filamentary structure and exponential growth...
TRANSCRIPT
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Filamentary Structure andExponential Growth of
Nonlinear Ballooning Instability 1
Ping Zhu
in collaboration withC. C. Hegna and C. R. Sovinec
University of Wisconsin-Madison
Plasma Physics SeminarMadison, WI
February 16, 2009
1Research supported by U.S. Department of Energy.
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A ballooning instability in a tokamak (NIMROD simulation)
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Filamentary structure persists during (type-I) edgelocalized modes (ELMs) in the MAST tokamak
[Kirk et al., 2006]
Both linear and nonlin-ear ELM phases.
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ELM filaments resemble linear ballooning modestructure
MAST [Kirk et al. , 2006] ELITE [Snyder et al. , 2005]
NIMROD [Sovinec et al. , 2006]
I Characteristics of ELM filamentsI Toroidal mode number n ∼ 15− 20I Elongated structures aligned with field linesI Persist well into nonlinear phase
I Questions to address in theory:I Why do nonlinear ELM and ballooning filaments resemble
linear ballooning structure?I What is the temporal evolution in the nonlinear regime?
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Different nonlinear regimes of ballooninginstability are characterized by the relativestrength of the nonlinearity with powers of n−1
I Nonlinearity and ballooning parameters
ε ∼ |ξ|Leq
� 1, n−1 ∼k‖k⊥
∼Ly
Lz� 1
I For ε � n−1, linear ballooning mode theory [Coppi, 1977; Connor,
Hastie, and Taylor, 1979; Dewar and Glasser, 1983]
I For ε ∼ n−1, early nonlinear regime [Cowley and Artun, 1997; Hurricane,
Fong, and Cowley, 1997; Wilson and Cowley, 2004]
I For ε ∼ n−1/2, intermediate nonlinear regime → this talk[Zhu, Hegna, and Sovinec, 2006; Zhu et al. , 2007; Zhu and Hegna, 2008; Zhu, Hegna, and Sovinec, 2008; ]
I For ε � n−1/2, late nonlinear regime; analytic theory underdevelopment.
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Intermediate nonlinear regime was previouslyidentified for line-tied g mode [Plasma Physics Seminar 2006]
z
x
y
g∇ρ0
Left: Equilibrium configuration; Right: ux contour of line-tied g-moded
dx0
(p0 +
B20
2
)= ρ0g · x , g = −gx , B0 = B0z
ρ0(x0) = ρc + ρh tanh (x0 + Lc)/Lρ, p0(x0) = ρ0(x0)Lρ → pedestal width; 2ρh → pedestal height
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Transition from early nonlinear regime ε ∼ n−1 tointermediate nonlinear regime ε ∼ n−1/2
[Zhu et al. , 2007]
0 10 20 30 40 50 60 70 80 90time (τ
A)
0.0001
0.001
0.01
0.1
1
(ux) m
ax
case a_rtp032105x01
ε=n-1
(CA)
ε=n-1/2
(Int)
t1 t2 t3
I (ux)max(t = 0) = 10−3
I t1 ∼ 46: ε ∼ n−1, ux ∼ 0.008, ξx ∼ 0.1 → mode width in y ;t2 ∼ 80: ε ∼ n−1/2, ux ∼ 0.3, ξx ∼ 3.6 → mode width in x .
I t1 <∼ t <∼ t2: finger formation initiates during the transition;t2 <∼ t <∼ t3: finger pattern becomes prominent as themode proceeds through the regime ε ∼ n−1/2.
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Exponential growth persists in intermediatenonlinear regime of the line-tied g-mode [Zhu et al. , 2007]
0 20 40 60 80 100 120time (τ
A)
1e-08
1e-07
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
velo
city
max
ux (theory)
ux (simulation)
uz (theory)
uz (simulation)
ε=n-1
(CA)
ε=n-1/2
(Int)
I (ux)max(t = 0) = 10−5
90 100 110 120 130time (τ
A)
0.001
0.01
0.1
1
velo
city
max
ux (theory)
ux (simulation)
uz (theory)
uz (simulation)
ε=n-1
(CA)
ε=n-1/2
(Int)
I Zoom-in of the left figure
I Simulation results agree with numerical solution of thenonlinear line-tied g mode equations.
I Both simulation and numerical solution of theory indicateexponential-like nonlinear growth. Why?
