Volume 84A, number 1 PHYSICS LETTERS 6 July 1981

WHISTLER DETRAPPING FROM A DENSITY CREST DUCT

V.I. KARPMAN and R.N. KAUFMAN IZMIRAN, Academic City, Moscow Region 142092, USSR

Received 27 April 1981

A theory of whistler wave leakage from a magnetic field aligned duct with enhanced plasma density is presented. The energy flux from the duct and the corresponding wave attenuation rate are calculated in the WKB approximation. Possible experimental confirmations of the theory are indicated.

In recent years, the wave guide propagation of whistler waves has been intensively studied in connec- tion with the concept of magnetospheric field aligned ducts, the problem of whistler nonlinear self-focus- ing, etc.

Ray theory predicts, in particular, a perfect trap- ping in density crests if 0 < wB/2 (OH is the elec- tron gyrofrequency) [ 11. The same conclusion fol- lows from the approximation based on the parabolic (Schrodinger) equation [2].

In contrast to these theories we show, on the ground of the full set of Maxwell equations, that the whistler wave escapes from the density crest even if w < wB/2. The leakage is the stronger the less the ratio of the duct width to the wavelength. This may explain why in narrow self-focused whistler beams only the density troughs have been observed [3], though the theories based on the non-linear Schro- dinger equation predict both dips and humps [4-61. Our results also give natural explanations to some phenomena observed in other experiments [7,8].

Assume that the external magnetic field is directed along the z-axis, “+, = const., and suppose, for sim- plicity, that the electron plasma frequency wp de- pends only on x. Considering, for brevity, only the y- component of the wave electric field, one can write

E,, = Z(x) exp [i(w/c)pz - iwt] . (1).

We also assume that the wave guide width is much greater than c/w and, therefore, we may use the WKB approximation. Then from the Maxwell equations

0 031-9163/81/0000-0000/$02.50 0 North-Holland Publishing Company 9

under the conditions

(me/mi)wB 4 w < WB < up,

we get four linearly independent solutions and the fol- lowing expressions for F(x), respectively:

F;(x) = @(p2 - 4&‘4Q(X) (2)

where

k= 1,2, (Y= w;(x)/w$,

Q(X) = [p - (-l)Q? - 4cr)l’2] 1’2) (3)

4; = $u-2[(1 - 2&p2 - 2a + (-l)kp(p2 - 4cY)l’2] )

and t( = w/tiB. One should bear in mind an important difference between the two wave branches designated by the subscript k: if I$) is the x-projection of the energy flux, then

sign II;‘) = (-l~sign qk,

i.e. I;;‘(x) and J-c(x) correspond to waves propagating in the positive and negative directions, respectively.

It follows from eq. (3) that for u < l/2: &o) > 0 if o. Q CY < p2/4, and q;(o) > 0 if (Y < p2/4: where cru = p%(l - u); otherwise 4; < 0. For u > l/2: &cr) <Oforallcr,and&o)>Oifa<~.

Now, suppose that u < l/2 and the density profile (w(x) is a symmetric hump: o(x) = a(-~), o(0) = h.

Volume 84A, number 1 PHYSICS LETTERS 6 July 1981

Then, if

maxt4%,, a(m)lu(l - u)) <p2 < s,/u(l - u),

(4)

the plots of qk(~) look as shown in fig. 1, and

a(+~~) = 06. Thus, the branches q1 and q2 are iso- lated [because (Y,,, < p2/4, according to eq. (4)], and ray theory predicts wave guide propagation for the branch ql. However, the branch isolation breaks due to deviations from ray propagation. This may be described as a partial transformation 41 + q2 and it leads to whistler detrapping from the duct.

To investigate this phenomenon, consider F(x) in three different regions:

F(x)= C&(x)+ C,F,-(x) (-x0 <x<x,), (5)

F(x) =A,lqIl -U2(p2 _ 4&)-l/4

(6)

F(x) = B,lq1l -I/2(p2 _ 4@/4

(7)

The termsA2FF(x) andBzFl(x), in eqs. (6) and (7) describe the mentioned transformation. They

may be considered as small as far as the WKB condi- tions are fulfilled. The leakage leads to wave attenua- tion in the z-direction. This is described by the ap- pearance of a positive imaginary part of p which gives an imaginary part of qk. Designating quantities calcu- lated in the ray approximation by the index 0, we write p = p” + ip’ and q1 = q$r + iq;, where p’ and 4; may be considered as small. Therefore

4; = (Q@P)P = -bg,(&Y~gx(4?M (C/W)P ) (8) where ug is the group velocity [we used the relation aq/dp = -ugZ/ugxwhich follows from the dispersion equation o = o(kx, k,)], and p = (o/c)p’ is the atten- uation rate. The latter can be calculated if one knows the relations between the coefficients in the system of equations (5)-(7).

To find them, we first continue eq. (6) in the inter- val 1x1 <x0, then into the region x < -x0, and then we compare the results with eqs. (5) and (7). As far as

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Fig. 1.

the points fxo are singular, they must be rounded in the complex plane. One should also take into account the branch points x where q1 (x) = 42(x), i.e. o(x) = p2/4. As is clear from fig. 1, they should be com- plex. It may be shown that for a rather broad class of symmetric profiles, the singularities of that kind, closest to the real axis, are x = +i[, where E; > 0. Let

us suppose, for simplicity, that other singularities are unimportant for our problem (this appears to be true in typical concrete examples which we have consid- ered).

