On the origin of “very-low-frequency emissions”

Journal of Atmospheric and Terrestrial Physics, 1962, Vol. 24, pp. 685 to 689. Pergamon Press Ltd. Printed in Northern Ireland

On the origin of "very-low-irequency emissions"* H. UNZJ" A n t e n n a L a b o r a t o r y , D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g , T h e ()hio S t a t e V n i v e r s i t y , ( ! o l m n b u s , Ohio, U.S.A. (Received 6 April

1962)

Abstract T h e m a g n e t o - i o n i c t h e o r y for d r i f t i n g p l a s m a is a p p l i e d to t h e t h e o r y o n t h e ol'igin o f v e r y - l o w - f r e q u e n c y emissions. T h e f r e q u e n c y a t w h i c h t h e r e will he i n t e r a c t i o n , a n d possible a m p l i f i c a t i o n , b e t w e e n t w o different, s t r e a m s of e l e c t r o n s is f o u n d . I t is s h o w n t,h a t ])hysical p h e n o m e n a are e x p l a i n e d b y i n t e r a c t i o n b e t w e e n several s t r e a m s of e l e c t r o n s of different p l a s m a f r e q u e n c y a n d different velocity. l . INTRODUCTION IN TWO recent papers b y GALLET (1959) and GALnET and HELLIWELL (1959), the origin of the very-low-frequency emissions generated in the earths' exosphere are discussed. Selective travelling-wave amplification in the outer ionosphere has been postulated to explain very-low-frequency emissions, a class of very-low-frequency (1-30 kc/s) natural noise. Several of the phenomena have been explained. Although the above postulate seems to be rather successful in general, the explanation of some of the phenomena by the above theory is not so easy (GALLET and HnLLIWELL, 1959) and "the model may look very artificial" (GALLET, 195q). Lately the magneto-ionic theory for drifting plasma has been developed by the author (U~-z, 1962). It has been decided to re-examine the theory presented previously and try to explain the physical phenomena in the light of the new theory, keeping intact the basis postulate about the travelling-wave amplification in the outer ionosphere. It has been found that the analogy with the small-signal theory of travelling-wave tubes (PIEaCE, 1950; HUTTER, 1960) exists in our case only under special conditions, and a more general approach to the problem is suggested. In the next section the general theory will be developed, and later a comparison between theory and observations will be given. 2. INTERACTION CONDITIONS

Let. us assume two streams of electrons of constant velocities vI and v,~with electron density X 1 and X 2 correspondingly, moving along a longitudinal magnet.ic field line Yj. The condition for the interaction of electromagnetic waves in the two streams is that the two phase velocities of the two waves ill the two electron strean,s will be equal: V9 - - V p l :

Vl) 2.

(la)

* T h e reseai'eh reportDd in t h i s p a p e r was s p o n s o r e d in p a r t b y t h e A s t r o s u r v e i l l a n e c Sci(,nc(,s L a b o r a t o r y , Elect.tonics R e s e a r c h ] ) i r e e t o r a t e , A i r F o r c e C a m b r i d g e R e s e a r c h Laboi'atori~,s, ()ffice o f A e r o s p a c e l~eseareh, U n i t e d S t a t e s Airforee, B e d f o r d , M a s s a c h u s e t t s , U.S.A. un
686

~[. UNZ

I n t h e p a r t i c u l a r case of small-signal t h e o r y (HUTTER, 1960) the p h a s e v e l o c i t y of the w a v e is equal to t h e v e l o c i t y of the s t r e a m , v, ---- v0. I t has been suggested in the l i t e r a t u r e (GALLET, 1959; GALLET a n d HELLIWELL, 1959) t h a t this is the condition for " v e r y - l o w - f r e q n e n c y emissions" signal amplification. I t should be p o i n t e d out t h a t the e q u a l i t y a b o v e exists o n l y u n d e r t h e condition (HUTTER, 1960) ( % . 4 "), which is usually t r u e in t r a v e l l i n g - w a v e t u b e s ; this condition is n o t generally t r u e in whistler-like p r o p a g a t i o n ( H E L L I W E L L a n d MORGAn, 1959), a n d therefore a new, m o r e general a p p r o a c h to t h e p r o b l e m is necessary. T a k i n g t h e r e f r a c t i v e index for the first s t r e a m to be n 1 - - c/V~l a n d the r e f r a c t i v e i n d e x for t h e second s t r e a m to be n,, = c/v,._,, e q u a t i o n ( l a ) m a y be r e w r i t t e n as: n -- n 1 = n 2

(lb)

where (lb) r e p r e s e n t s the i n t e r a c t i o n condition b e t w e e n the w a v e s in the t w o s t r e a m s of electrons. I t has been s h o w n b y the a u t h o r (UNz, 1962) t h a t for p l a s m a drifting w i t h c o n s t a n t v e l o c i t y ~70, the r e f r a c t i v e index will be g i v e n b y the following algebraic e q u a t i o n s : C [ A ~ - - ]=Le(n e _ 1) el - - A [ Y T e ( n

