Dusty plasmas in basic science, astronomy, industry & fusion
description
Transcript of Dusty plasmas in basic science, astronomy, industry & fusion
Dusty plasmas in basic science, astronomy, industry & fusion
John GoreeThe Univ. of Iowa
The growth of dusty plasmas as a field of research
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YEAR82 84 86 88 90 92 94 96 98 00 02 04
N = N0 e( T / 3.9 )
Inspec DatabaseSubject Heading: Dusty Plasma(s)
Outline
1. What is dust?2. Formation of dust
• Fusion• Industry• Astronomy
3. Dust charge4. Forces acting on dust5. Some physics experiments:
• Voids under microgravity conditions• Strongly-coupled vs. Weakly-coupled
Plasmas• Waves & Instabilities• Shear flow• Wakes
What is dust?“Dust” = small particles of solid matter, 10 nm – 1 mm, usually
dielectric
Astronomy: “dust”
M16 pillarCredit: NASA, HST, J. Hester & P. Scowen (ASU)
Semiconductor industry“particulates” or “particles”
G.S. Selwyn, Plasma Sources Sci. Tehcnol. 3, 340 (1994)
Safety Issues for fusion
RadiologicalDust:
• activated• retains tritium• ITER safety limit: 350 kg Tungsten dust
Fire & chemical explosionHydrogen:
• stored in dust• released during accidental exposure to:
• air• steam
• ITER safety limit: 6 kg dust allowed on hot surfacesPhil SharpeFusion Safety Program, Idaho National Laboratory
Dust in Fusion Plasmas Workshop2005
What is dust?
Formation of dust
1. What is dust?2. Formation of dust
• Fusion• Industry• Astronomy
3. Dust charge4. Forces acting on dust5. Some physics experiments:
• Voids under microgravity conditions• Strongly-coupled vs. Weakly-coupled
Plasmas• Waves & Instabilities• Shear flow• Wakes
Formation
Produced in the gas phase• Nucleation• Coagulation
Purchased from vendor
What’s the source of dust in a plasma?
Produced on surfaces• Flaking of deposited films• Bubbles & blistering of
surfaces
Fusion: various shapes of dust collected from the Tore Supra tokamak
Phil SharpeFusion Safety Program, Idaho National Laboratory
Dust in Fusion Plasmas Workshop2005
Formation: tokamaks
Composition is mainly: • carbon• constituents of stainless steel
Formation: tokamaksTungsten dust formation: flaking from He bubbles
N. Ohno, S. Takamura, Dai. Nishijima“Formation and Transport of Dust in the Divertor Plasma Simulators”
Dust in Fusion Plasmas Workshop2005
Divertor Plasma Simulator NAGDIS-II
2 m
Surface Temp.: 2200 KFlux: 8.3×1022 m-2s-1
Ion Energy: 15 eVTime: 104 s
Dust
Poloidal Limiter
High Z dust is emitted from the Mo poloidal limiter.
Observation of High Z Dust in TRIAM-1M by Fast Framing Camera, 4500 fps
Formation: tokamaks
N. Ohno, S. Takamura, Dai. Nishijima“Formation and Transport of Dust in the Divertor Plasma Simulators”
Dust in Fusion Plasmas Workshop2005
A lesson from the semiconductor industryParticles were always there, but you didn’t know it until you used the right diagnostics:
G.S. Selwyn, Plasma Sources Sci. Tehcnol. 3, 340 (1994)
cameraimagingin-situ
electron microscopy ex-situ
Formation: gas phaseGas-phase formation in
astrophysics:
• Vapor flowing outward from a carbon star cools & nucleates dust
• Dust grains then grow by “coagulation”
M16 pillar, Credit: NASA, HST, J. Hester & P. Scowen (ASU)
Formation: gas phaseGas-phase
formation
G. Praburam and J. GoreeAstrophys. J 1995
Formation: gas phaseCauliflower particles grow in the gas phase:
Gary Selwyn, IBM, 1989 Ganguly et al., J. Vac. Sci. Technol. 1993
intact fractured
Formation: gas phase