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Outline
1. Nonlinear ballooning equationsI FormulationI Analytic solution
2. Comparison with NIMROD simulationsI Simulation setupI Comparison methodI Comparison results
3. Summary and Discussion
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A Lagrangian form of ideal MHD is used to developthe theory of nonlinear ballooning instability
ρ0
J∇0r · ∂2ξ
∂t2 = −∇0
[p0
Jγ+
(B0 · ∇0r)2
2J2
]
+∇0r ·[
B0
J· ∇0
(B0
J· ∇0r
)]+
ρ0
J∇0r · g (1)
where r(r0, t) = r0 + ξ(r0, t),∇0 =∂
∂r0, J(r0, t) = |∇0r| (2)
The full MHD equation can be further reduced for nonlinearballooning instability using expansion in terms of ε and n−1
ξ(√
nx0, ny0, z0, t) =∞∑
i=1
∞∑j=0
εin−j2
(xξx{i,j} +
y√n
ξy{i,j} + zξz{i,j}
).
(3)
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Nonlinear ballooning expansion is carried out forgeneral magnetic configurations with flux surfaces
I Clebsch coordinate system (Ψ0, α0, l0)
B = ∇0Ψ0 ×∇0α0 (4)
I Expansions are based on intermediate nonlinearballooning ordering ε ≡ |ξ|/Leq ∼ n−1/2 � 1
ξ(√
nΨ0, nα0, l0, t) =∞∑
j=1
n−j2
(e⊥ξΨ
j2
+e∧√
nξα
j+12
+ Bξ‖j2
)(5)
J(√
nΨ0, nα0, l0, t) = 1 +∞∑
j=0
n−j2 J j
2(6)
where e⊥ = (∇0α0 × B)/B2, e∧ = (B×∇0Ψ0)/B2.I The spatial structure of ξ(Ψ, α, l) and J(Ψ, α, l) is ordered
to be consistent with linear ideal ballooning theory:Ψ =
√nΨ0, α = nα0, l = l0.
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The linear local ballooning operator will continueto play a fundamental role in the nonlineardynamics
Linear ideal MHD ballooning and g modes obey
ρ∂2t ξ = L(ξ) (7)
where
L(ξ) ≡ B · ∇0(B · ∇0ξ)− [∇0(B · ∇0B) +∇0(ρg)] · ξ
−BB · ∇0
{1
1 + γβ
[B · ∇0ξ
‖ − ρB · gB2 ξ‖ −
(2κ +
ρgB2
)· ξ⊥
]}−2B · ∇0B + ρg
1 + γβ
[B · ∇0ξ
‖ − ρB · gB2 ξ‖ −
(2κ +
ρgB2
)· ξ⊥
](8)
is an ODE operator, with ξ = ξ‖B + ξ⊥ [Hurricane, Fong, and Cowley, 1997].
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A set of nonlinear ballooning equations for ξ aredescribed using the linear operator [Zhu and Hegna, 2008]
ρ(|e⊥|2∂α∂2t ξΨ
12
+ [ξ 12, ∂2
t ξ 12]) = ∂αL⊥(ξΨ
12, ξ‖12) + [ξ 1
2,L(ξ 1
2)], (9)
ρB2∂2t ξ‖12
= L‖(ξΨ12, ξ‖12) (10)
L⊥(ξΨ, ξ‖) ≡ e⊥ · L(ξ) (11)= B∂l(|e⊥|2B∂lξ
Ψ) + 2e⊥ · κe⊥ · ∇0pξΨ
+2γpe⊥ · κ
1 + γβ
(B∂lξ
‖ − 2e⊥ · κξΨ)
, (12)
L‖(ξΨ, ξ‖) ≡ B · L(ξ) (13)
= B∂l
[γp
1 + γβ
(B∂lξ
‖ − 2e⊥ · κξΨ)]
, (14)
[A, B] ≡ ∂ΨA · ∂αB− ∂αA · ∂ΨB. (15)
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The local linear ballooning mode structure andgrowth continue to satisfy the nonlinearballooning equations in Lagrangian space
I The nonlinear ballooning equations can be written in thecompact form[
Ψ + ξΨ, ρ|e⊥|2∂2t ξΨ − L⊥(ξΨ, ξ‖)
]= 0, (16)
ρB2∂2t ξ‖ − L‖(ξΨ, ξ‖) = 0. (17)
I The general solution satisfies
ρ|e⊥|2∂2t ξΨ = L⊥(ξΨ, ξ‖) + N(Ψ + ξΨ, l , t), (18)
ρB2∂2t ξ‖ = L‖(ξΨ, ξ‖). (19)
I A special solution to the nonlinear ballooning equations isthe solution of the linear ballooning equations
N(Ψ, l , t) = 0, where Ψ = Ψ + ξΨ. (20)
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Implications of the analytic solution
I The solution is linear in Lagrangian coordinates, butnonlinear in Eulerian coordinates
ξ = ξlin(r0) = ξlin(r− ξ) = ξnon(r).
