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Interaction of upper and lower hybrid waves and generation of the downshifted maximum feature of stimulated electromagnetic emissions
Jourtud of Atmospheric
Pergamon
and Solar-Terrestrial 0
Physics, Vol. 59, No. 1997 Elsevier
Science
PII: 81364-6826(96)001365
18, pp.
X2-2429,
1997
Ltd. All riehts reserved. Printed in &eat Britain 136‘668x/97 $17.00+0.00
Interaction of upper and lower hybrid waves and generation of the downshifted maximum feature of stimulated electromagnetic emissions M. M. Shvarts and S. M. Grach Radiophysical
Research
Institute, B. Pecherskaya 25, 603600 Nizhniy (e-mail: [email protected])
(Received in final form 8 December
Novgorod,
1995; accepted 18 December
Russia
1995)
Abstract-The decay of an upper hybrid (UH) wave to a lower hybrid (LH) wave and a daughter
upper hybrid wave is considered. The linear stage of the process was analyzed for ionospheric conditions and compared with experimental data as well as with theoretical results of other authors. It was shown that the effective excitation of the prominent downshifted maximum feature of stimulated electromagnetic emissions in the ionospheric modification experiments could take place possibly due to interaction between two oblique UH waves and a heavily damped LH wave. @ 1997 Elsevier Science Ltd
a few seconds, which is close to that of the artificial small-scale (li 5 2mc/fo) magnetic field aligned irregularities (FAI) of plasma density. The subscript ‘I’ relates to the geomagnetic field direction. Below we analyse the dispersion relation for the parametric decay of UH pump wave depending on the dispersive properties of the interacting waves and on the plasma density in the vicinity of the pump wave UH point to find out the appropriate parameters, which could be responsible for the observed DM peak downshifts AfoM in the SEE spectra. The effects of a small but finite electron gyroradius, of LH wave Landau damping on electrons and ions, as well as a possibility of oblique propagation of UH waves are taken into account. The influence of cyclotron harmonic and regular plasma inhomogeneity effects are neglected.
1. INTRODUCTION
In the present paper we consider the three-wave (decay) interaction of two upper hybrid (UH) waves and a lower hybrid (LH) wave for conditions typical for the ionospheric F-region. The purpose of the treatment is to interpret the downshifted maximum (DM) feature in the spectrum of electromagnetic emission, which is nonlinearly stimulated in the ionosphere by a powerful electromagnetic pump wave (PW) and detected on the ground (ThidC et al., 1982; Stubbe et al., 1984). The DM is a prominent spectral structure of the stimulated electromagnetic emission (SEE) in the lower sideband of the PW with its peak downshifted approximately IO-18 kHz from the PW frequency fo = ~0127~ and reaching an intensity of a few tens of decibels above the background noise level. The DM is observed for a wide PW frequency range between 3.5 and 9.5 MHz, excluding narrow regions near electron cyclotron harmonics. The highest frequency components of the DM feature occur roughly at the same downshifts Afh for all Jo, the values Afh - 7.5-8 kHz are close to lower hybrid frequency in the F-region (Leyser et al., 1990, 1994). This fact allowed these authors to conclude that the LH waves play an important role in the DM generation. On the other hand, the typical DM growth time after PW turn on is
2. BASIC FORMULAS
The dispersion relation of the parametric decay of a UH pump wave into UH and LH daughter waves can be easily obtained from a set of equations for three-wave interaction and the general dispersion relation of parametric instability. Omitting the details of derivation (see e.g. Tsytovich, 1967; Silin, 1973; Kotik and Trakht2421
M. M. Shvarts and S. M. Grach
2422
engerts, 1973; Grach and Trakhtengerts, 1975; Grach, 1975; Sharma and Shukla, 1983) we give here the well-known dispersion relation of the parametric decay in the following form (yj << Wj and w& > LO&): (Y +
r”l)(r
+ Y8)
=
y:>
(1)
where y = Im w is the instability growth rate, rf the nonlinear growth rate
<,i
=
the squares of the thermal speeds,
Gil&i
eB/(cm,i) the cyclotron frequencies, 01 = L (kj, B) the angle between the wave vector and
W ce., --
the magnetic field B, (-e) the electron charge, me,i the particle masses, N the plasma density, and c the speed of light. Higher gyroharmonic effects (n 2 3) in (5) are neglected. For kj - 0 equation (5) exhibits well known solutions: 2
W”,“l
=
w&
sin’&i
+ w:
=
wtll- W$~02eu.ul, w: = w:h Wj, kj and yj (j = U, ~1 , 8) frequencies, wave vectors and damping rates of the interacting waves, respectively (where the subscripts u, ul and 4? correspond to the UH pump, UH daughter and LH waves, respectively), si the ion contribution to scalar dielectric permittivity E, ?”(w,) the electron contribution to the dielectric permittivity tensor i, E, the UH pump wave electric field, and aj = Ej/Ej = kj/kj. The matching conditions between the coupling waves are w,(k.)
=
w,l(kul)
+
wr(ktL
(3)
k. = k.1 + kc.
(4)
The dependencies wj(kj) in (3) are defined through the dispersion relation
$2
W2
l-
pe W;
CO?
Oj
-
(w! -w&)
sin’ t?j
wpi -
_
I
_
w4 PVe2
W4
I
(WZ
-
sin4 ej
w&)(w?
-
4w&)
+
1 w:bJ:e - 3&w: + 6~4) sin2 0,cos2 e, I I (w? - w2ce)3 3
+
COS4
ej
I
=
0,
(5)
which is valid for both UH and LH waves if k:ve2 << Wt.
Ws
>
Lo%,
I(wj-nw,,i)/JZkllv,,iI
B
1, and n = 0, +I,. . . In the expressions (2) and (5) as well as in the inequality before equation (l), W~,,i = 4rreZN/m,i are the squares of the electron (e) and ion (i) Langmuir frequencies,
1
me
YU.Ul =
y.&
wi
3&
2
1 + 3 cos2ee ,
(7)
where w& = W& + W& and wf,, = W,wci/(l + = wcewi are the squared upper hy&/wz,) brid and lower hybrid frequencies, respectively. For kp, 5 1, where pe = ve/ wee is the electron gyroradius, the thermal corrections have a marked effect on the dispersion properties of UH and LH waves and must be taken into account in (l)-(4). This is clear from Fig. l(a,b), in which sections of dispersion surfaces are displayed. In Fig. l(a) the spatial matching conditions (4) are also shown schematically. For UH waves at k, cos 8, - w,/c the corrections of order wi/cr omitted in (5) must be taken into account. For ionospheric conditions, assuming that the PW frequency is not close to a gyroharmonic, we can take the decay rate of UH waves to be purely collisional with an electron collisional frequency ve:
2 _
(
(6)
v,/2.
For the decay rate of LH waves, the Landau damping on electrons yLand and ions yLand have to be taken into account in addition to collisional damping .ymi. According to Ginzburg and Rukhadze (1975) Ye = Ycd +
YEand + YAnd
=- V, wf c0s2 ee + w: sin’ eg 2 W&(m,/mi + cos2 ed + ’
w:
Wf
k&3 1cos & 1 exp +:$exp
m (Me/mi+ cos2 ed
(-&)I,
- 2Gve2 COS2or >
(8)
Hybrid
waves and the generation
of the downshifted
maximum
d =
feature
in SEE
wfhwulkzep:cos’o( Wj
2423
E,' 16rrNT,’
where o( = L (k,, k.i), cos CI = sin 0, sin B,i cos Ap, Ap = pu and vu1 are the azimuthal angles vectors in the plane perpendicular field.
