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Ion heating, burnout of the HF field and ion sound generation under the development of a modulation instability of an intense Langmuir wave in a plasma

The development of one-dimensional parametric instabilities of intense long-wave plasma waves is considered in terms of the so-called hybrid models, when electrons are treated as a fluid and ions are regarded as particles. The analysis is performed for both cases when the average plasma field energy is lower (Zakharov's hybrid model -- ZHM) or greater (Silin's hybrid model -- SHM) than the plasma thermal energy. The efficiency of energy transfer to ions and to ion perturbations under the development of the instability is considered for various values of electron-to-ion mass ratios. The energy of low-frequency (LF) oscillations (ion-sound waves) is found to be much lower than the final ion kinetic energy. We also discuss the influence of the changes in the damping rate of the high-frequency (HF) field on the instability development. Reduced absorption of the HF field leads to the retardation of the HF field burnout within plasma density cavities and to the broadening of the HF spectrum. At the same time, the ion velocity distribution tends to the normal distribution in both ZHM and SHM.


I. INTRODUCTION
Powerful fields that can be excited and are able to propagate in a plasma, i.e. in a nonlinear medium with a wide range of characteristic oscillations, often produce a great variety of instabilities. As a result, the new generally unsteady states come into existence in the medium. The theoretical description of all these successive relaxation processes in the form of cascades of instabilities requires solving very complex problems (in the mathematical sense) with the use of numerical methods and numerical experiments.
Investigating the after-effect of instabilities is important for a variety of applications in plasma physics and plasma electronics. Moreover, the employment of more and more powerful sources of energy leads in particular to the use of plasma in the conventional vacuum high-current electronic devices. The experimental and practical activities in space must also take into account the presence of plasmas. The very great variety of instabilities and even cascades of instabilities create problems for the implementation of the technologies developed up to now. On the other hand, the abundance of various unstable states provides opportunities for discovering and developing entirely new technologies.
Among the variety of plasma instabilities, the class of parametric instabilities should be especially emphasized. The modulation of velocities and densities of plasma particles by intense external fields results in the development of instabilities similar to those described by Mathieu's or Hill's equations. In the case when the external radiation is in resonance with plasma eigenmodes, such instabilities can be interpreted as decay or modulation ones.
The interest to the parametric instability of intense Langmuir waves, which can be easily excited in the plasma by various sources [1][2][3][4][5][6][7][8][9] was stipulated, in particular, by new possibilities in heating electrons and ions in a plasma. The correct methods for the description of the parametric instability of long-wave plasma waves were developed in the pioneer works of V.P. Silin 10 and V.E. Zakharov [11][12][13] . The theoretical concepts proposed by V.P. Silin 10 were confirmed by the early numerical experiments on the one-dimensional simulation of the parametric decay of plasma oscillations 14 (see also article 15,16 , and review 17 ). However, the greatest experimenters' interest was provoked by the mechanism of wave-energy dissipation discovered by V.E. Zakharov.
In Zakharov's model 11 that describes the instability of intense long-wave Langmuir waves in a non-isothermal plasma, namely the modulation instability, results in the excitation of a range of short-wave oscillations. In Silin's model, a strong Langmuir wave in a cold plasma leads to intense oscillations of the electron velocity with the amplitudes comparable to the wavelengths of the excited modes. In this case, the instability in the general case should be referred as parametric 10 . Nevertheless, both these processes are similar 36,37 . Hence, the term "modulation instability" can be applied to the instability of a strong Langmuir field within the framework of Silin's model.
