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Parametric excitation of alfvén wave by magnetosonic wave with oblique propagation
0032~33/82/020199-09$03.00/0 Pergamon Press Ltd.
Planet. Space Sci., Vol. 30, No. 2, pp. 199-207, 1982 Printed in Great Britain.
PARAMETRIC EXCITATION OF ALFVfiN WAVE BY MAGNETOSONIC WAVE WITH OBLIQUE PROPAGATION KIYOHUMI
Onagawa Magnetic Observatory
YUMOTO
and Geophysical
and TAKAO
SAITO
Institute, Tohoku University,
Sendai 980, Japan
(Received in final form 12 August 1981) Abstract-A parametric excitation of the Alfvtn wave (kA, wA) by the magnetosonic wave (kf’, w&. which propagates obliquely to the static magnetic field, has been analyzed. The theoretical model for a one-fluid with uniform, unbounded, ideally conducting and compressible plasma is employed. The resonance conditions are chosen such as, ky = kfs + kA and tilrs - o* = 6 - tizfs. The pump magnetosonic wave is assumed to be strong enough, so that the pump wave is given as a constant. In both the case of the standing and the propagating pump the growth rates of the excited waves depend on not only the pump power but also the p-ratio. In the standing pump the threshold pump intensity of the oscillating instability is zero at the perfect matching. It is found that we can obtain a larger growth rate of the parametric excitation of Alfvtn wave by the fast magnetosonic pump wave for fIlr- 70-80” and the occurrence regions of parametric excitations are localized at the resonance point in the magnetosphere (0 - mdmi). It is concluded that the parametric instability of Pc3 range HM-waves is the more possible theory than the linear resonance theory.
1. INTRODUCTION
(10-45 s) which originates outside the magnetosphere has been considered to be driven by several possible sources, e.g. quasi-parallel bow shock oscillations (Greenstadt and Olson, 1976), upstream waves associated with the bow shock (GulClmi, 1974; Kovner et al., 1976; Hoppe et al., 1981), Kelvin-Helmholtz instability for h B L (Southwood, 1968) and velocity shear instability for A Q L (Yumoto and Saito, 1980) at the magnetospheric boundary, where A and L stand for the ‘wavelength’ for the direction of plasma inhomogeneity and the characteristic length of plasma inhomogeneity. The linear resonance theory of long-period magnetic pulsation based on the idea of a steady state resonance coupling between a monochromatic surface wave excited at the magnetopause due to Kelvin-Helmholtz instability (and/or other mechanism) and a shear AlfvCn wave associated with local field line oscillations has been analyzed by many workers (Tamao, 1965 and 1978; Southwood, 1974; Chen and Hasegawa, 1974). On the other hand, the observed HM disturbances (Pc3 and 4 ranges) in the magnetosheath are controlled by the solar wind velocity and the direction of the IMF (Saito et al., 1979b). The power of the largeamplitude (SB 2 10 nT) HM disturbances in the magnetosheath is maximum near the subsolar point and decreases with increasing the distance from the point. The HM disturbances tend to be Geomagnetic
pulsation
Pc3
isotropic when the cone angle eXB is small. Pc3 magnetic pulsations observed in the outer magnetosphere are constituted with not only localized AlfvCn waves but also propagating compressional HM-waves (Arthur et al., 1977; Sato and Kokubun, 1980). The transverse events of Pc3 at synchronous orbit, ATS 6, have a distinct local time-frequency dependence, but the compressional events show no discernible local timefrequency dependence (Fig. 7 of Arthur et al., 1977). There are several Pc3 events in the Dodge data for which waves in the ground observation were not seen (Pate1 et al., 1979). The characteristics of Pc3 magnetic pulsations observed on the ground are mainly that the enlarged pulsation amplitudes are associated with the solar wind speeds and the direction of the IMF (Greenstadt et al., 1979; Saito et al., 1979c), and that the wave forms tend to be of the beating type (Kato et al., 1967). Further observation and analysis of Pc3 pulsation outside and in the magnetosphere and on the ground are needed to clarify the wave propagation mechanisms. A decay instability, i.e. nonlinear resonance theory of the pump compressional waves which should be coupling with another AlfvCn and a sound wave is considered to be the more possible theory of Pc3 pulsations than the linear resonance theory, because the large-amplitude compressional disturbances are maximum near the subsolar point in the magnetosheath and propagate in the outer mag199
200
K. YUMOTQ and T. SAITCI
netosphere, and the wave form of Fe3 pulsation observed on the ground tends to be of the beating type. It is well known that an ele~~omagnetic wave* above a certain threshold j~te~s~ty~ can excite other plasma waves through the parametric coupling rnech~~~srn. The case of a large-arn~l~~de Alfvbn wave acting as a pump to excite another Atfvitn wave and a sound wave has been studied by Hung (1974). Cramer (19771 employed as a pump a large-amplitude fast magnetoa~ust~~ wave propagating perpendicularly to the magnetic field. in this paper we study the interaction of a large-amp~itnde magn~toson~~ wave and another AIfvCn and a magnetoson~c wave. We wifE demonstrate a ~arametr~c excitation of the HMwaves which ~ru~agate obliquely to the magnetic Geld, because the maximum occu~en~e of the ~ompressiona~ events of PC? pu~sa~ons is at Jt = cos-’ @Jo* SB) - 70” (Arthur et al., 1977) and the direction of minimum variance of’ HM-waves in the magoetosh~ath is approximately normal to the magneto~ause (Fig. 4 of Saito et al+,1979a). In Section 2 we will derive the formula far a nonlinear system of a rna~e~son~~ pump wave and another Alfv6n and a mague~osoni~ wave which are propagating obliquely to the static field. Two distinct types of parametric instabilities, i.e. rhe oscillating and the purely growing waves will be described in Section 3, and conclusions and discussions will be presented in the final section.
Let us take a coordinate system of a plasma such that an ambient magnetic field & is parallel to the z-axis and hydromag~etic (A&&n and magnetosonicf waves propagate obliquely to the magnetic field, since the directions of rn~n~rnurn variance of compressional NM-waves in the magnetosheath and magnetosphere are oblique to the static field (Arthur et ab, 1977; Saito et ai., 1979af. The angles between B, and the wave vector k are jndicated by 6?--r~(k, z) and Cp= L(k,, x), where .L refers to the component perpendicular to the static magnetic field. The equiijbrium state for a one-fluid model with a uniform, unbounded, ideally conducting and compressible plasma, neglecting resistivity, is characterized by a density pO+a scalar pressure p% a zero drift velocity (V, = 0) and the steady magnetic field Ho, The first order perturba?~on fields are presented as pl? p,, V, and Hj, where j = X, y and z. We now consider the parametric excitation of the Alfven and mag~efosonic ~de~oted by index 2) waves by a larger-amplitude magnetosoni~ wave
{denoted by index I) pump which is frequetly observed in the magnetosheath (Saito at ai., 1979bf. Suppose there is a perturbation (p?, V?, H?l in the plasma of the form exp i(k? * r F wzfst); &is ~eriurbation can occur s~onta~eo~s~~ as one com~oneni of the NM noise near the ~~asmapause and the detached plasma region. To obtain some approximate solution of nonlinear coupled system, we employ the method of Hung (19743. The coupled equations between excited EM-waves and the pump wave are readily obtained,
4=HAx(vxa:s)) - &{(V:s * V)VA-i-(VAc (VA- V)V:5),
WI
where the superscripts and subscripts ‘A*, ‘f” and ‘s’ stand for the AIfvCn, the mag~etosoni~ fast and slow waves, respectively. NonIinear terms corresponding to the self-coupling of these waves are neglected. We assumed spatial and temporal dependences of the forms a”-exp i(k* *17 w,+t) and b” exp i{kf”. r 6 ufst) for the Alfv&n and magnetosoni~ waves, res~e~tjve~y, where wA for a given wave vector kA and Ofa for given wave vectors kf” are the frequencies of the linear waves and are defined as real positives. The superscripts ‘+’ and ‘-’ correspond to the forward and backward propagating waves with respect to the wave vector k. For simpii~cation, fet us consider the case of 4:” = 0 and #” 4 0, i.e. k:“, = H:“, = fi,, = 0 and k$ JF H”: f Vt $0. Using the ~orrna~ modes (A.3) and the perturbation fields (A.5 and A& in A~p~~~~x~~equations (1.1) and (1.2) then reduce to
Parametric excitation of AIfvCn wave
(2) The satisfied:
201
following
resonance
conditions
k,fs = k2fs + k A and wlfs- oA = 6 - O2fs.