(9)
cos 0” cos 0,i + mul, where mu of the UH wave to the magnetic
3. ANALYSIS OF THE DISPERSION
RELATION
The growth rate of the three-wave decay, obtained from (I),
y,=_2Y!!2B+ 2
0
0.2
0.4 koe
0.6
0.6
Fig. 1. Upper panel (a) shows sections of UH wave dispersion surfaces for wu = 6 MHz and wee = 1.35 MHz for different plasma frequencies 1: wpe/2rr = 5.845 MHz and 2: o+/2rr = 5.8 MHz. The matching condition (4) for the process UH - UH + LH is shown schematically. Lower panel (b) shows sections of LH wave dispersion surfaces for different frequencies WI 1: WI = l.t,5W]h, 2: W.f = 1.5Wl,,, and 3: Wp = hl,, where w]h/2n = 7.676 kHz and kll = kcosB.
where me/mi appears in the formula due to the ion contribution. In the expressions (5) and (8) we consider the ions to be unmagnetised. This approximation is valid if Re w >> wci and Im w = y > Wci/2n (Sizonenko and Stepanov, 1964). According to (8) the damping on magnetised electrons vanishes for 0~ - r-r/2, but prevails in ion Landau damping for cos2Qr > m,lmi, where we = We cos eg. Notice that, according to (2) and (7), in a dense plasma (w’,, B wL), the maximum values of the nonlinear growth rate yn occur for si = - W~i/w’, Ei > 1, and the following approximate formula is valid
J -
YeJ2+
(Kl
4
y2
n
(10)
is a function of four independent parameters of the coupling waves, which belong to an intersection of their dispersion surfaces in k-space [Fig. l(a,b)] defined by the dispersion relations (5) and the matching conditions (3) and (4). For the further analysis it is convenient to choose them to be w”, &, wg, and kt. In addition, we have to analyze y as dependent on plasma density N, because the properties of the UH waves exhibit an essential dependence on N in the vicinity of the UH resonance Nub, where wi, = wtul - wt. According to (2) and (9) the three-wave interaction is most effective for the WJ frequency range Wih 5 we < Wpi. In this case lower hybrid waves propagate almost perpendicular to the magnetic field, cos* l?e 5 m,/mi or, for the ionospheric Fregion, 89.5” 5 88 5 90”. Since we -+c w,, wul the change of the UH wave wavenumber in the interaction process is rather small, and we may consider k,ll z k,lll, lkUll = Ikulll, i.e. 6% = &I. On the other hand, small but finite values of cos 81 must be taken into account in (10) for the calculation of ~8 because of the important role of Landau damping. Let us begin our analysis of expression (10) from the dependence of y on kc. Figure 2 displays the growth rate y (solid lines) and the LH wave damping rate yt (dotted lines) versus kpp, for different frequencies of the lower hybrid wave we. For our calculations the following typical ionospheric F-region parameters were used: w,/(2rr) = 6.0 MHz, w,,/(2n) = 5.845 MHz, w,/(2rr) = 1.35 MHz, T, = x = 1500 K, v, = 300 SC’, and mi/m, = 3 x lo5 for O+ ions. The
2424
M. M. Shvarts and S. M. Grach
8
2 0 0
0.1
0.2
0.3 klpe
0.4
0.5
0.6
Fig. 2. The growth rate y (solid lines) and the LH wave damping rate ,yj (dotted lines) versus kgpe for different we. The LH wave frequencies are l:wr = l.O5Wth, 2:wJ! = l.l25wt,,, 3:wC = 1.25Wt,,, 4:WC = 1,5Wth, 5: We = 1.75Wlh, and 6: WJ? = 2W]h where W],,/2n = 7.676 kHz. electric field strength of the decaying UH wave is taken to be E, = 0.6 V/m, which is well above the threshold field defined from the expression yz = yulyC. From Fig. 2 it is seen that: (i) The curves y(kt) last only until ke = kt,, where
h? PnlPe
1_ 4 -
-
w:h/w;,
-
(4
w:h/w$
3 1 + 4(~/T,)(wf,/w$)
15 -
(11)
is the maximum wavenumber of the LH waves, defined by the dispersion relation (5) at Bp = 7r/2 (see Fig. 2). (ii) A maximum of the function y(ke) at ke = k;, with k;p, - 0.25-0.4, is attained when the essential growth of Landau damping with kc starts to dominate the increase of the nonlinear growth rate yn. This maximum occurs only for rather large frequencies (we > 1.12 Wth for the chosen parameters), when k; < ke,; if ktm < kz, then y(kj) is largest for kp = ke,. (iii) The largest value k; occurs for cosee m and wp - l.SSWlh. For this 84 the Landau damping is a minimum because of the decrease of yi with decreasing cos f3r and the decrease of & [due to the increase of wg, (7)] with increasing cos 0e. Consequently, the essential growth of yp with kg for these 198 and WP starts at larger kp (Fig. 2, curve 4). In the construction of Fig. 2 we supposed that cos2 (Y = 1, and neglected the dependence of k,
W *w 10 P 5-
0* I
I
I
I
I
I
e
0.4 -
W3
0
0.3 -
c/
4
2 5 :
b
0
’ 1
d
1 1.25
I
,
1.5 1.75 “‘C/Wlh
1 2
I
_
2.25
Fig. 3. Panels (a) and (b) show the largest y (ys) over ka, versus we for different 8” and wpe/2rr = 5.8 MHz.
The corresponding values of the LH wave damping rate yes and LH wave wavenumber kJg are shown in panels (c) and (d), respectively. Here 8, are 1: & = SO”, 7.Q.. = 75” ‘7.0. = 70° A.0 = h<‘= 5.62 - LO’= *n,i
Hybrid waves and the generation of the downshifted maximum feature in SEE and k,i on N. The account of a real mutual arrangement of k, and k.1, as well as of their real values obtained from (5), sets some limitations on k’Z and cos2 o( in (9) and reduces y in comparison with Fig. 2. Figure 3(a,b) shows ~a, the largest y over kg, versus we for different 0” near the upper hybrid resonance of the UH pump wave (wpe/27r = 5.845 MHz). Figure 3(c,d) displays the corresponding values of the LH wave damping rate and wavenumber keg, respectively. As is seen from Fig. 3, there are four types of JJ~(WI) curves. The first type [curves 1 and 2 in Fig. 3(a)] contains two growing parts, labelled a and c, and two decreasing parts (b and d). Curve 3 [Fig. 3(a)] has all these parts, but, in addition, there is a smooth maximum of y(we) labelled e. The curve 4 in Fig. 3(b) possesses the parts a, 6, e and d. Lastly, the curves 5 and 6 [Fig. 3(b)] contain only the parts a and b. Part a corresponds to rather small values of WI and kt,(wg) where ke,,, < k, = Jz77 k, (0,). According to item (ii) in the list above, for yg we must use here keg = kt,(ws) and cos2 1y = 1 - k~,(w~)/2k,(f?,) in the expressions (9) and (10). In this case the increase of ke,,, with we [see Fig. l(b)] defines the growth of yg, as well as the increase of k, with decreasing 8, defines the larger frequency width of the part a for smaller 0,. ’ In the curves 1 and 2 for larger we and ke,(we), when k, < kem(we) < kh = 2k0, y(kgm) < y(k,) because of the dependence of y on cos’ LX.Here, ys is obtained for ke = k,, cos2 a = 2/3 [see Fig. 3(d)], and decreases with increasing we due to the factor wi in the denominator of the expression (9) (part b). Further, for kh < kt,,,(we) < k, = 2k,sin8,, the largest y again occurs at ktg = kern(we), and ~a grows rapidly with we due to the increase of kern and 1cos a) (part c). For larger wd, when kl,,, > kc, the magnitude of the LH wave wavenumber is limited by the value 2k, sin 0,, because of the matching conditions (4) and in (9) and (10) we must use keg = 2k,, sin B,, Ap = rr, and o( = 28, for Ye. In this case, as in part b, for the curves 1 and 2, rf decreases as we-3 . Notice that the values keg for the whole curves 1 and 2 are found to be small enough (ka 5 2k,(0,) sin B, < k;) to
’ The last statement is valid until k,sinO, < k,,, where kum is the maximum possible value of k,,; see Fig. l(a).
