Wave-Particle Interaction in Collisionless Plasmas: Resonance and Trapping

Zhihong Lin

Department of Physics & AstronomyUniversity of California, Irvine

International Summer School on Plasma Turbulence and TransportChengdu, 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

http://gk.ps.uci.edu/zlin/zlin/pic1d/

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 0th order quantities

Particle Motion in a Propagating Wave

cos( )kx t

sin( )dv

m qk kx tdt

0

0

i

i i

v v

x x v t

• 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

1 sin[ ( ) ]i i

dvm qk kx kv t

dt

ivk

cos[( ) ] cos( )i i ii

i

qk kv t kx kxv v

m kx

ikv

• 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

ivk

sin( )i i

qk tv v kx

m

• 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: =pc

• Transit resonance: =pt

• Precessional resonance: =p

• Three resonances break three adiabatic invariants, respectively

• Nonlinear Landau resonance: 2-1=(k2-k1)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

( ) 0q

v ft x m v

2 4s

nq

0f

0f f f

0q fv f

t x m v

• Causality: response of a stable medium occurs after the impulse

• Fourier-in-space, Laplace-in-time transformation

• Inverse transformation

• -integration path C1 lies above any singularity so that

• k analytic at Im()>

Initial Value Approach

0

( , ) ( , ) i kx tk dt dx t x e

2

1( , ) ( , )

(2 )

i i kx t

it x d dk k e

( 0) 0t

• Perturbed distribution function

• Singular at resonance

• Poisson equation

Linearized Vlasov Equation

00

q fi f k

m vfkv

vk

20

20 0 0

0

4

4 /4

s

s s

k n q fdv

f n q f vi n q dv k dv

kv m kv

00

q fi ikv f f ik

m v

• Dielectric constant

• Inverse Laplace transform

Inverse Laplace Transform

002

4

s

i fD n q dv

k kv

22 4 s ss

s

n q

m

2

02

/1

/s

s

f vD dv

k k v

1

1

002

1( , ) ( , )

21 4

2

i t

C

i t

Cs

t k d e k

i fd e n q dv

Dk kv

• kwas originally defined at Im ()>

• For t>0, need to lower path C1 to C2 so that

• Deform the contour such that no pole is crossed

• kis now defined on the whole -plane

Analytic Continuation

1 2C C

Im( )

• 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,

nth root: nth branch, n=n(k)

Time Asymptotic Solution

t

2

002

1 4( , )

2i t

Cs

i ft k d e n q dv

Dk kv

ikvte

ni te

20

2

/( , ) 1 0

/s

s

f vD k dv

k k v

• 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

20 02

0 02

4 1( , )

2 ( ) ( , )

4

( , )

i t

Cs

ikvt

s

i dt k n q dv f e

k kv D k

en q dv f

k D kv k

22 /

0ev vf e

2 2 2/ ( / 2)e eikvt v v kv tdve e

• Long time evolution dominated by normal modes:

• As -contour is lowered from C1 to C2, the pole in the complex-v plane cross real-v axis

• To preserve C3 integral, C3 contour needs to be deformed into C4

P is principal value.

Normal Modes

( , ) 0D k

2 2'0

02

/1 ( )

/ | |s s

s

f v iD P dv f

k k v k k k

• Linear dispersion relation

• For weakly damped mode Dr >>Di, r>>i

0th order:

1st order:

Landau Damping

0r i

r i

D D iD

( , )( , ) ( , ) ( , )r r

r r i i rr

D kD k D k i iD k

( , ) 0

( )r r

r r

D k

k

( , ) / ri i r

r

DD k

• 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 Oscillation2 2/

0

1sv v

s

f ev

e iv vk

2 2'

021 , ( )

| |p p p

r iD D fk k k

2

2 2

21/ 20

3 4'

0 3 3

4( )

( )2 | | | |

p

e

r pe

p p p k vi

e

n e

m

f ek k k k v

• 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

0q q fv f

t x m v m v

0f f f

qf

m v

0| | | |f fv v

0q fk

m vfkv

• Assuming weakly damped normal modes

• At resonance

• Linear theory valid if

• Bounce frequency of trapped particles

Validity of Linear Theory

| | | |i r

20 0 0

2 2

1 1 | || | | || | | || |

i

qk f f qk f kf

v m v kv kv v m v

rvk

2

| || | 1

i

qk k

m

2 2i b

22 | |b

qk

m

• Given a plane wave

• Transform to wave frame:

• Near a potential valley (q<0):

• For deeply trapped resonant particle

• Simple harmonic oscillation

Nonlinear Trapping

/

/

v v k

x x t k

2

2sin( )

d xm qk kx t

dt

2 2

2sin( )bd x

k xdt k

0

| | 1

v

k x

2

22 b

d xx

dt

• 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

22

2

1( , ) cos( )

2bm

H x v m v k xk

2

2bm

Ek

2

2bm

Ek

• 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

( ) 0q

v ft x m v

( ) 0q

f ft m v

0q fk

m vfkv

0f f

0 0f D ft v v

2

,

1

k

qkD i

m kv

• Quasilinear diffusions: flattening of f0

• Relaxation to marginal stability

• Time irreversible

• HW: what approximation in QLT lead to time irreversibility?

Quasilinear Flattening