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Quasilinear theory of non-resonant electromagnetic instabilities
Phnet. .%UC Sci., Vol. 40, No. IO. pp. 1479-1482, I992 Printed in Great Britain.
0
00324633/92 S5.00+0.00 1992 Pergamon Press Ltd
QUASILINEAR THEORY OF NON-RESONANT ELE~TRO~A~NETI~ INSTABILITIES FRANK VERHEEST Instituut voor theoretische mechanika, Universiteit Gent, Krijgslaan 281, B-9000 Gent, Belgium and P. K. KAW Institute for Plasma Research, Bhat, Gandhinagar 382 424, India
Abstract-Our quasilinear theory includes explicitly macroscopic drifts, which are then modified, together with other equilibrium quantities, by the growth of unstable waves. We study the saturation of nonresonant electromagnetic instabilities, due to streaming effects besides pressure anisotropies. The resulting equations governing the equilib~um changes can be integrated exactly, so that more precise estimates are given for the ultimate levels of magnetic field fluctuations. This is applied to the quenching of streaming low-frequency instabilities in the mass-loaded solar wind near comets.
1. INTRODUCI’ION
In quasilinear theory one describes the modification of equilibrium quantities, such as plasma densities, due to the growth of unstable waves. In the present paper we include equilibrium drifts, which are then similarly modified by the instability, and look in particular at non-resonant electromagnetic instabilities which generalize the classic firehose instability. Existing quasilinear treatments of beam-plasma instabilities have usually dealt with resonant wave-particle effects or with resonant modes, in situations where a rather weak beam interacts with an ordinary plasma (see e.g. Gary, 1978 ; Gary and Feldman, 1978 ; Sagdeev et al., 1986; Lee and Ip, 1987; Gaffey et al., 1988; Yoon and Ziebell, 1991). Many of the more recent papers were motivated by the pickup by the solar wind of newly created ions of interstellar or cometary origin, following the successful satellite encounters with comets P/Ciacobini-Zinner and P/Halley.
In the kinetic treatment of quasilinear theory by Barnes (1968), equilibrium drifts figured in the general theory but were omitted in the detailed calculations leading to the stabilization of the ordinary firehose instability. Likewise, Davidson and Viilk (1968) looked at this problem from a macroscopic point of view, but omitting the equilibrium drifts altogether, as did Kennel and Scarf (1968) from a kinetic theory point of view. Later, Barnes (1970) briefly discussed inter aliu the stabilization of a beam induced firehoselike instability, by assuming that the beam effects could be treated as an equivalent parallel pressure. As
we wiI1 see further on, this is not quite correct in the details of the stabilization process. We will start from a multispecies fluid description and take fully anisotropic pressure effects into account, besides the beam effects. As an application we look at the saturation of low-frequency instabilities occurring in the solar wind near comets, as referred to already in the first paragraph, but at larger wavelengths than addressed by Winske and Leroy (1984). We would also like to mention that beam-plasma effects in an astrophysical context were recently reviewed by Gary (1991), since successful missions to planets and comets have emphasized their utmost importance in many heliospheric applications. The structure of the paper is as follows. In Section 2 basic results are recalled from our more extensive work (Verheest and Kaw, 1992). How the equilibrium saturates is briefly discussed in Section 3, and in Section 4 we mention a possible application to the massloading of the solar wind near comets. Finally, Section 5 contains the conclusions.
2. BASIC RESULTS The starting fluid equations
point
is the well known
set of basic
per species, namely the continuity equations, the equations of motion, and the pressure equations (Davidson and Volk, 1968; Verheest and Kaw, 1992). We look for simplicity at transverse electromagnetic fluctuations propagating parallel to the external magnetic field and split all variables according to 1479
1480
F. VERHEE~T
f=
(J)+Sf=Cfk
I
exp (ikz-iio,t).
and
(1)
The fluctuations themselves will be determined from the linearized form of the fluid equations, together with Maxwell’s equations, as customary in quasilinear theory. The growth of unstable waves to larger amplitudes will then in turn modify the equilibrium quantities, so as to quench the instability. When going through the algebra, the full details of which will be reported elsewhere (Verheest and Kaw, 1992), one encounters the linear dispersion law
.
