Shear Alfvén Wave (SAW) Spectra in a Periodic Magnetic ... · •B-dot probe scans for B ......
Transcript of Shear Alfvén Wave (SAW) Spectra in a Periodic Magnetic ... · •B-dot probe scans for B ......
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Shear Shear AlfvAlfvéénn Wave (SAW) Spectra Wave (SAW) Spectra in a Periodic Magnetic Mirror Arrayin a Periodic Magnetic Mirror Array
B-dot Probe Triple
Langmuir Probe
x
y
z
λm
SAW
SAW Antenna
Spectral Gap
SAW frequency
B~
Y. Zhang, W.W. Heidbrink, H. Boehmer, R. McWilliams (UC Irvine)Guangye Chen, Boris N. Breizman (UT Austin)S. Vincena, T. Carter, D. Leneman, W. Gekelman, B. Brugman (UCLA)http://hal9000.ps.uci.edu/Presentations.htmWork supported by DOE DE-FG02-03ER54720
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Periodic Structures Influence Wave Propagation Periodic Structures Influence Wave Propagation in Many Physical Environmentsin Many Physical Environments
nmrq m /~)(qRvATAE 2/~ω
TE/TM TE/TMReflection
l
Fig. 2. Waveguide Bragg ReflectorFig. 3. Optic Fiber Bragg Grating
Cladding
Core
High Index of Refraction Zones
Fig. 1. TAE structure in DIIID(E. M. Carolipio et. al. PoP 8(7) 3391, 2001) Waves in a periodic medium
(or boundary condition)
)]cos(1[0 rKM vv⋅+= εε
?
ToroidalMode Number
,
B0
Fig. 5. Mirror Alfvén Experiment in LAPD
SAW
//kv
Fig. 4. Solid - state Electron Probability Density in Periodic Ion Core Potential
L λmGiven Bragg reflection condition 2k// =km or λ = 2 λm, a propagation gap is centered at fB=vA///2 λm
xkx n22 sin~)(+ψ xkx n
22 cos~)(−ψ
qRL = ,
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Similarities: TAE and Multiple Mirror Similarities: TAE and Multiple Mirror AlfvAlfvéénn ExperimentExperimentTAE Multiple Mirror
Alfvén ExperimentPeriodicity Magnetic field:
B0 (θ)Magnetic field:
B0 (z)Periodic length 2πqR Mirror cell length: λm
Mirror Depth
Wave function ~exp(imθ) ~exp(ik// z)
Physical quantity Plasma displacement: SAW field energy:
Bragg Condition k//=n/2qR kn=nπ/λm
Spectral Gap ExperimentSimulation
Eigenmodes ?
2)(θξ2)(~ zB
Rr /=εminmax
minmaxBBBB
+−
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•Various mirror array configurations are powered by 10 independent magnet power supplies; B0~ 0.5 kG – 2.0 kG.
•Helium plasma column density FWHM ~0.60 m, (1 shot per second cathode discharge)
Mirror Array Configuration in Well Diagnosed LAPD PlasmaMirror Array Configuration in Well Diagnosed LAPD Plasma
23
•Microwave interferometers for column plasma density calibration (port 23) (npeak ≈ 1 x 1012 /cm3)
•B-dot probes for local BSAW measurements (port 14, 16, 18, 20, 36, 38)
14 16 18 20 36 38
•Triple probes for local Te, ni, Vf measurements (port 13, 15, 19, 35)
13 15 3519
•SAW antennas: small disk (p51);
5146 49
copper rod (p46 to p49); rectangular loop (p47)
47
rectangular loop
South North
•LAPD chamber length 20.70 m, diameter 1.0 m; Experimental plasma length 16.54 m.
20.70 m
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MIRROR ARRAY ALFVMIRROR ARRAY ALFVÉÉN EXPERIMENT N EXPERIMENT (UC IRVINE, UCLA)(UC IRVINE, UCLA)
Shear Alfvén Wave (SAW) propagation in a mirror array field
• Local Alfvén speed (vA) varies periodically with mirror magnetic field• SAW refraction in perpendicular direction• Standing wave modes observed at gap frequency—standing wavelength (~λm/2)
observed
Alfvén wave field spectral gaps are observed in various mirror configurations:
• Spectral gap is observed with at least 4 mirror cells in a series • Spectral gap width is linearly proportional to mirror depth• Spectral gap depth is proportional to mirror depth
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v A(m
/s)
v te
(m/s
)Plasma Parameters Vary Between: Plasma Parameters Vary Between: BBmaxmax ((IAWIAW) and ) and BBminmin ((KAWKAW))
IAW
KAW
Inertial Alfvén Wave (IAW)—Electrons respond inertially to E// of the wave.
