Effect of thermal ions on fluid nonlinear frequency shift of ion acoustic waves in multi-ion species plasmas

TL;DR

This paper develops a comprehensive fluid model for ion acoustic waves in multi-ion plasmas, incorporating ion temperature effects to improve the accuracy of nonlinear frequency shift predictions relevant to space physics and fusion.

Contribution

It introduces a new model that accounts for thermal ions and multiple ion species, extending previous cold ion models for better accuracy in nonlinear frequency shift calculations.

Findings

The model predicts more accurate frequencies for large amplitude IAWs.

Thermal ion effects significantly influence the slow mode in multi-ion plasmas.

Applicable to space physics and inertial confinement fusion scenarios.

Abstract

A model of the fluid nonlinear frequency shift of ion acoustic waves (IAWs) in multi-ion species plasmas is presented, which considers the effect of ion temperature. Because the thermal ion exists in plasmas in inertial confinement fusion (ICF) and also solar wind, which should be considered in nonlinear frequency shift of IAWs. However, the existing models [Berger et al., Physics of Plasmas 20, 032107 (2013); Q. S. Feng et al., Phys. Rev. E 94, 023205 (2016)] just consider the cold ion fluid models. This complete theory considering multi-ion species and thermal ions will calculate the frequency of the large amplitude nonlinear IAWs more accurately, especially the slow mode with high ion temperature, which will have wide application in space physics and inertial confinement fusion.

Equations

$${\partial }_t{n}_a+{\partial }_x(n_a{v}_a)=0,$$

$${\partial }_t{v}_a+v_a{\partial }_x{v}_a=-C_a^2{\partial }_x\varphi -{\gamma }_a{v}_{th,a}^2{\partial }_x{n}_a,$$

$$-{\partial }_x^2\varphi +4\pi e{n}_0\exp [\varphi ]=4\pi \sum _a{q}_a{n}_a,$$

$$(\varphi ,n_a,v_a)=$$

$$\times \exp [il(kx-\omega t)],$$

$${\varphi }_0+\frac{{\varphi }_0^2}{2}=-\frac{{\varphi }_1^2}{4}-\frac{{\varphi }_2^2}{4},$$

$$-i\omega v_{a1}+ik{C}_a^2{\varphi }_1+ik\frac{{\gamma }_a{v}_{th,a}^2}{n_{a0}}{n}_{a1}=-i\frac{1}{2}{v}_{a-1}{v}_{a2},$$

$$-2i\omega v_{a2}=-2ik\left[C_a^2{\varphi }_2+\frac{1}{4}{v}_{a1}^2+\frac{{\gamma }_a{v}_{th,a}^2}{n_{a0}}{n}_{a2}\right].$$

$$n_{a2}=\frac{k}{\omega }\left[n_{a0}{v}_{a2}+\frac{1}{2}{n}_{a1}{v}_{a1}\right],$$

$$v_{a2}=\frac{k}{\omega }\left[C_a^2{\varphi }_2+\frac{1}{4}{v}_{a1}^2+{\gamma }_a{v}_{th,a}^2\frac{n_{a2}}{n_{a0}}\right],$$

$$v_{a2}$$

$$=\frac{k/\omega }{1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}}[C_a^2{\varphi }_2+\frac{1}{4}{v}_{a1}^2+\frac{1}{2}\frac{k{\gamma }_a{v}_{th,a}^2}{\omega }\frac{n_{a1}}{n_{a0}}{v}_{a1}],$$

$$n_{a2}$$

$$[4k^2{\lambda }_{De}^2+(1+{\varphi }_0)]{\varphi }_2+\frac{1}{4}{\varphi }_1^2$$

$$=$$

$$n_{a1}$$

$$v_{a1}$$

$$[4k^2{\lambda }_{De}^2+(1+{\varphi }_0)]{\varphi }_2+\frac{1}{4}{\varphi }_1^2$$

