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Theory of longitudinal gradients in the equatorial electrojet
Journal of Atmo~'pher~and Terrestrial Physics, Vol. 38, PP. 279 to 286. Pergamon Press, 1976. Printed In Northern Ireland
Theory o~ longitudinal gradients in the equatorial electrojet J. GAGI,IEPAII,~ a n d M. C R O C H E T
Institut de Physique du Globe, Universit6 Pierre et Marie Curie, 75230 Paris Cedex 05, France and A. I). RIC~OZ~D* Laboratoire de Physique de l'Exosphbrc, Universitd Pierre et Marie Curie, 75230 Paris Cedex 05, France (Received 1 M a y 1975; in revised form 8 J u l y 1975)
Abstract--Longitudinal gradients in the equatorial electrojet are observed with scales ranging from several thousands of kilometers on down. Since previous theoretical models of the electrojet electric fields and currents have assumed that no longitudinal gradients exist, we examine theoretically the consequences of allowing for them. The vertical structures of the electric fields and currents are altered in predictable ways, the alterations being increasingly important as the electric field and current gradients increase. In contrast to the infinite-eleetrojet model, reversals of the electric field and current directions with height become possible when longitudinal gradients are present. We show how electric fields produced by hypothetical generating sources located at various latitudes are attenuated between the source and the Equator. For electric fields with small longitudinal scale lengths, the source must lie close to the magnetic equator in order to cause noticeable effects in the electrojet.
INTRODUCTION The e q u a t o r i a l electrojet is generally i m a g i n e d to be a r i b b o n of current a few h u n d r e d kilometers wide, flowing along t h e E a r t h ' s m a g n e t i c e q u a t o r a t a n a l t i t u d e of a b o u t 100 kin. Theoretical models w h i c h h a v e been d e v e l o p e d up to n o w (e.g. UNTIEDT, 1967; SU•IURA a n d PORES, 1969; KI~YLOV et al., 1973; RICHMOND, 1973) h a v e a s s u m e d for simplicity t h a t t h e longitudinal e x t e n t of t h e electrojet is so great in c o m p a r i s o n w i t h its w i d t h , t h a t longitudinal g r a d i e n t s of t h e electric field, c o n d u c t i v i t y a n d c u r r e n t can be c o m p l e t e l y n e g l e c t e d w h e n only local features are being considered. These models h a v e b e e n fairly successful in r e p r o d u c i n g several o b s e r v e d features of t h e eleetrojet, a n d t h e y p r o v i d e a useful s t r u c t u r e for organizing observations a n d for r e l a t i n g v a r i o u s physical p h e n o m e n a in t h e e q u a t o r i a l ionosphere, L o n g i t u d i n a l g r a d i e n t s of t h e eleetrojet, however, n e e d n o t always be so insignificant as these models assume. To be sure, t h e giobal-scale g r a d i e n t s associated w i t h t h e diurnal changes of t h e conduct i v i t y a n d t h e electric field were e s t i m a t e d b y I~ICHMOND (1973) tO cause errors o n l y on t h e order of 10~o in t h e features of his model. T h e r e exist variations of smaller scale, however, w h i c h can be * Present address: High Altitude Boulder, Colorado 80303, U.S.A.
