Low Frequency Alfven Eigenmodes in Toroidal Plasmas

Zhihong LinUniversity of California, Irvine

Fusion Simulation Center, Peking University

Yaqi Liu, Huasen Zhang, Wenlu Zhang, Jian Bao, Ge Dong

GTC TeamSciDAC ISEP Center, ITER-CN Simulation Project

IAEA-TM EP, PPPL, 2017

Low Frequency Alfven Eigenmodes

• Motivation– Loss of energetic particles due to low frequency BAAE & BAE– Strong coupling of BAAE & BAE with thermal plasmas: impact on

confinement of thermal plasmas, transfer of energy from energetic to thermal ions?

• Existence & excitation of BAAE– Discrete BAAE has not been predicted by MHD or kinetic analytic theory– Does BAAE exist despite heavy damping by thermal ions?– Can BAAE be excited by realistic fast ion density gradient?

• Nonlinear interaction between BAAE & BAE– Can BAAE be nonlinearly generated?– What is nonlinear dynamics of BAAE?

Low Frequency Modes Induced Fast Ion Loss in DIII-D

• Both high frequency Alfvén eigenmodes (AE) and low frequency modes (LF)

• Up to 45% neutron deficit observed; Half induced by AE; The other half caused by LF modes

• LF: Beta-induced Alfven eigenmode (BAE)? beta-induced Alfven-acoustic eigenmode (BAAE)?

G. Kramer & R. Nazikian

100~270kHz

40~100kHz

BAE like

• DIII-D LF 40~100kHz

• Unstable BAE-like mode, 𝜔𝜔 = 61 kHz (91kHz in lab frame)

• BAAE-like modenonlinearly generated𝜔𝜔 = 28kHz (58kHz in lab frame)

• LF modes also driven unstable by thermal plasma pressure gradient

n=2

- - - qmin

FFT IFFT

BAAE like n=2

- - - qmin

t𝜔𝜔 𝑣𝑣𝑖𝑖/𝑅𝑅

Initial Results from GTC Simulation for n=2 LF Modes

Existence of BAAE Verified by GTC

Gyrokinetic particle simulation of beta-induced Alfven-acoustic eigenmode,H. S. Zhang, Y. Q. Liu, Z. Lin, and W. L. Zhang, Phys. Plasmas 23, 042510 (2016)

𝛿𝛿𝛿𝛿GTC

• Ti<<Te to minimize BAAE damping• Frequencies and radial location of three excitation methods agree• Discrete frequency ~ nearby accumulation point: nonlocal effects• Direction of mode structure, frequency sweeping in reverse shear,

agrees with experiments

Transition from BAE to BAAE for Larger Device Ti=0.5Te

BAE&BAAE coexist

a/R=0.29B=1.91T

(n,m)=(4,6)βe=2.87%

Te=2keVTi=0.5TeTf=9Te

6

Polarization of Unstable BAAE and BAE Alfvenic

BAAE(R=400cm) BAE(R=260cm)

unstable damped unstable damped

• All poloidal harmonics of unstable BAAE and BAE are Alfvenic• Gradually decrease the drive → damped modes.

• Perturbative, radially local theory not valid for BAAE

⁄E∥ E∥ES (BAAE)� ≪ 1, m=6 Alfvén~1, m = 5,7Acoustic

⁄E∥ E∥ES (BAE) ≪ 1, m=5,6,7 Alfvén

Linear Wave-Particle Energy Exchanges• BAAE & BAE excited by transferring from fast ion perpendicular energy• BAAE damped by transferring to thermal ion parallel & perpendicular energy• BAE damped by transferring to thermal ion perpendicular energy• BAAE: ωr=0.50vi/R0 , γ=0.04, γEP=0.08, γD_unstable=-0.04, γD_stable=-0.22• BAE: ωr=2.5vi/R0 , γ=0.28, γEP=0.44, γD_unstable=-0.16, γD_stable=-0.05• BAAE excited even 𝛾𝛾EP ≪𝛾𝛾D_stable: perturbative theory not valid for BAAE

BAEBAAE

Excitation of Low Frequency Alfven Eigenmodes in Toroidal Plasmas, Yaqi Liu, Zhihong Lin, Huasen Zhang, Wenlu Zhang, Nuclear Fusion 57, 114001 (2017)

Nonlinear Generation of BAAE by BAE

• Linearly stableBAAE NL driven by BAE with 𝑛𝑛f = 9.5%𝑛𝑛𝑒𝑒

• 𝛾𝛾𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵~𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐when BAE saturates

• NL generation of BAAE by BAE in DIII-D?

