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Scattering of electromagnetic waves by drift modes
Volume 106A, number 4
PHYSICS LETTERS
3 December 1984
SCATI'ERING OF ELECTROMAGNETIC WAVES BY DRIFT MODES
P.K. SHUKLA, M.Y. YU and Adel EL-NADI 1 Institut ffir Theoretische Physik, Ruhr-Universitiit Bochum. 4630 Bochum 1, Fed Rep. Germany Received 10 September 1984
It is shown that high-frequency electromagnetic waves can parametrically excite the convection and ion drift waves in a slightly inhomogeneous magnetized plasma. The growth rates of the nonlinear decay instabilities are obtained analytically.
A number of authors [ 1-4] has investigated the nonlinear decay of electromagnetic radiation in a uniform magnetized plasma. Stimulated scattering of strong high-frequency radio waves off ion waves has been observed [5] in the earth's ionosphere. On the other hand, in a nonuniform plasma, the high-frequency waves can also efficiently scatter off the low-frequency (w < Wci = ion gyrofrequency) drift waves [6-8]. Recently, the high-frequency (co ~ Wci) electrostatic ion drift [9] and convection waves [10] have been shown to play an important role in a weakly inhomogeneous plasma. Since the high-frequency electromagnetic waves are employed for heating as well as diagnostic purposes, it is of practical interest to investigate the coupling of radiation with convection modes in a nonuniform plasma. In this note, we show that a highfrequency electromagnetic wave can excite nonthermal ion drift as well as convection waves. It is well known that the excitation of low-frequency electrostatic waves (oA k) by a high-frequency (much larger than the electron plasma frequency ~Ope, the electron gyrofrequency 6Oce, as well as w) electromagnetic wave (c~0, k0) is governed by the equation [2,7] 1
1
- - + - - ~
Xe
1 + Xi
k2tr~0kb2 2oa0(~- 6)'
(1)
where 6 is the frequency shift [2], xp = sin ¢ cos 0 (in
general, ~ ~ 1 and k ~ 2k0) , 0 is the angle between k and k0, q~is the angle between k (=k - k0) and u 0 (-qeE/me6O 0 is the constant amplitude of the electron quiver velocity in the electromagnetic pump field E0). Here, qe ( = - e ) and m e stand for the electronic charge and mass, respectively. Eq. (1) describes the three-wave decay interaction process in which the high-frequency electromagnetic Stokes sideband ( c o , k _ ) and the low-frequency electrostatic mode are excited at the expense of the pump. The low-frequency dielectric susceptibility X/of species/(=e for the electrons, =i for the ions) for flute modes is given by [7,11]
X~ k2X----~n/- 1 -
X
1
--~
-
~ _ tl C..,3cj 1
-
b/t~c/
)11
2
2
-
"
2
-
,
(2)
2
where wo/.. = 4nnoe~ /rn/,. XDi. = T//4nnoe. , A = In(b'/) X exp(-b/), b/= k~.T//m/6o2), COc/= q/Bo/m/c, o~ = - k y gcTj/qjBo, tc = -n61 dno/dx represents the inverse length of the inhomogeneity, B 0 is the stre ~gth of the external magnetic field directed along the z axis, T~ (m/) is the temperature (mass) of species/, c is the velocity of light, and I n is the modified Bessel function of order n. In the zero Larmor radius ( b / ~ 1) approximation, eq. (2) yields the well-known cold plasma result [12],
t Permanent address: Electrical Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt.
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
169
Volume 106A, number 4
3 December 1984
PHYSICS LETTERS T D ~ ( 1 + 4k0Pa) 2 2 -1 ")'i,
602-~o2" l 60 ky " (3) c/ On the other hand, for 60 >> 60ci, Ti ~ Te, and b i >> 1, one finds Xj =
Xi = (kXDi)-2 .