Kinetic Alfvén Wave (KAW)—Electrons respond adiabatically to E// of the wave.
Port 19 (Bmax)
Port 13 (Bmin)
Port 35 (Bmin)
Port 13 (Bmin)
Port 19 (Bmax)
Electron Thermal Speed
B-dot probe scan Range
Antenna4 Mirror Configuration B0
zPort 13 Port 19 Port 35
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Examples of SAW Antennas and Wave PatternsExamples of SAW Antennas and Wave Patterns
Fig. 1. Dia.=1.0 cm disk antenna [1] / Fig. 2. Rectangular Loop antenna (T. Carter, B. Brugman)
-
-
m=0m=0 m=1m=1
10 cm
10 c
m
[1] W. Gekelman, D. Leneman, J. Maggs, and S. Vincena, Phys. Plasmas 1 (12), 3775 (1994)[2] C. C. Mitchell, J. E. Maggs, S. T. Vincena, and W. N. Gekelman, J. Geophys. Res., VOL. 107, NO. A12, 1469 (2002)
Dia.=0.96 cm, l=96 cm rod antenna. [2]
Bθ
r
PeakAntenna
z
Cone Angle
x-y plane
x-y plane
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170 kHz
50 kHz
90 kHz130 kHz
210 kHzBθ(m
G)
FrequencyFrequency--Spatial SAW Field Scans Show Spatial SAW Field Scans Show ““ConeCone”” PatternPattern
•Left figure shows radial profiles for various SAW launching frequencies. (B=1.0 kG)
The spatial profiles resemble the theoretical prediction below:
250 kHz
z=11.8 m•B-dot probe scans for Bθ (Byalong y=0 line) : 100 MHz maximum sampling rate; 5 to 10 shots averaging at each frequency and each spatial position; 5 kHz frequency scan stepping.
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nmin nmaxnmax
Single Mirror Cell is a Diverging Lens for SAWSingle Mirror Cell is a Diverging Lens for SAW
iA nBv /∝Since , vA maxima correspond to the locations of By maxima and the SAW index of refraction (n) minima.
The mirror throat resembles a biconvex lens with incoming SAW travels FASTER insidethe lens (opposite to a regular optical biconvex lens).
The SAW index of refraction contour at mirror throat
A virtual Diverging lens causing SAW to diverge
B-dot #1 Z=11.80 m
B-dot #2 Z=5.12 m
1 MirrorAntenna Z=0 m
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Multiple Mirrors Disperses Wave Pattern Multiple Mirrors Disperses Wave Pattern RadiallyRadially( z=11.8 m, f=200 kHz)( z=11.8 m, f=200 kHz)
4 Mirror ConfigurationR0~ 5.50 cm“Cone angle”~0.24º
Flat Field 1.0 kGR0~ 3.95 cmCone angle~0.17º
Antenna Z=0 m
B-dot #1 Z=11.80 m
LAPD Cathode Z=16 m
4 Mirror
By (RMS) By (RMS)(m=0)(m=0) (m=0)(m=0)
R0R0
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Spectral Gap Appears with Maximum Number of MirrorsSpectral Gap Appears with Maximum Number of Mirrors
•The peak amplitude of the Alfvén-cones are plotted against frequency.
•From Bragg reflection condition
fB=vA///2 λm =120 kHz, λm = 3.6 m.•(Shaded area corresponds to large under-estimate of intensity due to truncated radial profiles.)