(2)
Global quantities occurring here include the mean bulk drift 0 of the whole plasma, the total parallel and perpendicular pressures and the normalized kinetic energy in the relative parallel motions
Unstable fluctuations are possible when
P. K. KAW
These equations have to be considered in conjunction with (9) The underlying invariant p u+2p L +“p 2
is 0
W+(SB2)=C PO
(10) ’
up to terms of order yi. So one sees, also from a combination of equations (6) and (8), that the streaming effects &, W occur as a kind of additional parallel pressure, except for changes in the perpendicular pressure. The development of the instability means that W decreases, while P, increases. Since the reason for this electromagnetic instability is no longer the pressure anisotropy alone, one could even have that initially P,, = P,. In that case, equations (7) and (8) show that a pressure anisotropy is created, even when none existed to start with, provided that initially W > Vi. Thus, not all the free-streaming energy can go into magnetic field fluctuations.
Pl. PI, uo_+v:<-&+w, 3. SATURATION
and then
In order to analyti~lly study the saturation of the instability and determine the ultimate levels of magnetic field fluctuations, we use dimensionless variables
Re wk = kO,
At W = 0 one recovers the criterion for the usual firehose instability, whereas at W # 0 an instabiIity is possible even with isotropic pressures at the outset, provided W is large enough. From the quasistatic part of the equations of motion one deduces the changes in normalized kinetic energy (Verheest and Kaw, 1992) as
(11) and eliminate II, ya&[ exp (2y,t) in equations (9). This yields
For the changes in the total pressure (Verheest and Kaw, 1992) one recovers analogous results as in nonstreaming plasmas (Davidson and Vijlk, 1968 ; Barnes, I968 ; Davidson, 1972)
f P,, = -
4P, -2P,
(6)-
#,, = -(28,, -P&
(12)
81 = 12P,,&
(13)
*=
(6)
~PL =$CykIB,12 exp (2yk0, ok
OF THE INSTABILITY
-2w6,
(14)
the dot denoting time derivatives. Rather than starting the discussion from the approximate invariant (lo), we analytically integrate equations (12)-(14) and obtain /T,,= e-’
(PI,0 cd&b +
Non-resonant
BL
electromagnetic
instabilities
1481
TABLE l.V.4Lw.s FOR p AND &COMPUTED FRO~~EQUATIONS (19) AND (21), RESPECTIVELY, AT&, = PLO = 1 AND M = 3
=e-+,,corh~b
br computed from equation
P
+&(/?lO+%)sinh&b),
w=w
II
(17)
Having expressed everything in terms of b and of the initial values pliO,ljlo and w0 at t = 0, with b, = 0, we proceed to determine the final level of fluctuations br from the quenching of the instability, which occurs if in our dimensionless notation
Consequently,
1 = 0.
(18)
br satisfies the equation
+ezhf = wO, which can be evaluated
4. APPLICATION
(19)
numerically.
TO COMETARY
PLASMAS
We shall apply the previous results to estimate the level of magnetic field turbulence which could be reached due to firehose-like non-resonant modes in the neighbourhood of comets, where the solar wind picks up ions of cometary origin. For such applications we can take a two-beam model, one beam modelling the undisturbed solar wind and one beam the cometary ions of the water group. One beam has velocities of typically 400 km s- ’ or less, depending on the direction of the solar wind flow with respect to the interplanetary magnetic field, while the other is almost stationary. Then the quantity w,, is
WO
v:
w()=-=--_=
NJ; V&
b, computed from equation (21)
(16)
emZb.
