difference is on the order of the diamagnetic drift velocity.
14
Chapter 3 Vortex Turbulence in the LAPD
3.1
E × B Vorticity and its relevance in Plasma Turbulence It is widely known that vorticity plays a significant role in the nonlinear dynamics
of neutral fluids and plasmas. For instance, Navier-Stokes equation has alternative formulations in terms of the fluid vorticity [16, 17, 18], as defined in the previous chapter ω ≡ ∇ × v.
(3.1)
The same definition follows in fluid plasma equations, like Magnetohydrodynamics (MHD), and generalized magneto-fluid models, from which reduced sets of equations can be obtained for specific plasma conditions. The most well know of these models are the Hasegawa-Mima equation [19], the Hasegawa-Wakatani [20] two-field equations and the Hamaguchi-Horton model [21]. The common feature of these models, the rate of change of vorticity, has its origin in the divergence of the plasma polarization current that provides the charge balance in the quasi-neutral plasma. Therefore, measuring vorticity is essential for the validation and quantitative understanding of these models. A statement of local vorticity conservation, due to the incompressibility of the E × B flow is d ω= dt
µ
¶ ∂ + vE · ∇ ω = Sω (x, t) ∂t
(3.2)
where the sources and sinks Sω depend on the specific model and can act as a coupling term to other relevant dynamical fields. Measurements of vorticity are usually obtained from fluid velocimetry, that is, vorticity is calculated by a process of finite differences on measured flow 15
fields. One of the main disadvantages of this process is that it is prone to inherent errors in the numerical schemes used in obtaining the curl of the velocity field. In this chapter, we present experimental results from the vorticity probe, introduced in the previous chapter. This vorticity probe takes advantage of the fact that in a strongly magnetized plasma, the stream function associated with the dominant E × B flow is proportional to the plasma electrostatic potential ϕ as expressed by the equation vE =
E×B ez × ∇ϕ = B2 B
(3.3)
where we have taken B = Bez . In terms of the stream function, ϕ/B, the magnetic-fieldaligned vorticity is ωz =
1 2 ∇ ϕ. B ⊥
(3.4)
Equation 3.4 is valid as long as the turbulence is dominated by electrostatic fluctuations or magnetic fluctuations can be neglected. In this case, plasma measurements have two advantages over measurements in neutral fluids: (1) the stream function field can be obtained by measuring plasma potential, (2) the finite difference scheme is to be applied to a scalar rather to a vector field. Measuring vorticity under this conditions is both simpler and more accurate, provided equation Eq. (3.4) holds. The vorticity probe design and use are discussed in [22] in the context of Kelvin-Helmholtz turbulence generated in the LArge Plasma Device (LAPD) facility at UCLA.
3.2
Vorticity fluctuations in Kelvin-Helmholtz turbulence The Kelvin-Helmholtz instability is excited in the LAPD experiment by biasing the
floating anode-cathode source with respect to the chamber wall of the device. This results in a sharply localized radial electric field as shown in Fig. 3.1. This electric field, along with the axial magnetic field, creates a sheared poloidal E × B jet stream at the edge of 16
the plasma column. This jet stream flow seems to form as one of the natural self-organized states after the K-H instability in which the vorticity ω as function of the potential ϕ is a steady state solution of the vorticity equation vE · ∇ω = 0
(3.5)
Various types of solutions exist with or without embedded vortices. The simplest solution is a jet stream with localized vθ (r). In section 3.5 we show a nonlinear simulation that illustrates this feature. Measurements were made using the vorticity probe at 51 radial locations. At each spatial location, data is taken from 25 different plasma discharges. The data is acquired using 14-bit, 100 MS/s digitizers. An effective sampling rate of 1.56 MS/s is obtained by averaging 64 consecutive samples at 100 MS/s. The averaging is done in hardware and provides an effective anti-aliasing filter. A total of 28672 samples are taken during each discharge, resulting in a time record 18.35 ms long. Fig. 3.2 shows probability distributions functions (PDFs) associated with vorticity at every measured radial location as compared to a Gaussian distribution represented by the solid line. The vorticity pdfs at a given radii are obtained from 102400 vorticity samples as a result of combining 25 experimental shots with 4096 time points each during the bias pulse divided into 100 equally spaced bins of fluctuating vorticity amplitudes, normalized to the standard deviation. Vorticity PDFs show the intermittent character of vorticity through heavy tails, which we associate to the existence of coherent structures present in the shear layer. The heavy tails of the vorticity probability distributions functions shown in Fig. 3.2 signify that large values of the vorticity occur frequently. The situation here for vorticity fluctuations
17
Figure 3.1: Top: Typical cathode/anode bias voltage pulse with respect to the chamber wall for establishing the rotation jet. Bottom: Radial electric field measured with the triple probe in the stationary section of the bias pulse.
