Example Sheet No. 3 Exercise 3.1 - Calculation of the Debye length ...

Fluids without Collisions - Introduction to Kinetic Plasma Physics 09/06/2009
Example Sheet No. 3
Exercise 3.1 - Calculation of the Debye length
a) Using the Vlasov equation, calculate the general dielectricity constant ɛ(k) for
ω = 0.
Hint: The plasma wave approximation used in the lecture will not be applicable here.
Also, you might use the discussion concerning adiabatic potentials at the end of the 2nd lecture.
Calculate the potential distribution around a predetermined stationary point charge in
a plasma. An advisable course of action might be:
b) Transform the charge density into Fourier space.
c) Calculate - in Fourier space - φext, φ, φint and ρint.
d) Transform these quantities back into real space.
e) Interpret your results! Which characteristic screening length can be observed in
the resulting potential?
Exercise 3.2 - Energy transmission between particles and potential
a) How does the transferred power density between particles and potential transform
between different reference frames (non-relativistic Galilei transformation).
A suitable starting point for decreasing and increasing solutions of the Vlasov
equation (ω ± iɛ) might be the following equation for the transmitted power:
P (ω) � = ℜ � δφ(−iω)qδn � = ℜ � iδφωk 2 χδφ �� = −|φ| 2 k 2 ℑ(ωχ)
b) Calculate the force density exchanged between particle and potential!
c) Under which circumstances is the transmitted energy independent of the velocity
of the reference frame? What about the momentum density?
d) Calculate the energy balance of the bump-in-the-tail instability in the reference
frame of the resonant particles (v = ω/k)!
e) How is the energy of the resonant particles split between the non-resonant particles
and the field? [difficult]
Contact: Andreas.Kammel@ipp.mpg.de - http://www.ipp.mpg.de/∼Klaus.Hallatschek/ 1 of 2

Fluids without Collisions - Introduction to Kinetic Plasma Physics 09/06/2009
Exercise 3.3 - BGK modes
Construct the BGK solutions of the Vlasov equation for the following electrical potentials
(as usual, you may assume a fusion plasma, thus T ≫ qφ)
Recommended course of action:
a) φ/φ0 =
�
x
�2 − 1 for |x| ≤ l
l
(0 otherwise)
b)
c)
φ/φ0 =
φ/φ0 =
sin(kx) − 1
�
− exp − x2
2σ2 �
• Utilize the following ansatz for the distribution function: ftotal(v) = f0(ɛ + qφ) +
f(ɛ + qφ) where f(ɛ + qφ) = 0 for ɛ + qφ > 0 and f0 marks the Maxwellian
distribution.
Hint: Use constant phase space density for the trapped particles
• Use this to construct the equation for the particle density.
• Carry out the transformation from v to z ≡ ɛ + qφ.
• Use � b
a
1
√ b−z
√ 1 dx = π for the calculation of the particle density - you should now
z−a
be able to solve the particle density for the perturbed distribution function f.
Hint: You might have to expand the term of the particle density by an appropriate integral
• Thus, calculate f(φ) for the different φ given.
Contact: Andreas.Kammel@ipp.mpg.de - http://www.ipp.mpg.de/∼Klaus.Hallatschek/ 2 of 2