Modulation of electron-acoustic waves

Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 355 – 358

www.elsevier.com/locate/jastp

Modulation of electron-acoustic waves P.K. Shuklaa;∗;1 , M.A. Hellbergb , L. Sten/oc a Institut

fur Theoretische Physik IV, Fakultat fur Physik und Astronomie, Ruhr-Universitat Bochum, D-44780 Bochum, Germany b School of Pure and Applied Physics, University of Natal, Durban 4041, South Africa c Department of Plasma Physics, Umea* University, SE-90187 Umea, * Sweden Received 15 April 2002; received in revised form 17 September 2002; accepted 28 November 2002

Abstract FAST observations have indicated signatures of large amplitude solitary waves in the auroral zone of the earth’s ionosphere. Our objective here is to propose a model for the generation of density cavities by the ponderomotive force of electron-acoustic waves. For this purpose, we derive a nonlinear Schr8odinger equation for the electron-acoustic wave envelope as well as a driven (by the electron-acoustic wave ponderomotive force) ion-acoustic wave equation. Possible stationary solutions of our coupled equations are obtained. c 2003 Elsevier Science Ltd. All rights reserved.  Keywords: Electron and ion acoustic waves; Ponderomotive force; Nonlinear phenomena; Modulational instabilities; Density cavities

In interpreting =eld and particle observations by the FAST satellite, it has been speculated in a somewhat questionable way that intense packets of electron-acoustic waves in association with two-electron populations in the earth’s auroral ionosphere (Pottelette et al., 1999) may be present. The two-electron populations should then comprise a hot (∼keV) component and a minority cold (¡ 60 eV) component. In such a two-component electron plasma, there are electrostatic electron-acoustic waves (Watanabe and Taniuti, 1977; Yu and Shukla, 1983; Tokar and Gary, 1984; Mace and Hellberg, 1990) in which the restoring force comes from the pressure of the hot electrons, and the mass of the cold electron component provides the inertia. The electron-acoustic wave frequency is typically ! = kD !pc =(1 + k 2 D2 )1=2 , where k is the wave number, D = (Te =4nh0 e2 )1=2 is the Debye radius involving the temperature (Te ) and the number density (nh0 ) of the hot electron component, !pc = (4nc0 e2 =me )1=2 is the ∗

Corresponding author. E-mail addresses: [email protected] (P.K. Shukla), [email protected] (M.A. Hellberg). 1 Also at the Department of Plasma Physics, Umea I University, SE-90187 UmeaI , Sweden.

plasma frequency involving the number density (nc0 ) of the cold electron component, e is the magnitude of the electron charge and me is the electron mass. The phase speed of long-wavelength (in comparison with D ) waves is (nc0 =nh0 )1=2 VTh , where VTh = (Te =me )1=2 is the hot electron thermal speed. Since the electron-acoustic wave phase velocity is much smaller (larger) than the hot (cold) electron thermal speed, one requires that nc0 =nh0
1 and that the ratio of the hot to cold electron temperatures is very large. Furthermore, the electron-acoustic wave frequency is much larger than the ion plasma frequency !pi [!pi = (4ni0 e2 =mi )1=2 , where ni0 = nh0 + nc0 ≡ ne0 and mi is the ion mass], as the ions remain stationary on the time scale of the electron-acoustic wave period. Hence, for electron-acoustic waves, we have kD (me =mi )1=2 . In a magnetized plasma, the electron-acoustic wave may also couple to the electron cyclotron mode (Mace and Hellberg, 1993). Finite amplitude electron-acoustic waves can be generated due to numerous processes (Tokar and Gary, 1984; Singh and Lakhina, 2001). Recently, the properties of EA solitary waves in a magnetized plasma have been investigated (Berthomier et al., 2000; Mace and Hellberg, 2001). The nonlinear propagation of electron-acoustic waves has also been considered by Singh et al. (2001).

c 2003 Elsevier Science Ltd. All rights reserved. 1364-6826/03/$ - see front matter
doi:10.1016/S1364-6826(02)00334-6

356

P.K. Shukla et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 355 – 358

Furthermore, in multicomponent space plasmas there also appear the usual ion-acoustic waves whose phase velocity (frequency) is much smaller than VTh (!pi ). In the ion-acoustic waves, the restoring force comes from the pressure of the inertialess hot electrons, while the ion mass provides the inertia. The ion-acoustic wave frequency is

