Plasma 2 Lecture 17: Collisional Drift Waves - APPH E6102y Columbia University - Columbia ...
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Plasma 2
Lecture 17:
Collisional Drift Waves
APPH E6102y
Columbia University
1
1
Drift Wave Instability (and Transport)
2
2UJ 0, 2 8- a
LIJ
sn
I— /8t+u Vn+u Vn +nV g =0, values of magnetic field only one single mode
(ld)
lei
~ ~
gth UJ
es 1 )) le li
is dete cte d, but in the mode transition regions
0.1
~ 4—
UJ
= -Vy x BcB s+KTc(eB n) 'Vn x B, (le)
two separate modes are observed. The rapid
u
le&0.0 1 1 rise of y =-Im(a) with increasing magnetic
(b) field, once the condition b&b~ is satisfied, is
evident. The second point is that the maximum
VOLUME 18, NUMBER 12 PHYSICAL REVIEW LETTERS
growth rate is found to correspond to frequen- 20 MARGH 1967
u. =ETc(+eBan) iVn &B.
3— i, e cies Re(a) = —0. 5k vd. In particular, we find
Perturbed quantities are indicated by the sub-
that for b(b,
the growth rate y is maximized
(2)
UJ COLLISIONAL
script 2
1.~0 Ion EFFECTS
m=2 motion and electron inertia along
for
IN PLASMAS DRIFT-WAVE
the —
dimensionless parameter ZO EXPERIMENTS
=ALII
KT AND INTERPRETATION*
Al=3 x slightly above unity, me
— (mev&&kyvdb)
CF
the lines of force have been ignored.
LU
m=4 0 The
being the electron mass. At this point, the
B. f F. Perkins, P.
U
effects of ion-ion diffusion acrossH.
DRIFT FREQUENCY
the W.f ield Hendel,
e n- g
magnitude Coppi,
of y is -0.2k&gd and is comparable
and A. Politzer
ter through the coefficient which is given
Plasma Physics Laboratory,
0 I I p& I
Princeton
with the instability frequency.University, Figure 1(b)Princeton, shows New Jersey
T/ conditicns, "
by pi 0= —,'(nETvf/Qi 2
k~3 ) for4 our experimental
MAGNETIC FIELD, kG
with Qi the ion gyrofrequency and
that the observed VOI. UMZ 18, NUMsZR 12
frequencies
(Received 25 January 1967) are proportion- PHYSICAL RKVIKW LKTTKRS
al to, but less than, k vd. These considerations
20 MARCH 1967
hz a dimensionless coefficient tabulated in Ref. indicate that the criterion" g =&~, which can
FIG. 1. (a)
11. For simplicity we Observed oscillation amplitudes are com-
assume a WEB-type be written as
1000—
+ =5
—m STABILIZATION
8 =7xi
m=4
pared with theoretical growth rates as a function of mag-
results Torr pressure is
kg
field report on low- 10 (0)
We
solution
netic
=
forthe
(pi epx(ifk
ystrength various azimuthal of
dxx+ikyy+ikiiz+i~t) mode experiments
and
num-
2
+k 2)l/2
and the residual m=2
N=) X
main-
1. consider
bers.
temperature
implies
for
absolute such
The modes value that
mode localization
the theoretical
kx»n
of the
alkali plasmasmagnetic ~(dn/dx).
the x direction
(slab model)in.curves
This
field strength
in strong magnet-
has been scaled
o
0.5—
-- GROWTH
X
B I/
tained /
/
2.0-
at approximately
E
cA
" 10 ' Torr,(
so that the
~ '
RATE k ( 4e4k
I )i.6 —u (Z' 1
ic fields
anda factor
by -1.
of and
will restrict 5 to their
to modes
us give a good interpretation
fit
withto the data. The in terms of
azimuthal 0.4 —
Bplasma,
(M'c KTm v, v. . f fully is kn ionized. The plasma column
M ' 500—
st, relative
mode number
collisional
amplitudem is 1, )defined
driftto themodes,
the ratio
since asthose with ofmthe = 1,maxi- C3
—0.3 over is
8 8Z CL
2'E
and 128 cm long. Plasma
mum density to
equivalent fluctuation
an off-axis central
shift of the
in
density. which
whole (b) The diffusion
plas-
describes the cmof in
3 onset diameter
the single modes. In
Fig. 2(a) we plot the experimental values at
5x10" 5x10"
Cl
oscillation frequency (after subtraction of the rotational
thecolumn,
ma transverse wavelength,
are less localized and cannot resulting
be from ion- instability onset of B/k&densities
(center) vs8-n'~'LIJaandand thisfind theoret- ranged from
agree-
to
- simulated
ionexperiments,
=kyvd/2m
in
collisions, a plane
as a function of the '
Doppler shift) is compared with the drift frequency vd
geometry. Moreover,
magneticanfield
plays important in
strength.
UJ 0, 2
I—
role.
ment between the experiments
ical
cmIn addition, plasma
prediction, ',
including the numerical
(ionizer-plate)
coef- 0
0 temperatures
the drift
The which has where
frequency, k)i=m/L, L is theoflength
an uncertainty +0.5 UJ
ficient. other measurements have
A, ', I05
I/2 I/2
of the
kc/sec,Collisional
plasma
is computed column. drift
from the modes' J1&
expressing
By experimental values arise
andof in the presence
0. 1
from
ion mass 2100
shown dependence on plasma
as predicted by theto
~
UJ
2900'K,
4 —temperature
transition
and
crite- and magnetic fields from I I
- k,of
J1ti terms
T,in snd
a of cp, one
n ~(dn/dx).
—3.5&&10 gradient
density data, arefor
Thederives for Te
perpendicular= Tz the
a potassium
to the 0.0 mag- rion, Eq. (3). ln Fig. 2(b), the radial extent
to is5shown
of the2oscillation 6. Electric
kG.as a function of mag- fields 0- 2.
are
not 0 =3applied to
ost asma, cm
pdispersion relation in its simplest form,
no 3, T =2800'K. }r X
netic field. The extent (b) of the oscillation in the
netic fieM andIKTresult from the combined effects the plasma, and the thermionic voltage between
radial direction was found to increase for de-
rk v, .bmi creasing values of ~, as expected from the }(,t:m
of ion inertia, electron-ion collisions, and end plates is maintained below 5 mV. The waves
full analysis of the normal-mode equation and
ydmv. e ei
4 3—
corresponding to (a/Bg) -k&. The position of
I.O-
mean electron kinetic energy along the magnet- are detected as either ion-density or plasma-
bi.
the amplitude maximum does not coincide with
0.5—
the position of maximum density gradient and L o rior Rod ius
ic-field lines.
I on
iv . 2potential fluctuations with Langmuir probes.
may be determined by the radial dependence
of the growth rate, taking into account both the
(u-k v — Ibu), (2)
UJ ~ m=2
Our experiments y determine frequencies, —
d 2 / am- CF
LU
U
0 Al=3
m=4
DRIFT FREQUENCY
variations of n and dn/dx.
Special effort was directed towards
It is evident from our treatment that the lin- conclusive B, kQ
FIG. 2.
earized approximation is inadequate to explain ratio field strength to
(a) The of magnetic
plitudes, and azimuthal
where vd -(1/n)(dn/dx)(cKT/eB) is the elec-
= mode numbers of steady- identification of the drift wave,
the large experimental amplitudes (typically
root for
since is
itversuswassever-
perpendicular wave number
stabilization
of the density the
square
plotted the
points of
n, /no-ep/KT-20%) at which the growth of the
statediamagnetic
tron drift waves velocity, as '(kx +k ')aL of magnetic field
5 =-,functions
0
recognized that a plasma
0 2
I I I
9.rotation
al modes. Theory
3 waves 4 ceases. However, there are good argu-
FIELD, kG
MAGNETIC ments
tor 10 .
(3)] gives a proportionality fac-
comparable
measuredof radial7X
Eq'.