w,+B,,r-PN-
(19)
PM23
(20)
where p = pe/(psw +p,) is the normalized mass density of the pickup cometary ions and M = U,JVA,sw is the Alfvenic Mach number of the cometary ion speed, viewed in the solar wind frame and using the AlfvCn velocity VA,sw of the undisturbed solar wind (- 80 km s- ‘) as a reference speed. Since the Alfvenic Mach number can be as high as five, we will not expect w,, - 1 to be always very small, but can then use equation (19) to evaluate br at different levels of p. For simplicity, we have assumed pl10 = fiLO = 1, so that the instability is purely beam driven. The results
0.11 0.15 0.20 0.30 0.40 0.50
0.00 0.15 0.29 0.50 0.64 0.75
0.00 0.13 0.22 0.32 0.36 0.39
are shown in Table 1, evaluated at M = 3 which corresponds to a solar wind which flows obliquely to the direction of the interplanetary magnetic field, at an angle of about 45”. The results would seem to indicate that one can use quasilinear theory in a regime which is somewhat outside the theoretical ranges where one would expect the theory to be valid! This point has also emerged from numerical simulations (Winske and Leroy, 1984). On the other hand, it is interesting to compare the expected levels predicted by our quasilinear theory with those obtained by Verheest and Lakhina (199 1) from rather different considerations, based upon Fowler’s theorem on the thermodynamics of unstable plasmas (Fowler, 1968). Fowler’s theorem essentially gives upper bounds based on linear growth rates, and is independent of the way the instability saturates. It has been used in the form (Verheest and Lakhina, 1991)
b = (6B2) ~
f
Bi
Pow,+~,,,--P,o-PP,Vi 2p,w0+4,-p,cl
wo+&o-810-l = 2w3+a,,o-Blll
(21)
The levels computed from this expression have also been included in Table 1, in the third column. At low levels of p, just above per = l/M*, both methods, one based on quasilinear theory and the other on Fowler’s theorem, are in surprising agreement on the order of magnitudes, given the totally different underlying philosophies. This comparison also indicates that quasiiinear estimates tend to be on the higher side, since they essentially neglect higher-order effects. Such levels of magnetic field turbulence raise the fundamental question of whether quasilinear theory is still appropriate. A cautious answer would appear to be yes, since numerical simulations of the full nonlinear system would seem to indicate levels of the same order (Winske and Leroy, 1984). This was also one of the reasons motivating us to seek exact expressions
1482
F. VERHJZEST amd P. K. KAW
for the fluctuation levels, rather than relying upon the approximate method of small fractional changes as has usually been done, and which is certainly not good at the present levels of turbulence.
We have given a quasilinear treatment of nonresonant electromagnetic instabilities, due not just to pressure anisotropies as usual, but also or even mainly to streaming effects. Not only did we obtain the equations governing the changes in the parallel kinetic beam energy and in the parallel and perpendicular pressures, for the case of unstable waves, but we were also able to carry out the exact integration of these. As one of the many possible applications we looked at the saturation of low-frequency instabilities occurring in the solar wind upstream of comets, due to the mass-loading by pickup ions of cometary origin. Our computations show that the levels of magnetic field turbulence are not really small, as would be required for a proper application of quasilinear theory. This also motivated us to find exact expressions for the fluctuation levels. The legitimacy of using quasilinear theory has a partial answer in some numerical simulations which predict similar levels. Finally, we included a brief comparison with and discussion of estimates based upon Fowler’s theorem on the thermodynamics of unstable plasmas and which indicate lower- levels of turbulence. Fowler’s theorem essentially gives upper bounds based on linear growth rates, and is independent of the way the instability saturates, in contrast to quasilinear treatments. This comparison showed that quasilinear estimates tend to be on the higher side. Acknowledgement-It
is a pleasure for F.V. to thank the Belgian National Fund for Scientific Research for a research grant.
REFERENCES Barnes, A. (1968) Quasilinear theory of hydromagnetic waves in collisionless plasma. Phys. Fluids 11, 2644. Barnes, A. (1970) Theory of generation of bow-shock-associated hydromagnetic waves in the upstream interplanetary medium. Cosmic Electrodyn. 1, 90. Davidson, R. C. (1972)Methods in Nonlinear Plasma Theory. Academic Press, New York. Davidson, R. C. and Volk, H. J. (1968) Macroscopic quasilinear theory of the garden-hose instability. Phys. Fluids 10, 2259.