Observatory,
e x p e c t e d to cause errors of correspondingly greater m a g n i t u d e . F o r example, t h e ' c o u n t e r electrojet', a d a y t i m e reversal in t h e direction of current flow, is u s u a l l y o b s e r v e d to be l i m i t e d in longitudinal e x t e n t to a few tens of degrees (Gouz~ a n d MAYAUD, 1967). On a n e v e n smaller scale, r a d a r measurem e n t s of t h e electron speed w i t h i n t h e electrojet h a v e r e v e a l e d significant longitudinal v a r i a t i o n s o v e r a distance of a few h u n d r e d kilometers (BALsLEY, 1970). T h e r e also exist v e r y i m p o r t a n t v a r i a t i o n s of t h e electric field a t sunrise a n d sunset ( B ~ s L E Y , 1970). T h e p h y s i c a l sources of these smaller scale v a r i a t i o n s in t h e electric field a n d cLurrent of t h e electrojet can be classified, for t h e purposes of our present discussion, as being either 'local' or ' r e m o t e ' . One local source w o u l d be t h e presence of eonduct i v i t y v a r i a t i o n s w i t h longitude, such as a t sunrise a n d sunset, which cause a disruption of the largescale current flow a n d t h e creation of smaller-scale electric fields. A second local source w o u l d be t h e presence of l o n g i t u d i n a l l y v a r y i n g local n e u t r a l winds, which can drive electric currents b y action of b o t h t h e d y n a m o electric field, V × B (where V is t h e w i n d v e l o c i t y a n d B t h e g e o m a g n e t i c field), a n d t h e electrostatic field, E, w h i c h in general exists in association w i t h t h e d y n a m o field due to a c c u m u l a t i o n of space charge, l~emote sources, t h a t is, sources existing outside of t h e e q u a t o r i a l 279
280
J. GAG~EPAI~r,M. CROCHET and A. D. RICHMOND
region itself, can affect the electrojet only b y means of the electric field they create, which can be transferred from the source region through the conducting ionosphere to the electrojet region in association with current flow between the source a n d eleetrojet regions. Possible remote sources are (1) ionospheric winds existing outside the equatorial region, (2) solar wind-magnetosphere interactions, which transfer electric fields to the ionosphere at high latitudes, and (3) conductivityinhomogeneities at non-equatorial latitudes. The present study has two goals. First, we wish to estabhsh quantitatively how the presence of longitudinal gradients in the electrojet electric field and current affect the vertical and latitudinal structure of eleetrojet features. Second, we wish to examine the 'transmissivity' of the ionosphere to electric fields produced by remote sources; i.e. we wish to be able to tell how remote from the E q u a t o r a source can be and still have a significant influence on the electrojet. We shall not examine how the electric fields are actually produced b y hypothesized sources. Thus our mathematical t r e a t m e n t neglects a n y longitudinal variation of the ionospheric conductivity, and also neglects the presence of winds within the latitudinal region under examination. For this reason, our results cannot be used to examine sunrise-sunset effects, nor can they be applied to effects due to winds existing within the electrojet region itself. I n spite of these limitations, we believe t h a t our results m a y be applicable to m a n y types of longitudinal variations in the electrojet, MATHEMATICAL DEVELOPMENT Following RICHMOND (1973), we assume the geomagnetic field to be dipolar, and use the dipolar coordinates a, ~, fl defined b y = r/sin ~ 0, = ~0, fl
