Wave propagation through an epstein density profile across an inhomogeneous magnetic field

Physica 32 74 l-748

Sluijter, F. W. Weenink, M. P. H. 1966

WAVE PROPAGATIO’N THROUGH AN EPSTEIN DENSITY PROFILE ACROSS AN INHOMOGENEOUS MAGNETIC FIELD by F. W. SLUIJTER

and M. P. H. WEENINK

FOM-Instituutvoor Plasmafysica, Rijnhuizen, Jutphaas, Nederland Synopsis The propagation of a plane monochromatic electromagnetic wave normally incident on a stratified plasma is considered. The stratification is of the transitional Epstein type, i.e. the density varies in space as a hyperbolic tangent. The static or slowly varying magnetic field is perpendicular to the density gradient. Only the extraordinary wave mode is considered here. The spatial dependence of the magnetic field is of the same type as that of the density, but the magnetic field is maximum where the density is zero and vice versa. The motion of the electrons is described by the ideal fluid equations. The ions are assumed to be infinitely heavy. The problem then leads to Heun’s equation. In a special case it reduces to the hypergeometric equation. For this case the reflection and transmission coefficients are given explicitly.

1. I&oduction. Electromagnetic wave propagation in stratified inhomogeneous media can be adequately treated in many cases by the WKBmethod. If, however, the gradient of the refractive index becomes steeper, the approximation

becomes

worse.

For

those

cases

more

precise,

if not

exact, solutions of the wave equation have to be considered, known as full wave solutions (cf., e.g., ref. 1, p. 353ff. and 458ff.). Full wave solutions are known for a number of density profiles. A physically realistic class of such solutions is known as Epstein profiles. An Epstein profile gives a smooth transition from low to high density. A special case of an Epstein profile is a hyperbolic tangent. The appreciable density change takes place within a limited region 2) 3) 4). One of us has generalized this treatment in the case of normal incidence, so as to include the effect of a uniform static magnetic field perpendicular to the direction of propagation, that is, parallel to the stratifications). This treatment leads for the extraordinary wave to the solution of Heun’s equation6) 7) instead of the hypergeometric equation, as did the original Epstein treatment. We will treat now a similar problem in which, however, both plasma density and static magnetic field depend in an Epstein way -

741

-

742

F. W.

SLUIJTER

AND

M. P. H. WEENINK

on one spatial variable such that the magnetic field is minimum where the density is maximum and vice versa. We will give transmission and reflection coefficients

in a special case. The model that is used is essentially

model. The ions, however, will be supposed to be infinitely only assure quasi-neutrality.

It is shown that

a two-fluid

heavy. They will

the effect on the wave pro-

pagation of finite compressibility can, in general, of collisions has also been neglected.

be neglected.

The effect

2. The eqtiations. For the electron plasma which we consider, the following hydrodynamic equations apply : Dv wz __ Dt

= --ne(E

+ v x B) -

Ofi,

(2.1)

$

+ v* (nv) = 0,

(2.2)

&

Wffw)

(2.3)

which have to be solved together

= 0,

with Maxwell’s

equations:

VxE=+

(2.4) (2.5)

where E is the electric field, B the magnetic field, EOand ,UOthe dielectric and magnetic permeability of vacuum respectively, -e the charge of an electron and m its mass, v the electron fluid velocity, n the electron particle density, fi the electron pressure and y the ratio of specific heats at constant pressure and constant volume ; the equations are written in Giorgi-units. As the static magnetic field is supposed variables we find from equation (2.5) :

to depend on one of the spatial

V x Bo=po_h,

(2.6)

j. = -noevo.

(2.7)

where The subscript 0 refers to the profile without high frequency perturbations, Bo thus being the static or slowly varying magnetic field. Choosing a Cartesian coordinate system and confining ourselves to the case of all dependent variables having a z dependency only, and Bo along the x axis, it follows that jo, and thus 00, has only a y component: j. =

t&q1

T = -noevo,

&, denoting the unit vector in the y direction.

(2.8)

WAVE

Keeping nlpn

au1

__at

PROPAGATION

THROUGH

AN EPSTEIN

this in mind we linearize + nlynvl.