Fowler, T. K. (1968) Thermodynamics of unstable plasmas, in Advances in Plasma Physics (Edited by Simon, A. and Thompson, W. B.), Vol. lipp. 201-225. Wiley, New York. Gaffev. J. D.. Jr.. Winske. D. and Wu. C. S. (1988) Time scales for formation and spreading of velocity shells of pickup ions in the solar wind. J. geophys. Res. 93, 5470. Gary, S. P. (1978) The electromagnetic ion beam instability and energy loss of fast alpha particles. Nucl. Fusion 18, 327.
Gary, S. P. (1991) Electromagnetic ion/ion instabilities and their consequences in space plasmas : a review. Space Sci. Rev. 56, 373.
Gary, S. P. and Feldman, W. C. (1978) A second-order theory for kJjB, electromagnetic instabilities. Phys. Fluia!s 21, 72. Kennel, C. F. and Scarf, F. L. (1968) Thermal anisotropies and electromagnetic instabilities in the solar wind. J. geophys. Res. 73, 6149.
Lee, M. A. and Ip, W.-H. (1987) Hydromagnetic wave excitation by ionized interstellar hydrogen and helium in the solar wind. J. geophys. Res. 92, 11,041. Sagdeev, R. Z., Shapiro, V. D., Shevchenko, V. 1. and Szegii, K. (1986) MHD turbulence in the solar wind-comet interaction region. Geophys. Res. Left. 13, 85. Verheest, F. and Kaw, P. K. (1992) Quasilinear theory revisited : firehose-type instabilities in multispecies beamplasma systems, in VIth Workshop on Nonlinear Stability and Waves (Edited by Callebaut, D. K. and Malfliet, W.). World Scientific, Singapore (in press). Verheest. F. and Lakhina. G. S. (1991) Nonresonant lowfrequency instabilities ‘in multi-beam plasmas : applications to cometary environments and plasma sheet boundary layers. J. geophys. Res. 96,7905. Winske, D. and Leroy, M. M. (1984) Diffuse ions produced by electromagnetic ion beam instabilities. J. geophys. Res. 89, 2673.
Yoon, P. H. and Ziebell, L. F. (1991) Quasilinear diffusion rates of cometary ions. Phys. Fluids B 3, 2124.
0
00324633/92 S5.00+0.00 1992 Pergamon Press Ltd
QUASILINEAR THEORY OF NON-RESONANT ELE~TRO~A~NETI~ INSTABILITIES FRANK VERHEEST Instituut voor theoretische mechanika, Universiteit Gent, Krijgslaan 281, B-9000 Gent, Belgium and P. K. KAW Institute for Plasma Research, Bhat, Gandhinagar 382 424, India
Abstract-Our quasilinear theory includes explicitly macroscopic drifts, which are then modified, together with other equilibrium quantities, by the growth of unstable waves. We study the saturation of nonresonant electromagnetic instabilities, due to streaming effects besides pressure anisotropies. The resulting equations governing the equilib~um changes can be integrated exactly, so that more precise estimates are given for the ultimate levels of magnetic field fluctuations. This is applied to the quenching of streaming low-frequency instabilities in the mass-loaded solar wind near comets.
1. INTRODUCI’ION
In quasilinear theory one describes the modification of equilibrium quantities, such as plasma densities, due to the growth of unstable waves. In the present paper we include equilibrium drifts, which are then similarly modified by the instability, and look in particular at non-resonant electromagnetic instabilities which generalize the classic firehose instability. Existing quasilinear treatments of beam-plasma instabilities have usually dealt with resonant wave-particle effects or with resonant modes, in situations where a rather weak beam interacts with an ordinary plasma (see e.g. Gary, 1978 ; Gary and Feldman, 1978 ; Sagdeev et al., 1986; Lee and Ip, 1987; Gaffey et al., 1988; Yoon and Ziebell, 1991). Many of the more recent papers were motivated by the pickup by the solar wind of newly created ions of interstellar or cometary origin, following the successful satellite encounters with comets P/Ciacobini-Zinner and P/Halley.
