Weakly Collisional Landau Damping and BGK Modes: New

Results on Old Problems

A. Bhattacharjee, C. S. Ng, and F. SkiffSpace Science Center

University of New HampshireDavidson Symposium, Princeton, June 11-12, 2007

What have I learned from Ron?

1. Be broad in your perspectives---plasma physics is diverse, and yet deeply interconnected.

2. Learn plasma theory by doing it----calculate, and do so as rigorously as possible. Don’t worry if the calculation is long---if you are careful, the terms will all cancel out in the end leaving a nice result.

3. If you want to serve your community and do your calculations, be disciplined and organized.

4. Advice from one Editor to another: choose your reviewers well, and let the review process of a paper be a dialogue between the authors and reviewers.

Introduction

High-temperature plasmas are High-temperature plasmas are nearlynearly collisionless. collisionless.

Two classic results in a collisionless plasma:Two classic results in a collisionless plasma:

Landau damping of linear electrostatic wavesLandau damping of linear electrostatic waves

Landau (1946)

Exact nonlinear 1D electrostatic solutions (BGK modes)Exact nonlinear 1D electrostatic solutions (BGK modes) ---- (undamped)---- (undamped) Bernstein, Greene & Kruskal (1957)

This talk: two new results on these old and classic problems.This talk: two new results on these old and classic problems.

Vlasov-Poisson equations

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∂fs

∂t+ v ⋅

∂fs

∂r−

qs

ms

∇ψ ⋅∂fs

∂v= 0

€

E = −∇ψ

€

∇⋅E = −∇2ψ = 4πρ = 4π qs dvfs∫s

∑

Landau Damping of Plasma Oscillations

• Vlasov (1938): Collisionless kinetic theory, asssuming normal modes of the form

• Landau (1946):

“…most of his (A. A. Vlasov’s) results turn out to be incorrect. Vlasov looked for solutions of the form and determined the dependency of the frequency on the wave vector k. Actually there exists no dependence of on k at all, and for given value of k, arbitrary values of are possible.”

• Landau solved the initial-value problem (using Laplace transforms) and obtained collisionless damping for monotonic distribution functions. Landau-damped solutions are not eigenmodes, but represent the linear response of the system in the asymptotic limit

€

exp(−iωt + ik ⋅r)

€

exp(−iωt + ik ⋅r)

€

ω

€

ω

€

ω

€

t → ∞

Vexing question

For a linear perturbation of the form if the distribution function is non-monotonic, there are unstable eignemodes with

. In this case Landau’s analysis coincides with Vlasov’s.

But if the distribution function is monotonic (such as a Maxwellian), the solutions with are not eigenmodes.

What is the physical reason for this strange asymmetry?

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exp(−iωt)

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Imω > 0

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Imω < 0

Case-Van Kampen modes

Vlasov-Poisson equation for a single Fourier mode:Vlasov-Poisson equation for a single Fourier mode:

These are the Case-Van Kampen eigenmodes. These are the Case-Van Kampen eigenmodes.

Landau-damped modes are Landau-damped modes are notnot eigenmodes, but eigenmodes, butlong-time remnants of an arbitrary smooth initial condition.long-time remnants of an arbitrary smooth initial condition.

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(u − Ω )g(u) = η(u) g(u ')du '−∞

∞

∫

Exact solution (singular and undamped):Exact solution (singular and undamped):

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gΩ (u) = Pη (u)

u − Ω

⎡ ⎣ ⎢

⎤ ⎦ ⎥+ δ(u − Ω) 1− P

η (u')

u'−Ωdu'

−∞

∞

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

[Van Kampen (1955), Case (1959)]

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f1(z, t,v) = e i(kz−ωt ) ˜ f 1(k,ω,v)

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g = v0 2 f1 /n0

Case-Van Kampen modes form a complete set

A general solution :A general solution :

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g(z, t,u) = e iz e−iΩ tc(Ω )gΩ (u)dΩ−∞

∞

∫

Phase-mixing of undamped Case-Van Kampen modes Phase-mixing of undamped Case-Van Kampen modes produces Landau damping of the plasma wave.produces Landau damping of the plasma wave.

Thus, the linear Vlasov-Poisson is completely solved.Thus, the linear Vlasov-Poisson is completely solved.

Question:Question: what is the effect of collision, even if it is weak? what is the effect of collision, even if it is weak?

A model equation :A model equation :

€

∂f

∂t+ v

∂f

∂z−

eE

m

∂f0

∂v= ν

∂

∂vvf + v0

2 ∂f

∂v

⎛

⎝ ⎜

⎞

⎠ ⎟

Lenard & Bernstein (1958)

Collision as a singular perturbation

--- normalized collision frequency--- normalized collision frequency

Collisions have to be included if there are sharp gradients in Collisions have to be included if there are sharp gradients in velocity space.velocity space.

Eigenmodes of the system change greatly even if collisions Eigenmodes of the system change greatly even if collisions are extremely weak.are extremely weak.