4 Mirror
Flat Field 1.0 kG
2 Mirror
1 Mirror
z= 5.12 m
1 Mirror z=11.8 m
fB
Bθ(m
G) Spectral
Gap
2 Mirror
Antenna Z=0 m
B-dot #1 Z=11.80 m
B-dot #2 Z=5.12 m
4 Mirror
1 Mirror
Flat1.0 kG
LAPD Cathode Z=19.52 m
TOP: Varying number of mirror cells
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Gap Width is Proportional to Mirror Depth (Gap Width is Proportional to Mirror Depth (BBmaxmax--BBminmin)/()/(BBmaxmax++BBminmin))
Normalized Bθ Versus Normalized Frequency (f/fB)
Shallow Mirror
1.05-1.35 kG
Antenna Z=0 m
B-dot #1 Z=10.24 m
Flat Field 1.2 kG
Base Mirror0.9-1.5 kG
Deep Mirror0.81-1.67 kG
LAPD Cathode Z=17.92 m
Maximum Error bar
•Bragg Frequency: fB=vA///2 λm , λm = 3.6 m; vA//is an error source for fB .
•Maximum error of normalized Bθ: ±0.04 (a.u.)
•Gap width (BTrough) is (inversely) proportional to mirror depth
TOP: Varying mirror depth
Gap Width
Gap Width
Gap Width
BTrough
BTrough
BTrough
BTrough
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Spectral Gap Features Spectral Gap Features vsvs Mirror Depth (BMirror Depth (Bmaxmax--BBminmin)/2B)/2Baverageaverage
Mirror Depth (%)0 10 20 30 40
Spe
ctra
l Gap
Wid
th (k
Hz)
0
20
40
60
80
100
120
Mirror Depth (%) vs Gap Width (kHz) Linear Fit
Mirror Depth (%)0 10 20 30 40
B T
roug
h (a
.u.)
0.0
0.2
0.4
0.6
0.8
1.0
Ref
lect
ivity
0.0
0.2
0.4
0.6
0.8
1.0
Mirror Depth (%) vs B Trough (a.u.) Mirror Depth (%) vs Reflectivity
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Standing Wave Mode Observed at Gap FrequencyStanding Wave Mode Observed at Gap Frequency
~λm/2 standing wavelength observed 2λm ?
Antenna4 Mirror Base ConfigurationB0
B-dot Signal (By) Port 14 to Port 20
z
50 cmmG mG
f= 75 kHzf= 75 kHz
f= 175 kHz (Gap frequency)f= 175 kHz (Gap frequency)
f= 275 kHzf= 275 kHz
B-dot Signal (By) Port 36 to Port 38
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MIRROR ARRAY ALFVMIRROR ARRAY ALFVÉÉN SIMULATION N SIMULATION WITH EXPERIMENTAL PLASMA PARAMETERSWITH EXPERIMENTAL PLASMA PARAMETERS
(UT AUSTIN, UC IRVINE)(UT AUSTIN, UC IRVINE)
m=0 SAW launching and propagation is calculated by a 2D (r,z) field solver for a density profile measured from experiment
Finite mirror array simulation shows similar spectral gap with electron-ion collision and electron Landau damping included
Infinite mirror array simulation shows sharpest spectral gap
The calculated gap location is found to be in good agreement with experiment
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Core Theory for SimulationCore Theory for Simulation
where E and B are the electric and magnetic fields, D is the electric displacement vector, and ja is the antenna current density.
where is the dielectric tensor. In a cold magnetized plasma, we have
where
Maxwell’s Equations
sν Includes: Electron-ion Coulomb CollisionElectron Landau Damping
ie−ν
Landauν
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Checking Simulation with Experiment DataChecking Simulation with Experiment Data
• Cold, axially-uniform plasma column with radius a=0.2m;
• Uniform B0=1kG;
2
624422222
// 41
21
ci
ckckckωωεεω ++−= ⊥⊥
)/( ak π=⊥
•Density radial profile theoretical fitpeak• Non-uniform radially;• Uniform axially.
311103.7 −×≈ cmne
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Infinite Number of MirrorsInfinite Number of Mirrors——Sharp GapSharp Gap
• Simulation uses one mirror with periodic boundary condition: the phase shift of electromagnetic field over the mirror is
• Use measured density profile;
• Electron-ion collision frequency:
• m=0 modes, • waves of frequency between
the two peaks of are significantly depressed.