In the kinetic treatment of quasilinear theory by Barnes (1968), equilibrium drifts figured in the general theory but were omitted in the detailed calculations leading to the stabilization of the ordinary firehose instability. Likewise, Davidson and Viilk (1968) looked at this problem from a macroscopic point of view, but omitting the equilibrium drifts altogether, as did Kennel and Scarf (1968) from a kinetic theory point of view. Later, Barnes (1970) briefly discussed inter aliu the stabilization of a beam induced firehoselike instability, by assuming that the beam effects could be treated as an equivalent parallel pressure. As
we wiI1 see further on, this is not quite correct in the details of the stabilization process. We will start from a multispecies fluid description and take fully anisotropic pressure effects into account, besides the beam effects. As an application we look at the saturation of low-frequency instabilities occurring in the solar wind near comets, as referred to already in the first paragraph, but at larger wavelengths than addressed by Winske and Leroy (1984). We would also like to mention that beam-plasma effects in an astrophysical context were recently reviewed by Gary (1991), since successful missions to planets and comets have emphasized their utmost importance in many heliospheric applications. The structure of the paper is as follows. In Section 2 basic results are recalled from our more extensive work (Verheest and Kaw, 1992). How the equilibrium saturates is briefly discussed in Section 3, and in Section 4 we mention a possible application to the massloading of the solar wind near comets. Finally, Section 5 contains the conclusions.
2. BASIC RESULTS The starting fluid equations
point
is the well known
set of basic
per species, namely the continuity equations, the equations of motion, and the pressure equations (Davidson and Volk, 1968; Verheest and Kaw, 1992). We look for simplicity at transverse electromagnetic fluctuations propagating parallel to the external magnetic field and split all variables according to 1479
1480
F. VERHEE~T
f=
(J)+Sf=Cfk
I
exp (ikz-iio,t).
and
(1)
The fluctuations themselves will be determined from the linearized form of the fluid equations, together with Maxwell’s equations, as customary in quasilinear theory. The growth of unstable waves to larger amplitudes will then in turn modify the equilibrium quantities, so as to quench the instability. When going through the algebra, the full details of which will be reported elsewhere (Verheest and Kaw, 1992), one encounters the linear dispersion law
.
(2)
Global quantities occurring here include the mean bulk drift 0 of the whole plasma, the total parallel and perpendicular pressures and the normalized kinetic energy in the relative parallel motions
Unstable fluctuations are possible when
P. K. KAW
These equations have to be considered in conjunction with (9) The underlying invariant p u+2p L +“p 2
is 0
W+(SB2)=C PO
(10) ’
up to terms of order yi. So one sees, also from a combination of equations (6) and (8), that the streaming effects &, W occur as a kind of additional parallel pressure, except for changes in the perpendicular pressure. The development of the instability means that W decreases, while P, increases. Since the reason for this electromagnetic instability is no longer the pressure anisotropy alone, one could even have that initially P,, = P,. In that case, equations (7) and (8) show that a pressure anisotropy is created, even when none existed to start with, provided that initially W > Vi. Thus, not all the free-streaming energy can go into magnetic field fluctuations.
Pl. PI, uo_+v:<-&+w, 3. SATURATION
and then
In order to analyti~lly study the saturation of the instability and determine the ultimate levels of magnetic field fluctuations, we use dimensionless variables
Re wk = kO,
At W = 0 one recovers the criterion for the usual firehose instability, whereas at W # 0 an instabiIity is possible even with isotropic pressures at the outset, provided W is large enough. From the quasistatic part of the equations of motion one deduces the changes in normalized kinetic energy (Verheest and Kaw, 1992) as
(11) and eliminate II, ya&[ exp (2y,t) in equations (9). This yields
For the changes in the total pressure (Verheest and Kaw, 1992) one recovers analogous results as in nonstreaming plasmas (Davidson and Vijlk, 1968 ; Barnes, I968 ; Davidson, 1972)
f P,, = -
4P, -2P,
(6)-
#,, = -(28,, -P&
(12)
81 = 12P,,&
(13)
*=
(6)
~PL =$CykIB,12 exp (2yk0, ok
OF THE INSTABILITY
-2w6,
(14)
the dot denoting time derivatives. Rather than starting the discussion from the approximate invariant (lo), we analytically integrate equations (12)-(14) and obtain /T,,= e-’
(PI,0 cd&b +
Non-resonant
BL
electromagnetic
instabilities
1481
TABLE l.V.4Lw.s FOR p AND &COMPUTED FRO~~EQUATIONS (19) AND (21), RESPECTIVELY, AT&, = PLO = 1 AND M = 3
=e-+,,corh~b
br computed from equation
P
+&(/?lO+%)sinh&b),
w=w
II
(17)
Having expressed everything in terms of b and of the initial values pliO,ljlo and w0 at t = 0, with b, = 0, we proceed to determine the final level of fluctuations br from the quenching of the instability, which occurs if in our dimensionless notation
Consequently,
1 = 0.