2425
provide a purely collisional damping rate of LH waves, M = ~~1 [Fig. 3(c)]. For smaller 8, the magnitude of the UH wave vector component perpendicular to the magnetic field, k, sin 8, and k,, increase [Fig. l(a)]. This fact leads, in particular, to the growth of the frequency size of parts a, b and c (and the values y(k,), y(kb)) with decreasing 0”. On the other hand, the magnitude of the LH wavenumber may become large enough to an essential increase of the Landau damping, ke > kt In Fig. 3 for the curve 3 the Landau damping rate does exceed the collisional damping rate even in part c, and YLand grows with wq in this part of the curve due to the growth of ke,,, [Fig. 3(c)]. Nevertheless, the largest y still occurs at keg = ke,,,(wp) because cos’ o( is maximum at this point. For kt, > k,., the ~a still occurs at ke = kc, but the value yg(kc) in this curve is significantly smaller than, e.g., in curve 2, where k,(&) is smaller, and Landau damping is not important. In this part of the curve for increasing we, ~a first grows due to the decrease of the ion Landau damping rate Ytand, and then decreases because of the growth of yEand with cos2 08. This part of the curve 3, labelled by e, (as well as the similar part of the curve 4) can be clarified as a movement of the displaying point in Fig. 2 over the curves y(ke) along the vertical line kg = 2k,sin 0, [which corresponds to the horizontal line in Fig. 3(d)] for kep, = 0.3 15 (curve 3) and ktp, = 0.385 (curve 4). For further increases of we, the y(k,) becomes smaller than y(k,), and kg = k,, wsa = 213 and ~1 = ~~~1must be used in (9) and (10) for calculations of ~a in this part of the curves 3 and 4 (part d, the extension of part b). Notice that the absence of the growing part c in the curve 4 (as well as in the curves 5 and 6) is due to strong Landau damping of LH waves with kb -Cke < kc.
Part e appears in curve 4 (0, = 65”) only in the region of minimum of the total Landau damping YLand = &nd
+ &and
at 01
-
1.5wlh
and
kept
-
0.375. Figure 4 shows y versus we and ke for 0, = 65”. As is seen from Fig. 4, there are two hills of the surface Y(we, ke) corresponding to ke - k, and we - 1.12Wrh and to the minimum of the Landau damping. For smaller 8, the last hill is rather low and cannot be seen in the curves 5 and 6 for yg6(we).
2426
M. M. Shvarts and S. M. Grach
1
Fig. 4. y versus we and kt for 0” = 65” and wf = 0.985w&,.
The variations of the plasma density N near the upper hybrid resonance lead to noticeable changes of the UH wave number k,,, extremely so for 0, close to 90” [see Fig. l(a)]. The curves ya(wp) for 0” = 90” and different N/N,h, with N < h$h, are shown in Fig. 5. It is seen that the influence of the plasma density decrease is similar to the influence of the 0, decrease, since both of them give a rise of k,:the behaviour of the curves for ye for mie = 0.999&h,, 0.997&,*, 0.994&*, and 0.985wi,,, where cuih* = UJ’,- c&) in Fig. 5 is analogous to that for 0, = 80”, 75”, 70”, and 65” in Fig. 3(a,b), respectively. Notice, finally, that for 8, = 65” (curve 4 in Fig. 3) and for wi, = 0.985w:,, (curve 4 in Fig. 5) yg -K yp in part e. At the same time the LH wave damping rate is still small in comparison with the wave frequency, ~4 << we. So, in this case the weakly damped LH wave could be considered as a heavily damped product of the decay process.
4. DISCUSSION
Models of the DM generation connected with the interaction between PW, FAI, upper hybrid and lower hybrid waves have been discussed in the
literature (Leyser, 199 1, 1994; Grach and Shvarts, 1991; Vas’kov and Bud’ko, 1993a,b; Murtaza and Shukla, 1984). Leyser (1991, 1994) considered a parametric decay of a UH wave into an electromagnetic (EM) wave and an LH wave [UH EM + LH, (I)]. The UH wave was supposed to be excited by electromagnetic pump wave scattering off the ASI. In contrast, Vas’kov and Bud’ko (1993a,b) considered a parametric decay of the electromagnetic PW into UH and LH waves [EM - UH + LH, (II)]; UH wave scattering off the FAI was supposed to be responsible for the SEE generation. In the present paper, as well as in Grach and Shvarts (1991) we study the decay of a UH mother wave into a UH daughter wave with lower frequency, and an LH wave [UH - UH + LH, (III)]; the mother UH wave and the SEE are considered as a result of the scattering off the AS1 of the PW and UH daughter waves, respectively [double transformation (Belikovich et al., 1981)]. Neither of these models contain any nonlinear saturation mechanisms. Nevertheless, some conclusions concerning the processes of the LH waves excitation in ionospheric heating experiments can be drawn. Below we discuss briefly the results obtained and compare them with the results of Leyser (1991, 1994) and Vas’kov and
Hybrid waves and the generation of the downshifted maximum feature in SEE
6f 2
4-
2-
0’
’ 1
I
I
1.25
1.5
I
1
,
1.75
2
2.25
1
wC/wlh
Fig. 5. us
for & = 90” and different N/N&.
Here
I- w2 = 0.999w2“h*, 2-W& = 0.997w&*, 3-w;, = 0.99L&*. 4-w2 Pe = 0.985w2uh*, where wii,* = fJJ;_ W&.
Bud’ko (1993a,b). Z
Notice that the growth rate of process (II) can be obtained from the expressions (1) and (2) in which, however, a. must be replaced by = EEM/~EEM~. and, for W&/W& >> 1, UEM I&~~~(W3fh I2 N l/2. For the process (I), UEM must be used instead of (1.1, ~&(o~)czEM~* E l/2, and s IwZweM must be used in (2) inwhere n2 = c?k2/w2 is the stead of $$lwZw,, refractive index squared. For a plasma density close to the upper hybrid resonance [and 8, close to 90” for the process (III)] the dependencies y(w~) for all three processes are similar. They exhibit a sharp peak at cop = w* n. 1.12wih and a decreasing part d. At the same time, for (III) the limitations on the mutual arrangement of k, and k.i, due to the matching conditions, affect the value of the growth rate at w - wih (parts b and c in Fig. 3). This value w* corresponds to A~DM - 9 kHz and is lower than Af& in the measurements. For lower plasma densities the maximum in the growth rate is found to be displaced to higher w*, up to w* - 1.5wih (Afo~ - 12 kHz) which is closer to the values observed experimentally (Fig. 5). This is connected with the increase of the UH wave wavenumber (and, due to (4), of the LH wave wavenumber), with decreasing N,
’ Here we do not discuss the excitation of high harmonic ion Bernstein waves, considered in Leyser (1991, 1994).