I Linear ballooning mode operator requires solution havingfilamentary structure ξ ∼ A(Ψ, α)H(l)
I Linear ballooning mode structure gives exponential growth
∂2t ξ = Lξ = Γ2A(Ψ, α)H(l) = Γ2ξ.
I Perturbation developed from linear ballooning instabilityshould continue to
I grow exponentiallyI maintain filamentary spatial structure
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Outline
1. Nonlinear ballooning equationsI FormulationI Analytic solution
2. Comparison with NIMROD simulationsI Simulation setupI Comparison methodI Comparison results
3. Summary and Discussion
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Simulations of ballooning instability are performedin a tokamak equilibrium with circular boundaryand pedestal-like pressure
I Equilibrium fromESC solver [Zakharov and
Pletzer,1999]
I Finite element meshin NIMRODsimulation.
0
0.02
0.04
0.06
0.08
p0
0.51
1.52
2.5
3
q
0 0.2 0.4 0.6 0.8 1
(Ψp/2π)
1/2
0
0.5
1
1.5
2
<J ||B
/B2 >
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Linear ballooning dispersion is characteristic ofinterchange type of instabilities
0 10 20 30 40 50toroidal mode number n
0
0.05
0.1
0.15
0.2
0.25γτ
A
f_050808x01x02
I Extensive benchmarks between NIMROD and ELITE showgood agreement [B. Squires et al., 2009].
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Simulation starts with a single n = 15 linearballooning mode
0 5 10 15 20 25 30 35 40 45time (µs)
1e-15
1e-12
1e-09
1e-06
0.001
1
1000
kine
tic e
nerg
y
n=0n=15n=30total
elecd=ndiff=0,kvisc=kperp=25,nhypd=1
mx=20,my=32,poly=5,lphi=7
f_es081908x01a
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Isosurfaces of perturbed pressure δp showfilamentary structure (t = 30µs, δp = 168Pa)
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For theory comparison, we need to know plasmadisplacement ξ associated with nonlinearballooning instability
ξ connects the Lagrangian and Eulerian frames,
r(r0, t) = r0 + ξ(r0, t) (21)
In the Lagrangian frame
dξ(r0, t)dt
= u(r0, t) (22)
In the Eulerian frame
∂tξ(r, t) + u(r, t) · ∇ξ(r, t) = u(r, t) (23)
where u(r, t) is velocity field, ∂t = (∂/∂t)r, and ∇ = ∂/∂r. ξ isadvanced as an extra field in simulations.
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Lagrangian compression ∇0 · ξ can be moreconveniently used to identify nonlinear regimes
I Nonlinearity is defined by |ξ|/Leq, but Leq is not specific.I Linear regime
∇0 · ξ = ∇ · ξ (24)
I Early nonlinear regime
∇0 · ξ ∼ λ−1Ψ ξΨ + λ−1
α ξα + λ−1‖ ξ‖ (25)
∼ n1/2n−1 + n1n−3/2 + n0n−1 ∼ n−1/2 � 1.
I Intermediate nonlinear regime
∇0·ξ ∼ λ−1Ψ ξΨ+λ−1
α ξα+λ−1‖ ξ‖ ∼ n1/2n−1/2+n1n−1+n0n−1/2 ∼ 1.
(26)I The Lagrangian compression is sensitive to nonlinearity:
matrix (I−∇ξ)−1 could become singular passing beyondintermediate nonlinear regime.
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The Lagrangian compression ∇0 · ξ is calculatedusing the Eulerian tensor ∇ξ
Transforming from Lagrangian to Eulerian frames, one finds
ξ(r0, t) = ξ[r− ξ(r, t), t ] (27)
∇ξ =∂ξ
∂r
=
(∂r∂r− ∂ξ
∂r
)· ∂ξ
∂r0
= (I−∇ξ) · ∇0ξ (28)
The Lagrangian compression ∇0 · ξ is calculated from theEulerian tensor ∇ξ at each time step using
∇0 · ξ = Tr(∇0ξ) = Tr[(I−∇ξ)−1 · ∇ξ]. (29)
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Exponential linear growth persists in theintermediate nonlinear regime of tokamakballooning instability [Zhu, Hegna, and Sovinec, 2008]
0 5 10 15 20 25 30 35time (µs)
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
100
|ξ|max
(m)
(div0ξ)
max
Int
Dotted line indicates the transition to the intermediate nonlinearregime when ∇0 · ξ ∼ O(1)
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Lagrangian compression may also be able toidentify other nonlinear ballooning regimes
0 5 10 15 20 25 30 35 40 45time (µs)
1e-06
0.0001
0.01
1
100
10000
1e+06
1e+08
|ξ|max
(m)
(div0ξ)
max
Int
20x32,poly=5,lphi=7elecd=ndiff=0,kvisc=kperp=25,nhypd=1f_es081908x01a
I Intermediate nonlinear regime is entered ∼ 28µs.I Large ∇0 · ξ indicates transition to nonlinear regimes.