= -en&

equations

DENSITY

(2.1)

PROFILE

to (2.5)

743

to obtain:

VvrJ = +

Vl

x Bo + vo x Bl) -

en1vo

x Bo - vp1,

an1

(2.9) (2.10) (2.11) (2.12) (2.13)

The subscript 1 refers to wave-type quantities. Assuming the time dependency of the form exp (- iot) where o is the angular frequency of an incident monochromatic, linearly polarized, plane wave; assuming further El to have no x component (i.e. considering the extraordinary wave mode), and, finally, elimination of Bi, ni, vi and 00 with the help of equation (2.8) yields:

d2E, --+ dz2

co; - ~2 c2

E,=--__

cl.0

dE,

co; dz

dz

iw

++($F)-;nz-)Ez,

(2.15) where E, and EC are the y and z components speed of light in vacuum

of El, respectively,

c is the

and 2 -

WP

$j =

noe2, cow -

2. m

Bo,

(2.16) (2.17) (2.18)

If v”, < cs, i.e. the sound speed for the electron gas small compared to the speed of light, and if v,” has no steep gradients, we can neglect the term containing v”,, A judgement concerning the possible neglect of other terms can better be postponed until the introduction of an explicit z dependence.

744 3. Reduction

F. W.

SLUIJTER

to Hem’s

to a special Epstein

AND

equation.

M. P. H. WEENINK

Let the density

vary with z according

profile such that 2

Lu2 _ _p P

%o

2

1 + tanh &

>

= u$, YIP, V-1

(3.1)

in which 7

=

-exp

4.

If we suppose that Bo varies with z according

(3.2) to (3.3)

we find for the z dependence

of Q (3.4)

Clearly, 0~0 is the plasma frequency for z -+ 00 and QO the cyclotron frequency for 2 + --00. Comparing the two terms of the coefficient of E, in equation (2.14) we find that the former can be neglected with respect to the latter, provided that

Comparing the coefficients of dE,ldz and E, in the same equation we find that the term with dE,/dz can be neglected with respect to the term with E, provided

if il is the local wavelength. Comparing _ - the coefficients we find a similar condition

of E, and dE,/d, on the left of equation for the neglect of the dE,ldz

-?
(2.15)

term:

(4

If cc)& < 02 condition (b) is the severest of the last two ; if depends on the spatial dependency of 1. In the optical range condition (b) is always severe and (c) may be and they are both certainly violated near a resonance L -+ 0. They have to be made, however, to keep the problem in tractable form. Elimination of E, from the remainder of equations (2.14) and (2.15)

WAVE

PROPAGATION

THROUGH

AN EPSTEIN

DENSITY

PROFILE

745

leaves us with

GE,

2

(09 - a$)2 - f&x?2 E

k. CdyflP -

F+

co?“, - J-22)

= o

(3.5)

’

y

where ko = w/c. Substitution dependent

of equations variable

(3. l), (3.2) and (3.4),

E, finally

= (-r)ikoz u = u exp ikoz,

yields Heun’s equations)

of the

(34

7) : 6

1+a+p-r-s

$+(++

and transformation

by means of

d”+ > dv

+p v-a

11-l

+

+u + b ~(7 _ l)(y _ a) u = O

(3’7)

with TV= iZ(k,, + k,);

y =

/3 = iZ(ko -

6 = 0;

a= where k,

02 m2_

is the wavenumber

k,);

l-2; 2

OPO

1 + ailk,,; (3.8) 2

; b = k;l2

OPO a2_

. 2

WPO

’

for z + co, given by

(3.9)

I.

This equation can then be solved in the usual way, and one thus finds reflection and transmission coefficients in terms of special solutions of Heun’s equation (cf. ref. 5). If conditions

(a) and (6) hold, but condition

(c) is violated,

the second

term on the left of equation (2.15) cannot be neglected. However, including this term does not change much. One is then left with Heun’s equation where 6 # 0 and b is slightly changed, the other parameters remaining unchanged. 4. A qbecial case. Instead of solving Heun’s equation we will solve the special case: 0~:~ = L$. This means that the profile is such that it can be characterized in the CMA-diagram by a straight line parallel to the resonance line (see fig. 1). Clearly, the profile does not contain a resonance. If L?~/os < 1 the profile has either two or no cutoffs. The two cutoffs might coincide. If L?;/os > 1 th e profile has always one cutoff (cf. ref. 9, p. 13/f.). Now a = 1 and so we have a confluence of the singular points a and 1 of equation

746

F. W.

SLUIJTER

AND

R’ 3

I

M. P. H. WEENINK

1-7

1.5-

0

1

0.5

1.5

Fig. 1. CMA-diagram for the extraordinary (w2 -

k2 = kg

042

co‘qw2 -

-

!iJ; -

028~ Q2)

mode;

, k being the local wavenumber.