2421
and the existence of the LH wave Landau damping minimum. For the process (III) at 8, = 90”, W* 1.5Wlh occurs at w& - 0.985w&,, or for N- 1.45% lower than at the upper hybrid resonance; for processes (I) and (II) this is at w& 0.945&,,, or for N - 5.5% lower than Nub. Such a difference is due to the different matching conditions: for our Fig. 5 we used ke = 2k, in (2), while for the processes (I) and (II) it is necessary to use ke = k,,. If oblique UH waves (0, f r-r/2) take part in the parametric decay, the maximum of y(wa) at W* 1.5Wlh does appear at wi, = w’, - wz&,i.e. near the upper hybrid resonance [Fig. 3(b), curve 4 for 8, = 65”]. Here both of the maxima w* 1.1 wih and w* - 1.5 wih are present. Notice that for the processes (I) and (II) the matching conditions (4) forbid the participation of oblique UH waves since kt,,, < k,, kt and k, z -kr. According to Grach et al. (1981), the presence of oblique UH waves in the ionospheric heated volume is due to multiple scattering of the UH waves off FAI. On the other hand, an effective interaction between EM and UH waves and FAI, responsible for the excitation of UH waves and FAI in the ionospheric modifications, occurs in a narrow layer wie - (1 + 0.003)$& for FAI scale lengths 1, - l-15 m (Grach et al., 1981, 1997). For lower plasma densities w& - (0.985-O. 945) wih, the effective interaction between EM and UH waves and FAI could occur for FAI with 1, - 0.50.25 m, which, however, have a very low intensity (Minkoff et al., 1974). Thus, the process UH -. UH + LH at w& - w$* for oblique UH waves with kulp, = ktp, / 2 - 0.15-O. 2 seems to be more probable to be responsible for the excitation of the DM feature in SEE spectra than the processes (I) and (II). Additional arguments in favour of the double transformation mechanism are presented in the paper by Leyser et af. (1994). First, the cascade type of 2DM generation (their Fig. 8) can be more easily explained in terms of a UH - UH + LH process. Then, the overshoot in the temporal evolution of the 3DM (their Fig. 9) is similar to the overshoot obtained for the broad continuum feature of the SEE at the same frequency shifts and interpreted in terms of the double transformation, with the UH wave spectrum due to induced scattering off ions (Grach et al., 1997). According to Stubbe et al. (1984) and Leyser
2428
M. M. Shvarts and S. M. Grach
et aI. (1994) the frequency shift of DM peak position A& increases with the pump frequency fo. We have not obtained this effect in our calculations. But we have found that an addition of a small fraction of light ions (i.e., a decrease of m,/mi) raises both the lower hybrid frequency Wth, the frequency position w; of the Ye maximum, and the ratio w: /u~th. Such an increase is not enough, however, to explain the values observed experimentally, if we assume that Afo~ = w: /2rr. Our results can be compared with the experimental data only qualitatively. For a complete interpretation of the DM feature of the SEE, a nonlinear theory of the saturation stage as well as a regular plasma inhomogeneity account are required. Notice, however, that according to Stenflo and Shukla (1992) and Yu et al. (1985) an influence of the convection mode, which exists due to the inhomogeneity in LH frequency range, is negligible for kep, 2 10e4. In our treatment above we supposed the UH pump wave to be monochromatic and took the reasonable value E, = 0.6 V/m. For the construction of a nonlinear theory it is necessary to take into account that in the experiment the real distribution of UH wave energy over the dispersion surface and the type of phase correlations of UH waves are unknown. Further, the two hills in the surface y( we, kc), corresponding to different ratios y/y/, must relate to different regimes of the saturation stage of parametric decay instability in the different parts of phase space. This fact can help with the construction of the nonlinear theory. Notice that recently Vas’kov et al. (1994) reported the discovery of VLF emissions from the ionospheric heated volume in the frequency range 7.5 kHz 5 f 5 9.5 kHz, which correspond, probably, to the ‘low frequency maximum’ of the surface y(we, kg), while the DM feature of the SEE is more likely to correspond to the ‘high frequency maximum’. Acknowledgements-The authors acknowledge the financial support from the International Science Foundation (Grant no. R87300) and Russian Foundation of Fundamental Investigations (Grant no. 94-02-03253-a).
REFERENCES
Belikovich, V. V., Benediktov, E. A., Grach, S. M. and Terina, G. I. (1981) Double transformation of waves off artificial ionospheric irregularities. In Abstracts of XIII Conference of USSR on Radio Propagation, July 1981, Moscow, vol. I, pp. 107-109. Nauka (in
Russian). Ginzburg, V. L. and Rukhadze, A. A. (1975) Waves in Magnetized Plasma. Nauka, Moscow (in Russian). Grach, S. M. (1975) Parametricinstability of VLF waves in the upper ionosphere. Izvestiay VUZ. Radiofizika, 11, 1627-1637 (in Russian). Grach, S. M. and Shvarts, M. M. (1991) Decay interaction of plasma waves in the region of artificial turbulence. In Proc. III Suzdal URSI Symp. Modif: Ionos. Powerful Radio Waves, September 1991, Moscow, 158-159. Grach, S. M. and Trakhtengerts, V. Yu. (1975) Parametric excitation of ionospheric irregularities extended along the magnetic field. Izvestiay VUZ. RadioJzika, 9, 1288-1296 (in Russian). Grach, S. M., Mityakov, N. A., Rapoport, V. 0. and Trakhtengerts, V. Yu. (1981) Thermal parametric turbulence in a plasma. Physica, 24 102-106. Grach, S. M., Shvarts, M. M., Sergeev, E. N. and Frolov, V L. (1997) Broad continuum feature of stimulated electromagnetic emission Journal of Afmospheric and Solar-Terrestrial Physics, (submitted). Kotik, D. S. and Trakhtengerts, V Yu. (1973) On decay interaction of VLF waves in Earth magnetosphere. Geomagnetism and Aeronomy, 4, 871-877 (in Russian). Leyser, T. B. (1991) Parametric interaction between upper hybrid and low hybrid waves in heating experiments. Geophysical Research Letters, 18, 408-41 I. Leyser, T. B. (1994) Electromagnetic radiation by parametric decay of upper hybrid waves in ionospheric modification experiments. Physics of Plasmas, 1, 2003-2007. Leyser, T. B., Thide, B., Derblom, H., Hedberg, A., Lundborg, B., Stubbe, I? and Kopka, H. (1990) Dependence of stimulated electromagnetic emission on the ionosphere and pump wave. Journal of Geophysical Research, 95, 17,233-l 7,244. Leyser, T. B., ThidC, B., Waldenvik, M., Veszelei, E., Frolov, V L., Grach, S. M. and Komrakov, G. P (1994) Downshifted maximum features in stimulated electromagnetic emission spectra. Journal of Geophysical Research, 99, 19,555-19,568. Minkoff, J., Kugelman, P. and Weissman, I. (1974) Radio frequency scattering from a heated ionospheric, volume I, VHF/UHF field-aligned and plasma-line
Hybrid waves and the generation of the downshifted maximum feature in SEE bdCkScdtter
measurements.
