Nonlinear dynamics of toroidal Alfvén eigenmodes in ... · Nonlinear dynamics of toroidal Alfvén...
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Nonlinear dynamics of toroidal Alfvén eigenmodes in presence of a
tearing modeZ.W. Ma
collaborators:J. Zhu and W. Zhang
Institute for Fusion Theory and Simulation, Zhejiang University
PPPL,8-15-2018
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Outline
1. Background2. A brief introduction of CLT-K code
and its benchmark3. Simulation resultsa. n=2 EPMb. TAE & Tearing mode
Summary
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Alfven spectrum in Tokamak
3
TAEs in gaps can be easily driven by energetic particles (EPs).
1. BackgroundToroidal Alfven eigenmodes (TAEs)
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4
In DIII-D experiments,it is found that TAE can lead to a large percentage of EP loss!
[E. Strait Nuclear Fusion 1993]
TAETAE
EPs(theory)
EPs(theory)
EPs(experiment)
EPs(experiment)
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Tearing mode instability Tearing mode instability driven by electric current is one of the most
important instabilities in magnetized plasmas. Many eruptive phenomena in both space and laboratory are believed to
be associated with the tearing mode instability. Tearing mode instability is regarded as the primary cause for
degradation of the plasma performance in Tokamak.
Kink driven Tearing modem/n=1/1
Tearing modem/n=2/1
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HL-2A J-TEXT
W. Chen at. el. EPL, 107 (2014) 25001
High frequency branch(TAE/BAE) and low frequency(TM) are always observed on HL-2A and J-TEXT experiments.
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Weakly chirping of TAEs Pitch-Fork of TAEs
P. W. Shi at. el. Physics of Plasmas 24, 042509 (2017)
Strong TM activities lead to nonlinear mode-mode couplings and TAEs are weakly chirping.
HL-2A
Pitch-Fork TAEs is observed without tearing modes.
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NIMROD (Takahashi 2009) M3D-K (Cai huishan2011)
R. Takahashi, D. P. Brennan, and C. C. Kim, Phys. Rev. Lett. 102, 135001(2009)H. Cai, S. Wang, Y. Xu, J. Cao, and D. Li, Phys. Rev. Lett. 106, 075002 (2011)H. Cai and G. Y. Fu, Phys. Plasmas 19, 072506 (2012).
Co-passing particles
Counter-passing particles
isotropic particles
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S. Wang and Z. W. Ma, Phys. Plasmas 22, 122504 (2015);S. Wang, Z. W. Ma, and W. Zhang, Phys. Plasmas 23, 052503 (2016)
2. A brief introduction of CLT-K code and its benchmarkThe MHD component of CLT-K is based on CLT (Ci-Liu-Ti means MHD in Chinese) which is an initial value code that resolves the full set of resistive Hall MHD equations in 3D toroidal geometries.
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W. Park, S. Parker, H. Biglari, M. Chance, L. Chen, C. Z. Cheng, T. S.Hahm, W. W. Lee, R. Kulsrud, D. Monticello, L. Sugiyma, and R. B. White, Phys. Fluids B 4, 2033 (1992).
The Hybrid Kinetic-MHD Equations
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Drift kinetic equations for energetic particles
where
Cary 2009
J. R. Cary and A. J. Brizard, Rev. Mod. Phys. 81, 693 (2009).
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Current JE from energetic particles
:Guiding center current
:Magnetization current
:Polarization current
EJ
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Parameters for n=1 TAE benchmark study
• Equruimbirum parameters
• Hot particle parameters
• Slowing down distribution
• For simplify
Benchmark for CLT-K
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Benchmark study on linear n=1 TAE
CLT-K
NOVA
TAE : ωNOVA=0.100 ωantenna=0.097 ωEP=0.093γ/ωβNOVA=1 γ/ωβEP=1.25
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Benchmark study on nonlinear n=1 TAE
Theoretical scaling law:Eφsat ~ γL2!
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Parameters for n=2 EPM simulation
n=2 q0 = 1.25Beta_av = 0.016
3. Simulation resultsa. n=2 EPM (PoP,2016)
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Time evolution of frequency n=2 mode driven by isotropic energetic particles
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Time evolution of mode structure for n=2 mode
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Time evolutions of peak values for two poloidal harmonics (m=2 and 3)
1. m=2 and m=3 modes are almost the same during the linear stage, which is TAE-like structure.