driven by the background plasma jet is similar to that of the density and density fluxes and the coherent intermittent structures measured in the linear PISCES-A [23] and the ADITYA
18
Background flow and vorticity (KH) 15
-1
10
Vorticity (105 /s) ExB flow (104 m/s)
-2
10
Probability
10
Vorticity PDF at r=37 cm
5
-3
10
-4
0
10
-5
-5
5
10
15
20 25 r(cm)
30
35
10
40
-10
-5
0
ω/σ
5
10
Vorticity PDF at r=30 cm
Vorticity PDF at r=22 cm -1
10
-2
10
-2
Probability
Probability
10
-3
10
-4
10
-3
10
-4
10
-5
-5
10
-10
10 -5
0
ω/σ
5
10
-10
-5
0
ω/σ
5
10
Figure 3.2: Vorticity probability distribution functions for representative radius in each reported regime. PDF are constructed with vorticity fluctuation measurements from 25 experimental shots at each radius during the bias pulse, for a total of 102400 samples. tokamak machine where the probability distributions showed heavy tails [24]. This suggests that new studies with the vorticity probe may be useful for analyzing the edge turbulence of tokamaks, especially in the L to H mode transition. In fig. 3.3 we show relevant statistical quantities for an ensemble of 25 experimental runs under identical conditions. The measurements of vorticity show a high degree of reproducibility over the experiments. First we consider the degree of correlation of the floating potential difference between the outermost probe pins 5 and 6 as the probe is
19
Background flow and vorticity (KH)
(a)
15
5
10
Correlation
Vorticity (10 /s) ExB flow (km/s)
5 0 -5 10
15
20 25 r(cm)
30
(c)
6
Skewness
Kurtosis
8
35
4 2 0
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65
20
25
30 35 r(cm)
40
45
1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5
(b)
20
25
30 35 r(cm)
40
45
(d)
20
25
30 35 r(cm)
40
45
Figure 3.3: Radial variation on the left side of cylinder for the correlation, the kurtosis and skewness. The antisymmetry of the skewness agrees with the change if vortex rotation direction across the layer and coincides with the larger values of kurtosis that occurs in for a field of vortices. √ scanned across the shear flow layer. The spacing between these probes is d = 10 2 mm = 14 mm. Frame (a) shows that the normalized correlation function hϕ 5 ϕ6 i is near unity except in the shear layer where it drops to about 0.7. Thus, we infer that the radial correlation length of the fluctuations drops from greater than 14 mm to about 10 mm in the center of the shear flow layer. Frame (c) shows the Kurtosis of the vorticity signal, giving indication of the strong non Gaussian statistic of vorticity turbulence data outside the core of the plasma jet. This is consistent with the heavy tails present in the PDFs of
20