Particles grownby sputtering tungsten
D. Samsonov and J. GoreeJ. Vac. Sci. Technol. A 1999
300 nm
Coagulated particles consisting of 3+ cauliflowers
Formation: gas phase
Particles grownby sputtering graphite
D. Samsonov and J. GoreeJ. Vac. Sci. Technol. A 1999
Formation: gas phase
Particles grownby sputtering aluminum
D. Samsonov and J. GoreeJ. Vac. Sci. Technol. A 1999
Polymer microspheres:• melamine-formaldehyde• diameter 8.09 0.18 m• used in basic science experiments• introduced into plasma with a “salt shaker”
Formation: purchased from vendor
Outline
1. What is dust?2. Formation of dust
• Fusion• Industry• Astronomy
3. Dust charge4. Forces acting on dust5. Some physics experiments:
• Voids under microgravity conditions• Strongly-coupled vs. Weakly-coupled
Plasmas• Waves & Instabilities• Shear flow• Wakes
Charging: mechanisms
Charging by collecting electrons and ions only negative charge
Ielectron collection
+ Iion collection
H+
e-
_
Goree, Plasma Sources Sci. Technol. 1994
Ielectron collection
+ Iion collection
+ Ielectron emission
H+
e-
e- +Electron emission
•secondary emission due to e- impact•photoemission•thermionic
positive charge
Charging: mechanismsParticles immersed in a plasma collect currents:
Itotal = Ielectron collection + Iion collection + Ielectron emission
Each of these currents depends on the potential V of the particle
Goree, Plasma Sources Sci. Technol. 1994
Equilibrium:
Itotal = 0 at the “floating potential” V:
Q = CV
C = a
is capacitance of sphere of radius a
H+
e-
e-
a
surfacepotential V
Charging: mechanismsCharging by collecting electrons & ions only
Consider a particle that is suddenly exposed to plasma:
• Initially it collects electrons more rapidly than ions, due to higher vte
• Eventually it reaches equilibrium “floating potential”:
Hydrogen, Ti = Te
V = -2.5 kTe
• Example:Parameters:
Te= 1 eV
a = 1 mCharge: Q = - 1737 e
Goree, Plasma Sources Sci. Technol. 1994
Ielectron collection
+ Iion collection
H+
e-
_
Forces
1. What is dust?2. Formation of dust
• Fusion• Industry• Astronomy• Pure physics
3. Dust charge4. Forces acting on dust5. Some pure physics experiments:
• Strongly-coupled vs. Weakly-coupled Plasmas
• Waves & Instabilities• Shear flow• Wakes
Forces
Forces acting on a particle
Ion drag a2 big for high-density plasmas
Radiation pressure a2 if a laser beam hits particle
Gas drag a2 requires gas
Thermophoretic force a2 requires gas
Coulomb QE a provides levitation
Lorentz Q v B a tiny except in astronomy
Gravity a3 tiny unless a > 0.1 m
Ion drag force
Orbit force: Ion orbit is deflected
Collection force: Ion strikes particle
_ _
Momentum is imparted to the dust particle
Void is due to ion drag
D. Samsonov and J. GoreeInstabilities in a Dusty Plasma with Ion Drag and IonizationPhysical Review E Vol. 59, 1047-1058, 1999
Dust (laser light scattering from a horizontal laser sheet)
Glow
Ion drag force
Plasma: • RF parallel-plate• glow discharge• argon gas
Dust:• nm size• carbon• grown by sputtering
graphite target
Void is due to ion drag
D. Samsonov and J. GoreeInstabilities in a Dusty Plasma with Ion Drag and IonizationPhysical Review E Vol. 59, 1047-1058, 1999
Dust (laser light scattering from a horizontal laser sheet)
Glow
Ion drag force
Plasma: • RF parallel-plate• glow discharge• argon gas
Dust:• nm size• carbon• grown by sputtering
graphite target
How ion drag produces a void:
J. Goree, G. E. Morfill, V. N. Tsytovich and S. V. Vladimirov, Theory of Dust Voids in Plasmas, Physical Review E Vol. 59, 7055-7067, 1999