=
cos
O/r~
= si.~ o/(~ + 3
h e = r sin
O,
h~
-I-
=
r8/(1
cos ~
=
e~((~zE ~
--
asE~)
+ e~(%E~ + ~IE~) - eB%E p,
(1)
1 FO(h~,hpJc,)+ -O(hph=J~) h~h~h# L ~ Oq~ ~(hahej#).~
V. ~
-F ~-fl V × E =
| J
= 0,
(2)
a(h~E~)I"~ ~-# .~
e~ ~ ~l.a(,~,~#) ~
h~h# L °T -F - e~ -
vO(haEa) a(hflE#)~ h#haL" ~-fl ~c¢ J
-F e# F~(h~E~) ~(h~E~)~ l~hq,L. ~ -~ J =
0,
(3)
where J is the current density, ~0, ~1, °2 are the parallel, Pedersen, Hall eonductivities, and ea, e~, e# are u n i t vectors. We follow a suggestion b y KRYr,OVet al. (1973) a n d RICHMOND (1973), who showed t h a t a reasonable solution for the electric fields a n d currents can be obtained b y assuming ~0 to be infinite, so that E~ = 0. (4) Then the a a n d ~ components of (3) tell us that (h~Ea) and (h~Ee) are constant along a magnetic line of force, i.e. these quantities are functions of only ~ a n d ~. The fl-component of (3) provides us with one relation between (h~Ea) and (h~E~). The second equation relating these two quantities is obtained b y using (1) in (2) a n d b y intregrating hahq~hpV . ~ with respect to fl along a line of force from one end, at the southern base of the conducting ionosphere, to the other end, a t the northern base. Nothing t h a t 3"# = 0 at each end, we then have :
0 = ZF ( fP'h_..T.~a~hp dfl ) (haEa)
where r, 0, ~ are the usual spherical coordinates. The associated metric coefficients are:
4
instabilities, we have the following set of equations to solve:
a=L\J~, ha
-- ||//*P~~h#r dfl
~#,
0)'~,
a F / I@'~
-F - - I I I
a~L\J#,
3 COS2 0) 1/~. -I-
Neglecting ionospheric winds and possible plasma
(bee ~) \
azhp dfl) (haEa)
( f flzha~ 61h~dfl)(hq)Eq~). \,]p~ h~
(5)
Theory of longitudinal gradients in the equatorial eleetrojet W e eliminate t h e ~ - d e r i v a t i v e s in (3) a n d (5) b y a s s u m i n g t h a t E~ a n d E~ v a r y in l o n g i t u d e as Ea, ~
=
Re
(ea.~ein~)
(6)
where R e denotes t h e real p a r t , i i s , v / - - ~ a n d n is the longitudinal wavenumber; t h e n a]a~ is r e p l a c e d b y in. I t is a s s u m e d t h a t t h e c o n d u c t i v i t ies are c o n s t a n t w i t h longitude. T w o b o u n d a r y conditions are required, as we n o w h a v e t w o firstorder o r d i n a r y differential e q u a t i o n s for (haea) a n d (h~e~) w i t h respect to ~. One c o n d i t i o n is t h a t t h e v e r t i c a l c u r r e n t d e n s i t y v a n i s h a t t h e base of t h e c o n d u c t i n g region (which we t a k e t o be at an a l t i t u d e of 80 kin). S e t t i n g J a = 0 a t 80 kin a t t h e E q u a t o r gives ale a -- ~2e~ = 0
at
fl = 0, ct =
(R E +
80km)
(7) w h c r o R E i S t h e radius of t h e E a r t h . F o r t h e second condition, we choose: e~ = 2 O m V . m
-1
at
fi = 0 , c ¢ = ( R E + 102 k m ) .
281
(11) to step u p w a r d to find s~ a n d % at each sueeessive l e v e l ; finally we n o r m a l i z e our results to t h e c o n d i t i o n (8). The c o n d u c t i v i t i e s al, o 2 a n d t h e l i n e - i n t e g r a t e d c o n d u c t i v i t i e s (9) are o b t a i n e d f r o m I:~ICB-~OND'S (1972, 1973) m o d e l w i t h t h e m a g n e t i c field s t r e n g t h B 0 = 3.21 × 10 -5 T, t h e solar flux Fxo. 7 = 90 × 10 -22 W . m -~ . Hz-~ a n d t h e zenith angle X = 0 °. Outside t h e e q u a t o r i a l region, we define a new v a r i a b l e 0* as t h e c o l a t i t u d e w h e r e a g i v e n ]ine of force intersects t h e 130 k m a l t i t u d e level, i.e. 0* ---- sin -1
{r*/~) 1/2.