B0
0φ Lφ
L=4m0φφφ −= Ls
Lk sz /φ=
πφ =s
Spectral GAP
)(102 16 −− ×= sieν
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Computation Setup With Computation Setup With Magnetic BeachMagnetic Beach
• Small Br approximation is used in the beach section so the beach section is set flatter than experiment.
• Due to limited simulation spatial grid size, the effective ion collision frequency is introduced in the beach to strengthen ion cyclotron resonance.
• Electron-ion collision frequency:
beach beach
)(102 16 −− ×= sieν
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Virtual Frequency Scans of Wave Virtual Frequency Scans of Wave Amplitude with Magnetic BeachAmplitude with Magnetic Beach
AntennaB0
Increasing effective collision frequencydue to Landau damping:
Gapωων /Landau∝
• Spectral gap is evident at distant mirror away from antenna as expected.
• better antenna coupling at high frequencies.
• Wave damping due to Landau damping is more significant at high frequencies.
Port 14 Port 36
)(102 16 −− ×= sieν
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Closer Look: Closer Look: Simulation Results and Experiment DataSimulation Results and Experiment Data
)(s102 -16×=−ieν
GAPLandau ωων /10=
AntennaB0
Port 14 Port 36
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Contour Plot of Contour Plot of PoyntingPoynting Flux in z Shows Flux in z Shows Standing Wave at Gap Frequency Standing Wave at Gap Frequency
||||)(BEBE××
= zS
• S=-1: left traveling wave;
• S=1: right traveling wave;
• S=0: standing wave;
B0 : 4 mirrors
Sbeach beach
Ion cyclotron resonance locations.Wave does not propagate beyond.
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CONCLUSIONS AND FUTURE WORKSCONCLUSIONS AND FUTURE WORKS
• SAW travels parallel through Kinetic and Inertial regime in the mirror array plasma;
• Multiple mirror configuration produces an observable spectral gap due to Bragg reflection; Gap width is proportional to magnetic mirror depth;
• SAW standing wave pattern is observed at gap frequencies, although the wavelength differs from what dispersion relation predicts;
• A finite difference code successfully simulates spectral gap features as well as wave propagation
→Acquire wavelength information for straight field and single mirror cell configuration;
→Use kinetic dispersion relation in simulation to include kinetic effect (damping, end reflection);
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Thank You!Thank You!
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Diverging RefractionDiverging Refraction Through Through Single Mirror CellSingle Mirror Cell ObservedObserved
• The peak intensity locations of By Radial profiles are plotted versus wave frequency;
• Single mirror cell (mirror III) causes SAW to diverge perpendicularly;
• The radial profile spreadsout more for higher wave frequencies.
Flat Field 1.0 kG z=11.80 m
By Radial Profile
Peak Amplitude
R0
R0
R0
(cm
)
Single Mirror Single Mirror ConfigurationConfiguration
z=5.12 m
Single Mirror Single Mirror Configuration Configuration
z=11.8 mz=11.8 m
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Small Br ApproximationSmall Br Approximation
•
• For B0zmax = 1500G, B0zmin = 900G, the approximation gives which underestimates the field lines at both ends of LAPD.
• From magnetic flux conservation:
For the B-dot probe radial scan range, maximum angle of rotation:
zBrB
zBB
rrB
rrB z
rz
r ∂∂
=⇒=∂∂
+∂∂
+∂∂
⇒=⋅∇ 00
0000 2
101)(10φφ
)(0 zfz
B z =∂∂
056.0min0
max0 ≈z
r
BB
Bmin Bmax
rmin
rmax
α
r
z
1.8 m
2min
2max
max
min
rr
BB
=
o38.12/
arctan 10max ≈⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
m
rrλ
α
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Computation SetupComputation Setup• The mirror section
follows experiment, i.e. four mirrors with m.
• Use actual plasma parameters measured at LAPD
• Use a buffer section with artificially high collision frequency to absorb waves, mimicking the magnetic beach.
4=ml
buffer buffer
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Frequency scan of wave amplitudeFrequency scan of wave amplitude
• Stopping gap is evident at distant mirror away from antenna as expected.
• better antenna coupling at high frequencies.
• Wave damping due to Landau damping is more significant at high frequencies.
Antenna positionB0
Increasing effective collision frequencydue to Landau damping ( ).1/s102 6×=eν LDν