i~adiophysics and Quantum Electronics, "Col. 37, No. 5, 1994

PLASMA WAVES NEAR THE DOUBLE RESONANCE LAYER

IN THE IONOSPHERE

S. M. Grach, B. Thidd, and T. B. Leyser

Dispersion properties of plasrna waves with frequencies w close to the upper hybrid frequency wu and the multiple electron cyclotron frequency nwse (double resonance) are considered for an inhomogeneous plasma with opposite gradients of the electron density and magnetic field magnitude. We show that a region of possib!e solutions of the dispersion relation decreases in real space as well as in wave vector space as of the wave frequency approaches double resonance. The results are applied to an interpretation of experiments on ionospheric modification by high-power radio waves with frequencies

close to nOgBe.

t. INTRODUCTION

At present it is commonly accepted that in the action of high-power SW radio transmission on the plasma of an ionospheric F-region, the greater part of pump wave (PW) energy is lost in the upper hybrid resonance (UHR) region of the PW, where w0 ~ wu, as a result of the thermal parametric instability (TPI) [1, 2, 3]. At TPI development, small-scale geomagnetic field aligned irregularities of plasma density (~.Pe << 1, n. l B) as well as plasma (upper hybrid) waves with a frequency w equal to PW frequency w0 and a wave vector lkl ~ ]~] are excited in the ionosphere.

Here the following notations are used: wu = ~/W2pe + w2e is the upper hybrid frequency, O3pe : ~/4~e2N/m and WBe = e B / m c the electron plasma and cyclotron frequencies respectively, e and m the electron charge and mass, c the light speed, N the plasma density, B the geomagnetic field, I B I = B, ~, the wave vector of the small-scale

irregularity, Pe = ~/Te/mW2e the radius of cyclotron rotation of thermal electrons. The experiments on the action on the ionosphere with PW frequencies close to harmonics of electron cyclotron frequency wo ~ nwB, have shown that the efficiency of the interaction of high-power electromagnetic waves with a plasma at wo ~ nwBe is significatly lowered. In particular, the amplitude of a~'tificial ionospheric radiation [4, 5], the plasma expulsion from the UHR region [6, 7], and the anomalous absorption of PW [7, 8] decrease. In order to construct a physical picture of artificial ionospheric turbulence excitation at w0 ~ nWBe it is necessary to study dispersion properties of plasma waves in the ionosphere in the double resonance region, where the multiple cyclotron frequency is close to the upper hybrid one, i.e., wo ~ w ~ wu ~ nwBe. The present paper is devoted to this.

Below the dispersion properties of plasma waves, transverse to the magnetic field, are considered in homoge- neous plasma. The dependence of the wave vector of the transverse plasma wave with fixed frequency on an altitude near the double resonance level is studied under conditions similar to those of the ionospheric F-region (opposite gradients of plasma density and magnetic field magnitude). The influences of the medium inhomogeneity and longi- tudinal component of the wave vector on the plasma wave propagation in the ionospheric conditions are considered. The relationship of the results obtained to experimental data on artificial ionospheric radiation measurements at ~0 "~ nWBe is discussed.

2. DISPERSION RELATION IN HOMOGENEOUS MEDIUM

The dispersion relation for potential plasma waves in a Maxwellian plasma is [9]

= I k2V:2 1 - I s ( k ~ p 2 ) e - k l P ~ J + ( f l s ) = O, s = O O J - - 8~Be

(i)

Radiophysical Research Institute, Nizhny Novgorod, Russia. Swedish Institute of Space Physics, Uppsala Division, Sweden. Swedish institute of Space Physics, Kiruna Division, Sweden. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 5, pp. 617-633, May, 1994. Original article submitted February 21, 1994.

392 0033-8443/94/3705-0392512.50 �9 Plenum Publishing Corporation

where ~s = (w - SWBe)/( ik l l[VTe) , V~e = T e / m w ~ e is the thermal electron speed, Is(z) the modified Bessel function,

J+(x) the so-called plasma dispersion function, which, for x >> 1, can be expanded as

i 3 . ~ - ~ / 2 j + ( x ) ~ l + ~ + x _ ~ + ' v 2 x e ~ . 2 .

