1. Introduction

The Weibel instability ^[1] is a micro-convective instability. It corresponds to the excitation of electromagnetic modes in plasma characterized by anisotropy in temperature which corresponds to a plasma described by an anisotropic velocity distribution function. This anisotropy can be generated by several mechanisms, namely: the heat transport, the expansion of the plasma and the inverse bremsstrahlung absorption ^{[2, 3, 4, 5, 6]}.

The present work aims to the investigation of the Weibel instability induced by inverse bremsstrahlung absorption in the laser fusion plasma. This requires the investigation of the dispersion relation for low-frequency electromagnetic modes in plasma heated by a laser pulse.

The spatio-temporal dependence of the high-frequency laser pulseis supposed to be a normal mode. It results highlight new terms in the dispersion relation which can contribute to the Weibel modes instability. We will keep all terms and study their influence on the growth rate of the Weibel instability.

The present paper is organized as follows: in the section 1, we present the used theoretical model where the basic equation is the Fokker-Planck equation. Weconsider homogeneous plasma in the presence of a high frequency and low magnitude laser electric field. In order to calculate the distribution function from the Fokker-Planck equation, we use the method of time scales separation and the iterative method. In section 2, we solve the linear part of the Fokker-Planck equation associated with the disruption of the distribution function and establish the dispersion relation of the Weibel modes. Solving the dispersion relation leads to the calculation of rates of instabilities. Finally, a brief conclusion and a summarization of the main obtained results are presented.

2. Basic Equation

To describe the laser fusion plasma corona which is fully ionized plasma where the interactions between particles are dominated by the Coulomb interaction, by Braginski notation the suitable kinetic equation is that of Fokker-Planck equation ^[7]. For electrons, the Fokker-Planck equation is written in the laboratory frame as:

(1)

Where f is the distribution functions of electrons, is the electron mass, e is the elementary charge, and are respectively the electric and the magnetic fields present in the plasma, and are respectively of electron-electron collision and electron-ion operators ^[8].

and , where and represent the high-frequency fields associated to the laser wave, and mean low frequency fields associated to the disturbance in the plasma.We point out that has a small magnitude compared to that of .In addition, The spatio-temporal evolution of the laser wave is of the form:

(2)

where and are respectively the wave electric field magnitude and the wave frequency. In order to solve the equation (1), we consider two time scales, a low-frequency hydrodynamic timescale and a high-frequency (laser field) one. Therefore, the electronic distribution function f can be written as the sum of a quasi-static distribution function , which varies slowly in time and a high-frequency distribution function , following the temporal variation of high frequency laser electric field , so:

(3)

The rating on the indices «s» and «h» refers to secular time scales low frequency and high frequency respectively and will be used throughout this work.

The separation of time scales in equation (1) leads to the following high-frequency and low-frequency kinetic equations:

(4)

(5)

The symbol <> denotes the average over the laser wave cycle time, . These equations make a system of two coupled equations. Note here that the terms in the electric field and magnetic field reflect the inclusion of the low frequency electromagnetic field effect in our study. In particular, the first term on the right hand side of the equation (4) reflects the coupling of quasi-static fields with the laser field. Let us recall here that this field present in the plasma is generated by the mechanism of the Weibel instability. These terms do not appear in the work reported in reference ^[3].

We consider the high frequency approximation, where the laser wave frequency is greater than the collisions frequency .

This approximation is largely justified in the laser fusion plasma experiments. For example of typical Parameters: (electron temperature T_e=1 KeV; overage free scaling of particles λ_ei=1 µm and laser wave length λ_l=1.06 µm respectively) we have .

In this case, the high frequency electronic distribution function is calculated as follow:

(6)

where the used Einstein notation means that the repeated indices represent the summation over the indices The expression (6) represents the high-frequency component of the electronic distribution function which depends on its low-frequency component .

We will now solve the low frequency Fokker-Planck equation (5) using the expression of the distribution function of high frequency established in the previous paragraph. Substituting (6) in (5), we obtain then after some algebra:

(7)

where

(8)

The next step of our calculation is to linearize the equation (7) by setting

(9)

We calculated the second anisotropic equation using the same approximation as applied by projection of the equation (7) on Legendre polynomial p₂(v) ^[9].

1. The terms proportional to the second anisotropic distribution function , are ignored. This is justified by the fact that: which correspond to the low magnitude laser wave approximation largely fulfilled in the laser-plasma interaction experiments ^[6].

2. We have neglected the terms proportional to and those proportional to .

3. The expression of the second anisotropic distribution function is obtained in the stationary approximation is .

The evolution of the second anisotropy equation is obtained by:

(10)

This result shows that the second anisotropy is positive , which corresponds to a temperature anisotropy , where denotes the temperature in the direction ox and in the direction perpendicular to ox. Indeed, the plasma is heated preferentially along the laser wave field direction: ox.

3. Analysis of the Weibel Instability

This paragraph is devoted to the analysis of the Weibel instability. We determine the dispersion relation of the Weibel modes and deduce the growth rate of Weibel instability. For this it is necessary to calculate the perturbed distribution function due to an electromagnetic disturbance.

