Comparison with Linear Theory

**Figure 3:** Temporal development of |B_x| for the wave number $k c / \omega_p =1.32$ .
$\begin{figure} \begin{center} \resizebox {8.8cm}{!}{ \includegraphics{maxmode.eps} } \end{center}\end{figure}$

To identify the wave mode responsible for the growing field $B_\perp$ , the growth rates of different wave numbers are compared to the values obtained by WHAMP (Rönnmark, 1982). This program allows to evaluate the plasma dispersion function numerically for arbitrary temperature ratios.

**Figure 4:** Growth rate of B_x determined from simulations with m_p/m_e = 49 (asterisk) and m_p/m_e = 81 (boxes) compared to results obtained from linear theory for m_p/m_e = [49, 81, 1836] (dotted, dashed, dash-dotted) for the initial electron temperature ratio $T^e_\Vert/T^e_\perp\approx 20$ . In addition, the growth rate for $T^e_\Vert/T^e_\perp\approx 6$ and m_p/m_e=49 is shown (dotted, thin).
$\begin{figure} \begin{center} \resizebox {8.8cm}{!}{ \includegraphics{growth.eps} } \end{center}\end{figure}$

Figure 4 shows a comparison of the growth rates between linear theory and PIC simulation. The linear theory results are determined for the initial plasma parameters. The growth rates of the PIC simulation are measured by a linear fit in the time series of spatial Fourier transforms of B_x in the time interval $2-8\Omega_p^{-1}$ .The three growing waves and the suppressed growth at $k c/\omega_p = 1.76$ are well in accordance with linear theory. However, the growth rates for $kc/\omega_p=[0.44,0.88]$ are too low, whereas the one for $k c / \omega_p =1.32$ is too high. This will be investigated in Sect. 5.1.

In order to check the influence of the artificial mass ratio, the growth rates are determined for an increased mass ratio of m_p/m_e=81 and otherwise identical plasma parameters (see Fig. 4). Due to the larger mass ratio and the fixed simulation box size of $100 c/\omega_e$ , the resolution in wave numbers decreases. Again, the waves grow in accordance with linear theory, but the determined growth rates are lower than expected for $kc/\omega_p=0.59$ and a bit larger for $kc/\omega_p=1.18$ . For comparison, the growth rates for the real mass ratio m_p/m_e=1836 is included in Fig. 4. The hypothetical light weight protons inhibit wave growth at large k.

**Figure 5:** Comparison of the dispersion relations obtained by the simulation (asterisk) and by WHAMP (dashed). For comparison, the dispersion relation for the real mass ratio m_p/m_e=1836 is included (dash-dotted).
$\begin{figure} \begin{center} \resizebox {8.8cm}{!}{ \includegraphics{dispersion.eps} } \end{center}\end{figure}$

Figure 5 shows a comparison of the dispersion relation determined both by WHAMP and from the simulation. Due to the limited resolution in k-space, only 3 samples lie within the range of non-zero growth (see Fig. 4). According to linear theory, the period of the sample with smallest wave number, $kc/\omega_p =0.44$ , is $45 \Omega_p^{-1}$ and thus much longer than the saturation time of the instability. The frequency can therefore not be measured in the simulation and only an upper bound can be given. For the other two samples, the frequencies are determined by removing the exponential growth and measuring the time between the first two amplitude minima, thus the first half period. Like the growth rates, the wave frequencies agree well with the values determined from linear theory. However, they are both too low.

**Table:** Comparison of growth rates $[\gamma_{0.44},\gamma_{0.88}, \gamma_{1.32}]$ between simulation (bold) and linear theory at $kc/\omega_p = [0.44, 0.88, 1.32]$ and wave number of maximum growth k_m for different electron temperature ratios $T^e_\Vert/T^e_\perp$ .The other plasma parameters are identical to Sect. 2.
$T^e_\Vert/T^e_\perp$	$\gamma_{0.44}/\Omega_p$	$\gamma_{0.88}/\Omega_p$	$\gamma_{1.32}/\Omega_p$	$k_mc/\omega_p$

10	${\bf 0.2}\;\;\;0.2$	${\bf 0.4}\;\;\;0.4$	${\bf 0.3}\;\;\;0.4$	1.6
15	${\bf 0.2}\;\;\;0.4$	${\bf 0.6}\;\;\;0.7$	${\bf 0.7}\;\;\;0.7$	1.2
20	${\bf 0.4}\;\;\;0.4$	${\bf 0.7}\;\;\;0.7$	${\bf 0.5}\;\;\;0.4$	1.0
30	${\bf 0.6}\;\;\;0.7$	${\bf 0.6}\;\;\;0.8$	${\bf 0.0}\;\;\;0.0$	0.7

Additional simulations with varying initial temperature ratio have been carried out to test the agreement with linear theory. Table 1 shows the growth rates for varying temperature ratios $T^e_\Vert/T^e_\perp$ , but otherwise identical parameters as in Sect. 2. A larger anisotropy leads to a shift of the maximum growth rate towards smaller wave numbers (see also Fig. 4).

The good agreement of both dispersion relation and growth rate indicates that the instability under investigation is indeed the Electron Firehose Instability.