In the works 38,39 an attempt was made to compare these models, which have similar physical nature, by the example of one-dimensional description. The choice of the onedimensional approach, as was noted by J. Dawson 40 , "often keeps the main features of the processes, but simplifies their description and leads to a fuller understanding of what the important phenomena are" (see also 41 ). Of particular interest is the process of ion heating, so in this paper we use the particle (or finite-sized particle) description for ions because the account of inertial effects can be significant just at the nonlinear stage of the process 42 . It was observed 42,43 that simulation in terms of the so-called hybrid model (incorporating one of Zakharov's equations for the HF waves and using particle simulation for ions) demonstrates that fluctuations of ion density are rather significant and favor the development of parametric instability. The non-resonant interaction between ions and HF plasma oscillations, along with ion trapping by the potential wells produced by these oscillations, lead to an instability of density cavities resulting from the modulation instability and produce fast particle groups. In papers 38,39 the simulation of one-dimensional ion dynamics was performed in terms of the particle method 45,46 . The number of particles used in numerical calculations was 2 · 10 4 , which is equivalent to the number of ions, about (2 · 10 4 ) 3 ∼ 10 13 , in the three-dimensional case, in agreement with the conditions of most experiments. Thus, the interaction between modeling particles and plasma oscillations in this simulation is in rather good accordance with the interaction between real particles and plasma waves, naturally with regard to the inherent limitations of the one-dimensional description. Nevertheless, there is reason to believe that the description of field energy transfer to ions within the framework of the When the intensity of the external electric field is much greater than the specific thermal energy of plasma electrons W = |E 0 | 2 /4π ≫ n 0 T e , it is reasonable to explore the approach presented by V.P. Silin 47 .
Let consider a one-dimensional plasma system where an intense plasma wave with the wavelength λ 0 and frequency ω 0 is excited by an external source. This intense wave will be referred to as the pumping wave. Since the parametric instability results in the growth of oscillations with rather small wavelength λ ≪ λ 0 , the pumping wave can be suggested to be spatially uniform within the region of interaction: where |E 0 |and φ are the slowly varying wave amplitude and the phase respectively, ω 0 is the external wave frequency, n 0 and T e are the density and temperature of plasma electrons.
Charged particles of plasma oscillate under the action of the electric field and their velocities can be written as The equations, governing the nonlinear dynamics of the parametric instability of an intense plasma wave, were derived in the work 36 . The equations for the HF plasma field spectrum modes E = n E n (t) · exp(ink 0 x) (plasma electrons are treated as fluid and described by hydrodynamic equations) are given by Here ω pe = 4πe 2 n 0 /m e is the background electron plasma frequency, e and m e are the electron mass and charge, M is the ion mass of an ion, E n = |E n |·exp(iψ n ) is a slowly varying complex amplitude of the electric field of electron plasma oscillations whose wave-number is k n = nk 0 , k 0 = 2π/L, where L is the characteristic dimension of the plasma system, is the ion charge density, a n = a · n, n, m are nonzero integers, and ±1, e.g. a n = nk 0 b = n (ek 0 E 0 /m e ω 2 0 ). The term E n θn 6 /n 6 M in Eq. (2) simulates the HF plasma wave damping by electrons, with n M = 20.
The equations of ion motion are given by and the ion density can be determined from The slowly varying electric field strengthĒ n , acting on the ions, is equal tō The equation for the uniform component of the electric field E 0 = |E 0 | exp(iφ) can be written as Note, that the values with subscripts with unlike signs are independent. In Eqs. (2)-(6), we have used the formula 48 where J m (x) is the Bessel function.

