# Presentation: 425889

## Weakly Collisional Landau Damping and BGK Modes: New Results on Old Problems

- A. Bhattacharjee, C. S. Ng, and F. Skiff
*Space Science Center*
*University of New Hampshire*

- Davidson Symposium, Princeton, June 11-12, 2007

## What have I learned from Ron?

- Be broad in your perspectives---plasma physics is diverse, and yet deeply interconnected.
- 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.
- If you want to serve your community *and* do your calculations, be disciplined and organized.
- 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 *nearly* collisionless.

- Two classic results in a collisionless plasma:

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

## Vlasov-Poisson equations

## 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.”
- _x000c_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

## 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?

## Case-Van Kampen modes 

- Vlasov-Poisson equation for a single Fourier mode:

- These are the Case-Van Kampen eigenmodes.
- Landau-damped modes are *not* eigenmodes, but
- long-time remnants of an arbitrary smooth initial condition.

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

## Case-Van Kampen modes form a complete set

- A general solution :

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

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

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

## Collision as a singular perturbation

** --- normalized collision frequency

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

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

## Discrete eigenmodes in experiments

*Skiff et al. *(1998)

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

## A complete set of discrete eigenmodes

*Ng et al., PRL, *(1999, 2004)

*g**n* found in closed form involving incomplete gamma function.

*c**n* 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 spectrum**

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

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

## Calculating the eigenvalues

## Shape of the eigenfunctions

## What is a BGK mode?

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

*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*

- 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 

*Ergun et al. *(1998)

**B**

## 3D BGK mode for finite B?

- 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

- 1D:

- In 2D/3D,

## 3D solution depending on energy as well as angular momentum

## 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.