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Kinetic Alfvén waves in turbulent plasmas
Volume 60A, number 3
PHYSICS LETTERS
21 February 1977
KINETIC ALFVEN WAVES IN TURBULENT PLASMAS P.K. SHUKLA Institutefür Theoretische Physik, Ruhr Universitiit Bochum, 453 Bochum, West Germany Received 6 October 1976 The anomalous damping of a coherent kinetic Alfv~nwave in turbulent plasmas is investigated.
The kinetic Alfvén waves are expected [1] to play an important role in large scale plasmas. In this note, we consider the propagation of a kinetic Alfven wave in a turbulent plasma. It is found that the nonlinear interaction of the high-frequency randomly distributed plasmons with the low-frequency kinetic Alfvén wave leads to a damping. The analytical expression for the damping rate is presented. Consider a low-13 (me/mi 3 8irn0Te/B~‘~ 1) plasma embedded in an external magnetic field B02, The dynamics of stationary high-frequency shortwavelength electron plasma wave turbulence is governed by the Liouville equation [2,31 ‘~
ai~
ai~ a.~k alk
(1) 2>/47rw* is the plasmon distribution where ‘k = (~ Ek function, = k24e, and vg = i(awk/ak) = ~k4e/wp~ is the group velocity of the plasmons. In the presence of a kinetic Alfvén wave (I2,q), the plasmon distribution changes. The perturbed distribution is obtained by linearizing (1). Assuming that the perturbations vary as ‘k = + ‘k exp (—iW+ iq ‘r), = ~o+ ~e exp(—i~2t+ iq r), we find
~ 4+
~ q.(~l4/lJk) ‘kTn~ (~2—q~u) (2) where awk/ar = ~ has been used. Here o.,.~= —
(4lrnoe4/me)112, and 1e is the electron density perturbations associated with a kinetic Alfvén wave. I)~is the unperturbed plasmon distribution function. The modified plasmon distribution reacts back on the kinetic Alfvén waves through the averaged ponderomotive force term Ek(uk V)Uk in the electron momentum equation. Here, Uk = —eEk/mewk is the velocity of the electrons in the fields of the plasmons. Neglecting the electron inertia, one [3] obtains from the z-compo-
nent of the electron momentum equation n /n0=eiJi/T [1—A/4n0T e
e
],
(3)
where (2) has been used. In (3), ti is the ambipolar potential, and q (a4lak) A = w~ ~ (4) k ‘~ q g’ To study the propagation of the kinetic Alfvén wave in a low-j3 plasma, we [2] use the two classical potential fields 0 and i~to describe the electric field E~= —açb/ax and E5 = —a~/az.They ptoduce only shear perturbations in the magnetic field. That is to say B~= B0(const), B~= 0, and ~=ca~(0a_~1). (5) t X z The z-component of the Ampèr law reads a4~ ~ ~ a2.i
E
~.
—
‘‘
ax
‘‘=
aZ
~
(6)
C
where (5) has been used. The contribution of the ions to the current density J~is negligible because of the low-(3 assumption. Thus, J~is given by
a~ a (7) The plasma wave turbulence does not affect the ion dynamics. The reason is that the direct action of the ponderomotive force on the ion motion is usually small by a factor me/mi as compared to other terms. The ions move in a plane perpendicular to the external magnetic field and are coupled to the electrons through the space charge fields. In a low-j3 plasma, the ion density for T~ ~0 is obtained from the ion continuity and momentum equations. We find z~t~e~
223
Volume 60A, number 3
an1
PHYSICS LETTERS
e
=0, (8) where for ~2~ w~(wd= eB 0/m1c) the parallel ion inertia is neglected. We now linearize the above equations in the usual manner. Eliminating 0~and ~i, and using the charge neutrality condition ~e = ~i1,we fmally obtain a dispersion relation. The latter governs the propagation of a kinetic Alfvén wave in the ~resence of Langmuir turbulence. We have q~v’~ = —(q~pjq~u~)2A/(4nØT~),
—
where u~ VA(1+ q~p~)1/2, p =c
/w~j
(9)
UA =
21 February 1977
3/2iT~2(6flOTe)~’/2 cificially,wehave[5J W 2 XeWpeL , (12) X (ii tanh ~ —In cosh ii)/k where i~= 6’I2kXe( Win 1I’2, and W is the total 0 TeY energy density of the system. Inserting (12) into (11) we obtain the damping of the kinetic Alfvén waves. For W/flOTe ~ 6 v~IVIe, the result is
j~
F
=
—(4 In ~
X
(UTe/ VA)3 (W/8lrfloTe)3”2.
