RELATIVISTIC NONLINEAR EFFECTS IN PLASMAS

P.K. SHUKLA, N.N. RAO and M.Y. YU Institut fr Theoretische Physik, Ruhr.-Universitt Bochum, D-4630 Bochum 1, Fed. Rep. Germany and N.L. TSINTSADZE Institute of Physics, Academy of Sciences of the Georgian S.S.R., Tbilisi, U.S.S.R.

NORTH-HOLLAND - AMSTERDAM

I

PHYSICS REPORTS (Review Section of Physics Letters) 138, Nos. 1 & 2 (1986) 1149. North-Holland, Amsterdam

RELATIVISTIC NONLINEAR EFFECTS IN PLASMAS P.K. SHUKLA, N.N. RAO* and M.Y. YUInstitut fr Theoretische Physik, Ruhr-Universitdt Bochum, D-4630 Bochum 1, Fed. Rep. Germany

and N.L. TSINTSADZEInstitute of Physics, Academy of Sciences of the Georgian S.S.R., Tbilisi, U.S.S.R. Received 7 November 1985 Contents: 1. Introduction 2. Basic equations 3. Relativistic effects in unmagnetized plasmas 3.1. Electron plasma waves 3.2. Electromagnetic waves 3.3. Stimulated Raman scattering 3.4. Stimulated Brillouin scattering 3.5. Finite amplitude electromagnetic solitons 3.6. Weakly relativistic self-trapped laser beams 3.7. Three-dimensional relativistic solitons 3.8. Relativistic self-focusing 3.9. Intense self-trapped laser beams 3.10. Stability of supersonic solitary waves 3.11. Wave breaking 4. Relativistic nonlinear effects in magnetized plasmas 4.1. Electrostatic upper-hybrid waves 4.2. Nonlinear ordinary mode radiation 4.3. Ultrarelativistic cyclotron waves 4.4. Circularly polarized waves of weakly relativistic amplitudes 4.5. Finite amplitude envelope solitons 3 9 12 12 18 23 26 28 39 41 42 44 46 48 52 53 58 65 79 81 5. Profile modifications, shock structures, and magnetic field generation 5.1. A survey of the nonrelativistic results 5.2. Relativistic profile modifications 5.3. Compressionrarefaction wave-plateau shocks 5.4. Rarefaction wave-plateau shocks 5.5. Self-generation of magnetic fields 6. Nonlinear waves in electronpositron plasmas 6.1 Characteristics of an electronpositron plasma 6.2. Finite amplitude waves and solitons 6.3. Particle acceleration and heating 6.4. Weakly nonlinear theories 7. Some applications associated with relativistic effects 7.1. Free electron lasers 7.2. Cyclotron resonance masers 7.3. Particle acceleration by electromagnetic waves 7.4. Beat-wave accelerators 7.5. Interaction of bunched electron beam with a plasma 8. Conclusions Appendices References

93 93 94 94 99 104 113 114 116 121 124 130 130 136 137 138 140 140 142 144

*

Permanent address: Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India.

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P. K. Shukia et al., Relativistic nonlinear effects in plasmas Abstract:

3

The purpose of this article is to present a review of the nonlinear effects associated with relativistic electron-mass variation and the ponderomotive force in unmagnetized as well as magnetized plasmas. Many high-frequency waves can become unstable with respect to the electron-mass modulation and the excitation of low-frequency density fluctuations. The nonlinear equations which govern the evolution of the modulationally unstable waves are derived. The phenomena of soliton formation, radiation collapse, and profile modification are investigated. Finite amplitude theories of the envelope solitons are reviewed. In a multidimensional situation, the electromagnetic waves can undergo self-focusing. The use of the variational methods allows one to calculate the nonlinear wavenumber and radius of the self-focused laser beams. Analytical solutions for the self-trapped radiation and the three-dimensional relativistic solitons are obtained. It is found that magnetized plasmas can support the propagation of new types of ultrarelativistic electromagnetic waves. The modulational instability of the latter is analyzed. Furthermore, it is shown that the relativistic pondefomotive force in a magnetized plasma can produce large amplitude field-aligned electrostatic potentials which can effectively accelerate particles to very high energies. Finally, we consider the nonlinear propagation of intense electromagnetic waves in electronpositron plasmas. Possible applications in inertial fusion, beat-wave particle accelerator, rf heating of magnetically confined plasmas, and pulsar radiation are pointed out.

1. Introduction The interaction of electromagnetic waves With charged particles is of practical interest in the study of laboratory as Well as astrophysical plasmas. Linear theory shows that electromagnetic waves with frequencies less than the electron plasma frequency cannot propagate in an unmagnetized plasma. However, for large field intensities, nonlinear effects [116]such as the relativistic electron-mass variation [110] and the ponderomotive force [1315] can lead to a downshift of the local electron plasma frequency. This results in the possibility of electromagnetic wave energy penetration into the overdense region as defined classically. Phenomena such as this are very important in the studies of laserpellet interaction, the rf heating of magnetically confined fusion plasmas, the radar-induced modification of the ionosphere, the free electron lasers, the gyrotrons, as well as some wave-driven particle accelerators. When the wave intensity is sufficiently large, the directed component of the electron velocity can become quite large. For example, the ratio of the quiver velocity, v0 = eE/meo w to the velocity of light c is [17]v0/c=

6 x 10A\/7,

(1.1)

where e is the electronic charge, E is the wave electric field, w is the wave frequency, meo is the electron 2 and A is the laser wavelength in microns. For a laser with rest mass, I is the laser intensity in W/cm A = 10.6 p.m and I = 1016 WI cm2, one finds v 0 0.6c. In such cases, the effects of relativistic electron-mass have to be included in any investigations involving such waves. Akhiezer and Polovin [1] presented a general theory of nonlinear wave motion in an electron plasma. Taking into account the electron-mass variation as well as the second-order density perturbation effects, they derived a set of nonlinear equations and also the corresponding dispersion relation for the waves. Specifically, they showed that in a cold plasma an intense plane electromagnetic wave of large amplitude can drive electrons to relativistic velocities and produce a strong nonlinear Lorentz force in the longitudinal direction, thus leading to a coupling between the transverse and the parallel (with respect to the wavevector) momenta. Hence, the electromagnetic waves are coupled to the Langmuir waves. If the phase velocity of the coupled waves is greater than the velocity of light, the motion is bounded. Approximate analytical results for the coupled electromagnetic and Langmuir waves have been discussed by several authors [1,310]. Decoster [10] reviewed the recent progress in the study of wave motion including relativistic effects. He has also discussed a weak coupling analysis, and the existence of nonlinear resonances.

4

P.K. Shukia et a!., Relativistic nonlinear effects in plasmas

There have been several [110] derivations of the dispersion relations for the relativistic propagation of electromagnetic wave in an electron plasma. In the weakly relativistic limit v0 ~ c, one finds the electron plasma wave dispersion relation [6] w=w~0(13u~/16), (1.2)

where Urn = eE/meow 0c, and w~,0is the unperturbed electron plasma frequency. On the other hand, for Um~ 1, one obtains f6] w~s2 where eEw=

3/2

1TWp0(l~Um)

2

1/2,

(1.3)4, so that

\r2meocwpo(1

u~~)~

irmeocw~oI2eE.

(1.4) 0 can propagate

Equations (1.2) and (1.4) show that pure longitudinal waves with frequencies less than w~ in an electron plasma. The relativistic electron-mass variation as well as the harmonic generation nonlinearities were included by Sluijter and Montgomery [3] in their study of linearly polarized electromagnetic waves. In the weakly relativistic limit, their dispersion relation is2 ~0 2

2

=

kc +

22

2

2c

(~3

22

4w

kc 2

2

)][71

(1.5)

where k is the wavenumber. In the strongly relativistic limit U~~ 1, one finds2 22 1 2 22 +~TWpo/WUrn~

~kc

(1.6)

where kc ~ w has been assumed. On the other hand, for circularly polarized electromagnetic waves, harmonic generation does not occur. Relativistic electron-mass variation effects lead to [6] 2c2 + w~ 2 (1.7) 2 = k 0(1 + U~,).

Thus, waves in the frequency range+ ~~2)_1/4 K 0. Next, we analyze eq. (3.77) for the case in which collisions dominate. Assuming 1 to,,0 Kc, i. 4 11, < 4, one finds from eq. (3.77) ~ I v Kc 4t--~i 4 \to0 to0 ~ 2c2, then K2c2 I w~0> (1~ K lKc ImQ= i-(00 2 2 1/2 =

(3.81)hr +

ill1,

Kc. 3qto~ 0w~/4to~, and the growth rate turns out to be

(3.82)

(_)ii

toe.

(3.83)

C

=

Let us now discuss the steady-state solutions of the system of equations (3.73) and (3.74) in the limit 0, and ~ 9~.We then find from eq. (3.74)

26on

P.K. Shukia et a!., Relativistic nonlinear effects in plasmas

e2=

n0

4meotootopo 2 2 2

oZJEI

.

(3.84)

Substituting the last expression into eq. (3.73), one finds the nonlinear Schrdinger equation with a nonlocal nonlinearity, 2 = 0, 1Ed~JEf where r = w 2w~ 2m~ 0t, ~ = z Vgt~ a1 = 3qe 0/16w,~c and 0, /3, following integrals of motion [95]:+

i~~E + ~E

a

2E 13

(3.85) 2/4m~=

1JEI

2. Equation (3.85) has the 0w~c

e

ii

2 =

=f f

dx El2,

(3.86)

dx [J~EJ2

~a 1JEJ

4 ~/31(~JEJ

2)2],

(3.87)

which allow a study of the dynamics of arbitrary initial distribution. From eqs. (3.86) and (3.87), without taking into account the local (a 1 = 0) nonlinear term, we obtain [951, d~ x

f

2JEJ2 dx

=

812

2/3~ dx (a~IEl2)2.

f

(3.88)

From this equation, it easily follows that the evolution of any distribution with 2 24 when 4 is positive. Thus, eqs. (3.310) and (3.315) together with the appropriate ion number density expressions (eqs. (3.311), (3.312) and (3.118)) complete the description of the low-frequency response of the plasma.

On the other hand, when the phase velocity of low-frequency oscillations is larger than the electron thermal velocity, then the electron number density is obtained from+ t9z(hieLe) +=

0,

(3.119) (3.120)

vezve

=

~~(4) + 41.,,),=

where

41,, /3 /3(1 + A2)~2,~s=

m 1lm~0, and the normalizations are as above. In the stationary frame,

one gets 2 2(41 + 41,,)]~ (3.121) M~[~sM where v~ = 0, 2e = 1, A = 0 at J ~J = The case of cold electron response (eq. (3.121)) was used in the numerical study of the circularly polarized electromagnetic solitons by Kozlov et al. [60]. Substituting the expressions for ne and n 1 obtained above into eqs. (3.108) and (3.115), we obtain [64, 66] 2)2 exp[41 /3(1 + A2)~2+ /3], (3.122) /3 d~A = )tA + A(1 + A d~41 = n 2)~2+ /3], (3.123) 1(41) + exp[41 /3(1 + A where n 1(41) is given by=~.

32

P.K. Shuk!a et a!., Relativistic nonlinear effects in plasmas

n.(41) =

11~

I exp( a-41)

M(M2

241) 1/2

for interaction; for Raman Brillouin interaction; for quasi-static interaction.