I.
azi-
(b) The (~z) and
for predicting that the saturation stage perturbation are dis-
is assumed to be smaller than unity, al = (2KT/ muthal (~0} wavelengths of the
strength and ion density. andTheir interpretation
M)imam/Qf is the ion Larmor radius,
with the electron diamagnetic
is reached when the perturbed (ExB)c/B' drift
as a
3 velocity becomes of the order of the diamagnet-velocity field.is pres-
played function of the magnetic
ve&
is based vlf,
=2(M/ms)iI a theory
on taking for v&f thewhichdefinitionincludes the effects 1. FIG. ent in Q machines for certain
(a) Observed oscillation our case -2x10'
ic velocity (inamplitudes ranges of plas-
com- This
are cm/sec).
of
of Ref. 11. Here
ion-ion collisions
we have neglected
on terms
the ionof motion. Although
pared with theoretical growth
ma temperature
netic field strength for various
point isrates
confirmed the experiment
as a byfunction
density. '
lation is l~i-k&vd. In this connection
of mag-
addition, it is shown [see Fig. 1(a)] that the
and
azimuthal
that more detailed
mode num-
and, in
Our experiments
measurements than those
notice we
order k)) ET(mevefkyv&) i in comparison with 1. bers. The absolute value closely of the thatmagnetic
given in Fig. 1(b) have
pattern of the measured amplitudes follows
field strength
= —~. shown &u/k&vd
the observed relative wave
This dispersion relation is a quadratic equa- amplitudes are not differ from most previous
for the theoretical (slab model)a functioncurves
of the calculated
has been
growth rates a, s
The observation of one single mode at a giv-
drift-wave
field.scaled
en magnetic field is explained, within
of the magnetic If we consid- work the lim-in
small, the linearized approximation is expect-
tion whose roots are easily evaluated numer- -1. by a factor of 5 to give era this
that the neutral beam was goodas anfitindicationits of the linearized theory, by the fact that
that at the
to the data. Thesaturation
only one collimated
stage the amplitude is proportional to a pow-
mode has appreciable growth to reduce rate out-
ically, and reveals two important points. First, relative amplitude is defined er of asthe the
growthratio
rate, of thecanmaxi-
'4 we
side the mode-transition regions.
associate the
ed and
there existsfound
a critical predict
to value of b givencorrectly
by bz frequencies the 'ion density at the edge
mum density fluctuation toobserved the central
frequencydensity. Several of
with that of (b)theThe
important
maximum ionizerof general
the considerations plate
' oscillation frequency growth
(after rate.
subtraction of the rotational nature arise from this work. First, the agree-
and the abrupt such
=4k')'KT(meve;vlf) that the linearized
appearance of certain Dopplermodes 3
shift) is comparedlinearwhere
the other hand, the
with
if
the drift frequency
approximation"
temperature
makeOnuse of a quasi-
in treating vd
the problem
gradient
ment
we is predicted
of the theoretically large.
observations, coupled with the fact that
with
frequenciesThis2428 HENDEL, CHU, AND POLITZER
THE PHYSICS OF FLUIDS VOLUME II, NUMBER 11 1968
ion-ion collisions and finite ion Larmor radius) and while ion-fluid viscosity is stabilizing. Equation (7)
resistivity (due to electron-ion collisions). We con- can be expressed
Collisional in terms of density
Drift Waves-Identification, wave and
Stabilization, nand
Enhanced Plasma Transport
sider low-frequency (Re w « ni ) localized waves. Ion potential wave cp. The vorticity can be calculated
motion along lines of force (kllVi ,th « Re w), electron from Eq. Plasma(2),
H.
rLaboratory,
W.
!,hysics
* T. K. CHU,
= -bn.(n/no
HENDEL, AND P.
+ A.
POLI'l'ZER
ecp/KT)
Princeton University, Princeton, , where
New Jersey
inertia (ne = eB/mec » Re w), and perpendicular (Received 24 January 1967; final manuscript received 3 July 1968)
terms whose contributions to Eq. (7) are of the order
resistivity (V'i « ne) are neglected. The convective of bVidni are neglected
?enslty,
drift waves are identified by the dependences of wand k on
field, and ionin comparison
and by.comparisons with those
with a linear of
theory which
mcludes res18tiVlty and vIscosity. Abrupt stablhzatlOn of aZlmuthal modes is observed when the sta-
term and the collision-free part of the ion-fluid stress unity. The axial current gradient can be related 2430
bilizing ion diffusion over the transverse wavelength due to the combined effects of ion Larmor radius to HENDEL, CHU,
and ion-ion (viscosity) balances the destabilizing electron-fluid expansion over the parallel
tensor may be omitted because their contributions wavelength,cp through
nand the equation
electron-i.on of parallel
collisions (resistivity). motion
The finite-amplitude (n/no for 10%)
18 cohere?-t oSClllatlOn, entire plasma body, shows a phase difference between density and
to the equation of continuity for ions cancel. S • The the electrons
waves (whlCh 18 predlCted by lmear theory for growing perturbations). The wave-induced
radial transport ex,ceeds classical diffusion, but is below the Bohm value by an order of magnitude.
o I 2 3 4
·3 -I
first-order linearized equations thus become Although observations have been extended to magnetic fields three times those for drift-wave onset 5 1m Wx10 I sec
Uell = -(ikIlKT/mevei)(n/no - ecp/KT).
turbulence has not been encountered. (8)
' 1500
+ Uil. XcB)
4
no M aUil.
at -_ - K TV l.ni + noe ( - V l.CP Substituting into Eq. (7), we obtain
1. INTRODUCTION a loss mechanism based on the electric field fluctua-
Drift waves can arise in fully ionized, magnetically tions of turbulent plasma instabilities. Although this
a (n ecp )
1 (n ecp ) (n 1 ecp )
_
- nieVd x-
B + J.l.l.
,,2
V l.Uil.,
plasma
b at no + KT fu
confined, 10w-!3 (13 = 87rp/B 2 which is the ratio of view currently prevails and the Bohm diffusion
= no - KT - tl. no
(2) pressure to magnetic pressure) plasmas as a coefficient appears to be close to an experimental and + KT
C
Diamagnetic
result of the combined effects of density gradient, ion theoretical upper limit, enhanced plasma losses have
(1.til _ J:)( n+ _
inertia, and electron parallel motion. According to been 2related to specific instabilities in only a few
Doppler Shift Max= Growth
linear theory, the driving mechanism depends on the cases, and the main loss mechanism for many plasma
1000
o= -KTV l.n, - noe( - V l.CP + Uel. tl. no KT
existence of a phase difference between wave electric devices remains obscure. In low-!3 plasmas,
cancels… 8f ky
til KT'
therefore,
I
o
field and plasma density oscillations. In the collision- special interest is attached to drift (9) waves, which:
gauss-em
1
less regime such phase difference is produced by inter- (1) can occur in the absence of currents; (2) are of
sufficiently low frequency (Re w « Qi = eB/Mc) to
- n,eVd, (3) where l/tll =
action between resonant electrons and the wave, and /m,vei
and l/tl. = lb2vii'
For
in the collision-dominated regime by electron-ion convect ions; (3) are driven only by density (or
temporalThevariation
collisions (resistivity). mechanism forof exp
stabiliza- (-iwt),
temperature) onset of the over-
gradients; (4) have large instability
500
w+
growth rates (I' = 1m w Re w for collisional drift
(4) stable wave
tion, according to linear theory, is provided
(w = in the
Re and in ilm waves); andw) thus depends
(5) have on a
ro../
large perpendicular wave-
collisionless regime by ion Landau damping
lengths (A.l comparable to the density gradient scale
ani
Ti + noVl.'Uil. + ( 't"7)
Uil.· v 1. no - Vd ani
ay =
the strongly
0 effects(5) of ion
,
necessary
This
phase
collisional regime
Damping
by difference
the combined between nand waves.
Larmor radius and ion-ion collisions length no/V no). Drift-wave experiments in thermally
requirement
(viscosity). The specific damping mechanismof phasewhich difference
ionized alkali plasmasis provided
cp
by since ion
are significant
limits the wave amplitude to its saturation level can Larmor radius, wavelength, and growth rate can be
nontrivial values of U,II, Eq. (8). It can be verified
only be deduced from nonlinear theory, and is out- made comparable to those in fusion plasmas.
ane + noV 1. ·Uel. + (
Ti 't"7)
U,i' V 1. no + Vd ay
an, side the from
scope of theEq.present that, for
(9)discussion. Here,positive
"col- Early theoretical
growth work on instabilities
rate'Y =
Imw, in inhomo- o 500 1000
lisional drift waves" designate drift waves in the geneous 10w-!3 plasmas led to the prediction of drift
All' em
the density regime
strongly collision-dominated wavefor which leads n
both the potential
instability cp.
waveand in resistive plasmas. ,5
in collisionless 3 4
+ nOVlluel10, = (6)
parallel According to Eq. (9), the wave
resistivity
considered.
and transverse viscosity must be The effect
motion was
of viscosity
is stabilized associated
discussed as favoring
with
at FIG.
both transverse ion
1. Stability characteristics of density-gradient-driven
stability.6,7
collisional An
drift waves. Contour lines are lines of constant
small ofand
drift large til' For
from large til' the stability criterion
7 equations; 7 unknowns: u , u , u , n , n The importance waves derives their interpretation of observed stabilization
growth rate. of azimuthal
Potassium plasma, T = 2800 oK, no = 1011 em-a,
where Uil.' ni' Uel.' Uell' n. and are perturbed velocities
i e e|| l/tlli < l/tl.' e
possible causal relation to enhanced modes, based on ion diffusion over the transverse
'\lno/no = -1 em-I.
is 10w-!3 plasmas, i.e., and 4 particleonset
losses
and densities of ions cp
and electrons, is the perturbed observed in losses at
above the the propagation velocity
2 For reviews see: F. Boeschoten, J. Nuc!. Energy, Pt. C6,
339 (1964); V. E. Golant, Usp. Fiz. Nauk 79, 377 (1963) [Sov.
electric potential, J.l.l. is the transverse viscosity, 19 and w/k
lower limitReset by classicalisbinary-collision
y nearly zero. diffusion.Invoking
Phys.-Usp.6,the equations
161 (1963)]; ofMod.(1957).
