Instability in Nonequilibrium and Nonthermal Plasma Discharges
Plasma Instability - Seoul National University
81
Plasma Instability • MHD instabilities – Basic MHD instabilities and stability analysis – Tokamak instability theory – Beta limit • Experimental Tokamak instabilities – Basic MHD instabilities – Disruptions – Other instabilities • Micro-instabilities : finite Larmor radius and kinetic dissipation effects, where wavelength comparable to ion gyro radius – Drift wave instabilities – Trapped particle instabilities – Micro-tearing modes
Transcript of Plasma Instability - Seoul National University
Plasma Instability
• MHD instabilities
– Basic MHD instabilities and stability analysis
– Tokamak instability theory
– Beta limit
• Experimental Tokamak instabilities
– Basic MHD instabilities
– Disruptions
– Other instabilities
• Micro-instabilities : finite Larmor radius and kinetic dissipation effects, where wavelength comparable to ion gyro radius
– Drift wave instabilities
– Trapped particle instabilities
– Micro-tearing modes
MHD Instability
• Basic destabilizing forces
– Current gradients
– Pressure gradients combined with adverse magnetic field curvature
• Resulting instabilities
– Ideal modes : instabilities even with perfectly conducting plasmas
– Resistive modes : dependent on the finite resistivity of the plasma
Magnetic Islands (Tearing Modes)
Stabilizing Effects
• Magnetic field line bending / compression
– stronger stabilizing effect for high-m modes
– high-m modes are localized around a resonance surface
of rational surface on which m=nq
– limited q operation excludes most global low-m modes
• Good magnetic field curvature
– the center of curvature being in the opposite direction to
the pressure gradient
Theoretical Stability Analyses
• Normal mode analysis
– calculation of the eigenfunctions and corresponding
eigenvalues for the frequency --> sign of Im()
• Marginal stability equation
– solution of the marginal stability equation : Im()=0
• Energy principle
– potential energy change resulting from a plasma
displacement is examined
Wo
W1
Wo
W1Energy
ForceF
F
Stable
Unstable
Single Fluid Equations : MHD
Equation of motion
Equation of mass/charge conservation
EBjpvvt
vm
)(
Ohm’s law )(1
epBjen
jBvE
Conservation of entropy 0)(
mpdt
d
Maxwell’s equationst
B)(E
Bv jB
o
Ideal MHD
0 BvE
Mass density i
s
ssm nmnm Charge density 0s
ssne
Quasi neutralityCurrent density )( eis
s
ss uuneunej
Center of mass velocity e
i
ei
ei
eeii
s
ss
s
s
ss
um
mu
mm
umum
mn
umn
v
0)(
v
tm
m
0
j
t
0 vpdt
dp or
Linear Stability Theory : perturbation theory
Linearize ideal MHD equations about a static equilibrium
100111
0 )()( BBBBpt
v
t
B)v( 1
01
B01001
1
vppv
t
p
0)( 101
v
t
10 ppp 10 m 1vv
10 BBB
Introduce Lagrangian displacement vector ,t
v
1
)(B 01 B
001 ppp
)( 01
100112
2
0 )()()( BBBBpt
F
Normal Mode Analysis
Perturbations in normal modes tiertr )(ˆ),(