#(b&a’)
I
(2.2)
and
4aeG2r,2(1+ tan fbs cm 4~s
= F,(w&, &)B;~,B:,, + F,(w&, ~fs)B$s~fs
47rC Ho
exp I- i(* 8 - tizrs)t] exp I- $2 6 + 02fs)fl,
B& = constant, a%, -= at We shall restrict our consideration to the parametric excitation which satisfies the following assumptions; (1) For solutions in the form of plane waves with “slowly” varying amplitudes, i.e. a’ = A’(t) *exp [i(k* . r T oA . t)] + complex conjugate, b& = B%,(t) *exp [i(k’,” . r T mnfs * t)] + complex conjugate
(3.1)
(4.1-2)
A frequency mismatch 8 has been introduced. The coupling terms on the right-hand sides of equations (1.1-2) result from spatial resonance and temporal resonance between the pump wave and an excited wave. (3) A pump wave is strong enough, i.e. IB&( Z+ /B&l - (A’/, therefore, the amplitude of the pump wave should be constant. Substituting the solution (3) into equations (2.1), (2.2) and (2.3), only terms with approximately the same time-scale oscillating exponentials on both sides of equation (2) are selected. Thus, by taking a space average of equation (2) we obtain a simple set of linear equations describing the parametric coupling of A’ and B’, s
cot2 ,$A)
are
(5.1)
(5.2)
T~~F~(&,,~)B;,,A’ X exp [- $8 Z mZfr)t]+ F,(w&
o;3BLf,A-
X exp [- i(- 6 7 WZfs)t]l,
(5.3)
where a barred quantity stands for its complex conjugate, and the functions F, and Fz are expressed in Appendix B. Equation (5) can readily be solved using the transformation, X’ = A+,
(3.2)
X- = A- exp (- i2St),
(6.1-2)
Y’ = B;,, exp [- i(S - w2fs)t], Y- = B& exp I- i(S + w2&],
with n = 1,2 and
(6.3-4)
with X’(t), Y’(t) 0: exp (-id). On substituting these new variables into equation (5), we obtain the dispersion law,
io 0 + MB,t, - MB;,,
0 $0 - 28) MB n, - Mzf,
:ts
KB :rs JB - J&r, KB,, i(0 - 8 + 0~~~~) 0 0 i(w - 6 - 0~3
-
.I-
X'
1
11 x-
Y’
Y-
=o,
202
K. YUMOTOand T.
i.e. o(o - 26)(w - 8 + 6&)(0 - 8 - W2fJ + MlBlr,j260[K(6J - 8 f WZf*) - .I(@ - s - ozr,)l + MIIG*12(W - 2S)EK(@ - 6 - UZfJ - ftu - 8 + WZ,)l ZZ0, (7)
SAITO
where the coefficients b and c are restricted within the following limits, i.e. b < 0 and b2 > 4ac. We can obtain a growing solution whenever the wave vectors (k*, k?, kY) and the frequencies (oA, wlfr, W& satisfy the limits. The threshold pump intensity for excitation is approximately, II&l:,, = c/lb].
where the functions Appendix B.
3.EXCITATIONOF ALFVkN MAGNETOSONIC
AND
WAVES
In the previous section we derived the dispersion law (7) of the parametric coupling of the magnetosonic pump wave and another Alfven and a magnetosonic wave which are propagating obliquely to the magnetic field. Note that there are separately the two cases of the standing and the propagating pump. 3.1 S~~~~i~g plop (/B:,s/ = /B&j) In this case we assume the magnetosonic pump wave as, lB;r,1= II&l. The dispersion relation (7) becomes
It is noticeable that the threshold pump intensity is zero at the perfect matching. Parametric instability will occur at any amplitude if there is no damping, but in practice even a small amount of either effective collisional damping or resistivity will prevent the instability unless the pump wave is rather strong. At perfect matching the growth rate is y2= ]I&/“. MK and the frequency becomes fig,,, = (& + MJ]B&I*), where the function MK(kA, k’,“, WA,us) is restricted within a positive value. 3.1.2 A purely growing instability (tiRea,= 0). This is the case where StResi= 0. We can readily obtain the growth rate as a following form,
g =
(n” - s”)(Q’-
O&J + 2MIB;,,12[(K - J)
- B + (B2 - C)“2 ( - B ri: (EZ - C)“L
for for
(BsO,C Oh (10.2)
with
x (n” - SO*fs)+ 2K6w,,,l = x2 - [(SZ-t &,) -2M(K-
(9.3)
K, .I and M are given in
J)lB:,,j”l. X-t [S2&,
B = ; Ifs” -I-u&) - 2M(K - J)&I”]
+ S~xslB:~sl~* 2M(K + J)] = 0,
(8)
with (w - a)2 = n* = $&;;,,, - r3 + 2i@nfal = x, where fiReat and y stand for the real and imaginary parts of R. Then the growing instability of the AlfvCn wave is distinguished by the following two cases; 3.1.1 An o~c~ffari~gi~~~ub~f~~y (fZReal f 0). If the discriminating equation of equation (8) is negative, i.e. D = a/B&/“ + b/B&l’ + c < 0, we have agrowing instability, where a =4M*(R-.Q2>0, b= - 4M[(K - Jf(6” + o&J + 2(K + J)t%& and c = (6’ - o&#. In this case, the frequency and growth rate are given by G,,
= Y2+ 0.51(&
+ S2) - 21MfK - J)js;J] (9.1)
and
and C = [S*&, + Soz,,]B:,,/2 . 2M(K + J)]. The threshold power for excitation jB:,/t,
is (10.3)
=/b/la.
At the perfect matching the growth rate is y22M(K - J)(B:rr12, and the threshold power intensity is flsL& 0~&llMKI, where we can obtain a growing instability when the functions are restricted within equations (lO.l-2), i.e. M(K T J)lB:t,l*/& > 1. 3.2 Propaguting plop (/B;&/ = 0) If jB&] = 0, from equation (7), one sees that the backward mode a- is not excited, and the dispersion relation is given by
y* = - 0.25f(w&, + S2) - 2M(K - J)]Bt,/*] + 0.5[S2&
f So,f$3:fs~” * 2M(K + J)l”z,
,_ _. (Y.2)
(0 + S)(sn.+ w&n -.I@+
- 02fs) + MjB:$ x3+ px+
CO&J=
. rmst q =o,
02d
(11)
Parametric excitation of Alfvt% wave where X = R - S/3, with
and 4 = 2a3/27 - (6/3)[M(K-
J)IB:,,J’-
- CMN + J@G12%
&I + ~&,I.