Equations (3.122) and (3.123) constitute a coupled set of equations governing the propagation characteristics of circularly polarized electromagnetic solitons for three types of low-frequency responses. Note that under the quasi-neutrality assumption, they reduce exactly to the system of equations obtained by Yu, Shukla and Spatschek [59]. Before discussing the localized solutions of the governing equations, it is interesting to note some of the symmetry properties apparent in them. Firstly, both equations are invariant with respect to the transformation fs Secondly, while A A leaves the equations unchanged, no such property exists with respect to the transformation 41 ~ 41. This implies [97]that the solutions for A( ~)have both symmetric as well as the anti-symmetric structures, whereas 41(e) is always symmetric. Clearly, the symmetric (anti-symmetric) solutions will have an anti-node (node) with respect to ~ at the place of the symmetry axis. The number of maxima and minima at other places will depend on the relative magnitudes of the free parameters. In the next section, we discuss the Brillouin case. The case of quasi-static interaction is considered in section 3.5.4.~.

3.5.2. Localized solutions for Brillouin interaction While exact analytical solutions of the governing equations (3.122) and (3.123) do not seem to exist, numerical solutions may be obtained by solving them as an eigenvalue problem with appropriate boundary conditions. However, it is interesting to note that they are structurally similar to the governing equations for modulated nonlinear Langmuir waves derived by Varma and Rao [99, 100]. In the following, we obtain approximate solutions by using the method of solution developed by these authors. We note that the governing equations can be derived (for the Brillouin interaction case) from a variational principle through a Lagrangian L(A, 41, A~,41~)given by L=

~/32(d~A)2 ~(dk41)2 + ~A/3A2

M(M2

241)1/2+

exp[41

13(1 +

A2)2

+

/3],

(3.124)

yielding the constant of motion ~/32(d~A)2 ~(d~41)2 ~A/3A2 + M(M2 _241)1/2

exp[41

/3(1 + A2)112 + /3]

=

1

+

M2,(3.125)

where localization boundary conditions have been used to evaluate the constant on the right-hand side. Equations (3.122)(3.125) can be combined to eliminate the independent variable ~ resulting in the following equations for ~I A2 in terms of 41 [641: 8f3~IH d~1+ 133F(d 4 ~P) 3 213(2H+

/3G)(d~ ~[,)2

4/3~PF(d

4 ~t)+ 8~IG= 0,

(3.126)

where the quantities F, G and H are given by 2 241)~/2 + exp[41 /3(1 + 1J/)U2 +

/31,/3],/3] + (1 + M2).(3.127)

F= M(M

G = MI

+

~P(1 +

1P)2

exp[41

/3(1

+ ~J1)1/2 +

H = ~A/31P M(M2

241)1/2

exp[41

/3(1

+ .jf)1/2 +

P. K. Shukla et a!., Relativistic nonlinear effects in plasmas

33

Equation (3.126) can be solved by looking for a solution of the form (3.128) where the coefficients a~ are functions of free parameters, and are to be determined self-consistently. Note that eq. (3.128) preserves the symmetry properties mentioned above. The coefficient a0 is zero by virtue of the boundary conditions. The to eq. find(3.128) the restbecomes of a~ is straightforward: For reasons of 2, inprocedure which case convenience, one defines 0 = 41/M (3.129) where b~ = M2~a,,. Equations (3.127) and (3.129) are substituted into eq. (3.126) and the various quantities expanded in terms of powers of 0. The coefficient b~(for any n) can then be explicitly (and uniquely!) determined by equating the coefficient of 0~to zero. Some of these coefficients are listed in appendix B. Equation (3.129) provides an implicit solution of eqs. (3.122) and (3.123) for any degree of nonlinearity in 0. The explicit solutions for 0( ~)and, therefore, for 1P( ~)can be obtained by numerically integrating eqs. (3.123) and (3.129) after retaining terms up to the desired order in 0. However, analytic solutions are indeed possible if terms up to third order in 0 are retained. The equations for 0( ~) is obtained from eqs. (3.123) and (3.129) as M (d~0) = a 102 2 2 3 4

+

a20

+

a30

,

(3.130)

where as are given by a1y11, a2~(y2)y~~), (3.131) and the quantities [99, 100]0(e)= =

71,

y~ and 73 are given in appendix B. The localized solution of eq. (3.130) is given by 2[K(~~o)]I{$1(2)

/31(2/32(1

sech

/32(1) tanh2[K(~ ~)]},

(3.132)

where

2

K

(1

+

A)1f3, and f3~and [a 2~ (a~

/~2are

defined by (3.133)

/31(2) =

12]. 4a1 a3)

In the above equation, ~ is defined with the upper sign and $2 with the lower sign of the radical. In eq. (3.132), ~ denotes the initial phase of the soliton which can be taken, without any loss of generality, to be equal to zero. The choice of coefficients (/3k and /32) in eq. (3.132) is discussed below. Note that the solutions have four independent parameters, namely, M, ~,Li and /3 which determine them completely. In order to simplify the discussion of the explicit solutions, we consider a specific case where terms up

34

P. K. Shuk!a et a!., Relativistic nonlinear effects in plasmas

to quadratic in 0 are retained in eq. (3.129), that is

I= b10+ b202,

(3.134)K

where 0( ~)is still being given by eq. (3.132) except that y~ = 0 in eq. (3.131). For localized solutions should be real which implies that 1 + A >0, or,~ >

[/3(Li 1)

2M2].

(3.135)

This inequality sets a limit on the compatible values of the free parameters for physically acceptable solutions. In the small amplitude case, that is, I = b 1 0, we have~P=~(M2_1)41. (3.136)

Clearly, since ~I 0 by physical requirement, it follows that subsonic (M < 1) solitons have potential dips whereas supersonic (M> 1) solitons are associated with potential humps. Similar features were also observed by Yu, Shukla and Spatschek [59] for finite amplitude, quasi-neutral solutions. Consider the behaviour of 0(e) as given by eq. (3.132) (with ~ = 0). One immediately finds that 2(K~), 0(~0)=/32(l)sech 0(~~)= /3,(2~2(1) sech2(K~)./31(2) /32(1)

(3.137)

The sign of 0( ~)thus depends on the choice of the subscripts and also on the relative magnitudes of ~ and /32. By explicit evaluation of these quantities for typical parameter values, Rao et al. [64]find that /3~>0 and /32 0 and /~2 M~, the dip at the centre increases, reaching a maximum for certain cutoff Mach number (Mu) when ~P( ~ =0) =0. The equation determining the cutoff Mach number is obtainedfrom eqs. (3.134) and (3.138) as b1+b2f32=0. (3.141)

Clearly, we note that M~ ~ The equations

36

P. K. Shuk!a et a!., Re!ativistic nonlinear effects in p!asmas

determining the critical and cutoff Mach numbers are given in this case by b1+2b2/31=0, b1+b2/3,=0.(3.143)

The anti-symmetric solitons are obtained explicitly as 12sech(Kfltanh(K~)/[/3 2(K4)]. (3.144) A(~)=[b2f3~/32(/3,/32)] 2/31tanh In contrast to the potential-dip solitons which exist for M < 1, the potential-hump solutions are possible only for M> 1. Furthermore, the symmetric solutions in both cases have four independent parameters, whereas the corresponding anti-symmetric solutions have only three parameters. The fourth parameter is fixed by the condition that A = 0 at ~ = 0. Finally, the finite amplitude, quasi-neutral solutions can be recovered exactly by imposing the quasi-neutrality assumption on eqs. (3.122), (3.123) and (3.125). The details can be found in ref. [64]. Physically, the existence of potential-dip as well as the potential-hump solitons can be understood as follows. In both cases, the basic mechanism is due to a reduction in the local plasma frequency resulting in trapping of the high-frequency waves. In the former case, the reduction arises because of the relativistic electron-mass increase and the density depletion. On the other hand, for the potential-hump solitons, the effect of the relativistic electron-mass increase on the local plasma frequency dominates over the effect due to the associated density increase [64], and the net result being a decrease in the plasma frequency. 3.5.3. Parameter space analysis As discussed above, the symmetric and the anti-symmetric solutions for A( ~)exist for complementary regions of the parameters, say, M and 8. Specifically, for the potential-dip case, the symmetric (anti-symmetric) solutions of A (~)occur for M < M~ (M = M ~) whereas solitons with potential-hump have symmetric (anti-symmetric) structures when M> ~ (M = Mw). Furthermore, no solutions were found in the region M~0, A0 solutions discussed in section 3.5.2 are thus special cases of the more general class of solutions admissible to the governing equations. We close this section by remarking that, like in the anti-symmetric case, symmetric solutions for A(~)collapse to the null solutions in the limit A 0*0. 3.5.4. Localized solutions for quasi-static interaction As pointed out in section 3.5.1, quasi-static interaction occurs for time scales much larger than the ion plasma period when the ions establish a thermal equilibrium at a temperature T. The relevant

38

P. K. Shukla et a!., Relativistic nonlinear effects in plasmas

governing equations are

/3 d~Awhere

=

AA

+

A(1

+

A)~2exp[41 /3(1

+

A2)2

+

/3],

(3.154) (3.155)

d~41=exp(U41)+exp[41f3(1+A2)2+/3],U=

Tel T 1. The corresponding integral of motion is, 2 ~A$A2 + (1/u) exp(u41) + exp[41 /3(1 ~(df41)

+

A2)12

+

/3]

=

(1 + 1/u), (3.156)

where the constant of integration on the right-hand side has been evaluated at = The method of solution outlined in section 3.5.2 can be easily extended for the above equations. However, it is instructive to attempt a complementary expansion in the form(3.157)

where 1J = A2. Note that, like in the earlier case, the above equation preserves the symmetry properties of the governing equations. The coefficients c, are functions of the free parameters. To determine c,,, we combine eqs. (3.154)(4.156) to obtain [98] 8/3~PH(d~ 1,41) + 8G~P(d,,,41) where F=exp[41 /3(1G=

3 4/3~IF(d~,,41)2 + 2/3(2H $G)(d~,,q5) + /33F = 0,

(3.158)

+ ~f,)1/2 +~f,)~112

/3]exp(u41),+

MI

+

~I(1 +

exp[41 /3(1

~P)112 +

~],(3.159)

H= ~A/3~1(1+1/u)+(1lcr)exp(u41)+exp[41/3(1+~P)2+/3].

Equation (3.158) is structurally similar to eq. (3.126) except that the roles of ~I and 41 are interchanged. The coefficients c,, are easily obtained as described in section 3.5.2. The first coefficient c 0 = 0 because of the localized boundary conditions whereas c1 and c2 are given by c 1~/3 2[/3(1+u)4(1+A)]~, c 2 2(1+/3) ac~+bc,~/3 $(1+)16(1+A) ,

(3.160)

2) and b = ~(/32 3/3 3). Similarly all the other c,~can be explicitly and where a= 3 ~/3(1 u uniquely calculated. Explicit solutions for lI~(~) and, hence, for 41(4) can be obtained by using eq. (3.157) in eq. (3.154). This yields f3(d~W)2 =a,

~~P2 + a 2 ~j,3 + a3~

(3.161)

P.K. Shukla et at., Relativistic nonlinear effects in plasmas

39

where, a, =4(1+ A),a22[c12(1+/3)], a3=3[c2+2c12(1+13)c1+8$+8(1+/3)].