I n t h e following we d e n o t e w i t h a star t h e v a l u e of a q u a n t i t y e v a l u a t e d a t t h e coordinates r = r*, 0 = 0". A s s u m i n g t h a t
;
~2(h*/hct)±1~lha dfl =(h~*/ha*)± I
(8)
This choice has b e e n m a d e in order a p p r o x i m a t e l y to normalize all of our c o m p u t a t i o n s such t h a t t h e m a x i m u m h e i g h t - i n t e g r a t e d eleetrojet c u r r e n t d e n s i t y (and hence, a p p r o x i m a t e l y , t h e m a x i m u m g r o u n d - l e v e l m a g n e t i c effect) occurs a t ~ = 0 a n d has t h e s a m e v a l u e for a n y n. As t h e m a x i m u m e a s t - w e s t electrojet c u r r e n t d e n s i t y occurs approxi m a t e l y a t 102 k m a n d is a l m o s t e n t i r e l y d e p e n d e n t on E a (rather t h a n Ee) a t this altitude, t h e normali z a t i o n expressed b y (8) t u r n s o u t to be a good a p p r o x i m a t i o n to a n o r m a l i z a t i o n in t e r m s of g r o u n d - l e v e l m a g n e t i c effect, T e solve (3) a n d (5) numerically, we use a p p r o x i m a t e forms of these equations. I n t h e e q u a t o r i a l region, defined as
(12)
alh~dfl
(13)
d~ a n d l e t t i n g I be t h e inclination of t h e geomag,letie field below t h e horizontal, we o b t a i n t h e following e q u a t i o n s f r o m (3) a n d (5): d [sin 0*ee*] = --in sin l*¢a*, dO* d [sin O*(Exe * dO* -- Zse~*)] =
in
sin I*[Z2e * + Z x e * ]
(14)
(15)
w h i c h are solved in 1° i n c r e m e n t s of 0* b e t w e e n 80 ° a n d 20 ° colatitudo. T h e c e n d u c t i v i t i e s are comp u t e d b y lotting B a n d X v a r y w i t h c o l a t i t u d e as B = B0[1 + 3 c o s 2 0"] 1/2,
(R E + 8 0 k i n ) < ~ < (R E + 4 0 0 k i n ) , we assume t h e m e t r i c coefficients to be constant, w i t h h a = 1, h~ = r* ~ (R E + 1 3 0 k m ) . D e n o t i n g
Z = -~r -- 0". 2
f~x ~ L 2 h ~ dfl ~ Ex. 2
The line integrals are a p p r o x i m a t e d b y (9)
we t h e n h a v e _ (d/d~)[Xxe ~
de~/d~ = = E2s~]
(in/r*)ea,
--(in/r*)[E~e~ +
(10) Xle,] (11)
w h i c h are solved in 2 k m i n c r e m e n t s of ~. I n practice, we s t a r t w i t h an a r b i t r a r y v a l u e of s , a t 80 k m a n d find ea f r o m (7) ; we t h e n use (10) a n d
;~
~1,2h~ d~ =
2 fff s i n 1"
~1,~ dr
o
where r 0 = (R E + 8 0 k i n ) . T h e b o u n d a r y conditions are t h a t t h e longitudinal electric field and t h e m e r i d i o n a l c u r r e n t (E~s~* -- E2e** ) ~*be c o n t i n u o u s w i t h t h e values f o u n d f r o m t h e equatorial solution for t h e line of force passing t h r o u g h 130 k m a l t i t u d e at 10 ° l a t i t u d e (80 ° colatitude),
J. GAGNEPAIN,~VI. CROCHET and A. D. RICHMOND
282