F. C. Hoh, Rev.
F. F. Chen, Phys.diffusion,
Today 10, 115 it
Phys. 34,
may be stabilized as a result of increased
To explain anomalously high plasma losses in arc 267 (1962);
discharges,motion and thethecontinuity equation of electrons,
3 For a review see: B. B. Kadomtsev, Plasma Turbulence
I'd and Vii are the electron-ion and ion-ion collision
in 1949 Bohm 1
proposed existence of (Academic axial
Press Inc., New York, 1965). wavelength. In the limit of short All, the phase
S. S. Moiseev and R. Z. Sagdeev, Zh. Eksp. Teor. Fiz. 44,
address:cp
4
nand
* Permanent RCAcan be related
Laboratories, Princeton, by
New 763 (1963) [Sov. Phys.-JETP 17,difference 515 (1963)]. between density and potential waves
frequencies. Jersey. F. F. Chen, Phys. Fluids 7, 949 (1964).
6
A. A. Galeev, S. S. Moiseev, and approaches Nuc!. when the stability condition is
R. Z. Sagdeev, J.zero
D. Bohm, E. H. S. Burhop, H. S. W. Massey, and R. W.
1 6
For simplicity, consider solutions of spatial varia-
Williams, The Characteristics of Electrical n Discharges(l/t ll ikyVd
in M) ag-- Energy, Pt. ecp
C6, 645 (1964).
and O. approached. Dokl. It should be noted that stabilization at
netic Fields, A. Guthrie and R. K. Wakerling, Eds. (McGraw- A. B. Mikhailovskii (10)
P. Pogutse, Akad.
+ +
7
tion exp (ikxx ikyy Hill Book Company, New York, 1949).
ikllz) assuming the localiza- no = (l/t l ) -NaukiwSSSRKT' 156, 64 (1964) [Sov. Phys.-Dok!. 9,379 (1964)].
short All occurs before the wavelength becomes so
2426
tion condition20 kx » (dno/dx) (l/no). AddingThisEqs. which states that foroflarge
4 Reuse
article is copyrighted as indicated in the article.
tn the nand cp waves short that the assumption of neglect of ion parallel
are
AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions.
Downloaded to IP: 128.59.62.83 On: Thu, 05 Dec 2013 00:51:37 motion breaks down, i.e., at stabilization the parallelOutline
• Review Drift Wave Formalism (last lecture)
• Ion dynamics (⊥) including collisions
• Electron dynamics (||) including collisions
• Characteristics of the collisional drift wave
• Radial (“anomolous”) transport
• What’s next: Landau damping, (low-
frequency) EM: δB⊥
5
5Simple Drift Wave Description
es and transport
“Collisionless”
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
balance equation
S D
6
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
6
much of the low-frequency drift-wave dynamics falls inSimple Drift Wave Description: Parallel Dynamics
“Collisionless”
es and transport
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
balance equation
S D
7
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
7
much of the low-frequency drift-wave dynamics falls inSimple Drift Wave Description: Continuity
es and transport “Collisionless”
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
(only ion E×B drift motion)
balance equation
S D
8
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
8
much of the low-frequency drift-wave dynamics falls inBasic “Drift Wave”
“Collisionless”
es and transport
(only ion E×B drift motion)
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
balance equation
S D
9
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
9
much of the low-frequency drift-wave dynamics falls inIon Inertial Currents (Acoustic & Polarization Drift)
es and transport
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
balance equation
S D
10
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
10
much of the low-frequency drift-wave dynamics falls inIon Inertial Currents (Polarization Drift)
es and transport
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
balance equation
S D
11
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
11
much of the low-frequency drift-wave dynamics falls inis, |V| = |V′ |. In the center-of-mass frame, the scattering angle χ
D 0
defined by Eq. (12.1.2), can be estimated by replacing µ ss′ V 2 by its Maxwellian
average, 3κT ; see Eq. (2.1.4). With these substitutions it is then easy to show that
parameter b obey the relation
the logarithmic term in Eq. (12.2.7) simplifies to !χ" b
0
tan = ,
!2χ " b b
b λ m D
ln = ln
b b 0 0
! "
12πϵ κT 0
tan
0
= ln
e
λ = ln(12πN ) = ln Λ,
2 D (12.2.8) D
= ,
where
Ch. 12: Collisional Processes
where the quantity in the parentheses occurs sufficiently often that it is defined to
be a new parameter,
2
e s e s′
b
where Λ = 12πN , (12.2.9)
D
b0 =
called the plasma parameter. For most plasmas, ln Λ ranges from about 10 to 40.
4πϵ0 µ ss′ V 2
Based on the above analysis, we can now estimate the mean-free path, λ ,
required for multiple small-angle collisions to produce a deflection of the order e s e s′ m
◦
of 90 . Multiple small-angle collisions produce a change ⟨(∆υ ) ⟩/∆ℓ, given by b0 = ⊥
2
′ 2
is the distance of closest
Eq. (12.2.7). To produce a deflection of the order of 90 in a path length, λ , we
require that
approach, V 4πϵ
= |v−
◦
0 µ v
ss ′V | is the relative velo
m
particles
⟨(∆υ ) ⟩
λ = Vat , infinity, (12.2.10)
⊥
2
and 2
′
is the distance of closest approach, V |v− v | is the relative velo
m
∆ℓ
which, after substituting Eq. (12.2.7) for ⟨(∆υ ) ⟩/∆ℓ, gives 2
=
m s m s′
⊥
particles
λ =
1 at
.
infinity,
m 2
and
(12.2.11) µ ss′ =
8πn b ln Λ 0 0
m s + m s′
Next, we calculate the mean-free path for a single 90 large-angle collision, λ ,
which, using Eq. (12.1.6), is given by
◦
m s m s′ S
µ ss′ =
is λthe =
nσ
1
reduced
=
1
n πb
.
S
0
mass.S 0
In the case of electrons
(12.2.12) 2
0
m s + m sof ′ mass me scatter
mass mi , where mi ≫ me , the reduced mass becomes µ ss′ ! me . T
is the reduced mass. In the case of electrons of mass m scatter
cross section σ ss′ (χ), where χ is the scattering angle in thee center-
9C D-
4 4 45 8 4 D-
mass mi , where mi ≫ me , the reduced mass becomes µ ss′ ! me .
64 5C 7:8 C: 6 C8 2846 8CD . 8:8 1 5C4C
64 5C 7:8 C: 6 C8 8C D D- 7
.
