RADIO SCIENCE Journal of Research NBS/USNC- URSI Vol. 69D, No.6, June 1965

Phase Velocities and Attenuation Distances in the Ionosphere

D. R. Croley, Jr.

McDonnell Aircraft Corporation, St. Louis, Mo.

and

B. S. Tanenbaum

Case Institute of Technology, Cleveland, Ohio

(Received January 8, 1965)

Plane wave phase velocities and atte nu ation distances in an infinite , homogeneous, partly ionized gas composed of an interacting mixture of neutral molecul es, ions, and elec trons in the presence of a uniform magne ti c fi e ld a re determined for three sets of plas ma conditions correspo nding to the 80, 100, a nd 300 km le vels of the ionosphere. Propagation in th e parallel direc tion and the perpe ndicular direction relative to the magne ti c fi e ld in the freque ncy ra nge from approximately 10 - 5 Hz to 109 Hz is considered.

1. Introduction Tanenbaum and Mintzer [1962, 1963] have studied

the propagation of plane waves in an infinite, homo­geneous, partly ionized gas composed of an interac ting mixture of neutral molecules, ions, and electrons in the presence of a uniform magnetic field . This letter reports the application of the three fluid theory to gas conditions at the 80, 100, and 300 km levels of the ionosphere. Phase velocity and attenuation distances have been calculated on an EDP computer using the dispersion relations obtained by Tanenbaum and Mintzer. Calculations are made for two distinct cases: (1) transverse wave propagation parallel to the uniform magnetic fi eld (figs . 1 and 2); and (2) coupled longitudinal and transverse wave propagation perpen­dicular to the uniform magnetic fi eld (figs. 3 through 8). The characteristics of the curves differ significantly at each altitude in both cases. The phase velocity curves obtained using the 300 km data resemble figures 2 and 4 of Tanenbaum and Mintzer [1962], where colli­sional damping effects are ignored, while the curves

Considering firs t transverse waves propagating parallel to the magnetic field, we show in figures 1 and 2 the phase velocity, w/Rek, and attenuation dis­tance, (Imk)- ' , as calculated directly from the disper­sion relation [Tane nbaum and Mintzer, 1962],

i using the 80 and 100 km data are decidedly different (due to the increasing importance of the collision terms in the dispersion relations as the altitude dec reases from 300 to 80 km).

(1)

where

P = (Vin + Vni - iW)/(Vlli - iw). (2)

The (+) refers to left circular polarization (LCP), and the (-) refers to right circular polarization (RCP). Values of temperature, ion and electron density, neu­tral density, and ion mass, as well as the important frequencies and velocities for the 80, 100, and 300 km levels are summarized in table 1 [Chapman, 1956; Cowling, 1945; Minzner, Champion, and Pond, 1959]. The range of frequencies shown may be academic, but serves to display the important features contained in the dispersion relations.

819 764-883 0-6~- 3

---- ---- ---

o ., ~ ~ ., Q;

.5 >­~ (3 g w > w If) <l I Il.

Id' -

Ie!-- -

-

Id' -

I

Iq--

lu

lu

u

O' I ICY"

t---

-

~OO km

100 km ---80 km r-~--

----::: <~ ~.:-

-----~ ~ ----.;.:-~;"""' ---- - ---e.----

----

--

/ V I

r.P __ ~ ..- /1' - ~

~I-'" /- / I I ,.0 _/ G.-I

/ ~-:jJ:f1-:. _____ '-.,. ""- T ,.- -.-

L-~ -/:::::-:;..---- ReP ---e.---- ~ i -

-~ ;::::.; -;F-

~'i=-=----~---:::: ---.:-.-:::=-- -- --- r:..--- ' I -- - -e.---- "\

--~

"t-

L J 10" 10' 10' 10' 10'

FREQUENCY (Hz)

FIGURE 1. Phase velocities as a function of frequency for transverse waves traveling parallel to a uniform magnetic field at 80, 100, and 300 km.

Id',---,-- -,---,---,---,---,---,---,---,---.---.------.---.-

I ---~--~-_4--_+--_+--~--~--+_--}_--~-_4-----_+--

., 10·ILO~·--L---LIO~~--L---L10··--~--~10·o--~--~10·z--~--~1o~'--~--~IO~'~-- 10'

FREQUENCY (Hz)

I

i I

""1 I

r 1

+-= II -r---I i

I i IdO

FIGURE 2. Attenuation distances as a function of frequency for waves traveling parallel to a uniform magnetic field at 80,100, and 300 km.

