J o u r n a l of Atmospheric and Terrestrial Physics, 1968, Vol. 30, pp. 1219-1229. Pergamon Press. Printed in Northern Ireland
Whistler mode propagation in the ionosphere in the
presence oi a longitudinal static electric field*
H. C. HsI~.~
Electron Physics Laboratory, Department of Electrical Engineering,
The University of Michigan, Ann Arbor, Michigan, U.S.A.
(Reoeived 15 October 1967)
Abstract--The effect of a longitudinal electrostatic field on whistler mode propagation has been
studied in a region of the ionosphere where the influence of particle collisions and ion motion
may be significant. As an illustration, the propagation in two cases of interest has been considered; one deals with the E-region in the neighborhood of 100 km height, and the other with
the F-region in the neighborhood of 250 kin.
I t is shown that the presence of a weak static electric field E e tends to reduce the attenuation
of the forward whistler mode when the magnetostatic field B e and the electrostatic field E o a r e
in the same direction. Moreover for a sufficiently large value of E o the whistler mode may
experience an amplification. On the other hand, when E o and B 0 are in opposite directions an
increase in IE01 tends to increase the attenuation of the wave. The variation of the attenuation
constant, phase constant, and the phase velocity of the whistler mode with the signal frequency
and with the electrostatic field is also discussed in detail.
1. INTRODUCTION
THE WkLtSTLER m o d e p r o p a g a t i o n in a w a r m collisionless e l e c t r o - m a g n e t o - i o n i c
m e d i u m h a s b e e n s t u d i e d r e c e n t l y b y HSIEH (1967) using t h e coupled M a x w e l l V l a s o v e q u a t i o n s . T h e m e t h o d of analysis g i v e n b y HSIEH (1967) is suitable for t h e
a n a l y s i s of whistler m o d e p r o p a g a t i o n in t h e m a g n e t o s p h e r e b u t it is n o t a d e q u a t e
for c o n s i d e r a t i o n of t h e p r o p a g a t i o n in t h e lower i o n o s p h e r e w h e r e t h e p a r t i c l e
collision effects are k n o w n t o p l a y a n i m p o r t a n t role. H o w e v e r this m e t h o d of
a n a l y s i s can b e easily e x t e n d e d so t h a t t h e effect of collisions can be t a k e n i n t o
a c c o u n t . This is done b y using t h e B o l t z m a n n e q u a t i o n w i t h a p r o p e r l y a s s u m e d
collision t e r m i n s t e a d of t h e V l a s o v e q u a t i o n .
T h e p u r p o s e of t h e p r e s e n t p a p e r is to discuss t h e effect of a l o n g i t u d i n a l electros t a t i c field on whistler m o d e p r o p a g a t i o n in a region of t h e i o n o s p h e r e w h e r e t h e
effects o f collisons a n d ion m o t i o n m a y be of significance.
I n t h e p r e s e n t p a p e r t h e following a s s u m p t i o n s are m a d e :
1. All q u a n t i t i e s of i n t e r e s t are to be c o m p o s e d of t w o p a r t s , a t i m e - i n d e p e n d e n t
part and a time-varying part.
2. S m a l l a m p l i t u d e conditions are satisfied.
3. A o n e - d i m e n s i o n a l analysis is applicable.
4. All t i m e - d e p e n d e n t q u a n t i t i e s h a v e h a r m o n i c d e p e n d e n c e of t h e f o r m
e x p [j(oJt - - kz)], w h e r e ~ a n d k are t h e a n g u l a r f r e q u e n c y a n d t h e p r o p a g a t i o n
c o n s t a n t r e s p e c t i v e l y ; t a n d z are t h e t i m e a n d s p a t i a l v a r i a b l e s respectively.
* Sponsored by the National Aeronauticsand Space AdmirListration under Grant No. NsG 696.
1219
1220
H.C. HsmE
5. The effective collision frequency of electrons v~ and that of positive ions v~ are
independent of the particle velocity.
6. Electrical neutrality is satisfied.
7. The equilibrium distribution function of electrons and ions has the form of a
iKaxwellian.
8. The electrostatic field E0, which is directed parallel to the magnetostatic field
B0, is sufficiently weak so that the drift velocities of charged particles are much
smaller than the phase velocity of the whistler mode.