C:
543
,
8CD
, ,
04 4 - - D 5 86 8 .4 5C 7:8 . C8
12
cross section σ ss′ (χ), where χ is the scattering angle in the center-
x
∆υy
12 x−7
100 eV plasma J = σE,5 × 10 (12.4.23)
5 keV plasma 1 × 10−9
is then given, in the gas
Interstellar Lorentz gas approximation,
(1 eV) 5 ×by10the −7expression
Solar corona (10 eV) 32π1/2 ϵ 20 (2κT 10−5
5 ×e )3/2
(12.4.24)
Earth’s ionosphere σ(0.1 =
eV)
m1/2
e e 2
2 ln ×
Λ 10 . −2
Plasma Resistivity
It turns out that Eq. (12.4.21), which is derived assuming that electron–electron
collisions are neglected, overestimates the actual plasma conductivity. This occurs
not because electron–electron collisions contribute additively to the effective
number of collisions that electron suffer, but more subtly because electron–electron
collisions change the electron distribution function and hence, the electron–ion
resting to compare the resistivity of fully ionized plasmas
collisional interaction. Note that, except for the weak dependence through ln12.5
at different
Λ, Collision Operator for Maxwellian Distributions of Electrons an
es the
to conductivity
some common depends materials. For example,
only on the electron temperature,inandTable 12.1 we compare
is completely Table 12.1 Comparison of the resistivity of various
independent of the number density.
ties of 100 eV and 5 keV plasmas to those of copper and stainless steel.
types of plasmas with some common materials
The plasma resistivity η is the inverse of the conductivity σ. A more detailed
see, a 100
calculation eVresistivity
of the plasma thathas
takesvery low resistivity,
into account comparable to that of
the effect of electron–electron
Material Resistivity, η (ohm m)
eel,collisions,
but larger than out
first carried thatbyofSpitzer
copper.and HarmHowever, a 5 keV
(1953), yields plasma, typical of
the following
formula for the resistivity: Copper 2 × 10−8
atory fusion experiments, is an order of magnitude less resistive than
Stainless steel 7 × 10−7
−5 ln Λ
η ≃5.2 ×10 ohm m, (12.4.25) 100 eV plasma 5 × 10−7
(κT e )3/2
5 keV plasma 1 × 10−9
q. where
(5.6.8), we can make an estimate of the electron–ion
κT e is expressed in electron volts. To be precise, the above equation gives the
collision
Interstellar gas (1 eV) 5 × 10−7
, given byresistivity parallel to an equilibrium magnetic field. The resistivity
veiSpitzer–Harm Solar corona (10 eV) 5 × 10−5
perpendicular to the magnetic field is approximately twice the parallel resistivity. Earth’s ionosphere (0.1 eV) 2 × 10−2
n0 e 2 n0 e4 ln Λ
νei ≃ ≃ . (12.4.26)
me σ 32π1/2 ϵ 2 m1/2 3/2
0 e (2κT e )
D-
4 45 8 4 D-
64 5C 7:8 C: 6 C8 2846 8CD . 8:8 1 5C4C
64 5C 7:8 C: 6 C8 8C D D- 7
.
C:
543
,
8CD
, ,
04 4 - - It is 8interesting
D 5 86 .4 5C 7:8 . C8 to compare the resistivity of fully ionized plasmas
temperatures
13
to some common materials. For example, in Table 12.1
on was previously used in Section 2.5.2. the resistivities of 100 eV and 5 keV plasmas to those of copper and st
As one can see, a 100 eV plasma has very low resistivity, comparab
stainless steel, but larger than that of copper. However, a 5 keV plasm
large laboratory fusion experiments, is an order of magnitude less r
copper.
13ion plasma frequency fpi = ωpi /2π
= 2.10 × 102 Zµ−1/2 ni 1/2 Hz
ωpi = (4πni Z 2 e2 /mi )1/2
NRL Plasma Formulary
= 1.32 × 103 Zµ−1/2 ni 1/2 rad/sec
electron trapping rate νT e = (eKE/me )1/2
= 7.26 × 108 K 1/2 E 1/2 sec−1
https://www.nrl.navy.mil/ppd/content/nrl-plasma-formulary
ion trapping rate νT i = (ZeKE/mi )1/2
= 1.69 × 107 Z 1/2 K 1/2 E 1/2 µ−1/2 sec−1
electron collision rate νe = 2.91 × 10−6 ne ln ΛTe −3/2 sec−1
ion collision rate νi = 4.80 × 10−8 Z 4 µ−1/2 ni ln ΛTi −3/2 sec−1
Thermal Equilibration
Lengths If the components of a plasma have different temperatures, but no rela-
tive drift, equilibration is described1/2
electron deBroglie length λ̄ = h̄/(me kTe ) by = 2.76 × 10−8 Te −1/2 cm
e+ - e- energy exchange: ! −7 −1
classical distance of e2 /kT = dT1.44
α
= × 10 ν̄ϵα\βT(Tβ −
cmTα ),
minimum approach dt
β
electron gyroradius re = vT e /ωce = 2.38Te 1/2 B −1 cm
ion gyroradius where ri = vT i /ωci
−19 (mα mβ )1/2 Zα 2 Zβ 2 nβ λαβ
ν̄ϵα\β ==
1.81.02
× 10× 102 µ1/2 Z −1 T 1/2 B3/2
−1
cmsec−1 .
i
(mα Tβ + mβ Tα )
electron inertialFor
length
electrons andc/ω = 5.31
ionspewith Te ≈×Ti10
5
≡T ne, −1/2 cm
this implies
ion inertial length = 2.28 × 107 Z −1
e|i c/ωpi i|e −9 (µ/n
2
1/2
i ) 3/2 cm3 −1
ν̄ϵ /ni = ν̄ϵ /ne = 3.2 × 10 Z λ/µT cm sec .
2 1/2 2 1/2 −1/2
Debye length λD = (kT /4πne ) 14 = 7.43 × 10 T n cm
Coulomb Logarithm
For test particles of mass mα and charge eα = Zα e scattering off field
particles of mass mβ and charge eβ = Zβ e, the Coulomb logarithm is defined
28
as λ = ln Λ ≡ ln(rmax /rmin ). Here rmin is the larger of eα eβ /mαβ ū2 and
h̄/2mαβ ū, averaged over both particle velocity distributions,
" where mαβ =
mα mβ /(mα + mβ ) and u = vα − vβ ; r14max = (4π nγ eγ /kTγ )−1/2 , where
219. B. A. Df Trubnikov, “Particle Interactions in a Fully Ionized Plasma,” Re-
i
= ν (F̄ − f ) + ν (F − f ).
ie i i ii i i
viewsDtof Plasma Physics, Vol. 1 (Consultants Bureau, New York, 1965),
p.slowing-down
The respective 105. rates ν given in the Relaxation Rate section
α\β
s
above can be used for ν , assuming slow ions and fast electrons, with ϵ re-
αβ
placed by Tα . (For νee and νii , one can equally well use ν⊥ , and the result
Braginskii Equations
is insensitive to whether the slow- or fast-test-particle limit is employed.) The
20. J. M. Greene, “Improved Bhatnagar–Gross–Krook Model of Electron-Ion
Maxwellians Fα and F̄α are given by
Collisions,”
!
m
" Phys.
# $ Fluids %&
m (v − v )
16, 2022 (1973).
α
3/2
α α
2
F α = nα exp − ;
2πkTα 2kTα
21. S. I. Braginskii,
!
m
" #“Transport
$
m (v − v̄ )
α
%& Processes in a Plasma,” Reviews of Plasma
3/2
α α
2
F̄ = n exp − ,
Physics, 2πkT̄ Vol. 1 (Consultants Bureau, New York, 1965), p. 205.
α α
2kT̄ α α
where nα , vα and Tα are the number density, mean drift velocity, and effective
temperature obtained by taking moments of fα . Some latitude in the definition
η η
22. J. Sheffield, Plasma Scattering of Electromagnetic
species)
P Radiation
of T̄α and v̄α is possible;20 one choice is T̄e = Ti , T̄i = Te , v̄e = vi , v̄i = ve .
stress tensor (either
2 (Academic
= − (W + W ) −
2
(W − W ) − η W xx
0
xx yy
1
xx yy 3 xy ;
Press, New York, 1975), p. 6 (after J. W. Paul). P = − η2 (W + W ) + η2 (W − W ) + η W
Transport Coefficients
yy
0
xx yy
1
xx yy 3 xy ;
Transport equations for a multispecies plasma:
η3
Pxy = Pyx = −η1 Wxy + (Wxx − Wyy );
α 2
d n
23. K. H. Lloyd dt
+and G.= 0;Härendel, “Numerical Modeling
n ∇· v
α
α α of the Drift and De-
P = P = −η W
P = P = −η W + η W ;
xz zx 2 xz − η4 Wyz ;
yz zy 2 yz 4 xz
formation of Ionospheric
α $ Plasma
% Clouds and of Ptheir = −η W Interaction with zz 0 zz
d v 1
m n Other Layers
− ∇ · P +of theE Ionosphere,” J. Geophys. Res. 78, 7389 (1973).
α (here the z axis is defined parallel to B);
α = −∇p
α Z en + v ×B +R ;
α α α α α α
dt c 3nkT 6nkT i i i i i
ion viscosity η0 = 0.96nkTi τi ; η1 = ; η2 = ;
10ωci2 τi 5ωci2 τi
3 dα kTα nkTi nkTi
nα + pα ∇ · vα = −∇ · qα − Pα : ∇vα + Qα . η3i = ; η4i = ;
24. C. W. Allen, Astrophysical Quantities, 3rd edition (Athlone Press, Lon-
2 dt 2ωci ωci
nkT nkT
Here dα' e e e e e
Rα =
don, 1976), Chapt. 9.