In contrast to the previous case, one has only an oscillating solution. The frequency and growth rates are n Real=-OS(m+n)
and
Y=
Ji
;
>
Im-nl, (12-1.2)
with
203
if the function M(K - J)/B&/2 > (4/3)&, (i.e. P > 0). In the limiting case of alay = 0, i.e. kc = k:” = 0, the right side of equation C5.3) and the second term of dispersion equation (7) become zero, which means that the magnetosonic waves and the AlfvCn wave are completely uncoupled for each other. This result is consistent with the results of Cramer (1977) and Elfimov and Nekrasov (1973) that between the AlfvCn and magnetosonic waves only occur if, in a cylindrical geometry, the azimuthal mode number m does not vanish. For the limits of the pump wavenumber perpendicular to the magnetic field, i.e. k$ = k:“, = 0, the nonlinear term of equation (2.1) becomes zero. It is found that the magnetosonic pump wave which propagates normal to the ambient field cannot excite another AIfvCn and a magnetosonic wave through the parametric coupling mechanism. 4. CONCLUSION
where the wave vectors (k*, k:“, kS”) and frequencies (wA, wtfs, wzrS) are selected so that the discriminating equation of equation (11) is negative. We have for the threshold power, lB:&
= [- s/2+(sZf4+
,3/27)“2]“3
+ [- s/2 - (s*/4 + ,3/27)“2]“3 + f , (13) where t = 2~~127 - uv/3 + w,
s = v - $13, for
M’( K - J)3 > 0,
with ii = - [12(K - &*(o:f, + S2/3) i- 27[6{ K - 5)/3 + W&K + J)1*1/4M(K - J)3, tr= -[I- 12(K - J)(&, + wzr.(K + J)JS(S’w = - [-4(#&
+ S’/3)’ - 4[6(K - J)/3 9&5)~/4M*(K - J)‘,
-t 6z/3)3+482(s2-
- 9~:~.)~127]/4M~(K - J)3. At the perfect matching we obtain the growth rate of a growing instability y*- N(K -J)lB&12, and the threshold power jBG,l$ = u/3 - oi2M(K -4,
The parametric excitation of an Affv6n and a magnetosonic wave by a magnetosonic pump wave which propagates obliquely to the magnetic field has been analyzed. The nonlinear system of three waves in a uniform, unbounded, ideally conducting, compressible plasma has been studied. The major results of this analysis may be summarized as follows: (1) We can obtain growing parametric instabilities whenever the wavenumbers (kA, kf”, k?) and the frequencies (oA, aIfs, COG&satisfy appropriate limits (cf. Section 3), where the frequencies and growth rates of the excited waves and the threshold pump intensities are functions of the wavenumbers and the frequencies of three waves. (2) In both the case of the standing pump and the propagating pump the growth rates of the excited waves depend on not only the pump power but also the p-ratio. (3) In the standing pump the threshold pump intensity of the oscillating instability is zero at perfect matching. (4) The magnetosonic pump wave which propagates normal to the magnetic field cannot excite another AlfvCn and magnetosonic wave. The theoretical model presented here ignores the resistivity of plasma. In practice even a small amount of either effective collisional damping or resistivity will prevent the instability unless the pump waves is rather strong. For the appIicability of an improved theory including the resistivity term to magnetic pulsation Pc3, the frequency and
204
K. YUMOTOand T. SAITO
growth rate of excited waves and the threshold pump intensity in the {(o*, mlfs, o&, (k*, k:“, ky)} space should be solved by means of computer analysis in the near future. The standing and propagating pumps correspond to trapped oscillations (Tamao, 1978) of the fast mode in the trough between the two peaks of AlfvCn velocity in the inner magnetosphere and the earthward propagating oscillations in the outer magnetosphere, respectively, as shown in Fig. I. The perturbation (p:, VF, HF) of magnetosonic wave can occur spontaneously as one component of the HM noise near the plasmapause and the detached plasma region. The localization of the perturbation restricts to the equatorial extent of the parametric excitation of AlfvCn wave and filters out the frequency of the AlfvCn wave (oA = k:V,) due to the decay instability of the nonmonochromatic pump waves, excited at magnetopause due to Kelvin-Helmholtz instability and/or driven by other mechanisms outside the magnetosphere. In cold plasmas (p 4 l), we obtain the growth rate and the threshold power intensity as the following forms,
Km/s IO4
103PROPAGATINGPUMP -
IO4-
;--v;cc,_‘l
I I BC
I
I MP
I,
firme/nli ! I
and
with
GH.= PM + (2 - v’/(W)/cos em) - V:k::z/&)]-’
x (1 - &/V:k:“2)(1
For the fast magnetosonic pump wave, the gain factor Glf becomes as the following form, Glf = [cos* Blr+ d(p/2) tan & sin O,J’.
(16)
It is noteworthy that in both cases of the standing and propagating pump we can obtain larger growth rate of parametric excitation of AlfvCn wave by the fast pump wave, i.e. y2-P-‘G:,I%l*l~&&a - 10’0&(w for p > mJmi = 10d3 in the plasma trough and p < m,lmi in the plasmasphere, Gfr - 10’ for Blr - 70-80” as shown in Fig. 2, and IB~$/p~- lo-‘. It is physically reasonable that the growth rate of the AlfvCn wave excited by the fast pump wave becomes small near Blr = L(Bo, ki) = 90”, since the nonlinear term of equation (2.1) becomes zero when the k-vector is perpendicular to the ambient magnetic field (Fig. 2). The properties of HM waves in a uniform, unbounded plasma has been analyzed in the present paper. This approach is valid for oscillations with
&m,hi I
I
I
I
I 1
m. 3 nT IO -cm-3
FIG. 1. SCHEMATIC (A) Alfvtn
(B)
PLASMA PARAMETERS MAGNETOSPHERE.
velocity (V:(L)= velocity(G:(L)=
IN THE DAY-SIDE
01
I 0"
B$/4n&m,,)
and
sound
I
I
I
I
I
300 8,f
-
I
0
I
I ‘1 90”
2,
yPo/Nom,).
Plasma number density, N,(L), and ambient magnetic fieldintensity,B,(L).
FACTOR G,, VS INCIDENT ANGLE THEFASTMAGNETOSONICPUMPWAVEINCOLDPLASMA.
FIG. 2. GAIN
l$, FOR
Parametric excitation of AlfvCn wave
large growth rates in inhomogeneous ( Vi/r*) Q J%,~~
and
plasma,
A$,
i.e.
(17)
where Linhomis a characteristic length of the inhomogeneity in the magnetosphere and h,r is the wavelength of the pump wave. The equatorial extent of the parametric excitation is (Vi/r’)
- V~/(1030*r,a) - 10-3k,;: - 10-4A:f,
and Afr, since Linhom- (8 In B,/&)’ - 3R, and A,r - VA . T -4RB for Pc3 range magnetic pulsation. Therefore, the plasma model in general is not an over-simplified model for the parametric excitation of AlfvCn waves in the magnetosphere. The region of occurrence of the parametric excitation of AlfvCn wave is localized at the resonance point, i.e. (VA/y) - lo-*hztS 3 lO~*A,, - O.O4R,. This is consistent with that the equatorial extent of the observed Pc3 is estimated to be 5 1/2RE in the magnetosphere (Hughes et al., 1978). Sato and Fukunishi (1981) demonstrated that two types (QP) ELF-VLF emissions are generated by Pc3 range standing oscillations which have effective compressional components, and compressional propagating pulsations which have klh"T
55m
APR 6. 1981 CNAGAWA
I
40
30
2s
20
15
PERIOD -SEC FIG.
3.
AN
EXAMPLE
OF THE BEATING
TYPE GEOMAGNETIC
PULSATION.
(A) Ordinary magnetogram by means of a high-sensity ring-core magnetometer at the Onagawa Magnetic Observatory. (B) Pc3 magnetic pulsation composed of two periods is clarified by the MEM power spectral analysis.
205
vector in a radial direction toward the Earth in the magnetosphere, respectively. An example of the beating type geomagnetic pulsation on the ground is shown in Fig. 3. It is found that Pc3 magnetic pulsations have generally two periods which are denoted by a primary peak (TJ and a secondary peak (TJ. The observed beat period is inconsistent with the expected period ( Tbeat= T, T21T, - T2j-‘) obtained by the MEM power spectral analysis. Whence we conclude that the parametric instability of Pc3-range HM-waves is the more possible theory of the beating type geomagnetic pulsations than the linear resonance theory.
Acknowledgements-We are indebted to Prof. H. Oya, Tohoku University, for valuable discussions and criticisms. This study was partially supported by Grant-in-Aids for Scientific Research Project No. 56740169 and No. 00546039, by the Ministry of Education, Science and Culture of Japan.