At w ~ nwBe for exactly transverse propagation (k i B, J+ = t) in the long wavelength limit ( k i p e << 1)

the dispersion relation takes the form

3 k peWPeWBe _ , = 0, (2) ] Cd2e 2 2 2 2 ~2eOj2C,n(~, , / )e )2( 'n , - -1 >

_ _ ~ 2 W 2 ~ ,2

where cn =: 1 / (2n- in!) . Equation (2) has two solutions. At a sufficiently large difference between wu and nWBe

(w~,- nw~e) 2 >4W2peW2Bec~(kp~) 2(n-U,

these solutions correspond to upper hybrid

2 2 2 2 3 k peWPeWBe

~ ~ , = w,. + 2w,,.tw2 r _ 3 w 2 )

and electron cyclotron (Bernstein, w ~ ,~zBe) waves.

solutions of (2) are very close and

(3)

If the reverse inequality w ~ wu ~ nWBe is fulfilled, the

�9 ~ 3k'4p4 w2 ' ~ w~. 4- 2n2w2Be :h n~2pe 2 ( n 2 _ 4)n 2 + c~,(kpe)2(",-t) (4)

Notice tha t in the frequency region between the upper hybrid and multiple cyclotron frequencies there are no solutions

of Eq. (2), and even if w~, = nWBe the difference between w+ and w_ is finite: w+ - w_ = CnWpe(kpe) n - t (Gross gap

[10]) . The damping rate. of plasma waves at w ~ r~wBe is determined by electron collisions and electron cyclotron

damping 7 = Imw = 7 col + 3 ,cyct, ../cot ,~ b,e/2, (5)

71" W 4 k~ 2 - A 2 kzV, 2 ..fcyel ~ . Pe r /t.2.~2'~o- Pc.. / '11 Te

V o ti,,ln,JJIVTe

At the double resonance w .-~ w~,..-~ ,ZVB, we obtain from (2). (5) (fls >> 1)

),cycl ~ _ ~ A A u 2 2 2 _: e - A /kll V~.e

IklllVT,(1 + A J A )

where A := w - ~idBe , Av, : u2 -- Wu.

AA,~ > 0 (see next section).

(6)

Notice that real solutions of (2) in the double resonance region exist only if

3. INHOIVlOGENEOUS MEDIUM, PERPENDICULAR PROPAGATION

Let us consider the propagation of plasma waves at a fixed frequency in the double resonance region. In this

case it is convenient to represent the dispersion relation as a dependence of the wave vector value k on the plasma

density and magnetic field magnitude:

, - ~2(n-1) 11, 1 -- lZ -- V 1-4u ] Cn[~Pe) " = v(1 - ' (7)

where u = W2Be/W 2, v ---- W2pe/~ 2, n > 2. If the parentheses in the numerator of the r.h.s, of Eq. (7) have different signs (this corresponds to a frequency magnitude w between w+ and w_), the dispersion relation does not have solutions. Under ionospheric conditions the regular inhomogeneity of plasma density is of great influence on plasma wave properties and on the excitation of artificial plasma turbulence. In the double resonance region wu "~ nwBe

393

it is necessary also to take into account the magnetic field inhomogeneity [11]. In the ionosphere, below the F-

region maximum, the gradients of the magnetic field magni tude and the plasma density are opposite: the density N

increases and the magnetic field B decreases with alt i tude z. F o r t h e following t rea tment we use the following linear

approximat ion of the N and B dependences on z: N = N0(1 + z/Llv) , B = BoO - z /LB) , or

v = v o 1 + u = u 0 1 - . (8)

The Under typical conditions LB "." RE~3 ~ 2130 km (RE is the Ear th radius), a n d L N / L B ~ 0.015-0.035. double resonance point is chosen to be an origin of coordinates (z = 0) in (8). At this point w~,.(z) = nwBe(Z), or V0 = (n 2 -- 1)u0. After subst i tut ing (8) into the dispersion relation (7), it takes the following form:

where ~ = cn(n 2 - 1)/2,

= _ z ) ( z -

L N L B ( 9 )

