7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Analysis of the Distribution Function in the Landau Damping Process

Koki Maekaku1, Hideo Sugama2, Tomo-Hiko Watanabe3
1
Department of Advanced Energy, The University of Tokyo, 2 National Institute for Fusion
Science, 3Department of Science, Nagoya University
e-mail : maekaku.koki20@ae.k.u-tokyo.ac.jp
It is an interesting and useful topic to construct a fluid
model for describing kinetic effects in collisionlessplasmas, where particle distribution functions largely
deviate from the Maxwellian distribution. One of themost famous researches on such a fluid model is thedevelopment of closures for fluid moment equations
which can reproduce Landau damping[1].
As another related study, an analysis using a water-bag
model has been proposed to approximately represent
kinetic phenomena [2]. In the water-bag model, theone-dimensional distribution function 𝑓(𝑥, 𝑣, 𝑡) isapproximated to a step-function divided by 𝑁 + 1
velocity contours 𝑉𝑗 (𝑥, 𝑡) ( 𝑁 ∈ ℤ , 𝑗 = 1 ⋯ 𝑁 + 1 )(Fig.1). Then, multiple systems with different degrees offreedom can be constructed hierarchically according to
the value of the parameter 𝑁 corresponding to thedegrees of freedom. The water-bag model has
model-dependent invariants called Casimir invariants.
Using these Casimir invariants, we studied the kinetic
effects of one-dimensional Vlasov-Poisson systems[2].
In our research, we succeeded in showing therelationship between the system of the water-bag modelwith different degrees of freedom and the Casimirinvariants.
On the other hand, this result is limited to the casewhere the velocity contours 𝑉𝑗 (𝑥, 𝑡) that constitute thewater-bag model are single-valued functions. Thiscondition turns out to be inapplicable to such cases as inLandau damping, for which 𝑉𝑗 (𝑥, 𝑡) are multivalued.
Numerical simulation based on the contour dynamics [3]is a powerful means to investigate how the velocity
contours 𝑉𝑗 (𝑥, 𝑡) become multivalued functions. The
contour dynamics simulation is performed to clarify thestructure of the distribution function 𝑓(𝑥, 𝑣, 𝑡) in theLandau damping process, and the simulation results arecompared with the linear analytical solutions. In thiscomparison, we confirm that there are damped
oscillations corresponding to complex frequency
eigenvalues 𝜔 = 𝜔𝑟 + 𝑖 𝛾 obtained by the kinetic
dispersion relation [Fig. 2] in addition to the well-known
oscillations due to the ballistic mode. Then, using thefact that the contour dynamics simulation obeys
nonlinear Hamiltonian dynamics, we also examine thevariation of the distribution function due to thequasilinear effect.

References[1] G. W. Hammett and F. W. Perkins, Phys. Rev. Lett.
64, 3019 (1990).
[2] K. Maekaku and Z. Yoshida, Phys. Plasmas 29,082303 (2022).
[3] H. Sato, T.-H Watanabe, and S. Maeyama, J. Comput.
Phys. 445, 110626 (2021).

𝑓(𝑥, 𝑣, 𝑡)𝑣

𝑥

Fig.1 Schematic diagram of the distribution function
𝑓(𝑥, 𝑣, 𝑡) in the water-bag model𝛾𝑂

𝜔𝑟Fig. 2 Distribution of the complex frequency eigenvalues𝜔 = 𝜔𝑟 + 𝑖𝛾 in the complex plane obtained by thekinetic dispersion relation.

𝑣

𝑂𝑥

Fig. 3 An example of contours of 𝑓1 (𝑥, 𝑣, 𝑡)/𝑓0𝑀 (𝑣)obtained by the contour dynamics simulation of theLandau damping where the distribution function is givenby the sum of the background Maxwellian andperturbation parts as 𝑓(𝑥, 𝑣, 𝑡) = 𝑓0𝑀 (𝑣) + 𝑓1 (𝑥, 𝑣, 𝑡).