Ion drag force
Ionization sourcePositive plasma potl
Outward ion flow
dust void
Ion drag force
J. Goree, G. E. Morfill, V. N. Tsytovich and S. V. Vladimirov, Theory of Dust Voids in Plasmas, Physical Review E Vol. 59, 7055-7067, 1999
E. C. Whipple, Rep. Prog. Phys. 44, 1198 (1981)
Two contributions:
• Orbit force (this is the usual drag force for Coulomb collisions, except that ln is problematic)
• Collection force (ions actually strike the particle)
Depends on ion velocity ui
Force ni
Orbit forcefrom Rosenbluth potential
Collection forcefrom OML model
1
10
0.01 0.1 1 10 100io
n dr
ag fo
rce,
nor
mal
ized
ion velocity / ion thermal velocity
Te / Ti = 60, mi = 40 amu, D = 130 mIon drag is normalized by 4 ni a2 Te / (Ti/Te)0.5
V-2
V
V2
Ion drag force
0.1
1
10
100
1000
0.01 0.1 1 10
ion
drag
forc
e, n
orm
aliz
ed
ion velocity / ion thermal velocity
Ion drag force
Data computed March 2005 using the same code as in J. Goree, G. E. Morfill, V. N. Tsytovich and S. V. Vladimirov, Theory of Dust Voids in Plasmas, Physical Review E Vol. 59, 7055-7067, 1999
• Fusion edge plasma parameters:
• Te = Ti, deuterium mass
Te / Ti = 1, mi = 2 amu, D = 13 mIon drag is normalized by 4 ni a2 Te / (Ti/Te)0.5
Ion drag force
Gas drag
molecular-flow regime
Epstein:
• Ngas gas atom: number density• mgas mass• cgas mean thermal speed • V velocity of particle with respect to
the gas• dimensionless, ranges from 1.0 to
1.442
P. Epstein, Phys. Rev. 23, 710 (1924).
M. J. Baines, I. P. Williams, and A. S. Asebiomo, Mon. Not. R. Astron. Soc. 130,
63 (1965).
VacmNF gasgasgas2
34
Gas drag force
Stokes-flow regime
Radiation pressure force
Radiation pressure
B. Liu, V. Nosenko, J. Goree and L. Boufendi, Phys. Plasmas (2003).
cIaqF laser
laser
2
incident laser
momentum imparted to microsphere
transparent microsphere
cIaF laser
radiation
2
Physics experiments
1. What is dust?2. Formation of dust
• Fusion• Industry• Astronomy• Pure physics
3. Dust charge4. Forces acting on dust5. Some physics experiments:
• Microgravity conditions• Strongly-coupled vs. Weakly-coupled
Plasmas• Waves & Instabilities• Shear flow• Wakes
Physics experiments
Remainder of this talk:
All experiments performed with polymer microspheres
EquipotentialContours (RF glow discharge)
electrode
electrode
positive
potential
electrode
electrode
With gravity, particles sediment to high-field region 2-D layer
Without gravity, particles fill 3-D volume
QE
mg
Microgravity conditions
Microgravity conditionsTo obtain a 3D dust suspension, use zero g conditions:Parabolic flights, NASA KC-135
Parabolic flights, NASA KC-135
Microgravity conditions
Parabolic flights, NASA KC-135
video
Microgravity conditions
“strongly coupled”dusty plasma: Q bigstar interior: r smallpure-ion plasma: T small
Strongly-coupled vs. weakly-coupled plasmas
> 1 plasma is like a solid or a liquid << 1 plasma is like a gas
TkrQ
B
02 4/
energy kinetic particleenergy potential cleinterparti
Coulomb coupling parameter:
Physics experiments
Next:
Waves in a weakly-coupled dusty plasma
Parameter:gas: Argonp = 1.0 .. 2.5 Pani = 1015 m-3
B = 20 .. 80 mT
dust:MF-spheresd = 1 µmnd = 0.5 .. 3 x 1011 m-3
particles
anodic plasma
anode
3 cm
RF-discharge
camera
dust tray
plasma column probes
Anode:UA = 50 .. 100 VIA = 3 .. 12 mA
Dusty Plasma Research, A. Piel, 2005 41
Courtesy Alexander Piel, Kiel University, Germany, 2005
Dust acoustic wave experiment: Kiel Univ.
