Filamentary Structure and Exponential Growth of Nonlinear Ballooning Instability 1 Ping Zhu in collaboration with C. C. Hegna and C. R. Sovinec University of Wisconsin-Madison Plasma Physics Seminar Madison, WI February 16, 2009 1 Research supported by U.S. Department of Energy.
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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.
A ballooning instability in a tokamak (NIMROD simulation)
Filamentary structure persists during (type-I) edgelocalized modes (ELMs) in the MAST tokamak
[Kirk et al., 2006]
Both linear and nonlin-ear ELM phases.
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?
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.
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
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.
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?