820

let[-...

~ Iei' ~ ~ 1! ] 1<1->­~

-· u c g 10 w > w (f)

• --

~ 10 D- ,Lle

I Va

• I ~

'--~

l

l I

--l-

I

~ r --r + + I

f--

I

1-- --- -,,---"

1- J- MOD,

+ -I--

T 1 1

f - -/

/

+ /

( / /

-i- t / i -/

/ ---- ....... / " I ~ / -'

-' ~~ / -'

Mdo, TRANS / / /

'- - j- r / - tEQ5 -/ // /

1/ t

--r;---f-S6 I /f ---t- -

EO 5 ~---~I --.MQI nECT , - jEo, ---- / J --- /

VERSE

_ ~Q 7 -' ~ -- MOD, 10N~EO 6~ EUTRAL ( EO 3) I + -_.

I - -

IJ- -d . I 'Vni 10

2n

1 1 10' I

Vin ffi

,---.- ~ Ie! Ie!

1-'-. -"-fo 10' ,

-let

r -10'

FREQUENCY (Hz)

F I GURE 3, Phase velocities as aJunction oj Jrequency Jor waves traveling pelpendicular to a uniJorm magnetic field at 800 km. The solid lines identify the wave modes described by Tane nbaum and Mintzer [1962]. The dO lled li nes are the stro ngly attenuated approxi. mations.

w u z ~

Id'

10'0

Ic!~-+--

~ 10'

I !

I 1

-----

l --

.~ -I

I t--I I C'n "

T 1 1 t-- I t

T I

I M(iO, I\t:. I KAl i t.

FREQUENCY (Hz)

+- I

- - -1 j-

I

I

I .~

I

I

EO 6

.1

-- #' Ml -----A,~'Y';

~o<:J ' ~o

/

MOD, EL'ITTRO'f'.J EO 7

-

-

-

-

-

10' 10~

FIGURE 4. Attenuation distances as a Junction oj Jrequency Jor waves traveling perpendicular to a uniJorm magnetic field at 300 km.

821

-----------

-

cf

~I

§ w >1

c!

c. 0

O· 'a-

O·

~v eX Ue if l

U I V d a

if

---f---

-I - ------ ~

!93 ..=-:::::-:::: V- EQ

------~

IMUL . ~[ I tlAI It. 1-----EQ~ __ --------

-~'---.

10' 10' 10'

F REQUE NC Y (H z )

--

/l / ~ i\... MOO. TR ~NS'v'ERSE

FD ~

/ ~

,/ ,/

./ / --- / \MOO EL EcreDIII ,// EQ 7 - 1--'.

1

I • o • 10 7 ) \ 10, 101 10 10'

F IGURE 5. Phase velocities as a function of frequency fo r waves traveling perpendicular to a uniform magnetic fie ld at 100 km .

., ~ 10 -; E

w • ~ 10 eX 1-. (/)

0 10 z o i= eX • ::> 10 z W l­I-

0 eX 10

• 10

• 10

MOD. NEUTRAL m 13)

I--- I-- --I---t-- --I--- [Q 5 -I ~--- --- ---- I- - r--__

-[~~

-------- ---- ---- --- ..;~(--

• 10 10' 10 FREQUENCY (H z)

. -

_d f .J

~ -if b7

~ /

~ /\

-------j

j ~ ) 1 I

MOD. EL ~CTRON(E Q.7 .- ---- ----- -- EQ- 6-- ---j

I ~ --- --- 1--------1--/

J

• • T 10 10. 10 10 ,101 10 • 10

F IGURE 6. Attenuation distances as a function of frequency for waves traveling perpendicular to a uniform magnetic field at 100 km.