15 mm
Time lapse 1:10 p = 2.5 PaIA = 10 mAB = 20 mT
Dust acoustic wave experiment: Kiel Univ.
Dusty Plasma Research, A. Piel, 2005 42
Courtesy Alexander Piel, Kiel University, Germany, 2005
Physics experiments
Next:
Shear flow in a strongly-coupled dusty plasma (plasma crystal).
Shear flow in a 2D dusty plasma
two Ar+ laser beams:• 0.61 mm width• rastered into vertical sheets
vide o cam e ra(top v iew)
lower elec trod e
RF
m icros pheres
VCR
yx
Ar laserbeam 1
Ar laserbeam 2
m od ulator
scanningm irrors
m od ulator 1
scanningm irrors
vide o cam e ra(sid e view)
Ar+ laser pushes particles
low power: slow deformation, rotationmedium power: plastic deformation, flow high power: melting the lattice
undisturbed monolayer
Transport: radiation pressure
Shear flow in a strongly-coupled dusty plasma
Zoom-in view
A 2D liquid, observed at an atomistic level
Shear flow in a strongly-coupled dusty plasma
Video data:
Data recorded: x & v for each particlei.e., a kinetic approach
Next step in analysis: convert to a continuum approach, by spatial averaging
Shear flow in a strongly-coupled dusty plasma
particle’s x,y position measured in each video frame
Velocity profiles
-6
-4
-2
0
2
4
6
-5 0 5
distance y (mm)
parti
cle
velo
city
vx (m
m/s
)
Plaser
= 3.41 W
2.00 W
1.44 W
1.04 W0.82 W
Shear flow in a strongly-coupled dusty plasma
Navier-Stokes equation
vv)(/3)/(vvvvEp
t
21
v fluid velocity areal mass density
(2D)p pressure (2D)/ kinematic viscosity
(2D) second viscosity (2D)E gas drag
0)(v)(v2
2
ydy
ydx
Ex
Navier-Stokes equation reduces to:
kinematic viscosityE gas drag coefficient
Our experiment:
2D
/t = 0
/x = 0
vy = 0
Navier-Stokes equation
Velocity profiles fit to Navier-Stokes
-6
-4
-2
0
2
4
6
-5 0 5
distance y (mm)
parti
cle
velo
city
vx (m
m/s
)P
laser = 3.41 W
2.00 W
1.44 W
1.04 W0.82 W
Results: viscosity vs. inverse temperature
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1
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4
100 1000
kine
mat
ic v
isco
sity
(
mm
2 /s)
coupling parameter
y
0.36
0.53 0.42
r / w Dwater at STP (3D)
high temperature
viscosity hasa minimum
low temperaturey 1/Ty
Physics experiments
Next:
Waves in a strongly-coupled dusty plasma
Waves: two modes in a lattice
S & P waves in seismology
Only the P wave passes through the core of Earth – because the core is liquid
Freq
uenc
y
Theory for a triangular lattice, 0°Wang, Bhattacharjee, Hu , PRL (2000)
0
1
2
3
0 2 4wavenumber ka/
acoustic limit
compressional
shear
Wave dispersion relation – 2D triangular lattice
Longitudinal wave
4mm
k Laser incident here
f = 1.8 Hz
Nunomura, Goree, Hu, Wang, Bhattacharjee Phys. Rev. E 2002
Random particle motion
No Laser!
4mm
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
Wave spectrum
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
6.0
4.0
2.0
0.0
Longitudinal mode6.0
4.0
2.0
0.0
k (mm-1)
f (H
z)f (
Hz)
ka/-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0
/
0
3.0
2.0
1.0
0.0
4.0
/
0
3.0
2.0
1.0
0.0
4.0
5
10
15
Ene
rgy
dens
ity /
k BT
(10-
3 mm
2 s)
k
a
= 0°
Transverse mode
& sinusoidally-excited waves
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
Mach Cones
Courtesy of Dan Dubin, UCSD
Mach cone angle
C = U Sin U
kSupersonic disturbance
Acoustic wavefronts
cone
Courtesy of Dan Dubin, UCSD
Lateral wakeTransverse Wake
Ship’s wake
k
Courtesy of Dan Dubin, UCSD
water
airWake pattern is
determined bydispersion relation
Mach cone
Lateral & transverse wakes
k
kHas both features:
• Mach Cone• Lateral & transverse wakes
plasma crystal
Dan Dubin, Phys. Plasmas 2000
Wakes in a dusty plasma
V/CL = 1.17
Mach cone excitation
Nosenko et al. PRL 2002
Speed map for compressional Mach cone
particle speed v (m/s)
The Early History of Dusty Plasmas
• The first observations of a dusty plasma in the laboratory were made by Langmuir.