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206
YUMOTO
Kovner, M. S., Lebedev, V. V., Plyasova-Bakunina, T. A. and Troitskaya V. A. (1976). On the generation of low-frequency waves in the solar wind in the front of the bow shock. Planet. Space Sci. 24,261. Pat& V. L., Greaves, R. J. and Wahab, S. A. (1979). Dodge satellite observations Pc3 and Pc4 magnetic pulsations and correlated effects in the ground observations. J. geophys. Res. 84,4257. Saito, T., Takahashi, K. and Sakurai, T. (1979a). Examination of the resonance theory on PC’S by means of an analysis of magnetic fluctuations in the magnetosheath and the magnetosphere. Planet. Space Sci. 27, 809. Saito, T., Yumoto K., Takahashi, K., Tamura, T. and Sakurai, T. (1979b). Solar wind control of Pc3. Mugnetospheric study 1979, Proc. International Workshop on Selected Topics of Magnetospheric Physics, Tokyo, March, 1979, pp. 155-1591 _ Saito, T., Yumoto, K. and Tamura, T. (1979~). Roles of the solar wind and the plasmasphere on Geomagnetic Pc3 and Pi2 pulsations. Japanese IMS Symp., ISAS, University of Tokyo, November, 1979, pp. 11-18. Sato, N. and Kokubun, S. (1980). Interaction between ELF-VLF emissions and magnetic pulsations; Quasineriodic ELF-VLF emissions associated with Pc3-4 magnetic pulsations and their geomagnetic conjugacy. J. geophys. Res. 85, 101. Sato. N. and Fukunishi, H. (1981). Interaction between ELF-VLF emissions and magnetic pulsations: Ciassification of auasi-oeriodic ELF-VLF emissions based on frequency-time spectra. L geophys. Res. 86, 19. Southwood, D. 1. (1968). The hydromagnetic stability of the magnetospheric boundary. Planet. Space Sci. 16, 581. Southwood, D. J. (1974). Some features of field line resonances in the magnetosphere. PInnet. Space Sci. 22,483. Tamao, T. (1965). Transmission and coupling resonance af hydromagnetic disturbances in the non-uniform earth’s magnetosphere, Sci. Rep. Tohoku University. Ceophys. 17,43. Tamao, T. (1978). Coupling modes of hydromagnetic oscillations in non-uniform, finite pressure plasmas: two-fluids model. Planet. Space S&26, 1141. Yumoto. K. and Saito. T. (1980). Hydroma~netic wave driven by velocity. shear instability in the magnetospheric boundary layer. Planet. Space Sci. 28,789.
and T. SAITO
&q+ vgg=o, with j = x and y, respectively, where the superscripts and subscripts ‘A’, ‘f’ and ‘s’ stand for the Alfven, the magnetosonic fast and slow waves. The Alfvtn and sound speeds are defined as Vi = H$4?rp and c = (8P/@)lp,p,. We assume spatial and temporal dependences of the forms a” -exp i(kA - rro,t) and b’exp i(k’” . r 7 q,t) for the Alfvdn and magnetosonic waves, respectively, where uA for a given wave vector kA and tin for given wave vectors kf” are the frequencies of the linear waves and are defined as real positives. The superscripts and subscripts ‘+’ and ‘-’ correspond to the forward and backward propagating waves with respect to the wave vector k. Our consideration in this paper is restricted to the Alfven wave which has foilowing conditions; V *VA=V.HA=O,
VAiHO and
HAi&, (A.2.1)
i.e. we assumed H’: = 0 from the outset. For the magnetosonic waves we also assumed the following conditions; 0. HM = 0 and VM is in the He - kM plane. (A.2.2) By taking linear combinations of fA.1) and (A.21, we can introduce the normal modes of the Alfvin and magnetosonic waves, respectively, defined as;
APPENDIX A Before going on to a nonlinear study a system of linearized equation. The linearized forms netosonic waves can be expressed
g&H at
EL0
and
o dz
analysis, we shall first magnetohydrodynamic of Alfven and magas
hq-z$$=O,
(A.l.l-2)
sf
= dfs
. [(Vfi
v$j-$](
v+qz
cos &fs. tan et,
and where the coefficients C,-C, and gl-g6 are chosen such that the resulting dispersion laws are satisfied as the following equations,
$$ + po(V . V’S)= 0,
ahJ_Hfl_O at
o a2
-
’
0: = V:kp
(A.4.1)
for the AlfvCn wave, characterized and (V x H,)l, and
by non-zero (V x V,),
(A.1.3-5) a@ Po-g+cz
Parametric excitation of Alfvtn wave
OJ:, = 0.5[( vi + v$ /P+ {(Vi + _ 4 vi
vy
k”4
~~~2kfE2}1/2]
(A.4.2)
for the fast and slow magnetosonic waves, characterized by non-zero V. V,, H,,, p, and V,,. Let USexpress the perturbed quantities p,, V, and H, in terms of the normal modes a and b. By means of (A.l), (A.3 and tA.4), one immediately gets for kc + 0, i.e. tan @ + 0 Ht’ = C4a’,
vef = -
WA
Hc’ = -cot
C4a’
vy
‘CH,’
=
4A C4a’, -
cot
207
$A
(AS.l-4)
(B.1)
. [Gf: - k::I1] and F~(&, 2)
=
f : 4~V?G2tJU
+ cot*
[i(kf: - kt)
9a)
vy
and pfS’ = Gf,b&, H’S’ I
=
_
f
&&&,
(8.2)
Ht’ = fu tan ,#,rs b’fs3
Hr”’ = f”b&,
~;‘&&,f 5
z
0
fsr
(A.6.1-7) +_F$&an$f,& I
v’, (* o,/k:)
e’=0
.G”b’ PO
The coefficients K, J, M, f’.” of the dispersion equation are expressed as functions of (kf’, kr, kA, wlfr, urn), i.e.
(7)
fs,
K=[i~[~(k~Ytan42fS+k$~) with c, = 031+
-p
cot* f$*)_‘, @= 2 e/v:,
Gf~=[(l+\/(~)cos8,.)+~(l-~)(l-~)
(
k$tan&f,-k:*
)I
-ik~~~(~G2t~+~G,f~~,)~.2~l+cot2~~~3~
kff tan 4*fs - k: -tan @Ifs cos &is ) +k&&G ff” Z po Or& 2fr ’ I
and
+-(,-~)~Gf,.
M
=
i
@2fs
R
.4i(l
ff’0 + cot* 4J’ grCG2rS
+ cot* I$~),
tan h 4rcV, Gus K cos f#%r Hoomlkf: 7 ) x kf: + k”22
APPENDIX
The functions follows;
FdGt,,
B
F, and F2 of equation
(B.4)
1 9
(JW
(5) are given as and
o% f~=-~(l-~)~G~rS,
=2,1+cot2~,&tanm2f,[(~$j-(T~)]
n=land2. (B.6)
Planet. Space Sci., Vol. 30, No. 2, pp. 199-207, 1982 Printed in Great Britain.
PARAMETRIC EXCITATION OF ALFVfiN WAVE BY MAGNETOSONIC WAVE WITH OBLIQUE PROPAGATION KIYOHUMI
Onagawa Magnetic Observatory
YUMOTO
and Geophysical
and TAKAO
SAITO
Institute, Tohoku University,
Sendai 980, Japan
(Received in final form 12 August 1981) Abstract-A parametric excitation of the Alfvtn wave (kA, wA) by the magnetosonic wave (kf’, w&. which propagates obliquely to the static magnetic field, has been analyzed. The theoretical model for a one-fluid with uniform, unbounded, ideally conducting and compressible plasma is employed. The resonance conditions are chosen such as, ky = kfs + kA and tilrs - o* = 6 - tizfs. The pump magnetosonic wave is assumed to be strong enough, so that the pump wave is given as a constant. In both the case of the standing and the propagating pump the growth rates of the excited waves depend on not only the pump power but also the p-ratio. In the standing pump the threshold pump intensity of the oscillating instability is zero at the perfect matching. It is found that we can obtain a larger growth rate of the parametric excitation of Alfvtn wave by the fast magnetosonic pump wave for fIlr- 70-80” and the occurrence regions of parametric excitations are localized at the resonance point in the magnetosphere (0 - mdmi). It is concluded that the parametric instability of Pc3 range HM-waves is the more possible theory than the linear resonance theory.