(u−Ω)g(u) −η(u) g(u' )du'−∞

∞

∫ =−iμ∂∂u

ug+12∂g∂u

⎛

⎝ ⎜

⎞

⎠ ⎟

Each Case-Kampen mode has an infinitely sharp gradient in Each Case-Kampen mode has an infinitely sharp gradient in velocity space:velocity space:

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gΩ (u) = Pη (u)

u − Ω

⎡ ⎣ ⎢

⎤ ⎦ ⎥+ δ(u − Ω ) 1− P

η (u')

u'−Ωdu'

−∞

∞

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

Discrete eigenmodes in experimentsSkiff et al. (1998)

measured distribution measured distribution functions with great functions with great accuracy in a weakly accuracy in a weakly collisional plasma using collisional plasma using laser-induced florescencelaser-induced florescence

Eigenmodes decay exponentially, instead of or Eigenmodes decay exponentially, instead of or

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exp −aμt 3( )

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exp −aμx 3( ) as expected from classic theory as expected from classic theory [[Su &

Oberman (1968)]]

A complete set of discrete eigenmodesNg et al., PRL, (1999, 2004)

For an initial value problem: For an initial value problem:

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g(u,t) = cngn(u)exp(− iΩnt)n

∑ Θ ( t)

gn found in closed form involving incomplete gamma function.

cn can be found by integration involving the initial data.

Properties of the complete set of eigenmodes

It is discrete, unlike the Case-Van Kampen continuous It is discrete, unlike the Case-Van Kampen continuous spectrumspectrum

Dispersion relation tends to Landau’s roots in the limit of weak Dispersion relation tends to Landau’s roots in the limit of weak collisions: collisions:

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D(Ω ) μ →0 ⏐ → ⏐ ⏐ 1+ α 1+ ΩZ(Ω )[ ] = 0

except near the collisional modes except near the collisional modes

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Ω =−i[nμ +1/(2μ )]

All eigenfunctions are non-singular, unlike Case-Van Kampen modes.

The new set replaces the Case-Van Kampen modes.The new set replaces the Case-Van Kampen modes.

Calculating the eigenvalues

Shape of the eigenfunctions

Solid curves: eigenfunctions for the least-damped eigenvalue

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Ω0. : Dashed curves the functio n

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η(u)/(u−Ω0). ( )a and (b)

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=0.025. (c) and (d)

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=0.000391.

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−∂f

∂t= v ⋅

∂f

∂r+∇ψ ⋅

∂f

∂v= 0

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∇2ψ = dvf∫ −1

Normalized; uniform ion background (for simplicity)

1D solution: Bernstein, Greene & Kruskal (1957)

Traveling wave solution by change of reference frame:

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f (x − v0t,v− v0)

x x

vv

What is a BGK mode?An exact undamped nonlinear solution of the steady-state Vlasov-Poisson system of equations.

What is a BGK mode

BGK (1957) Construction method of 1D BGK mode

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v∂f (x,v)

∂x+

dψ (x)

dx

∂f (x,v)

∂v= 0

€

d2ψ (x)

dx2= dvf (x,v)∫ −1

€

f = f (w)1st equation can be solved by:

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w = v 2 /2 −ψ (x)

Integro-differential equation:

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d2ψ (x)

dx 2= 2

dwf (w)

2[w +ψ (x)]−ψ (x )

∞

∫ −1

Can be solved by given f(w) or (x).

Physical picture of 1D BGK mode

Electron velocity increases in the center, so electron density decreases

3D features in BGK mode observations

bipolar

unipolar

unipolar€

E ||

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E ||€

E⊥

Traveling direction

Ergun et al. (1998)

B

3D BGK mode for finite B?

[Chen 2002]Conjecture: No solution for B = 0?

Proof: conjecture is true if f = f(w), i.e., f depends only on energy

[Ng & Bhattacharjee 2005]

For infinitely strong B: basic assumption: electrons moving along B only

back to 1-D problem

Chen & Parks (2002)

No 2D/3D solution if f depends only on w

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−ρ =d2ψ (x)

dx 2= 2

dwf (w)

v−ψ (x )

∞

∫ −11D:

2D:

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−ρ =∇2ψ = 2π dwf (w)−ψ

∞

∫ −1+

+++

++ +

-

--

-

-

-

-

-

3D:

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−ρ =∇2ψ = 4π v dwf (w)−ψ

∞

∫ −1

In 2D/3D,

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dρ / dψ ≥ 0

3D solution depending on energy as well as angular momentum

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f = f (w,l) with

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l = v⊥r is also a

solution of the Vlasov equation.

For a spherical potential

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=(r )

++++

++ +

-

--

-

-

-

-

-Possible to support the electric field self-consistently --- satisfying the Vlasov-Poisson equation.

Conclusion In Landau damping, even weak collisions can have profound implications, and change completely the nature of the spectrum of eigenmodes.

A new complete spectrum of discrete eigenmodes is found that replaces the Case-Van Kampen continuous spectrum, where Landau solutions now become the true eigenmodes.

2D/3D BGK modes cannot exist if the distribution function depends only on energy.

3D BGK modes for B=0, and 2D BGK modes for finite B are constructed when f also depends on angular momentum.

See following papers for more details:C. S. Ng, A. Bhattacharjee, and F. Skiff, Phys. Rev. Lett. 83, 1974 (1999).C. S. Ng, A. Bhattacharjee, and F. Skiff, Phys. Rev. Lett. 92, 065002 (2004).C. S. Ng and A. Bhattacharjee, Phys. Rev. Lett. 95, 245004 (2005).