/dt ≡ ∂/∂t + vα · ∇;'
Rαβ and Qα =
pα = nα kTα , where k is Boltzmann’s constant;
electron viscosity η = 0.73nkT τ ; η = 0.51
Qαβ , where Rαβ and Qαβ are respectively
ω τ
; η = 2.0
ω τ 0 e e 1 2
ce e
2 2
ce e
;
β β nkTe nkTe
η3e = − ; η4e = − .
the momentum and energy gained by the αth species through collisions with 2ωce ωce
the βth; Pα is the stress tensor; and qα is the heat flow.
25. G. L. Withbroe and R. W. Noyes, “Mass and Energy Flow in the Solar
For both species the rate-of-strain tensor is defined as
Chromosphere36 and Corona,” Ann. Rev. Astrophys. W = 15,
∂v
∂x
+ 363
∂v
∂x
− δ (1977).
2
3
∇ · v. jk
j
k
k
j
jk
15
When B = 0 the following simplifications occur:
26. S. Glasstone and R. H. Lovberg, Controlled Thermonuclear Reactions i
R = nej/σ ; R = −0.71n∇(kT ); q = −κ ∇(kT ); ∥ T e i ∥ i
(Van Nostrand, New York, 1960), Chapt. 2.
u
qeu = 0.71nkTe u; qeT = −κe∥ ∇(kTe ); Pjk = −η0 Wjk .
For ωce τe ≫ 1 ≫ ωci τi , the electrons obey the high-field expressions and the
27. References to experimental measurements
ions obey of branching
the zero-field expressions. ratios and cross
Collisional transport theory is applicable when (1) macroscopic time rates
15
sections are listed in F. K. McGowan, et al., Nucl. Data Tables A6,
of change satisfy d/dt ≪ 1/τ , where τ is the longest collisional time scale, andIon Collisional (⊥) Dynamics
16
16Ion Collisional Dynamics (w Viscous Damping)
17
17Ion Continuity
18
18Drift Wave Dynamics (Adiabatic Electrons)
19
19Electron (||) Dynamics w Collisions
20
20Electron (||) Dynamics w Collisions
21
21Strong Collisions (yields unstable drift wave)
22
22How Much Transport from Drift Waves?
es and transport
a ra llel
l o ng p gth
a v e len of the
FIG. 2. Three-dimensional configuration drift-wave
fields in a cylinder. w
,kive) the dominant terms in the electron fluid force
balance equation
S D
23
du ei n em ej i
m en e 52en e E i 2π i p e 1 (11)
dt e
are the electric field E i 52π i f and the isothermal pres-
sure gradient π i p e 5T e π i n e . The temperature gradient
is small compared with density gradient due to the fast
electron thermal flow associated with k i v e . v . Thus
23
much of the low-frequency drift-wave dynamics falls incities involved and because of changes difficulties in the measurement of plasma losses o -
o 3 4
a density and its gradient in the pres- noted above. B (kG I
s. Indirect measurements of wave- FIG. 8. Plasma density in relation to drift-wave onset for a
ced fluxes are complicated by problems A. Enhanced Plasma Loss vs Magnetic Field and varying mixture of potassium and cesium ions. no 1011 em-a,
T = 2800°K.
ion of these fluxes from other losses Ion Mass
e exchange, dc drifts, and volume or In Fig. 7, plasma density is shown as a function of creases abruptly. In the outer part and at positions
mbination; and in the measurement of the magnetic field for different radial locations, with outside the plasma column, the density increases at
ameters from which enhanced plasma DRIFT
COLLISIONAL
constant WAVES
ion and electron influx2435 from the end plates. onset, signifying radial transport associated with the
termined.
(Larmor
and
does
ence.
In
beradius
M3/8
addition,
shown
Observing Drift-Wave Transport
for a
does not allow us to distinguish between an
t must effect),that (resistivity
/
MI 3
conclusive
the presence
MI/2
effect only),
Starting with low magnetic field, the plasma confine- wave. The dependence of this density reduction on
1500,--------,-------.----r-::--r-----,------,
of ment increases with B. When B. is reached, the wave onset, for varying average ion mass, is given in
not cause changes in the bound- destabilizes to large amplitude and the plasma den- Fig. 8.
(resistivity and viscosity effects) depend-
and VI.
concomitant losses.
MEASUREMENTS OF ENHANCED PLASMA sity in the central part of the plasma column de-
Blkl
measurements of
TRANSPORT wave-induced
CAUSED
DRIFT WAVES
BY COLLISIONAL radial Gauss-em
/
(0)
ort are based on the detailed indentifi-
Drift (universal) instabilities are potentially peril-
pertinent
ous to instability, and benefit
plasma confinement as they may
enhanced loss in such plasmas. However, the mech-
from
cause
500
3.0
--:/"
"ow
--
one anism
azimuthal
responsiblemode of the
for enhanced plasmadrift wave
transport due DRIFT-WAVE
r=O mm
and tocan be stabilized
a finite-amplitude abruptly.
instability The
cannot, in general,
be deduced from linearized calculations, and a rig-
rimental result is the observation of an
orous nonlinear theory of collisional drift-wave I t STABLE
2.5 ( MlI'>,{amu},/.
SET
FIG. 6. Ratio of magnetic field to perpendicular wavenumber
y reduction
induced plasma inside,
losses, and an experimental
predicting increase versus average ion mass for a varying mixture of potassium r'Bmm
amplitudes and phase difference, has not been re-
smaported.
column32 coincidingthewith theof simul-
and cesium ions.
for AU = 2L, T =
n .,0" 2.0gives B /kJ.. = 1.0 X 10 (M3/s> G-cm
Theory
, oK, no
o 2800 3•
2
1011 cm- Experimental points
Experimentally, relation enhanced are averages for ml = 2 to 6.
(em- )
bilization
plasma loss of tothe drift
specific wave istodifficult
instabilities largeto 1.5
establish due to problems in the identification of sidered in the theory but which does behave similarly
ether with a phase difference between
instabilities and to the simultaneous presence in to the higher modes, can be "turned on and off"
tential plasma devices ofBy
many oscillations. varying
different unstableany of abruptly. Plasma loss
modes.
--------
1.0 rates can be determined and
Furthermore, direct measurements of classical and compared for stable and wave states under otherwise
ameters in the stability criterion, the
enhanced radial fluxes are complicated because of identical conditions, thus obviating some of the V>
r=16mm
m =the 1small
mode, Fig.
velocities 5(a),
involved andwhich
because ofischanges
less difficulties in the 05measurement of plasma losses a:
100
in local plasma density and its gradient in the pres- noted above.
mparison with the higher modes con-
ence of probes. Indirect measurements of wave-
induced enhanced fluxes are complicated by problems
L I
A. Enhanced Plasma Loss vs Magnetic Field 5and
..;
s ofinnonlinear
the separationtheories
of these of collisional
fluxes from other drift
losses Ion Mass B, kG
given
such recently: T. H. Dupree,
as charge exchange, dc drifts, andBull. Am.or
volume In Fig. 7, plasma density is shown as a function of
3 (1968);
end-plateT. recombination;
H. Stix, Phys. and inRev. Letters 20,of the
the measurement FIG. 7. Plasma density at different radial positions in relation
magnetic field for different radial locations, with
local wave parameters from which enhanced plasma constant ion to drift-wave onset. Cesium, T = 2800 oK.
and electron influx from the end plates.
FIG. 9. Measured density profiles and wave parameters
versus radius for a potassium plasma at T = 2760°K.
transport is determined. In addition, for a conclusive Starting with low magnetic field, the plasma confine- Density profiles in the stable (no" B = 1964 G) .and
measurement it must be shown that the presence of ment increases with B. When B. is reached, the wave wave regimes (now, m =1, B = 2050 G). (b) Relative ampli-
ed as
theindicated
instability in thenot
does article. Reuseinofthe
cause changes AIP content
bound- is subject
destabilizes to to theamplitude
large terms at: andhttp://scitationnew.aip.org/termsconditions.
the plasma den- tude of density (n/no) and (e/kT) oscillations. (c)
ary conditionsDownloaded
and concomitant to losses.