( n 2 u 0 - 1)LB ,,, A(O)LB (10) Zc ---- 2n2u0 -- w

z'u = zu -- 6zu " Au(O) LN, CO

,,~ (n2u0 - 1)LN ,.., 2A(0)n2LN 3LN(n2-- 1)(kpe)2 (11) z , , - - u0(n 2 - 1 ) -- ~ ( n 2 - 1 ) ' 6Zu= n2(n 2 - 4 ) '

where A(z) = ~ - nWSe(Z), Au(z) = ~ -- ~u(z), Zc is the multiple cyclotron level, at this level w = nwBe(Z), z~, the

upper hybrid level (w = wu(z)), 2v, the . "local" upper hybrid level, where w = &u(z). We have also omi t ted small

te rms of order Iz~. - zc l /LB compared to unity. From (9) one can see, in particular, tha t in a p lasma with oppositely

directed V N and VB, plasma waves with k _1_ B near the double resonance level (z = 0, where w~, = nwBe) can exist

only between the local upper hybrid and the multiple cyclotron levels, where A(z)A, , ( z ) > 0. Outside this region

the fl'equency w falls in the interval between solutions w+ and w- of dispersion relation (2) (see (4)).

Let us begin the analysis of the dispersion relation (7), (9) fox' the case n = 3. Here we obtain the simple

analytic solution (1 - 4u ..~ 5/9)

( k ' P e ) 2 - 1 - 9 U ( - l • l + 2 0 ( l - u - v ) ( 1 - 4 u ) ) l O u -3v-~=gu)" (12)

or, using (8), (10), (1t)

(hpe)2 - 9 ( z - z c ) ( - l + ~ I + 5 0 ( z ~ ' - z ) ( L B / L N - 4 ) ) 5 - L ; 2~(z-- '-~) " (13)

If ( l~-gu) > 0 (in this case A(0) = ~ - 3~Be(0) > 0, z,~. > zc) there is one positive solution ((kpe) 2 > 0) of the

dispersion relation. I t corresponds to the upper sign before the radical in (12) and (13), and exists if the parentheses

(zu - z), (z - ze) under the radical in (13) have the same sign. In th is case o~ - 3oJBe > 0, ~ - w~,. > O, z~ < z < z~, i.e., the solution corresponds to the + sign in expression (4); plasma waves can exist between the mult iple cyclotron

level and the upper hybrid level and have a normal dispersion (OoJ/Ok > 0). If 1 - 9u < 0 (A(0) < 0, ze > z~,) there also exists one positive solution of (9) with w < 3WBe, w < w~ in the

region z~, < z < zc . It corresponds to the "-" sign in (3) and has an anomalous dispersion. But, below the upper

hybrid level a t z3 < z < z,,., (1 - 4LN/LB)Z~. -- 0 .54zr A(0)

z3 = 1-- 4 .54LN/LB ~-- w LB, (14)

there exist two solutions of Eq. (9) with fl'equency values w~. < w < o3u, w < 3~Be. One of them (the + sign before the radical in (12)) corresponds to an upper hybrid mode with a normal dispersion, while the other one corresponds

to a Bernstein mode with anomalous dispersion. At z = z3 the dispersion equation (9) has a multiple root

(kpe)~ -- 9u - 1 _ 9(zc - z3). lO'u, 5LB ' (15)

394

at z < z3 no solutions of the dispersion equation exist. Thus, in a plasma with \TN T,L V B , plasma waves with

co ~-- 3coB~ and' k '_1_ B can exist only between the "multiple root" and multiple cyclotron levels when A(0) < 0

and between upper hybrid and multiple cyclotron levels when A(0) > 0. If A(0) decreases, the upper hybrid (11),

multiple cyclotron (10), and multiple root (14) levels approach each other and the region of plasma wave existence

decreases in real space as well as ~n wave vector space ((kp~)22x ~ (A(O)/w)v/6LB/L--NN at z ~-- (zu + zc)/2) . If the

levels zv, and Zc coincide exactly (A(0) = 0, co = 3WBe(0) = co.,(0)) in such a plasma there is no possibility of plasma

wave existence anywhere. The behavior of the dependence kp~(z) for n > 3 in the double resonance region is similar on the whole to

the n = 3 case. So, if the value A(0) is positive and sufficiently large (in this case z~,, > zc, w > -*os~), there is one real solution of dispersion relation (9). It has a normal dispersion and is localized in the region zc < z < z~,.