Fig. 3.2. Frame (d) shows the Skewness of the signal, another useful statistical quantity that gives a measure of the asymmetry of the PDF. The skewness of the vorticity PDF in Fig. 3.3 with positive, counterclockwise rotating plasma vortices inside the maximum of the jet, vmax ∼ 20 km/s at r = 30 cm shown in Fig. 2.3a and clockwise rotating vortices outside the maximum of vθ (r) confirms the existence of patches of alternating sign vortices from the localized jet. This configuration follows from the conservation of angular momentum and mass. Simulations in section 3.5 show that like regions of vorticity with the same direction of rotation are strongly merging. Thus we may expect larger scale vortices to form in the nonlinear fluctuation spectrum. The size of the vortices can be estimated using the width of the Skewness plot on each side of the jet, under the assumption that when the Skewness reaches a local maximum the vorticity probe is located at the center of the vortex. With this assumption, we obtain a vortex diameter of dv = 10 ± 2 cm or dv = (10 ± 2)ρs . There are two types of aliasing errors: (i) the usual single probe sampling error for signal frequencies higher than the Nyquist fN = 1/2δt and (ii) spatial aliasing errors for short wavelength signals. For each pin there are anti-aliasing filters in frequency. This helps reduce spatial anti-aliasing under the assumption that small scale structures exist at higher frequency. However, the sample rate (and anti-aliasing filter) is at much higher frequency than the frequency of the observed fluctuations (we sample at a few MHz, and the fluctuations are at around 10-20 kHz). For spatial aliasing problem we consider the effect of the finite probe space h on a single wavelength signal, the ratio of vorticity measured with the probe to the actual value of vorticity is given by sin2 R(k, h) = 4
¡ kx h ¢
+ sin2
2 h2 (kx2
21
³
+ ky2 )
ky h 2
´
(3.6)
For kh < 1 we have R ≈ 1. For kx0 = kx + 2πn, ky0 = ky + 2πm, the probe picks up the same signal for k and k0 which is the aliasing error. However, the probe value is smaller as given by R(k, h) at the spatial Nyquist frequency kN = π/h where the ratio function is R = 4/π 2 ≈ 0.444. So, keeping data for k < kN ≈ 628 m−1 will largely suppress the spatial aliasing errors. Using a range of probe sizes would allow an accurate assessment of the power in the high k components of the vorticity field.
3.3
Two point spectral analysis Using standard one and two point correlation techniques [25, 26] with selected tips
of the vorticity probe, wave characteristics can be obtained at different radial positions. Fig. 3.4a shows the auto power spectrum for the plasma potential from the central tip 7 (see Fig. 2.2) at probe position r = 28.4 cm, corresponding to the center of the shear layer. The turbulent spectrum during the bias pulse is dominated by frequencies in the range from 5 to 50 kHz. Cross phase between two angularly separated probes, shown in figure 3.4b, can be used to identify the azimuthal mode numbers m dominant in the power spectrum. Cross phase vs. frequency
m=5 m=4
m=7
m=2
Cross phase (rad)
Auto Power (arb)
Potential Power Spectrum
m=8
10
20
30 40 f(kHz)
50
60
0.3
∆ y=7.1 mm -6 slope=4.7x10 rad/Hz
0.2 0.1
Vg=2 π∆ y/slope=9.5 km/s
10
20
30 40 f(kHz)
50
60
Figure 3.4: Left: Auto power spectrum for plasma potential at the center of the shear layer in the KH regime. Right: Cross phase between two angularly separated probes for the KH regime.