ˆ)ˆ( 2
0F
--> an eigenvalue problem
02 stable 02 unstable
eigenvvector
eigenvalue
• Marginal stability equation 0)ˆ( F
q1
stable
unstable2/2 Bpo
pIq /1~1
• Full dynamics
• Eigenmode analysis
2
2
0)(t
F
dFdWKL
2
1
2
1 22
0
Energy Principle
dFW
2
1Potential energy change due to an arbitrary displacement
Linearized force 10011 )()()( BBBBpF
with )(B 01 B
001 ppp
dBBBB
pp
W
o
)()())((1
)(
2
1
0000
00
SdBBp
dBBBppW
o
oo
)1
(2
1
)(11
)()(2
1
101
10
2
1
2
00
for conducting boundary 0ˆ n
vacuum o
v
plasma
o
o
dB
dBjB
ppW
2
1
2
12
00 )()()(2
1
0Wunstable
Tokamak Instabilities
• Ideal modes occur even if the plasma is perfectly conducting
• For ideal MHD stable plasmas, resistive modes can be unstable
• Ideal and resistive modes are paired usually
• Tokamak MHD instabilities
– Kink instabilty : driven at low by the current gradient, at high
by pressure gradient as well
– Tearing mode : the resistive form of the kink instability
– Internal kink : m=1, plasma core where q<1, driven by the
pressure gradient at that region
– Resistive m=1 instability : similar to internal kink affecting the
plasma core, but with a different energy source
– Ballooning modes : localized, driven by the pressure gradient
– Vertical instability : arising from plasma elongation
Large Aspect-Ratio Tokamak
• Analytic calculation for MHD stability using large aspect-ratio
approximation instead of numerical one --> still contains some
toroidal effects
• Tokamak ordering
– large aspect-ratio limit with q ~ 1 :
– low approximation
RaBB /~/
02
2
rB
dr
d
r
BB
dr
d
dr
dp
oo
oBp 2/~ 2
1~p
a
B
a
B
dr
dp
Bdr
dBo
2~~~
BB ~
ojj ~
2~ 1~q
jBBjBdr
dpBjj oo ~/~/)(
)(W
Sausage and Kink Modes
LBaB z /2/
Stabilizing Condition :
Kruskal-Shafranov condition
Kink Instabilities
vacuum o
v
plasma
o
o
dB
dBjB
ppW
2
1
2
12
00 )()()(2
1
For a circular, large aspect-ratio tokamak with low beta,
/)/1(/)/1(~/ Rrr ~r 111 ~ BBBr
rdrdB
RrdrdBBjB
RWWWb
ao
va
rrzo
o
vp
2
011
2
1 )(
Perturbations in a Fourier analyzed in the form of
)( rrdr
d
m
i
)(B 01 B
rrqm
n
R
imBB
)
1(1
rr
qm
n
dr
d
R
BB
)1
(1
0
0
)( nmie
rdr
rqm
n
dr
dr
dr
d
qm
n
B
RrB
dr
d
r
rqm
n
dr
d
qm
nm
R
BW
a
rrrr
rr
o
p
0
2
222
22
)1
()()1
()(1
)1
()1
(
222
2
0
222
22
)1
()1
(2
)1
()1()(
a
aaa
a
o
p
aqm
n
qm
n
q
rdrqm
nm
dr
dr
R
BW
/)/1(1 rBr rB /1
22
2
22
1
2
1
2 )(dr
d
r
mBBB rv
b
ao
b
a
b
ao
b
av
o
v
dr
dr
R
dr
drrdr
dr
dr
dr
d
rr
mRrdrB
RW
)(
)()(
2
2
2
222
2
flux function
0
0 B
0)(1
2
2
r
m
dr
dr
dr
d
r
0 B
0)(1
2
2
r
m
dr
dr
dr
d
r
mm rr
0)( br
aa
a
ra
im
aqm
n
R
imBarB
1)
1()(1
aa
aa
a
am
nqB
qm
n
R
aB
)1()
1(
amm
mm
aa
a
b
b
ar
b
b
r
m
nqB
)()(
)()(
)1(
})1
)(1()1
(2
)1
()1()({
222
2
0
222
22
a
aaa
a
o
vp
aqm
nm
qm
n
q
rdrqm
nm
dr
dr
R
BWWW
m
m
ba
ba2
2
)/(1
)/(1
222
22
)1
( a
ao
v aqm
nm
R
BW
boundary conditions
For conducting wall 0a
0W stable
Growth Rate for Various Modes
For any conducting wall at b>a, 0W for aqnm /stable
aqnm / can be unstable !