822

I

�:q--+---+--I--+---+---+------jI�_-+----+---I------l---t----1---I-_d ~ o ~~ ~ -~ --~----~----_+----_+----_+--~ r~~--~----4_----4_----~----~----~-

~~o _ ' D6.NS'-JE~~ ~ I ~~p .su ~ ~ I~.~ ~ :- - - i--=-~-----'-f=~'------_+j---------__+-------__+-------_+--------_+i-=-E-Q.-L---_+-------_-_+----------If---------~------------1-------I-I_-- - - ---

u ~j----~ U --- ~ - ---

> 10'~----+_----+_----+_----,' ----_+----_+----__+----__I----__I-_~~~-~-~---~----~----~-----~----~ Ml • Nt. CAL E

UJ

~V :I: -a.. E..9Y-- --

"':tr= --c~--- -~c- --~=-----~-------I-------+-----I-----+-----I-----+-------1-----+---~--------+-------~ I - --~--

L -10' la' la' la' 10' 10' 10'

FREQUENCY (Hz)

FI GU RE 7. Phase velocity as a function of frequency for waves traveling perpendicular to a uniform magnetic field at 80 km.

Id',-----,-----,-----,-----,-----,-----,-----,-----,-----,-----,-----,-----,-----,----

MOD. N UTRAL EO 3 l ~

r----_I_----~--__+----_+----_I_---~----+-----~----+----~'_-_-_-_--I+_-_-_-_-_~f-------_--I+ _____ +_--~

~ld~~-~-4--4--4--+--+--~-~---~-~~-~I--- v1 ~ -~~ 1----+1- j UJ 10' ----

~ 1- +-----+------t------t------------+------"'--!1!?~D. :R··'WSV·-d--+------t----+--- I- - - / =-' ~ ---+--~--+--+_--+~--£+~~£~(~£Q~)' --+---f-----~ Id _I ~-- _______ 7 __ --_ ~Icf --

~ f - - - - - ------,'-::.:-=~l::___----II------+-----+-----+------+------I-----+-- - -- I----J ~Io'l -- -~6 _ j -----I 1--__ ~--

16'~I ----_+~~~----_+----__+----_+----_+----_+----_+--~-~~----__I----__I----__I----__II__----1_ - l ____ ~.E:~ - ___ '-- _________ ----I--------~_c::..~- ___ ____ _ _ I __ __ _ _ __ _ _ - -r-- - __ --

Id

I c--------'---------'-o--------'------'-~---'------'------'----------'------'------'+c------'-+----__JL__- - - - c- ~'cc - - -- - - -- J

16' 16' 16' 16' Id laf• tiH- f.j O· f. 10' 10'

FREQUE NCY (Hz)

FIGURE 8. Attenuation distance as a function of frequency for waves traveling perpendicular to a uniform magnetic field at 80 km.

823

TABLE 1. Ionospheric data at 80, 100, and 300 km

Ionospheric data (mks units)

Temperature ................................ k· ... Magnetic field ..... .............. .. .. ... ... .... . . .... weber/m2 ••

Neutral particle density., ................ particles/m3 ...

Jon density ..... ................... ..... .. ............ part icles! m3 ...

Ion. neutral mass ... .................. ..kg .. Electron plasma freq .............. rad/sec .. Ion plasma freq ...................... ........ rad/sec .. Electron cyclotron freq ................... ............... rad/sec ... Ion cyclotron freq .. ........ ........ .. . ...... rad/sec o.

Electron-neutral collision freq ...... . .. sec- t ••

EJect ron-ion collision freq .. .. ..... ......... .... sec- I ...

Ion-neutral collision freq ............................ sec- I ..

Neutral-electron col1ision freq ....... sec- I .. ,

lon-electron collision freq ..................... sec- I ..

Neutral-ion collision freq .. . . ........................ sec- I ..

lon-neutral sound speed .......... m/sec .. Alfvcn velocity. .......................... m/sec .. Ion Alfven velocity ............... ••.... .... m/sec ... Electron sound speed ... ........... m/sec ..

From the 300 km curves it can be seen that the phase velocity starts out at the Alfven velocity, VA, for W« Vni, and increases to the ion Alfven velocity, V~, for Vin < < W < < w~. Just below the ion cyclotron frequency, w~, the LCP wave and the RCP wave sep­arate. The LCP wave resonates at w~, and is strongly damped for frequencies between w~ and WOI = [w~ + 0 w~) 2 + W~W i)1 /2 - 1 w~. The phase velocity in this frequency band is high, since k is almost purely imaginary.) Above W01, the attenuation is again slight, and the phase velocity decreases towards the speed of light, c, for W» W01. The RCP wave resonates at w~ and is strongly damped for frequencies between w~ and W02 = WOl + w~; then, above W02 the attenuation is again slight, and the phase velocity decreases to·

. wards c, for W > > W02. The phase velocity in the strongly damped frequency bands, w~ < W<WOl for the LCP and w~ < W < W02 for the RCP waves, is infinite (Rek = 0) in the limit of no collisions. That the phase velocity is large but finite at 300 km is a consequence of small but nonzero collision frequencies.