22
The power spectrum is mostly localized at the shear layer region, although important levels of fluctuations are also encountered at large values of r in the lower density plasma. As the fluctuation energy appears in low mode numbers, the cylindrical geometry becomes important. The vorticity gradient term ky vy00 (x) in the Rayleigh equation (Kelvin£ ¤ d 1 d 2 Helmholtz instability) is replaced by (m/r) dr r dr (r Ω) in the cylindrical plasma with
angular rotation frequency Ω = vθ /r. The stability analysis [27, 28] shows that the cen-
trifugal force acts as an effective gravity g0 r/a giving rise to the Rayleigh-Taylor instability partially controlled by the Coriolis force effect 2mΩ and shear flow. In the simplest case of solid body rotation and a Gaussian density model n(r) = n0 e−r
2 /a2
the stability is given by
A˜ ω2 + B ω ˜ + C = 0 where ω ˜ is the Doppler shifted frequency ω − mΩ. The coefficients are A = νm,n (b/a) + Ap where b is the radius of the surrounding conducting wall and Ap ∝ kk2 measures the divergence of the parallel electron current, B = 2mΩ−A p ω∗e from the Coriolis d effect and the density gradient ω∗e = −(mTe /eBr) dr ln n0 (r) and C = m2 (Ω2 + c2s /Rc a) the
total interchange instability. In a straight, axial Bz laboratory plasma, only the centrifugal force is present. The stability of the m = 1 and m = 2 modes depend on how close the conducting wall is to the plasma density radius and the details are contained in the eigenvalue νm,n = m + 2n + f (b/a) where m, n are the azimuthal and radial eigenmode numbers and f (b/a) → 0 as the conducting wall moves to infinity, b/a → ∞. Examples of the eigenfunctions and eigenvalues as function of the wall-to-plasma radius and the shear in the rotation frequency Ω(r) for the modes m = 1, 2 and 3 are shown in [27]. This stability analysis suggests that the large density fluctuation observed in the edge is associated with lower m modes driven by the centrifugal force acting on the density gradient.
23
3.4
KH Instability in the LAPD Plasma version of the Kelvin-Helmholtz instability can be obtained from the elec-
trostatic response in a two-fluid model. The reduced model equation for the dynamics for the electrostatic potential is obtained from the assumption of quasi-neutrality, ∇ · j = 0.
(3.7)
in which j is the electric current. Assuming the plasma to be immersed in a constant background magnetic field B = Bz ez we can split the current contributions into the perpendicular and parallel parts j = j⊥ + jk . The perpendicular part is comprised of the currents associated to perpendicular particle and fluid drifts, namely, the polarization drift, diamagnetic drift, etc. The currents from electron and ion currents from E × B motion cancel each other in the quasi-neutral state. Using the expression for the polarization current m i ni d m i ni jp = − 2 ∇ϕ = − 2 B dt B
µ
¶ ∂ + vE · ∇ ∇ϕ, ∂t
(3.8)
with vE =
1 E×B = ez × ∇ϕ B2 B
(3.9)
in (3.7), we obtain µ
¶ ∂ ∂ + vE · ∇ ∇2 ϕ = ∇2 ϕ + [ϕ, ∇2 ϕ] = 0, ∂t ∂t
(3.10)
where we have introduced the Poisson bracket between two 2D functions f, g: [f, g] ≡
∂f ∂g ∂f ∂g − ∂x ∂y ∂y ∂x
24
(3.11)
For the KH regime, the shear layer the scale length Ln , calculated from the profile of the mean ion saturation current, gives Ln ∼ 15 cm which is large enough to justify neglecting the density gradients in the shear flow modeling. This density scale length is approximately 15ρi where ρi = 3 mm. The vorticity equation (2.4) is the statement that the divergence of the cross field current vanishes. The simple KH model used here has canceling electron and ion currents from the E × B motion. Considering a slab approximation for this plasma we take x as the radial coordinate and y to be the periodic coordinate. We then linearize Eq. (3.10) around a steady-state poloidal flow like that in the LAPD plasma rotation. By doing so we obtain Rayleigh’s eigenvalue equation, ϕ00k (x) − k 2 ϕk (x) −
kvy00 (x) ϕk (x) = 0 kvy (x) − ω
(3.12)
for the plasma in the uniform magnetic field. During the rotation bias pulse, the maximum velocity in the flow corresponds to an electric field E ≈ 1.5 kV/m which occurs approximately at r ≈ 0.4 m. Taking the magnetic field to be B = 0.075 T along the z direction we obtain vmax =
E ≈ 2 × 104 m/s. B
(3.13)
We solve the eigenvalue problem for an ideal triangular basic flow that resembles the equilibrium electric field profile in Fig. 3.1, i.e., having 0 (1 − |x|) vy (x) = 0
the form x ≤ −1 |x| ≤ 1 x≥1.
