Nonlinear Consequences of Weakly Driven Energetic Particle Instabilities

2008 International Sherwood Fusion Theory Conference March 30 - April 2, 2008, Boulder, Colorado Nonlinear Consequences of Weakly Driven Energetic Particle Instabilities Boris Breizman Institute for Fusion Studies UT Austin In collaboration with H. Berk, A. Ödblom, M. Pekker, N. Petviashvili, P. Sandquist, S. Sharapov, Y. Todo, R. Vann, and JET-EFDA contributors Page 1

Outline Examples of Experimental Data Near-threshold Technique Non-rigid Modes Multiple Modes and Global Transport Thoughts on Next Steps Page 2

NONLINEARITY IN ACTION (experimental observations) Page 3

Nonlinear Splitting of Alfvén Eigenmodes in JET Page 4

Rapid Frequency Chirping Events Hot electron interchange modes in Terrella (Courtesy of Michael Maul, Columbia University) Alfvén modes in MAST (Courtesy of Mikhail Gryaznevich, Culham laboratory, UKAEA) The ms timescale of these events is much shorter than the energy confinement time in the plasma Page 5

Alfvén Wave Instability and Particle Loss in TFTR Saturation of the neutron signal reflects anomalous losses of the injected beams. The losses result from Alfvénic activity. Projected growth of the neutron signal Page 6 K. L. Wong, et al., Phys. Rev. Lett. 66, 1874 (1991)

THEORY AT THE THRESHOLD (building blocks) Page 7

Near-threshold Nonlinear Regimes Why study the nonlinear response near the threshold? Typically, macroscopic plasma parameters evolve slowly compared to the instability growth time scale Perturbation technique is adequate near the instability threshold Single-mode case: Identification of the soft and hard nonlinear regimes is crucial to determining whether an unstable system will remain at marginal stability Bifurcations at single-mode saturation are of interest for diagnostic applications Long-lived coherent nonlinear structures can emerge Multi-mode case: Multi-mode scenarios can keep the system near marginal stability Resonance overlap can trigger avalanche-type events in phase space Page 8

Key Element in Theory Interaction of energetic particles with unstable waves Pendulum equation for particles in an electrostatic wave: m x = ee cos(kx!"t) Wave-particle resonance condition:! " kv = 0 Phase space portrait in the wave frame: m(v-!/k) x Page 9 A

Wave-Particle Lagrangian Page 10 Perturbed guiding center Lagrangian: ( ) L = ( $ % P! &! + P " &" # H P! ;P " ;µ & ' + ( &)A 2 + 2 Re particles Dynamical variables: P!,!, P ", " are the action-angle variables for the particle unperturbed motion A is the mode amplitude! is the mode phase Matrix element linear mode structure Mode energy: W =! A 2 Resonance condition:! " n! # µ;p # ;E ( ) V l P! ;P " ;µ modes ( ) " l! $ ( µ;p # ;E) " s! % ( µ;p # ;E) = 0 ( AV l ( P! ;P " ;µ)exp(#i) # i*t + in" + il!) particles, modes, l is a given function, determined by the The quantities n, l, and s are integers with s = 0 for low-frequency modes.

Dynamical Equations Unperturbed particle motion is integrable and has canonical action-angle variables I i and ξ i. Unperturbed motion is periodic in angles ξ 1, ξ 2, and ξ 3. Single resonance approximation for the Hamiltonian: H = H 0 ( I ) + 2Re $% A( t)v ( I )exp(i! " i#t) &' Kinetic equation with collisions included:!f!t + "( I )!f $ 2 Re[ ia(t)exp(i# $ i%t) ]!f!#!i = & 3 eff (!" /!I ) $2! 2 f!i 2 Equation for the mode amplitude with background damping included: da dt =!" d A + i# G & d$v * exp(!i% + i#t) f Page 11

Transition from Steady State Saturation to the Explosive Nonlinear Regime (1)-saturated mode (2)-limit cycle Mode Amplitude (a. u.) (3)-chaotic nonlinear state (4)-explosive growth Mode Amplitude (a. u.) Time (a. u.) Time (a. u.) Page 12 Instability drive increases from (1) to (4)

LESS THAN RIGID MODES (robust behavior of soft entities) Page 13

Alfvénic Frequencies and Mode Structure In a cold uniform plasma, shear Alfvén waves have no group velocity across the magnetic field. There is no communication between neighboring field lines. Some of the observed Alfvén perturbations in tokamaks do not represent true eigenmodes (standing waves in radial direction). Page 14

Eigenmode Equation for Alfvén Cascades Offset from Alfvén continuum! *!r " 2 # " 2 G # V 2 A $, n # m R 2 % & +, q ' ( ) 2 - /./!0!r # m2 r 2 * " 2 # " 2 G # V 2 A $, n # m R 2 % & +, q ' ( ) 2 2 # " PG 2 # " fast ion - / 0 = 0./ Geodesic acoustic frequency: IFS-JET collaboration, PoP 12, 112506 (2005) Pressure gradient offset: G.Y. Fu and H.L. Berk, PoP 13, 052502 (2006) Fast ion offset: IFS-JET collaboration, PRL 87, 185002 (2001)! G 2 " 2 MR 2 2! PG 2! fast ion # T e + 7 4 T & i $ % ' ( " # 2 MR r d ( 2 dr T e + T i ) " #! m eb Mc r n plasma d dr n fast ion Low frequency boundary:! 2 =! 2 2 G +! PG 2 +! fast ion Page 15