= KCs =(1 + K 2 D2 )1=2 , where K is the wave number and Cs = (Te =mi )1=2 is the ion-acoustic speed. Clearly, electron and ion acoustic waves appear on two distinct time scales that are longer than those of the Langmuir waves. We note that the phase speeds of the latter are much larger than VTh and that their frequencies are slightly above the electron plasma frequency. The nonlinear coupling between Langmuir and ion-acoustic waves in two-electron temperature plasmas has been considered by several authors (Khirseli and Tsintsadze, 1980; Hanssen et al., 1994). In this paper, motivated by the speculations of Pottelette et al. (1999), we consider the nonlinear coupling between electron- and ion-acoustic waves in space plasmas. Speci=cally, we show that the ponderomotive force of the electron-acoustic waves provides the possibility of reinforcing the ion-acoustic waves, which, in turn, cause amplitude modulation of the electron-acoustic waves. This process can describe the experimentally observed scenario in the auroral zone of the Earth’s ionosphere. The dynamics of one-dimensional electron-acoustic waves in the presence of ion-acoustic waves is governed by Poisson’s equation 92  = 4e(nh1 + nc1 ); 9x2

(1)

the inertialess equation of motion for hot electrons nh1 = (nh0 + nhs )

e ; Te

(2)

the continuity and momentum equations for the cold electrons 9nc1 9 + [(nc0 + ncs )vc ] = 0 9t 9x

(3)

and 9vc e 9 ; = me 9x 9t

(4)

where  is the electron-acoustic wave potential and nhs (ncs ) is the density perturbation of the hot (cold) electron component involved in the ion-acoustic wave dynamics. The divergence of the /ux nc1 vcs and the advection nonlinearities, viz (vcs 9vc =9x) + (vc 9vcs =9x), are negligible since the frequency of the ion-acoustic perturbation is much smaller than the electron-acoustic wave frequency.

Combining Eqs. (1)–(4), we readily obtain the electron-acoustic wave equation
 2  2  9 9 92 2 D2  + ! − pc 9t 2 9x2 9t 2     9 ncs 9 92 nhs  − Ce2 ≈ 2 ; (5) 9t nh0 9x nc0 9x where Ce = D !pc is the electron-acoustic speed. The right-hand side of Eq. (5) exhibits the coupling between the electron-acoustic wave potential and the electron density /uctuations of the ion-acoustic waves. Next, we derive the relevant equation for the ion-acoustic waves in the presence of the ponderomotive force of the electron-acoustic waves. The governing equations for the latter are the ion continuity equation 9nis 9vis + ni0 = 0; 9t 9x

(6)

the ion momentum equation mi

9vis 9 i Ti 9nis = −e − 9t 9x ni0 9x

and the inertialess electron momentum equation  2   nh0 9  e  9 me nh0 2 Te 9nhs Ce 1 + =e − ; 4 ni0 nc0 9x  Te  9x ni0 9x

(7)

(8)

where nis ( ) is the ion number density perturbation (electrostatic potential) of the ion-acoustic waves, vis is the ion /uid velocity perturbation, i is the adiabatic index, and Ti is the ion temperature. The left-hand side of Eq. (8) comes from   9 2 9 2 me nh0 (9) vh  + nc0 vc  ; 9x 9x 2ni0 where vh = (!=k)e=Te and vc = −(e=me !)k are the quiver velocity of the hot and cold electrons in the electron-acoustic wave =eld, and the angular bracket denotes averaging over 2=!. The =rst (second) term in Eq. (9) represents the contribution of the hot (cold) electrons to the ponderomotive force. To derive Eq. (9) we also used the approximation kD
1. Furthermore, the contribution of the ion ponderomotive force in Eq. (7) has been neglected because it is me =mi times smaller than the electron ponderomotive force [the left-hand side of Eq. (8)]. Besides, we also neglected the self-interaction nonlinearities in Eqs. (6)–(8) as we focus on small amplitude ion-acoustic waves that are driven by the ponderomotive force of the high-frequency electron-acoustic waves. Invoking the quasi-neutrality condition, nis ≈ nhs , which is justi=ed since the cold electrons do not participate in the ion-acoustic wave dynamics due to their small random motion, we combine Eqs. (6)–(8) to obtain  2  2 9 nhs e2 (1 + )Cs2 92 2 9 − C = ||2 ; (10) a 9t 2 9x2 nh0 9x2 4Te2