(3.14)
Here the velocity is normalized to vmax , and all lengths to a = 0.1 m, corresponding to the half width of the shear layer. 25
Differentiation with respect to x gives vy00 (x) = δ(x + 1) − 2δ(x) + δ(x − 1).
(3.15)
Therefore the eigenfunctions obey the simple equation ϕ00k (x) − k 2 ϕk (x) = 0,
(3.16)
except at the points x = 0, ±1. The jet flow has a steep vorticity gradient, modeled by suitable boundary conditions that can be derived by integrating Eq. (3.12) around the singular points. Doing this integration results in the three boundary conditions k ∆[ϕ0k (−1)] = − ϕk (−1) ω 2k 0 ∆[ϕk (0)] = − ϕk (0) kv0 − ω k ∆[ϕ0k (1)] = − ϕk (1). ω
(3.17) (3.18) (3.19)
where we have defined the operator − ∆[f (x0 )] ≡ f (x+ 0 ) − f (x0 )
(3.20)
In practice, the profiles are continuous, but vary rapidly over a small scale δ. The jump conditions are well satisfied if kδ ¿ 1; for k = m/r this is equivalent to m ¿ r/δ. We can also see that Eq. (3.12) is invariant under the transformation x → −x. That means that if ϕk (x) is a solution to Eq. (3.12) with eigenvalue ω, then ϕ(−x) is also a solution with the same eigenvalue. This fact allows us to find eigenfunctions with a definite parity. Let us first look for odd eigenfunctions, i.e., eigenfunctions such that ϕk (x) = −ϕk (−x). The general solution to the Rayleigh equation, Eq. (3.12), in this case is of the form ϕodd k (x) =
½
Ae−k|x| sgn(x) |x| > −1 B cosh (kx)sgn(x) + C sinh (kx) |x| ≤ 1 26
(3.21)
where we have chosen the solution to decay at x → ±∞. By using the matching conditions at the singular points, we obtain the dispersion relation ωk =
´ 1³ 1 − e−2k . 2
(3.22)
We thus see that all the odd modes are neutrally stable. These modes, however, are important in the nonlinear analysis. Now we proceed to consider the even modes, which satisfy ϕk (x) = ϕk (−x). The general solution to the Rayleigh equation in this case is ϕeven (x) k
=
½
Ae−k|x| |x| > 1 B cosh (kx) + C sinh (k|x|) |x| ≤ 1 .
(3.23)
Applying the conditions at the singular point we obtain ωk =
´ p i 1 h³ 2k + e2k − 1 ± G(k) , 4
(3.24)
where G(k) ≡ 9 − 10e−2k − 12k + e−4k − 4e−2k k + 4k 2 .
(3.25)
The function G(k) is negative for 0 ≤ k ≤ kc , with kc ≈ 1.833. Hence, in this range, the eigenvalues are ωk± = ωR ± iγk ,
(3.26)
where ωR = γk =
´ 1³ 2k + e2k − 1 4 1p −G(k). 4
(3.27) (3.28)
In this case, modes with eigenvalue ωk+ are unstable and grow according to |ϕk (x, t)| = |ϕk (x)| eγk t . 27
(3.29)
Linear dispersion relation
Unstable (even) modes Stable (odd) modes
40
0.0 -0.5
20 0
Growth rates
0.5
γ (105 /s)
f(kHz)
60
Growth and decay rates vs. mode number
1.0
80
Decay rates 0
4
2
6 8 10 Mode number m
12
-1.0
14
0
2
4
6 8 10 Mode number m
12
14
Growth and decay rates (Horton-Tajima-Kamimura)
Linear dispersion relation (Horton-Tajima-Kamimura)
Growth Rate (103 /s)
Frequency (kHz)
15 30 20 10 0
0
2
4
6 8 10 Mode number m
12
10 5 0 -5 -10 -15
14
0
2
4
6 8 10 Mode number m
12
14
Figure 3.5: Frequency and growth rate as a function of kθ a for the model of the equilibrium radial electric field measured in the plasma. There are two modes: the unstable Kelvin Helmholtz mode and the neutrally stable modes with odd symmetry corresponding to a wavy motion of the jet. On the other hand we see that for every growing mode there corresponds a damped mode, due to the Hamiltonian structure of the system, namely, modes with eigenvalue ω k− . If we include dissipation from viscosity and resistivity, the Hamiltonian symplectic structure is broken. However, dissipation is more important for the high k modes. The general wave function is of the form " # X ϕ(x, y, t) = ϕ0 (x) + < eγk t ϕk (x)ei(ky−ωk t) k
28
(3.30)
Figure 3.6: Stream function ψ = φ/B from the iso-potentials of the unstable eigenmode of Eq. (3.12). The last frame in the saturated state shows the alternation of the vortex directions across the jet.