(3.162)

The localized solution of eq. (3.161) is given by (ref. [98], or eq. (3.132)) (3.163) sech2(K~)/[f32 /3, tanh2(K~)], where K =(1 + A)2 and the initial phase ~ is taken to be zero. Since /3~>0 and /32 1, the choice of the subscripts in eq. (3.163) is dictated by the physical requirement that 1~ A2 must be positive. The solution for 41(e) is then given by= /32/~1 =

c, ~i+ c 2.~2

(3.164)

By explicit evaluation, it turns out that [98] W( ~)has always a single-hump structure whereas q5( ~)has both the single- and double-hump structures, the transition occurring at a certain critical value of /3 to be evaluated by the equation c, +2c2/32=0. (3.165)

While the electron number density perturbation has the usual inverted bell-shaped structure, the ion number density perturbation has a density-dip structure. The fine structure of the latter is identical with the mirror image of 41( fl. A comparison between the two complementary expansions, namely, eqs. (3.129) and (3.157), together with eqs. (3.154) and (3.155), reveals that eq. (3.157) leads always to symmetric localized A(~),while eq. (3.129) gives both the symmetric and anti-symmetric localized solutions for A( ~) which is consistent with the requirement of the governing equations based on their symmetry properties. Equation (3.161) does possess an anti-symmetric solution which is not localized and, hence, does not satisfy the boundary conditions incorporated in the analysis. On the other hand, 41( ~)has, in both the cases, only a symmetric structure consistent with eq. (3.123). Finally, we mention that the existence of localized solutions for the Raman interaction can also be investigated using the method of solution described earlier. It is expected that the results may be qualitatively similar to those obtained for the Brillouin as well as the quasi-steady interactions. 3.6. Weakly relativistic self-trapped laser beams Let us now introduce the multidimensional effects in the study of nonlinear electromagnetic wave propagation in plasmas. In multispace dimension, the electromagnetic waves undergo self-focusing [37,38]. Specifically, when an intense electromagnetic beam propagates through a plasma, the ponderomotive force as well as the relativistic electron-mass increase can reduce the plasma frequency near the beam axis. Consequently, there appears a stationary rarefied density channel that acts as wave guide for the laser beam. The lower plasma density on the beam axis shows the phase velocity of the wave

40

P. K. Shuk!a et a!., Relativistic nonlinear effects in plasrnas

on axis, imparting to the phase fronts a curvature that counters diffraction. However, if the laser beam is strong, nonlinear self-focusing effects balance diffraction preventing the beam from diffracting out. Then, the beam can propagate in a self-trapped manner [48]. Chiao et al. [37] were the first to show that in a nonlinear optical medium in which the index of refraction is a function of the field intensity, an electromagnetic wave can produce its own dielectricwave guide that can overcome the diffraction of the beam and cause the beam to self-focus. Accounting for the ponderomotive force nonlinearity only, Akhmanov et al. [38]and others [395 1] have used the paraxial ray approximation, moment methods, or the variational principles in order to investigate self-focusing of cylindrical laser beams. Note that the paraxial ray approximation puts heavy emphasis on the central parts of the beam. While the paraxial ray theory gives a qualitatively correct picture of the variation of the beam width with laser intensity, the predictions for the nonlinear frequency seems to be incorrect [46]. According to Anderson [46], the paraxial ray theory overestimates the waveguide effect associated with the finite transverse beam width, even to the point of making the nonlinear wavenumber decrease with increasing wave amplitude. Use of the moment [44]and the variational methods [47, 50] shows that the wavenumber should decrease. If this is the case, a high-power laser beam can penetrate farther into an overdense region. For waves with infinite transverse dimension, this is a well-known result [6]. In what follows, we briefly present the derivation of the nonlinear wavenumber shift of a circularly polarized self-trapped cylindrical laser beam propagating in a nonlinear medium. The analyses are from Anderson et al. [47]. Note that the complex scalar wave equation for a circularly polarized wave in the weak relativistic limit is(V2

?

~)~~i=

~T~

~

(3.166)

where ~P= eAlmeoc2, and A is the vector potential. Assuming cylindrical symmetry and self-trapped propagation, one can write W(r, z, t) = 1I(r) cos(kz w 0t), where z is the coordinate in the direction of the wave propagation. After averaging over time and defining ~P*= (w,,0l\/~c)1P, eq. (3.166) reduces to d 2~* + r d=

~[~

F2~ +

~

=

0,

(3.167)

where p2

k2

k~,and k

0 = w0lc is the linear wavenumber. Anderson et al. [47] have investigated eq. (3.167), using a variational technique involving trial functions. According to them, eq. (3.167) is equivalent to

J

L(~, dr ~*) dr = 0,

(3.168)

where the Lagrangian is given by 2 + ~rV1*4. (3.169) L = ~r(d~~i~*)2 ~rF1P* Following the variational Ritz procedure, we can minimize the variational integral within a set of certain trial functions. On choosing=

B sech(r/a),

(3.170)

P.K. Shukla et a!., Relativistic nonlinear effects in plasmas

41

as a trial function, one can show that (L)J0=

L

~

sech(r/a) dr

0F where a 2 x tanh2 x~ = ~ + ~ln 2, ~ = (x sech2 x) = ln 2, and ~ = (sech4 x) = ~ In 2 = (x sech The 0 variational equations of (L) with respect to a and B yield aF= (a 112 ~s0.76, BIF 01/30) 2.17. Thus, the radial intensity profile~.

~a0B

2 ~/3

2a2B2 +

(3.171)

1~1i*(r).217Fsech(rF/076)

(3.172)

turns out to be in good agreement with the numerical solutions of eq. (3.167). Secondly, we take the conventional Gaussian trial function, i.e., 1I*(r) = B exp(r2/2a2). For (L) one finds then (L)=

(3.173)

~B2 ~F2a2B2+ ~B4a2.= 4p2

(3.174) and a212=

The variational equations now give B2 ~Ir*(r) = 21 exp(12r2/2)

1. It emerges that the resulting Gaussian (3.175)

is indeed an approximate solution of eq. (3.167). 3.7. Three-dimensional relativistic solitons [36] In this section, we shall consider the multidimensional propagation of arbitrarily large amplitude electromagnetic waves. Here, the circularly polarized waves satisfy the complex scalar equation A (3.176) (1 + e2A2Im~ 2)112 0c Let us now look for the steady propagation of the wave in the frame z vgt. Then, eq. (3.176) becomes(V2_~~~ o2)A_toPO

c

c

~ d~W+ and

~

T~Y (1+~I2)112

(3.177)

(3.178) where b 2 =[1(k 0clw0)2 1

]

42

P. K. Shuk!a et a!., Relativistic nonlinear effects in p!asmas

21w From eq. (3.178), one observes that v~ = k 0c 0, as it should be. Introducing a new variable z = bz, we can rewrite eq. (3.177) as 2 (V where V2=

+

Q2)~I = V~ +

c

2

(1+11)Q

2 1/2

(3.179)

0/bc. Equation (3.179) is of the form of a nonlinear Schrdinger equation. The solution of the latter can be found by noting that ~I is a solution of the usual Schrdinger problem with a potential 2)(1+ ~I2(r))2] , (3.180) V(r) = ~[(to~0/c satisfying[~V2+ V(r)]~P ~Q211I.

a ~, and

=

w

(3.181)

Pulse-like solutions will exist, corresponding to the bound-state solutions of eq. (3.181). Hence, the bound-state regime is given by Q2> 0 which implies that the group velocity of the wave packet is smaller than the velocity of light. For S-states, V(r) remains spherically symmetric and the equation may be simplified to2 2 2 (~10

d~,U0+QU02 where U

U 2)2 , 0 (1+U~lr

(3.182)

0(r) = rI, with U0(0) = 0 = U0(oo). The wave energy is given byw~Q (t~)2J ~2

dr.

(3.183)

4irk0

e

Equation (3.182) has been numerically solved by Gersten and Tzoar [36]. According to them, three-dimensional localized solutions can propagate for 2)12 , (3.184) k0c ~ to0 ~ (w~0 + k~c and beyond a critical energy which is given approximately by~cr60meoc5~2to~o.

(3.185)to,, 0 -~ 101651,

As an illustration, we take

and find ~cr

525 erg.

3.8. Relativistic self-focusing

In this section, we consider the stationary self-focusing of an axially symmetric high-frequency circularly polarized electromagnetic Gaussian beam in a collisionless electron gas. The distribution of the