Passing from eq. (6) to eq. (5) along the contour L, (fig. 2) we see that the first term of eq. (6) turns into the term CI F[ of eq. (5) which gives

Cl =AIexp[$i*-i: T,Idx].

0

(9)

During this transition, the second term of eq. (6) proves to be negligible and, therefore, drops out (cf. with similar situations in ref. [9], $52). Now we pass along the contour L2 (fig. 2) turning eq. (6) into eq. (7). Then the first term of eq. (6) transforms into the last one of eq. (7) and we obtain

(10)

Volume 84A, number 1 PHYSICS LETTERS 6 July 1981

Fig. 2.

In a similar way, moving from x = - along the line L3 and then along L4, we have

c2 =Bl exp[iin - i:y qldx], -X0

(11)

A2=B1exp[?in-irPqldx

-X0

+iy j’(r72 -ql)dx .

0 1

Substituting into eqs. (9) -( 12) Q = & we have

(12)

(13)

IA212 = lC212T, IB212 = lCI12T, (14)

where

1 is

T=exp -2:ImJ (42-41)dx 1 0

(15)

is the coefficient of the transformation q1 + q2. This process is very similar to quantum tunneling, because a part of the ray trajectory is complex. Therefore, we use the term “tunnel transformation” for such type of phenomena. The general theory of tunnel transfor- mation for the propagation of a whistler in a stratified plasma is described in ref. [lo], where expressions like (15) have been obtained for some types of reflec- tion and refraction * ’ . In particular, the general argu-

*l In the parabolic approximation the branch (7.~ is lost. This is the reason why our results do not follow from the S&r&linger equation.

ments given in ref. [lo] show that T Q 1 in the cases where the WKB approximation is applicable. There fore, in the zeroth approximation we may neglect the terms with A2 and B2 in eqs. (6) and (7). Then, the standard reasoning, which is used in the absence of leakage, gives the relation I C112 = I C212 and, also, the trapping condition

x0 0 - c s

qydx=n(n+;), n=0,1,2 ,.... (16) -X0

To calculate the attenuation rate ~1, let us write the conditions of the conservation of energy fhrx at the points *x0, taking into account the leakage in the first approximation:

IC~12exp[-2~ Jo 4; dx] 0

- lC212exp kFf” 4; ti] = IA212,

0

I C212 exp [ 2: Jxo 4; dx

0 I (17)

- IC,12exp [ -2: Jxo *iti] = I&l2

0 (as far as A2 and B2 are small quantities, we omit in the rhs of (17) the exponential factors containing &). Then from eqs. (17), (8) and (14) we have, to first order in q;,

(18)

Formulas (14)-( 16) and (18) present the complete solution of the problem for the plane duct.

As a typical concrete example, consider the profile

o(x) = b1 - b2 tanh2(x/a), (19)

where bl > b2 > 0. Then

[/a = arctan [(p2 - 4b1)/4b2] li2. (20)

Computing the integral in (1 S), we come to the ex- pression

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Volume 84A, number 1 PHYSICS LETTERS 6 July 1981

T = eXp [-(n&+&k)(S2 - Sl)],

Sk = Ip2(1 - 4U2) + (p2 - 4b,) (21)

+ 2(-l)‘p(p2 - 4!1$‘~]~‘~, k = 1 2 3 *

It follows from (21) that T is a decreasing function of

p. Then Tmin = T(Pmax), where pLax = ot,,/~(l - U) [see eq. (4)]. This gives

Tmin = exp inu7,ax[~;l_,:,5’2) . (22)

One sees that T + 1 if u + l/2. For u > l/2, whistler

trapping in density crests is absolutely impossible. Attenuation of a whistler wave in the density crest

for u < l/2 is seen from fig. 7 of ref. [7]. The effect of wave radiation from a density crest at the initial

stage of the whistler self-focusing has been observed in ref. [8].

From the above results, especially from eq. (22) it follows that detrapping is strong if the beam radius

is of the order of the wavelength. Therefore, narrow beams are possible only when they propagate in den-

sity troughs which may trap whistler waves both for u < l/2 and u > l/2.

A more detailed account of the considered topics will be given elsewhere.

References

[ 1 ] R.A. HelliweIl, Whistlers and related ionospheric phe- nomena (Stanford Univ., 1965).

[2] H. Washimi, J. Phys. Sot. Japan 41 (1976) 2098.

[3] R.L. Stenzel, Phys. Fluids 19 (1976) 857. [4] H. Washiml, J. Phys. Sot. Japan 34 (1973) 1373. [5] V.I. Karpman and H. Washimi, J. Plasma Phys. 18

(1977) 173. [6] K.H. Spatschek, P.K. Shukla, M.Y. Yu and V.I.

Karpman, Phys. Fluids 22 (1979) 576. [7] H. Sugai, H. Niki, S. Takeda and M. Inutake, Phys.

Fluids 23 (1980) 2134. [8] A.A. Ralmashnov, Phys. Lett. 79A (1980) 402. [9] L.D. Landau and E.M. Lifshitz, Quantum mechanics

(Pergamon, New York, 1977). [lo] V.I. Karpman and R.N. Kaufman, Tunnel transforma-

tion of whistler waves in an inhomogeneous plasma, pre- print IZMIRAN (1980); Zh. Eksp. Teor. Fiz. 80 (1981), to be published.

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