P.K. Shukla et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 355 – 358

where Ca2 =(Te +i Ti )=mi and =nc0 =nh0 . We have assumed that for 92 nhs =9t 2
Ca2 92 nhs =9x2 , Eq. (10) gives nhs e2 (1 + )Cs2 =− ||2 ; nh0 4Te2 Ca2

(11)

which exhibits the generation of a density cavity by the ponderomotive force of the electron-acoustic waves. Eqs. (5) and (10) are the desired pair describing the nonlinear coupling between electron- and ion-acoustic waves in space plasmas. In the following, we consider the parametric excitation of ion-acoustic perturbations on account of the nonlinear coupling between the electron-acoustic pump and its sidebands. Accordingly, we decompose the wave potential as  = 0 exp(ikx − i!t) + complex conjugate  + ± exp(ik± x − i!± t);

(12)

±

D± ± ≈ −!2

nhs 0± nh0

(13)

which exhibits an oscillatory instability. Next, we focus on the modulational instabilities (Anderson et al., 1999). Here we have D± ≈ 0 and s = 0. Accordingly, we obtain from Eq. (15) ( 2 − s2 )(1 + k 2 D2 )[(! − KVg )2 − 2 ] =−

e2 K 2 Cs2 !|0 |2 ; 4Te2

(19)

which can be numerically analyzed to obtain the growth rate of the modulational instabilities. However, for ; 
KVg , we have from Eq. (19)

2 = s2 −

e2 Cs2 !|0 |2 : 4Te2 Vg2 (1 + k 2 D2 )

nhs e2 K 2 Cs2 (1 + ) = (0+ − + 0− + ); s nh0 4Te2

(14)

2 2 2 2 where D± = (1 + k± D )!± − k± Ce2 , s = 2 − K 2 Ca2 ≡

2 − s2 , 0+ ≡ 0 , 0− = ∗0 , and the asterisk denotes the complex conjugate. In deriving Eq. (13) we have assumed that nhs =nh0 ncs =nc0 , which is consistent with the large density variations of the hot electrons that control the dynamics of the ion-acoustic perturbations. Combining Eqs. (13) and (14) and assuming 
1 we obtain the nonlinear dispersion relation

e2 K 2 Cs2 !2 |0 |2  −1 D± : 4Te2 +;−

(15)

For
! and K
k, we have D± ≈ ±2!(1 + k 2 D2 )( − KVg ∓ );

(16)

where Vg = kCe2 =! and  = k 2 Ce2 =2!. Assuming that D+ ≈ 0, we obtain from Eqs. (15) and (16) ( 2 − s2 )(1 + k 2 D2 )( − KVg − ) e2 K 2 Cs2 !|0 |2 : 8Te2

|0 |2 ¿

4Te2 Vg2 s2 (1 + k 2 D2 ) : e2 Cs2 !

(20)

(17)

(21)

Let us now enquire possible nonlinear states of modulationally unstable waves. Supposing that the nonlinear couplings between the electron- and ion-acoustic waves produce envelope of waves, we represent  = ’() exp(ikx − i!t)

and

=−

For  s , KCe , , Eq. (17) admits a complex frequency √  1=3 1 + i 3 e2 K 2 Cs2 !|0 |2

= ; (18) 2 2 2 8Te (1 + k 2 D )

Eq. (20) admits a purely growing mode if

where 0 (± ) is the pump (sideband) wave potential, and !± = ± ! and k± = K ± k are the frequency and the wave number of the upper and lower sidebands, respectively. Inserting Eq. (12) into Eqs. (5) and (10) and matching the phasors we obtain

2 − s2 = −

357

(22)

and take 9’=9
!’. Accordingly, we obtain from Eqs. (5) and (10) a system of equations which resembles that of the Zakharov equations (Zakharov, 1972; Tskhakaia, 1982) 2i!(1 + k 2 D2 )

92 ’ 9’ + Ce2 2 + !2 N’ = 0 9 9x

and  2  2 9 e2 (1 + ) 2 92 |’|2 2 9 N= − C Cs ; a 2 2 9 9x 9x2 4Te2

(23)

(24)

where N = nhs =nh0 . In a stationary frame  = x − U, we seek solutions of the form ’() exp(i), and obtain from Eqs. (23) and (24) 92 ’ − ’ + Q|’|2 ’ = 0; 92

