Wave-Particle Interaction in Collisionless Plasmas: Resonance and Trapping Zhihong Lin Department of Physics & Astronomy University of California, Irvine International Summer School on Plasma Turbulence and Transport Chengdu, 8/16-8/18, 2007

In high temperature plasmas, collisional mean free path is much longer than wavelength Wave-particle energy exchange depends on ratio of wave phase velocity to particle velocity: kinetic effects WPI plays key roles in ► Excitation and damping of collective modes ► Diffusion in velocity space: thermalization, heating, acceleration ► Transport of particle, momentum, and energy Studies of WPI ► Coherent WPI: resonance, trapping ► Chaos, quasilinear theory ► Weak & strong turbulence theory Wave-Particle Interaction (WPI)

Linear resonance: how does particle responds to a given wave Linear Landau damping Nonlinear trapping 1D particle-in-cell code is used for illustrations Outline

Given an electrostatic 1D plane wave: Motion of a particle with mass m and charge q What is particle energy gain/loss? Linearization: assume that wave amplitude is small; use unperturbed orbit when calculating particle acceleration Lowest order equation of motion i represents initial value, 0 represents 0 th order quantities Particle Motion in a Propagating Wave

First order velocity perturbation For particle with Doppler shifted frequency Particle sees changing phase of wave Response is oscillatory; no net energy transfer to particle over a complete Doppler shifted wave period Non-Resonant Particle

For particle with Doppler shifted frequency is zero; particle ride on the wave Particle sees constant phase of wave & static potential Response is secular; particle gain/lose energy Phase space volume of resonant particle is zero Resonant Particle

Particle with infinitesimally smaller velocity will be accelerated If there are more slower particles, particles gain energy Energy exchange between wave and particle depends on the velocity slope at resonant velocity v=  /k: Landau resonant Transit time resonance in magnetized plasma: mirror force Resonances in tokamak plasmas Cyclotron Resonance:  =p  c Transit resonance:  =p  t Precessional resonance:  =  p Three resonances break three adiabatic invariants, respectively Nonlinear Landau resonance:  2 -  1 =(k 2 -k 1 )v Landau Resonant

Linear resonance Linear Landau damping: what is the feed back on wave by particle collective response? Nonlinear trapping Outline

1D electrostatic Vlasov-Poisson equations collisionless plasmas Summation over species s Conservation of probability density function (PDF) in phase space Time reversible Assume uniform, time stationary plasmas Small amplitude perturbation at t=0 +, expansion Vlasov-Poisson Equations

Causality: response of a stable medium occurs after the impulse Fourier-in-space, Laplace-in-time transformation Inverse transformation  -integration path C 1 lies above any singularity so that  k  analytic at Im(  )>  Initial Value Approach

Perturbed distribution function Singular at resonance Poisson equation Linearized Vlasov Equation

Dielectric constant Inverse Laplace transform Inverse Laplace Transform

 k  was originally defined at Im (  )>  For t>0, need to lower path C 1 to C 2 so that Deform the contour such that no pole is crossed  k  is now defined on the whole  -plane Analytic Continuation

As,  t  is dominated by contributions from poles Ballistic modes:  =kv, continuous spectrum, Damped quickly by phase-mixing Normal modes: D( ,k)=0, discrete spectrum, n th root: n th branch,  n =  n (k) Time Asymptotic Solution

Ballistic (Van Kampen) modes: Assuming a smooth initial perturbation Initial phase-space perturbation propagates without damping Perturbed potential decays in t~1/  e for k ~1 Phase mixing: destructive phase interference BGK mode: finite amplitude Van Kampen modes; singular  f Phase Mixing of Ballistic Mode

Long time evolution dominated by normal modes: As  -contour is lowered from C 1 to C 2, the pole in the complex-v plane cross real-v axis To preserve C 3 integral, C 3 contour needs to be deformed into C 4 P is principal value. Normal Modes

Linear dispersion relation For weakly damped mode D r >>D i,  r >>  i 0 th order: 1 st order: Landau Damping

Uniform Maxwellian Assuming ion fixed background Dielectric constant Dispersion relation Damping depends on velocity slope of distribution function Instability due to inverted shape of distribution function Landau Damping of Plasma Oscillation

Linear resonance Linear Landau damping Nonlinear trapping: What is the back-reaction on distribution function? validity of linear theory? Transition to chaos? Outline

Expansion Linearization: ignore nonlinear term Valid if Linear solution of normal modes: Linear theory breaks down most easily at resonance Validity of Linear Theory

Assuming weakly damped normal modes At resonance Linear theory valid if Bounce frequency of trapped particles Validity of Linear Theory

Given a plane wave Transform to wave frame: Near a potential valley (q  <0): For deeply trapped resonant particle Simple harmonic oscillation Nonlinear Trapping

Hamiltonian Passing particle: mechanical energy Trapped particle: Phase space trajectory of trapped particle: closed island Passing particle: open Boundary: separatrix Assumption of unperturbed orbit invalid when trapping occurs: upper bound of wave amplitude for linear Landau damping Phase Space Island & Separatrix

Phase space island & separatrix: integrable system Oscillation of wave amplitude Small dissipation lead to chaotic region near sepatrix: non-integrable system Phase Space Island & Separatrix

Island size is set by wave amplitude Island separation is set by number of modes, i.e., mode density Islands overlap for densely populated modes: island size > separation Particles jump between resonances before complete a bounce motion Large degree of freedom: onset of stochasticity Quasilinear theory for small amplitude fluctuation Multi-modes: Island Overlap

Vlasov equation Slow evolution of distribution function, spatial average over wavelength and time average over wave period Use linear solution of perturbed distribution function Quasilinear diffusion Quasilinear Theory

Quasilinear diffusions: flattening of f 0 Relaxation to marginal stability Time irreversible HW: what approximation in QLT lead to time irreversibility? Quasilinear Flattening