i.e. t h e field line p e a k i n g a t 330 k m at t h e E q u a t o r . T h e electric field c o m p o n e n t %* is n o t e x a c t l y c o n t i n u o u s because of slight discontinuities in t h e c o n d u c t i v i t i e s ZL2. VERTICAL STRUCTURE OF ELECTRIC FIELDS AND CURRENTS
F i g u r e s 1 a n d 2 show t h e h e i g h t s t r u c t u r e s of t h e electric field, c o m p u t e d f r o m (7), (8), (10), a n d (11), a n d of t h e e a s t w a r d c u r r e n t density, c o m p u t e d f r o m (1), a t t h e E q u a t o r , for f o u r different l o n g b t u d i n a l phases of t h e sinusoidal v a r i a t i o n ( n ~ 0 - - - 4 5 °, 0 °, 45 °, and 90°) a n d for longitudinal w a v e n u m b e r s (wavelengths) of 0 (oo km), 1 (40,000 kin), 2 (20,000 kin), 4 (10,000 km), 8 (5000 km), 16 (2500 km), 32 (1250 kin), a n d 64 (625 km). N o t e t h a t t h e scale for E e in Fig. 1 is e x p a n d e d b y a factor of 10 o v e r t h a t for Ea. F o r l o n g i t u d i n a l phases of n~0 = 135 °, 180 °, 225 °, a n d
_@o
270 °, t h e v e r t i c a l structures of the electric field a n d c u r r e n t are identical to those shown in Figs. 1 a n d 2 for nW = 45 °, 0 °, 45 ° a n d 90 °, respectively, e x c e p t t h a t t h e signs are reversed. R e c a l l t h a t nW = 0 ° represents a p p r o x i m a t e l y t h e m a x i m u m of t h e h e i g h t - i n t e g r a t e d e a s t w a r d eleetrojet c u r r e n t density. T h e phase nq0 = --45 ° is w e s t w a r d of t h e m a x i m u m , a n d hence represents an eleetrojet c u r r e n t whose s t r e n g t h is increasing t o w a r d t h e east. Conversely, n~0 ~ ÷ 4 5 ° represents a c u r r e n t w h i c h is decreasing t o w a r d t h e east. T h e phase n ~ = 90 ° represents a t r a n s i t i o n f r o m an e a s t w a r d to a w e s t w a r d clectrojet. The ease n = 0 is identieM to RIOHSIOND'S (1973) % = CO m o d e l ; for longitudinal phases of --45 °, 45 °, a n d 90 °, we h a v e simply t a k e n the ~0 -- 0 ° solutions a n d multiplied t h e m b y cos( 45°), cos(45°), a n d cos(90°), respectively. (It should be n o t e d t h a t n -- 0 represents an u n p h y s i c a l s i t u a t i o n in t h e sense t h a t
,oo
,r
J o
16
~ :
_/o
1
4__
~ 6 4 /
/
~-
/
8
-
--
/'
/
,/ 100 16
0
1
2
I o,,I
l
0
/
!'
,,
,
o,t2_
I
. "
/" 32
i,l,: .'
0
/
/
.~/_
/ /
32
l
1 O0 L "~-~-
1
2
2
-1
O
" i; L__o
1
s._
2
_
0
t
/
1 .. ~
32
~
!
0
1
3oo /
!(~
0
1 2 E ~ ( l O - 2 v m -1 )
,oo ~
8
_
- 64
I
2
_
-1
64
0
1
-2
-I
E~( 10 - 3 v . r n - l l
Fig. 1. Height profiles of the E= (upper) and E~ (lower) components of ~he electric field for wavenumbcrs n = 0, 1, 2 . . . . . 64. For each value o f n, the profiles are drawn at four longitudes: from left to right, n~0 = --45 °, 0 °, 45 °, 90 °.
0
o
Theory of longitudinal gradients in the equatorial electroje~
283
200 !