Radio
Science,
9, 941-
956.
Murtaza, G. and Shukla, P. K. (1984) Nonlinear generation of electromagnetic waves in a magnetoplasma. Journal of Plasma Physics, 31, 423436.
Sharma, R. P and Shukla, P K. (1983) Nonlinear effects at the upper hybrid layer. Physics of Fluids, 26, 87-99.
Silin, V. P. (1973) Parametric Interaction of Powerful Radiation with Plasma. Nauka, Moscow (in Russian). Sizonenko, V. L. and Stepanov, K. N. (1964) On instability of plasma with current in magnetic field. Ukrainsky Fizicheskiy Zhurnal, 16,438-445 (in Russian). Stenflo, L. and Shukla, P. K. (1992) Generation of radiation by upper hybrid waves in non-uniform plasmas. Planetary and Space Science, #,473-476. Stubbe, P., Kopka, H., Thide, B. and Derblom, H. (1984) Stimulated electromagnetic emission: A new technique to study the parametric decay instability in the ionosphere. Journal of Geophysical Research, 89, 752337536.
ThidC, B., Kopka, H. and Stubbe, P. (1982) Observations of stimulated scattering of a strong high-frequency radio wave in the ionosphere. Physical Review Let-
2429
ters, 49, 1561-1564. Tsytovich, V. M. (1967) Nonlinear Effects in Plasma. Plenum Press, New York. Vas’kov, V. V. and Bud’ko, N. I. (1993a) Decay of high-power radio wave into a lower-hybrid and upper-hybrid or electron
Union
1993, Kyoto, p. 398. Vas’kov, V. V. and Bud’ko, N. I. (1993b) Decay of Powerful Radio Wave on Upper Hybrid or Electron Cyclotron and Lower Hybrid Plasma Waves. Technical
Report 95, IZMIRAN (in Russian). Vas’kov, V. V., Bud’ko, N. I., Kapustina, 0. V., Mikhailov, Yu. M., Ryabova, N. A., Komrakov, G. P, Maresov, A. N. and Gdalevich, G. L. (1994) Detection of VLF and ELF waves stimulated by the heating facility ‘Sura’ in topside ionosphere from data of satellite Intercosmos-24. In Proc. IV Suzdal URSZ Symp. Modif: Zones. Powerful
Radio
Waves,
August 1994, Uppsala, Sweden, p. 26. Yu, M. Y., Shukla, P. K. and El-Nadi, A. (1985) Linear and nonlinear instability of electron convection mode. Physics Reviews A, 32, 613-616.
Pergamon
and Solar-Terrestrial 0
Physics, Vol. 59, No. 1997 Elsevier
Science
PII: 81364-6826(96)001365
18, pp.
X2-2429,
1997
Ltd. All riehts reserved. Printed in &eat Britain 136‘668x/97 $17.00+0.00
Interaction of upper and lower hybrid waves and generation of the downshifted maximum feature of stimulated electromagnetic emissions M. M. Shvarts and S. M. Grach Radiophysical
Research
Institute, B. Pecherskaya 25, 603600 Nizhniy (e-mail: [email protected])
(Received in final form 8 December
Novgorod,
1995; accepted 18 December
Russia
1995)
Abstract-The decay of an upper hybrid (UH) wave to a lower hybrid (LH) wave and a daughter
upper hybrid wave is considered. The linear stage of the process was analyzed for ionospheric conditions and compared with experimental data as well as with theoretical results of other authors. It was shown that the effective excitation of the prominent downshifted maximum feature of stimulated electromagnetic emissions in the ionospheric modification experiments could take place possibly due to interaction between two oblique UH waves and a heavily damped LH wave. @ 1997 Elsevier Science Ltd
a few seconds, which is close to that of the artificial small-scale (li 5 2mc/fo) magnetic field aligned irregularities (FAI) of plasma density. The subscript ‘I’ relates to the geomagnetic field direction. Below we analyse the dispersion relation for the parametric decay of UH pump wave depending on the dispersive properties of the interacting waves and on the plasma density in the vicinity of the pump wave UH point to find out the appropriate parameters, which could be responsible for the observed DM peak downshifts AfoM in the SEE spectra. The effects of a small but finite electron gyroradius, of LH wave Landau damping on electrons and ions, as well as a possibility of oblique propagation of UH waves are taken into account. The influence of cyclotron harmonic and regular plasma inhomogeneity effects are neglected.
1. INTRODUCTION
In the present paper we consider the three-wave (decay) interaction of two upper hybrid (UH) waves and a lower hybrid (LH) wave for conditions typical for the ionospheric F-region. The purpose of the treatment is to interpret the downshifted maximum (DM) feature in the spectrum of electromagnetic emission, which is nonlinearly stimulated in the ionosphere by a powerful electromagnetic pump wave (PW) and detected on the ground (ThidC et al., 1982; Stubbe et al., 1984). The DM is a prominent spectral structure of the stimulated electromagnetic emission (SEE) in the lower sideband of the PW with its peak downshifted approximately IO-18 kHz from the PW frequency fo = ~0127~ and reaching an intensity of a few tens of decibels above the background noise level. The DM is observed for a wide PW frequency range between 3.5 and 9.5 MHz, excluding narrow regions near electron cyclotron harmonics. The highest frequency components of the DM feature occur roughly at the same downshifts Afh for all Jo, the values Afh - 7.5-8 kHz are close to lower hybrid frequency in the F-region (Leyser et al., 1990, 1994). This fact allowed these authors to conclude that the LH waves play an important role in the DM generation. On the other hand, the typical DM growth time after PW turn on is
2. BASIC FORMULAS
The dispersion relation of the parametric decay of a UH pump wave into UH and LH daughter waves can be easily obtained from a set of equations for three-wave interaction and the general dispersion relation of parametric instability. Omitting the details of derivation (see e.g. Tsytovich, 1967; Silin, 1973; Kotik and Trakht2421
M. M. Shvarts and S. M. Grach
2422
engerts, 1973; Grach and Trakhtengerts, 1975; Grach, 1975; Sharma and Shukla, 1983) we give here the well-known dispersion relation of the parametric decay in the following form (yj << Wj and w& > LO&): (Y +
r”l)(r
+ Y8)
=
y:>
(1)
where y = Im w is the instability growth rate, rf the nonlinear growth rate
<,i
=
the squares of the thermal speeds,
Gil&i
eB/(cm,i) the cyclotron frequencies, 01 = L (kj, B) the angle between the wave vector and
W ce., --
the magnetic field B, (-e) the electron charge, me,i the particle masses, N the plasma density, and c the speed of light. Higher gyroharmonic effects (n 2 3) in (5) are neglected. For kj - 0 equation (5) exhibits well known solutions: 2
W”,“l
=
w&
sin’&i
+ w:
=
wtll- W$~02eu.ul, w: = w:h Wj, kj and yj (j = U, ~1 , 8) frequencies, wave vectors and damping rates of the interacting waves, respectively (where the subscripts u, ul and 4? correspond to the UH pump, UH daughter and LH waves, respectively), si the ion contribution to scalar dielectric permittivity E, ?”(w,) the electron contribution to the dielectric permittivity tensor i, E, the UH pump wave electric field, and aj = Ej/Ej = kj/kj. The matching conditions between the coupling waves are w,(k.)
