Stability of relativistic surfatron acceleration

In this paper we consider the surfatron acceleration of relativistic charged particles by a strong electrostatic wave propagating in a transverse direction relative to the background magnetic field. We investigate how high-frequency fluctuations of the background magnetic field affect the process of the resonant acceleration. We show that the presence of fluctuations leads to particle escape from the surfatron resonance and illustrate that fluctuations of different components of the magnetic field have quite a distinct effect on the energy gained by particles. In the case of the same power density, the strongest effect corresponds to fluctuations of the component directed along the background magnetic field, while the effect of the component along the wave front is substantially weaker. This is more important for particles with a large velocity component along the background magnetic field. We demonstrate that the dynamics of particles can statistically be described in terms of the adiabatic invariant diffusion. We derive the corresponding diffusion equation and compare solutions of that equation with results obtained by the explicit particle tracing.

Figure 1

(a) Schematic view of the system. (b) Phase portraits on the resonance plane.

Figure 2

A characteristic particle trajectory in the state of capture in the absence of fluctuations. (a) Projection on the $(x,p_x)$ plane. (b) The adiabatic invariant $I_{\varphi }$ for time interval in capture. (c) The particle energy $\gamma$ as a function of time. (d) Three fragments of the particle trajectory in the $(\varphi ,p_{\varphi })$ plane.

Figure 3

(a) Profiles of the function ${\Gamma }_z\left(x\right)$. (b) The values of ${\overline{\Gamma }}_z$ as a function of ${\sigma }_z=\mathrm{Var}\left(b_z\right)$.

Figure 4

A characteristic particle trajectory in the state of capture in the presence of fluctuations. (a) Projection on the $(x,p_x)$ plane. (b) The adiabatic invariant $I_{\varphi }$ for the time interval in capture. The solid line is value of the area under separatrix $S$. (c) The particle energy $\gamma$ as a function of time. The parameters are ${\sigma }_z=0.03$, $u=0.5$, $k{\phi }_0=20$, $k=100$.

Figure 5

Three fragments of the particle trajectory in the $(\varphi ,p_{\varphi })$ plane for the $A$, $B$, and $C$ moments of time indicated in Fig. 4.

Figure 6

Energy distributions for four values of ${\sigma }_z=\mathrm{Var}\left(b_z\right)$. The grey strips indicate the initial energy range.

Figure 7

(a) Profiles of the function ${\Gamma }_y\left(x\right)$. (b) The values of ${\overline{\Gamma }}_y$ as a function of ${\sigma }_y=\mathrm{Var}\left(b_y\right)$.

Figure 8

Energy distributions for four values of ${\sigma }_y=\mathrm{Var}\left(b_y\right)$. The grey strips indicate the initial energy range.

Figure 9

Profiles of function ${\Gamma }_y(x,p_z)$ for $p_z=0.25$ (left panel) and $p_z=1$ (right panel). Black dotted curves correspond to the function ${\Gamma }_y\left(x\right)$ for $p_z=0$.

Figure 10

Energy distributions for four values of ${\sigma }_y=\mathrm{Var}\left(b_y\right)$ and two values of $p_z$. The grey strips indicate the initial energy range.

Figure 11

The distribution function $F(I,t)$ for six moments of time: $t_1=1.1t_0$, $t_2=1.5t_0$, $t_3=2t_0$, $t_4=6t_0$, $t_5=10t_0$, and $t_6=30t_0$. System parameters correspond to magnetic field fluctuations $b_y\ne 0$ with ${\sigma }_y=0.08$. The initial distribution is shown by grey color.

Figure 12

Energy distributions $Q\left(\gamma \right)$ for three systems. $Q\left(\gamma \right)$ is renormalized for comparison with Figs. 6 and 8.