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Three views on Landau damping

A. Burov

AD Talk, July 27, 2010

Three views on Landau damping:

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1. Originally, Landau found a decay of oscillations in collisionless plasma, when he solved Vlasov equation with Laplace transform (1946).

2. Later, Bohm and Gross pointed out that Landau damping results from energy transfer from the oscillating coherent field  to its resonant particles (1949).

3. The third point of view on the Landau damping was developed by van Kampen, who built a theory of eigenfunctions of the Vlasov equation (1955).

All the three approaches were developed for Langmuir waves in classical plasma. In this talk, instead, all the three approaches are discussed for longitudinal oscillations of a coasting beam.

Jeans-Vlasov Equation

• A Hamiltonian flow of particles in the phase space {z,p} with a density f :

3

0 ;

;

0 .

df fz f p f dN f dzdp

dt t z p

H Hz p

p z

df f H f H f

dt t p z z p

continuity

No cooling, no diffusion, no collisions

Vlasov [1937] (Jeans [1915])

Coasting beam Hamiltonian

• For a case of longitudinal motion in a coasting beam with the space charge

4

2

2 3 2 20

( , ) ( , ) ;2

2 / ; 1/ 1/ ; ln( / ) 1/ 2t

pH z p f z p dp

gr c g a b

Steady state solution

• General approach to the Vlasov equation: – Find a steady state solution :

– Superimpose a tiny perturbation and check its stability

• For the coasting beam a simplest family of steady state solutions is (other families were found by S. Koscielniak et al., 2001, and V. Danilov et al, 2004)

5

),(),,( pzFtpzf

)),((0 pzHFFp

F

z

H

z

F

p

H

),,(~

),(),,( tpzfpzFtpzf

2 2

0 0( ) ; ;2 2

p p NF F p H

C

z and p are the coordinate and velocity relatively the reference particle

Jeans theorem

Linearization

• Linearization of the Vlasov equation over the small perturbation leads to:

• Due to the space homogeneity, the solution is looked for as

• Assuming a given initial condition , the Laplace transformation can be applied:

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f~

dpdFFFdpcrg

dpfFzz

fp

t

f

/;1;/2

~~;0

~~~

3200

nkCtpfikzf 2;),(~

)exp(~

)(~

)0,(~

0 pftpf

)0,(~~

)()(~

)(

)(~~

;),(~

)(~

0

tpfipFkpfkp

dppfdtetpfpf ti

General solution

• From here, the density perturbation is found as

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( , 0);

( )1 ;

( ) exp( )2

k

k

G

i f p t dp

kp

F p dpk

kp

dt i t

G

. .

poles

Laplace rule: Contour G goes above all the singularities

Solution in time domain

• Let it be assumed the delta-function initial condition:

• For that case, the solution follows:

8

)()(~

10 ppApf

0:functiondielectrictheofrootsare

)(

)exp()exp()(

~

;)(

~

1,

1

1

1

k

kkpk

k

kp

tiA

tikpAt

kp

iA

Solution in time domain

• Let it be assumed the delta-function initial condition:

• For that case, the solution follows:

9

)()(~

10 ppApf

0:functiondielectrictheofrootsare

)(

)exp()exp()(

~

;)(

~

1,

1

1

1

k

kkpk

k

kp

tiA

tikpAt

kp

iA

dispersion equation

Solution in time domain

• Let it be assumed the delta-function initial condition:

• For that case, the solution follows:

• For a smooth initial perturbation with a width , the terms

sums to , and can be neglected if

10

)()(~

10 ppApf

0:functiondielectrictheofrootsare

)(

)exp()exp()(

~

;)(

~

1,

1

1

1

k

kkpk

k

kp

tiA

tikpAt

kp

iA

p )exp( 1tikp)||exp( ptk

pk ||))Im((min

dispersion equation

Dielectric function

• Dielectric function is defined in the upper half-plane, where it is regular analytic function.

• At the real axis:

• For significantly small interaction parameter 1,

has no zeros in the upper half-plane – for any distribution shape F(p) – so the beam is stable.

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)/()sign(/

)(.V.P1

0

)(1 kFki

pk

dppF

ikp

dppFkk

20 / p

k

Landau damping

• In the opposite case, the roots of the dispersion equation :

12

| | | | pk

2

23

if 0

1 sign( ) ( / ) 0

| | ( ) , otherwise2

| |

k

iku

ki k F k

iku k u F u

u

Landau damping

• In the opposite case, the roots of the dispersion equation :

13

pk ||

||

otherwise,)(||2

0if

)/()sign(13

2

2

u

uFuki

ku

iku

kFkik

k

negative mass instability

Landau damping

• In the opposite case, the roots of the dispersion equation :

• Below transition, , the beam is stable or unstable depending on the sign of derivative of the steady state distribution . Landau anti-damping is possible as well !

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pk ||

||

otherwise,)(||2

0if

)/()sign(13

2

2

u

uFuki

ku

iku

kFkik

k

Landau damping

0)(uF

)(2||

2 uFuL

Example for various distributions

• For three examples:

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1

4

4

34

4

22

2

1 7.114.0)(;

8.8exp

32.0)(;

2exp

2

1)(

ppppp

ppF

ppF

ppF

|| L

pu /

The dependence is normally steep!

The considered case assumes the pure space charge impedance:

Ohm377/4;// 02

0

cZgiZnZ

ZZ

sc

sc

2 3 4 5 6

0.01

0.1

1

1F2F 3F

Stability threshold

• With an additional small impedance , the stability threshold is determined by the same plot, since :

• Thus, the threshold temperature is an extremely shallow function of the real part of impedance ReZ.

• Since the beam stability is determined by the poorly measured and poorly reproducible tiny tails of the distribution, the threshold can normally be predicted only with low accuracy, like factor of 1.5 or 2. This is the case of many machines with space charge or inductive dominated impedances. Similar situation is for transverse oscillations as well.

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||Re scZZ

||2

Re

scZ

Z

pu /p

|)|2/(Re)/Im( scZ ZZ

2 3 4 5 60.01

0.1

1

1F2F 3F

p

Intermediate Conclusions

• Landau damping was derived as a formal consequence of the Jeans-Vlasov equation for collisionless plasma or beam.

• The damping rate is proportional to the distribution function of the particles moving at the wave velocity, . These particles are in a resonance with the wave, they are called as resonant particles.

• For other geometry, Landau damping may be proportional to the distribution function itself, , as it is for the transverse oscillations of a coasting beam with chromatic tune spread.

• For space charge dominated impedances, the stability threshold is determined by far tails of the distribution. It is sensitive to the distribution shape, and barely sensitive to the real part of the impedance.

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)(uFL

)(uFL

Physical reason for Landau damping

• Where energy of the decaying wave goes to?

• Since the damping rate , it is natural to guess that the wave energy goes to the resonant particles. This was shown by Bohm and Gross in 1949.

• For the considered longitudinal motion, slightly slower particles (p=u-0) are accelerated by the wave, taking its energy, while slightly faster ones (p=u+0) are decelerated. That is why

• For the transverse oscillations in a coasting beam with chromatic tune spread: , so it is always positive, there is no anti-damping.

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)(uFL

)(uFL

)(uFL

Nonlinear case: no Landau damping

• In principle, the resonant particles may be bunched by the wave, oscillating inside its buckets with a “synchrotron” frequency . In the linear approximation, this does not happen, since . It means that the linear approximation requires . In the opposite case particle-wave energy exchange goes back and forth. After the relaxation, the wave energy remains almost the same. Generally, there is no damping in nonlinear cases.

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|| waveEs0wave E

sL

0 1 2 30

0.1

0.2

0.3

0.4u u

)( pF

p

Nonlinear relaxation: no Landau damping

- width of the bucket

|| waveE

Landau damping and eigenfunctions

• What are the eigenfunctions and eigenvalues for the Jeans-Vlasov equation?

• With , it leads to

• Eigenfunctions can be normalized as . With that, an infinite set of solutions follows:

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dpfFzz

fp

t

f ~~;0