1)2 ~- flT2X2n2(1 - - n f i L ) 2]

e -

-- 2 XYL(~T

"

YT)n(1 - -

nfiL)(n

2 --

1) e -- 0

(2a)

A = (n 2 -- 1)(1 -- i Z ) q-X(1 -- nflL) °"

(2b)

C = (n ~ -- 1)(1 - - X

(2c)

iZ) + neXflT e

where f i t VoT/C, flL = VoL/C, VOT a n d VoL are the t r a n s v e r s e a n d l o n g i t u d i n a l c o m p o n e n t s of the c o n s t a n t drift v e l o c i t y v e c t o r of the p l a s m a . All the rest of t h e s y m b o l s are defined as in the m o n o g r a p h b y ~ATCLIFFE (1959). T a k i n g in our p a r t i c u l a r case fiLl = fix = VI/C, ilL2 = f12 = Ve/C a n d a s s u m i n g no t r a n s v e r s e c o m p o n e n t s of either p l a s m a drift or m a g n e t i c field, a n d neglecting collisions Z -- 0, e q u a t i o n (2) will b e c o m e for the two electron s t r e a m s : (n.12 -- 1)(1 -- X1)([(nl 2 -- 1) + XI(1 -- nl/~1)2] ~ -- YL2(nl " -- 1) 2} ---- 0

(3a)

(n22 -- 1)(1 -- X2)([(n.~ 2 -- 1) ~- X2(1 -- n2fi,z)2] 2 - - YL2(n,~ 2 - 1) 2} = 0

(3b)

I t m u s t be r e m e m b e r e d t h a t the two electron s t r e a m s h a v e different electron densities X1, X2; b u t t h e y drift along t h e s a m e longitudinal m a g n e t i c field YL" E x c e p t for t h e trivial solution of p r o p a g a t i o n in free space n,.._, - 1 , the solutions of (3a) are X 1 : 1, (o <%1 (4a) (1 -

1 -[- X 1

n l fi,)2

nl 2

-

-

• }'L

1

(4l))

a n d those of (3b) are X 2 -

1~

o) z

1 • X 2 (1 - - n 2 f12) 2 n2 ~ -- 1

o)p2

~Z }r/_

(Sa) (Sb)

w h e r e w~l is t h e p l a s m a f r e q u e n c y o f t h e first s t r e a m only, a n d ~%2 is the p l a s m a frequency of the second stream only.

On the origin of' "very-low-frequency (,missions"

687

Using the i n t e r a c t i o n condition ( l b ) i n (4b) and (5b), two sinmltaneous equations for the e o m m o n refractive index n are found" (1 - - n i l 1 ) 2 = . - - 1 ± n °= -- 1 X 1 (1 -- nfi2)2 n 2 -- 1

YL

((~a.)

- - 1 ~ YL Xo

I t has been shown b y STOREY (1953) t h a t for whistlers n 2 >> 1, and t h a t : YL should be used in (6) for the e x t r a o r d i n a r y mode of propagation. Since the lowf r e q u e n c y emissions b e h a v e like whistlers (GAI,H~T, 1959), equation (6) could be r e w r i t t e n as : 1 9~,

fll

~

(7a)

,

1

,

fi2 = ±

i!

(7b)

S u b t r a c t i n g (7b) from (7a) and squaring, one gets

( / ~ 2 - ~1) 2 [Faking

(YL

X i = \~7,~/ '

x,vhere

(9p ---

1 -. ~=X; z~ 7~?:~1/')

])

X2 = \~7~,)/ '

Ne z

,

YL

(S)

o>

/~oHo le] (0 H

---

one gets after rearranging: ",(o~,= ± 70-1/

(o =: (on.

(9)

E q u a t i o n (9) gives the f r e q u e n c y (o at which interaction conditions exist between b e a m No. 1 of plasma f r e q u e n c y o9~1 and relative velocity fit and b e a m No. 2 of plasma f r e q u e n c y m:~ and relative velocity fi2 drifting in the direction of magnetic field of g y r o f r e q u e n c y (oH" :For m e d i u m and low latitude (HELLIWELLand MORGAN,1959), and at frequencies below a b o u t l0 kc/s, oo < (oH and (9) becomes:

(0

(0~°12 0)~°22 (OH

- ~ \fOp2

(10)

~q U92o1

E q u a t i o n (9) or (10) gives the condition for interaction for a certain f r e q u e n c y (o" this interaction could bring amplification according to the travelling-wave t u b e mechanism. :For the p a r t i c u l a r case /5'1 ~ fie, (o~1 <~ "o.2, (10) will l>eeome: 0)~12 ~22 ~,) -(l l ) (OH

where (11) is the result derived b y GALLET (1959).