(4)
Thus, the unmagnetized ions follow the straight-line orbit in the electrostatic wave field. First, we consider the excitation of the electron convection mode. Accordingly, we assume 60 "~ R e and write eq. (3) as
60 e (1 + 160.1
where L n 1 = -to. For 60 >~ 60pi, 60ci, one can set Xi 0. Substituting (5) into eq. (1), letting 60 = iTe + 6or, 2 2 2 ,~ • 60r = 60pe/[ WcelkyLn (1 + 60pe/60ce) 6, one obtains the growth rate
Ik0o0t
160oe11/2
7e ~ [k0--~n il/2 [ -~0-'0[
60 e
6027602e,
(6)
where [koyL n I > 602e/(602e + 60c2e). Secondly, we consider the excitation of the ion convection wave [ 10]. Here, we take 60ci '~ co "~ Iwce60ci 11/2 and 1 @ IkyLn[ "~ (m~me)l[2. Thus, the appropriate ×i for our purposes are Xe = 602i/6060v ,
(7)
×i = -Jpi/602 ,
(8)
where 60v = 60ciky ILn 1(~"60ci) is the frequency of the convection wave [ 10]. Physically, the latter arises when the ion polarization current exactly balances the electron current generated by the coupling of the cV¢ X Bo/B 2 flow with the density gradient Vn 0. Inserting eqs. (7) and (8) into eq. (1), letting 60 = Wv + iTi and w v ~ 6, one finds the growth rate Ik0o0 1
60pi
7i - [koyLn 11/2 (60060ei)1/2
where Pa2 = Ti/me602e" Clearly, TD < ~i for k2p2a >>, 1. The threshold is determined by [7] v0>F
c2
Ve
60 600 600
co
~)
1 +
1
'
(ll)
where F is the linear damping rate of the excited lowfrequency mode, and v e is the electron ion collision frequency. It is easy to verify that 2 2 -1 ")'e/Ti = (mi/me)l/2(Ok/C)( 1 + 60ce/60pe) ,
(12)
where v A = (~7,i/60pi)C is the Alfv6n speed. Thus, 7e Ti for OA/C ~ (me/mi) I/2. Taking some typical values for plasmas produced by Nd glass lasers, e.g., 6o0 ~ 1.8 X 10 15 rad s -1 , Wce 3 X 10 13 rad s -1, COpe ~ 3 X 10 14 rad s -1, Oo/C 10 -3, L n ~ 10/lm, mi/m e = 1836, one finds 7i 10 11 rad s -1 , which is of the same order as the growth rate obtained by Stenflo [7]. In summary, we have presented a nonlinear mechanism by which enhanced convection and ion drift waves can be created in a nonuniform plasma. The present nonlinear effects can have important consequences to plasma confinement. For example, nonthermal high-frequency drift-type fluctuations can give rise to anomalous cross-field particle transport [9]. On the other hand, the scattering of electromagnetic waves by density fluctuations can provide useful information on various plasma parameters in a nonuniform magnetized plasma. This research was performed under the auspices of the Sonderforschungsbereich 162 Plasmaphysik Bochum/Jiilich. One of the authors (A. EI-Nadi) would like to thank G. Ecker for hospitality during his stay in Bochum.
References
(9)
Finally, we consider the excitation o f the ion drift wave in a plasma with T i ~, T e. Here, we substitute eqs. (4) and (5) into eq. (1), let 60 = 60D + i~'D, assume 6 ~ 60D = 1607 I/(1 + k2p2a), and find the growth rate well above threshold
170
(10)
[11 D.W. Forslund, J.M. Kindel and E.L. Lindman, Phys. Rev. Lett. 29 (1972) 249; P.K. Shukla, M.Y. Yu and K.H. Spatschek, Phys. Fluids 18 (1975) 265. [21 M.Y. Yu, P.K. Shukla and K.H. Spatscbek, Z. Naturforsch. 29A (1974) 1736; S. Bujarbatua, A. Sen and P.K. Kaw, Phys. Lett. 47A (1974)464; P.K. Shukla and S.G. Tagare, J. Geophys. Res. 84 (1979) 1317.