IP: 128.59.62.83 sity
On: inThu,
the 05 central
Decpart
2013 of 00:51:37
the plasma column de- Phase angle", by which the density wave leads the potential
The present measurements of wave-induced radial 24 wave.
plasma transport are based on the detailed indentifi-
cation of the pertinent instability, and benefit from
the fact that one azimuthal mode of the drift wave 3.0
DRIFT-WAVE
This article is copyrighted as indicated in the article. Reuse of AIP con
r=O mm
is dominant and can be stabilized abruptly. The
principal experimental result is the observation of an
2.5
I tSTABLE SET
Downloaded to IP: 128.59.62.8
abrupt density reduction inside, and an increase r'Bmm
no .,0", 2.0
outside the plasma column coinciding with the simul-
(em- l )
taneous destabilization of the drift wave to large 1.5
amplitude, together with a phase difference between
density and potential oscillations. By varying any of 1.0 24growth
results are interva1.
in agreement 32 In this with connection,
the growth acharac- theory38 phenomenoncollisional drift wave suitable
is analogous to heat toconvection
cause enhanced
in a
teristics
including of the unstable perturbation
higher-order terms showspredicted that close by to horizontal
plasma transport.
layer of fluid Such heated
causal relation
from below,between wherecolli-
theonset
respectivethe amplitude linear theories: the instability
A is proportional to a power occur- of periodic sional fluid
drift cells
wavesare and enhanced
formed. plasmathere
Although transport
is no is
ringthebetween
linear growth two rotating rate, Acoaxial2 ex: 'Y. cylinders/o
For drift waves, ,41 the it average observed in the there
velocity, present is an experiment. The plasma
average convective
has beeninstability
convective conjectured of that nonlinearfluid
a horizontal interaction
layer transport transportofinduced heae 2 by ,43; the
thewave electricmotion
outward field is larger
in-
heated
limits from the wave below,42,43
growthand whenthetheinstability perturbed inazi- volves thanhotter
that due fluidto classical
than thebinary-collision
inward motion.diffusion by
44
muthalIndensity
flames. gradient becomes
these experiments, the perturbedcomparable motion to the To oneconclude,
order of the magnitude for a on
present results 10% the relative
parametric wave
2438 HENDEL, CHU, AND POLITZER COLLISIONAL DRIFT WAVES 2439to dependence
zeroth-order radial
is characterized density gradient;
as a coherent oscillation, i.e.,similar
amplitude amplitude.ofThis transport
frequency wand is awavenumber
direct consequence
k cons- of
32 In this thesaturation
one discussed may here. be expected to occur when the per- titute the aphase difference identification
comprehensive between the of coherent density
the density-
plasma 10ss35 in the stable regime cannot be ac- growth VII. DISCUSSION
interva1. AND CONCLUSION
connection, a theory38 collisional turbed
The drift
present wave
xcB/Bsuitable
Ewave experimental
2 to
drift results
velocitycause show enhanced
becomes that the and potential waves,
com- gradient-driven collisional whichdriftis predicted
wave at the for aoscilla-
growing
counted for by classical diffusion, charge exchange, including
The present higher-order
experiment terms shows that
is conducted in a regionclose to of plasma parable transport.
to theSuch causalamplitudes
diamagnetic relation
velocity. between
39isIndeed, colli-these wave by linear theory. The coherent
observed pattern of mode similar to tory finite-amplitude saturation stage, wave,
based which
on, andhas
and volume or end-plate recombination. Theoreti- onset
the plasma the amplitude
where theAfree-energy is proportional reservoirs power of sional
to a available drift
velocities waves are and
found enhanced
to be plasma
roughly transport
comparable is the in no
in spatial- with,
or temporal-average velocity,
that of the calculated linear growth rate, and that agreement results from linear theory. induces
The
Conclusions
cally, radial and longitudinal distributions of wave 2 it observed in the(,....,2 present 3 are
the
for linear growth rate,
self-sustained A ex: 'Y.areFor
instabilities drift waves,
limited; i.e., the theexperiment
observed frequenciesX 10experiment.
cm/sec), thoseat The theplasma
predicted saturation radiallywave,
for coherent outwardinvolving transport
the entire because
plasmaofbody, the isii-
amplitudes require solution of a complicated bound- has been conjectured that nonlinearnointeraction current is transport induced by therates. wave
plasma is close to thermal equilibrium, stage.linear
highest In addition,growth we Ifelectric
note wethetake field
existencetheseis larger of other shown
results phaseto difference;
produce enhanced the outward plasmaoscillatory
transport. motion
This
ary-value problem. Experimentally, the edge oscilla- limits
applied,the andwave growth when by thedensityperturbed azi- than thatindication
due to classical binary-collision diffusion
regions dominated and tem- as fluid-dynamics
an experiments
that whose large-amplitude
at the saturation stageby the finding involves denser plasma
of plasma transport thaninduced
the inward motion.
by the This
single-
tion, which has not been considered in the loss 36 are becomes
muthal
peraturedensity gradients gradient separated. comparable
In this plasma to the one order ofare
results
amplitude ismagnitude
in agreement
proportional a10%
for toawith the
power relativeof thewave
growth charac- mode
linear phenomenon
coherent is analogous
drift to heat convection
wave represents a departure in a
measurements, has maximum amplitude at the edge zeroth-order
region, whereradial density gradient; amplitude amplitude.
i.e., mechanism This transport is with
aperturbation
direct
the only known excitation teristics
growth rate, ofinthe unstable
agreement the consequence
theorypredicted consider- of by from horizontal layer of concept
the prevalent fluid heated from below,
of plasma loss fromwhere
of the plasma column, Fig. 2. This unidentified saturation
for instability mayisbetheexpected density to occur when
gradient, we observe the per-a the phase
ingthe difference
respective
higher-order linearbetween
terms the the
theories:
discussed coherent instability
above, density
we occur- can magnetic periodicconfinement,
fluid cells arewhich formed. Although therelossis no
associates plasma
ANDoscillation,
POLITZER found to affect the density in the central turbed
self-sustained, Ewave xcB/B coherent
2
drift velocity whose
oscillation becomes behavior com- and potential
ring between
associate thewaves, twowhich
observed rotating is predicted for modes
coaxial cylinders/o
finite-amplitude a growing ,41 the with
with average velocity, there Furthermore,
is an averagethis convective
plasma turbulence. work
part of the plasma column only negligibly, shows a parable
agrees with to the diamagnetic
results from the velocity.
linear theory 39 Indeed, of density- these wave byof
convectivelinear theory.
instability The coherent
of apredicted wave, which
horizontal fluid haslayer indicatestransport heae 2 ,43;
those highest growth rate from linear the ofpresence of anthe extensive
outward regimemotion ofin-
phaseVII. difference
DISCUSSION between AND its CONCLUSION
density and potential velocities are found to be roughly comparable in the notheory.
gradient-driven collisional
3
drift waves. spatial-
heated or
from temporal-average
below,42,43 and velocity,
the induces in coherent
instability volves hotter fluid
oscillations than thewave
beyond inwardonset (B » Be);
motion.
oscillations and is expected to cause additional experiment
The identification (,....,2 X 10of cm/sec), this waveatisthe basedsaturation on its radially flames. outward
44 transport
In plasma-confinement
these experiments, because the of the isii-
perturbed motion
The present experiment is conducted in a region of The crucial experiment the although To conclude,
azimuthal themodes
presenthave results
been on observed
the parametric
up
plasma transport in the edge region. Other losses, stage. In addition, weofnote the existence of other phase difference;
the plasma where the free-energy reservoirs available measured dependence frequency and wavenumber is characterized
measurement of the as outward
anomalous a coherent oscillatory
losses, oscillation,
i.e., losses motion
similar m = 7, turbulence,
above to to dependence of frequency wand wavenumber
potentially important kand cons-
such as dc convective flow of plasma due to the fluid-dynamics
on all experimentally experiments whose parameters.
accessible large-amplitude The involves the one
classical denser plasmaDrift
discussed
diffusion. than the
here. waves inward are motion. of particular This
for self-sustained instabilities are limited; i.e., the titute afor
desirable comprehensive
further study,identification of the density-
has yet to be encountered.
temperature gradient l5 • 29 of the end plates, may also results
wave isare in agreement
observed to showwith abruptthe growthstabilization charac- of phenomenon
interestThesince isthe
present analogous
only to heat
required
experimental convection
sustaining
results mechanism
show in a the gradient-driven collisional drift wave at the oscilla-
that
plasma is close to thermal equilibrium, no current is
be present near the plasma edge. Complete identifi- teristics
azimuthal of modes
the unstable occurring perturbation
when the stabilizing predicted ion by horizontal
is aobserved
generally layer of fluid
present
pattern heatedamplitudes
ofdensity
mode from
(or below,is where
temperature) similar
ACKNOWLEDGMENTS
gra- to tory finite-amplitude saturation stage, based on, and
applied, and regions dominated by density and tem-
cation of these losses, although of interest to the the respective linear theories:wavelength
the instability due occur-
perature gradients 36 are separated. In this plasma diffusion over the transverse to the periodicdient.