For A(0) < 0 (ze > z~,,, w < nwBe) in the region z~, < z < z~ there is one real solution of Eq. (9), corresponding

to a Bernstein. mode (B) with anomalous dispersion; at Zn < z < z,,. we have two solutions: an upper hybrid mode

(UH) with normal dispersion and a Bernstein mode (B) with anomalous dispersion. At z = z,, these two solutions

coincide, i.e., there is a multiple root of the dispersion relation (9). The value of the multiple root (kp~(z)), , can be

obtained from Eq. ( 9 ) a t z ,= z,:

1 = - -

an(., - 2 ) L N L B (t6)

The value z.. can be obtained from the equation

(z , , , -z . . ) '~'-2 ( 3 ) " - t ( n - : 2 ) " ' 2 t (17)

Of course, zn depends on the value A(0) (see (10), (11)). In particular, at z,,, = ze = 0 (A(0) = 0), i.e., exactly at

the double resonance, we obtain ( n > 3)

Thus; in contrast t o n = 3, :at higher harmonics plasma waves, pexpendicuiar to the magnet ic field, can exist at

z < zc = z,,~ Besides, at 0 < A ( 0 ) < A.., O < z~ - ~ < AZn, wtiem

Az . . = L g (b,~nN'~ ~-'~-3 , bn = ( n - 3 ) n - a ( 1 9 ) ' k LB ] a . ( n - 2) ~-2

n--2

A,, = wb~, -~-3 1 + 2 L N / L n '

the re are two regions, separated in space, where solutions t o the dispersion relation (9) exist: 1) ze < z < zu with

one solution with sufficiently small values of kpe, where zc < ~u, A > 0 and 2) a region with sufficienty large values

kpe, (kpe) 2 > ((kpei2) * "~ A ( O ) L B n 2 / ( w L N ) , zc > ~,,,, A < 0; here two solutions of Eq. (9), corresponding to UIt

and B modes, are obtained. In the region z < zc, z ~ zc, kpe ~ (kpe)* there is no solution of Eq. (9).

The dependences of kpe on z a t 60 ~-. 5WBe at two different signed values A(0) for a plasma with V N T J. VB, LN << LB are shown in Figs. 1 and 2. In out' calculations we have. used the values WBe = 27r x 1350 td-Iz, L N / L B = 0.022, LB = 2240 kin, arid p~ = 3 cm. These parameters are typical for the ionospheric F.region. The value kpe(z)

was calculated from the dispersion relation [9]

60 2

e = - k2 d + - = 0 ( 2 1 )

with rega~:d to electromagnetic corrections of order w2/k2C 2. The components of the dielectric permit t iv i ty tensor eij were calculated in the kinetic approximation for a Maxwellian distribution function with kll = 0. The electromagnetic corrections at z > z,~ lead to the appearance of an additional solution of (21) in comparison with (9), corresponding to the electromagnetic Z-mode. The parts of the dispersion curves corresponding to the Z-mode, the upper hybrid

mode (UH), and the Bernstein mode (B) are depicted in Figs. t and 2. The dotted lines marked by ze, zu and zu show the multiple cyclotron, upper hybrid, and local upper hybrid levels respectively. Dot ted lines marked by ~B,

395

0 . 2 - - T - - T 1 r

o.1 z iv li .: (s

. Zt= o

�9 1o ~ ......... '

/.,'--~ • -0.1 i"..t 1 " i" '"'" !i : . "." :". .".' .. '.i'

-0.3 " "

-0.4 - - - - - - - - - ~ ~ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

kp,

Fig. 1. kpe versus z for A ( 0 ) = 2~ x 2 kHz for n = 5. The dotted lines labeled zc, z~,

~,u, and ~B correspond to the multiple cyclotron, upper hybrid, local upper hybrid l eve l s ,

and .~ = .~B position, respectively; 7, UH, B correspond to different dispersion laws for perpendicular propagation; I-IV, for oblique propagation; vertical line, to ~B -- 1.

{B• as well as regions marked by I -V will be explained in the next section. The dependence of kpe on z for the same values IA(0)I and different n are shown in Figs. 3 and 4. Here the double resonance levels for different n are combined to facilitate visual comparison. The curve labeled "6" in Fig. 3 exhibits the two separate space regions of existence of solutions of the dispersion relation (9).