Volume 106A, number 4
PHYSICS LETTERS
[ 3 ] K.B. Dysthe, E. Leer, J. Trulsen and L. Stenflo, J. Geophys. Res. 82 (1977) 717; L. Stenflo and J. Trulsen, J. Geophys. Res. 83 (1978) 1154. [4] V.K. Tripathi and R.R. Sharma, Phys. Fluids 22 (1979) 1799; G. Grebogi and C.S. Liu, Phys. Fluids 23 (1980) 1330; V. Stefan and N.A. Krall, Phys. Rev. Lett. 52 (1984) 41. [5] B. Thide, H. Kopka and P. Stubbe, Phys. Rev. Lett. 49 (1982) 1561. [6] S. Bujarbarua, A. Sen and P.K. Kaw, Plasma Phys. 18 (1976) 171; P.K. Shukla and S. Bujarbarua, J. Geophys. Res. 85 (1980) 1773; P.K. Shukla and M.Y. Yu, Phys. Lett. 82A (1981) 18.
3 December 1984
[7] L. Stenflo, Phys. Rev. A23 (1981) 2730; Radio Sci. 18 (1983) 1379. [8] S.N. Antani and D.J. Kaup, Phys. Fluids 27 (1984) 1908. [9] P.K. Shukla, M.Y. Yu, H.U. Rahman and K.H. Spatschek, Phys. Rep. 105 (1984) 227. [101 A. EI-Nadi, Phys. Fluids 25 (1982) 2019; A.L. Berk and R.R. Dominguez, Phys. Fluids 26 (1983) 1825. [ 11 ] K. Miyamoto, Plasma physics for nucleax fusion (MIT, Cambridge, MA, 1980) ch. 12, p. 283. [ 12 ] A.B. Mikhaiiovskii, Theory of plasma instabilities (Consultants Bureau, London, 1974) VoL 2.
171
PHYSICS LETTERS
3 December 1984
SCATI'ERING OF ELECTROMAGNETIC WAVES BY DRIFT MODES
P.K. SHUKLA, M.Y. YU and Adel EL-NADI 1 Institut ffir Theoretische Physik, Ruhr-Universitiit Bochum. 4630 Bochum 1, Fed Rep. Germany Received 10 September 1984
It is shown that high-frequency electromagnetic waves can parametrically excite the convection and ion drift waves in a slightly inhomogeneous magnetized plasma. The growth rates of the nonlinear decay instabilities are obtained analytically.
A number of authors [ 1-4] has investigated the nonlinear decay of electromagnetic radiation in a uniform magnetized plasma. Stimulated scattering of strong high-frequency radio waves off ion waves has been observed [5] in the earth's ionosphere. On the other hand, in a nonuniform plasma, the high-frequency waves can also efficiently scatter off the low-frequency (w < Wci = ion gyrofrequency) drift waves [6-8]. Recently, the high-frequency (co ~ Wci) electrostatic ion drift [9] and convection waves [10] have been shown to play an important role in a weakly inhomogeneous plasma. Since the high-frequency electromagnetic waves are employed for heating as well as diagnostic purposes, it is of practical interest to investigate the coupling of radiation with convection modes in a nonuniform plasma. In this note, we show that a highfrequency electromagnetic wave can excite nonthermal ion drift as well as convection waves. It is well known that the excitation of low-frequency electrostatic waves (oA k) by a high-frequency (much larger than the electron plasma frequency ~Ope, the electron gyrofrequency 6Oce, as well as w) electromagnetic wave (c~0, k0) is governed by the equation [2,7] 1
1
- - + - - ~
Xe
1 + Xi
k2tr~0kb2 2oa0(~- 6)'
(1)
where 6 is the frequency shift [2], xp = sin ¢ cos 0 (in
general, ~ ~ 1 and k ~ 2k0) , 0 is the angle between k and k0, q~is the angle between k (=k - k0) and u 0 (-qeE/me6O 0 is the constant amplitude of the electron quiver velocity in the electromagnetic pump field E0). Here, qe ( = - e ) and m e stand for the electronic charge and mass, respectively. Eq. (1) describes the three-wave decay interaction process in which the high-frequency electromagnetic Stokes sideband ( c o , k _ ) and the low-frequency electrostatic mode are excited at the expense of the pump. The low-frequency dielectric susceptibility X/of species/(=e for the electrons, =i for the ions) for flute modes is given by [7,11]