True Modes and Quasimodes in Alfvén Cascade Observations on JET Alfvén Cascades concentrate at the point of shear reversal, where the system is most uniform. Their structure is less robust than their frequencies. Alfvén Cascades with downward frequency sweeping represent quasimodes rather than true eigenmodes. The theory explains the mystery of downward sweeping. Normalized wave equation:! 2 "!z 2 = # $ %z2 $ z 4 ( )"! < 0 IFS-JET collaboration, 2007 Quasimode Potential well for true modes! > 0 True mode Potential hill for quasimodes Radiative damping can be balanced by fast particle drive Page 16

Fishbone Onset Linear responses from kinetic (wave-particle) and fluid (continuum) resonances are in balance at instability threshold; however, their nonlinear responses differ significantly. Resolution requirements are are prohibitively demanding for nonlinear codes, especially in simulations of near-threshold instabilities. G.Y. Fu et al., PoP 13, 052517 (2006) Questions: Which resonance produces the dominant nonlinear response? Is this resonance stabilizing or destabilizing? Approach: Analyze the nonlinear regime near the instability threshold Perform hybrid kinetic-mhd simulations to study strong nonlinearity Page 17

Linear Near-threshold Mode Frequency ω and growth rate γ Double resonance layer at ω = ± Ω(r)! ~ " Energetic ion drive A. Ödblom et al., PoP 9, 155 (2002)! = r"# A s $1 Page 18

Early Nonlinear Dynamics Weak MHD nonlinearity of the q = 1 surface destabilizes fishbone perturbations Near threshold, fluid nonlinearity dominates over kinetic nonlinearity As mode grows, q profile is flattened locally (near q=1) continuum damping reduced explosive growth is triggered Page 19

Speculation on Downward Chirping U(x) Consider particle motion in a quartic potential well In the presence of dissipation, the oscillation amplitude will decrease in time together with the oscillation frequency x Is this a sensible model for the decay of a fishbone pulse? Page 20 Why quartic potential well Local Alfvén frequency is proportional to the distance from the q=1 surface The characteristic distance is of the order of plasma displacement for strongly nonlinear perturbations The resulting scaling law for frequency downshift is that for a quartic well

Unexplained Fishbone Features Transition from explosive growth to slowly growing MHD structure (i.e., island near q=1 surface) Modification of fast particle distribution in presence of magnetic island Mode saturation and decay Quantitative simulation of frequency sweeping Frequency increases during explosive phase and decreases during pulse decay, consistent with behavior of damped nonlinear oscillator Burst repetition rate in presence of injection Need to include sources/sinks/collisions Page 21

Spontaneous Chirping of Weakly Unstable Mode Simulation of near-threshold bump-on-tail instability (N. Petviashvili, 1997) reveals spontaneous formation of phase space structures locked to the chirping frequency Chirp extends the mode lifetime as phase space structures seek lower energy states to compensate wave energy losses due to background dissipation Clumps move to lower energy regions and holes move to higher energy regions Time evolution of normalized mode amplitude phase space clump Mode frequency sweeping phase space clump Interchange of phase space structures releases energy to sustain chirping mode Page 22

Phase Space Structures and Fast Chirping Phenomena Validation of nonlinear single-mode theory in experiments and simulations Reproduction of recurring chirping events on MAST Multi-mode simulation (Sherwood 2006) Nonlinear excitation of stable mode Frequency (khz) Frequency [khz] 140 120 100 80 Alfvén modes in shot #5568 64 66 68 Time [ms] (ms) 70 72 Velocity (a.u.) Time (a.u.) Mode overlap enhances wave energy Frequency (a.u.) Bump-on-tail simulations (collaboration with R.Vann) Particle distribution function Wave energy Time (a.u.) Velocity (a.u.) Time (a.u.) Page 23

MULTIPLE MODES (puffs from a pressure release valve) Page 24

Effect of Resonance Overlap The overlapped resonances release more free energy than the isolated resonances Page 25

Particle Transport Mechanisms of Interest Neoclassical: Large excursions of resonant particles (banana orbits) + collisional mixing Convective: Transport of phase-space holes and clumps by modes with frequency chirping Quasilinear : Phase-space diffusion over a set of overlapped resonances Important Issue: Individual resonances are narrow. How can they affect every particle in phase space? Page 26

Intermittent Quasilinear Diffusion A weak source (with insufficient power to overlap the resonances) is unable to maintain steady quasilinear diffusion Bursts occur near the marginally stable case f Classical distribution Metastable distribution Marginal distribution Sub-critical distribution RESONANCES Page 27

Simulation of Intermittent Losses classically slowed down beam stored beam energy with TAE turbulence Page 28 co-injected beam part counter injected beam part Numerical simulations of Toroidal Alfvén Eigenmode (TAE) bursts with parameters relevant to TFTR experiments have reproduced several important features: synchronization of multiple TAEs timing of bursts stored beam energy saturation Y. Todo, H. L. Berk, and B. N. Breizman, Phys. Plasmas 10, 2888 (2003).

Issues in Modeling Global Transport Reconciliation of mode saturation levels with experimental data Simulations reproduce bursty losses and particle accumulation level Wave saturation amplitudes appear to be larger than the experimental values (however, this discrepancy may not affect the fast particle pressure profiles) Edge effects in fast particle transport Sufficient to suppress modes locally near the edge Need better description of edge plasma parameters Creation of transport barriers for fast particles Page 29

Page 30 CONCLUDING REMARKS

Items of Interest for Further Study Diagnostic applications of benign modes (q min, T e, T i, ν eff ) Quantitative interpretation of nonlinearly saturated isolated modes (TAEs, cascades, high-frequency modes, etc.) Lifecycle of fishbones and EPMs, starting from instability threshold Intermittent strong bursts versus stochastic diffusive transport Marginal stability constraints on fast particle profiles in burning plasmas Transport barriers for fast particles Page 31