$$=$$

$$[4k^2{\lambda }_{De}^2+(1+{\varphi }_0)-\sum _a\frac{Z_a{n}_{a0}/n_{e0}}{1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}}\frac{k^2{C}_a^2}{{\omega }^2}]{\varphi }_2$$

$$=$$

$$C_{2s}^2\equiv \sum _a\frac{Z_a{n}_{a0}/n_{e0}}{1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}}{C}_a^2,C_{4s}^4\equiv \sum _a\frac{Z_a{n}_{a0}/n_{e0}}{(1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}{)}^3}{C}_a^4,$$

$$[4k^2{\lambda }_{De}^2+1-\frac{k^2{C}_{2s}^2}{{\omega }^2}]{\varphi }_2=[\frac{3}{4}\frac{k^4{C}_{4s}^4}{{\omega }^4}-\frac{1}{4}]{\varphi }_1^2.$$

$${\varphi }_2=A_{2\varphi }{\varphi }_1^2,A_{2\varphi }=\frac{[\frac{3}{4}\frac{k^4{C}_{4s}^4}{{\omega }^4}-\frac{1}{4}]}{[4k^2{\lambda }_{De}^2+1-\frac{k^2{C}_{2s}^2}{{\omega }^2}]}.$$

$$A_{2\varphi }=\frac{C_{s2}^2}{4{\omega }^2-C_{s2}^2}\left[\frac{3k^2{C}_{4s}^4}{4{\omega }^2{C}_{2s}^2}-\frac{{\omega }^2}{4k^2{C}_{2s}^2}\right].$$

$$n_{a1}$$

$$v_{a1}$$

$$n_{a1}$$

$$v_{a1}$$

$$n_{a2}$$

$$v_{a2}$$

$$v_{a1}=$$

$$+\frac{1}{2}\frac{k{\gamma }_a{v}_{th,a}^2}{\omega }\frac{n_{a2}}{n_{a0}}{v}_{a1}].$$

$$v_{a1}=$$

$$+\frac{\frac{k^4{C}_a^4}{{\omega }^4}(1+\frac{6k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}+\frac{2k^4{\gamma }_a{v}_{th,a}^4}{{\omega }^4})}{8(1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}{)}^4}{\varphi }_1^2]{\varphi }_1.$$

$$n_{a1}=$$

$$+\frac{k^4{C}_a^4/{\omega }^2}{8(1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}{)}^4}(5+\frac{4k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}){\varphi }_1^2]{\varphi }_1.$$

$$[k^2{\lambda }_{De}^2+1+{\varphi }_0]{\varphi }_1+\frac{1}{2}{\varphi }_2{\varphi }_1=\sum _a{Z}_a\frac{n_{a1}}{n_{e0}}.$$

$$[k^2{\lambda }_{De}^2+1+{\varphi }_0-\sum _a\frac{Z_a{n}_{a0}/n_{e0}}{1-\frac{k^2{\gamma }_a{v}_{th,a}^2}{{\omega }^2}}\frac{k^2{C}_a^2}{{\omega }^2}]{\varphi }_1$$

Full text

††preprint:

I Introduction

The nonlinearities of ion acoustic waves (IAWs) are of fundamental interest to plasma physics. Understanding the fluid effects and kinetic effects of nonlinear IAWs is of key significant in space physics such as solar wind Vecchio_2014JGR ; Valentini_2014APJL ; Gurnett_1977JGR ; Gurnett_1978JGR ; Gurnett_1979JGR and also stimulated Brillouin scattering (SBS) in inertial confinement fusion (ICF) He_2016POP ; Glenzer_2010Science ; Glenzer_2007Nature ; LanKe_2017PRE ; Lan_2016MRE ; Huo_2016PRL ; Huo_2016MRE .