4. I N H O M O G E N E O U S M E D I U M , O B L I Q U E P R O P A G A T I O N

As one can see fl'om the previous section, the variations of the refractive index with z are ra ther significant.

The characteristic value of the scale of such variations can be l ~ 100-200 m or even less. If z _L B, the influence of the regular inhomogeneity is negligible because 1-1 << k, and there is a possibility of plasma wave propagation along

z. But for the opposite case, when z II ~ N II V B II B, the influence of the regular inhomogeneity is significant, and the exact transverse direction is not accessible for plasma wave propagation. In this case it is necessary to take into account parallel wave vector component kll _= kz. For the analysis of properties of oblique propagating plasma waves we use the dispersion relation in the form (1) in the long wave limit (k2p 2 << 1; #s >> 1; J+ ~ 1 + 1//7~). Omitt ing

the imaginary part of c, the high order terms ~ k6VT6e, and keeping the resonant term (s =- n) and using (8), (10), and (11), we obtain for the double resonance region

6nLB(k • 2(n-1) - - E'~--z

= z - z o L N (22) 1 - ClnL~(kd 'pe)2"

(z-zcp

To consider plasma wave propagation with k L = const, it is convenient to represent the dispersion relation in the following form:

k2n2 ~ 2 ( r - a)(1 - a - ~) k~T (23)

where the following dimensionless quantities are used:

~ '= (Zc- z)/Az; CB = (~n(k• ~N = L N / A z

396

0.4 T '~ ~ ~ T r T l - Z,a 1

1 0.2 ".(e " ./B~ ~ B IV "'. ",..

" V ' .

• 103

-0.2

-0.4

"'. "=u

-0.6 J . a _ _ "- _ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

kp,

Fig. 2. kpe versus z for A(0) = -27r x2 kHz for n = 5. The dotted lines labeled zc, z~,, 5%. ~B, and .(B_L correspond to the multiple cyclotron, upper hybrid, local upper hybrid levels,

= ~B, and ~( = fiB• positions, respectively; Z. UH, B correspond to different dispersion taws forperpendicular propagation; I, IV, and V, for oblique propagation; vertical lines, to (_~ = 1 and k = k,,.

1 i l ~LsLlv(k• A z = z c - ~,,; ,~ = ~ - - A z 2

At z = zc (multiple cyclotron level) (~ = 0, at z = ~, (local upper hybrid level) ( = 1. Real solUtions of Eq. (23) are possible only for (. > 0, i.e., for z > zc, ~, > zc when A > 0, or for z < Zc, 5,~ < zc when A < 0.

Here it is also useful to represent the formulas for parallel and transverse group velocity components (lktl] << tk•

lk• ~ 1~,1) a s f o l l o w :

VIt~= n2k 2 ,2 ~/3(kp~)~-/3-2~I (24) ..-7=f-I + ~

n2 e~ /3{~ , )2~ /3 -2r ( 2 5 )

In formulas (23), (24), (25), the upper sign must be used for A < 0, and the lower sign must be used when

A > 0. The possibility of plasma wave propagation (existence) should be estimated from the inequality

= l~zl . / ktldC >~ rr/2, (26) J

where the integral should be taken over the region k~ > 0. Let us consider different relations between the parameters

in Eq. (23).

1. A > 0, ~B > 1. Here there are two regions of-existence of real solutions of Eq. (23): ~ < ff < t - ~ and

> (B- In the first one, labeled by I in Fig. t , the plasma wave has turning points at ff = c~ (a little above the multiple cyclotron level) and at ~ = 1 - ~ (a little below the local upper hybrid level) and can propagate "inside" the dispersion curve for kll + = 0; the turning points are the limits of the integration in (26). In this case the plasma wave belongs to the Bernstein branch: Vil+kll + > 0, V• > 0. Notice that the dashed line in the figure marked by "~B" shows the ~ = fib position.

397

z L~ x I0 ~

0.2

0

-0.2

-0.4

-0:6 0

i i I i i i

3 I I I , I I I I

0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.8 k~

Fig. 3. kpe versus z for A (0 ) = 21r x 2 kHz for n = 3, 4. 5, and 6, respectively.