• He reported these observations on September 18, 1924 at an address at the Centenary of the Franklin Institute in Philadelphia.
• “. . . we have observed some phenomena of remarkable beauty which may prove to be of theoretical interest.”
Langmuir, Found and Dittmer, Science, vol. 60, No. 1557, p 392 (1924)
A
2 – 4 Torr Argon
tungstenglobules
0.01 -0.1 mm
negativeparticles
C
S
Langmuir’s Streamer Discharge
Langmuir’s Observations
• small tungsten ‘globules’ were sputtered into the discharge from the filament
• these globules could be made to move upward and their motions could easily be followed visually
• by concentrating a beam of sunlight into the tube, he could see a ‘very intense scattering’ from the particles
Langmuir’s conclusions
• Langmuir concluded that since the walls of the tube are negatively charged, the particles must also be negatively charged because they do not deposit on the walls
• the negatively charged particles is surrounded by a positive ion shielding cloud
• the negative particles can lose their charge when moving through an ion sheath, and the resulting neutral particles can condense into larger solid particles
Formation: gas phaseGas-phase
formation resulting from graphite sputtering:
• Graphite targets were sputtered by Ar+ in a glow discharge
• Particles grew in the gas phase
• Particles (white) are imaged here resting on the graphite lower electrode
G. Praburam and J. GoreeCosmic Dust Synthesis by Accretion and CoagulationAstrophysical Journal Vol. 441, pp. 830-838, 1995
Formation: gas phaseGas-phase
formation resulting from graphite sputtering:
• Graphite targets were sputtered by Ar+ in a glow discharge
• Particles grew in the gas phase
• Particles (white) are imaged here resting on the graphite lower electrode
G. Praburam and J. GoreeCosmic Dust Synthesis by Accretion and CoagulationAstrophysical Journal Vol. 441, pp. 830-838, 1995
Formation: gas phaseGas-phase
formation resulting from graphite sputtering:
• Graphite targets were sputtered by Ar+ in a glow discharge
• Particles grew in the gas phase
• Particles (white) are imaged here resting on the graphite lower electrode
G. Praburam and J. GoreeCosmic Dust Synthesis by Accretion and CoagulationAstrophysical Journal Vol. 441, pp. 830-838, 1995
Formation: gas phase
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
Log lambda used in code log_lambda = max([3.,alog(debye_length / max([b_c,b_pi]))]) ; John's ad-hoc Coulomb logarithm ; corresponds to impact parameters ranging from ; the one that causes pi/2 scattering or collection on grain ; whichever is bigger, to the Debye length ; the outermost max function assures a nearly zero log lambda if the ; Debye length is shorter than the other length ; minimum value of 3 is suggested by Tsytovich (private communication)
Formation: gas phaseExplanation proposed for cauliflower shape:
The origin of the bumpy shape has been attributed to columnar growth.
If true, column size will depend on temperature
J.A. Thornton, J. Vac. Sci. Technol. A 11, 666 (1974).
columnar growth, for thin films on a planar surface, using sputter deposition
Formation: gas phase
lo we r e le c tro d e
up p e r e le c tro d e
Gas-phase formation resulting from sputtering:
• Targets were sputtered by Ar+ in a glow discharge• Particles grew in the gas phase
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
tu rb op um p
g rou nd ed p la te
v acu um v esse lin s id e w all
g as in le t
p ow e rede lec trod e
g ro u n d edelec trod e
la se r sh ee t= 4 88 n m
4 c m
g ro u n dsh ie ld
Formation: gas phaseGas-phase
formation resulting from sputtering:
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
dust
partic
le dia
met
er (n
m)
100
400
300
200
00 50 100 150 200 250 300
tim e (se c )
10 6
10 7
10 8
10 9
10 10
dust
num
ber d
ensity
(cm-3
)
Growth of carbon particles, from sputtering graphite in an rf discharge
Formation: gas phase
Particles grownby sputtering titanium
Spherical-shaped primary particles that have coagulated into aggregates consisting of a few spheres.