=
w,l(kul)
+
wr(ktL
(3)
k. = k.1 + kc.
(4)
The dependencies wj(kj) in (3) are defined through the dispersion relation
$2
W2
l-
pe W;
CO?
Oj
-
(w! -w&)
sin’ t?j
wpi -
_
I
_
w4 PVe2
W4
I
(WZ
-
sin4 ej
w&)(w?
-
4w&)
+
1 w:bJ:e - 3&w: + 6~4) sin2 0,cos2 e, I I (w? - w2ce)3 3
+
COS4
ej
I
=
0,
(5)
which is valid for both UH and LH waves if k:ve2 << Wt.
Ws
>
Lo%,
I(wj-nw,,i)/JZkllv,,iI
B
1, and n = 0, +I,. . . In the expressions (2) and (5) as well as in the inequality before equation (l), W~,,i = 4rreZN/m,i are the squares of the electron (e) and ion (i) Langmuir frequencies,
1
me
YU.Ul =
y.&
wi
3&
2
1 + 3 cos2ee ,
(7)
where w& = W& + W& and wf,, = W,wci/(l + = wcewi are the squared upper hy&/wz,) brid and lower hybrid frequencies, respectively. For kp, 5 1, where pe = ve/ wee is the electron gyroradius, the thermal corrections have a marked effect on the dispersion properties of UH and LH waves and must be taken into account in (l)-(4). This is clear from Fig. l(a,b), in which sections of dispersion surfaces are displayed. In Fig. l(a) the spatial matching conditions (4) are also shown schematically. For UH waves at k, cos 8, - w,/c the corrections of order wi/cr omitted in (5) must be taken into account. For ionospheric conditions, assuming that the PW frequency is not close to a gyroharmonic, we can take the decay rate of UH waves to be purely collisional with an electron collisional frequency ve:
2 _
(
(6)
v,/2.
For the decay rate of LH waves, the Landau damping on electrons yLand and ions yLand have to be taken into account in addition to collisional damping .ymi. According to Ginzburg and Rukhadze (1975) Ye = Ycd +
YEand + YAnd
=- V, wf c0s2 ee + w: sin’ eg 2 W&(m,/mi + cos2 ed + ’
w:
Wf
k&3 1cos & 1 exp +:$exp
m (Me/mi+ cos2 ed
(-&)I,
- 2Gve2 COS2or >
(8)
Hybrid
waves and the generation
of the downshifted
maximum
d =
feature
in SEE
wfhwulkzep:cos’o( Wj
2423
E,' 16rrNT,’
where o( = L (k,, k.i), cos CI = sin 0, sin B,i cos Ap, Ap = pu and vu1 are the azimuthal angles vectors in the plane perpendicular field.
(9)
cos 0” cos 0,i + mul, where mu of the UH wave to the magnetic
3. ANALYSIS OF THE DISPERSION
RELATION
The growth rate of the three-wave decay, obtained from (I),
y,=_2Y!!2B+ 2
0
0.2
0.4 koe
0.6
0.6
Fig. 1. Upper panel (a) shows sections of UH wave dispersion surfaces for wu = 6 MHz and wee = 1.35 MHz for different plasma frequencies 1: wpe/2rr = 5.845 MHz and 2: o+/2rr = 5.8 MHz. The matching condition (4) for the process UH - UH + LH is shown schematically. Lower panel (b) shows sections of LH wave dispersion surfaces for different frequencies WI 1: WI = l.t,5W]h, 2: W.f = 1.5Wl,,, and 3: Wp = hl,, where w]h/2n = 7.676 kHz and kll = kcosB.
where me/mi appears in the formula due to the ion contribution. In the expressions (5) and (8) we consider the ions to be unmagnetised. This approximation is valid if Re w >> wci and Im w = y > Wci/2n (Sizonenko and Stepanov, 1964). According to (8) the damping on magnetised electrons vanishes for 0~ - r-r/2, but prevails in ion Landau damping for cos2Qr > m,lmi, where we = We cos eg. Notice that, according to (2) and (7), in a dense plasma (w’,, B wL), the maximum values of the nonlinear growth rate yn occur for si = - W~i/w’, Ei > 1, and the following approximate formula is valid
J -
YeJ2+
(Kl
4
y2
n
(10)
is a function of four independent parameters of the coupling waves, which belong to an intersection of their dispersion surfaces in k-space [Fig. l(a,b)] defined by the dispersion relations (5) and the matching conditions (3) and (4). For the further analysis it is convenient to choose them to be w”, &, wg, and kt. In addition, we have to analyze y as dependent on plasma density N, because the properties of the UH waves exhibit an essential dependence on N in the vicinity of the UH resonance Nub, where wi, = wtul - wt. According to (2) and (9) the three-wave interaction is most effective for the WJ frequency range Wih 5 we < Wpi. In this case lower hybrid waves propagate almost perpendicular to the magnetic field, cos* l?e 5 m,/mi or, for the ionospheric Fregion, 89.5” 5 88 5 90”. Since we -+c w,, wul the change of the UH wave wavenumber in the interaction process is rather small, and we may consider k,ll z k,lll, lkUll = Ikulll, i.e. 6% = &I. On the other hand, small but finite values of cos 81 must be taken into account in (10) for the calculation of ~8 because of the important role of Landau damping. Let us begin our analysis of expression (10) from the dependence of y on kc. Figure 2 displays the growth rate y (solid lines) and the LH wave damping rate yt (dotted lines) versus kpp, for different frequencies of the lower hybrid wave we. For our calculations the following typical ionospheric F-region parameters were used: w,/(2rr) = 6.0 MHz, w,,/(2n) = 5.845 MHz, w,/(2rr) = 1.35 MHz, T, = x = 1500 K, v, = 300 SC’, and mi/m, = 3 x lo5 for O+ ions. The
2424
M. M. Shvarts and S. M. Grach
8
2 0 0
0.1
0.2
0.3 klpe
0.4
0.5
0.6
Fig. 2. The growth rate y (solid lines) and the LH wave damping rate ,yj (dotted lines) versus kgpe for different we. The LH wave frequencies are l:wr = l.O5Wth, 2:wJ! = l.l25wt,,, 3:wC = 1.25Wt,,, 4:WC = 1,5Wth, 5: We = 1.75Wlh, and 6: WJ? = 2W]h where W],,/2n = 7.676 kHz. electric field strength of the decaying UH wave is taken to be E, = 0.6 V/m, which is well above the threshold field defined from the expression yz = yulyC. From Fig. 2 it is seen that: (i) The curves y(kt) last only until ke = kt,, where
h? PnlPe
1_ 4 -
-
w:h/w;,
-
(4
w:h/w$
3 1 + 4(~/T,)(wf,/w$)
15 -
(11)
is the maximum wavenumber of the LH waves, defined by the dispersion relation (5) at Bp = 7r/2 (see Fig. 2). (ii) A maximum of the function y(ke) at ke = k;, with k;p, - 0.25-0.4, is attained when the essential growth of Landau damping with kc starts to dominate the increase of the nonlinear growth rate yn. This maximum occurs only for rather large frequencies (we > 1.12 Wth for the chosen parameters), when k; < ke,; if ktm < kz, then y(kj) is largest for kp = ke,. (iii) The largest value k; occurs for cosee m and wp - l.SSWlh. For this 84 the Landau damping is a minimum because of the decrease of yi with decreasing cos f3r and the decrease of & [due to the increase of wg, (7)] with increasing cos 0e. Consequently, the essential growth of yp with kg for these 198 and WP starts at larger kp (Fig. 2, curve 4). In the construction of Fig. 2 we supposed that cos2 (Y = 1, and neglected the dependence of k,
W *w 10 P 5-
0* I
I
I
I
I
I
e
0.4 -
W3
0
0.3 -
c/
4
2 5 :
b
0
’ 1
d
1 1.25
I
,
1.5 1.75 “‘C/Wlh
1 2
I
_
2.25
Fig. 3. Panels (a) and (b) show the largest y (ys) over ka, versus we for different 8” and wpe/2rr = 5.8 MHz.