P.K. Shukla

et al., Relativistic nonlinear effects in p!asmas

43

beam intensity is taken to be 2), (3.186) E . E*15,0 = E~ exp(r~ Ia where r~= + y2, and a is the initial effective beam width. The wave equation for the complex amplitude of the electric field is given by V2E+~[1_~!~

~~~_. 1]E=0, (3.187)

2E E* Im~ In 2. an electron gas, the ponderomotive force driven electron number density where I = e 0to~c ~eis given by [61]=

no[1

+ ~2

V2(1 + 1)1/2]

(3.188)

where the ions are assumed to be immobile and the radiation pressure is assumed to dominate over the electron thermal pressure. Inserting eq. (3.188) into eq. (3.187), we find [51](v2+~)E=~0~+~v2vi-:c1.

(3.189)

We now seek a solution of eq. (3.189) in the form E= E 0(z, r1)[~ cos ~(z, r~) + ~ sin ~(z, r1)] 2f2(z)j,=J

(3.190)

k~(z) dz + ~r~O(z), I = I~(z) exp[r~Ia

and study the relativistic self-focusing under the geometrical optics approximation (r~ 4 a2f2). Here, f(z) is a dimensionaless beam width. Substituting eq. (3.190) into eq. (3.189) one finds for the axial region [51]2 2 2

to 0

ki(z)=j2

2~

~+j,,

1

w,,0 1 20 ~ \/1+I~~ =d~ln(ki1I0), a4f4 + j)2 1~ (1 4+I~ a4f4 1

2

k11 d50

+

0

~

2c

2 a2f2 j)3/2 1 (1 + I~ [0 k

0,

(3.191)

d~f + (d~0) d~(ln 1~)

11 d~(ln f)]

=

0.

The boundary conditions are as follows: 10(z = 0) = 100,f(z = 0) = 1, w~0(z = 0) = w~0,and 0(z = 0) = 00. 2a2f2(1 + Io)3!2 is caused by the relativistic In the second equation of the set (3.191) the term w~010I2c mass variation while the term I~(4 + 1~) 1a4f4(1 + J~)2 represents the nonlinear electron striction. For

44

P.K. Shukia et a!., Relativistic nonlinear effects in plasmas

narrow beams (a -= 10 to bOA), these terms are of the same order. The strictional term becomes dominant in the focal region. On the other hand, for sufficiently broad beams (a 2~100A) the strictional nonlinearity is insignificant as compared to the relativistic mass variation nonlinearity. The latter plays an essential role in determining the minimum dimensions of the channels. In this case, one [43]observes that in the ultra-relativistic limit (1~ ~ 1), one has k(z)~k0 and the focal length is found to be ~a(V0/c)2(w0l w~0) ~1 a, demonstrating that of the beam undergoes quasi-channeling with f~ /2( A la)23 and the length thegeometrical channel is (V,,a/cA)23L 3 L; L a is the characteristic scale length 2(101c) of the inhomogeneity. The absorption of the beam energy in the channeling region (to,, 0 4 to0) is [43]

Q

1

I I ,..L \i av0 \ exp[_l,~__)(\j)

r

,

~i

,,

2/3

(3.192)

where ~ is the collision frequency. Clearly, the relativistic nonlinearity may be responsible for the enhanced absorption [101]. Garuchava et al. [51] have numerically investigated the set of equations (3.191). In what follows, we briefly summarize their results. Reference [51]imposed the following restrictions in order to preserve the accuracy of the numerical calculations: k11(z)a10,neIrO=

f0.1, 2

zthe is the initial position, 00 is the initialin phase. It is seen from eq. exist. (3.211) 1 +amplitude, 1 I2cx, the x0 compressional (~n >0) and envelope solitons the supersonic regime that if V We now discuss the two-dimensional stability of the supersonic soliton solution given by eqs. (3.211) and (3.212). Since we are restricted by the consideration of those perturbations that are slowly varying in both space and time variables, we can use the method in ref. [54].We then assume the perturbed soliton solution in the form E=V~A[2a(V21)1]2g(~)exp[i0+i~V(xx=

2[2a(V2 6A

1) 1]1f(~),

0)],

(3.213)

(3.214)

with=

~A2 + ~V2,

a,x 0=

V,

~ = A(x

x0).

(3.215)

We consider the perturbations oblique to the propagation direction, that is, x-axis. By introducing new variables n and 41 = arctan( y Ix), we consider only those solutions with fixed 41 being functions of n and time. For example, the perturbation of 41 = i~are perpendicular as has been discussed previously by Wardrop [54] and Hojo [54]. Dividing A, V, g and f into the unperturbed and perturbed parts as A=A0+a(n,t), V=V0+0(-q,t),f(~)=f0(~)+f,(~),

(3216)

g(~)=g0(~)+R(~)+iI(~),

2(~), and substituting eqs. (3.213)(3.216) into eqs. (3.209) and (3.210), with g0 = sech(~) andf0 = equations sech we obtain three linearized with respect to the perturbed quantities a, 0, R, I and f,. Following the analysis of Hojo [54],we consequently obtain the coupled equations for the perturbed amplitude a and the perturbed velocity 0 as a~a+V 0cos41a~a+qA~o~a ~ ~ (1 ~8)V0cos41a,~,,0=0, (3.217)

482i

P. K. Shuk!a et a!., Re!ativistic non!inear effects in p!asmas

6~

2

~

A

Al

~

6A0V0(1

+

~)a~o + [.~1o 24

A0

6Ao]v cos 4th~,,0 = 0,

(3.218)

where 8 = 2aV~ 1 (>0) and it is assumed that V~ ~ 1 and a 4 1. For the perturbations in the form exp(ik~ itot), we obtain the algebraic equation of fourth-order with respect to Q(= w/A0k):

AQwith A=

4+Ei~13+d2+bQ+E=o,

(3.219)

8/A + 12Al6+

18A,

B = 2v(61A C D=

12AI6

12A), 27A) +

6(8/A

1/A

36A18

,2(61A + 12A15 6A

9A6),

6v(6/A1IA9A36A/6),

E=54(1+ ~6)lA. Here, A = A 0/V0 and i. = V0 cos 41/A0. For the perpendicular perturbations (41 = ir/2), eq. (3.219) can easily be solved since B = D = 0, and simple expressions of growth rates in several limiting cases can be obtained. For highly supersonic case (i.e., 6 ~ 1), the growth rate is given by-

Im

to

3A0k,

(3.220a)

in which case the amplitude perturbation dominates in driving the instability. For low supersonic case (i.e., 8 0), the growth rate is proportional to the wave amplitude A0 (for small A0), and saturates at 2V Im to ~8 0k (3.220b) at large A0, where the velocity perturbation dominates. These features are in good agreement with numerical results of eq. (3.219) for 41 = ir/2 shown in figs. 4 and 5. On the other hand, the numerical results for the case of stability of oblique perturbation for fixed 41 are exhibited in fig. 6. It is seen that the growth rate for these perturbations is reduced as compared with the perpendicular one. In conclusion, the supersonic Langmuir solitary wave is unstable against the slowly varying perturbations oblique to the propagation direction. The growth rate has a maximum for the perpendicular perturbation (41 = uI 2). For this case, the growth rate increases with the wave amplitude in highly supersonic case (8 1), and saturates for large wave amplitude in low supersonic case (6 0).~

3.11. Wave breaking In the present section, our approach differs to some extent from the preceding discussions of the

P. K. Shuk!a et a!., Relativistic nonlinear effects in plasmas

49

102

icr2

1

6

io2

ic.f4

102

1

A

0

Fig. 4. The growth rate vs. S (=2aV~ 1) for the perpendicular perturbations (~ = ir/2). (a) a =0.02,A0 = 0.1; (b) a = 0.002,A0 = 1; (c) a=0.02,A0=1.

Fig. 5. The growth rate vs. A0 for the perpendicular perturbations. (a) a = 0.02, 5 = 10; (b) a = 0.002, 8 = 0.01; (c) a = 0.02, 5 = 0.01.

relativistic modulational instability, soliton formation, etc. Here, we present a summaryof what is known as the relativistic wave breaking [25] which takes place before the parametric processes develop significantly. In particular, the spatial variation of the local frequency of the wave causes the wave breaking of both the free and mode-converted waves. The phase of the wave advances more rapidly in the high-frequency region so the electron fluid in this region overtakes the fluid in the low-frequency region, causing the wave to break. The increase of the electron mass in the high-intensity region of the wave produces the frequency variation in the free wave while a combination of these effects and the spatial dependence of the plasma frequency in a density gradient causes the frequency variation of the mode converted wave. It appears that one cannot analytically follow the evolution of the wave after it has broken. However,

hflWiIi~N\\\\~~

0Fig.

2

4

6

V0cost~/A0

6. The growth rate vs. i 1, eq. (4.20) becomes 2). 0(X M Note that for a(X M2) >0 ( X and a >0, eq. (4.23) predicts spatial collapse in one space dimension. 4.1.2. Magnetohydrodynamic modulations Let us now introduce the magnetohydrodynamic response [95]to the upper-hybrid waves. Assuming cold ions (T 1 4 Te) and plasmas with to~, Q, one finds for the low-frequency (~1 4 (1k) driven MHD perturbations [87]~

(a~ v~a~)~

=

~-~

v~a~

,

(4.24)

n0 where2Va

B()=

~

X

3E~I

425

VA +

2

2 ~ c~, ~~e = ~n1 = ~n, and p;a~ 4 1 has been assumed.

P. K. Shukla et a!., Relativistic nonlinear effects in plasmas

57

In the stationary frame ~ = x top0_[El2 2

Vgt~eqs.

(4.24) and (4.25) become

B0

(4.26)v~3E~

~

to~o

~

V~ V~ ~

El

2+

~2

2

El2C

i- a~ i

.

(4.27)

Inserting eqs. (4.26) and (4.27) into eq. (4.3), we get2

2ito

0a~E+ aa~E + (to~0+ 2(l~~)[3 m,(v~v~)1 m~0c ~ 3E~E

11~o3E~ ~ cE a~j El

2 = 0.

(4.28)

Equation (4.28) is a nonlinear Schrodinger equation. For 3O~~ ~ w~ 0,eq. (4.28) becomes 211e0 I m~ 2 1 El 2ia~E ~ ~pO PeE 2 2 ~ i3 2 0c 2 I E=0. eO L m.(vgva)J E~~-

(4.29)

This equation has soliton solutions provided that~ 2 Va)

2

~ eq. (4.28) isto,, a 2ia0E+ ~ 0 ~

2E+ ~ [~,_ m~ 22 ] El2 E ~- ~ a~lEl2=0. 0c 3 m1(v~ Va) E~ f1~~ 3~

(4.31)

2/mj, the relativistic nonlinearity dominates over the ponderomotive nonlinearity, If 3(v~ v~)> meoc and we find a compressional (~n >0) cusp-shaped soliton (a~lEl2 >0). Furthermore, for 3v~ 3v ~+ meoc2/mj, the relativistic and ponderomotive nonlinearities cancel each other. We find then2 2

2ia

0E +=

to~, ~-

0

a~E

~-

2 eO ~ (12 3E

2lEl2 0. ak=

(4.32)

Letting E P

g(x) exp[iq(~) + iO(t)] in eq. (4.32), and assuming stationarity, one finds

0a~gB0gq0ga~g=0, where P0=

(4.33) 2],and q2If1~ 0=

3v~/2to~0, B0

=

[0~ + P0(a~)

w~0c

0E~.

58

P.K. Shuk!a et al.. Relativistic nonlinear effects in plasmas

Equation (4.33) can be integrated once, and the result is+ IP(B (a~g)2 0,

g, P0, q0)

=

0,

(4.34)

where 21(P 2). (4.35) 1= B0g 02q0g In deriving eq. (4.34), we used the boundary conditionsg, a~g-+0 at l~l~~*ca. A solution of eq. (4.34) is [95,96] g2=

g~ sech2[(B 0/P0)

12l~l

+

(1 g2Ig~)2],

(4.36)