1. INTRODUCTION
(10-45 s) which originates outside the magnetosphere has been considered to be driven by several possible sources, e.g. quasi-parallel bow shock oscillations (Greenstadt and Olson, 1976), upstream waves associated with the bow shock (GulClmi, 1974; Kovner et al., 1976; Hoppe et al., 1981), Kelvin-Helmholtz instability for h B L (Southwood, 1968) and velocity shear instability for A Q L (Yumoto and Saito, 1980) at the magnetospheric boundary, where A and L stand for the ‘wavelength’ for the direction of plasma inhomogeneity and the characteristic length of plasma inhomogeneity. The linear resonance theory of long-period magnetic pulsation based on the idea of a steady state resonance coupling between a monochromatic surface wave excited at the magnetopause due to Kelvin-Helmholtz instability (and/or other mechanism) and a shear AlfvCn wave associated with local field line oscillations has been analyzed by many workers (Tamao, 1965 and 1978; Southwood, 1974; Chen and Hasegawa, 1974). On the other hand, the observed HM disturbances (Pc3 and 4 ranges) in the magnetosheath are controlled by the solar wind velocity and the direction of the IMF (Saito et al., 1979b). The power of the largeamplitude (SB 2 10 nT) HM disturbances in the magnetosheath is maximum near the subsolar point and decreases with increasing the distance from the point. The HM disturbances tend to be Geomagnetic
pulsation
Pc3
isotropic when the cone angle eXB is small. Pc3 magnetic pulsations observed in the outer magnetosphere are constituted with not only localized AlfvCn waves but also propagating compressional HM-waves (Arthur et al., 1977; Sato and Kokubun, 1980). The transverse events of Pc3 at synchronous orbit, ATS 6, have a distinct local time-frequency dependence, but the compressional events show no discernible local timefrequency dependence (Fig. 7 of Arthur et al., 1977). There are several Pc3 events in the Dodge data for which waves in the ground observation were not seen (Pate1 et al., 1979). The characteristics of Pc3 magnetic pulsations observed on the ground are mainly that the enlarged pulsation amplitudes are associated with the solar wind speeds and the direction of the IMF (Greenstadt et al., 1979; Saito et al., 1979c), and that the wave forms tend to be of the beating type (Kato et al., 1967). Further observation and analysis of Pc3 pulsation outside and in the magnetosphere and on the ground are needed to clarify the wave propagation mechanisms. A decay instability, i.e. nonlinear resonance theory of the pump compressional waves which should be coupling with another AlfvCn and a sound wave is considered to be the more possible theory of Pc3 pulsations than the linear resonance theory, because the large-amplitude compressional disturbances are maximum near the subsolar point in the magnetosheath and propagate in the outer mag199
200
K. YUMOTQ and T. SAITCI
netosphere, and the wave form of Fe3 pulsation observed on the ground tends to be of the beating type. It is well known that an ele~~omagnetic wave* above a certain threshold j~te~s~ty~ can excite other plasma waves through the parametric coupling rnech~~~srn. The case of a large-arn~l~~de Alfvbn wave acting as a pump to excite another Atfvitn wave and a sound wave has been studied by Hung (1974). Cramer (19771 employed as a pump a large-amplitude fast magnetoa~ust~~ wave propagating perpendicularly to the magnetic field. in this paper we study the interaction of a large-amp~itnde magn~toson~~ wave and another AIfvCn and a magnetoson~c wave. We wifE demonstrate a ~arametr~c excitation of the HMwaves which ~ru~agate obliquely to the magnetic Geld, because the maximum occu~en~e of the ~ompressiona~ events of PC? pu~sa~ons is at Jt = cos-’ @Jo* SB) - 70” (Arthur et al., 1977) and the direction of minimum variance of’ HM-waves in the magoetosh~ath is approximately normal to the magneto~ause (Fig. 4 of Saito et al+,1979a). In Section 2 we will derive the formula far a nonlinear system of a rna~e~son~~ pump wave and another Alfv6n and a mague~osoni~ wave which are propagating obliquely to the static field. Two distinct types of parametric instabilities, i.e. rhe oscillating and the purely growing waves will be described in Section 3, and conclusions and discussions will be presented in the final section.
Let us take a coordinate system of a plasma such that an ambient magnetic field & is parallel to the z-axis and hydromag~etic (A&&n and magnetosonicf waves propagate obliquely to the magnetic field, since the directions of rn~n~rnurn variance of compressional NM-waves in the magnetosheath and magnetosphere are oblique to the static field (Arthur et ab, 1977; Saito et ai., 1979af. The angles between B, and the wave vector k are jndicated by 6?--r~(k, z) and Cp= L(k,, x), where .L refers to the component perpendicular to the static magnetic field. The equiijbrium state for a one-fluid model with a uniform, unbounded, ideally conducting and compressible plasma, neglecting resistivity, is characterized by a density pO+a scalar pressure p% a zero drift velocity (V, = 0) and the steady magnetic field Ho, The first order perturba?~on fields are presented as pl? p,, V, and Hj, where j = X, y and z. We now consider the parametric excitation of the Alfven and mag~efosonic ~de~oted by index 2) waves by a larger-amplitude magnetosoni~ wave
{denoted by index I) pump which is frequetly observed in the magnetosheath (Saito at ai., 1979bf. Suppose there is a perturbation (p?, V?, H?l in the plasma of the form exp i(k? * r F wzfst); &is ~eriurbation can occur s~onta~eo~s~~ as one com~oneni of the NM noise near the ~~asmapause and the detached plasma region. To obtain some approximate solution of nonlinear coupled system, we employ the method of Hung (19743. The coupled equations between excited EM-waves and the pump wave are readily obtained,
4=HAx(vxa:s)) - &{(V:s * V)VA-i-(VAc (VA- V)V:5),
WI
where the superscripts and subscripts ‘A*, ‘f” and ‘s’ stand for the AIfvCn, the mag~etosoni~ fast and slow waves, respectively. NonIinear terms corresponding to the self-coupling of these waves are neglected. We assumed spatial and temporal dependences of the forms a”-exp i(k* *17 w,+t) and b” exp i{kf”. r 6 ufst) for the Alfv&n and magnetosoni~ waves, res~e~tjve~y, where wA for a given wave vector kA and Ofa for given wave vectors kf” are the frequencies of the linear waves and are defined as real positives. The superscripts ‘+’ and ‘-’ correspond to the forward and backward propagating waves with respect to the wave vector k. For simpii~cation, fet us consider the case of 4:” = 0 and #” 4 0, i.e. k:“, = H:“, = fi,, = 0 and k$ JF H”: f Vt $0. Using the ~orrna~ modes (A.3) and the perturbation fields (A.5 and A& in A~p~~~~x~~equations (1.1) and (1.2) then reduce to
Parametric excitation of AIfvCn wave
(2) The satisfied:
201
following
resonance
conditions
k,fs = k2fs + k A and wlfs- oA = 6 - O2fs.