On the 100 km curves there is no resonance at w~, the LCP and RCP waves have widely separated phase velocities and attenuation distances, for all save very high frequencies, and the phase velocity never levels off at VA or V~. This occurs because Vin> > w~. while Ve < < w~. Inspection of the quantity in brackets in the dispersion relation shows for a wide range of frequencies ( w~ pdp" < W < w~), that ± w~/w is the largest term, which accounts for the wide split in the LCP and RCP waves and the absence of the resonance at w~. It is this term, too, which is responsible for the steady increase in the phase velocity, since k is pro· portional to Wl /2; hence the phase velocity is propor­tional to W 1/2 • Although the resonance at w~ is absent, the attenuation distance is very small for the LCP wave between w~ and WOl and the RCP wave between w~ and W02 as in the 300 km case. For wii < < Vin, the "mag­neto·hydrodynamic" term, w~w~/w;P ,does not dominate until W < < w~(p;/ Pn), which is well below the smallest

80 km 100 km 300km

T 205 280 1428 Bo 3.0 X 10-' 3.0 X 10-' 2.75 X 10-' N, 4.38 X 10"' 5.4X lOll 1.035 X 10" No LOX 10' 1.3 X 10" 1.8 X IOIZ m, 4.82 X 10-" 4.32 X 10- 26 3.570 X 10-" w, 1.783 X 10' 2.033 X 10' 7.568 X 10' w, 7.754 X 10' 9.339 X 10' 3.822 X 10> w~ 5.276 X 10' 5.276 X 10' 4.836 X 10' w; 9.97 X 10' 1.112 X 10' 1.234 X 10' v" 6.517 X 10' 9.39 X 10' 4.064 X 10' /It'i 1.073 X 10' 8.743 X 10' 1.051 X 10' v" 1.022 X 10' 1.406 X [(}' 3.261 X 10- ' v" 2.811 X 10- >0 4.766 X 10-' 1.803 X 10-' v" 2.028 X 10- ' 1.843 X 10-' 2.681 X 10- ' v" 2.334 X 10-' 3.385 X 10-' 5.672 X 10- ' Wo 5.569 X 10' 2.101 X 10' 7.583 X 10' w" 5.465 X 10' 1.787 X 10' 7.330 X 10' W02 5.822 X 10' 2.314 X 10' 7.814X 10' W03 7.346 X UP 2.345 X 10' 2.438 X 10' Uf,/I 3. 12 X IO' 3.86 X 10' 9.6 X 10' V, 5.83 5.53 X 10' 4.02 X 10' VA 3.86 X 10' 3.57 X 10' 9.68 X 10' U, 7.2 X 10' 8.4x 10' 1.895 X 10'

frequency used in our calculations. The phase ve­locity would, however, approach VA for both the LCP and RCP waves if the calculations were extended to these very low frequencies.

At 80 km the collisional terms in the dispersion relation are still more important, since w~ < < Vin and w~ and Ve are of comparable magnitude. Hence, forw~pd pn < < W < < w~, the expression in brackets approxi­mately equals (± w~ + iVe)/w. Thus the LCP and RCP curves are separated, but not as much as in the 100 km case, there is no resonance at w~ or w~, and the phase velocity, over a wide range of frequencies increases at a rate proportional to W 1/2•

The remaining figures all show the phase velocities and attenuation distances for the four coupled longi­tudinal and transverse waves able to propagate per- J

pendicular to the magnetic field. The situation is complicated by the fact that several approximate dispersion equations have to be pieced together ac­cording to the scheme outlined in table 1 of Tanen­baum and Mintzer [1962]. Moreover, although there are four solutions to the dispersion equation, one or more of these usually represents a highly attenuated wave. Hence, the solid and broken curves in figures 3 to 8 represent, respectively, waves with large or small attenuation distances .