X~ k2X----~n/- 1 -
X
1
--~
-
~ _ tl C..,3cj 1
-
b/t~c/
)11
2
2
-
"
2
-
,
(2)
2
where wo/.. = 4nnoe~ /rn/,. XDi. = T//4nnoe. , A = In(b'/) X exp(-b/), b/= k~.T//m/6o2), COc/= q/Bo/m/c, o~ = - k y gcTj/qjBo, tc = -n61 dno/dx represents the inverse length of the inhomogeneity, B 0 is the stre ~gth of the external magnetic field directed along the z axis, T~ (m/) is the temperature (mass) of species/, c is the velocity of light, and I n is the modified Bessel function of order n. In the zero Larmor radius ( b / ~ 1) approximation, eq. (2) yields the well-known cold plasma result [12],
t Permanent address: Electrical Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt.
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
169
Volume 106A, number 4
3 December 1984
PHYSICS LETTERS T D ~ ( 1 + 4k0Pa) 2 2 -1 ")'i,
602-~o2" l 60 ky " (3) c/ On the other hand, for 60 >> 60ci, Ti ~ Te, and b i >> 1, one finds Xj =
Xi = (kXDi)-2 .
(4)
Thus, the unmagnetized ions follow the straight-line orbit in the electrostatic wave field. First, we consider the excitation of the electron convection mode. Accordingly, we assume 60 "~ R e and write eq. (3) as
60 e (1 + 160.1
where L n 1 = -to. For 60 >~ 60pi, 60ci, one can set Xi 0. Substituting (5) into eq. (1), letting 60 = iTe + 6or, 2 2 2 ,~ • 60r = 60pe/[ WcelkyLn (1 + 60pe/60ce) 6, one obtains the growth rate
Ik0o0t
160oe11/2
7e ~ [k0--~n il/2 [ -~0-'0[
60 e
6027602e,
(6)
where [koyL n I > 602e/(602e + 60c2e). Secondly, we consider the excitation of the ion convection wave [ 10]. Here, we take 60ci '~ co "~ Iwce60ci 11/2 and 1 @ IkyLn[ "~ (m~me)l[2. Thus, the appropriate ×i for our purposes are Xe = 602i/6060v ,
(7)
×i = -Jpi/602 ,
(8)
where 60v = 60ciky ILn 1(~"60ci) is the frequency of the convection wave [ 10]. Physically, the latter arises when the ion polarization current exactly balances the electron current generated by the coupling of the cV¢ X Bo/B 2 flow with the density gradient Vn 0. Inserting eqs. (7) and (8) into eq. (1), letting 60 = Wv + iTi and w v ~ 6, one finds the growth rate Ik0o0 1
60pi
7i - [koyLn 11/2 (60060ei)1/2
where Pa2 = Ti/me602e" Clearly, TD < ~i for k2p2a >>, 1. The threshold is determined by [7] v0>F
c2
Ve
60 600 600
co
~)
1 +
1
'
(ll)
where F is the linear damping rate of the excited lowfrequency mode, and v e is the electron ion collision frequency. It is easy to verify that 2 2 -1 ")'e/Ti = (mi/me)l/2(Ok/C)( 1 + 60ce/60pe) ,
(12)
where v A = (~7,i/60pi)C is the Alfv6n speed. Thus, 7e Ti for OA/C ~ (me/mi) I/2. Taking some typical values for plasmas produced by Nd glass lasers, e.g., 6o0 ~ 1.8 X 10 15 rad s -1 , Wce 3 X 10 13 rad s -1, COpe ~ 3 X 10 14 rad s -1, Oo/C 10 -3, L n ~ 10/lm, mi/m e = 1836, one finds 7i 10 11 rad s -1 , which is of the same order as the growth rate obtained by Stenflo [7]. In summary, we have presented a nonlinear mechanism by which enhanced convection and ion drift waves can be created in a nonuniform plasma. The present nonlinear effects can have important consequences to plasma confinement. For example, nonthermal high-frequency drift-type fluctuations can give rise to anomalous cross-field particle transport [9]. On the other hand, the scattering of electromagnetic waves by density fluctuations can provide useful information on various plasma parameters in a nonuniform magnetized plasma. This research was performed under the auspices of the Sonderforschungsbereich 162 Plasmaphysik Bochum/Jiilich. One of the authors (A. EI-Nadi) would like to thank G. Ecker for hospitality during his stay in Bochum.