The nonlinear saturation of SBS Froula_2002PRL ; Berger_1998POP ; Neumayer_2008PRL ; Giacone_1998POP ; Vu_2001PRL ; Albright_2016POP in plasmas relevant to ICF is closely related to the ion-acoustic wave saturation. Therefore, studying the nonlinearities of IAWs is important to understand the underlying physics of the saturation of SBS and to interpret scattered light levels from current ICF experiments. The nonlinear frequency shift (NFS) of IAWs induced by trapping Froula_2002PRL ; Giacone_1998POP ; Vu_2001PRL ; Albright_2016POP and harmonic generation Bruce_1997POP ; Rozmus_1992POP is suggested to be a possible saturation mechanism of SBS. Therefore, the theory to calculate the nonlinear frequency shift of IAWs is vital to ICF.

The nonlinear effects on the frequency of the nonlinear IAWs are hot topics as a result of their potential role in determining the saturation of SBS in ICF. Cohen_1997POP ; Albright_2016POP The fluid NFS of IAWs resulting from harmonic generation is obtained by the isothermal cold ion fluid equations where ion is considered to be cold Berger_2013POP ; Chapman_2013PRL ; Feng_2016PRE . However, in fusion plasmas, ion temperature might be almost comparable to electron temperature. Therefore, the effect of ion temperature on the fluid NFS should be considered.

In this paper, a multi thermal ion fluid model is proposed to calculate the fluid NFS of IAW in multi-ion species plasmas. This model considers the thermal ions and is verified to be consistent to Vlasov simulation data better, especially for the slow IAW mode with high ion temperature.

II Theoretical Model

From the isothermal hot ion fluid equations:

[TABLE]

the fluid nonlinear frequency shift resulting from harmonic generation is derived. Where the electrostatic potential is normalized by $T_{e}/e$, i.e., $\phi\to e\phi/T_{e}$; $C_{a}=\sqrt{Z_{a}T_{e}/m_{a}}$ and $v_{th,a}=\sqrt{T_{a}/m_{a}}$ are the sound speed and the thermal velocity of ion $a$. The thermal ion effect, i.e., the term of $-\gamma_{a}v_{th,a}^{2}\partial_{x}n_{a}$ in Eq. (2) is considered in this paper. Following Pesme et al. Pesme_2005POP , Berger et al. Berger_2013POP and Feng et al. Feng_2016PRE the variables in a Fourier series are given by

[TABLE]

where $(\phi_{a-l},n_{a-l},v_{a-l})=(\phi_{al},n_{al},n_{al})^{*}$ and $\exp\phi\simeq 1+\phi+\frac{1}{2}\phi^{2}$ by keeping terms for $l=0,\pm 1,\pm 2$ up to $2^{\mathrm{nd}}$ order. From Eqs. (1)-(3) for $l=0$ by retaining terms with matching exponents in the Fourier series, one obtains

[TABLE]

where conservation of charge, $n_{0}=\sum_{a}Z_{a}n_{a0}$, has been used. The equation of Eq. (2) for $l=1$ is

[TABLE]

where the left hand are the linear terms and the right hand are the nonlinear terms. The corresponding equation for $l=2$ is

[TABLE]

Keeping terms only to second order in $\phi_{2}$, we will calculate for $\phi_{2}$ in the following. Firstly,

[TABLE]

which further gives

[TABLE]

Substituting above results into Eq. (7), one obtains

[TABLE]

Since $\phi_{2}$ is estimated to second order in $\phi_{2}$, $n_{a1}$ and $v_{a1}$ in Eq. (10) are simply estimated to first order in $\phi_{1}$. Thus, from Eq. (6), one obtains

[TABLE]

Substituting these results into Eq. (10), one obtains

[TABLE]

This result can be rewritten to

[TABLE]

Due to $\phi_{0}\sim|\phi_{1}|^{2}$, this term in the left hand of the above equation is neglected and

[TABLE]

is defined. Where $\frac{k^{2}\gamma_{a}v_{th,a}^{2}}{\omega^{2}}$ is the term considering the thermal ion effect, which is zero in cold ion model Berger_2013POP ; Feng_2016PRE . One obtains