~~~

)(~

)exp(~

pftiikzf

kvdppfpFpfpv vv /;')'(~

)()(~

)(

)(~

pf 1)(~

dppf

'

')'(P.V.1)(;)()(

)(P.V.)(

~

pv

dppFvApvvA

pv

pFpfv

( ) Re( ( ))A v v

Van Kampen Modes

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'

')'(P.V.1)(;)()(

)(P.V.)(

~

pv

dppFvApvvA

pv

pFpfv

Here v is an arbitrary real number inside the distribution. The eigenfrequency =kv .

For a finite-width distribution F(p), the set of eigenvalues v includes the continuous spectrum of inside particles, and a set of discrete real numbers with A(v)=0. These discrete modes not necessarily exist.

If the beam (or plasma) is stable, this continuous set of eigenmodes is complete. Any smooth initial condition can be expanded over that singular basis. The Landau damping results as phase mixing of the continous van Kampen modes. The discrete modes of the finite-width distribution are smooth and do not decay (loss of Landau damping).

This infinite set of eigenmodes was found for plasma oscillations in 1955 by a Dutchphysicist N. G. van Kampen.

Unstable modes

• In case of unstable beam, the continuous spectrum is not complete. On top of that, there is a finite number of mode pairs (just one pair in our space charge example):

• Contrary to the continuous spectrum, these discrete modes are smooth functions.

• What is good in van Kampen approach? Well, for that simple

system it may look as a useless intellectual game. However, for more complicated beam or plasma configurations, may not be found. Then, the first step in the analysis is a numerical calculation of the eigenfunctions. And that directly leads to van Kampen modes.

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)Im(

are rootsof thedielectric function: 0k

k

Summary

• Laplace transform to the Jeans-Vlasov equation leads to the dielectric function , describing a beam response to external perturbations. Beam stability is determined by roots of the dispersion equation:

• Resonant particles: in their frame the wave field does not oscillate. Dissipation of the wave energy in collisionless beam (or plasma) is due to transfer of the wave energy into motion of the resonant particles.

• Landau damping can be considered as phase mixing of the singular eigenmodes of continuous van Kampen spectrum. Unstable oscillations appear as discrete van Kampen eigenmodes.

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k

)(;0 kk