Landau Damping Simulation Models - hsxie.me

Introduction Linear simulation model Dispersion relation Particle-in-cell Vlasov continuity simulation Nonlinear

Landau Damping Simulation Models

Hua-sheng XIE (�u)§[email protected])

Department of Physics, Institute for Fusion Theory and Simulation,Zhejiang University, Hangzhou 310027, P.R.China

Oct. 9, 2013

Present at: Sichuan University


Content

Introduction

Linear simulation model

Dispersion relation

Particle-in-cell

Vlasov continuity simulation

Nonlinear


Introduction

Landau damping1 is one of the most interesting phenomena foundin plasma physics. However, the mathematical derivation andphysical understanding of it are usually headache, especially forbeginners.

Here, I will tell how to use simple and short codes to study thisphenomena. A shortest code to produce Landau dampingaccurately can be even less than 10 lines!

1I think I can safely say that nobody understands Landau damping fully.


Linear simulation model: equation

We focus on the electrostatic 1D (ES1D) Vlasov-Poisson system(ion immobile).

The simplest method to study Landau damping is solving thefollowing equations

∂tδf = −ikvδf + δE∂v f0, (1a)

ikδE = −∫

δfdv , (1b)


Example code

1 k =0.4 ; dt =0.01; nt =8000; dv =0.1 ; vv=−8:dv : 8 ;2 df0dv=−vv .∗ exp(−vv . ˆ 2 . / 2 ) / s q r t (2∗ p i ) ;3 d f =0.∗ vv +0.1.∗ exp (−(vv−2.0) . ˆ 2 ) ; t t=l i n s p a c e (0 , nt ∗

dt , nt+1) ;4 dE=ze r o s (1 , nt+1) ; dE (1 ) =0.01;5 f o r i t =1: nt6 d f=df+dt .∗(−1 i ∗k .∗ vv .∗ d f+dE( i t ) .∗ df0dv ) ;7 dE( i t +1)=(1 i /k ) ∗sum( d f ) ∗dv ;8 end9 p l o t ( t t , r e a l ( dE) ) ; x l a b e l ( ’ t ’ ) ; y l a b e l ( ’Re (dE) ’ ) ;


Simulation result

0 10 20 30 40 50 60 70 80−0.4

−0.2

0

0.2

0.4

t

Re(

dE)

Figure 1: Linear simulation of Landau damping.


Exercise

Exercise 1: Solving the following fluid equations

∂tδn = −ikδu, (2a)

∂tδu = −δE − 3ikδn, (2b)

ikδE = −δn, (2c)

using the above method to reproduce the Langmuir wave

ω2 = 1 + 3k2. (3)


Dispersion relation

D(k, ω) = 1− 1

k2

∫C

∂f0/∂v

v − ω/kdv = 0, (4)

where C is the Landau integral contour. For Maxwellian

distribution f0 = 1√2π

e−v2

2 , we will meet the well-known plasma

dispersion function (PDF)

ZM(ζ) =1√π

∫C

e−z2

z − ζdz . (5)

Hence, (4) is rewritten to

D(k, ω) = 1− 1

k2

1

2Z ′

M(ζ) = 0. (6)


Numerical solutions

Table 1: Numerical solutions of the Landau damping dispersion relation

kλD ωr/ωpe γr/ωpe

0.1 1.0152 -4.75613E-15

0.2 1.06398 -5.51074E-05

0.3 1.15985 -0.0126204

0.4 1.28506 -0.066128

0.5 1.41566 -0.153359

0.6 1.54571 -0.26411

0.7 1.67387 -0.392401

0.8 1.7999 -0.534552

0.9 1.92387 -0.688109

1.0 2.0459 -0.85133

1.5 2.63233 -1.77571

2 3.18914 -2.8272


Comparison of DR and linear simulationAdding some diagnosis lines to the code. Perfect agreement:ωtheory = 1.28506− 0.066128i and ωsimulation = 1.2849− 0.06627i

0 10 20−0.4

−0.2

0

0.2

0.4

0.6(a) k=0.4, dv=0.019531, dt=0.01

t

δ E

0 10 20−25

−20

−15

−10

t

ln|δ

E|

(b) ωS=1.2849, γS=−0.066265

Re[δ E]

Im[δ E]

simulationtheory

Figure 2: Linear simulation of Landau damping and compared withtheory.


More

Note: Several related topics have been omitted here, e.g.,Case-van Kampen ballistic modes (eigenmode problem),non-physical recurrence effect (the Poincare recurrence) atTR = 2π/(k∆v)[1], solving the dispersion relation with generalequilibrium distribution functions (not limited to Maxwellian),advanced schemes (e.g., 4th R-K), and so on. One can referRef.[2] and references in for more details.


PIC simulation: equations

Normalized equations (Lagrangian approach)

dtxi = vi , (7a)

dtvi = −E (xi ), (7b)

dxE (xj) = 1− n(xj), (7c)

where i = 1, 2, · · · ,Np is particle (marker) label andj = 0, 1, · · · ,Ng − 1 is grid label. The particles i can be everywhere, whereas the field is discrete in grids xj = j∆x . ∆x = L/Ng .Domain 0 < x < L [note:

∫n(x)dx = L], periodic boundary

conditions n(0) = n(L) and⟨E (x)

⟩x

= 0. Any particle crosses theright boundary of the solution domain must reappear at the leftboundary with the same velocity, and vice versa.