[TABLE]

Finally, the relation between $\phi_{2}$ and $\phi_{1}$ is

[TABLE]

Defining $C_{s2}^{2}=4k^{2}C_{2s}^{2}/(1+4k^{2}\lambda_{De}^{2})$, then one obtains

[TABLE]

In the following, $\phi_{1}$ will be estimated to less than third order in $\phi_{1}$ from equations for $l=0$. First, the density and velocity are

[TABLE]

The first order and the second harmonic of density and velocity are

[TABLE]

Now, we first derive velocity $v_{a1}$ by substituting result of first order into the term of high order

[TABLE]

Inserting the fist order result of the velocity $v_{a1}$, one obtains

[TABLE]

Next, the density perturbation is

[TABLE]

The Possion equation for $l=1$ is

[TABLE]

Substituting the expression of Eq. (27) into this Possion equation gives

[TABLE]

Applying the relation $\phi_{0}\approx-\frac{1}{4}|\phi_{1}|^{2}$, and defining

[TABLE]

one obtains

[TABLE]

Substituting $\phi_{2}=A_{2\phi}\phi_{1}^{2}$ and defining $C_{s1}^{2}=k^{2}C_{2s}^{2}/(1+k^{2}\lambda_{De}^{2})$, one can obtain

[TABLE]

where

[TABLE]

Since $\omega_{\mathrm{harm}}=\omega+\delta\omega_{\mathrm{harm}}$, and $\omega^{2}-C_{s1}^{2}(\omega)=0$, $\omega_{\mathrm{harm}}$ is the effective fundamental IAWs frequency after accounting for harmonic effects. Due to the inclusion of the second harmonic terms, the frequency shift of the fundamental mode in multi-ion species plasmas is given by

[TABLE]

II.1 single-ion species plasmas

Applying our result for single ion plasmas

[TABLE]

where $\alpha=k^{2}\lambda_{De}^{2}+1$ and $\beta=\gamma_{a}v_{th,a}^{2}/C_{s}^{2}=\frac{\gamma_{a}T_{a}}{ZT_{e}}$. Therefore, the fluid nonlinear frequency shift in single ion species plasmas is given by

[TABLE]

where $c=\frac{4k^{2}\gamma_{a}v_{th,a}^{2}}{5\omega^{2}}$. For cold ion, $\beta=0,c=0$, the nonlinear frequency shift is

[TABLE]

which is consistent with the result of Berger et al. Berger_2013POP from cold ion assumption. Where $\tilde{k}=k\lambda_{De}$ is defined. Comparing Eq. (42) with Eq. (43), the effect of ion temperature on the frequency shift is presented by these terms including $\beta$. It is obvious that ion temperature can enhance the frequency shift. More detail, ion temperature will make a significant effect if $\gamma_{a}T_{a}/Z_{a}T_{e}\gtrsim 0.1$.

As shown in Fig. 1(a), the ratio of the fluid NFS including ion temperature to that with cold ion $\delta\omega^{h}/\delta\omega^{c}$ will decrease with $ZT_{e}/T_{i}$ increasing. When $ZT_{e}/T_{i}$ is very large, such as $ZT_{e}/T_{i}=30$, the fluid NFS including ion temperature will be close to the fluid NFS with cold ion. Especially, when $ZT_{e}/T_{i}=2$, the ratio $\delta\omega^{h}/\delta\omega^{c}$ will reach $3-4$. That is to say, the effect of ion temperature on fluid NFS is obvious in higher ion temperature. On the other hand, with $k\lambda_{De}$ increasing, $\delta\omega^{h}/\delta\omega^{c}$ will decrease. When $ZT_{e}/T_{i}$ is very large, such as $ZT_{e}/T_{i}=30$, $\delta\omega^{h}/\delta\omega^{c}$ will be the same in different wave numbers $k\lambda_{De}$, which is because the effect of thermal ions on fluid NFS will not be obvious when $ZT_{e}/T_{i}$ is very large. As shown in Fig. 1(b), the fluid NFS will decrease obviously with $ZT_{e}/T_{i}$ increasing. Especially, when $ZT_{e}/T_{i}=\infty$ or $T_{i}/T_{e}=0$, the NFS from single thermal ion fluid theory will be consistent to NFS from single cold ion fluid theory Berger_2013POP ; Chapman_2013PRL . However, the higher the ion temperature is, the effect of thermal ions on fluid NFS will be more obvious.

