Longitudinal and Transverse Instability of Ion Acoustic Waves

Ion acoustic waves are found to be susceptible to at least two distinct decay processes. Which process dominates depends on the parameters. In the cases examined, the decay channel where daughter modes propagate parallel to the mother mode is found to dominate at larger amplitudes, while the decay channel where the daughter modes propagate at angles to the mother mode dominates at smaller amplitudes. Both decay processes may occur simultaneously and with onset thresholds below those suggested by fluid theory, resulting in the eventual multidimensional collapse of the mother mode to a turbulent state.

Figure 1

$Z{T}_e/T_i=8$, $k{\lambda }_{\mathrm{De}}=0.4$, $Z{m}_e/m_i=1/1836$ (H plasma). (a) Evolution of the electrostatic ($U_{\mathrm{ES}}$), electron kinetic ($K_e$) and ion kinetic ($K_i$) energies. The driven (undriven) period is shown in gray (white). (b)–(e) Snapshots of the IAW potential $\mathrm{\Phi }$ before, during, and after the crash in electrostatic energy. The half harmonic ($\lambda =2{\lambda }_1$) is the dominant mode during the crash. The distribution functions $f_{e,i}$ in (f)–(k) are snapshots shown as deviations from Maxwellian $f_{e,i}^M$, taken (f),(h),(j) before and (g),(i),(k) after the crash in $U_{\mathrm{ES}}$ at times indicated by the leftmost and rightmost red dashed lines, respectively, in (a). Shown are (f),(g) $\mathrm{\Delta }{\tilde{f}}_e=(v_{te}/n_{e0})\int (⟨f_e{⟩}_y-f_{e0})d{v}_y$; (h),(i) $\mathrm{\Delta }{\tilde{f}}_i=(v_{ti}/n_{i0})\int (⟨f_i{⟩}_y-f_{i0})d{v}_y$; and (j),(k) $\mathrm{\Delta }{\tilde{f}}_i=(v_{ti}^2/n_{i0})(⟨f_i{⟩}_{x,y}-f_{i0})$. $⟨\cdots {⟩}_a$ indicates an averaging over the dimension(s) $a$. The white dashed lines in (f)–(k) indicate the measured nonlinear phase velocity $v_1={\omega }_1/k_1$ of the fundamental mode, while black dashed lines show analytic estimates of the species trapping widths, $v_1\pm v_{\mathrm{tr},j}$.

Figure 2

Fourier space analysis of the case presented in Fig. 1. (a) Fourier mode evolution of the mother and fastest-growing daughter modes corresponding to TID and OAI. (b) Linear (preturbulence) growth rates $\gamma =\gamma (k_x,k_y)$. (c) Mode frequencies $\omega =\omega (k_x,k_y)$. (d) TID growth rate, $\gamma =\gamma (k_x,0)$. (e) Slices showing $\gamma$ for various fixed $k_x$’s. Arrows indicate identical modes in (d) and (e).

Figure 3

Analysis of cases with the same parameters as Fig. 1 but varying mother wave amplitudes (the horizontal axis). (a) Scaling of $\gamma$ with $|{\varphi }_1|$ for TID and OAI. The gray horizontal line indicates the linear Landau damping rate of the $l=1/2$ mode, ${\nu }_{1/2}$. (b) The scaling of $k_y$ for the fastest-growing mode for OAI with $|{\varphi }_1|$ (the left vertical axis) and the scaling of the nonlinear frequency shift $\mathrm{\Delta }{\omega }_1^{\mathrm{NL}}={\omega }_1^{\mathrm{NL}}-{\omega }_1^{\mathrm{L}}$ with $|{\varphi }_1|$ (the right vertical axis).