Since q ~ r2, modes with resonance rational surface
outside the plasma has
For growth rate, the eigenmode equation need to be solved
0)()()(2
22222
dr
dFrFmr
dr
drF
dr
d
2/1/)( orBnqmF
1 mr
aao
a
nqmnqmBa
nqmmr
dr
d 2
)()/(
)()(
2222
2
BC‟s
r=ar=0
))/(1( 2arjj o
Stability Diagram for External Kink Modes
))/(1( 2arjj o
Higher current
density gradient
at plasma surface
Internal Kink Mode
• Resonant surface : q = 1 from m=1/n=1 mode
• Sufficient condition for stability : qo > 1
• Potential energy for large aspect-ratio approximation
)()1
()1()( 22
0
222
22
Ordr
qm
nm
dr
dr
R
BW
a
o
0a no surface term
0)()(~ 22 dxxqW
o
o
r
r
.)( 1 constrr 0)( 1 rr 1)( 1 rrq
0as
leading order : marginally stable
T
o
oC WR
rBRW
nOW
~)(2)
11()( 41
2
22
2
2
then n=1
)144
13()1(3
~ 2
1poT qW 2
1
2
1
01
1)2/(
2)(1
rB
rdrpp
o
r
p
Tokamak Stability Diagram for Ideal Kinks
))/(1( 2arjj o
Rayleigh-Taylor instability : It occurs when a heavy incompressible
fluid is supported against the force of gravity by a light
incompressible fluid.
Plasma analogy :
plasma <-----------> “heavy” fluid
magnetic field <-----------> “light” fluid
general plasma forces <-----------> gravity
p Force : Unstable !
Minimum-B configuration
for stable condition
Interchange (Flute) Mode in a Simple Mirror
Localized High-n Modes
rdrdr
dp
r
k
dr
dBk
kW r
zr
o
22
2
2)(
2
11
Relevant part of the potential energy in a cylindrical plasma
stabilize high-n modes
0 at resonant surface (q=m/n) localized high-n modes
• Suydam criterion)()(
8
22
pq
qrB
o
z
rBB
rBB
q
q
z
z
/
/
• Mercier criterion for large aspect-ratio circular tokamak
zkrmk zˆˆ)/(
)1)(()(8
22
2
qpq
qrB
o
Stabilizing contribution of the average
curvature of the toroidal magnetic field
• Mercier criterion is only a necessary stability condition
Complete treatment of the destabilizing bad curvature
on the outer side of the torus Ballooning modes
magnetic
shear
2/~
c
dR
drdpW
Balooning Modes (high-n modes)Destabilizing energy available from the pressure gradient
Energy required for line bending
222
// )/(~ os BkW
qRk /1~// oc RR ~
• Ballooning modes become important
only when the pressure gradient is
sufficiently large that
o
o
Rq
B
dr
dp2
2 /~
2~
q
• Infernal Mode : low mode number pressure-driven instability at low
magnetic shear region close to rational surface with low mode number
Balooning Modes
Potential energy in the limit of large mode number in an
orthogonal coordinate system (, , ) : Eq. (6.13.1)
Minimization leads to an Euler equation of (6.13.2) with
pRBX
y
RdJBinFX /expEikonel transform
For large aspect-ratio circular tokamak with low beta 2~
Instability occurs for RaBdrdp o /~)//()/( 2
dF
d
dFW 222 )cossin())(1(
sin sdr
dq
q
rs
dr
dp
B
Rqo
2
22 average
magnetic shear
stabilizing effect of line bending destabilizing effect of the pressure gradient
shear dependent contribution
0cossin)sin())sin(1( 2
Fs
d
dFs
d
d
cos
sd
d
S- diagram
dr
dp
B
Rq
dr
dq
q
rs
o
2/2
2
s
Balooning Modes
Second stability region
dF
d
dFW 2222 )(sin
sinsin s
• Pressure and q profiles control (DIII-D)
• Bean shaped plasma boundary (PBX-M)
Axisymmetric Modes
• Elongated tokamak plasmas are susceptible
to an axisymmetric instability (n=0)
– circular cylindrical plasmas are neutrally
stable to a rigid displacement without
conducting wall