Although the terminology cannot be used very pre­cisely, we can classify the waves with large attenuation distances as' "modified" neutral, ion, electron, or transverse waves. The modified neutral wave is described by the dispersion relation

(for IEI« 1), (3)

with

iVni [ vinX + iw ] E=~ v in( l +X)+2Vni+ iw ' (4)

824

For a weakly ionized gas (3) can be used (i.e., lEI is small) provided W exceeds v"iX/(l + X). Since this is an extremely low frequency, (3) could be applied for our entire range of w. As shown in the figures this modified neutral wave has a phase velocity of U" and an attenuation distance which decreases from

, 2U,,(1 + X)/vniX, for low frequencies, to 2U,,/V"i, for high frequencies. At 300 km X == 101; consequently there is no noticeable difference. At 100 km X == 4, so the change is slight. At 80 km X == 10- 3, so the attenuation distance increases by about 103.

The behavior of the remaining curves is determined in large measure by the single "high phase velocity" dispersion relation

(5)

which is valid for Iw2fk21» U~. When collisional damping effects are slight, as at 300 km, the phase velocity obtained from (5) starts out at the Alfven ve­locity, VA, for W < < V"i and increases to V~ for Vi" < < W < < W03 = wi(wgwDl/2/(wr + wgw~)1 /2 . At W03 this wave resonates and continues on for higher fre­quencies as a modified ion wave described by the dispersion relation

(6)

with !V=w2+iwVin, D~,i= W~,i [1+i(ve/w)] -1/2, and Dr = WI + iwve(me/mi). (For frequencies below W03, the attenuation distance obtained from (6) is always

I very short.) For W > W03 the attenuation distance computed from

(5) is very short and the phase velocity very large until W reaches the first cutoff frequency, W01; then, for WOl < W < Wo= [w~+(wg)2]1/2, the attenuation distance is again large and the phase velocity decreases toward c. At Wo there is a second resonance, and the wave continues for higher frequencies as a modified electron wave described by the dispersion relation

(7)

[For frequencies below Wo, the attenuation distance obtained from (7) is always short.] For W > Wo (5) again describes a wave with very short attenuation distance until W reaches the second cutoff frequency W02. Thereafter this modified transverse wave is only slightly attenuated and has a phase velocity that de­creases towards c for W > > W02.

All the features described above appear only in the 300 km curves shown in figures 3 and 4. At 100 km (figs. 5 and 6) the phase velocity obtained from (5) does reach V~ for Vin < W < W03, but the resonance at W03 is absent, although the attenuation distance is still short for W03 < W < W01. This occur since Ve exceeds W03; hence the term ive/w dominates the bracketed expression in (5) near W03. In addition the ion wave [as described by (6)] is greatly attenuated for all w, since its attenuation distance, which approaches 2Ui/vin for large w, falls from 6,000 m at 300 km to 55 cm at 100 km and 6.2 mm at 80 km.

At 80 km (figs. 7 and 8) collisions are of still greater importance in (5), since· Vin exceeds W03, eliminating the possibility of Alfvim waves with phase velocity V~; and Ve exceeds Wo, eliminating the resonance at wo0 The term ive/w now dominates the term in brackets in (5) for a wide range of frequencies, resulting in curves that are almost identical to the LCP and RCP waves of figures 1 and 2 at 80 km. In addition, the modified electron wave [as described by (7)] is greatly attenu· ated for all w, since its asymptotic attenuation distance, 2Ue/ve, falls from 380 m at 300 km and 1.8 ill at 100 km, to 2.2 cm at 80 kill.

2. References Chapman, S. (1956), The electrical cond uctivity of the ionosphere:

A review, Nuovo Cimento 4,1385. Cowling, T. G. (1945), The electrical conductivity of an ionized gas

in a magnetic field with appUcations to the solar atmosphere and ionosphere, Proc. Roy. Soc. (London) I83A, 453.

Minzner, R. A., K. S. W. Champion, and H. L. Pond (1959), The ARDC model atmosphere, 1959 AFCRC TR-59-267 (Geop hysics Research Directorate , Air Force Cambridge Research Center, Bedford, Mass.).

Tanenbaum, B.S., and D. Mintzer (1962), Wave propagat ion in a partly ionized gas, Phys. Fluids 5, 1226.

Tanenbaum, B.S., and D. Mintzer (1963), Wave propagation with a transverse magnetic field, Proc. Intern. Conf. Ionosphere , London, July 1962 (Inst. Phys. Soc. London).

(Paper 69D6-514)

825