References
(9)
Finally, we consider the excitation o f the ion drift wave in a plasma with T i ~, T e. Here, we substitute eqs. (4) and (5) into eq. (1), let 60 = 60D + i~'D, assume 6 ~ 60D = 1607 I/(1 + k2p2a), and find the growth rate well above threshold
170
(10)
[11 D.W. Forslund, J.M. Kindel and E.L. Lindman, Phys. Rev. Lett. 29 (1972) 249; P.K. Shukla, M.Y. Yu and K.H. Spatschek, Phys. Fluids 18 (1975) 265. [21 M.Y. Yu, P.K. Shukla and K.H. Spatscbek, Z. Naturforsch. 29A (1974) 1736; S. Bujarbatua, A. Sen and P.K. Kaw, Phys. Lett. 47A (1974)464; P.K. Shukla and S.G. Tagare, J. Geophys. Res. 84 (1979) 1317.
Volume 106A, number 4
PHYSICS LETTERS
[ 3 ] K.B. Dysthe, E. Leer, J. Trulsen and L. Stenflo, J. Geophys. Res. 82 (1977) 717; L. Stenflo and J. Trulsen, J. Geophys. Res. 83 (1978) 1154. [4] V.K. Tripathi and R.R. Sharma, Phys. Fluids 22 (1979) 1799; G. Grebogi and C.S. Liu, Phys. Fluids 23 (1980) 1330; V. Stefan and N.A. Krall, Phys. Rev. Lett. 52 (1984) 41. [5] B. Thide, H. Kopka and P. Stubbe, Phys. Rev. Lett. 49 (1982) 1561. [6] S. Bujarbarua, A. Sen and P.K. Kaw, Plasma Phys. 18 (1976) 171; P.K. Shukla and S. Bujarbarua, J. Geophys. Res. 85 (1980) 1773; P.K. Shukla and M.Y. Yu, Phys. Lett. 82A (1981) 18.
3 December 1984
[7] L. Stenflo, Phys. Rev. A23 (1981) 2730; Radio Sci. 18 (1983) 1379. [8] S.N. Antani and D.J. Kaup, Phys. Fluids 27 (1984) 1908. [9] P.K. Shukla, M.Y. Yu, H.U. Rahman and K.H. Spatschek, Phys. Rep. 105 (1984) 227. [101 A. EI-Nadi, Phys. Fluids 25 (1982) 2019; A.L. Berk and R.R. Dominguez, Phys. Fluids 26 (1983) 1825. [ 11 ] K. Miyamoto, Plasma physics for nucleax fusion (MIT, Cambridge, MA, 1980) ch. 12, p. 283. [ 12 ] A.B. Mikhaiiovskii, Theory of plasma instabilities (Consultants Bureau, London, 1974) VoL 2.
171