where g~ = P 2 ~ g~.Thus, there occur super-magnetoacoustic compressional envelope and 0 ~ g solitons with 01 a 2q0, cusp-shaped electric field profile. Berezhiani and Paverman [87]have presented some examples for the width and the amplitude of the solitons when the upper-hybrid waves are modulated by the magnetohydrodynamic perturbations. They have chosen some typical low-density (n 3 cm3), high magnetic field and high temperature 0 = iO (B 0 = 3050 kG, and Te -= 100 eV1 keV) plasma and have found that the narrow wave packets (cusp solitons) would have the filling wavelength Am 4 x i0~cm, the width L 2 x 10-2 cm, and the characteristic upper-hybrid wave electric field E - 10 V/cm. On the other hand, for the broader wave packet (L4~V. to~0c/Q~0), the wavelength Am is in the interval 1 cm> Am >0.1 cm, and one finds that (LE) -~ i0 In order to investigate one dimensional collapse, we write eq. (4.32) in the familiar form-~

iarE + a~E p

0Ea~lEl

2

=

0,

(4.37)

where r = P 0t, and /3o = q0/P0. Equation (4.37) is exactly of the same form as eq. (3.85) with a = 0. Thus, the results of section 3 are applicable here. 4.2. Nonlinear ordinary mode propagation The ordinary (0) mode electromagnetic waves are being utilized for electron-cyclotron resonance heating of magnetic fusion devices, as well as for the ionospheric modification experiments. At high-power levels, parametric instabilities set in. For example, recent work [107] has conclusively demonstrated the occurrence of a three-wave decay interaction during the ionospheric modification experiments. The ordinary electromagnetic wave is characterized by the alignment of the wave electric field in the direction of the equilibrium magnetic field B0~. The wave magnetic field will then be in the y-direction. For purely perpendicular propagation, the linear dispersion relation for nonrelativistic ordinary waves in a warm plasma is given by [108] 2c2/w2 = 1 to~,e/to2 W~eA/to(to (1), (4.38) k

where A = I 1(b) exp(b), b = ~ Pc= t~teL

0eis

the electron Larmor radius,

to

is the wave frequency,

P. K. Shuk!a eI a!., Relativistic nonlinear effects in plasmas

59

+ k2c2 which does and k(=kI) is the wavevector. For b 4 1 and to ~ (la, eq. (4.38a) becomes to2 = not contain resonance and to the lowest order does not even involve B 0 explicitly. to,,~, According to the linear 2 ~ w~,the mode will propagate (k2 >0), whereas as to approaches the propagation of theory, for is to cut off. On the other hand, the linear propagation and absorption of electromagnetic the mode radiation near the electron-cyclotron harmonics can, even in a weakly relativistic plasma, be strongly affected by the velocity-dependent relativistic correction to the electron mass [109]. Let us now derive the nonlinear evolution equation for the ordinary mode radiation (to ~ propagating perpendicular to the external magnetic field B 01. Since the electric field of the 0-mode radiation is polarized along B0, the first-order density perturbation associated with the mode is identically zero. However, due to the v0 x B1 (B1 is the wave magnetic field, and v0 is the quiver velocity in the electromagnetic field) force, there appears a second-order density perturbation which depends on the strength of the external magnetic field. The second-order density perturbations are computed in the following manner. Let the wave electric and magnetic field be

E=fEcoso,

B1=9~Ecosw,

where

~

=

tot

kx. The lowest-order quiver velocity is given by (4.39)

=

meoto

eE~. z s~n~,

whereas the change 6 v ~ of the quiver velocity due to the relativistic mass variation is3~ =

sin3

~

~,

(4.40)

e~to)

The second-order electron density perturbations are determined by

a

2~In 0(6n~ + 0)axvex = 0,e= ~

(4.41)]

Ii XB0

.~

c 2~.

+

6E~j (1~0v~~,

(4.42) (4.43) (4.44)

atvey

=

(leoVex,

= 4ue 6n~ Combining eqs. (4.41) with (4.42) and Fourier analyzing, we obtain the second-order density perturbation

(2)

cka2

2 2

2

0 n02

cos2~,

(4.45)

4w

to~0

where a0

= eE/m~0toc and w~,0 = to~0+ (1~O. The parallel component of the nonlinear electron current density which is in phase with the wave

60

P. K. Shuk!a et a!., Relativistic nonlinear effects in plasmasis3 2322

electric field.NL j.I

3n0e ~ eE \ . n0e E c k =--t\ , sin~+ s1nq~. 2 2 8c meow / 2m~0to(4to toH~)

(4.46)

Perturbations which are out of phase shall be ignored. The wave equation then becomes2

22

2

12

2

21

c a~ + to~0)E = 2to~OqBaOE wpone/no, (4.47) where n ~ is the low-frequency electron density fluctuations caused by the ponderomotive force, andqB=4(to3 2

By using the WKB approximation, E = E(x,t)

to~0)/(4to to~0).

2

2

2

exp(ik0x

ito0t) + c.c.

,

(4.48)

where E(x, t) is the slowly varying (a0E 4 to0E) amplitude of the modulated wave, we obtain from (4.47) a nonlinear Schrodinger equation 2k 2a~E 2ito0a,E + 2ic 0a~E + cw~ 0~

n0

E+

q~

m~0w0c

=

0,

(4.49)

2k~has been used. where w~ = to~0+ c 4.2.1. Slow electrostatic response [1101 (A) Quasi-static limit. When the parallel phase velocity of the low-frequency modulation is much smaller than the electron and ion thermal velocities, then, from the z-component of the particle momentum equations, we findo~1~41pe,

(4.50)

(4.51) where P = e41/Te, E141pe =

V41, a-

=

T~/ T~, and the ponderomotive potential is given by(4.52)

=

(to~oIto~)lEl2I161TfloTe.

In deriving (4.50) and (4.51), we have assumed an isothermal equation of state for both the electrons and the ions. The ponderomotive potential originates from the averaging (over one period of the 0-mode radiation) of the convective term (the z component of the nonlinear Lorentz force is identically zero) in the z component of the electron momentum equation. Here, as well as in the following, the contribution of the ion ponderomotive potential is generally small, and is, therefore, neglected. On using the quasi-neutrality condition (ne = n,1), we get

P.K. Shukla et a!., Relativistic nonlinear effects in plasmas

61

T

1E216un 0T(4.53)

~

where T = Te + T1. Note that P is established by the wave pressure only when T1 Eliminating ~Ifrom (4.50) and (4.53), we obtain 2/16irn = (to~0/to~)lEl 0T.

0.

(4.54)

(B) Driven response. In the cold plasma, the plasma response can be driven nonlinearly. If the modulation frequency (Li) is much smaller than the electron gyrofrequency (1eo, the slow electron response is given by

a0n~ + n0a1V~~ = 0,2I

(4.55)2T ~

C~e00

7?

~~pO 2 Pe~xt 2~2 E (/~U;~(~ 1~to0 iolTfl0i

e

where the second term on the right-hand side of (4.56) comes from the nonlinear Lorentz force (V~ZB~. On the other hand, for (1~ we have

a0n~+n0a~V~ =0,= ~

(4.57)(4.58)

a0a~41.

From (4.57) and (4.58), we get the slow ion density perturbation n~/n0 = (c/B0I11)a~41. Inserting (4.56) into (4.55), eliminating condition, we obtain=

(4.59)

41 with the aid of (4.59), and making

use of the quasi-neutrality (4.60)

(to~o/to~)p~a~lEl

2/16unoTe,

where m~ 0/m1 4 1 has been used. (C) Finite frequency density modulations. First, we consider the slow plasma motion on a (fast) time scale such that the ions do not find time to respond to the wave field. Thus, treating ions as an immobile fluid, we obtain2 2

(a~ +

to~, 0

3i~ea~)n0

n

to = ~0 2Vtex

16 E T irn0

(4.61)

where the right-hand side is the contribution of the nonlinear Lorentz force.

62

P. K. Shuk!a rt a!., Relativistic nonlinear effects in plasonas

Second, for u1~~ (1~~e0 one may introduce a slow plasma response on the time scale of the lower-hybrid oscillations. Here, the slow electrons are magnetized and move in a plane perpendicular to B0. The electron density perturbation is given by 1 (1p~a~)~ n Bflo

a~41+~p~a~ E TeQto 0

2

-

(4.62)

uno

For (1 ~- (li, unmagnetized ions follow a straight-line orbit. Thus, combination of the ion continuity and momentum equations gives (a~

V~1a~)

I fl~

e

~ =

a~41.

(4.63)

Combining (4.62) and (4.63) with the Poisson equation a~41=4u(n~n~), we obtain1

(4.64)

2=

22

+

toLH

V~a~) ~fl0

2

Vte

to0(1

~2/m

+

v~a~)a~ El

+ Lieoitopo)

un0

,

(4.65)

where toLH = (l~~Q~/(1 + (l~0/to~0), to~ = 4urn0e 1, V~, = a1T1/m1, and p~a~ 41. Third, we consider the low-frequency electrostatic response such that the parallel (to B0) phase velocity of the modulation is much smaller than the electron thermal velocity. Thus, the electron density perturbation is given by eq. (4.50). The slow ion density perturbation is determined by means of the following equations:

a0n~ + n0V1 v~+ n0a~V~ = 0,~2 2 1

(4.66)

e

T

I

~i

e

T.

fl1

I

-

(4.67)

a0V~

= -~-

~

~.

a2

.

(4.68)

Combining (4.66), (4.67) and (4.68), we readily get 2)n~/n + 2~ Q~a~), (a~ + Q~)(~~ V~V = 0 c~(a~V where c~ = Telmi and V2 =V2~.+ a~. Eliminating P from (4.50) and (4.69), we finally obtain

(4.69)

[a~+ ((l~where V~ = C~+

V~v2)a~ Li~c~a~]n~In=

+ 2(l~a~)lEj2/16un

ac~(a~V 0

0Te,

(4.70)

V~and

a = w~0/to~.

P. K. Shukla et al., Relativistic nonlinear effects in plasmas

63

a~~ a ~,we get

Two kinds of responses across the external magnetic field can be derived from (4.70). For the driven ion-cyclotron oscillations

a~ ll~, and-

(a~ + l1~

2/16irnoTe. v~a~)n~/n = 0 ac~a~lEI

(4.71)

On the other hand, for

a~ ll~ and a~ ~ a~, eq.~ =

(4.70) reduces to (4.72)

2/16unoTe. 0 ac~a~lEl Finally, we note that for a 0o. 0, eq. (4.70) reduces to eq. (4.54), provided that y1

(a~ t~a~)n/n

=

1 (isothermal ions).

4.2.2. Slow magnetosonic responseHere, we consider driven magnetosonic oscillations in the presence of the 0-mode radiation. For the magnetosonic waves, E = (0, E, 0), B = iB2 and k = 1k. The frequency of the oscillation is taken to be much smaller than the ion gyrofrequency l1~. Thus, finite density perturbations are created due to the E X B0 drift. For low-frequency (Li 4 ll~) magnetosonic modulations, one can neglect the displacement current in eq. (2.4). The y component of eq. (2.4) is then given byaXBZ=

4irn0e

(V~

vt).

(4.73)~ 1

In the drift approximations=_____

(I a0 I ~ a

fl~), we find2

ey

c B0lleo ~

2

(1

V te

lfl E I e + Ct) pe xL n0 ~ l6un0Te

,

(4 74

= ~

a~E~ +

aX

to

2~

~ aX l6lTfl 0Te (4.75)

From eq. (2.3) we have

a0B~= ca~E~.Combining eqs. (4.73)(4.75) we obtain2 2 2 2 2

VA

to0

a~ 16 E

irn0 e

T

~

~

,

(4.76)

where b = B~/B0, v~ = B~ /4irn0m1, 1~,,= v~/V~, n, = n~, and the nonlinear Lorentz force acting on the ions is smaller by a factor of meo/ m1. For p~a ~4 1, the electron density perturbation is given by 2/16irnoTe. (4.77) n~/n0 = b ap~a~IEI

64

P.K. Shukia et a!., Relativistic nonlinear effects in plasmas

Eliminating b from (4.76) and (4.77), we obtain 2(a~ v~2a~)]a~lEl2/16unoTe, (a~ v~,a~)n~/n,, = ac~[1 + A

(4.78)

where v~ = (1

+