• Threshold 𝑛𝑛f =9%𝑛𝑛𝑒𝑒

Inverse Wavelet Transform

9

BAAE NL Frequency Chirping Down

• Frequency chirping down

• Amplitude oscillation:

saturation not due to

profile relaxation

Y. Q. Liu, PhD Thesis

Predictive Capability Requires Integrated Simulation of Nonlinear Interactions of Multiple Kinetic-MHD Processes

Microturbulence

Macroscopic MHD instability

Energetic particle effects

Neoclassical transport

• Neoclassical tearing mode (NTM) is the most likely instability leading to disruption,NTM excitation depends on nonlinear interaction of MHD instability, microturbulence, neoclassical transport, and EP effects

• GTC gyrokinetic turbulence simulation of EP thanks to SciDAC GSEP (2008-2017)

NTM DisruptionThreshold?

NL dynamics?

RF control

Gyrokinetic Toroidal Code (GTC)• GTC: first-principles, integrated simulation capability for

nonlinear interactions of kinetic-MHD processes

• Current capability in a single version: Global 3D toroidal geometry (via EFIT, VMEC, M3D-C1) Microturbulence: 5D gyrokinetic thermal & fast ions, drift-kinetic &

hybrid electrons, electromagnetic fluctuations MHD and energetic particle (EP): Alfven eigenmodes,

kink, resistive & collisionless tearing modes Neoclassical transport: Fokker-Planck operators Radio frequency (RF) waves: 6D Vlasov ions Scalable to whole titan computer with GPU,

ported to MIC

Phoenix.ps.uci.edu/GTC[Lin, Science1998]

• Parallel electric field 𝑬𝑬|| = −𝛻𝛻||𝛿𝛿 −𝜕𝜕𝐵𝐵||

𝜕𝜕𝑡𝑡

• Need to calculate accurately scalar φ and vector potentials A||

• Ideal MHD E||~0. |E|||/|𝛻𝛻||𝛿𝛿|∼|E|||/|𝜕𝜕𝐵𝐵||

𝜕𝜕𝑡𝑡| ~(𝑘𝑘⊥𝜌𝜌𝑠𝑠)2 ≪ 1

• Physics: cancellation between electrostatic and inductive E||

• Canonical momentum used as independent velocity variable to avoid explicit time derivative operation:

• LHS: |1st term|/|2nd term|~(𝑘𝑘⊥𝑑𝑑𝑒𝑒)2 ≪ 1. Small numerical error of RHS leads to large error of A||

• Infamous “cancellation problem”: calculate 2nd-order 𝒪𝒪 𝑘𝑘⊥2𝜌𝜌𝑠𝑠𝑑𝑑𝑒𝑒 2

term from 0th–order 𝒪𝒪 1 equation!

Multiscale Gyrokinetic Simulation of MHD Modes

∑−=−s

ssneZπφφ 4~s

sss unZ

ceA ∑−=∇⊥π4

||2

[Lin, SciDAC PMP meeting, GA, 2001]

• E|| from electron parallel force balance. Calculate 2nd-order term from 2nd–order equation!

• For ω/k||<<ve, electron response mostly adiabatic (isothermal). First, estimate E|| using massless fluid electron𝐸𝐸|| = −𝛻𝛻|| 𝛿𝛿 + 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 = − 𝑇𝑇𝑒𝑒

𝑒𝑒𝑖𝑖0𝛻𝛻||𝛿𝛿𝑛𝑛𝑒𝑒

Vector potential A|| calculated from 𝐸𝐸||

Perturbed flow 𝛿𝛿𝑢𝑢𝑒𝑒 from Ampere’s law

Perturbed density 𝛿𝛿𝑛𝑛𝑒𝑒 from continuity equation

Electrostatic potential φ from Poisson equation using perturbed density 𝛿𝛿𝑛𝑛𝑒𝑒

• Then, E|| is corrected by kinetic electron non-adiabatic response using split-weigh scheme to reduce noise

• No tearing mode

Fluid-Kinetic Hybrid Electron Model

geff eind TeMe δδ φφ += + /)(

A Fluid-Kinetic Hybrid Electron Model for Electromagnetic Simulations, Lin and Chen, Phys. Plasmas 8, 1447 (2001)