where 1
(3.31)
(3.32)
In the turbulent flow layer there is a broad band of fluctuation frequencies that are difficult to associate with the linear eigenmode frequencies shown in table 3.1. We have computed the power spectrum of the floating potential and the vorticity for different time series ∆T = NT ∆t from NT = 256 to 2048 and averaged the results of the spectra over the M ensembles defined by dividing the stationary section of the bias pulse into M records of length NT ∆t. The result shows a broad band frequency spectrum where the power decreases as f −3 from f1 = 1/NT ∆t ∼ 1kHz to fmax = π/∆t = 1.6M Hz in the case where NT = 1024. Thus, the bulk of the power is at the low frequency end of the spectrum.
30
3.5
Nonlinear simulations Nonlinear simulations are performed using a pseudo-spectral code that solves a non-
linear system of two field equations. The solution domain corresponds to a two dimensional slab in which the x direction corresponds to the radial LAPD direction and the y axis correspond to the azimuthal coordinate, that is y → rφ. The pseudo-spectral method uses a Chebyshev basis along x to resolve the equilibrium profiles and to allow for general boundary conditions, like Dirichlet or Neumann. For the y dependence of the solution field, a Fourier decomposition is naturally used, due to the periodicity of the fields along y. Details of this method and the code are given in appendix A Fully nonlinear simulations with similar parameters for the actual experiment are performed. The initial state is taken to be a localized jet plasma flow with some small random noise perturbation smeared out by a Gaussian filter in transform space. Contour plots of selected times in the nonlinear evolution, for electrostatic potential and its associated vorticity, are shown in Fig. 3.7. Initially, there are two vortex layers associated with the initial flow field in an unstable configuration. Short wavelength perturbations, m =6,7,8,9 growth faster in the linear stage, in accordance with the linear model, as shown in the two top frames of Fig. 3.4. As the instability saturates, the nonlinear term acts to generate a chain of counter rotating vortices by the self advection of vorticity in the plasma. Merging of same sign small scale vortices gives rise to bigger vortex structures of opposite signs across the shear layer, transferring the energy to larger scales, demonstrating an inverse cascade process which explains the dominance of low mode numbers in the measured spectrum of Fig. 3.4. Simulation also shows that in the saturated state, typical vortex diameters are around dv = 5 − 10 cm in agreement with the estimates from the vorticity probe measurements.
31
Figure 3.7: Proceeding clockwise from the upper left, the figures show four stages in the growth and evolution of the vortices from the jet flow modeled for the LAPD. (Click on figure for animation)
32
Chapter 4 The Helimak Configuration
In this work we attempt to understand different basic plasma phenomena and instabilities in two university-scale plasma confinement devices, namely, the Helimak and the LArge plasma Device. This chapter is devoted to a description of the Helimak geometry, typical plasma parameters, MHD equilibrium and MHD stability.