II.2 multi-ion species plasmas

In multi-ion species plasmas, the linear dispersion relation of IAWs can be calculated from Williams_1995POP ; Berger_2013POP ; Feng_2016POP ; Feng_2016PRE

[TABLE]

where $Z(\xi_{j})=1/\sqrt{\pi}\int_{-\infty}^{+\infty}e^{-v^{2}}/(v-\xi_{j})dv$ is the plasma dispersion function, and $\lambda_{Dj}=\sqrt{T_{j}/4\pi n_{j}Z_{j}^{2}e^{2}}$, $v_{tj}=\sqrt{T_{j}/m_{j}}$ are the Debye length and the thermal velocity of specie $j$. And $m_{j},Z_{j},T_{j},n_{j}$ are the mass, charge number, temperature and density of specie $j$, respectively. In this paper, CH plasmas will be chosen as a typical example of multi-ion species plasmas due to its potential applications in ICF He_2016POP ; Glenzer_2007Nature ; Glenzer_2010Science . In CH plasmas, calculated by Eq. (44), the frequency of the fast mode in the condition of $k\lambda_{De}=1/3,T_{i}/T_{e}=1/15$ is $Re(\omega_{s})=6.102\times 10^{-3}\omega_{pe}$, and the frequency of the slow mode in the condition of $k\lambda_{De}=1/3,T_{i}/T_{e}=1/2$ is $Re(\omega_{s})=5.664\times 10^{-3}\omega_{pe}$.

Figure 2 gives the relation between $\phi_{2}/\phi_{1}$ and $\tilde{\phi_{1}}$ from three theories. It is shown that $A_{2\phi}$ calculated by the multi thermal ion fluid theory in this paper is obviously larger than those calculated by the single cold ion fluid theory Berger_2013POP ; Chapman_2013PRL and multi cold ion fluid theory Feng_2016PRE . This result may give an explanation of that the Vlasov simulation data is much larger than the single cold ion fluid theory in Figure 3(b) in the research of Chapman et al. Chapman_2013PRL , when the effect of multi-ion species and thermal ions are considered. The multi thermal ion fluid theory given by this paper will be closer to the Vlasov simulation data than multi cold ion fluid theory Feng_2016PRE and single cold ion fluid theory Berger_2013POP ; Chapman_2013PRL .

Figure 3 gives the fluid NFS calculated by three theories. Compared with single cold ion fluid theory and multi cold ion fluid theory, the effect of the thermal ions will give a larger fluid NFS. The Vlasov simulation results of the slow mode in Figure 3(d) in Chapman et al.’s work Chapman_2013PRL is obviously larger than the single cold ion fluid theory, especially when $|e\phi/T_{e}|^{1/2}$ is large, because only the single species and cold ions are considered in Chapman et al.’s research, while the system researched by Chapman et al. is CH plasmas but not single-ion species system and the ion temperature with $T_{i}/T_{e}=1/2$ could not be neglected. However, for the fast mode, the ion temperature is very low related to the electron temperature, thus the thermal ion effect can be neglected and the multi cold ion fluid theory can be applied. The multi thermal ion fluid theory in this paper will give a correction to the single cold ion fluid theory Berger_2013POP ; Chapman_2013PRL and multi cold ion fluid theory Feng_2016PRE . The effect of multi-ion species and thermal ions is considered in this paper.