+q~p~)2
(13)
(B~/41rn 0m1)h/2,and c~= (Te/mi)~2are respectively the Alfvén and ion-acoustic speeds. For Im f2 ~ Re ~, one can use the well-known Plemelj’s formula. In the resonance approximation (9) becomes 2u’~2 i (q~p ~ 2 rai~i k — q~ 1q~v~) 8flOTe 4e L —
(10) —
The author wishes to acknowledge the benefit of
(qzv~~i~~)
where we used the relation ~k -÷ L/2irfdk, L being the size of the system. Letting &2 = q~v’~ + ir, where F ~ ~ we obtain the damping rate ~
~=
8?2OTe e =
VA(VTeXe)
,
Wi,eL
ra4i [~]
useful discussions with Prof. Akira Hasegawa. This work is supported by the Sonderforschungsbereich 162 Plasmaphysik Bochum/JUlich.
References
IC
(11)
where Xe = VTe1CO is the electron Debeye length. Clearly, dampi~goccurs if ~1~/~k <0. As an illustration, we use a model spectrum [4,5] for the Lang. muir turbulence. This spectrum was first obtained by Kingsep et a!. [4j to describe a gas consisting of an ensemble of randomly distributed plasmons. For large k, they showed that the spectrum goes like k2. Spe-
224
In summary, we have investigated the anomalous damping of a kinetic Alfvén wave in turbulent plasmas. It has been shown that the nonlinear interaction of the latter with the low-frequency kinetic Alfvén wave leads to a damping. Our results are expected to be useful in large scale wave turbulent plasmas in which a coherent tic Alfvén is used for heating purposes. One kineshould then take into account the anomalous damping discussed in this note.
[1] A. Hasegawa and L. Chen, Phys. Rev. Lett. 36 (1976) 1362. [2] B.B. Kadomtsev, Plasma Turbulence (Academic Press, New York, 1965) p. 82. [3] A.A. Vedenov and L.I. Rudakov, Soc. Phys. Dokiady 9 (1965) 1073; A.A. Vedenov, A.V. Gordeev and L.I. Rudakov, Plasma Phys. 9 (1967) 719. [4] A.S. Kingsep, L.I. Rudakov and R.N. Sudan, Phys. Rev. Lett. 31(1973)1482. [5] M.Y. Yu and K.H. Spatschek, Phys. Fluids 19 (1976) 705.
PHYSICS LETTERS
21 February 1977
KINETIC ALFVEN WAVES IN TURBULENT PLASMAS P.K. SHUKLA Institutefür Theoretische Physik, Ruhr Universitiit Bochum, 453 Bochum, West Germany Received 6 October 1976 The anomalous damping of a coherent kinetic Alfv~nwave in turbulent plasmas is investigated.
The kinetic Alfvén waves are expected [1] to play an important role in large scale plasmas. In this note, we consider the propagation of a kinetic Alfven wave in a turbulent plasma. It is found that the nonlinear interaction of the high-frequency randomly distributed plasmons with the low-frequency kinetic Alfvén wave leads to a damping. The analytical expression for the damping rate is presented. Consider a low-13 (me/mi 3 8irn0Te/B~‘~ 1) plasma embedded in an external magnetic field B02, The dynamics of stationary high-frequency shortwavelength electron plasma wave turbulence is governed by the Liouville equation [2,31 ‘~
ai~
ai~ a.~k alk
(1) 2>/47rw* is the plasmon distribution where ‘k = (~ Ek function, = k24e, and vg = i(awk/ak) = ~k4e/wp~ is the group velocity of the plasmons. In the presence of a kinetic Alfvén wave (I2,q), the plasmon distribution changes. The perturbed distribution is obtained by linearizing (1). Assuming that the perturbations vary as ‘k = + ‘k exp (—iW+ iq ‘r), = ~o+ ~e exp(—i~2t+ iq r), we find
~ 4+
~ q.(~l4/lJk) ‘kTn~ (~2—q~u) (2) where awk/ar = ~ has been used. Here o.,.~= —
(4lrnoe4/me)112, and 1e is the electron density perturbations associated with a kinetic Alfvén wave. I)~is the unperturbed plasmon distribution function. The modified plasmon distribution reacts back on the kinetic Alfvén waves through the averaged ponderomotive force term Ek(uk V)Uk in the electron momentum equation. Here, Uk = —eEk/mewk is the velocity of the electrons in the fields of the plasmons. Neglecting the electron inertia, one [3] obtains from the z-compo-
nent of the electron momentum equation n /n0=eiJi/T [1—A/4n0T e
e
],
(3)
where (2) has been used. In (3), ti is the ambipolar potential, and q (a4lak) A = w~ ~ (4) k ‘~ q g’ To study the propagation of the kinetic Alfvén wave in a low-j3 plasma, we [2] use the two classical potential fields 0 and i~to describe the electric field E~= —açb/ax and E5 = —a~/az.They ptoduce only shear perturbations in the magnetic field. That is to say B~= B0(const), B~= 0, and ~=ca~(0a_~1). (5) t X z The z-component of the Ampèr law reads a4~ ~ ~ a2.i
E
~.