thatThe fluid
of the cells
growth are
rates
calculated formed.
predicted
linear Although
by linear
growth there theory
rate, is
and nofor that It in isagreement
a pleasurewith, to thank
resultsDr. F. F.
from Chen,
linear Dr. B.
theory. The
study of plasma confinement in Q devices, is difficult ring
combined between two of
effects rotating
ion-ioncoaxial collisions cylinders/o
and finite
COLLISIONAL ,41 the
ion average
DRIFT
collisional velocity,
the observed WAVES
drift waves thereareis large,
frequencies an are average ,...., Reconvective
2'Y those predicted 2439for Coppi,
,...., kyVd/2,
region, where the only known excitation mechanism w coherentDr. wave,
F. W. involving
Perkins, and Dr. T.plasma
the entire H. Stix for is
body,
and has not been reported. In the present case it is convective
Larmor radius instability
(viscosity) of balances
a horizontal fluid layer transport
the destabilizing as highest oflinear
is the transverse heae 2
,43; the
wavelength,
growth rates.outward
IfAiwe »TL, take motion these in-
making the valuable
results
for instability is the density gradient, we observe a shown discussions.
to produce enhanced We also thank Gereg and
plasmaL.transport. This
not necessary to account for these losses in detail to heated
effect
growth of from
electron
interva1. 32 In over
below,42,43
motion and
this the the
parallel
connection, instability
wavelength
a theory38 in volves hotter
collisional
as an fluid wave
drift
indication than that the inward
suitable
at the to motion.
cause
saturation enhanced
stage the C. finding
self-sustained, coherent oscillation whose behavior 38 L. D. Landau and E. M. Lifshitz, Fluid Mechanics
A. Johnson for help
of plasma in the experimentation
transport induced by the single- and
demonstrate drift wave enhanced plasma transport flames.
dueincluding
44
to, but In these experiments,
also limited
higher-order by,terms the
resistivity.perturbed
shows that motion Toamplitude
plasma
close to (Pergamon conclude,
transport. isthe present
Such
proportional results
causal aonp.power
relation
to the parametric
betweenof the 3, colli-
p.linear
46 . H. mode Fishman for programming the numerical cal-
agrees with results from the linear theory of density- Press, Ltd., London, 1959), 104; Ref. coherent drift wave represents a departure
since the measurement is based on a comparison of is onset
characterized
The interpretation
the amplitude as a coherent
of
A the oscillation,
present
is proportional to similar
a powertoof dependence
finite-amplitude .. B. drift
sional
growth Coppi,
rate, M. N.and
ofwaves
frequency
in Rosenbluth,
agreement wand with
enhanced andplasma
R. Z. Sagdeev,
wavenumberthe k cons-
transport
theory Inter-
consider- is culations.
gradient-driven collisional drift waves. national Centre for Theoretical Physics, Trieste, Report from the prevalent concept of plasma loss from
stable and wave states, utilizing the experimental
The identification of this wave is based on its the theone
experimental discussed
linear results
growth here.is based
rate, A 2 onex: linear
'Y. Fortheory. drift waves, Linearit titute inga comprehensive
observed
IC/66/24, in the present
higher-order
1966. identification
terms experiment.
discussed of the The
above, density-
plasma
we can This magneticworkconfinement,
was performed which under the auspices
associates plasmaofloss
evidence that other losses are approximately pro- The been
theory,
has present
however, experimental
conjectured thatresults
only considers nonlinear show interaction
disturbances that the 40 G. I. T!1ylor, Phil. Trans. Roy. Soc. A223, 289 (1923).
of gradient-driven
transport
associate induced
the collisional
by theofdrift wave wave
electric at fieldthe modesoscilla-
is larger with thewith United States Atomic Furthermore,
Energy Commission,
measured dependence of frequency and wavenumber
portional to density. 41 C. C. Lm, Theobserved
Theory finite-amplitude
Hydrodynamic Stability (Cam- plasma turbulence. this work
on all experimentally accessible parameters. The observed
infinitesimal
limits the patternamplitude.
wave ofgrowth
mode If theamplitudes
when perturbation is similar
the perturbed is found to
azi- tory
than finite-amplitude
those
bridge that ofdue
University toPress,
highest saturation
classical
growth stage,
binary-collision
Cambridge, rate predicted
England, based on,
diffusion
1955), from p.and 15. by Contract
linear indicates AT(30-1)-1238.
the presenceUse of was made of computer
an extensive regime of
42 M. Bernard, Ann. Chim. Phys. 23, 62 (1901).
wave is C. observed
Enhanced toLoss
show abrupt stabilization
as Function of Density of that beof unstable,
tomuthal the calculated
density i.e.,
gradient linear
growing growth
becomes without rate,
comparablelimit, andlinear that
to the inone
agreement
order ofwith,
theory. results from
magnitude
43 Lord Rayleigh, Sci. Papers 6, 432 (1916).
for a linear10% theory. relative The wave facilities coherent supported
oscillations in beyond
part bywave National
onset (B » Be);
Science
azimuthal modes occurring when the stabilizing ion the
theory observed
zeroth-orderis expected frequencies
radialto density
be valid aregradient;
thoseduring
only predicted
i.e., aamplitude
limited for coherent
amplitude.
44 G.
The wave,
H. This
Markstein,
crucial involving J. Aeron.
transport theis entire
plasma-confinement 18, plasma
a direct
Sci. 199consequence
(1951).
experiment body, isisofthe Foundation although Grant azimuthal NSF-GP579.
modes have been observed up
In Fig.
diffusion 11 the
over the transverse
measured wave amplitude
wavelength due to andthethe highest
initial
saturation linearinterval
time maygrowth inrates.
be expected which weoccur
If to
the take
amplitude these results
when of the
the per- shown to produce
themeasurement
phase difference enhanced
of between
anomalous plasmathe
losses, transport.
coherenti.e., losses This
density above to m = 7, turbulence, potentially important and
plasma density-decrease
combined effects of ion-ionwith onset and
collisions of the m ion
finite = 1 as an indication
growing
turbed wave
Ewave is that 2 at
small.
xcB/B The theagreement
drift saturation
velocity stage the
between
becomes com- finding
and of plasma
potential
classical waves,
diffusion. transport
which
Driftis induced
predicted
waves byarefor the ofa single-
growing
particular desirable for further study, has yet to be encountered.
mode radius
Larmor are shown as a function
(viscosity) balancesoftheplasma density.
destabilizing amplitude
present tois the
parablefinite-amplitude proportional
diamagnetic to velocity.
results a power
and the 39ofIndeed,
the linear
predictions mode
theseThis wave coherent
interest
article sincedrift
bycopyrighted
is linear theory.
theas onlywaveThe
indicated represents
coherent
required wave,
sustaining
in the article. aReusedeparture
whichof AIPhas
mechanism content is subject to the terms at: http://scitationnew.aip.org/termsconditions.
growthlinearrate, in found
agreement ACKNOWLEDGMENTS
Values
effect of amplitude
of electron motion overand theplasma density
parallel for the
wavelength ofvelocities theory
are bewith
must,totherefore, roughly the theory consider-
be comparable
considered in the from
signi- the
noisspatial- prevalent
a generally concept
or temporal-average
present densityof plasma
Downloaded (orvelocity, loss128.59.62.83
temperature)
to IP: from gra-On: Thu, 05 Dec 2013
induces 00:51:37
unstable
due to, butcase also are obtained
limited adjacent to onset when
by, resistivity. ing
ficant higher-order
experimentin its implications. X 103discussed
(,....,2 terms cm/sec), above, at the saturation we can magnetic dient. confinement,
radially outward
The growthtransport which
rates associates
because
predicted by linearplasma
of the loss
theory ii-for It is a pleasure to thank Dr. F. F. Chen, Dr. B.
maximum saturation
The interpretation amplitude
of the is reached. Also
present finite-amplitude associate
Nonlinear
stage. Intheaddition,
observedwefinite-amplitude
interactions, as inthe
note theexistence
cases modes of of with
strongother with plasma
collisional
phase difference; turbulence.