The initial probability distribution function [e.g., f0 = 1√2π

e−v2

2 ] is

generated by Np random numbers.


Key steps

Two key steps for PIC are: 1. Field E (xj) on grids to E (xi ) onparticle position; 2. Particle density n(xi ) to grids n(xj). Supposethat the i-th electron lies between the j-th and (j + 1)-thgrid-points, i.e., xj < xi ≤ xj+1. Usually, the below interpolationmethod is used

nj = nj +xj+1 − xi

xj+1 − xj

1

∆x, (8a)

nj+1 = nj+1 +xi − xj

xj+1 − xj

1

∆x. (8b)

The above procedure is repeated from the first particle to the lastparticle. Similar procedure are used to mapping E (xj) to E (xi ).


PIC result

0 5 100

2

4

6

(a) k=0.7, ωtheory

=1.6739−0.3924i

t

Ene

rgy

Ek

Ee

Etot

2 4 6 8 10

−4

−3

−2

−1

(b) ωS=1.7028, γS=−0.43894

Figure 3: PIC simulation of Landau damping (pices1d.m code).

The energy conservation is very well. Real frequency and dampingrate agree roughly with theory. A main drawback of PIC is thenoise. Usually, very large Np is required.


Vlasov continuity simulation: equations

Euler approach.Vlasov equation

∂t f (x , v , t) = −v∂x f − ∂xφ∂v f , (9)

Discrete

f n+1i ,j − f n

i ,j

∆t= −vj

f ni+1,j − f n

i−1,j

2∆x−

φni+1,j − φn

i−1,j

2∆x

f ni ,j+1 − f n

i ,j−1

2∆v,

(10)gives

f n+1i ,j = f n

i ,j−vj

f ni+1,j − f n

i−1,j

2

∆t

∆x−

φni+1,j − φn

i−1,j

2∆x

f ni ,j+1 − f n

i ,j−1

2

∆t

∆v.

(11)Poisson equation

∂2xφ =

∫fdv − 1. (12)


Discrete

φi+1 − 2φi + φi−1

∆x2=

∑j

fi ,j∆v − 1 ≡ ρi , (13)

i.e.,−2 1 0 · · 0 11 −2 1 · · · 00 1 −2 1 · · ·· · · · · · ·1 0 · · · 1 2

φ1

φ2

··

φN

=

ρ1

ρ2

··

ρN

∆x2, (14)

where we have used the periodic boundary condition φ(0) = φ(L),i.e., φ1 = φN+1 and φ0 = φN .


Simualtion results

0 10 20−0.05

0

0.05(a) Momentum v.s. Time

t

P−

<P

>

0 10 20−1

0

1(b) Density v.s. Time

t

n−<

n>

0 10 2010

−10

100

1010(c) k=0.6, ω

theory=1.5457−0.26411i

t

Ene

rgy

Ek

Ee

Etot

5 10 15 20 25−8

−6

−4

−2

(d) ωS=1.535, γS=−0.26977

Figure 4: Vlasov continuity simulation, history plotting (code fkvl1d.m).


x

v

(a) f(x,v), t=24(25)ωpe−1, k=0.6

0 2 4 6 8 10

−5

0

5

−5 0 50

2

4

(b) f(v,t)

v

f

−5 0 5−0.1

−0.05

0

0.05

0.1(c) δf(v,t)

v

δf

0

0.1

0.2

0.3

0.4

Figure 5: Vlasov continuity simulation, distribution function (codefkvl1d.m).


Nonlinear simulations

The PIC and Vlasov codes provided in the above sections can beeasily modified to study the linear and nonlinear physics of thebeam-plasma or two-stream instabilities.

A PIC simulation of two-stream instability is shown in Fig. 6 andFig. 7. The linear growth and nonlinear saturation are very clear.

Exercise 2: Solving the kinetic or fluid dispersion relations forbeam-plasma or two-stream plasma and comparing the results withlinear and nonlinear simulations using the above models.


Simulation result

Figure 6: PIC simulation of the two-stream instability, phase spaceplotting.


Simulation result

0 20 40 600

1

2

3

4

5

t

Ene

rgy

Ek

Ee

Etot

20 40 60

−4

−3

−2

−1

t

ln(E

e1/2 )

Figure 7: PIC simulation of the two-stream instability, history plotting.


C. Z. Cheng and G. Knorr, The integration of the vlasovequation in configuration space, Journal of ComputationalPhysics, 22, 330 - 351, 1976.

H. S. Xie, Generalized Plasma Dispersion Function:One-Solve-All Treatment, Visualizations, and Application toLandau Damping, Phys. Plasmas, 20, 092125, 2013. Also,http://arxiv.org/abs/1305.6476. With codes.

http://arxiv.org/abs/1305.6476

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Landau Damping Simulation Models - hsxie.me