(3.7), and Heun’s equation reduces to the hypergeometric p. 91). If we apply, instead of transformation (3.6), E, to equation

= (_@kor

(1 _ r)(r+a’+k’)/2

(3.5) we find the standard

one (cf. ref. 7,

‘w,

(4.1)

form of the hypergeometric

1 + tc’ + ,!?’ -

y’ >

v-1

dw -+---dv

equation

d/?‘W = 17b?-

0,

(44

1)

where

I

CL’= g 1 + 2iZ(kfJ + k,)

+

1-

4 WPO

4@2 co2(w2

I

-

‘1 oEo)

>

I’ (4.3)

,!I’

=

fr (1

+

y’ = y =

2iZ(ko -

k,)

+ (1 -

4k;Z” ,2,w;2

w2 ) )*}, PO

1 + 2iZko.

With the help of the expression transformation (4.1) as to give

for y from equations

,?& = (+I’--1)/2(1

(3.8) one can rewrite

_ #r+a’+@‘-Y)/~ ‘w.

(4.4)

We confine ourselves to a wave incident from the left. The solution relative to the singular point -00, which reduces for Q --f -co (,z -+ CO) to a plane harmonic wave propagating to the right, is found to be E, = (-#v-r)/*

(1 -r)

(r+a’-B’-I’)/2 F

@‘, y _ B’; 1 _ cL+ 8;

1 1-V

>

;

(4.5)

WAVE PROPAGATION THROUGH AN EPSTEIN DENSITY

the other solution

reducing

because of Sommerfeld’s Analytic continuation

to a wave propagating

radiation condition. to q + O(z -+ -co)

747

PROFILE

to the left is precluded

gives

us (cf. ref.

8, p. 107

form. (39)) (_9)+1)12(l

_ q)‘1++P’-Y)/2

.F

+

p’,

F

(a’,j+l’;y;~

(_#p-1J/2(1

_

y _

p’;

1 _

>

r-1

r)(-1-a’+B’+y)/2.

a’

1i

,

r(l

-

)=

+

r(Y)

r(l w

1 +a’--,1

b’;

+

-

-

y)

Y)

F(Y

-

a’

a’)

+

@‘)

.

W)

7

-/?‘;2-_;p

11-l As the left-hand side can easily be identified with the transmitted, wave, the first member on the left with the incident wave and the second one with the reflected wave, we can write down at once the transmission coefficient T and the reflection coefficient R, viz. T

=

‘(’ - a’) ‘(’ + p’ - 7) r(l

R =

-

‘(d

y) r(l r(l

W Finally,

we investigate

-

+ b’ a’) r(l

Y) r(y

the behaviour

-

+ B’ 4

(4.7)

a’) ’ y)

(4.8)

*

W’)

of T and R for i + 0, i.e. a sharp

vacuum-plasma boundary. With the help of expressions recurrence relation for Y functions we find: r(l lim T = lim z+o z-to Ql

-

a’) = lim

1-

y

-

y)

z-0 1 -

a’

lim R = 1:~ g:i z+o

1 ::{

= lim ‘5 z-+0 1 -

a’

=---

(4.3)

and

the

2Ro ho + h ’

= k-_ k + ko

(4.9)

(4.19)

which are exactly the Fresnel formulae for the case under consideration. It should be noted that these limit relations are found although conditions (u), (b) and (c) are violated for I + 0. Acknowledgement. The authors are indebted to Professor H. Bremmer for his criticism of the manuscript. This work was performed as part of the research program of the association agreement of Euratom and the

748

WAVE

“Stichting

PROPAGATION

voor Fundamenteel

cial support

AN

EPSTEIN

Onderzoek

from the “Nederlandse

pelijk Onderzoek” Received

THROUGH

DENSITY

der Materie”

Organisatie

PROFILE

(FOM) with finan-

voor Zuiver Wetenschap-

(ZWO) and Euratom.

12-l O-65

REFERENCES

1) 2)

Budden,

K. G., Radio Waves in the Ionosphere,

Epstein,

P.S.,

3)

Eckart,

C., Phys.

4)

Rawer,

K., Ann.

5)

Sluij ter, F. W., Proc. 7th Int. Conf. Phen. Ionized

6)

Heun,

Proc. Nat. Acad.

Sci. U.S.A.

Rev. 35 (1930) Physik

ErdBlyi, Stix,

A., &a., Higher

T. H., The Theory

Press, Cambridge

(1961).

1303.

functions,

Transcendental of Plasma

Gases Beograd

1965, to be published.

161.

and Legendre 7) Snow, C., Hypergeometric Printing Office, Washington (1952).

8)

University

35 (1939) 385.

K., Math. Ann. 33 (1889)

9)

Cambridge

16 (1930) 627.

Waves,

N.B.S.

Functions,

Appl. Math. Ser. 19, U.S. Government

Vol.

McGraw-Hill,

I, McGraw-Hill, New York

(1962).

New York

(1953).