– elongated plasmas are unstable to a motion
along the direction of elongation
– in the presence of conducting wall, there is a
critical elongation for instability
– complicated in reality from toroidal effects
and adjacent structural effects
• Elongation of the plasma due to currents in
external conductors
Stability Criteria of Axisymmetric Modes
• circular large aspect-ratio plasmas
without stabilizing conductors
in terms of the equilibrium magnetic field from external currents
0dR
dB
B
Rn z
z
• cylindrical elliptical plasmas with uniform
current density surrounded by a perfectly
conducting confocal elliptical shell2
ab
ab
ab
ab : Instability in
inertial time-scaleOtherwise, instability in resistive time-scale
1)( RD 1
2
ab
ab
ab
abD
• Toroidal stabilizing effect confirmed
in numerical calculations
Beta Limit
Basic MHD parameter measuring the confined pressure, related
to the MHD stability driven by pressure gradients
• Ballooning modes
/)/(~~ 2
2aBI
q
- Correct scaling from optimized
stability calculations for JET
aBI /~
- turnover at high beta is due to
the instability of low-m modes at
low q included in the calculations
Approximate Linear Scaling of Beta LimitFor large aspect-ratio circular plasmas,
a
o
o
a
drrdr
dp
BaBa
prdr
0
2
2222
0 2
)2/(
2
dr
dp
B
Rq
dr
dq
q
rs o
2
2267.167.1
a
drrqdr
d
Ra 0
3
22
113.0
)1(2.1 2/1
2 a
a
m qq
m%
aq/1
aq/28
Radial dependence
optimization
Troyon Beta Limit
a
mq
28%
aB
MAIm
][6.5% or
unrealistically optimistic !
With )2/1(
122 ar
q
a
mq
15%
aB
MAIm
][3% or
and 2aq
Computer calculation by Sykes and Troyon
aB
MAIgm
][% Troyon factor
aBMAIN
/][
% Normalized
8.2 gN Troyon limit
experiments
aB
MAIli
][4%
Tearing Modesdriven by the radial gradient of the equilibrium toroidal current density
jE ~
• Tearing and rejoining of magnetic field lines due to finite resistivity --> tearing instability– slow growth : inertial effects negligible
– vxB negligible at the resonance surface
– matching two solutions of both resistive layer and outer layer
Magnetic Islands (Tearing Modes)
Stability of Tearing Modes
0lim0
s
s
rr
rr
• Tearing mode stability criterion
unstable
0)/1(
/)(
12
2
mnqB
drdj
r
m
dr
dr
dr
d
r
o
)(1 s
r
rrq
q
m
nq
s
0)(/
/1
2
2
s
ro
rrq
q
B
drdj
dr
d
s
...)(...ln)(1
sss rrArrrrk
...)(...ln)(1
sss rrArrrrk
AA
srr
srr
numerical solutions with b.c.‟s
0)0( r )()()( ara
mar o
k
Stability of Tearing Modes
• Stability diagram for the m=2 kink and tearing modes
for current distributions ))/(1( 2arjj o
• The tearing instability
is continuous with the
kink instability as the
resonant surfaces pass
through the plasma
boundary
• stability for q0>2 results
from the absence of a
q=2 resonant surface
Conducting Wall Boundary Condition• Mode frequency due to either natural frequency or plasma rotation by
NBI affects the influence of conducting shell
– low frequency : penetration with resistive time scale
– high frequency : no penetration due to eddy current shielding
2
2
)(1
r
m
dr
dr
dr
d
r
bos ~
1s
tr
mE
r
m
dr
dr
dr
d
roo
2
2
2
2
)(1
vacuum region (a<r<b)
m
m
abaf
baf
a
m2
2
)/(1
)/(1
bm
bmf
b
b
/)/(
/)/(
)(2
2
o
bb
ir
m
b
m
b
simf
/1
1
2/ bos
shell region (b<r<b+)
1s 1f b
/ 0b• Vacuum vessel lower the frequency of the instability --> at large amplitude
the mode locks to the vessel (mode locking), bringing the frequency to zero.