0 is the collisionless skin depth. For short-wavelength perturbations, the dispersion as well as nonlinear terms in the dynamics of magnetosonic waves can become important Equations (4.49), (4.54), (4.60), (4.61), (4.65), (4.71), (4.72) and (4.78) are the fundamental coupled equations which describe the nonlinear interaction of the ordinary mode radiation with a great variety of low-frequency oscillations in a magnetoplasma. We note that the structure of some of the present coupled equations is similar to those found earlier in connection with the study of nonlinear effects at the upper-hybrid layer.-

f3~)v~ and A =

c/to,,

4.2.3. Stationary localized solutions (A) Quasi-static cases. Combining (4.49), (4.54) and (4.60), we obtain it9~E+iVgaxE+Pai~E+ QIEI

2E+RIEI2E0,+ 2RIEI2E=

(4.79)

ia

0E + iVg a~E + P~~E q0Ea~E~

0,2=

(4.80) c2/2w 0, 0/w0, P =V~I

where the electric field is normalized by (16unoTe)2. We have defined Vg = c2kQ=

to~/2to0(1 + a-

), q0

=

(top0/2to0)pe, and R

=

topoqBVte/tooc.

(B) Propagating solitons. Introducing ~ = x Vt, we find from eqs. (4.61), (4.65), (4.71), (4.72) and(4.78)

2 3v~e)a~ + w~.,]N = av~~a~EI2 , [(V [(V2 v~)a~ + to~~]N = 6[(V2 V~)a~ + w~~]a~lEl2 ,

(4.81) (4.82)

[(V2 V~)a~~ + Q~]N ac~a~El2, N ac~lEl2/(1 M~)v~2,

(4 83)(4.84)

(M~ - 1)N

a

~-

[i

+ A2(1

-

Y~)a~]lEl2,

(4.85)

where Vis the soliton velocity, M~ = V/v 5, Ma = V/Vm, and 6 = V~/w~(1 + Q~0/to~0). 2 ~ v~,~ v~[(V2 3V~)/w~.,]a~ 41; [(V2 V~ 2 V~)/Q~]a~ 41, For V to (4.83) yield 1)/to~~]a~ 41; and [(V eqs. (4.81)

N=

A

N= B

0a~lEl

2,

(4.86) (4.87) (4.88)

N= C where A()=

0a~lEl0a~lEl~ B

2,

2,=

0

6to~IIw~H, and C0

=

ap~ with p5 = c5/u11 and p14

=

P. K. Shuk!a et a!., Relativistic nonlinear effects in p!asmas

65

On the other hand, for (1 N

4 1, eq. (4.85) gives V2Iu~5jA2a~

a(c~/V~,)lEl2/(1 Mt).

(4.89)

Inserting (4.84) and (4.89) in eq. (4.49), we obtain

iawhere

0E + Pa~E + (R + Q)IEI

2E = 0,

(4.90)

Q 5=

Q(T, Q(T1

=

0)c~/(1

M~)V~,

(4.91)(4.92)

Qa

=

=

0)c~/(1 M~)v~

are the coefficients of nonlinear terms associated with the ion-sound and magnetosonic perturbations, respectively. Here, V= Vg~ we insert eqs. (4.86)(4.88) into eq. (4.49), we then get 2EIf q 2 = 0, (4.93) ia0E + Pa~E + RIEI 1Ea~lEl where p~q 1 = (q0A0, q0B0, q0C0) are the coefficients of nonlinear terms involving the upper-hybrid, the lower-hybrid, and the ion-cyclotron modulations, respectively. Note that eqs. (4.79), (4.80), (4.90) and (4.93) are similar in form to that of eqs. (3.49) and (3.85), respectively. Correspondingly, nonlinear solutions are given by eqs. (3.61) and (3.89). One encounters subsonic (Alfvnic) envelope solitons when the modulations are due to the ion-sound (magnetosonic) waves. The corresponding nonlinear density perturbations are rarefactional. Equations (4.80) and (4.93) admit sharply localized radiation electric fields.

4.3. Ultrarelativistic cyclotron waves The problem of supplementary heating of plasmas by electromagnetic waves in the regime of electroncyclotron resonance frequency is of much interest recently. It was pointed out some time ago that the relativistic modification of the electron motion can change the character of the cyclotron resonance. In the field of an intense circularly polarized wave, the so-called gyromagnetic autoresonance may take place [78]. It has been shown that for a sufficiently smooth increase of the external magnetic field with time, the electrons are trapped in the regime of the autoresonance and are accelerated 2). with the increasing magnetic to ultrarelativistic energies (much larger thanet the energy meoc On the other hand, it has beenfield pointed out by Kotsarenko [73]and Papuavishvili al.rest [75]that two types of circularly polarized electromagnetic waves with different refractive coefficients may propagate in such ultrarelativistic collisionless cold plasmas. The purpose of this section is to discuss the linear as well as nonlinear properties of the field-aligned ultra-relativistic cyclotron waves. We consider a homogeneous collisionless cold plasma in the presence of a circularly polarized electromagnetic wave of frequency to and wavevector k directed along the external magnetic field B 0~. Using Maxwells equations and taking into account the relativistic electron-mass increase, it is easy to obtain the following expression [7279]:

6622= =

P. K. Shuk!a et a!., Relativistic non!inear effects2

in plasonas

~

Q)

(4.94)

where to~ = 4irn1e2/m10 and 11g. = q,B0Im10c. The relation between and the pump wave amplitude E0 is given by the equation of motion for the electrons and Amperes law [79]Yl+toEJYJ/(WYJt2 2 2

1J)2 ,

(4.95)

where toE

=

10c. Note that (4.94) is valid for kt~1.4 toy, l2~.in a uniform plasma. Here, signs in front of (1~correspond to the right- and left-hand circularly polarized waves E = E0(1 ij) exp(itot + ikz). The system of equations (4.94) and (4.95) describes the dependence of the wave refractive index and the particle energy ~i on E0 and B0 at fixed values of the frequency to and the plasma density. Relation (4.95) represents a fourth-order equation for ~,iand it can be solved in general numerically. However, in the ultra-relativistic limit (y1 ~ 1), it takes a simple form y1~=(~Q1 toEJ)/w~1. On the other hand, in the weak relativistic limit, y~ in (4.95) becomes=

leEo I /m

(4.96)

1

+ to~1/(to (1)2

(4.97)

The corresponding dispersion relation is given by2 = to

k2c2

+

~~P1Q,

2

-

(4.98)

4.3.1. Ultrarelativistic waves Consider high-frequency (u1~4 to 4 Li~) intense waves such that the electrons remain ultrarelativistic and the ions form a neutralizing background. Then, eqs. (4.94)(4.96) yield k2c2/to2 = 1 ~to~O/totoEe~

(4.99)to:

Equation (4.99) can be solved forto = (to~o/4to~ +

k2c2)2

to~o/2toEC.

(4.100)

Clearly, in the ultra-relativistic limit, there exist two types of waves with different wavelengths [A 2]but with same frequency and polarization. For 2kctoEC 4 to~ 0= (2irc/to)(1 ~ to~o/totoEC) 0, eq. (4.100) gives two roots/~

,22

~

+

~~

to~)

toEe

to~,0

,

(4.lOla) (4.lOlb)

2c2w~/to~ k 0 -

P.K. Shukla et a!., Relativistic nonlinear effects in plasmas

67

Expression (4.lOla) was first found by Papuashvili et al. [75],whereas eq. (4.lOlb) appears in ref. [73]. There are several methods [79]to reach the state with y~ ~ 1 (the main difficulty is a compensation of the electron-mass increase and hence of cyclotron resonance detuning). One possible way is by choosing the incident waves in the form of a broad band (in to) wave packet with frequencies continuously changing from tomin to tomax = Li,,~,(at constant B0). Then, the electrons with relativistically increasing mass can find a corresponding wave from the packet with a frequency coinciding with the relativistically decreased 2. Thus, since electron-cyclotron frequency. Asreach a result, the electron energy may reach (wmax/tomin)meoc tomax> tomin~ the electrons may an ultra-relativistic state. However, this method is impractical because of the smallness of the energy gained in comparison with the energy carried by the wave packet. The method suggested by Golovanivsky [78]seems to be more effective. Here, one varies the external magnetic field strength. It may either be a variation with the coordinate (in this case it is necessary to produce plasma flow along the increasing B 0) or with the time (for example, from discharging a capacitor bank through coils). The increase of B0 is followed by a corresponding growth of y~, so that the ratio 11e0~Ye remains constant and a continuous resonance can occur. Papuashvili, Paverman, and Tsintsadze [79] studied this situation using (4.94) and (4.95) for an electron plasma. In particular, all the real solutions of (4.95) were calculated (in general, there are three real values of ye), and the corresponding n2 were obtained from (4.94). The dependence of Ye and n2 on the varying magnetic field is shown in fig. 9. If Bmin is smaller than B 1, where (4.102a) meoc ~to

then an increase of the magnetic field gives rise to a growth of y~,and the wave ~ (1) propagates in a plasma. When B> B1, this wave is shielded (its refractive index becomes imaginary). When the magnetic field reaches the value B2, where

leB2/m~ocl

213= ii~~ = to2/3 +

~

,

(4.102b)

the second wave w2) appears. Clearly, the inequality B 1 > B2 is a necessity for the simultaneous propagation of both waves. We note that the wavelengths of the ~ and ~ waves are very different.

10a/7

///

~2

b,.-

245

16 0.8

I

0

5

:~~:

__) ~~21 mode; 2/2 = 7 x 10_2, WE/W = 5 x 10_2. [79] Fig. 9. Typical shapes of ultra-relativistic dependence of y,(a) and n(_

2 (b) on the magnetic field:

()

nonrelativistic branch,

(.

. . )

w~~ mode,

68

P. K. Shukla et a!., Relativistic non!inear effects in p!asmas

b

~e

2~I

0

W~/W

/(.-)

Fig. 10. Typical shapes of ultra-relativistic dependence of y~(a) and n mode, ( ) w~mode; w~/w2 =7 x ~ Q~_Iw = 3. [79]

2 on the pump-wave amplitude:

nonrelativistic branch, ( .

.

.) ~

For example, in a plasma with n 3, Bmax = i0~ G, E 3 V/cm, and to = 2 X 1011 s~, the = iO cm = 3 X i0 wavelengths of to~ and ~ 0 modes are Aw=25cm and 0A~2~=7cm, respectively. In spite of the relatively small E 0, the magnitude of Ye was found to be quite large (Ye =9) due to the resonance interaction of the electrons with the pump wave. 2 on the pump wave amplitude E The dependence of y~ and n 0 was also investigated using (4.94) and (4.95). From fig. 10 and eq. (4.96), one sees that a wave can propagate only if E0 is limited from above by the inequality B2 < B1, or23)312 -

(4.102c)9 3 101

C1fl,~0 ((12/3

to

Note that in obtaining fig. 10, we have used n

0 = 10 cm , Bmax = 10 kG, and to = 2 X 10 S - It is interesting to point out that A (1) decreases with increasing E0, whereas A (2) is an increasing function of E0. This fact allows the identification of the waves. 4.3.2. Ultrarelativistic ion-cyclotron waves Stenflo and Tsintsadze [74] have demonstrated the existence of ultrarelativistic ion-cyclotron waves. For this purpose, they assumed nonrelativistic electrons and ultrarelativistic ions. For to 4 ~ Ye = 1, = (~Q~ toEI)/to, eq. (4.94) becomes 2c2/w2 = 1 + to~o/wQC() toPI/totoEl (4.103) k For E 0 4 B11 (or WEI 4 11), eq. (4.103) yields .

2to. 2WE

/

2kctoE.\ 222to,~ 1

to. 22toEi

(4104)

~

/

where 2k2c2w~4

to~.

Clearly, the dispersive ion waves

P. K. Shukla et a!., Relativistic nonlinear effects in p!asmas

69

=

~toE1

(i

+to 1

WEI),

(4.105)

