Fluid nonlinear frequency shift of nonlinear ion acoustic waves in multi-ion species plasmas in the small wave number region

The properties of the nonlinear frequency shift (NFS), especially the fluid NFS from the harmonic generation of the ion-acoustic wave (IAW) in multi-ion species plasmas, have been researched by Vlasov simulation. Pictures of the nonlinear frequency shift from harmonic generation and particle trapping are shown to explain the mechanism of NFS qualitatively. The theoretical model of the fluid NFS from harmonic generation in multi-ion species plasmas is given, and the results of Vlasov simulation are consistent with the theoretical result of multi-ion species plasmas. When the wave number $k{\lambda }_{De}$ is small, such as $k{\lambda }_{De}=0.1$, the fluid NFS dominates in the total NFS and will reach as large as nearly $15\%$ when the wave amplitude $|e\varphi /T_e|\sim 0.1$, which indicates that in the condition of small $k{\lambda }_{De}$, the fluid NFS dominates in the saturation of stimulated Brillouin scattering, especially when the nonlinear IAW amplitude is large.

Figure 1

(a) The coefficient of the second harmonic $A_{2\varphi }$ and (c) the fluid NFS coefficient $L$ as a function of the wave number $k{\lambda }_{De}$. In the condition of $k{\lambda }_{De}=0.1$, (b) the second harmonic amplitude $e{\varphi }_2/T_e$ and (d) the fluid NFS $\delta {\omega }_{\mathrm{harm}}/{\omega }_0$ as a function of $|e{\varphi }_1/T_e|$. The black solid line presents for single-ion specie plasmas and the red dot-dash line for CH (1:1) plasmas.

Figure 2

Time evolution of the electric field, calculated at a fixed point $x_0=5{\lambda }_{De}$, for (a) $T_i/T_e=0.1$, the fast IAW mode (the frequency of the driver ${\omega }_d=1.965\times {10}^{-3}{\omega }_{pe}$, the maximum amplitude of the driver $e{E}_d^{\max }{\lambda }_{De}/T_e=0.03$), and (b) $T_i/T_e=0.5$, the slow IAW mode (${\omega }_d=1.75\times {10}^{-3}{\omega }_{pe},e{E}_d^{\max }{\lambda }_{De}/T_e=0.0533$) in the condition of $k{\lambda }_{De}=0.1$. The red line is the envelope of the driver.

Figure 3

The frequency spectra of the electric field at $x_0=5{\lambda }_{De}$ and ${\omega }_{pe}t\in [0,2\times {10}^5]$ (black dashed line), ${\omega }_{pe}t\in [3\times {10}^5,4.9\times {10}^5]$ (red solid line) for (a) the fast mode and (b) the slow mode. This corresponds to Fig. 2.

Figure 4

The relation of the second harmonic amplitude and the first harmonic amplitude for the fast mode and the slow mode. The simulation results are compared to the fluid theory including the cold ion fluid model of single-ion specie and multi-ion species.

Figure 5

The phase pictures and the corresponding distributions of electrons, H ions and C ions for the fast mode and the slow mode at the specific time ${\omega }_{pe}{t}_0=4\times {10}^5$, $\tilde{E}\approx 0.01$. $v_{\varphi D}$ is the phase velocity of the driver.

Figure 6

The nonlinear frequency shift (NFS) of (a) the fast mode, $T_i/T_e=0.1$ and (b) the slow mode, $T_i/T_e=0.5$ compared to fluid theory in the multi-ion species plasmas (Sec. 2a) and the kinetic theory (Sec. 2b) in the condition of $k{\lambda }_{De}=0.1$, where “sudden” and “adiabatic” represent for the excitation condition of the ions (C or H ions), and the excitation condition of the electrons is adiabatic, i.e., ${\alpha }_e={\alpha }_{ad}$.

Figure 7

The analytical results of the total nonlinear frequency shift of (a) the fast mode, $T_i/T_e=0.1$ and (b) the slow mode, $T_i/T_e=0.5$ for various values of $k{\lambda }_{De}$, where “adiabatic” is considered as the excitation condition of the ions (C or H ions). Here “total” represents the total NFS, i.e., the kinetic NFS plus the multi-ion species fluid NFS $\delta {\omega }_{\mathrm{NL}}^{\mathrm{kin}}+\delta {\omega }_{\mathrm{NL}}^{\mathrm{flu}}$, multi-ion species.

Figure 8

The comparison of the multi-ion species fluid NFS (the total NFS: $\delta {\omega }_{\mathrm{NL}}^{\mathrm{kin}}+\delta {\omega }_{\mathrm{NL}}^{\mathrm{flu}}$, multi-ion) and the single-ion specie fluid NFS (the total NFS: $\delta {\omega }_{\mathrm{NL}}^{\mathrm{kin}}+\delta {\omega }_{\mathrm{NL}}^{\mathrm{flu}}$, single-ion) for the slow mode in the condition of Ref. [32], $T_i/T_e=0.5,k{\lambda }_{De}=1/3$, corresponding to Fig. 3 in Ref. [32].