The surface of theparticles appears smoother than that of the graphite.
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
Formation: gas phase
Particles grownby sputtering stainless steel
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
Formation: gas phaseParticles grownby sputtering copper
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
Formation: gas phase
D. Samsonov and J. GoreeParticle growth in a sputtering dischargeJ. Vac. Sci. Technol. A 1999
Particles grownby sputtering
Growth rate varies tremendously, depending on the material
Charging: electron depletion
0.01
0.1
1
10
0.01 0.1 1 10 100 1000
pote
ntia
l (n
orm
aliz
ed b
y el
ectro
n te
mpe
ratu
re)
normalized particle number density P
floating potentialof particle
plasma potential
-
-
-
-
Electron depletion
When the density N of negatively-charged dust is high:
• Dust potential is reduced
• Dust charge is reduced
• Plasma potential is altered
33 /695 cmcmmeV nNaTP
Goree, Plasma Sources Sci. Technol. 1994
Charging: secondary emission
)/2exp()/(4.7)( mmm EEEEE
Secondary electron emission (electron impact)
For mono-energetic electrons:
Yield
Graphite in bulk:m = 1
Em = 400 eV
For small particles, yield is bigger than for bulk, because of bigger solid angles for secondary electrons to escape particle
Goree, Plasma Sources Sci. Technol. 1994 0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10
yiel
d
/ m
E / Em
Charging : secondary emissionSecondary electron emission (electron impact)
For Maxwellian electrons:
Meyer-Vernet* provides formulae for electron current, result:
• Polarity of particle’s charge switches from negative to positive
• Occurs for Te in range 1 – 10 eV, depending on m
Other electron emission processes:• photoemission due to UV (very common in
space)• thermionic emission (uncommon?)
*Meyer-Vernet, Astron. Astrophys. 105,98 (1982)
Ielectron collection
+ Iion collection
+ Ielectron emission
H+
e-
e- +
Charging: charging time
Typically 1 sec for a 1 micron grain in a glow discharge
3
cmm
eV
naT
K
Goree, Plasma Sources Sci. Technol. 1994
K= -1510 secfor hydrogen, Te = Ti
Charging time
A particle’s charge:• Can change at a finite rate, as plasma conditions change• Fluctuates randomly as individual electrons & ions are collected
Characteristic time scale is called “charging time, can be defined as:• charge / current of one of the two incident species“floating potential V
Charging: stochastic fluctuationsCharge fluctuations
Stochastic, due to collection of individual electrons and ions at random times
Q 0.5 (Q/e)1/2
Goree, Plasma Sources Sci. Technol. 1994
-20
-15
-10
-5
0
0 0 .5 1 .0 1.5 2 .0 2 .5 3 .0
t (m sec)
continuous d iscre te
char
ge n
umbe
r N
charging tim e
Navier-Stokes equation
vv)(/3)/(vvvvEp
t
21
v fluid velocity areal mass density
(2D)p pressure (2D)/ kinematic viscosity
(2D) second viscosity (2D)E gas drag
Comparison: experiment & MD simulation
●equilibriumsimulation
▲ non-equilibriumexperiment
this talk
aQ
0
2
2
normalized by0.1
1
1 10 100 1000
norm
aliz
ed k
inem
atic
vis
cosi
ty
coupling parameter inverse temperature 1/T
Wave spectrum
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
6.0
4.0
2.0
0.0
Longitudinal mode6.0
4.0
2.0
0.0
k (mm-1)
f (H
z)f (
Hz)
ka/-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0
/
0
3.0
2.0
1.0
0.0
4.0
/
0
3.0
2.0
1.0
0.0
4.0
5
10
15
Ene
rgy
dens
ity /
k BT
(10-
3 mm
2 s)
k
a
= 0°
Transverse mode
& theory
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
Formation: tokamaksDust formation: flaking
J. Winter, Phys. Plasmas 7, 3862 (2000)