and the ion helicon wavesto~~~toEi,

top1

(4.106)

appear as new branches of the ion-cyclotron modes. The solutions (4.105) and (4.106) have the same form as those for the electrical helicon waves, since tEe/W~0 = to~1/to~~. In the present case, however,to4Li~.

The dampings of these new modes are given by [76] 2I2k2V~}. Im k = For l1~4

(4.107)

(~)1/2

~

~

~~

exp{(y1w

Li10)

to 0

1eo

the last expression becomes2

~/

~\1/2

Imk=~)

!7T1

~j~

to

exp(toEe/2k Vie).

p0

2

22

(4.108)

Clearly, the modes are damped only when toEc ~ kVte Similarly, one can show that the ordinary ion modes mentioned above exist when toEj k~ 1. When the particle temperature is sufficiently small the damping can be neglected. There exists an interesting difference between the usual nonrelativistic small amplitude waves and the ultra-relativistic solutions (4.lOla) and (4.105). In the nonrelativistic case, the damping is large if to l2~ < kr~1and there is no limit when to approaches 11~. In the ultrarelativistic case, the damping is large only when y1to (1~ < kv~1 as the difference between y1to and (1~ cannot be smaller than toe,. Thus, strong cyclotron resonance is absent in the relativistic limit [76].~

4.3.3. Parallel modulation of the electron helicons Tsintsadze et al. [76]have investigated the parametric instabilities of the high-frequency extraordinary strong relativistic electron modes. They have derived a general dispersion relation which shows how the pump wave (to, k) couples to the low-frequency mode (Li, K) and with the sideband modes (to (1, k K). In the ultra-relativistic limit, where eEolmeo(toye+

(1~) c,

(4.109)

the coupling of the pump to density fluctuations is enhanced. For stimulated forward scattering (K 4 k), one obtains [76] ~wK2V~] _(Q2 W~i)K4C2toEeV~/2(to~O + K2V~IYe), (4.110)2

(~2

to~)[(Li

KVg)

where vg=atolak, V~=a2toIak2,and

70 toas3K 2V~

P.K. Shukla et al., Relativistic nonlinear effects in p!asmas

+ K2C~/(Ye + K2A~e).

(4.111)

Note that the + or sign in eq. (4.110) is to be used when the wave (4.lOla) or (4.105) is to be studied, respectively. Considering the wave (4.102a), and assuming Kc> to,, 1, one obtains from eq. (4.110) (1=iKc.~

(4.112)2

On and the KV other hand, with eq. (4.lOlb) and the assumptions KADC 41 and to,~13~(1 Kr~e(meo/mj)~ 5, eq. (4.110) yields 112 (4.113) (1 =iKc(toEIIWPI) Furthermore, for toR Kvte(meo/mj)2 and 11 KVg 4 to,, 1, one gets from eq. (4.110):

(1 ~iKc[~2m1

(~!

112

(4.114)

to

y~c

It follows from the last three equations that an instability occurs in each case. 4.3.4. Parallel modulation of the ion helicons Stenflo and Tsintsadze [74] have found equations describing the nonlinear interaction of ultrarelativistic ion-cyclotron modes with low-frequency plasma motions along B0~.The evolution of the slowly varying wave envelope E is governed by2 IIEIE0 6n\ i(a0+V5a2)E+~V~a2Eto~ E0 _)E=0,

(4.115)

where the plus and the minus signs correspond to (4.105) and (4.106), respectively, and El E0 4 E0 has been used. Here, 6n is the density perturbation associated with the electrostatic plasma slow motion. On using a linearized hydrodynamic description for the ions, one obtains an equation for 6n in quasi-neutral limit:

(a~ V~a~) ~

= ~-

~ a~ (IEIE0) ,

(4.116)

where the right-hand side of eq. (4.116) comes from the averaging of the nonlinear Lorentz force in the momentum equations. Writing E = (E0 + 6E) exp(i~),where 6E(4E0) and p(4l) have space and time variation exp(iKz illt), we find from the linearized versions of eqs. (4.115) and (4.116) the dispersion relation which describes modulational instabilities: 2 [(Li KVg) Considering first the wave in eq. (4.105) and assuming Li/K 2- V5

~

K2V~](l12 K2V~)=

toKC

~!

V~.

(4.117)

and Vg~one gets from eq. (4.117):

P.K. Shukla et a!., Relativistic nonlinear effects

in p!asmas

71

(1

iKc(1 + toEI/41l1),

(4.118)

whereas for the ion helicon wave (eq. (4.106)) one finds, for Li/K ~- v5 and vg: 2c2\ 1/4 Li~iKc( / ito~ k.

(4.119)

~2Li~

to~

and, for LiIK~Vg, Li~Kc(V,/c2 2

toEl/Lil)

1/2-

(4.120)

From the last equation it follows that an instability occurs when toEl/(ll > 2V ~/ c2. For the Crab Nebula, taking some typical parameter values to 200 s~ (close to the rotation frequency of the Crab pulsar), n 3, one finds from eq. (4.106) the amplitude E 0 iO~cm 0 of the wave as E0 cn0e/4irw 30 V/cm, which agrees with the electric field values used by Sweeney and Stewart [111] where eEo/meowc 100. Supposing that the ion helicon wave exists in the Crab Nebula, we find that the wavelengths of the radiation would be of the order of 10 m. 4.3.5. MHD modulations of electron helicons Nishikawa, Tsintsadze and Watanabe [77] have considered the nonlinear interaction of the fieldaligned ultrarelativistic electron helicon and high-frequency waves with the magnetohydrodynamic (MHD) perturbations [112]. This interaction gives rise to a high-frequency wave envelope which has a structure across the magnetic field. In the present section, we discuss the multidimensional modulational instability, as well as the formation of three-dimensional solitons. When the modulation frequency is much smaller than the ion gyrofrequency ll~, one may use the magnetohydrodynamic equations to describe slow plasma motion [1121 d~v+V-(nv)=0, (a0+vV)v~-~Vn 1 Bx(VxB)+_L_, n 4irnm~ nm1

,B=Vx(vxB),

(4.121)

wheref is the ponderomotive force, which incorporates the relativistic motion of the electrons. In fact,fis given by,

f~ = ~-~ f~ = .~whereJ~

[i

{w(~

1)}]a~lEI2,.~ ~_{w2(1

(4.122) 1)}o,1E12]

[(~

1)a5IEI

2+

,

(4.123)

=1 to~ 0/to(to\/1+p

2(1)

72 P

P.K. Shuk!a et al., Relativistic nonlinear effects in p!asmas2=(p~+p~)/m~Oc2=toEC(1+ v2)/(toVT~~2~~e).

We consider a plane electromagnetic wave E = E(x, z, t) exp[i(kz tot)] propagating along the external magnetic field B 01. Let the characteristic dimensions for changes in the field amplitude and plasma density in the longitudinal and transverse directions, L~ and L2, respectively, be large as compared to the wavelength, i.e., kL~ ~ 1 and kL~ 1. It follows from the last inequalities that the longitudinal field component E2 is small in comparison with the transverse components (E2 4 E~, Er), and it may be found from the Maxwell equation:~

E2 =to

2

Eli

whereE11=1_to~0/to2\/1+

2

2 Furthermore, have two values: we will consider the ultra-relativistic case (v n2 c2k2/to2 = 1 ~ to~O/totoEe~

~

1). Then, the refraction index n may

(4.124)

Here, the minus sign refers to spectrum (4.lOla), and the plus sign to the helicon wave (4.lOlb). Taking into account that the electromagnetic waves are circularly polarized, i.e., E 1 = E~ in the case of longitudinal propagation, we obtain from the Maxwells equation the following expression for the slowly varying envelope of the electric field component E = E~ iE~ of the high-frequency spectrum (4. lOla):

i(a0 + ~a2)E + ~v~a~Ewhere

+

T1a~E ~

l~HE0]E

=0,

(4.125)

Vg=aw/ak,

V=aVg/ak,g

T

2 2\ k 1=~~i+$---),2~

(0]=

and E0 is the initial electromagnetic wave amplitude, before the modulation. Here, 6n/n0 n0)/n041, and (lEl E0)/E041 have been assumed. The equation for the amplitude of the helicon wave (4. lOib) is of the form

(n

lIE HE i(a0 + Vgaz)E_ + ~V~a~E + T2a~E toL E0

0

-,~---]E_= 0,

6n1

(4.126)

where/

22

toi

kcto11

k

P. K. Shukla et a!., Relativistic nonlinear effects in plasmas

73

The system (4.121), (4.125) and (4.126) is the basic equations describing both the modulational instabilities and wave self-focusing. To study the instability, it is generally assumed that the low-frequency modulation caused by the ponderomotive force is small, and that the system of equations (4.121), (4.125) and (4.126) can be linearized. We also assume that all quantities describing the magnetohydrodynamic motions depend on the coordinate through ~(=x~x + x~x [it), and write the electric field amplitude E as E=

a(~)exp[ip(~)].

Taking into account the above, we obtain from eqs. (4.125) and (4.126) and the system of MHD equations (4.121) a dispersion relation for the wave (4.lOla): [(Xztg)2 mc 4~(V~,y~+

T 1~~)+ ~to(V~+ T1X~)]D kLi+)(Q2

=

)(11to toEO{ ((1+ ~E0)

222

X~(l

X2v~i}(v~ +

Ti~~),

(4.l27a)

where 2f1~)(Li2Q~), D=(Li 2 ~ 2 2 2 2 22 2 2 1/2 }.

Q=~X{VA+cS[(VA+cS)4vAcScosO]

On the other hand, for the electron helicon (4.lOlb), one finds

[(11

2

1

p2

XzLg)

4(VgXz +

T

2

1

,2

2

2x~) 2 222

~to(VgXz+

T2x~)]D-

=

4

to~o{ (Q+

~o)2

+

X~(1 k

~)(Q2

22)}(P2

+

T2~~).

(4.l27b)

Let us analyze the dispersion relation (4.l27a). For the wave (4.lOla), we have 2/to >0c

and Vg 4 V~= to/k, c. Thus, one may easily show now that the second term in the dispersion equation is much smaller than the third one; and, therefore, it is omitted hereinafter. For the longitudinal modulational instability (x~ = 0), the dispersion relation (4.127a) becomes2

(11

2

+

x c )(Li

22

2

x

22

c,)

=

meo ~ m 1

x

44toE0

c i-topO

,

(4.128)2c~and an

where x x~. Equation meoto~o42mjto~o, and (4.128) has two complex solutions at (12 ~ x Im 11=

increment Im (1 = xc if

(meo/2m 1)

4(toEo/wpo)Xc, if meo ~2mIto~O/toEO -

74

P. K. Shukla et a!.,

Relativistic nonlinear effects in plasmas

In case of transverse modulations, (i.e. x2 rate: ImLi=~[1+~ ~ m1 (Li~ + ~E0)

=

0), we find from eq. (4.127a) an expression for the growth

].

One can easily show from the general dispersion relation (4. 127a) that there is no self-focusing of the wave (4.lOla). We now consider both the modulational instability and the self-focusing of the helicon wave (4.lOlb). In this case, we have Vg >0 and Vg = 2v~,and T2 may change sign: 2/to if to~ c 0 4 to1Li~ ~EOI 2Qeo/to~ if WpQ tolOeO ~E0I 1. ~C To study the self-focusing (Li = 0) of the electron helicon, we consider two cases: m 1 ~ and finite m1. In the first case (m1 so) self-focusing takes place if T2 3Xu)g/2~xc~, one finds\/~ /meo toEO\ Im[i=~) xc. 2 2m 1 kc

We inequalities now consider various limiting cases with x5 equation 0 and x~ ~ 0. Let us suppose that 114 x2 Vg~X these are fulfilled, then the dispersion (4.127b) takes the form:(2LixzVg

t~A

If

g)(Li2

x~c) =

~S~I

x~c2toEOT 2x~, (4.129)

where 2 T g=(x2V5) 2x~to/2.

P.K. Shukla et a!., Relativistic

nonlinear effects in p!asrnas

75

Equation (4.129) has two interesting solutions for=

I gI ~ 2Lixzvg:

x~c~ +

~

x~c2T 2x~.

