Parametric Landau damping of space charge modes

Landau damping is the mechanism of plasma and beam stabilization; it arises through energy transfer from collective modes to the incoherent motion of resonant particles. Normally this resonance requires the resonant particle’s frequency to match the collective mode frequency. We have identified an important new damping mechanism, parametric Landau damping, which is driven by the modulation of the mode-particle interaction. This reveals new possibilities for stability control through manipulation of both particle and mode-particle coupling spectra. We demonstrate the existence of parametric Landau damping in a simulation of transverse coherent modes of bunched accelerator beams with space charge.

Figure 1

Comparison between the off-resonance and the coupling resonance cases for the first space charge mode: (a) Landau damping rate $\frac{\lambda }{{\omega }_s}$ versus space charge parameter $q$, where ${\omega }_s={\omega }_0{Q}_s$ is the synchrotron frequency. For intermediate and strong space charge the damping is larger at the coupling resonance. (b) The mode tune $\nu$ at off-resonance and at coupling resonance are nearly the same. (c) The spatial overlap, Eq. (15), of the DMD extracted mode $X_1(z,u)$ with the space charge harmonic function $Y_1(z)$. The off-resonance and the coupling resonance mode shapes are nearly the same.

Figure 2

(a) Bunch tune footprint at off-resonance for $q=7.94$. The white dot corresponds to the bare betatron tunes. (b) $\sum _{i\in S}\mathrm{\Delta }{J}_x=\sum _{i\in S}(J_{xi}-J_{xi\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})$ and $\sum _{i\in S}\mathrm{\Delta }{J}_{yi}=\sum _{i\in S}(J_{yi}-J_{yi\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})$ of the 0.05% and 0.2% largest increasing energy particles versus turn number, normalized by the product of emittance and the number of the particles in the sum. (c) The same as (b) but for the largest decreasing energy particles. (d) Tune footprint for the 0.5% largest changing energy (increase and decrease) particles. The tunes are in the proximity of the coherent tune $Q_x=\nu -Q_s$. The color dimension scale in (a) and (d) differs by one order of magnitude.

Figure 3

(a) Bunch tune footprint at coupling resonance for $q=7.94$. The white dot corresponds to the bare betatron tunes. (b) $\sum _{i\in S}\mathrm{\Delta }{J}_{si}=\sum _{i\in S}(J_{si}-J_{si\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})$ of the 0.05% and 0.2% largest increasing energy particles versus turn number, normalized by the product of emittance and the number of the particles in the sum. (c) The same as (b) but for the largest decreasing energy particles. (d) Tune footprint for the 0.5% largest changing energy (increase and decrease) particles. Large part of the spectral weight is along the resonance line $2Q_x-2Q_y=0$, with the horizontal tune well below $\nu -Q_s$. The color dimension scale in (a) and (d) differs by one order of magnitude.

Figure 4

Coupling resonance, $q=7.94$. (a) Poincare plots, $J_d$ versus ${\mathrm{\Phi }}_x-{\mathrm{\Phi }}_y$, for randomly selected particles belonging to the 0.5% largest changing energy particles. Most of these particles are trapped in the resonance islands. (b) The horizontal tune density $\rho (Q_x)$ (black), the $Q_t$ shifted tune density $h(Q_x)$ (blue) and the one-tune-per-particle tune density ${\rho }_1(Q_x)$ (green) with the corresponding $Q_t$ shifted tune density $h_1(Q_x)$ for the 0.5% largest changing energy particles. $h(Q_x)$ and $h_1(Q_x)$ are strongly peaked at the resonant mode tune $Q_x=\nu -Q_s$, showing mode-particle resonance via the $B{J}_d\overline{x}$ term [see Eq. (12)].

Figure 5

The horizontal and the vertical emittances versus the turn number for the off-resonance and the coupling resonance cases for $q=7.94$. The emittances are nearly constant, the relative change being smaller than $5\times {10}^{-4}$.

Figure 6

Landau damping rate $\frac{\lambda }{{\omega }_s}$ of the horizontal first space charge mode versus $\mathrm{\Delta }{Q}_0/Q_s=(Q_{0x}-Q_{0y})/Q_s$ for different values of the space charge parameter $q$ and the synchrotron tune $Q_s$ where ${\omega }_s={\omega }_0{Q}_s$ is the synchrotron frequency. The width of the enhanced damping region in the proximity of coupling resonance $W$, defined at half maximum, scales linearly with the space charge tune shift, $W\approx 0.3Q_{sc\mathrm{m}\mathrm{a}\mathrm{x}}$, as shown in the inset. The enhanced damping region is asymmetric with respect the to $\mathrm{\Delta }{Q}_0=0$, extending predominantly on the negative side of $\mathrm{\Delta }{Q}_0$. Two maxima of the damping rate can be noticed, one at $\mathrm{\Delta }{Q}_0=0$ and the other one at $\mathrm{\Delta }{Q}_0\approx -M$.