The other region (4 > .~B, region II in Fig. 1) corresponds to the upper hybrid branch: Vll+kll + < 0, V•177 > O. Here the plasma wave propagation is possible "above" the dispersion curve for kll + = 0. But unlike usual upper

hybrid waves, in this case, the turning point is located at ~ = ~B, where VII+ = 0, kil + ---* cx~ (see (23), (24)). So, strong cyclotron damping (A ~o kliVTe) should suppress upper hybrid waves. Here it is necessary also to

take into account the higher order terms 3//3s 4 for s = 0, 1, 2, and n for the plasma function J+ representation.

. A > 0, @ < 1. In this casek~+ > 0 a t c~ < r < ~ B and at ( > 1 - a . The Bernstein wave can propogate

between the turning points ( = c~ and ~ = ~B (at .( = (B VII+ = 0, k[i + --* co). Cyclotron damping is significant in this region, denoted by III in Fig. i. The upper hybrid wave propagates above the turning point r = 1 - a, (IV in Fig. 1) and feel the neighboring cyclotron harmonic weakly.

. A < 0, (B < 1 (region IV, Fig. 2). This case also corresponds to the usual upper hybrid waves propagating "above" the upper hybrid branch of the dispersion curve, denoted by UH. However, during the propagation

upwards the value kliVTe becomes larger than IAI. This is beyond the long wave limit assumed. But upper

hybrid waves are weakly damped at A << kllVTe << w, i.e., for a short .wave limit. (J+(/3n) ~ iv/~/3n) in the

resonant term s = n in (1). The propagation of upper hybrid waves via the region A ~ k[iVTe should be investigated separately.

. A < 0, ~B > 1, k• > k,. (see (16)). This case corresponds to the Bernstein branch (Vll_kjl_ < 0, V•177 < 0). The plasma wave has turning points at .~ = a (below the multiple cyclotron level) and at ~ = I - a (above

the local upper hybrid level) and propagates "inside" the dispersion curve, so this region should be noted by I (Fig. 2).

. A < 0, @ > 1, k• < kn (region V, Fig. 2). As in the previous case, the turning points are located at ~ = c~

and at r = 1 - c~. But unlike the previous case, here during the propagation between turning points with k,• = const the plasma wave passes the point where V• changes sign. So, the plasma wave group path looks like a

cycloid, or even like a ring: it can return to the emitting point from the opposite direction. At the point V• = 0

(dotted line in the Fig. 2, marked by .~B) the plasma wave changes from the upper hybrid mode (Vll_kll_ < 0,

V•177 > 0) to the Bernstein mode (Vll_kll_ < 0, V•177 < 0) during propagation upward and from the Bernstein mode to the upper hybrid mode during propagation downward.

Thus the plasma wave propagation in the double resonance region is possible with the upper hybrid dispersion law (IV) above the tm'ning point ( = 1 - a, and with the Bernstein dispersion law (I) between turning points .( = c~

and at ( = I - a. In the last case the change in the dispersion law is possible during plasma wave propagation (V). In regions I, V the possibility of plasma wave propagation dependent on k• and A(0) should be investigated by (26) with integration limits ( = c~ and ~ = 1 - a. At @ >> 1 from (26) we obtain a simple expression

398

0.4

L~ • 103 -0.2

,0.8

*1 I i t I

o oi, o:2 o3 o., o s o.6 o., o8 ~o,

F~. 4. kpe versus z ~:or A (0 ) -- - 2 ~ • 2 kHz for n. = 3, 4, 5, and 5, respectively.

. , k • - 2a) J = 16 ~3/2/ .1/2 (27)

~B ~N

The position of the point J = ~r on the (~• A(0)) plane is shown in Fig. 5 for n = 6 for the same plasma parameters~ The dot ted line at A(0) > 0 corresponds to the position ~B = t. In Fig. 5 the upper par t of the lower

curve, si tuated above t he zero-level of A(0), corresponds nevertheless to A < 0 (see the "Perpendicular Propagation" sect ion, thetext~af ter formulas (18), (20) and Fig. 5)~ The numerals I -V placed in Fig. 5 show the regions of plasma

wave dispel~io a laws. The regions above the upper curve and below the lower curve at Fig. 5 correspond to J > r ,

i.e., here plasma wave propagation is available. The region between the upper and lower curves is forbidden for plasma wave propagation. The J = ~- positions for n = 3-7 are shown in Fig. 6 and Fig~ 7.