The corresponding values of the LH wave damping rate yes and LH wave wavenumber kJg are shown in panels (c) and (d), respectively. Here 8, are 1: & = SO”, 7.Q.. = 75” ‘7.0. = 70° A.0 = h<‘= 5.62 - LO’= *n,i
Hybrid waves and the generation of the downshifted maximum feature in SEE and k,i on N. The account of a real mutual arrangement of k, and k.1, as well as of their real values obtained from (5), sets some limitations on k’Z and cos2 o( in (9) and reduces y in comparison with Fig. 2. Figure 3(a,b) shows ~a, the largest y over kg, versus we for different 0” near the upper hybrid resonance of the UH pump wave (wpe/27r = 5.845 MHz). Figure 3(c,d) displays the corresponding values of the LH wave damping rate and wavenumber keg, respectively. As is seen from Fig. 3, there are four types of JJ~(WI) curves. The first type [curves 1 and 2 in Fig. 3(a)] contains two growing parts, labelled a and c, and two decreasing parts (b and d). Curve 3 [Fig. 3(a)] has all these parts, but, in addition, there is a smooth maximum of y(we) labelled e. The curve 4 in Fig. 3(b) possesses the parts a, 6, e and d. Lastly, the curves 5 and 6 [Fig. 3(b)] contain only the parts a and b. Part a corresponds to rather small values of WI and kt,(wg) where ke,,, < k, = Jz77 k, (0,). According to item (ii) in the list above, for yg we must use here keg = kt,(ws) and cos2 1y = 1 - k~,(w~)/2k,(f?,) in the expressions (9) and (10). In this case the increase of ke,,, with we [see Fig. l(b)] defines the growth of yg, as well as the increase of k, with decreasing 8, defines the larger frequency width of the part a for smaller 0,. ’ In the curves 1 and 2 for larger we and ke,(we), when k, < kem(we) < kh = 2k0, y(kgm) < y(k,) because of the dependence of y on cos’ LX.Here, ys is obtained for ke = k,, cos2 a = 2/3 [see Fig. 3(d)], and decreases with increasing we due to the factor wi in the denominator of the expression (9) (part b). Further, for kh < kt,,,(we) < k, = 2k,sin8,, the largest y again occurs at ktg = kern(we), and ~a grows rapidly with we due to the increase of kern and 1cos a) (part c). For larger wd, when kl,,, > kc, the magnitude of the LH wave wavenumber is limited by the value 2k, sin 0,, because of the matching conditions (4) and in (9) and (10) we must use keg = 2k,, sin B,, Ap = rr, and o( = 28, for Ye. In this case, as in part b, for the curves 1 and 2, rf decreases as we-3 . Notice that the values keg for the whole curves 1 and 2 are found to be small enough (ka 5 2k,(0,) sin B, < k;) to
’ The last statement is valid until k,sinO, < k,,, where kum is the maximum possible value of k,,; see Fig. l(a).
2425
provide a purely collisional damping rate of LH waves, M = ~~1 [Fig. 3(c)]. For smaller 8, the magnitude of the UH wave vector component perpendicular to the magnetic field, k, sin 8, and k,, increase [Fig. l(a)]. This fact leads, in particular, to the growth of the frequency size of parts a, b and c (and the values y(k,), y(kb)) with decreasing 0”. On the other hand, the magnitude of the LH wavenumber may become large enough to an essential increase of the Landau damping, ke > kt In Fig. 3 for the curve 3 the Landau damping rate does exceed the collisional damping rate even in part c, and YLand grows with wq in this part of the curve due to the growth of ke,,, [Fig. 3(c)]. Nevertheless, the largest y still occurs at keg = ke,,,(wp) because cos’ o( is maximum at this point. For kt, > k,., the ~a still occurs at ke = kc, but the value yg(kc) in this curve is significantly smaller than, e.g., in curve 2, where k,(&) is smaller, and Landau damping is not important. In this part of the curve for increasing we, ~a first grows due to the decrease of the ion Landau damping rate Ytand, and then decreases because of the growth of yEand with cos2 08. This part of the curve 3, labelled by e, (as well as the similar part of the curve 4) can be clarified as a movement of the displaying point in Fig. 2 over the curves y(ke) along the vertical line kg = 2k,sin 0, [which corresponds to the horizontal line in Fig. 3(d)] for kep, = 0.3 15 (curve 3) and ktp, = 0.385 (curve 4). For further increases of we, the y(k,) becomes smaller than y(k,), and kg = k,, wsa = 213 and ~1 = ~~~1must be used in (9) and (10) for calculations of ~a in this part of the curves 3 and 4 (part d, the extension of part b). Notice that the absence of the growing part c in the curve 4 (as well as in the curves 5 and 6) is due to strong Landau damping of LH waves with kb -Cke < kc.
Part e appears in curve 4 (0, = 65”) only in the region of minimum of the total Landau damping YLand = &nd
+ &and
at 01
-
1.5wlh
and
kept
-
0.375. Figure 4 shows y versus we and ke for 0, = 65”. As is seen from Fig. 4, there are two hills of the surface Y(we, ke) corresponding to ke - k, and we - 1.12Wrh and to the minimum of the Landau damping. For smaller 8, the last hill is rather low and cannot be seen in the curves 5 and 6 for yg6(we).
2426
M. M. Shvarts and S. M. Grach
1
Fig. 4. y versus we and kt for 0” = 65” and wf = 0.985w&,.
The variations of the plasma density N near the upper hybrid resonance lead to noticeable changes of the UH wave number k,,, extremely so for 0, close to 90” [see Fig. l(a)]. The curves ya(wp) for 0” = 90” and different N/N,h, with N < h$h, are shown in Fig. 5. It is seen that the influence of the plasma density decrease is similar to the influence of the 0, decrease, since both of them give a rise of k,:the behaviour of the curves for ye for mie = 0.999&h,, 0.997&,*, 0.994&*, and 0.985wi,,, where cuih* = UJ’,- c&) in Fig. 5 is analogous to that for 0, = 80”, 75”, 70”, and 65” in Fig. 3(a,b), respectively. Notice, finally, that for 8, = 65” (curve 4 in Fig. 3) and for wi, = 0.985w:,, (curve 4 in Fig. 5) yg -K yp in part e. At the same time the LH wave damping rate is still small in comparison with the wave frequency, ~4 << we. So, in this case the weakly damped LH wave could be considered as a heavily damped product of the decay process.