III Numerical Results

One dimension in space and velocity (1D1V) Vlasov-Poisson codeLiu_2009POP ; Liu_2009POP_1 is taken to excite the nonlinear IAW in CH plasmas. The form of the external driving electric field (driver) is

[TABLE]

where $\tilde{E}_{d}=eE_{d}\lambda_{De}/T_{e}$. $\omega_{d}$ and k are the frequency and the wave number of the driver. The envelope of the driver $\tilde{E}_{d}(t)$ is

[TABLE]

where the maximum amplitude of the driver is $\tilde{E}_{d}^{max}=eE_{d}^{max}\lambda_{De}/T_{e}$. And the duration time of the peak driving electric filed is $t_{0}$. The driver frequency chooses the fundamental frequency of the linear IAW, i.e., $\omega_{d}=\omega_{L}$. As shown in Fig. 4(a), the duration time of external driving electric field is $t_{0}=1\times 10^{5}\omega_{pe}^{-1}$ and turns off at $\omega_{pe}t=2\times 10^{5}$. After the driver is off, the BGK BGK mode is established, and the electric field oscillates at almost constant amplitude. As shown in Fig. 4(b), when the driver is on, the frequency of the electric field among the time scope of $\omega_{pe}t\in[0,2\times 10^{5}]$ is close to the linear frequency of the slow mode $\omega_{0}$. However, when the driver is off, the electric field will keep constant since the steady BGK BGK mode will be established through particle trapping. The nonlinear frequency shift of the slow mode will occur due to particle trapping and harmonic generation. During $\omega_{pe}t\in[3\times 10^{5},4.9\times 10^{5}]$, the slow mode with a larger frequency of $\omega_{0}+\delta\omega_{NL}$ than the linear frequency $\omega_{0}$ will be established as shown in Fig. 4(b). The nonlinear frequency shift $\delta\omega_{NL}/\omega_{0}$ is related to the amplitude of nonlinear IAW excited by the driver.

The amplitude of the driver $\tilde{E}_{d}^{max}$ varies to excite different amplitudes of nonlinear IAW. Therefore, the nonlinear frequency shift $\delta\omega_{NL}/\omega_{0}$ of the slow mode in different IAW amplitudes under the condition of $k\lambda_{De}=0.3,T_{i}/T_{e}=1/2$ can be obtained by Vlasov simulation. As shown in Fig. 5, the total NFS $\delta\omega_{NL}^{kin}+\delta\omega_{NL}^{flu}$ from multi thermal ion fluid theory is closer to the Vlasov simulation data than that from multi cold ion fluid theory Feng_2016PRE and single cold ion fluid theory Berger_2013POP ; Chapman_2013PRL . That the Vlasov simulation data is also larger than the total NFS from multi thermal ion fluid theory may be because that only the second harmonic terms are considered in the theory in this paper. However, the effect of the thermal ions on fluid NFS would give a better correction to the multi cold ion fluid theory Feng_2016PRE and the single cold ion fluid theory Berger_2013POP ; Chapman_2013PRL , as a result the thermal ion effect should be considered.

IV Conclusions

The effect of thermal ions is considered in the fluid nonlinear frequency shift model and a multi thermal ion fluid model is given. The multi thermal ion fluid model is verified to be better consistent to the Vlasov simulation results and will give a better correction to the previous theory, especially for the slow IAW mode with high ion temperature. It will give a complete theory considering multi-ion species and thermal ions to calculate the frequency of large amplitude nonlinear IAW. This theoretical model will have a potential application in space physics and ICF, since the large amplitude nonlinear IAW always be produced in solar wind and SBS.

Acknowledgements.

We are pleased to acknowledge useful discussions with C. Z. Xiao. This research was supported by National Postdoctoral Program for Innovative Talents (Grant No. BX20180055), the China Postdoctoral Science Foundation (Grant No. 2018M641274), the National Natural Science Foundation of China (Grant Nos. 11875091, 11575035, 11475030 and 11435011) and Science Challenge Project, No. TZ2016005.