5. DISCUSSION

From the above t reatment , the following conclusions can be made.

In an ionospheric plasma, for sufficiently small angles of the magnetic field to the vertical tin the middle and

high latitudes)~ the slight dependence of magnetic field on the altitude z ( ( 1 / B ) ( d B / d z ) = L~l) , in parallel with

the inhomogeneity of the plasma density ( ( t / N ) ( d N / d z ) = L N 1, LB >> LN) , exerts a significant influence on plasma wave properties in the double Iesonance region, where w ,.~ a~u(z) ,'.. nwBe(Z). As the wave frequency w approaches

to double resonance the region of plasma wave existence decreases both in reM space and in wave vector space.

For the third harmonic n = 3 at the exac t coincidence of the plasma wave frequency with the double resonance one, w = ~,~,,(0) = nwBe(O), the solution of the dispei~ion relation for plasma waves with k • B does not exist at

all; for n = 4 the region of the solution existence is limited by a few tens of meters (see (18)). For a realistic

estimation of the region of plasma wave existence, the longitudinal component of the wave vector has to be taken into account. A t rea tment for typical ionospheric F-region parameters (V N TJ. "~B, L N ~'~ LB , L N / L B = 0.022,

WSe ~--- 271- X 1350 kHz, LB = 2240 km and Pe = 3 cm) has shown the presence of forbidden zones in the plane (w,k• in which plasma waves cannot exist anywhere in such a plasma (see Figs. 6, 7). The zones are asymmetric relative to exact coincidence with the double resonance, i.e., they belong to plasma wave frequencies slightly above Wd = nwse(O) = wu (A(0) > 0); their sizes decrease with an increase of the harmonic number n and comprise approximately 30, 6, 1.8, 0.7, 0. t kHz for, respectively, n = 3, 4, 5, 6, 7 at the chosen plasma parameters.

399

6 i i 1 i i

5

4

3

2

!~.,1 1

0

-1

-2

-3 0.1 0.2 0.3 0.4 O.S 0.6 0.7

/,-p,

Fig. 5. Boundaries of propagation regions defined by inequality (26) for n = 6. The dotted line corresponds to the boundary between the regions of Bernstein (I) and upper hybrid (IV) dispersion laws (~B = i).

At the same time, the long wave limit consideration, neglecting the resonant ~ term contribution to the imagi-

nary part of the dielectric permittivity, may be insufficient. In particular, the inequality fin >> 1, An(z) >> IklflVTe) may be violated during the plasma wave propagation between the turning points (cases I, IV). Additional analysis is

also required for the case V, in which the plasm a wave has to pass the region An(z),,~ I klllVTe during the propagation.

Combining the condition fn >> 1 with the inequality (26) and using (22), (23), (27), we obta in the fotlwing crude estimation of the spatial size of the forbidden zone (see also [5])

where

Az2(1 - 2a) ~> 4peLB/n , (2s)

3klp (.. 2 - 1) IAz l ~ - ~ ) - . ( 2 9 ) t m J

IAzl depends, of course, on k• Here the frequency width of the forbidden zone exceeds the one obtained from the

above treatment. The last term in (29) exhibits the shift of the forbidden zone center to the positive A(0). The existence of the forbidden zones for the plasma waves in the double resonance region below the maximum

of the ionospheric F-region is one reason for the significant weakening of high-power radio wave interaction with

ionospheric plasma at ~0 ~ nwBe. Notice that the considered effect is due to the properties of the regular medium

(magnetic field and plasma density) inhomogeneity, and does not depend on the specific mechanism of the nonlinear interaction. Thus the influence of the medium inhomogeneity should be taken into account for alternative mechanisms

of the weakening of the interaction of the pump wave with the plasma waves at ~v 0 ~ nwBe. In particular, such an account was not taken in [13], in which the linear conversion of upper hybrid waves into Bernstein waves in an inhomogeneous magnetic field with VB _k B was considered, and in [12], in which it is shown tha t the efficiency of

upper hybrid wave trapping inside the field aligned irregularities of the plasma density decreases at w --~ nwBe. The effect considered does not explain all the peculiarities of the experimental data. In particular, in [4, 5, 15,

16] it was found that a decrease of the intensity and a narrowing of the spectrum of artificial ionospheric radiation emerges for a pump wave frequency width of a few tens of kilohertz, whereas the main spectra] maximum (the

400

2 *

[knz]

30 , i , i , , , , ,

20

10

5

0 ~ ~ ,5

-10

-20

-30 ' ' 0 0 : 0 5 0 . t 0.15 0.2' 0.25 0.3 0.35 0.4 0 .45 0 .5

kp.