Tokamak Macroscopic Instabilities
• Basic MHD instabilities– Magnetic islands
– Tearing modes
– Mirnov oscillations
– Sawtooth
• Disruptions– Causes of disruptions
– Physics of disruptions
– Mode locking,
error field instability,
vertical instability, ergodicity
• Other instabilities– Fishbone instability, TAE, MARFEs, ELMs
Magnetic Islands
• Formation of magnetic islands– Resistive MHD instability, particularly tearing modes
– non-linearly developed near the resonant surface in the plasma
Geometry of Magnetic IslandsEquilibrium configuration around a resonant surface nmqq s /
ime m
n
Angular coordinate orthogonal to the helix
Resonant perturbation
Equilibrium field in this orthogonal direction ))(1(* rqm
nBB
zq
qBB s)(*
srrz
zqqB
mrB
B
B
dr
dz
dr
dr
s
rr
ss )/(
sin)(ˆ
*
)cos(cos8
22
ommw
z
2/1ˆ
4
s
r
Bqm
Brqw
Magnetic field line equationintegrate
: island width
Tearing Modes
• Growth of islands– destabilizing effects from current gradients in the plasma– growth rate limited by resistive diffusion– island width ~ Br --> growth rate limited by Br diffusion
2
2
r
B
t
B r
o
r
Integrate over the island width with approx. constant Br
2/
2/
wr
wr
r
o
r
s
sr
B
t
Bw
)(22
1
2
2/
2/
2/
2/
wr
B
Bdt
dw
o
wr
wro
wr
wr
r
ro
s
s
s
s
2wBr
dwBdwwwdB rr 2
))((66.1 wwdt
dw
o
More accurately
plasma flow effect detailed treatment
of island region
ss ww )(Saturated island size
or approximately 0)( sw
w
Internally Driven Tearing Modes
• Tearing modes driven inside the islands
– modified resistivity within the island from enhanced impurity radiation or the effect of an injected pellet
– modified from the change of bootstrap currents within the island
hhhh jjjEt
)(
Faraday‟s law and Ohm‟slaw
wdr
dp
Bw o
wr
wr
s
s
)()(2/1
2/
2/
/jj hh
The associated flux perturbation defines the resulting magnetic island ho j 2
The resulting reduction of bootstrap current leads to a helical current bj
bo j 2
dr
dp
Bjb
2/1
If the pressure gradient were completely removed, bb jj bobo jj
dr
d
2
2
wq
q
B
drdpw
o
1
2/
)/(16)(
2
2/1
Mirnov Instabilities
Considered to be
tearing modes with m~qa
near the plasma surface
RI
Ba
RB
aBq
oa
a
22
Characteristics of Mirnov oscillations– frequency of 1-10kHz typically– propagation velocity has the sign of electron drift velocity– tearing mode theory with finite Larmor radius effects
mode frequency of the electron drift frequency
dr
dn
eBrn
mT e
e
ee *mode number
Current Penetration
(Double Tearing Mode)
Current Penetration (Double Tearing Mode)
dashed lines : before the instability
solid lines : after instabilityj
Tearing mode equation 0)/1(
/2
mnqB
drdjo
Helical flux of the equilibrium configuration associated with the
resonance surface ))(1(* rqm
nBB
dr
dB
**
The region between the resonances combines a destabilizing current gradient and a small value of 1-nq/m
Instability flatten the profiles
form islands
Sawtooth Oscillations
Heating raises central temperature
and makes current profile peaked
qo < 1 m=1/n=1 instability
Central temperature collapses
and thermal energy released to
the outer region
• Density and temperature
change typically a few percent
• Much shorter than the
characteristic resistive time
Sawtooth Models
Kadommtsev‟s Model
o
v 1
2
11*
2 ~~ rrB R
Ao
o
Collapse time
2/3
1
2/1
1
1 ~)(~~ rv
rARK
/~ *
1Bvj
o
Bv
2~
2*2
2
Collapse Time
oBj /~ *
211 ~ vvr
Ampere‟s law
Ohm‟s law
continuity
Discrepancy between the Model and Experiments
• Collapse time: the observed collapse time too short
JET (~10ms vs. ~100s) -->quasi-interchange instability? No qo~0.7
• Spontaneous onset: the time for equilibrium change to produce
a configuration having the observed growth rate is very much longer
than the sawtooth collapse time (JET: two orders of magnitude)
• qo remains below zero
• Stability during the ramp phase:
resistive kink unstable at ramp phase
expecting growth rate of less than 1 ms
based on q‟ at q=1 surface (~q‟2/3)
• Thermal collapse: temperature
flattening across whole q<1 region
on incomplete magnetic reconnection
TEXTOR
Disruptions
Causes of Disruptions
• Low-q disruptions
• Density limit disruptions
Murakami
Greenwald density limit2
/)()10( 320 aMAImn
Physics of Disruptions
• Tearing mode instability
• Non-linear growth of the tearing mode : mode locking
• The fast phase (Thermal quench)
• Current decay (Current quench)
• Runaway electron current
• Vacuum vessel current (Halo currents)
• Mode locking
• Error field instability
• Vertical disruption event(VDE)
• Ergodicity
Mode Locking
Thermal and Current Quench
Other Instabilities
• Fishbone instability
• Toroidal Alfven eigenmodes (TAE)
• ELM (edge localized mode)
• MARFE (Radiation instability, not MHD)
• Operational overview
Fishbone Instability
Fishbone Instability: Stability Boundary
Toroidal Alfven Eigenmodes (TAE)
continuum
(TAE)
• MHD wave equation of shear effect (large aspect ratio)
Instability growth rate γ
(Damping)
F(x)=x(1+2x2+2x4)exp(-x2);
predominantly associated
with Vα around or above VA
• Without damping, TAE is unstable to all m‟s.