(25)

where  = 2!(1 + k 2 D2 )=Ce2 is a constant and we have denoted Q = e2 (1 + )Cs2 !2 =4Te2 Ce2 (U 2 − Ca2 ). For U 2 ¿ Ca2 , Eq. (25) admits bright solitons (Hasegawa, 1975) consisting of bell-shaped structures for ’ and nhs . On the other hand, in the opposite limit, i.e. U ¡ Ca , we have grey and dark envelope solitons (Hasegawa, 1975) composed of a density cavity which traps the electron-acoustic wave envelope whose intensity distribution is rarefactive (a hole at the center and a constant value outside). To summarize, we have presented an investigation of the nonlinear coupling between electron- and ion-acoustic

358

P.K. Shukla et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 355 – 358

waves in space plasmas. For this purpose, we have employed a multi-/uid approach and have derived a pair of equations consisting of the electron-acoustic wave equation in the presence of density perturbations of ion-acoustic waves. The latter, which, in turn, is reinforced by the ponderomotive force of the electron-acoustic waves, is governed by a driven equation. It is found that the ponderomotive force of the electron-acoustic wave can produce a sub-sonic electron density cavity. Furthermore, we have used the nonlinear mode coupling equations to derive a general dispersion relation which admits non-resonant reactive and modulational instabilities. The nonlinear evolution of modulationally unstable waves is governed by Zakharov-like equations in which the electron-acoustic wave envelope varies on a time scale of the ion-acoustic wave period. The results of the present investigation should help to understand the phenomena of modulated electron-acoustic waves. Acknowledgements Manfred Hellberg acknowledges the support of the Alexander von Humboldt Foundation during his stay at Ruhr-Universit8at Bochum. This work was partially supported by the Swedish Research Council through the Contract No. 621-2001-2274 and by the European Commission through the Contract No. HPRN-CT-2001-00314 for carrying out the task of the Research Training Network entitled “Turbulent and Boundary Layers in Geospace Plasmas”. References Anderson, D., Fedele, R., Vaccaro, V., Lisak, M., Berntson, A., Johnson, S., 1999. Modulational instabilities in thermal wave model description of high energy charged particle beam dynamics. Physics Letters A 258, 244.

Berthomier, M., Pottelette, R., Malingre, M., Khotyaintsev, 2000. Electron-acoustic solitons in an electron-beam plasma system. Physics of Plasmas 7, 2987. Hanssen, A., PSecseli, J., Sten/o, L., Trulsen, J., 1994. Nonlinear wave interactions in two-electron temperature plasmas. Journal of Plasma Physics 51, 423. Hasegawa, A., 1975. Plasma Instabilities and Nonlinear ETects. Springer, Berlin, pp. 194 –203. Khirseli, E.M., Tsintsadze, N.L., 1980. Nonlinear waves in a two-temperature electron plasma. Soviet Journal of Plasma Physics 6, 595. Mace, R.L., Hellberg, M.A., 1990. Higher order electron modes in a two electron temperature plasma. Journal of Plasma Physics 43, 239. Mace, R.L., Hellberg, M.A., 1993. Electron-acoustic and cyclotron-sound instabilities driven by =eld-aligned hot-electron streaming. Journal of Geophysical Research 98A, 5881. Mace, R.L., Hellberg, M.A., 2001. The Kortweg-de VriesZakharov-Kuznetsov equation for electron-acoustic waves. Physics of Plasmas 8, 2649. Pottelette, R., Ergun, R.E., Treumann, R.A., Berthomier, M., Carlson, C.W., McFadden, J.P., Roth, I., 1999. Modulated electron-acoustic waves in auroral density cavities: FAST observations. Geophysical Research Letters 26, 2629. Singh, S.V., Lakhina, G.S., 2001. Generation of electron-acoustic waves in the magnetosphere. Planetary and Space Science 49, 107. Singh, S.V., Reddy, R.V., Lakhina, G.S., 2001. Broadband electrostatic noise due to nonlinear electron-acoustic waves. Advances in Space Research 28, 1643. Tokar, R.L., Gary, S.P., 1984. Electrostatic hiss and the beam driven electron acoustic instability in the dayside polar cusp. Geophysical Research Letters 11, 1180. Tskhakaia, D.D., 1982. Collapse of longitudinal waves in a plasma. Physical Review Letters 48, 448. Watanabe, K., Taniuti, T., 1977. Electron-acoustic mode in a plasma of two-temperature electrons. Journal of the Physical Society of Japan 43, 1819. Yu, M.Y., Shukla, P.K., 1983. Linear and nonlinear modi=ed electron acoustic waves. Journal of Plasma Physics 29, 409. Zakharov, V.E., 1972. Collapse of Langmuir waves. Soviet Physics—JETP 35, 908.