B. The hybrid model based on Zakharov's equations (ZHM)
As is shown in 36 , Eqs.
(2)-(6) reproduce the equations obtained in 37 after following sub- under the condition a n ≪ 1, that means that J 1 (a n ) ≈ a n /2, J 0 (a n ) ≈ 1, J 2 (a n ) ≈ a 2 n /8 Note that the damping of the HF modes (the term ∝ θ) was introduced in Eq. (9) phenomenologically, because we are only interested in how the changes in the absorption level influence the ion heating. More precise analysis of the effect of kinetic processes on the Landau damping and on the saturation of parametric instabilities in an electromagnetically driven plasma can be found in papers 49,50 .
In this case, the slowly varying electric field amplitude takes the form that makes it possible to describe ions as particles within the context of of Eqs. (3)-(4). The pump wave amplitude E 0 is governed by the equation In this case, the growth rate of the parametric instability normalized to the plasma frequency is given by 37

III. STATEMENT OF THE PROBLEM AND THE INITIAL CONDITIONS
The purpose of this paper is to clarify the efficiency of the energy transfer to ions and ion density perturbations in the course of development of modulation instability for the cases of both non-isothermal hot and cold plasmas in terms of the hybrid models. Both SHM and ZHM were considered for two cases of light and heavy ions.The parameters of the simulation are presented in Table 1. It is also interesting to elucidate the effect of HF spectrum damping and subsequent burnout of the Langmuir field within density cavities on the energy transfer to plasma ions.
The development of the instability was considered in terms of hybrid models Eqs.
where W 0 is the initial energy density of the intense Langmuir wave, I = s (dξ s /dτ ) 2 is normalized ion kinetic energy and δ is the rate of the linear instability. The portion of energy transferred from the intense Langmuir wave to ions is determined by the ratio W 0 /n 0 T e for the case of non-isothermic plasma (ZHM) and by the ratio (m/M) 1/3 for the case of cold plasma (SHM).
Below we consider more closely the nature of the energy redistribution with time and especially the process of energy transfer to the LF perturbations. We also discuss the  The particles outside the normal distribution (mainly in the so-called "tails") possess 13.8% of the total energy for light ions and 9,2% for heavy ions in ZHM model and much more in SHM: 25,6% for heavy ions and 13% for light ions, respectively. It means that in the case of instability of the intense wave in a cold plasma, a significantly greater proportion of fast particles should be expected.
We are interested not only in the ion kinetic energy distribution, but also in the collective excitation of ion oscillation (see Fig. 2), hence we define the frequency of the mode with the wave vector nk 0 associated with these oscillations, i.e., where the phases of LF modes can be found from the expression For both models, the ion kinetic energy in the assumed units can be written as 1 2 and the energies of collective excitations for ZHM and SHM, respectively, reduce to Note that in Zakharov's model these oscillations are referred to as ion-sound waves. Figure 3 demonstrates the time evolution of the ion kinetic energy and LF field energy.
It should be noted that the energy of the LF field is far smaller than ion energies in all cases.
Reducing the field energy with time is caused by the energy transfer to ions as well as by the destruction of plasma density cavities, 34 .
The rate of the HF field burnout within density cavities is determined by the value Θ = θ/δ. It is of interest how the simulation results depend on this parameter. Obviously, the decrease of this parameter not only inhibits the burnout of the HF field in the cavities, but also broadens the spectrum of HF modes, i.e. it increases the contribution of small-scale components that leads to the deepening of plasma density cavities and to the growth of the kinetic energy of ions ejected from the cavities.
Note that for both models the ion velocity distribution function approaches the Maxwellian curve with decreasing damping rate of HF modes, as may be seen from Fig. 4.     Figure (5) shows that in the case of a non-isothermal plasma the maximum energy of ionsound oscillations remains practically unchanged as the damping rate of HF field decreases, whereas the formation of the LF spectrum occurs with higher rate. In a cold plasma, on the contrary, the intensity of LF oscillations grows with the decrease of the damping rate of the HF field. After that, the LF spectrum is suppressed and its energy is transferred to ions. As might be expected, the energy, transmitted to ions, increases with the decrease of the damping rate of HF oscillations almost in the same proportion in both non-isothermal and cold plasmas (see. Fig. (6)).
It should be noted in conclusion that the ion-density perturbations with spatial scale smaller than the Debye radius r Di = v T i /ω pi do not contribute to the formation of LF electric fields by virtue of the screening effect. The Debye radius can be estimated from the expression 51 At the stage of developed instability this value is of the order of R Di ≤ 10 −3 and the number of spatial spectral modes of ion density does not exceed 1/R Di that is in agreement with the previous analysis.

V. CONCLUSION
The most important consequence of the instability of intense Langmuir waves in plasmas is the transfer of a portion of the field energy to ions and LF plasma density oscillations. It is reasonable to consider this problem within the framework of hybrid models, where electrons are treated as a fluid and ions are regarded as particles. The dynamics of the instability of intense long-wave oscillations in both hot and cold plasmas appears to be similar 36,37 .
It was noted earlier 38 that in the case of hot plasmas ions acquire a portion of field energy that is proportional to the ratio of the field energy to the plasma thermal energy. In the case of cold plasmas, ions get a portion of field energy that is proportional to the ratio of the instability growth rate to the plasma frequency or that is the same as to the cubic root of the ratio of electron to ion masses. The energy transferred to ions in the case of heavy ions is significantly smaller than in the case of light ions. Moreover, the amount of energy transferred to ions in the case of the cold plasma is inversely proportional to the cube root of the ion mass, and in the case of the hot plasma the portion of energy transferred to ions decreases with growth of the ion mass faster 39,51 . The kinetic energy distribution of ions in the SHM is characterized by a large fraction of fast particles.
In the case of a non-isothermic plasma (ZHM), the amplitudes of modes of the LF spectrum (ion-sound waves) are of the same order in a wide range of wave numbers. In a cold plasma (SHM), the long-wave oscillations dominate in the LF spectrum. The energy of the LF field is found to be much lower than the total kinetic energy of ions for all the cases discussed above. Reducing the energy of the LF field with time happens due to the energy transfer to ions.
The decrease of the damping rate of the HF field corresponds to the slowing of the HF field burnout in the cavities and leads to the broadening of the HF spectrum that causes the deepening of the cavities and increase of the kinetic energy of ions ejected from them. It should be noted that as the absorption rate of the HF field decreases, the ion velocity distribution function approaches the Maxwellian distribution in both models under consideration. In a cold plasma, the intensity of the long-wave LF oscillations is high and it increases with the decrease of the absorption of HF modes. It is important to note that the total energy transferred to ions increases as the absorption of the HF spectrum reduces.