Here, the first term is smaller than the second one, and the instability is insured provided that T2/g 0 and M = 0 or when a OFig. ii. Sagdeev potential V(R). Dotted lines: M0, and solid lines: M0. R=

1.5R,.

Multiplying eq. (4.137) by a~R = R, and integrating once, we get+

V(R) = H.=

(4.139) 0, we can immediately find the analytical solution. Noting that at (4.140)

For the case in which a >0 and M~iso,

R=

R

0 and R = 0 or V= 0 and hence H = 0, we find from eq. (4.139) ~R0 sech2(~\/~~),=

where we chose ~ = 0 as the instantaneous location of the solitary wave. For the case in which a 0, R remains nonzero at is. sowhere R vanishes. Analytical solutions can then be obtained when M2 41. In this case, R(~s so)can be approximated by R 0, and we can calculate 2/2R~ to obtain H by neglecting the term M H~aR~. (4.141) Equation (4.139) can then be put in a dimensionless form as (dsr)2 + m2/r2 ~r3 + r2=

,

(4.142)

where r = R/R 2 = M2/IaIR~,S = VIaIR 2(m~= 0, m 0~, a nd the critical value of m 0.08192. For m2 41, the solution in the region r2 ~ m2 can be written as r=[1 ~sech2(SI2)]-

M~/IaIR~) equals

Obviously, this solution is valid only in the region IS I > S 0 and 2IS I m2, not we very close S0, where S0 is 2(S can nowto ignore the terms determined by the relation sech 0/2) = or S0 = 1.303. For r 2r2/3 + r2 in eq. (4.142), in which case the solution is written as r = (3m2+

S2/3)~2,

78

P. K. Shukla et a!., Relativistic nonlinear effects in plasmas

~~>v,hh,__

_

Fig. 12. Numerical solution of eq. (4.142) for the case m = i0~.

Fig. i3. Numerical solution of eq. (4.142) for the case m = 10. The dashed line denotes the approximate solution (4.143) valid in the region 29 m.

or r=ISI/V~ in the region 9m24S24S~. Thus, near the origin, i.e., ISI=0, the solitary wave has a form of triangular dip with the minimum value r = \/~mat S = 0. Numerical solutions for m2 = 10 and 102 are shown in figs. 12 and 13. The triangular dip structure is clearly seen in the case m2 = ~ With varying m2, the value of r at S = 0 is plotted in fig. 14, which shows a linear dependence with the slope ~, i.e., r (S = 0) = m, except in the extreme vicinity of the critical point, m = Summarizing, we have discussed a new type of nonlinear Schrodinger equation which is characterized by a quadratic nonlinearity IEl E, in contrast to the usual cubic nonlinearity I El 2E. Although the plasma response is ultrarelativistic, the required field intensity is not necessarily extremely intense because of the collisionless cyclotron resonance of the electron motion. We have then shown some analytical solutions which describe solitary waves. For the modulationally unstable case where Pq >0, the solution for the

5-~-~-2LOG m2Fig. 14. The value of r at S=

0

vs. m2. The slope of the line is ], indicating that r, = r (S = 0) is proportional to m.

P. K. Shukla et a!., Relativistic nonlinear effects in p!asmas

79

amplitude is of a form similar to the KdV solution, whereas for the case where Pq 0

(

4176

.

where Qm = Q(P Pm) and Pm is the maximum value of p (at the centre). Since Qm (for a >0), A is always negative. Localized solutions are ensured provided that the potential V is negative between the two points (say, p = and p = Pm)- Furthermore, near p = Pm one requires aV/ap (for Pm ~ 0), which yields0 ~ ~ 0

2 [(H~Pm)

1/2

2 +a(1+pm)

2

]exp(KQm)

A~[1 +

a(1

+

2)1/2][l +0,

a(1 + p~,)~32]

.4 ~

(4.177)

On the other hand, near p V

one can Taylor expand V(p) taking p2+

/3~1+~)2 {(i +

a)[1

A(1 + a)]p2

~4[

1 4a

3a(l

+

A + Aa)]}.

(4.148)

It follows that V 0. The maximum electron density perturbation is N = exp( K Qm). (B) Hydrodynamic response. Next, we consider the hydrodynamic response to the radiation. In this case, eq. (3.118) gives~=

(4.179)

~M2(1n~),Ji

(4.180) from eqs. (4.165) and (4.180), one finds (4.181)

where n~ = ne is used. Eliminating ln tie=

~M2(1

n2)

f3Q

-

Note that eq. (4.181) represents the conservation of total energy, and gives a relationship between ne and p. From eqs. (4.173) and (4.181), one obtains

(1

+ p2)1/2

dfA

=

(~

i)

~_

dfn,

(4.182)

P. K. Shukla et a!., Relativistic nonlinear effects in plasmas

85

where n = tie One can now multiply eq. (4.169) by d~A, use eqs. (4.160) and (4.182), to write the energy integral + V(n, p, M; A; a) = 0, (d~n)2 where V= i+p2 (M2 ti2)2 M2+ ni

(4.183)

~A13p2[l+ a(1

+p2)h/2]2}.

(4.184)

In eq. (4.184), we need to use eq. (4.181) in order to relate p in terms of n. We have again assumed the plasma to be unperturbed at infinity. The nonlinear frequency shift is obtained by setting V= 0. The result is A22

(N1)(lM2/N)2

1/22

I3Pm [1+ a(1 +Pm)

]

(

4185-

)

where N is the maximum density perturbation at the centre. From eq. (4.181), the maximum density perturbation is found to be ~lnN_~(l_N2)=1_(l+p~)1I2_~ The maximum potential is then=

1)m2.

(4.186)

~M2(1

N2).

(4.187)

The condition that a VlanIfl,N ~ 0 yields N2 0.

0, the potential V is negative, provided that (4.189a)

On the other hand, p2 remains positive when (N1)(M2i)I(1+a)>O.

(4.189b)

It follows that for a >0, localized solutions with ~n = N 1 0) arise with M2 1). The existence regions for the finite amplitude solitons are given in the (N, Pm) parameter space shown in fig. 15. Clearly, for stronger magnetic fields, the corresponding density perturbations are smaller. On the other hand, as is clear from fig. 15, compressional as well as rarefaction solutions for the low-frequency density perturbations appear. Such solutions were also obtained for unmagnetized plasmas when the thermal effects were accounted for [31, 64].

86

P.K. Shukla et a!., Relativistic nonlinear effects in plasmasI I I

3.

-

N

>~v7

I

I

I

0Fig. 15. Existence regions in (N, p the corresponding line;

1

2

3

m(=

5

13

=

m) parameter space for moving solitons. For a given a 100. [67]

(l,,Iw,), the localized solutions exist in the region below

4.5.2. Relativistic radiation-pressure dominated limit [66] Here, we consider the limit when the incident waves are so strong that the ponderomotive pressure dominates the electron thermal pressure. Thus, the ambipolar potential is directly generated by the radiation pressure. In fact, from eq. (4.161), one finds 2)2 1 + ~ap2/(1 + (4.190)

=

/3[(i

+p

Since the ponderomotive force is balanced by the ambipolar field alone, the Poisson equation is essential for the establishment of finite density perturbation. Thus, for the forced Raman (FR), forced Brillouin (FB) and the force quasi-static (FQS) cases, the electron density perturbation is given respectively by n~=i+d~P,tie =

(4.191)+

M/(M2

2P)2+

d~I,

(4.192) (4.193)

=

exp(a4)

d~,

where a = Te/ T 1 and cP is given by eq. (4.190). In the following, we find conditions for the existence of localized solutions when the plasma low-frequency response is of FR, FB, and FQS type [66]. The latter are driven electrostatic fluctuations which are not the normal modes of the system when the radiation is turned off. From eq. (4.190), one

P. K. Shukla et a!., Relativistic nonlinear effects in p!asmas

87

finds d~cP= /3J 1/2d(A, (4.194)

(i+p)

where d(A is given by eq. (4.173). Multiplying eq. (4.169) by dkA and using eq. (4.194), we get 2 ~AI3d~A2 = tie d~P. (4.195) 1f32 dt(d(A)

On the other hand, eqs. (4.i91)(4.i93) yield, after multiplication by d~I,=

~e

~

n

5 dkI ,

(4.196)

where, 11 2 _2~)1/2 ~1exp(o~) M(M for FB, FR, for FQS. (4.197)

=

Subtracting eq. (4.196) from eq. (4.195), and integrating with respect to ~once, one obtains eq. (4.174) but with the effective potential V now being given by V(p)=

2B[D

+

X(cP)],

(4.198)

withD = Af3p2[a and for FR, for FB, for FOS.+

(1

+

p2)2]212(1 +

~

,

B = /32(1

+

p2)4[a

+

(1

+ p2)3t2]2

X(4)

= {

I.

M2[1 (1 2P1M2)2] o~[1 exp(crk)]

Here, cPis given in terms of p by eq. (4.190). Furthermore, we note that for a = 0, the expression (4.198) for V(p) reduces to that of previous works [65]. Localized solutions for p( ~) exist provided that the conditions given before eq. (4.177) hold. We therefore analyze V( p) near p = 0. Taylor expanding V(p), we get V=ap +bp where a b==

2

4

,

(4.199)

1

+

A(l

+

a)1f3(l

+ +

a), ~

f3~(l + a)2{K

[1 + A(i

+

a)](1

+

3a)}

88

P. K. Shuk!a et a!., Relativistic nonlinear effects in plasmas

K~ f3(1+cr)/2M2 for FB, (~of3(i+ a)2/4 for FQS. Hence for V(p) to be negative near p 1+A(1+a)>0. Also, using the condition V(pm)2(1+= =

10

for FR,

0, we require (4.200)

0 in eq. (4.198), one obtains(4.201)

p~)X(b =2

~m) 1122

A=

2

13Pm[~(l+Pm)

]

which relates A and Pm In eq. (4.201), we have defined=

f3[(i +p~)21 + ~ap~(1 +p1].

(4.202)

In order that p( ~)has appropriate profile at P = Pm we require V~( Pm) ~ 0 for Pm ~ 0. These conditions are satisfied provided that Z+10. It is easy to verify that these inequalities are consistent with the existence regions shown in fig. 16. Also, the value of A in the small amplitude limit is given by A= whereQ={ I.

-

1

a

fi _P~[4(l +

a)

-

Q]J~

(4.209)

fO FR, 2 for for FB, /3(1+a)/2M aj3(1+a)14 for FQS.

It follows from eq. (4.209) that eq. (4.200) is satisfied provided that

1>4(1

+

a)Q.

(4.210)

The expression for the electron density is obtained from eqs. (4.191) to (4.193) astie =til

+

R,

(4.211)

90

P.K. Shuk!a et a!., Relativistic nonlinear effects in plasmas

where= f3{2(1 +

a)ap2

[2a(1+ 4a) + 3b(1 + a)]p4}

and12

for FR,

i+f~T(i+a)_~-~-~(i+4a_3/3M2)p4 for FB,i_~p2(i+a)_~[1+4a_/3r2(i+a)2]p4 for

FQS.

It is clear that for all the three cases the solution for p2( 4~) has single-hump structures. While for the last two cases (FB and FQS), the ambipolar field ~P( ~) has a similar single-hump structure, the electron density profile has a density excess with a small depression at the centre. On the other hand, for the FR interaction, like in the unmagnetized case [61], the electron density profile consists of a depression at the centre and shoulders on the sides. Furthermore, the ion density perturbation has a hump (dip) for the FB(FQS) case.

The corresponding results for the right-handed circularly polarized waves can be obtained by letting a to be negative. However, because of the possibilityof gyroresonance, the fluid equations may not be valid everywhere. In fact, for a given a, there exists a lower limit for the field amplitude Pm (fig. 17) below which localized solutions for p( ~)do not exist. In contrast to the left circularly polarized waves, here the5CI

I

I