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Perturbation energy grows with the linear growthrate into the intermediate nonlinear regime(vertical line)
0 5 10 15 20 25 30 35 40 45time (µs)
1e-15
1e-12
1e-09
1e-06
0.001
1
1000
kine
tic e
nerg
y
n=0n=15n=30total
elecd=ndiff=0,kvisc=kperp=25,nhypd=1
mx=20,my=32,poly=5,lphi=7
f_es081908x01a
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Contours of plasma velocity and displacement ξ att = 5µs, linear phase
Re VR
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re VZ
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re VPhi
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiR
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiZ
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiP
2.0 2.5 3.0 3.5 4.0-1
.0-0
.50.
00.
51.
0R
Z
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Contours of plasma velocity and displacement ξ att = 30µs, intermediate nonlinear phase
Re VR
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re VZ
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re VPhi
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiR
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiZ
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiP
2.0 2.5 3.0 3.5 4.0-1
.0-0
.50.
00.
51.
0R
Z
![Page 29: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/29.jpg)
Contours of plasma velocity and displacement ξ att = 40µs, start of late nonlinear phase
Re VR
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re VZ
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re VPhi
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiR
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiZ
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re XiP
2.0 2.5 3.0 3.5 4.0-1
.0-0
.50.
00.
51.
0R
Z
![Page 30: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/30.jpg)
Distortions of flux surface are consistent withplasma displacement
Re P
2.0 2.5 3.0 3.5 4.0
-1.0
-0.5
0.0
0.5
1.0
R
Z
Re P
2.0 2.5 3.0 3.5 4.0-1
.0-0
.50.
00.
51.
0
R
Z
I Above: Total pressure contours.I Left: t = 30µs; Right: t = 40µs.
![Page 31: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/31.jpg)
Total pressure contours on 4 poloidal slices show3D view of distorted pedestal (t = 40µs)
![Page 32: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/32.jpg)
Total pressure isosurface shows nonlinearfilamentary structure (t = 40µs, p = 3602Pa)
![Page 33: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/33.jpg)
Distribution of plasma displacement vectors (ξ)aligns with pressure isosurface ribbons (t = 40µs)
![Page 34: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/34.jpg)
Zoom-in view of total pressure isosurface anddisplacement (ξ) vectors (t = 40µs)
![Page 35: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/35.jpg)
Distribution of velocity (u) vector is different fromthat of displacement (ξ) vector at t = 40µs
![Page 36: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/36.jpg)
Simulations started from multiple linear ballooningmodes also confirm theory prediction
1e-05
0.0001
0.001
0.01
0.1
1
10
100
|ξ|max
/a
(div0ξ)
max
0 10 20 30 40 50time (τ
A)
1e-8
1e-6
1e-4
1e-2
1
100
kine
tic e
nerg
y
Int
total n=15
n=30 n=0
1e-05
0.0001
0.001
0.01
0.1
1
10
100
|ξ|max
/a
(div0ξ)
max
0 10 20 30 40time (τ
A)
1e-8
1e-6
1e-4
1e-2
1
100
kine
tic e
nerg
y
n=0
n=1
n=6
n=11
n=13
n=15
total
Int
![Page 37: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/37.jpg)
Summary
I There is an exponential growth phase of nonlinearballooning instability
I The growth rate is same as the linear growth rate.I The spatial structure is same as the linear mode (in
Lagrangian space).I This nonlinear phase is defined by ordering ξΨ ∼ λΨ.
I ImplicationsI May explain persistence of filamentary structures in ELM
experiments.I Nonlinear ELM behavior of pedestal may be largely
controlled by its linear properties (which is determined bythe configuration).
![Page 38: Filamentary Structure and Exponential Growth ofhomepages.cae.wisc.edu/~pzhu/pub/pub09/wisc0216/nbg.pdf · 2009. 2. 16. · Filamentary Structure and Exponential Growth of Nonlinear](https://reader035.vdocument.in/reader035/viewer/2022071104/5fddc58766fa466ff401cdf6/html5/thumbnails/38.jpg)
Discussion
I What are missing?I Nonlinearity
I What is the “nonlinearity depletion” mechanism?I Marginal unstable configuration (Γ ∼ 0) need nonlinear
filament envelope equation (detonation?)I Late nonlinear regimes: saturation, filament->blob?
I 2-fluid, FLR (finite Larmor radius) effectsI Edge shear flow effectsI RMP (resonant magnetic perturbation) effectsI Geometry (non-circular shape, divertor separatrix/X-point)I Peeling-ballooning couplingI ......