4. DISCUSSION
Models of the DM generation connected with the interaction between PW, FAI, upper hybrid and lower hybrid waves have been discussed in the
literature (Leyser, 199 1, 1994; Grach and Shvarts, 1991; Vas’kov and Bud’ko, 1993a,b; Murtaza and Shukla, 1984). Leyser (1991, 1994) considered a parametric decay of a UH wave into an electromagnetic (EM) wave and an LH wave [UH EM + LH, (I)]. The UH wave was supposed to be excited by electromagnetic pump wave scattering off the ASI. In contrast, Vas’kov and Bud’ko (1993a,b) considered a parametric decay of the electromagnetic PW into UH and LH waves [EM - UH + LH, (II)]; UH wave scattering off the FAI was supposed to be responsible for the SEE generation. In the present paper, as well as in Grach and Shvarts (1991) we study the decay of a UH mother wave into a UH daughter wave with lower frequency, and an LH wave [UH - UH + LH, (III)]; the mother UH wave and the SEE are considered as a result of the scattering off the AS1 of the PW and UH daughter waves, respectively [double transformation (Belikovich et al., 1981)]. Neither of these models contain any nonlinear saturation mechanisms. Nevertheless, some conclusions concerning the processes of the LH waves excitation in ionospheric heating experiments can be drawn. Below we discuss briefly the results obtained and compare them with the results of Leyser (1991, 1994) and Vas’kov and
Hybrid waves and the generation of the downshifted maximum feature in SEE
6f 2
4-
2-
0’
’ 1
I
I
1.25
1.5
I
1
,
1.75
2
2.25
1
wC/wlh
Fig. 5. us
for & = 90” and different N/N&.
Here
I- w2 = 0.999w2“h*, 2-W& = 0.997w&*, 3-w;, = 0.99L&*. 4-w2 Pe = 0.985w2uh*, where wii,* = fJJ;_ W&.
Bud’ko (1993a,b). Z
Notice that the growth rate of process (II) can be obtained from the expressions (1) and (2) in which, however, a. must be replaced by = EEM/~EEM~. and, for W&/W& >> 1, UEM I&~~~(W3fh I2 N l/2. For the process (I), UEM must be used instead of (1.1, ~&(o~)czEM~* E l/2, and s IwZweM must be used in (2) inwhere n2 = c?k2/w2 is the stead of $$lwZw,, refractive index squared. For a plasma density close to the upper hybrid resonance [and 8, close to 90” for the process (III)] the dependencies y(w~) for all three processes are similar. They exhibit a sharp peak at cop = w* n. 1.12wih and a decreasing part d. At the same time, for (III) the limitations on the mutual arrangement of k, and k.i, due to the matching conditions, affect the value of the growth rate at w - wih (parts b and c in Fig. 3). This value w* corresponds to A~DM - 9 kHz and is lower than Af& in the measurements. For lower plasma densities the maximum in the growth rate is found to be displaced to higher w*, up to w* - 1.5wih (Afo~ - 12 kHz) which is closer to the values observed experimentally (Fig. 5). This is connected with the increase of the UH wave wavenumber (and, due to (4), of the LH wave wavenumber), with decreasing N,
’ Here we do not discuss the excitation of high harmonic ion Bernstein waves, considered in Leyser (1991, 1994).
2421
and the existence of the LH wave Landau damping minimum. For the process (III) at 8, = 90”, W* 1.5Wlh occurs at w& - 0.985w&,, or for N- 1.45% lower than at the upper hybrid resonance; for processes (I) and (II) this is at w& 0.945&,,, or for N - 5.5% lower than Nub. Such a difference is due to the different matching conditions: for our Fig. 5 we used ke = 2k, in (2), while for the processes (I) and (II) it is necessary to use ke = k,,. If oblique UH waves (0, f r-r/2) take part in the parametric decay, the maximum of y(wa) at W* 1.5Wlh does appear at wi, = w’, - wz&,i.e. near the upper hybrid resonance [Fig. 3(b), curve 4 for 8, = 65”]. Here both of the maxima w* 1.1 wih and w* - 1.5 wih are present. Notice that for the processes (I) and (II) the matching conditions (4) forbid the participation of oblique UH waves since kt,,, < k,, kt and k, z -kr. According to Grach et al. (1981), the presence of oblique UH waves in the ionospheric heated volume is due to multiple scattering of the UH waves off FAI. On the other hand, an effective interaction between EM and UH waves and FAI, responsible for the excitation of UH waves and FAI in the ionospheric modifications, occurs in a narrow layer wie - (1 + 0.003)$& for FAI scale lengths 1, - l-15 m (Grach et al., 1981, 1997). For lower plasma densities w& - (0.985-O. 945) wih, the effective interaction between EM and UH waves and FAI could occur for FAI with 1, - 0.50.25 m, which, however, have a very low intensity (Minkoff et al., 1974). Thus, the process UH -. UH + LH at w& - w$* for oblique UH waves with kulp, = ktp, / 2 - 0.15-O. 2 seems to be more probable to be responsible for the excitation of the DM feature in SEE spectra than the processes (I) and (II). Additional arguments in favour of the double transformation mechanism are presented in the paper by Leyser et af. (1994). First, the cascade type of 2DM generation (their Fig. 8) can be more easily explained in terms of a UH - UH + LH process. Then, the overshoot in the temporal evolution of the 3DM (their Fig. 9) is similar to the overshoot obtained for the broad continuum feature of the SEE at the same frequency shifts and interpreted in terms of the double transformation, with the UH wave spectrum due to induced scattering off ions (Grach et al., 1997). According to Stubbe et al. (1984) and Leyser
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M. M. Shvarts and S. M. Grach
et aI. (1994) the frequency shift of DM peak position A& increases with the pump frequency fo. We have not obtained this effect in our calculations. But we have found that an addition of a small fraction of light ions (i.e., a decrease of m,/mi) raises both the lower hybrid frequency Wth, the frequency position w; of the Ye maximum, and the ratio w: /u~th. Such an increase is not enough, however, to explain the values observed experimentally, if we assume that Afo~ = w: /2rr. Our results can be compared with the experimental data only qualitatively. For a complete interpretation of the DM feature of the SEE, a nonlinear theory of the saturation stage as well as a regular plasma inhomogeneity account are required. Notice, however, that according to Stenflo and Shukla (1992) and Yu et al. (1985) an influence of the convection mode, which exists due to the inhomogeneity in LH frequency range, is negligible for kep, 2 10e4. In our treatment above we supposed the UH pump wave to be monochromatic and took the reasonable value E, = 0.6 V/m. For the construction of a nonlinear theory it is necessary to take into account that in the experiment the real distribution of UH wave energy over the dispersion surface and the type of phase correlations of UH waves are unknown. Further, the two hills in the surface y( we, kc), corresponding to different ratios y/y/, must relate to different regimes of the saturation stage of parametric decay instability in the different parts of phase space. This fact can help with the construction of the nonlinear theory. Notice that recently Vas’kov et al. (1994) reported the discovery of VLF emissions from the ionospheric heated volume in the frequency range 7.5 kHz 5 f 5 9.5 kHz, which correspond, probably, to the ‘low frequency maximum’ of the surface y(we, kg), while the DM feature of the SEE is more likely to correspond to the ‘high frequency maximum’. Acknowledgements-The authors acknowledge the financial support from the International Science Foundation (Grant no. R87300) and Russian Foundation of Fundamental Investigations (Grant no. 94-02-03253-a).
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