4.1
The Helimak Configuration In simple terms, the Helimak configuration is a finite realization of the one-dimen-
sional, cylindrical, sheared slab, commonly used in theoretical calculations of plasma turbulence. This correspondence makes possible the comparison of well understood theoretical and numerical models with experimental data. The addition of magnetic curvature and shear are minimal additions to the cylindrical LAPD-type plasma geometry required to introduce the effects of a toroidal confinement geometry. The field curvature generates a charge-dependent drift of the guiding centers which separates electrons from ions. This charge separation is also mathematically equivalent to the one that occurs from the gravitational field, making this configuration suitable for the study of the Rayleigh-Taylor instability occuring in solar and astrophysical plasmas. In addition to magnetic curvature and magnetic shear effects, external electric fields can be applied to this configuration to generate sheared E × B flows, suitable for the study of the Kelvin-Helmholtz instability.
33
4.2
The Helimak Device The Helimak device is one of a class of basic plasma experiments in which selected
characteristics of a fusion plasma are retained in a simpler geometry and with better diagnostics than are possible in major confinement devices. A picture of the device is shown in figure 4.1. The Helimak is a toroidal device with a cross-section shown in Fig. 4.2. The dominant toroidal field, Bφ of order 0.1 T, is produced by a set of 16 toroidal field coils around the vacuum chamber. Three other poloidal field coils are used to produce a weaker vertical field, Bz , which may be varied up to 10 % of the toroidal field by changing the ratio of the current flowing through the toroidal coils (orange in Fig. 4.1) to the current flowing through the poloidal field coils (blue in Fig. 4.1). The magnetic field lines are thus helices spiraling from bottom to top and whose pitch varies with the radius as Figure 4.1: Picture of the Helimak device located in the Robert Lee Moore Hall at the toroidal field decreases as 1/r, where r is the University of Texas at Austin. measured from the vertical symmetry axis of the device. The field line length Lk may be varied from less the 20 m to more than 1 km. The height of the vacuum vessel is Lz = 2H = 2 m, the inner radius is rin = 0.6 m, and the outer radius rout = 1.6 m. Two sets of plates are mounted at the top and bottom, 180 ◦ apart, causing all low-pitched field lines to terminate on their surface, as shown in Fig. 4.2.
34
For the steepest pitches, some field lines terminate on the vessel. Although each plate is electrically isolated, all of them can be connected to the vacuum chamber. The field lines impinge nearly normal to the plates, which are dotted with more than 700 surface-mounted Langmuir probes. The inset illustrates one probe tip with its ceramic insulator protruding through the plate into the plasma.
8 / - + * / - + *
:
A!B CEDF
*,+.-0/ HG HG ;6? @
HG *1+2- /
3!465 +$7 ;6< = > 9
Figure 4.2: Left: Helimak cross section showing the magnetic field lines and probe geometry. Four sets of four conducting plates contain over 700 Langmuir probes as shown. Plates are isolated from each other and can be independently biased, however, in this experiment all of them are connected to the vessel. Right: Vacuum magnetic field lines spiraling from bottom to top with radially varying pitch.
The working gases, helium or argon, are ionized to form the plasma and they are heated up with PECH = 6 kW of microwave power at the electron cyclotron frequency. The power is admitted through an open waveguide on the inner side of the vacuum chamber, which we call the high field side (HFS). Since the single-pass absorption is small in this experiment, the vacuum chamber forms a highly over-moded, low Q cavity. In this sense, the heating system is similar to a microwave oven. In argon at a neutral density of 4×10 11 cm−3 , typical plasma parameters are ne = 1011 cm−3 , Te = 10 eV, and Ti < 0.1 eV. The full set of parameters is listed in Table 4.1. 35