#(b&a’)
I
(2.2)
and
4aeG2r,2(1+ tan fbs cm 4~s
= F,(w&, &)B;~,B:,, + F,(w&, ~fs)B$s~fs
47rC Ho
exp I- i(* 8 - tizrs)t] exp I- $2 6 + 02fs)fl,
B& = constant, a%, -= at We shall restrict our consideration to the parametric excitation which satisfies the following assumptions; (1) For solutions in the form of plane waves with “slowly” varying amplitudes, i.e. a’ = A’(t) *exp [i(k* . r T oA . t)] + complex conjugate, b& = B%,(t) *exp [i(k’,” . r T mnfs * t)] + complex conjugate
(3.1)
(4.1-2)
A frequency mismatch 8 has been introduced. The coupling terms on the right-hand sides of equations (1.1-2) result from spatial resonance and temporal resonance between the pump wave and an excited wave. (3) A pump wave is strong enough, i.e. IB&( Z+ /B&l - (A’/, therefore, the amplitude of the pump wave should be constant. Substituting the solution (3) into equations (2.1), (2.2) and (2.3), only terms with approximately the same time-scale oscillating exponentials on both sides of equation (2) are selected. Thus, by taking a space average of equation (2) we obtain a simple set of linear equations describing the parametric coupling of A’ and B’, s
cot2 ,$A)
are
(5.1)
(5.2)
T~~F~(&,,~)B;,,A’ X exp [- $8 Z mZfr)t]+ F,(w&
o;3BLf,A-
X exp [- i(- 6 7 WZfs)t]l,
(5.3)
where a barred quantity stands for its complex conjugate, and the functions F, and Fz are expressed in Appendix B. Equation (5) can readily be solved using the transformation, X’ = A+,
(3.2)
X- = A- exp (- i2St),
(6.1-2)
Y’ = B;,, exp [- i(S - w2fs)t], Y- = B& exp I- i(S + w2&],
with n = 1,2 and
(6.3-4)
with X’(t), Y’(t) 0: exp (-id). On substituting these new variables into equation (5), we obtain the dispersion law,
io 0 + MB,t, - MB;,,
0 $0 - 28) MB n, - Mzf,
:ts
KB :rs JB - J&r, KB,, i(0 - 8 + 0~~~~) 0 0 i(w - 6 - 0~3
-
.I-
X'
1
11 x-
Y’
Y-
=o,
202
K. YUMOTOand T.
i.e. o(o - 26)(w - 8 + 6&)(0 - 8 - W2fJ + MlBlr,j260[K(6J - 8 f WZf*) - .I(@ - s - ozr,)l + MIIG*12(W - 2S)EK(@ - 6 - UZfJ - ftu - 8 + WZ,)l ZZ0, (7)
SAITO
where the coefficients b and c are restricted within the following limits, i.e. b < 0 and b2 > 4ac. We can obtain a growing solution whenever the wave vectors (k*, k?, kY) and the frequencies (oA, wlfr, W& satisfy the limits. The threshold pump intensity for excitation is approximately, II&l:,, = c/lb].
where the functions Appendix B.
3.EXCITATIONOF ALFVkN MAGNETOSONIC
AND
WAVES
In the previous section we derived the dispersion law (7) of the parametric coupling of the magnetosonic pump wave and another Alfven and a magnetosonic wave which are propagating obliquely to the magnetic field. Note that there are separately the two cases of the standing and the propagating pump. 3.1 S~~~~i~g plop (/B:,s/ = /B&j) In this case we assume the magnetosonic pump wave as, lB;r,1= II&l. The dispersion relation (7) becomes
It is noticeable that the threshold pump intensity is zero at the perfect matching. Parametric instability will occur at any amplitude if there is no damping, but in practice even a small amount of either effective collisional damping or resistivity will prevent the instability unless the pump wave is rather strong. At perfect matching the growth rate is y2= ]I&/“. MK and the frequency becomes fig,,, = (& + MJ]B&I*), where the function MK(kA, k’,“, WA,us) is restricted within a positive value. 3.1.2 A purely growing instability (tiRea,= 0). This is the case where StResi= 0. We can readily obtain the growth rate as a following form,
g =
(n” - s”)(Q’-
O&J + 2MIB;,,12[(K - J)
- B + (B2 - C)“2 ( - B ri: (EZ - C)“L
for for
(BsO,C
with
x (n” - SO*fs)+ 2K6w,,,l = x2 - [(SZ-t &,) -2M(K-
(9.3)
K, .I and M are given in
J)lB:,,j”l. X-t [S2&,
B = ; Ifs” -I-u&) - 2M(K - J)&I”]
+ S~xslB:~sl~* 2M(K + J)] = 0,
(8)
with (w - a)2 = n* = $&;;,,, - r3 + 2i@nfal = x, where fiReat and y stand for the real and imaginary parts of R. Then the growing instability of the AlfvCn wave is distinguished by the following two cases; 3.1.1 An o~c~ffari~gi~~~ub~f~~y (fZReal f 0). If the discriminating equation of equation (8) is negative, i.e. D = a/B&/“ + b/B&l’ + c < 0, we have agrowing instability, where a =4M*(R-.Q2>0, b= - 4M[(K - Jf(6” + o&J + 2(K + J)t%& and c = (6’ - o&#. In this case, the frequency and growth rate are given by G,,
= Y2+ 0.51(&
+ S2) - 21MfK - J)js;J] (9.1)
and
and C = [S*&, + Soz,,]B:,,/2 . 2M(K + J)]. The threshold power for excitation jB:,/t,
is (10.3)
=/b/la.
At the perfect matching the growth rate is y22M(K - J)(B:rr12, and the threshold power intensity is flsL& 0~&llMKI, where we can obtain a growing instability when the functions are restricted within equations (lO.l-2), i.e. M(K T J)lB:t,l*/& > 1. 3.2 Propaguting plop (/B;&/ = 0) If jB&] = 0, from equation (7), one sees that the backward mode a- is not excited, and the dispersion relation is given by
y* = - 0.25f(w&, + S2) - 2M(K - J)]Bt,/*] + 0.5[S2&
f So,f$3:fs~” * 2M(K + J)l”z,
,_ _. (Y.2)
(0 + S)(sn.+ w&n -.I@+
- 02fs) + MjB:$ x3+ px+
CO&J=
. rmst q =o,
02d
(11)
Parametric excitation of Alfvt% wave where X = R - S/3, with
and 4 = 2a3/27 - (6/3)[M(K-
J)IB:,,J’-
- CMN + J@G12%
&I + ~&,I.