2. DISPERSION
EQUATION
Using the procedure of HsIE~ (1967) and b y combining the time-varying parts of
the coupled Maxwell-Boltzmann equations the dispersion equation of the right-hand
circularly polarized (whistler) mode for the system under consideration can be
obtained and expressed as follows:
C2k2
1
X1 (
V19'k12
aft -- W: \1 + . 2~w
1)
V~:
- - ~ + ~Vz 1 +
20~
1)
Wz ~ ,
where
Wl = [ ( 1 - Y : ) - j Z : ] ,
W, = [(1 - Y , ) - j Z ~ ] ,
V1
( 2 ~ e l I/2
k: = (k + j K I ) ,
k~. = (k + j K , ) ,
(eE0
~1 = ~ / ,
( e ~ ) 1/2
v2 =
\-~-/
,
x~ = \k-~,/'
and
~
in which c = the
80 = the
e ---- the
~n = the
M = the
Te = the
T~ = the
= the
=
\---M-/'
~'
=
\Msol
'
speed of light in free space,
dielectric constant of vacuum,
electron charge,
mass of electron,
mass of positive ion,
electron temperature,
ion temperature,
Boltzmann constant,
PB = the effective collision frequency of the electron and
= the effective collision frequency of the positive ion.
(1)
Whistler mode propagabion in the ionosphere
1221
I t should be noted t h a t equation (1) is a quadratic in k and can conveniently be
written as follows:
+ ~
[
where
-
(1
1 +
(2a)
1)]
,
1)
t~ = j 2 ~ X ( ~ l
3
~=
~ = o,
~23
'
I1--X(~--j.-f~-~)l'
(2b)
in which
W1 ---- W1,
~" ~ \ m c ~ /
~
~
W2,
~ ~ \-~ccw]'
X ----X1,
~~
Y ~
Yl,
0 ~- \ T J
,
(2e)
The symbol N appearing in equation (2) is introduced to emphasize the fact t h a t the
quantity under consideration is a complex quantity. I n the present discussion the
wave angular frequency m is regarded as a real quantity and the propagation constant
is regarded as a complex quantity, which can be written as
= (8 - J~),
(3)
where ~ and fl are the amplitude and phase constants respectively. I n view of the
fact t h a t the time and spatial dependence is assumed to be of the form
e x p [j(mt -
kz)] = e x p [ - ~ z
+j(~t
-
~z)],
a positive fl represents the forward wave, while a negative fl denotes the backward
wave. I t is apparent t h a t positive ~ represents attenuation and negative ~ represents
amplification of the wave. The two roots of equation (2a) give the propagation
constants of the forward and backward waves. Once the system parameters are
specified, equation (2a) can be solved for ( k / # 0 ) = [(B/ri0)- J(~/fl0)]- Thus the
investigation of the behavior of roots of equation (2a) provides information with
regard to the variation of the amplitude and phase of the whistler mode with the
various system parameters.
3. W H I S T L E R
i~ODE PROPAGATION IN THE Eo A N D F-REGIONS
IONOSPHERE
OF THE
The E- and F-regions of the ionosphere are regarded as consisting of a partially
ionized gas in which electrons, positive ions and neutral particles m a y interact with
each other. To study the propagation of a whistler mode in such regions a knowledge
of the plasma parameter is required. For example, in order to calculate the net
1222
H.
(3. H s r ~ . r r
growth or decay in amplitude of a whistler wave which propagates through a slowly
varying medium, knowledge of variation of such quantities as n, N, rio, E0, Te, T~,
re and ~ along the whistler path is required.
A theoretical analysis made b y NICOLV.T (1953) shows that the electron collision
frequency in the ionosphere depends on the neutral particle concentration in D- and
E-regions and on the electron concentration in F-region. The collision frequency
also depends on the temperature. For example, above the F2-1ayer, re~, the collision
frequency of electrons with ions, can be approximately given b y [NIcOL~T, 1953]
~e~ = [34 ~- 8"36 1ogx0 (Ta/~/nl/~)]nT -a/2,
(4a)
where n is the electron density and T is the temperature of the plasma.
In the case of a plasma consisting of electrons and protons, the effective collision
frequency of protons with electrons should be (see e.g., FERRARO and Pr.vm, TO~,
1961)
~2, = (m/m,,)l/2~e~,
(4b)
where m~ is the mass of the proton.
Some experimental data with regard to the profiles of number density and temperature of electron and positive ions in the ionosphere can be found in VAT.T.EY
(1965).
There is little information available in the literature as to the strength and spatial
variation of the electrostatic field E0 in the ionosphere. However, MozE~ and
BRUSTO~ (1967) recently reported sounding rockets measurements of E o ---- 20 mV/m
in the auroral ionosphere.
As an illustration of the method of analysis, the propagation characteristics of a
whistler mode with frequency in the range between 1 and 20 kc/sec, and with
]E01 < 10 mV/m, are examined for two special cases of interest; Case I deals with
the E-region in the neighborhood of height h ---- 100 km, and Case I I deals with the
F-region in the neighborhood ofh ---- 250 km. Thoparameters chosen for this example
are given below.