The evolution equation of the perturbed is obtained from the low-frequency Fokker-Planck equation by considering the first term order, so:

(11)

where

After some manipulation the mathematical equation (11) becomes.

(12)

Note also, that the right-side of equation (12) is zero. The result is a recurrence relation between the components:

(13)

We will now solve the system of equations (13) using mathematical techniques based on continued fractions. Solving kinetic equations with continued fractions ^[9] was used for the first time in ^[4], where the collisional propagator in the Fokker-Planck equation has been explicitly reversed on the basis of spherical harmonics. By applying these results to equation (13), and after some mathematical manipulations, we obtain:

(14)

where is the continued fraction defined the following recurrence relation:

(15)

We must note here that equation (14) is the exact solution of the infinite hierarchy of equations (13). It gives a relationship between components and that includes the contribution of all components with l>3. The explicit expressions of the components are then obtained as:

(16)

The calculation of the dispersion relation of electromagnetic Weibel modes in the semi-collisional approximation ^{[2, 4, 5, 10]} can be calculated using the perturbed Fokker-Planck equation coupled with Maxwell's equations presented as follows:

(17)

and

(18)

whereis the current density defined by

(19)

By considering that the spatio-temporal dependence of the field and as a Fourier mode , equations (18) and (19) can be represented as:

(20)

(21)

By developing the function , on the spherical harmonics basis , the equation (21) reads as:

(22)

We deduce then the dispersion relation of the Weibel modes as:

(23)

Here, , is the frequency of Weibel mode and being the growth rate which is derived in the linear approximation from equation (23) as:

(24)

4. Discussion

The first term of equation (24) is a loss term; it corresponds to a loss due to collisions between particles in the collisional limit while in the non-collisional limit , it describes the Landau damping of electromagnetic modes. The second term, corresponds to the Weibel instability source. Equation (24) gives explicitly the growth rate of the Weibel instability excited by inverse bremsstrahlung absorption in laser fusion plasma. This expression contains continuous fractions which are numerically calculated ^[5]. We present on Figure 1 and Figure 2, the spectra of the growth rate of Weibel unstable mode for typical parameters of plasma and laser.

We point out that the profile of the spectrum gamma(k) has present a maximum. This can be interpreted by the competition between the loss effects (collisions and Landau damping) and the inverse bremsstrahlung Weibel instability growing.

However, the values of the growth rate remain in the order of compared by reference ^{[11, 12]}. it is found that this reduction is independent of the values of the electron density. We believe that this is due to the choice of the Krook collisions operator where the collision frequency is considered constant. This approximation is not realistic since it is known that collisions reduce the anisotropy of the plasma it letting it to a steady state described by an isotropic distribution function.

Figure 1. Growth rate of Weibel instability γ as function of the collision parameter kλ_ei for typical physical parameters: n_e=10²⁷m^-3, T_e=2KeV, v₀/ v_t=0.3 and Z=4

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Figure 2. Growth rate of Weibel instability γ as function of the collision parameter kλ_ei for typical physical parameters: n_e=9×10²⁷m^-3, T_e=2KeV, v₀/ v_t=0.3 and Z=4

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5. Conclusion

In this work we have presented a theoretical investigation and a numerical analysis (Figure 1 and Figure 2) of the Weibel instability in the laser fusion plasma using the Fokker-Planck model. The stabilization effect due to generated static magnetic field is computed. It has been shown that by taking into account the coupling of the self-generated magnetic field by the Weibel instability with the laser wave field leads to a significant decreasing in the spectral range of unstable Weibel modes. Also this coupling undergoes a reduction in the growth rate values of Weibel instable modes. For the most unstable mode, this reduction is of two ranges.

References

[1]	E. S. Weibel. Phys. Rev. Lett. 2, pp 83-84 (1959).
	In article		CrossRef
[2]	K. Bendib, and al.Laser and Particle Beams 16, 3, pp 473-490 (1998).
	In article		CrossRef
[3]	A. Bendib, and al. Phys. Rev. E. 55, pp 7522 -7526 (1997).
	In article		CrossRef
[4]	A. Bendib and al, Phys. Fluids 30 (5), pp 1353-1361 (1987).
	In article		CrossRef
[5]	J. P. Matte and al, Phys. Rev. Lett. 58, pp 2067-2070 (1987).
	In article		CrossRef
[6]	A. Bruce Langdon. phys. Rev. Let. 44, pp 575-579 (1980).
	In article
[7]	S. I. Braginski, in (M. A. Leonvitch, Consultant Bureau) N. Y. 1965
	In article
[8]	I. P. Shkarofsky, and al. (Addison-Wesley) 1966.
	In article
[9]	M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York 1970).
	In article
[10]	A. Sid, physics of plasmas. 10, pp 214-219.2003.
	In article
[11]	Sid, A. and al. Plasma and Fusion Research Volume: 5 pp 007-1-6 (2010).
	In article
[12]	S.Peter Gary And al. Journal of geophysical research, Vol. 111, A11224. 1-5,2006.
	In article