(18)
br satisfies the equation
+ezhf = wO, which can be evaluated
4. APPLICATION
(19)
numerically.
TO COMETARY
PLASMAS
We shall apply the previous results to estimate the level of magnetic field turbulence which could be reached due to firehose-like non-resonant modes in the neighbourhood of comets, where the solar wind picks up ions of cometary origin. For such applications we can take a two-beam model, one beam modelling the undisturbed solar wind and one beam the cometary ions of the water group. One beam has velocities of typically 400 km s- ’ or less, depending on the direction of the solar wind flow with respect to the interplanetary magnetic field, while the other is almost stationary. Then the quantity w,, is
WO
v:
w()=-=--_=
NJ; V&
b, computed from equation (21)
(16)
emZb.
w,+B,,r-PN-
(19)
PM23
(20)
where p = pe/(psw +p,) is the normalized mass density of the pickup cometary ions and M = U,JVA,sw is the Alfvenic Mach number of the cometary ion speed, viewed in the solar wind frame and using the AlfvCn velocity VA,sw of the undisturbed solar wind (- 80 km s- ‘) as a reference speed. Since the Alfvenic Mach number can be as high as five, we will not expect w,, - 1 to be always very small, but can then use equation (19) to evaluate br at different levels of p. For simplicity, we have assumed pl10 = fiLO = 1, so that the instability is purely beam driven. The results
0.11 0.15 0.20 0.30 0.40 0.50
0.00 0.15 0.29 0.50 0.64 0.75
0.00 0.13 0.22 0.32 0.36 0.39
are shown in Table 1, evaluated at M = 3 which corresponds to a solar wind which flows obliquely to the direction of the interplanetary magnetic field, at an angle of about 45”. The results would seem to indicate that one can use quasilinear theory in a regime which is somewhat outside the theoretical ranges where one would expect the theory to be valid! This point has also emerged from numerical simulations (Winske and Leroy, 1984). On the other hand, it is interesting to compare the expected levels predicted by our quasilinear theory with those obtained by Verheest and Lakhina (199 1) from rather different considerations, based upon Fowler’s theorem on the thermodynamics of unstable plasmas (Fowler, 1968). Fowler’s theorem essentially gives upper bounds based on linear growth rates, and is independent of the way the instability saturates. It has been used in the form (Verheest and Lakhina, 1991)
b = (6B2) ~
f
Bi
Pow,+~,,,--P,o-PP,Vi 2p,w0+4,-p,cl
wo+&o-810-l = 2w3+a,,o-Blll
(21)
The levels computed from this expression have also been included in Table 1, in the third column. At low levels of p, just above per = l/M*, both methods, one based on quasilinear theory and the other on Fowler’s theorem, are in surprising agreement on the order of magnitudes, given the totally different underlying philosophies. This comparison also indicates that quasiiinear estimates tend to be on the higher side, since they essentially neglect higher-order effects. Such levels of magnetic field turbulence raise the fundamental question of whether quasilinear theory is still appropriate. A cautious answer would appear to be yes, since numerical simulations of the full nonlinear system would seem to indicate levels of the same order (Winske and Leroy, 1984). This was also one of the reasons motivating us to seek exact expressions
1482
F. VERHJZEST amd P. K. KAW
for the fluctuation levels, rather than relying upon the approximate method of small fractional changes as has usually been done, and which is certainly not good at the present levels of turbulence.
