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

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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
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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. Dont 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.

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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.
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Vlasov-Poisson equations
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Landau Damping of Plasma Oscillations

• Vlasov (1938) Collisionless kinetic theory,
  asssuming normal modes of the form
• Landau (1946)
• most of his (A. A. Vlasovs) 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

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Vexing question

• For a linear perturbation of the form
  if the distribution function is
  non-monotonic, there are unstable eignemodes with
• . In this case Landaus analysis
  coincides with Vlasovs.
• 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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Case-Van Kampen modes
Van Kampen (1955), Case (1959)
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.
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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?
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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.
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Discrete eigenmodes in experiments
Skiff et al. (1998)
measured distribution functions with great
accuracy in a weakly collisional plasma using
laser-induced florescence
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A complete set of discrete eigenmodes
Ng et al., PRL, (1999, 2004)
gn found in closed form involving incomplete
gamma function.
cn can be found by integration involving the
initial data.

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Properties of the complete set of eigenmodes
The new set replaces the Case-Van Kampen modes.
It is discrete, unlike the Case-Van Kampen
continuous spectrum?
All eigenfunctions are non-singular, unlike
Case-Van Kampen modes.
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Calculating the eigenvalues
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Shape of the eigenfunctions
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What is a BGK mode?
What is a BGK mode
An exact undamped nonlinear solution of the
steady-state Vlasov-Poisson system of equations.
Normalized uniform ion background (for
simplicity)
1D solution Bernstein, Greene Kruskal (1957)
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BGK (1957)
Construction method of 1D BGK mode
Can be solved by given f(w) or ?(x).
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Physical picture of 1D BGK mode
Electron velocity increases in the center, so
electron density decreases
?
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3D features in BGK mode observations
Ergun et al. (1998)
B
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3D BGK mode for finite B?
For infinitely strong B basic assumption
electrons moving along B only
Chen Parks (2002)
? back to 1-D problem
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No 2D/3D solution if f depends only on w
1D
In 2D/3D,
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3D solution depending on energy as well as
angular momentum
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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 B0, and 2D BGK modes for
finite B are constructed when f also depends on
angular momentum.