• Stability depends on magnitude and m dependence of damping.
• Finite orbit effects limit the increase of destabilizing term with m
• Small m‟s, typically 2-6 are most unstable.
Four main contributions to damping
Coupling to continuum damping
is predominate at low m.
(e-1/m)
(m-3/2)
(m2)†
†
ELM
(edge localized mode)
Types of ELMs
Type Ⅱ ELMs (Grassy ELMs)
Type Ⅰ ELMs (Giant ELMs)
MARFEs
(Multifaceted Asymmetric Radiation from the Edge)
• Plasma locally cooled by radiation in circumstances where the cooling itself leads to increased radiation and hence further cooling• Radiation instability of MARFE occurs on the inner side of tokamak or the X-points of diverted plasmas
Stability Diagram
Operational Overview
aB
MAIlic
][4[%]
)3
21)(1(
2
52
2
2
22
95R
a
a
b
RI
Baq
95
15[%]q
lic
3/ aR 3/5/ ab
Highest beta achieved : 12.5% (DIII-D)
Micro-Instabilities
Finite Larmor radius and kinetic dissipation effects,
where wavelength comparable to ion gyro radius
--> fine scale plasma turbulence
--> anomalous plasma transport in tokamaks
Different roles due to different mass
: drive or damp instabilities
• Dissipative microinstability : caused by dissipation
either from collisions or from Landau resonance
• Reactive microinstability : ion temperature gradient mode
Simple Model for Micro-Instabilities
• Toroidal geometry : large aspect ratio, circular cross
section, low tokamak equilibrium
• independent poloidal and toroidal mode numbers
• curvature drifts couple the poloidal harmonics
• Slab model when curvature drifts are not important
• Electrostatic instabilities : drift wave instabilities
• Passing particle instabilities
• Trapped particle instabilities
• Electromagnetic instabilities : Micro-tearing modes
Microinstabilities
Plasma waves and their associated instabilities
• Electron drift wave : „Universal‟, trapped electron
• Sound wave : Ion temperature gradient
• Alfven wave : Micro-tearing
Electron Drift Wave
0//// epEne
eee Tenn //~
Force balance
linearize
ExB drift wave
from diamagnetic drift2/ BnpBv jjdj
diii vn
t
n
dx
dn
yBdx
dnvni exi
~1~~
electron diamagnetic frequency
dr
dn
eBn
Tk ey
e *Electron drift mode
• destabilized by electron disspations
either by collisions or Landau damping
• stabilized by shear damping
• destabilized by trapped electrons
universal instability
Passing Particle Instabilities
• Electron drift mode
• Ion temperature gradient (ITG) mode : i mode
• e mode
Dispersion relation from gyro-kinetic equation
bj Negligible trapped particle effects TiTe vkvk ////
eee Tenn //~ Boltzman relation
endi *2 Slow ion magnetic curvature drift
RLnn / with the density scale length1)/ln( drndLn
Perturbed ion distribution function with finite ion Larmor radius effects
]sinexp[~
g
ci
M
i
i
vikgf
T
ef
gyro-kinetic equation
0])[(~
)(//
dK
df
dr
dfkzJ
m
eigvki
l
gv MM
ci
o
i
d
Passing Particle Instabilities
M
T
io
i
di fzJT
eigi
l
gv ))((
~)( *//
With equilibrium drift velocity of grad B and curvature, G-K eqn
ddi vk
])2
3
2(1[
2
2
** i
Ti
i
T
iv
v nTnT iii
/
1ifor • for weak toroidal coupling3/1
*
22
// )(~ iiTivk