—
‘‘
ax
‘‘=
aZ
~
(6)
C
where (5) has been used. The contribution of the ions to the current density J~is negligible because of the low-(3 assumption. Thus, J~is given by
a~ a (7) The plasma wave turbulence does not affect the ion dynamics. The reason is that the direct action of the ponderomotive force on the ion motion is usually small by a factor me/mi as compared to other terms. The ions move in a plane perpendicular to the external magnetic field and are coupled to the electrons through the space charge fields. In a low-j3 plasma, the ion density for T~ ~0 is obtained from the ion continuity and momentum equations. We find z~t~e~
223
Volume 60A, number 3
an1
PHYSICS LETTERS
e
=0, (8) where for ~2~ w~(wd= eB 0/m1c) the parallel ion inertia is neglected. We now linearize the above equations in the usual manner. Eliminating 0~and ~i, and using the charge neutrality condition ~e = ~i1,we fmally obtain a dispersion relation. The latter governs the propagation of a kinetic Alfvén wave in the ~resence of Langmuir turbulence. We have q~v’~ = —(q~pjq~u~)2A/(4nØT~),
—
where u~ VA(1+ q~p~)1/2, p =c
/w~j
(9)
UA =
21 February 1977
3/2iT~2(6flOTe)~’/2 cificially,wehave[5J W 2 XeWpeL , (12) X (ii tanh ~ —In cosh ii)/k where i~= 6’I2kXe( Win 1I’2, and W is the total 0 TeY energy density of the system. Inserting (12) into (11) we obtain the damping of the kinetic Alfvén waves. For W/flOTe ~ 6 v~IVIe, the result is
j~
F
=
—(4 In ~
X
(UTe/ VA)3 (W/8lrfloTe)3”2.
+q~p~)2
(13)
(B~/41rn 0m1)h/2,and c~= (Te/mi)~2are respectively the Alfvén and ion-acoustic speeds. For Im f2 ~ Re ~, one can use the well-known Plemelj’s formula. In the resonance approximation (9) becomes 2u’~2 i (q~p ~ 2 rai~i k — q~ 1q~v~) 8flOTe 4e L —
(10) —
The author wishes to acknowledge the benefit of
(qzv~~i~~)
where we used the relation ~k -÷ L/2irfdk, L being the size of the system. Letting &2 = q~v’~ + ir, where F ~ ~ we obtain the damping rate ~
~=
8?2OTe e =
VA(VTeXe)
,
Wi,eL
ra4i [~]
useful discussions with Prof. Akira Hasegawa. This work is supported by the Sonderforschungsbereich 162 Plasmaphysik Bochum/JUlich.
References
IC
(11)
where Xe = VTe1CO is the electron Debeye length. Clearly, dampi~goccurs if ~1~/~k <0. As an illustration, we use a model spectrum [4,5] for the Lang. muir turbulence. This spectrum was first obtained by Kingsep et a!. [4j to describe a gas consisting of an ensemble of randomly distributed plasmons. For large k, they showed that the spectrum goes like k2. Spe-
224
In summary, we have investigated the anomalous damping of a kinetic Alfvén wave in turbulent plasmas. It has been shown that the nonlinear interaction of the latter with the low-frequency kinetic Alfvén wave leads to a damping. Our results are expected to be useful in large scale wave turbulent plasmas in which a coherent tic Alfvén is used for heating purposes. One kineshould then take into account the anomalous damping discussed in this note.
[1] A. Hasegawa and L. Chen, Phys. Rev. Lett. 36 (1976) 1362. [2] B.B. Kadomtsev, Plasma Turbulence (Academic Press, New York, 1965) p. 82. [3] A.A. Vedenov and L.I. Rudakov, Soc. Phys. Dokiady 9 (1965) 1073; A.A. Vedenov, A.V. Gordeev and L.I. Rudakov, Plasma Phys. 9 (1967) 719. [4] A.S. Kingsep, L.I. Rudakov and R.N. Sudan, Phys. Rev. Lett. 31(1973)1482. [5] M.Y. Yu and K.H. Spatschek, Phys. Fluids 19 (1976) 705.