688

H. U~z 3. COMPARISON OF THEORY A~I) OBSEaVATIOXS

E q u a t i o n s (9) or (10) give as t h e i r m a i n result t h e r e s o n a n t f r e q u e n c y ~o a t which t h e r e is i n t e r a c t i o n b e t w e e n t w o electron b e a m s and, therefore, a possibility of amplification. F o r t h e ease w h e n the s a m e b e a m of electrons is discussed, fit .... fi~, ~'~,1 = %2 a n d (o is i n d e t e r m i n e d (all frequencies). I n case the two electron b e a m s are quite different,/~1 ¢=/~2, (%1 @ r%2, a n d oJ is a signal f r e q u e n c y ((o < (oH), t h e r e is a possibility of amplification at only t w o frequencies, o). I n ease t h e r e are several b e a m s of electrons, there will be several single fl'equeneies o), which could be amplified, comprising all different c o m b i n a t i o n s b e t w e e n the different b e a m s . I n t h e ease of an electron b e a m w i t h v e l o c i t y d i s t r i b u t e d continuously, i.e. fi = fi ((,J~,) t h r o u g h o u t the b e a m , f r o m fil ((')~1) t o / L (%2), E q u a t i o n (9) could be rewritten as: o,((oH -- (o)== L~i-/;~)~.(i}O~-l-ij

(12)

a n d the result will be a b r o a d - b a n d t r a v e l l i n g - w a v e t u b e amplifier for the frequencies :

kd(l/o)~) e)~n

> co(o)H -- (o),~ -

-

: kd(lle~,)io)~, 2

(13)

p r o v i d e d t h a t o~ < (Oli. I n s u m m a r y , the following kinds of amplification b y i n t e r a c t i o n b e t w e e n electron b e a m s are possible: (a) I n t h e ease of the s a m e b e a m of electrons: i n d e t e r m i n e d results (no interaction). (b) I n the case of several b e a m s of distinct velocities a n d m a g n i t u d e s amptifieat.ion in several distinct frequencies: ('J = (el; (~ = ~'~2 . . . oJ = (,>,,; for O) <~ (O H .

(c) I n t h e case of a b e a m of continuous velocity v a r i a t i o n : a b r o a d b a n d amplification: co2 < o) < (')1
On the origin of "very-low-frequency emissions"

~i89

(b) Quasi-constant tone: a h n o s t c o n s t a n t t o n e of discrete frequency. Sometimes several different tones of discrete f r e q u e n c y could be observed simultaneously. These are suggested to be case (b) for constant ~oH. (e) T r a n s i e n t p h e n o m e n a (i.e. dawn chorus): The incoming particles arrive in bunches, and the f r e q u e n c y changes due to change of o~rr as the b u n c h moves along. I t might be caused b y high-energy particles coming from the sun. (d) H o o k s : A plot in the f r e q u e n c y - t i m e d o m a i n has the shape of a hook. P r o b a b l y case (b) when the bunch of electrons is moving in a special direction relative to the magnetic field so t h a t the r e s o n a n t fl'cquency versus time will v a r y like a hook. (e) I n t e r a c t i o n between whistlers and. v.l.f, emissions: The whistler will I)e amplified and becomes a source for several v.l.f, emissions according re, either process. The e x p l a n a t i o n of some of the above eases b y this t h e o r y is simpler and more acceptable t h a n in the previous t h e o r y (GALLET, 1959" GaLLET and HELLIWELL. 1959), although in some of t h e m the ideas of the e x p l a n a t i o n arc quite similar (GALLET and H ELHWELL, 1959). B y using the above t h e o r y and t a b u l a t i n g the e x p c r i m e n t a l results, we couhl get; some q u a n t i t a t i v e results of the different electron streams, their velocities, etc. B y using e q u a t i o n (2) in the more general case, a complete analysis could b(, developed for the case when the electron streams are in a n y general direction, not just. along the magnetic field line. a u t h o r is grateful to Professor H. C. K o and Professor C. T. TAI of the Ohio State U n i v e r s i t y for their c o m m e n t s and several discussions on this paper. t~EFERENCES (;ALLET R. ~1. 1959 Prec. I . R . E . 47, 211. (x~ALLET :R. ~I. a n d HELLI~,VELL 1~. A. 1959 J . Res. Nat. Bur. Stand. ]) 63, 2l. HELLI\VELL R,. A. a n d MORG.a~ M. G. 1959 1"roe. I . R . E . 47, 200. HUTTER 1/. G. E. ] 960 B e a m a~ul Wave Electro~ics iy~ Microw~lv~; Tubes, Chaps 9. and 10. Van Nostrand. New York. ]hE~CE ,I. R. 1950 Travelling IVave Tubes. Van Nostrand. New York. 1~ kTCLIFFE .I.A. 1959 The Magneto-Ionlc Theory and its Acknowledgement--The

Applicotio++s to the Ionosphere.

('ambridge University Press. STOREY L. l~. O. UNz H.

1953 1962

P h i l . :Trat~s. A 246, 113. Trat~s I . R . E . AP-10, In i)ress.