Cases
Region
I
h = 100 k m
II
h=250km
OJ~
rad./see
2.3 X 107
4×
107
OJs
rad.]see
10 v
8"8×
To°K
270 °
l0 s
T~
~
rad.]see
~
rad.]see
~ ~ ~_ •
M
270 °
3"14 × 105
6"28 X l 0 s
2600 ° 1300 °
3"14 × 104
1'83 × l 0 S 0.34 × 10 -4
4"0 X 10 -4
The values of parameters w~, wz, Te, and T~ are taken from the data given b y
VALLEY (1965). The value of ~e for Case I is taken from SB~AROFSXY (1961), and the
value of ~e for Case I I is derived with the aid of equation (4a). Moreover for both
cases it is assumed that (v~/~e) ---- (re~M) 112. For Case I I the positive ion under
consideration is taken to be an oxygen ion.
Having specified the physical parameters of the region under consideration, for
example, the variation of the amplitude and phase of a whistler mode with the wave
angular frequency for different values of E 0 can be examined. The plots of ~ vs. ~o,
fl vs. w and (v~h/c) vs. ~, with v~h denoting the phase velocity of the whistler mode are
Whistler mode propagation in the ionosphere
1223
14
12
lO-
ft
- 8 mV/meter
4
2--
Eo=03
0
-2
0
4
8
12
16
20xlO 4
~,, rad/sec
Fig. la. ~ vs. oJ with E 0 as parameter for Ease I.
shown in Figs. la, lb, and lc respectively for Case I, and the corresponding plots are
shown in Figs. 2a, 2b, and 2c respectively for Case II.
4. DISCUSSIOI~ O F RESULTS
I t is well known t h a t particle collisions lead to the attenuation of the whistler
mode in the ionosphere. I t is noted from Figs. l a and 2a t h a t when E0 = 0 the
attenuation constant a is essentially invariant with respect to the signal frequency.
However, with the presence of an electrostatic field ~ does vary with the signal
1224
H.C.
Hsr~.E
16
16
: - 8 mV/meter
H-E°:-4
12
I0
E
4
8
i2
16
20xlO 4
~, rad/sec
Fig. l b . ~ vs. m w i t h E o as p a r a m e t e r for Case 1.
f r e q u e n c y . F o r E o > 0, t h e presence of E o leads t o a r e d u c t i o n of t h e a t t e n u a t i o n
a n d i r e o is sufficiently large t h e n ~ m a y b e c o m e n e g a t i v e ; t h u s t h e w a v e is amplified.
I t should be n o t e d t h a t w h e n E o > 0, for t h e p a r a m e t e r s chosen, ~ is n e g a t i v e e x c e p t
in t h e r a n g e o~ > 15 × 10 4 rad/sec w i t h E 0 ---- 4 m V / m in Case I. This suggests t h a t
w h e n Eo a n d B 0 are b o t h in t h e direction of t h e w a v e v e c t o r , t h e whistler m o d e
experiences a n amplification. M o r e o v e r w h e n E o ~ 0, ~ is p o s i t i v e so t h a t t h e w a v e
experiences a n a t t e n u a t i o n w h e n E o a n d Bo are o p p o s i t e l y directed. I~] increases
w i t h JEeR in t h e r a n g e co ~ 4 × 10 4 r a d / s e c for Case I a n d in t h e r a n g e co ~ 3 × 10 a
Whistler mode propagation in the ionosphere
1225
7 x l O -2
Eo: 8 mY/meter
Eo= 4 --
4
Eo=O~
Eo:-8
E
=-
-
2
4
6
i
8
IOxlO 4
~, rad/sec
Fig. lc.
(v~a/c) v s .
co w i t h E o a s
parameter for Case
I.
rad/sec for Case II. The maximum of l a[ for both Cases I and II appears to lie
somewhere in the low frequency range of the spectrum; w ~ 5 × 10 4 rad/sec for
]E0] ~ 8 mV/m. The comparison of Fig. la with Fig. 2a shows that the extent of the
effect of E 0 on ~ is greater in Case II than in Case I. However it should be noted that
in Case I it is assumed that T, = T~, whereas in Case II T, = 2T i.
On the other hand, from Figs. lb and 2b, it is observed that when E o = 0 the
phase constant fl increases monotonically wi.'th the signal frequency. When E 0 > 0,
the group velocity vg----d~o/dfl is positive and varies with the signal frequency.
However, when E o < 0, do~/dfl < 0 in the low frequency range and do~/dfl > 0 in the
11
1226
H.C.