slab branch of i mode
for a sheared slabiin qsi *
2/1)/(~
• for strong toroidal coupling2/1
* )(~ diii toroidal branch of i mode1~/ *efor
• slab branch of electron drift mode e*~
evolves from sound wave
0~)})1(
1()
sin(cos
2
)()({ 22
2/12
22
bxk
i
sxikqbx
i
n
ni
Energy Flow for the Electron Drift Mode
Electron Drift Mode
Electron drift wave is generally
more unstable in a toroidal
plasma than in the slab
Potential hill and
strong shear damping
Local well and
less effective shear damping
Confined mode structure with a potential
well --> suppress shear damping
1~ici stable
n
icqs
)/21)(1(
3
4
2.1
1
ncn
ncn
)/21)(1(
9.01 qs
nc
For flat density, ion temperature gradient determine the instability (ITG mode)
i or ITG mode
Stability diagram for the toroidal ITG mode
Slab and toroidal branches
Growth rate depends on the value of i
(i mode)
Trapped Particle Instabilities• Collisionless trapped particle mode
• Dissipative trapped particle mode : collision dependent, dangerous
Physical mechanism of collisionless trapped particle mode :
trapped particles are trapped between magnetic mirrors and spend
most time in the bad curvature region --> flute-like modes
• charge polarization due to the curvature drift of the trapped particles
• local electric fields from polarization
drive small scale ExB flows
• enhance the initial perturbations
and lead to instability
From drift kinetic equation,
)2( *
2
edede
: unstable for 0de1/2 nLR
Dissipative Trapped Particle Instabilities• Effective collision frequency be the same order as the relevant drift
frequency : trapped particle may be scattered into passing, etc
• trapped ion mode : dissipative version of collisionless mode,
stabilizing ion collisions and destabilizing electron collisions
• dissipative trapped electron mode : stability only by trapped electrons
i.e. the trapped electron collisions and an electron temperature gradient
• collision frequency not too high, i.e. bjj
Growth rate for the trapped ion mode
e
eie ii
2
*
2
2
*)1(1
2
Stabilizing ion
Destabilizing electron
Micro Tearing Modes
Short wavelength with high poloidal
mode number 0/2 rm Stable?
Two effects result in growth of magnetic
perturbation to a saturated island
• nonlinear effect of island structure can
modify equilibrium and particle drifts
• kinetic effects : electron temperature
gradient in a sufficiently collisional plasma
//2
//
2
jx
Ao
ti eeA ~
//
//// AE //)/( ArimBr
////// ikAE
//// / kv Electron „short out‟ //// kAi
1/ //// vkCurrent only flows for sLxrmk /)/(//
)/( sqm
Linear Micro Tearing Modes
r
m
A
dj
x
A
x
A
A
out
o
rr
out
rr
out
out
ss
2
1
//
//
////
//
//2
//
2
jx
Ao
mout rA ~//
Growth rate calculated numerically
In „semi-collisional‟ regime
nkvnkDnv eTee //
2
//// )/(
esTeyeTe Lvdkvkkv 222222
//// //
// ////// AEj
3/13/2~ eo
2
~
pes
Tey
o
c
L
vk
Growth rate
Collisionless growth rate
Growth Rate for Micro Tearing Modes
An approximate analytical form for the growth rate in
the „semi-collisional‟ regime
2/1
2/1
*
2
4/1
**
)4/51(
)/(
)4/11(28
3
/
)4/51(
)4/11(
)4/17(2
e
ee
s
n
pee
ee
ee
e
L
Lc