We have given a quasilinear treatment of nonresonant electromagnetic instabilities, due not just to pressure anisotropies as usual, but also or even mainly to streaming effects. Not only did we obtain the equations governing the changes in the parallel kinetic beam energy and in the parallel and perpendicular pressures, for the case of unstable waves, but we were also able to carry out the exact integration of these. As one of the many possible applications we looked at the saturation of low-frequency instabilities occurring in the solar wind upstream of comets, due to the mass-loading by pickup ions of cometary origin. Our computations show that the levels of magnetic field turbulence are not really small, as would be required for a proper application of quasilinear theory. This also motivated us to find exact expressions for the fluctuation levels. The legitimacy of using quasilinear theory has a partial answer in some numerical simulations which predict similar levels. Finally, we included a brief comparison with and discussion of estimates based upon Fowler’s theorem on the thermodynamics of unstable plasmas and which indicate lower- levels of turbulence. Fowler’s theorem essentially gives upper bounds based on linear growth rates, and is independent of the way the instability saturates, in contrast to quasilinear treatments. This comparison showed that quasilinear estimates tend to be on the higher side. Acknowledgement-It
is a pleasure for F.V. to thank the Belgian National Fund for Scientific Research for a research grant.
REFERENCES Barnes, A. (1968) Quasilinear theory of hydromagnetic waves in collisionless plasma. Phys. Fluids 11, 2644. Barnes, A. (1970) Theory of generation of bow-shock-associated hydromagnetic waves in the upstream interplanetary medium. Cosmic Electrodyn. 1, 90. Davidson, R. C. (1972)Methods in Nonlinear Plasma Theory. Academic Press, New York. Davidson, R. C. and Volk, H. J. (1968) Macroscopic quasilinear theory of the garden-hose instability. Phys. Fluids 10, 2259.
Fowler, T. K. (1968) Thermodynamics of unstable plasmas, in Advances in Plasma Physics (Edited by Simon, A. and Thompson, W. B.), Vol. lipp. 201-225. Wiley, New York. Gaffev. J. D.. Jr.. Winske. D. and Wu. C. S. (1988) Time scales for formation and spreading of velocity shells of pickup ions in the solar wind. J. geophys. Res. 93, 5470. Gary, S. P. (1978) The electromagnetic ion beam instability and energy loss of fast alpha particles. Nucl. Fusion 18, 327.
Gary, S. P. (1991) Electromagnetic ion/ion instabilities and their consequences in space plasmas : a review. Space Sci. Rev. 56, 373.
Gary, S. P. and Feldman, W. C. (1978) A second-order theory for kJjB, electromagnetic instabilities. Phys. Fluia!s 21, 72. Kennel, C. F. and Scarf, F. L. (1968) Thermal anisotropies and electromagnetic instabilities in the solar wind. J. geophys. Res. 73, 6149.
Lee, M. A. and Ip, W.-H. (1987) Hydromagnetic wave excitation by ionized interstellar hydrogen and helium in the solar wind. J. geophys. Res. 92, 11,041. Sagdeev, R. Z., Shapiro, V. D., Shevchenko, V. 1. and Szegii, K. (1986) MHD turbulence in the solar wind-comet interaction region. Geophys. Res. Left. 13, 85. Verheest, F. and Kaw, P. K. (1992) Quasilinear theory revisited : firehose-type instabilities in multispecies beamplasma systems, in VIth Workshop on Nonlinear Stability and Waves (Edited by Callebaut, D. K. and Malfliet, W.). World Scientific, Singapore (in press). Verheest. F. and Lakhina. G. S. (1991) Nonresonant lowfrequency instabilities ‘in multi-beam plasmas : applications to cometary environments and plasma sheet boundary layers. J. geophys. Res. 96,7905. Winske, D. and Leroy, M. M. (1984) Diffuse ions produced by electromagnetic ion beam instabilities. J. geophys. Res. 89, 2673.
Yoon, P. H. and Ziebell, L. F. (1991) Quasilinear diffusion rates of cometary ions. Phys. Fluids B 3, 2124.