Table 4.1: Helimak important typical plasma parameters Electron gyro-frequency Ion gyro-frequency Electron plasma frequency Ion plasma frequency Electron collision rate Ion collision rate Debye length Electron gyro-radius Ion gyro-radius Ion-Acoustic gyro-radius Electron thermal velocity Ion thermal velocity Ion sound velocity Alfven velocity Electron drift velocity
fce = 2.80 × 109 Hz fci = 3.80 × 104 Hz fpe = 2.84 × 109 Hz fpi = 1.05 × 107 Hz νe = 1.31 × 105 sec−1 νi = 2.08 × 106 sec−1 λD = 7.43 × 10−3 cm ρe = 7.53 × 10−3 cm ρi = 1.12 × 10−1 cm ρs = 2.05 cm vT e = 1.32 × 108 cm/s vT i = 2.68 × 104 cm/s cs = 4.9 × 105 cm/s vA = 1.09 × 108 cm/s vE = 1.50 × 106 cm/s
This configuration has a simple stable MHD equilibrium (see Ref. [29, 30, 5]) in which the charge separation from vertical drifts is largely neutralized by a return j k . A small j⊥ , required for force balance, flows to the end plates and returns through the conducting vessel. The configuration has been used in other experiments like in TORPEX at Ecole Polytechnique F´ed´erale de Lausanne, Switzerland [31] and the Blaamann device at the University of Trompso in Norway [32]. The Helimak experiment differs from those principally in size, having dimensions large compared with all scale lengths, including those for density and temperature gradients. This experiment operates in a steady state with stationary conditions for tens of seconds, giving excellent statistics for the turbulence data. The heated plasma flows along the field lines, establishing a pressure profile with B · ∇pe = 0 on the time scale H/cs ∼ 1 − 10 µs forming nested cylindrical surfaces of constant pressure. The pressure gradient ∇p = dp/drˆ er is balanced by the predominant force j × B ≈ jz Bφ . 36
4.3
Helimak Equilibrium For the Helimak configuration described in the previous section an MHD stable equi-
librium exists, as extensively described in reference [5]. We will present the most important results of this reference regarding the MHD equilibria on which we operate in the reported experiment. It is generally well known that in a purely toroidal magnetic field no MHD equilibria exists. The charge dependent magnetic curvature drift and gradient drift gives rise to a charge separation that can quickly establish large vertical electric field producing radial E × B plasma flows towards the wall. However, in the Helimak there is a small but finite vertical magnetic field component that allows for a vertical current shorting out the polarization fields due to the gradient B drift. This vertical current finds its closed path on the inner and outer conducting walls of the vacuum chamber. As a first approximation we model the Helimak to be uniform along the vertical direction with standing mode boundary conditions at the top and bottom plates. Then we assume the equilibrium has ∂/∂φ = ∂/∂z = 0 and equilibrium fields are p = p(r)
(4.1)
ρ = ρ(r)
(4.2)
v = vφ (r)eφ + vz (r)ez
(4.3)
B = Bφ (r)eφ + Bz (r)ez .
(4.4)
The force balance equation ρv · ∇v = j × B − ∇p takes the form d dr
µ ¶ ρvφ2 B2 µ0 Iz dIz p+ z + 2 2 − =0 2µ0 4π r dr r
where µ0 Iz (r) = 2πrBφ is the net vertical plasma current inside the radius r. 37
(4.5)
The linear ideal MHD stability of this equilibrium, given by Luckhardt [5], leads to the local Suydam criterion with the magnetic shear kk = kz (r − rs )/Ls · ¸ ¯ 2µ0 p0 (r) r q 0 (r) 2 ¯¯ + >0 ¯ Bz2 (r) 4 q(r) ¯
(4.6)
r=rs
near the resonant surface r = rs , defined as the surfaces for which the helical mode (l, m) follows the pitch of the magnetic field. Here q is equivalent to the tokamak safety factor
q(r) =
HBφ πrBz .
The shear length Ls defined through the local kk is 1/Ls = 2Bz /rBφ while
the connection length is given by Lc = 2HB/Bz . The current Iz = Izext + ∆Ip in Ampere’s Law 2πrBφ = µ0 Iz (r) is composed of the external current provided by the poloidal coils Izext and the induced plasma current ∆Ip . The current ∆Ip ∼ 100 A can be neglected when compared to the external current of I zext ∼ 10 kA. Thus, in the rest of the paper we use the vacuum magnetic field as the dominant field in the Helimak, so that the MHD stability criteria becomes βp =
2µ0 hpi Bz2