2. The m=2 mode is overtaken than the m=3 mode for a short period.
3. The m=3 mode becomes dominant at the late time.
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Mode power struture
EPM (initially TAE-like or rTAE) is linearly unstable and one component of EPM remains the frequency unchanged while another component of EPM chirps down to a BAE gap during the nonlinear phase, which can be called BAE-like EPM.
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Comparisons of δf for Λ=0.50
Linear stage Nonlinear stage
The same movement is also observed for Λ=0.00,0.25 cases, but it is more evident for Λ=0.50 cases
resonant line along a constant phase :
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• Equruimbirum parametersR0=4m, a=1m, βthermal=0, η=10-5~-7
• Hot particle parameters
• Slowing down distribution
• For simplify
1.6%
b. n=1 TAE and tearing mode NF,2018
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Assumption:1. The effect of fast ions on the equilibrium are neglected.2. Only n=1 component for Jh is considered .
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Time evolution of the mode with different βh
For small βh: weakly stabilization effectFor large βh : destabilization effect
Final nonlinear saturation levels are almost same.
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Time evolution of the mode(βh=0.008)
0 2000 4000 6000 800010-12
10-10
10-8
10-6
10-4
Saturation due to wave-particle interaction
Saturation due to nonlinear MHD mode coupling
TAE Tearing modes
Multiple tearing mode saturation
MHD modes
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Time evolution of different components of mode
0 2000 4000 600010-30
10-20
10-10
100
MHD modes
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Snapshots of the mode structure
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0 0.2 0.4 0.6 0.8 1-10
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n=0, m=0n=0, m=1n=1, m=1n=1, m=2n=2, m=2n=2, m=3n=2, m=4n=3, m=4n=3, m=5n=3, m=6n=3, m=7
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MHD modes
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Evolution of power spectrum for different mode
Primary mode
Secondary mode High frequency branch : TAELow frequency branch : tearing mode
n=1 TAE is dominant at the linear stage, then other TAEs (n=0,2,3) are excited during the nonlinear stage.
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Radial structure of mode power spectrum
m/n=2/1 poloidal harmonic first split into two branches
TM: low frequency around the q=2 rational surface(r=0.5)TAE: high frequency peaked near r=0.4 and splits into two branches
m=1
m=2
total
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Comparisons for different resistivity
Linear:Small resistivity: damping(TAE)Large resistivity: destabilize(tearing
mode)
Nonlinear:η↑ frequency chirping ↓
strong TMweak TM
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Comparisons of radial frequency structure for different resistivity
Tearing mode make the frequency chirping weaker!
Weak TM
Strong TM
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Comparisons of phase space structure for different resistivity for passing particle
η=1e-7
η=1e-5
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Comparison w/o EP
total ke low frequency part of Eφ
The low frequency part ofEφ is reduced by the TAE
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Snapshot of low frequency mode structure
Three figs are correspond to three peaks in evolution of kinetic energy.Here the mode structure is time averaged with 78ωA
-1 region
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TM effect on TAE
Here omega=0.06±0.02
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With inclusion of source and sink
TAE re-bursts are observed with increase of collisional frequency.
Three successive TAE bursts are detected in the distribution of the power spectrum
pitch angle scattering and slowing down are included in the collisional operator
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4. Brief Summary EPM (initially TAE-like or rTAE) is linearly excited. During the nonlinear phase, one poloidal component (m=2) of EPM
remains the frequency unchanged while another poloidal component (m=3) of EPM chirps down to a BAE gap. The m/n=3/2 mode becomes dominant at the late time
The n = 1 TAE is first excited by isotropic energetic particles at the linear stage and reaches the first steady state due to wave-particle interaction. After the saturation of the n = 1 TAE, the m/n = 2/1 tearing mode grows continuously and reaches its steady state due to nonlinear mode-mode coupling.
The enhancement of the tearing mode activity with increase of the resistivity could weaken the TAE frequency chirping through the interaction between the p = 1 TAE resonance and the p = 2 tearing mode resonance for passing particles in the phase space, which is opposite to the classical physical picture of the TAE frequency chirping that is enhanced with increase of dissipation.
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Thank you for attention!