drift waves the outward Furthermore,
are large,oscillatory
2'Y ,...., Re thisw motionwork
,...., kyVd/2, Coppi, Dr. F. W. Perkins, and Dr. T. H. Stix for
shown is the
experimental calculated
results is based growth
on linearrate from
theory. linear
Linear those
mode-mode of highest
fluid-dynamics coupling growth or rate
experiments predicted
of turbulence,
whose large-amplitudefrom distort
may linear indicates as is the
involves the presence
transverse
denser of
plasmawavelength,
than anthe extensiveAi »TL,
inward regime
motion. making of the valuable discussions. We also thank L. Gereg and
This
theory, for measured values of
theory, however, only considers disturbances of temperature, density, theory.
an incipient
results are instability
in agreement so that withthe thefinal growth saturation- charac- coherent
phenomenon oscillations is beyond to
analogous wave heat onset convection(B » Be); in a C. A. Johnson for help in the experimentation and
38 L. D. Landau and E. M. Lifshitz, Fluid Mechanics
magnetic field,
infinitesimal and density
amplitude. gradient scale
If the perturbation length.
is found The behavior
stage
teristics crucial of the plasma-confinement
bears
unstable no resemblance
perturbation experiment
to predicted
the charac- is theby although
horizontal
(Pergamon azimuthal
layer
Press,ofLtd., modes
fluid have1959),
heated
London, been
fromp.observedbelow,
104; Ref.where up
3, p. 46 . H. Fishman for programming the numerical cal-
toThe
be wave amplitude dependence on density
limit, islinear
found measurement
teristics
the respective of the of linear
anomalous
initialtheories: losses,
perturbation thei.e., losses above
predicted
instability occur- m =
by toperiodic .. B.7,Coppi, turbulence,M. N. Rosenbluth,
potentially andimportant
R. Z. Sagdeev,
there and Inter-
unstable, i.e., growing without nationalfluid Centre cells forareTheoretical
formed. Although
Physics, Trieste, isReport
no culations.
to be iscomparable
theory expected totobethat validof only
the linear
duringgrowth
a limited rate. classical
linear diffusion.
theory.37
ring between two Drift waves
However,
rotating other
coaxial are
nonlinearof particular
cylinders/o ,41 the desirable
mecha- average
IC/66/24, for further
1966. study,
velocity, there has is anyet to be encountered.
average convective This work was performed under the auspices of
interest sincelimit the only 40 G. I. T!1ylor, Phil. Trans. Roy. Soc. A223, 289 (1923).
The density
initial time interval decrease as function
in which of no, which
the amplitude of the can nisms
convective may therequired
instability of a sustaining
amplitude so thatmechanism
horizontal the
fluidmode layer transport 41 C. C.of Lm, heae
The
2
,43;
Theory the
of outward
Hydrodynamic motion
Stability in-
(Cam- the United States Atomic Energy Commission,
ACKNOWLEDGMENTS
be viewed
1 growing wave roughly
is small.as The
a measure of plasma
agreement betweenloss,thealso isphenomena
a generally
heated from present
observed below,42,43 density
at (or temperature)
the finite-amplitude
and the instability gra-in volves
satura- bridgehotter University fluidPress, thanCambridge,
the inward motion.
England, 1955), p. 15. Contract AT(30-1)-1238. Use was made of computer
bears similarity
present to the growth
finite-amplitude resultsrateandasthea function
predictionsof no. dient.
tion
flames. The
stage 44 growth
arethese
In closelyrates predicted
related
experiments, to theby linear
the unstable
perturbed theory pertur- for
motion ItTois 4243conclude,
aM. Bernard,toAnn.
pleasure
Lord Rayleigh,
thank
the present
Chim.
Sci. Papers
Dr.Phys.
F. F.
results
23, Chen,
6, 432on(1916).
62 (1901).
the parametric Dr. B. facilities supported in part by National Science
ofThese
linear observations
theory must, are consistent
therefore, with thosesigni-
be considered of the collisional
bation
is characterized drift waves
predicted byaare
as large,theory
linear
coherent 2'Yoscillation,
,...., Re
for wthe kyVd/2,
initialto Coppi,
,....,similar 44Dr.
dependence G. H. F.of W.
Markstein,Perkins,
frequency and Sci.
J. Aeron.
wand Dr. T.199
18,
wavenumber H.(1951).
Stix k cons-for Foundation Grant NSF-GP579.
astheis the Ai »TL, making the valuable
25
modeinamplitude
ficant dependence on magnetic field, Fig. 4
its implications. onetransverse
discussed wavelength,here. titute a discussions.
comprehensive We identification
also thank L.of Gereg the density- and
36 It has been suggested that the equilibrium radial electric
and strongly suggest thatasthe nonlinear
cases mechanism 38 The D.present experimental results that the C.gradient-driven
A. Johnson forcollisional help in the experimentation and
Nonlinear interactions, in the of strong field L.may Landau
play a role and
in E. M. Lifshitz, onset Fluidofshow Mechanics
low-fre9.uency- drift wave at the oscilla-
responsible for the results
r mode-mode coupling or of turbulence, may observed at the large
distort (Pergamon
oscillations
observed Press,
[L.
pattern Ltd., of
Enriques, London,
A.
mode M. 1959),
Levme,
amplitudesp. and104;G.Ref.
isB.similar3, p. 46 .to H. Fishman
tory
This finite-amplitude
article for programming
is copyrighted indicated inthe
assaturation stage,
the numerical
article. basedReuse on,ofcal- and
AIP content is subject to the terms at: http://scitationnew.aip.org/termscondition
.. B. Coppi, M. N. Rosenbluth, and R.
Plasma Phys. 10, 641 (1968)]. In the present experiment, culations. Z. Sagdeev, Inter-
ansaturation amplitude sois that
incipient instability suchtheasfinalto preserve
saturation- the that ofCentre
national
however, theeffect
the calculated
forof Theoretical
the radiallinear growth
Physics,
electric rate,
fieldTrieste,
E or, other and
Report that in agreement with, results from
than linear theory.
Downloaded to IP: 128.59.62.83 The On: Thu, 05 Dec 2013 00:51:37
characteristics of the unstable perturbation predicted IC/66/24,
Doppler
the 1966.of the
shift
observed observed
frequencies are Soc.those to be neg-
predicted This work
for coherent wave, was involving
performedtheunder entirethe plasma auspices body, of is
stage behavior bears no resemblance to the charac- 40 G. When
ligible. I. T!1ylor,
platePhil. Trans. Roy.
temperature IS mcreased, A223,Eor 289mcreases (1923). as the United States Atomic Energy Commission,
by linear theory. highest
41 aC. ?:C. 2.linear
teristics of the initial perturbation predicted by Ta, Lm,As Thethegrowth
plate rates.
Theoryteml?erature. If we
of Hydrodynamic take
is varied, these
Stabilitywe observe results shown to produce enhanced plasma transport. This
(Cam-
linear35 S.theory.37 However, other nonlinear mecha- bridge
drift-wave
as M.
We42 do
anUniversity
mode Press,
indication
Bernard,
not observe
changes
Ann.
Cambridge,
that
m exChim.
conSIstent
at asthe
IjTa,Phys.
England,
With Eq. 1955),
23,saturation
expected 62 (1901).
(17), stage mp. 15.
ex Tl.
if a cE or X BjB' facilities
the Contract
finding AT(30-1)-1238.
of plasma transport Use was made by
induced of computer
the single-
von Goeler and R. W. Motley, in Proceedings of Con- supported in part by National Science
n nisms
ferencemay limit the
on Physics amplitude
of Quiescent so that
Plasmas the mode
(Laboratori. Gas amplitude
Lord is
Rayleigh, proportional
Sci. Papers 6,to432
mechanism would playa role, as suggested in the above refer-
43 a power
(1916). of the linear mode coherent drift wave represents a departure
Ionizzati, Frascati,
phenomena observed Italy, 1967),
at the Pt. 1, p. 243; and
finite-amplitude private
satura-
44 G. H. Markstein, J. Aeron. Sci. 18, 199 (1951).
ence.
growth rate, in agreement with the theory consider- Foundation
from the Grant
prevalent NSF-GP579.
concept of plasma loss from
communication with F. F. Chen and G. Grieger. 37 N. A. Krall, Phys. Rev. 158, 138 (1967).
tion stage are closely related to the unstable pertur- ing higher-order terms discussed above, we can magnetic confinement, which associates plasma loss
bation predicted by linear theory for the initial associate the observed finite-amplitude modes with25 with plasma turbulence. Furthermore, this work
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