F ig . 6 . Boundaries of propagation regions d e f i n e d by i n e q u a l i t y

(26)-for n = 3, 4, and 5; respectively.

"downshifted maximum" according to the nomenclature of [14, 15]) disappears completely in the widths Awo/27r 5, 3, 1 and 0.1 kHz for n = 4,5, 6, 7, respectively ~5, 16]. The last values are close to those obtained in the above calculations. A more detailed comparison of the results obtained with experimental data as well as a discussion of the functkm of vertical movements andof horizontal inhomogeneities in the ionosphere air presented in [16].

S. M. Gi"ach acknowledges the financial support from the Russian Foundation of Fundamental Investiga- tions (grant No. 94-02-03253:a) and from the International Science Foundation (grant No. R87000). B. Thid~ and T. B. Leyser acknowledge the financial support from the Swedish Natural Research Council (NFR).

R E F E R E N C E S

.

.

.

4. 5.

.

7.

.

9. 10.

11. 12. 13. 14.

S. M. Grach, N. A, Mityakov, V. O. Rapoport, and V. Yu. Trakhtengerts, in: Thermal Nonlinear Phenomena in Plasma [in Russian], Gorky (1979), pp. 46-80. V. V. Vas'kov and A. V. Gurevich, in: Thermal Nonlinear Phenomena in Plasma [in Russian], Gorky (1979), pp. 8t-t38. T. R. Robinson, Phys. Rep., 179, No. 2 & 3 (1989). T. B. Leyser, B. Thid~, et al., J. Geophys. Res., 93, 8681-8688 (1990). T. B. Leyser, B. Thid~, S. Goodman, M. Waldenvik, E. Veszetei, S. M. Grach, A. N. Karashtin, G. P. Kom- rakov, and D. S. Kotik, Phys. Rev. Lett., 68, 3299-3302 (1992). L. A. Lobachevsky, Yu. V. Gruzdev, et at., J. Arm. Terr. Phys., 54, 75-85 (1992). G. P. Komrakov, S. M. Grach, M. A. Yurishchev, A. N. Karashtin, G. P. Komrakov, D. S. Kotik, B .Thid~, T. B: Leyser, M. Waldenvik, and E. Veszelei, X X I Y General URSI Assembly, Kyoto, 1993, Abstracts, p. 398. A. J. Stocker, F. Honary, T. R. Robinson, and P. Stubbe, J. Geophys. Res., 98, 13627-13634 (1993). V. L. Ginzburg, and A. A. Rukhadze, Waves in Magnetized Plasma [in Russian], Moscow, Nauka (1975). A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N. Stepanov, Electrodynamics of Plasma [in Russian], Nauka, Moscow (1974). S. M. Grach, Izv. Vysshikh Uchebn. Zaved., Radiofiz., 22, No. 5, 521-530 (1979). N. N. Rao and D. J. Kaup, J. Geophys. Res., 97, 6323-6341 (1992). E. Mjolhus, J. Atmos. Terr. Phys., 55,907-918 (1993). P. Stubbe, H. Kopka, B. Thid~, and H. Derblom, J. Geophys. Res., 89, 7523-7536 (1984).

401

6 ! i i . i ~ i g i i !

:t / / / pn,!

"3 f ? 0 01 02 03 04 05 06 07 08 09 k~

Fig. 7. Boundaries of propagation regions defined by inequality (26) for n = 5, 6, and 7, respectively.

15.

16.

T. B. Leyser, B. Thid~, M. Waldenvik, S. Goodman, V: L. Frolov, S. M. Grachl A. N. Karashtin, G: P. Kom- rakov, and D. S. Kotik, J. Geophys. Res., 98, 17597-17606 (1993). T. B. Leyser, B .Thid~, M. Waldenvik, E. Veszelei, V. L. Frolov, S. M. Grach, and G. P. Komrakov, J; Geo- phys. Res, 99, 19555-19568 (1994).

402