HsIE~
12
I0--
8--
6--
- E o = - 2 mV/meter
4--
2
0
-2
--
-4
--
0
I
I
I
I
2
4-
6
8
lOxlO 4
% rod/sec
F i g . 2 a . ~ v s . co w i t h E o a s p a r a m e t e r
for Case IL
high frequency range. Furthermore it is observed t h a t in Case I, when E o > 0, the
wavelength of the whistler mode increases with ]Eel and when E o < 0, it decreases
as ]Eel increases. For Case I I the wavelength increases with IEol regardless of the
algebraic signs of E o. Finally it is observed t h a t in Case I (see Fig. lc) the phase
velocity v~h of the whistler mode increases with E o when E o > 0 and decreases with
IEol when E o < 0. On the other hand, in Case I I (see Fig. 2c), v~h increases with ]Eel
regardless of the algebraic sign of E o.
W h i s t l e r m o d e p r o p a g a t i o n in t h e i o n o s p h e r o
1227
I0
Eo:O
8
=2
2_
4
6
8
IOxlO 4
~, rad/sec
Fig. 2b. ~ vs. co w i t h E o as p a r a m e t e r for Case I I .
I t is also of interest to note that Figs. 1 and 2 both suggest that the extent of
influence of E 0 on the change in the amplitude, wavelength and phase velocity of the
whistler mode under consideration in both Cases I and II is much greater in the low
frequency range of the spectrum, e.g., o) < 6 × 104 rad/sec, than in the high frequency range, e.g., o) > 6 x 104 rad/sec.
5. COHCLUDrSe Rv,~a~xs
In the present paper it is assumed that the collision frequencies of the electron and
positive ions are independent of the particle velocity. The classical AppletonHartree equation used for propagation in the ionosphere has this inherent assumption.
Because the electron elastic collision frequency with nitrogen molecules in the
atmosphere varies as the square of the electron speed (PHELPS and PAc~, 1959) a
large discrepancy between classical theory and ionospheric experiments can be
expected.
However, Sn~A~O~SEY (1961) has derived a generalized Appleton-Hartree
equation applicable to any variation of electron collision frequency with electron
1228
H.G.
HSL~K
22x10-2
I-
Eo=2 mV/meter
/
!
2.0
/
1.8
~:a. 1.6
/
1.4
Eo:-1~
/
/
1.2
/
/
1.0
/
/
/
/
/
I
2
/
/
/ •/-
Eo=0
I
I
4
6
~,
Fig. 20.
(V~h/C)VS.
I
8
IOxlO4
rad/sec
co w i t h E e as p a r a m e t e r for Case I I .
speed and any degree of ionization. His analysis suggests that the classical AppletonHartree equation should be applicable with no corrrection necessary for the region of
the ionosphere and the range of parameters under consideration in the present paper.
Thus the velocity-independent collision frequency model considered here is reasonable.
Most of the uncertainty in the calculation arises from a lack of knowledge of the
electrostatic field strength and the collision frequency of positive ions in the region
investigated. Consequently the numerical illustration shown in Figs. 1 and 2 only
provide information with regard to the order of magnitude and a general behavior of
the propagation constant. Although only the forward whistler mode has been considered in these figures, it is not difficult to analyze the propagation characteristic of
the backward whistler mode, which is also characterized by equation (2a).
The frequency range used for the illustration lies within the range over which a
class of naturally occurring radio noise in the ionosphere and magnetosphere such as a
whistler and VLF emission is observed ( H v , ~ v ~ . ~ , 1965). The result presented in
~Vhistler mode propagation in the ionosphere
1229
tiffs paper should be useful for the study of the propagation of a whistler or V L F
emission in the region of ionosphere where the presence of electrostatic ][ield m a y be of
significance.
REFERENCES
FElCR~O V. C. and PLU~a~TON C.
1961
I-IELL~w~:LL l~. A.
1965
HSmH H . C .
~¢IOZERF. S. and BRUSTO:," P.
NICOLET M . J .
Sm~OFSKY I.D.
VALLEY S . L .
1967
1967
1953
1961
1965
A n Introduction to iVIagneto-Fluid .Medeanice. Clarendon Press, Oxford, England.
Whistlers and Related Ionospheric Phenomena. Stanford University Press,
Stanford, Calif.
J. Atmosph. Terr. Phys. (in press).
J. Geophys. l~es. 72, 1109.
J. Atmosph. Terr. Phys., S, "200.
Prec. I.R.E. 49, 1857.
Handbook el Geophysics and Space Environments, McGraw-]~ill, New York,
Chap. 12.