Formation of Phase Space Holes and Clumps

It is shown that the formation of phase space holes and clumps in kinetically driven, dissipative systems is not restricted to the near threshold regime, as previously reported and widely believed. Specifically, we observe hole-clump generation from the edges of an unmodulated phase space plateau, created via excitation, phase mixing and subsequent dissipative decay of a linearly unstable bulk plasma mode in the electrostatic bump-on-tail model. This has now allowed us to elucidate the underlying physics of the hole-clump formation process for the first time. Holes and clumps develop from negative energy waves that arise due to the sharp gradients at the interface between the plateau and the nearly unperturbed, ambient distribution and destabilize in the presence of dissipation in the bulk plasma. We confirm this picture by demonstrating that the formation of such nonlinear structures in general does not rely on a “seed” wave, only on the ability of the system to generate a plateau. In addition, we observe repetitive cycles of plateau generation and erosion, the latter due to hole-clump formation and detachment, which appear to be insensitive to initial conditions and can persist for a long time. We present an intuitive discussion of why this continual regeneration occurs.

Figure 1

Fourier spectrogram of the electric field amplitude $E_1$, showing frequency chirping far from (a) and near to (b) the instability threshold. Simulations performed using BOT [10, 20].

Figure 2

Snapshots of the resonant fast particle distribution function for ${\gamma }_d/{\gamma }_L=0.1$ that display (a) the initial phase mixing followed by (b) the almost spatially uniform plateau with sideband trapping regions forming close to the edge, and finally (c) a detaching hole-clump pair. Obtained using BOT [10, 20].

Figure 3

Model “shelf” distribution for the fast particles.

Figure 4

Motion of the two nontrivial roots of Eq. (3) for ${\gamma }_d=0$ as the shelf width increases from below (red dashed) to above (blue dotted) the critical value $k\mathrm{\Delta }v=4{\gamma }_L/\pi$.

Figure 5

Spatially averaged distribution function evolved using the BOT code [10, 20] for ${\gamma }_d/{\gamma }_L=2$, $k\mathrm{\Delta }v/{\gamma }_L=10$ and initial normalized amplitude ${\omega }_B^2/{\gamma }_L^2={10}^{-6}$. The unstable plateau generates holes and clumps that eventually completely erode the plateau state.