𝑒𝑒 𝛿𝛿 + 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖𝑇𝑇𝑒𝑒

=𝛿𝛿𝑛𝑛𝑒𝑒𝑛𝑛0

𝜕𝜕𝐴𝐴||

𝜕𝜕𝑐𝑐= 𝛻𝛻||𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖

𝑛𝑛0𝑒𝑒𝛿𝛿𝑢𝑢𝑒𝑒 = −𝛻𝛻⊥2𝐴𝐴||

𝜕𝜕𝛿𝛿𝑖𝑖𝑒𝑒𝜕𝜕𝑡𝑡

= −𝛻𝛻||𝑛𝑛0𝛿𝛿𝑢𝑢𝑒𝑒

• Calculate exact E|| by separating A|| into adiabatic and non-adiabatic parts: 𝐴𝐴|| = 𝐴𝐴||

𝐵𝐵 + 𝐴𝐴||𝑁𝑁𝐵𝐵

• Define electron adiabatic responses using adiabatic E||

• Mixed-variable gyrokinetics [Mishchenko et al, PoP (2014)] separates A||into “symplectic” and “Hamiltonian” parts, and define ideal MHD as “symplectic” φind=-φ : Alfven wave

A Conservative Scheme Solving Exact DKE

Alfven wave, IAW, drift wave

𝜕𝜕𝐴𝐴||𝐵𝐵

𝜕𝜕𝑐𝑐= 𝛻𝛻||𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖

A conservative scheme of drift kinetic electrons for gyrokinetic simulation of kinetic-MHD processes in toroidal plasmas, J. Bao, D. Liu, Z. Lin, Phys. Plasmas, 2017

𝛿𝛿𝑓𝑓𝐵𝐵 =𝑒𝑒(𝛿𝛿 + 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖)

𝑇𝑇𝑒𝑒𝑓𝑓0 + 𝛿𝛿𝛿𝛿𝐵𝐵 𝜕𝜕𝑓𝑓0

𝜕𝜕𝛿𝛿

𝑒𝑒𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖𝑇𝑇𝑒𝑒

= −𝑒𝑒𝛿𝛿𝑇𝑇𝑒𝑒

+𝛿𝛿𝑛𝑛𝑒𝑒𝑛𝑛0

−𝛿𝛿𝛿𝛿𝐵𝐵

𝑛𝑛0𝜕𝜕𝑛𝑛0𝜕𝜕𝛿𝛿

A Conservative Scheme Solving Exact DKE• Non-adiabatic part of E|| calculated via electron parallel momentum

equation (generalized Ohm’s law) using non-adiabatic response• Recover collisionless tearing mode with current sheet at de scale • Non-tearing modes: current screened by collisionless skin depth de• No “cancellation problem”

𝛻𝛻⊥2 −1𝑑𝑑𝑒𝑒2

𝜕𝜕𝐴𝐴||𝑁𝑁𝐵𝐵

𝜕𝜕𝑐𝑐=

1𝑑𝑑𝑒𝑒2𝜒𝜒|| − 𝑐𝑐𝛻𝛻⊥2 𝛻𝛻||𝛿𝛿𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖

Compressible Magnetic Perturbations via Perpendicular Force Balance for Slow Modes

Gyrokinetic particle simulations of the effects of compressional magnetic perturbations on drift-Alfvenic instabilities in tokamaks, Dong et al, Phys. Plasmas 24, 081205 (2017)

• Effects of δB|| on TAE linear dispersion very small

Zonal Fields Generation and Nonlinear MHD

TAE

ITG

KBM

• ZF in TAE & KBM generated via 3-wave coupling, in contrast to modulational instability in ITG

• Explosive growth of localized current sheet in TAE& KBM by NL ponderomotive force: MHD forward cascade?

Conclusions and Plan

• BAAE can be excited by realistic fast ion density gradient despite strong damping by thermal ions– Non-perturbative and nonlocal theory needed for BAAE

• BAAE can be nonlinearly generated by BAE– Beyond linear eigenmode paradigm for AE physics and EP transport

• Plan: Integrated Simulation of Energetic Particles in Burning Plasmas– SciDAC ISEP (UCI, GA, PPPL, ORNL, LBNL, LLNL, PU, UCSD, UT), 2017– First-principles simulations (GTC, GYRO, M3D-K, TAEFL) of AE turbulence

and EP transport with multi-physics (coupling with microturbulence & tearing modes) and at long time scale (intermittency & transient)

– Reduced EP transport models (CGM, RBQ)– Verification & validation– EP module for whole device modeling: ISEP framework– Computational partnership: workflow, solvers, optimization, portability– (2 postdocs @ UCI)

AE Saturation & EP Transport Benchmark

• 8 GK & MHD codes• 3 reduce EP transport

models• Linear benchmark

nearly done• ~ 6 months: AE

saturation benchmark• ~ 1 year: EP

transport benchmark