In contrast to the previous case, one has only an oscillating solution. The frequency and growth rates are n Real=-OS(m+n)
and
Y=
Ji
;
>
Im-nl, (12-1.2)
with
203
if the function M(K - J)/B&/2 > (4/3)&, (i.e. P > 0). In the limiting case of alay = 0, i.e. kc = k:” = 0, the right side of equation C5.3) and the second term of dispersion equation (7) become zero, which means that the magnetosonic waves and the AlfvCn wave are completely uncoupled for each other. This result is consistent with the results of Cramer (1977) and Elfimov and Nekrasov (1973) that between the AlfvCn and magnetosonic waves only occur if, in a cylindrical geometry, the azimuthal mode number m does not vanish. For the limits of the pump wavenumber perpendicular to the magnetic field, i.e. k$ = k:“, = 0, the nonlinear term of equation (2.1) becomes zero. It is found that the magnetosonic pump wave which propagates normal to the ambient field cannot excite another AIfvCn and a magnetosonic wave through the parametric coupling mechanism. 4. CONCLUSION
where the wave vectors (k*, k:“, kS”) and frequencies (wA, wtfs, wzrS) are selected so that the discriminating equation of equation (11) is negative. We have for the threshold power, lB:&
= [- s/2+(sZf4+
,3/27)“2]“3
+ [- s/2 - (s*/4 + ,3/27)“2]“3 + f , (13) where t = 2~~127 - uv/3 + w,
s = v - $13, for
M’( K - J)3 > 0,
with ii = - [12(K - &*(o:f, + S2/3) i- 27[6{ K - 5)/3 + W&K + J)1*1/4M(K - J)3, tr= -[I- 12(K - J)(&, + wzr.(K + J)JS(S’w = - [-4(#&
+ S’/3)’ - 4[6(K - J)/3 9&5)~/4M*(K - J)‘,
-t 6z/3)3+482(s2-
- 9~:~.)~127]/4M~(K - J)3. At the perfect matching we obtain the growth rate of a growing instability y*- N(K -J)lB&12, and the threshold power jBG,l$ = u/3 - oi2M(K -4,
The parametric excitation of an Affv6n and a magnetosonic wave by a magnetosonic pump wave which propagates obliquely to the magnetic field has been analyzed. The nonlinear system of three waves in a uniform, unbounded, ideally conducting, compressible plasma has been studied. The major results of this analysis may be summarized as follows: (1) We can obtain growing parametric instabilities whenever the wavenumbers (kA, kf”, k?) and the frequencies (oA, aIfs, COG&satisfy appropriate limits (cf. Section 3), where the frequencies and growth rates of the excited waves and the threshold pump intensities are functions of the wavenumbers and the frequencies of three waves. (2) In both the case of the standing pump and the propagating pump the growth rates of the excited waves depend on not only the pump power but also the p-ratio. (3) In the standing pump the threshold pump intensity of the oscillating instability is zero at perfect matching. (4) The magnetosonic pump wave which propagates normal to the magnetic field cannot excite another AlfvCn and magnetosonic wave. The theoretical model presented here ignores the resistivity of plasma. In practice even a small amount of either effective collisional damping or resistivity will prevent the instability unless the pump waves is rather strong. For the appIicability of an improved theory including the resistivity term to magnetic pulsation Pc3, the frequency and
204
K. YUMOTOand T. SAITO
growth rate of excited waves and the threshold pump intensity in the {(o*, mlfs, o&, (k*, k:“, ky)} space should be solved by means of computer analysis in the near future. The standing and propagating pumps correspond to trapped oscillations (Tamao, 1978) of the fast mode in the trough between the two peaks of AlfvCn velocity in the inner magnetosphere and the earthward propagating oscillations in the outer magnetosphere, respectively, as shown in Fig. I. The perturbation (p:, VF, HF) of magnetosonic wave can occur spontaneously as one component of the HM noise near the plasmapause and the detached plasma region. The localization of the perturbation restricts to the equatorial extent of the parametric excitation of AlfvCn wave and filters out the frequency of the AlfvCn wave (oA = k:V,) due to the decay instability of the nonmonochromatic pump waves, excited at magnetopause due to Kelvin-Helmholtz instability and/or driven by other mechanisms outside the magnetosphere. In cold plasmas (p 4 l), we obtain the growth rate and the threshold power intensity as the following forms,
Km/s IO4
103PROPAGATINGPUMP -
IO4-
;--v;cc,_‘l
I I BC
I
I MP
I,
firme/nli ! I
and
with
GH.= PM + (2 - v’/(W)/cos em) - V:k::z/&)]-’
x (1 - &/V:k:“2)(1
For the fast magnetosonic pump wave, the gain factor Glf becomes as the following form, Glf = [cos* Blr+ d(p/2) tan & sin O,J’.
(16)
It is noteworthy that in both cases of the standing and propagating pump we can obtain larger growth rate of parametric excitation of AlfvCn wave by the fast pump wave, i.e. y2-P-‘G:,I%l*l~&&a - 10’0&(w for p > mJmi = 10d3 in the plasma trough and p < m,lmi in the plasmasphere, Gfr - 10’ for Blr - 70-80” as shown in Fig. 2, and IB~$/p~- lo-‘. It is physically reasonable that the growth rate of the AlfvCn wave excited by the fast pump wave becomes small near Blr = L(Bo, ki) = 90”, since the nonlinear term of equation (2.1) becomes zero when the k-vector is perpendicular to the ambient magnetic field (Fig. 2). The properties of HM waves in a uniform, unbounded plasma has been analyzed in the present paper. This approach is valid for oscillations with
&m,hi I
I
I
I
I 1
m. 3 nT IO -cm-3
FIG. 1. SCHEMATIC (A) Alfvtn
(B)
PLASMA PARAMETERS MAGNETOSPHERE.
velocity (V:(L)= velocity(G:(L)=
IN THE DAY-SIDE
01
I 0"
B$/4n&m,,)
and
sound
I
I
I
I
I
300 8,f
-
I
0
I
I ‘1 90”
2,
yPo/Nom,).
Plasma number density, N,(L), and ambient magnetic fieldintensity,B,(L).
FACTOR G,, VS INCIDENT ANGLE THEFASTMAGNETOSONICPUMPWAVEINCOLDPLASMA.
FIG. 2. GAIN
l$, FOR
Parametric excitation of AlfvCn wave
large growth rates in inhomogeneous ( Vi/r*) Q J%,~~
and
plasma,
A$,
i.e.
(17)
where Linhomis a characteristic length of the inhomogeneity in the magnetosphere and h,r is the wavelength of the pump wave. The equatorial extent of the parametric excitation is (Vi/r’)
- V~/(1030*r,a) - 10-3k,;: - 10-4A:f,
and Afr, since Linhom- (8 In B,/&)’ - 3R, and A,r - VA . T -4RB for Pc3 range magnetic pulsation. Therefore, the plasma model in general is not an over-simplified model for the parametric excitation of AlfvCn waves in the magnetosphere. The region of occurrence of the parametric excitation of AlfvCn wave is localized at the resonance point, i.e. (VA/y) - lo-*hztS 3 lO~*A,, - O.O4R,. This is consistent with that the equatorial extent of the observed Pc3 is estimated to be 5 1/2RE in the magnetosphere (Hughes et al., 1978). Sato and Fukunishi (1981) demonstrated that two types (QP) ELF-VLF emissions are generated by Pc3 range standing oscillations which have effective compressional components, and compressional propagating pulsations which have klh"T
55m
APR 6. 1981 CNAGAWA
I
40
30
2s
20
15
PERIOD -SEC FIG.
3.
AN
EXAMPLE
OF THE BEATING
TYPE GEOMAGNETIC
PULSATION.
(A) Ordinary magnetogram by means of a high-sensity ring-core magnetometer at the Onagawa Magnetic Observatory. (B) Pc3 magnetic pulsation composed of two periods is clarified by the MEM power spectral analysis.
205
vector in a radial direction toward the Earth in the magnetosphere, respectively. An example of the beating type geomagnetic pulsation on the ground is shown in Fig. 3. It is found that Pc3 magnetic pulsations have generally two periods which are denoted by a primary peak (TJ and a secondary peak (TJ. The observed beat period is inconsistent with the expected period ( Tbeat= T, T21T, - T2j-‘) obtained by the MEM power spectral analysis. Whence we conclude that the parametric instability of Pc3-range HM-waves is the more possible theory of the beating type geomagnetic pulsations than the linear resonance theory.
Acknowledgements-We are indebted to Prof. H. Oya, Tohoku University, for valuable discussions and criticisms. This study was partially supported by Grant-in-Aids for Scientific Research Project No. 56740169 and No. 00546039, by the Ministry of Education, Science and Culture of Japan.
REFERENCES Arthur, C. W., McPherson, R. L. and Hughes, W. J. (1977). A statistical study of Pc3 magnetic pulsations at synchronous orbit. ATS-6, J. geophys. Rex 82, 1149. Chen, L. and Hasegawa A. (1974). A theory of long-period magnetic pulsations. 1. Steady state excitation of field line resonance. J. geophys. Res. 79, 1024. Cramer, N. F. (1977). Prametric excitation of magnetoacoustic wave by a pump magnetic field in a high-p plasma. J. Plasma Phys. 17, 93. Elfimov, A. G. and Nekrasov, F. M. (1973). Decay instability of magnetohydrodynamic oscillations in a plasma cylinder in the field of a direct magnetosonic wave. Nucl. Fusion 13, 653. Greenstadt, E. W. and Olson, J. V. (1976). Pc3, 4 activity and interplanetary field orientation. .I. geophys. Res. 81, 5911. Greenstadt, E. W., ‘Russell, C. T., Formisano, V., Hedgecock, P. C., Scarf, F. L., Neugebauer, M. and Holzer, R. F. (1977). Structure of a quasi-parallel, quasi-laminar bow shock. J. geophys. Res. 82, 651. Greenstadt, E. W., Singer, H. J., Russell, C. T. and Olson, J. V. (1979). IMF orientation, solar wind velocity, and Pc3-4 signals: A joint distribution. .I. geophys. Res. 84, 527. Gultlmi, A. V. (1974). Diagnostics of the magnetosphere and interplanetary medium by means of pulsation. Space Sci. Rev. 16, 331. Hoppe, M., Russell, C. T., Frank, L. A., Eastman, T. E. and Greenstadt, E. W. (1981). Upstream hydromagnetic waves and their association with backstreaming on populations: ISEE 1 and 2 observations. .I. geophys. Res. 86, 447 1. Hung, N. (1974). Parametric excitation of Alfvtn and acoustic waves. J. Plasma Phys. 12, 445. Kato, Y., Tanaka, M. and Sato, Y. (1967). On the characteristics of beating type geomagnetic pulsation in Pc3 range. Sci. Rept. Tokoku University, Ser. 5, Geophys. 8,45.
K.
206
YUMOTO
Kovner, M. S., Lebedev, V. V., Plyasova-Bakunina, T. A. and Troitskaya V. A. (1976). On the generation of low-frequency waves in the solar wind in the front of the bow shock. Planet. Space Sci. 24,261. Pat& V. L., Greaves, R. J. and Wahab, S. A. (1979). Dodge satellite observations Pc3 and Pc4 magnetic pulsations and correlated effects in the ground observations. J. geophys. Res. 84,4257. Saito, T., Takahashi, K. and Sakurai, T. (1979a). Examination of the resonance theory on PC’S by means of an analysis of magnetic fluctuations in the magnetosheath and the magnetosphere. Planet. Space Sci. 27, 809. Saito, T., Yumoto K., Takahashi, K., Tamura, T. and Sakurai, T. (1979b). Solar wind control of Pc3. Mugnetospheric study 1979, Proc. International Workshop on Selected Topics of Magnetospheric Physics, Tokyo, March, 1979, pp. 155-1591 _ Saito, T., Yumoto, K. and Tamura, T. (1979~). Roles of the solar wind and the plasmasphere on Geomagnetic Pc3 and Pi2 pulsations. Japanese IMS Symp., ISAS, University of Tokyo, November, 1979, pp. 11-18. Sato, N. and Kokubun, S. (1980). Interaction between ELF-VLF emissions and magnetic pulsations; Quasineriodic ELF-VLF emissions associated with Pc3-4 magnetic pulsations and their geomagnetic conjugacy. J. geophys. Res. 85, 101. Sato. N. and Fukunishi, H. (1981). Interaction between ELF-VLF emissions and magnetic pulsations: Ciassification of auasi-oeriodic ELF-VLF emissions based on frequency-time spectra. L geophys. Res. 86, 19. Southwood, D. 1. (1968). The hydromagnetic stability of the magnetospheric boundary. Planet. Space Sci. 16, 581. Southwood, D. J. (1974). Some features of field line resonances in the magnetosphere. PInnet. Space Sci. 22,483. Tamao, T. (1965). Transmission and coupling resonance af hydromagnetic disturbances in the non-uniform earth’s magnetosphere, Sci. Rep. Tohoku University. Ceophys. 17,43. Tamao, T. (1978). Coupling modes of hydromagnetic oscillations in non-uniform, finite pressure plasmas: two-fluids model. Planet. Space S&26, 1141. Yumoto. K. and Saito. T. (1980). Hydroma~netic wave driven by velocity. shear instability in the magnetospheric boundary layer. Planet. Space Sci. 28,789.
and T. SAITO
&q+ vgg=o, with j = x and y, respectively, where the superscripts and subscripts ‘A’, ‘f’ and ‘s’ stand for the Alfven, the magnetosonic fast and slow waves. The Alfvtn and sound speeds are defined as Vi = H$4?rp and c = (8P/@)lp,p,. We assume spatial and temporal dependences of the forms a” -exp i(kA - rro,t) and b’exp i(k’” . r 7 q,t) for the Alfvdn and magnetosonic waves, respectively, where uA for a given wave vector kA and tin for given wave vectors kf” are the frequencies of the linear waves and are defined as real positives. The superscripts and subscripts ‘+’ and ‘-’ correspond to the forward and backward propagating waves with respect to the wave vector k. Our consideration in this paper is restricted to the Alfven wave which has foilowing conditions; V *VA=V.HA=O,
VAiHO and
HAi&, (A.2.1)
i.e. we assumed H’: = 0 from the outset. For the magnetosonic waves we also assumed the following conditions; 0. HM = 0 and VM is in the He - kM plane. (A.2.2) By taking linear combinations of fA.1) and (A.21, we can introduce the normal modes of the Alfvin and magnetosonic waves, respectively, defined as;
APPENDIX A Before going on to a nonlinear study a system of linearized equation. The linearized forms netosonic waves can be expressed
g&H at
EL0
and
o dz
analysis, we shall first magnetohydrodynamic of Alfven and magas
hq-z$$=O,
(A.l.l-2)
sf
= dfs
. [(Vfi
v$j-$](
v+qz
cos &fs. tan et,
and where the coefficients C,-C, and gl-g6 are chosen such that the resulting dispersion laws are satisfied as the following equations,
$$ + po(V . V’S)= 0,
ahJ_Hfl_O at
o a2
-
’
0: = V:kp
(A.4.1)
for the AlfvCn wave, characterized and (V x H,)l, and
by non-zero (V x V,),
(A.1.3-5) a@ Po-g+cz
Parametric excitation of Alfvtn wave
OJ:, = 0.5[( vi + v$ /P+ {(Vi + _ 4 vi
vy
k”4
~~~2kfE2}1/2]
(A.4.2)
for the fast and slow magnetosonic waves, characterized by non-zero V. V,, H,,, p, and V,,. Let USexpress the perturbed quantities p,, V, and H, in terms of the normal modes a and b. By means of (A.l), (A.3 and tA.4), one immediately gets for kc + 0, i.e. tan @ + 0 Ht’ = C4a’,
vef = -
WA
Hc’ = -cot
C4a’
vy
‘CH,’
=
4A C4a’, -
cot
207
$A
(AS.l-4)
(B.1)
. [Gf: - k::I1] and F~(&, 2)
=
f : 4~V?G2tJU
+ cot*
[i(kf: - kt)
9a)
vy
and pfS’ = Gf,b&, H’S’ I
=
_
f
&&&,
(8.2)
Ht’ = fu tan ,#,rs b’fs3
Hr”’ = f”b&,
~;‘&&,f 5
z
0
fsr
(A.6.1-7) +_F$&an$f,& I
v’, (* o,/k:)
e’=0
.G”b’ PO
The coefficients K, J, M, f’.” of the dispersion equation are expressed as functions of (kf’, kr, kA, wlfr, urn), i.e.
(7)
fs,
K=[i~[~(k~Ytan42fS+k$~) with c, = 031+
-p
cot* f$*)_‘, @= 2 e/v:,
Gf~=[(l+\/(~)cos8,.)+~(l-~)(l-~)
(
k$tan&f,-k:*
)I
-ik~~~(~G2t~+~G,f~~,)~.2~l+cot2~~~3~
kff tan 4*fs - k: -tan @Ifs cos &is ) +k&&G ff” Z po Or& 2fr ’ I
and
+-(,-~)~Gf,.
M
=
i
@2fs
R
.4i(l
ff’0 + cot* 4J’ grCG2rS
+ cot* I$~),
tan h 4rcV, Gus K cos f#%r Hoomlkf: 7 ) x kf: + k”22
APPENDIX
The functions follows;
FdGt,,
B
F, and F2 of equation
(B.4)
1 9
(JW
(5) are given as and
o% f~=-~(l-~)~G~rS,
=2,1+cot2~,&tanm2f,[(~$j-(T~)]
n=land2. (B.6)