"•
150
F-,
1 6 ~
8
4,2,1,0
32__ ~ 644 . ,'~ f / ",'1
"r
16 B ¢
100
64 32 16 8 4~2,1,o
64 32
16,84,2fl,o
50~_
0
5
10
I
0
!;
5 10 JJ/( 10 -6 arnp.m -2 |
15
_
5
10
-5
0
5
Fig. 2. I-teight profiles of the eastward current d e n s i t y J ~ for wavenumbers n = 0, 1 . . . . . 64. For each n value, the profiles are drawn a t four longitudes: from left to right, ng0 ~ --45 °, 0 °, 45 °, 90 °. t h e l o n g i t u d i n a l i n t e g r a l of E e a r o u n d t h e E a r t h is n o t zero, as r e q u i r e d b y t h e c o n d i t i o n V × E = 0). W h e n n is small, t h e electric field a n d c u r r e n t do n o t differ g r e a t l y f r o m t h e case n = 0. O u r r e s u l t s c o n f i r m I~ICHMONDS'S (1973) e s t i m a t e o f t h e influence of w e a k l o n g i t u d i n a l g r a d i e n t s o n t h e r a t i o .E=(lO2km)/E,p(350km): f e r n = l a n d ~ = ~=45 °, c o r r e s p o n d i n g t o L E = =~6490 k m i n R i c h m o n d ' s t r e a t m e n t , we find E=(102 k m ) / E e ( 3 5 0 kin) t o b e a b o u t 10~o s m a l l e r (~0 = --45 °) or 1 1 % l a r g e r (~ = 45 °) t h a n for n = 0. U p t o a b o u t n = 4, t h e d e p a r t u r e s of E=, E e , a n d J e f r o m t h e n = 0 s o l u t i o n s a r e a p p r o x i m a t e l y l i n e a r w i t h n. F i g u r e 1 s h o w s t h a t , for l a r g e n, t h e m a g n i t u d e of t h e electric field t e n d s t o g r o w r a p i d l y w i t h a l t i t u d e . C o n c e p t u a l l y , i t is p r o b a b l y m o r e u s e f u l t o c o n s i d e r t h i s t e n d e n c y as o n e of s t r o n g a t t e n u a tion with decreasing altitude. We have assumed i m p l i c i t l y t h a t t h e m e c h a n i s m r e s p o n s i b l e for g e n e r a t i n g t h e electric fields a n d c u r r e n t s is l o c a t e d o u t s i d e of t h e e q u a t o r i a l region, b e y o n d t h e field line w h i c h p e a k s a t 400 kin. T h e m o r e r a p i d t h e l o n g i t u d i n a l v a r i a t i o n s of t h e s e c u r r e n t s a n d fields, t h e m o r e difficult i t is for t h e m t o p e n e t r a t e d e e p l y i n t o t h e e q u a t o r i a l region. I n t h e n e x t s e c t i o n we e x t e n d o u r t r e a t m e n t of t h i s p h e n o m e n o n t o still l a r g e r v a l u e s o f ~. T h e i n t r o d u c t i o n of l o n g i t u d i n a l g r a d i e n t s c a u s e s c e r t a i n q u a l i t a t i v e differences o f t h e electric field a n d c u r r e n t profiles f r o m t h e n ~ 0 case. T h e e a s t w a r d electric field E ~ is n o l o n g e r c o n s t a n t w i t h a l t i t u d e , a n d r e v e r s a l s o f t h e electric field a n d c u r r e n t d i r e c t i o n s w i t h h e i g h t b e c o m e possible, For J~, the reversed current (with respect to the c u r r e n t d i r e c t i o n a t 102 k m ) occurs b d o ~ t h e m a i n
current layer when the eleetrojet strength increases t o w a r d t h e e a s t (e.g. n T = --45°), a n d i t occurs above t h e m a i n c u r r e n t l a y e r w h e n t h e e l e c t r o j e t s t r e n g t h d e c r e a s e s t o w a r d t h e e a s t (e.g. ng0 = 45°). T h e s e r e v e r s a l s are a c c o m p a n i e d b y a s l i g h t u p w a r d or d o w n w a r d d i s p l a c e m e n t of t h e m a i n c u r r e n t l a y e r itself. I n p r a c t i c e , i n o r d e r to o b s e r v e a r e v e r s a l of t h e electric field or c u r r e n t w i t h a l t i t u d e , it would probably be necessary that the longitudinal variation be very strong, corresponding to a w a v e l e n g t h less t h a n 2000 k m , a n d t h a t t h e obscrr a t i o n b e m a d e n e a r a n e a s t - t o - w e s t or w e s t - t o e a s t e l e c t r o j e t t r a n s i t i o n a t n ~ = -~90 °. F i g u r e 3 s h o w s h o w t h e l o n g i t u d i n a l p o s i t i o n s of t h e m a x i m u m (or m i n i m u m , or zero) v a l u e s of Ea a n d E e v a r y w i t h a l t i t u d e . I n t h e i o n o s p h e r i c _F-region, t h e m a x i m u m E~ occurs t o t h e e a s t of ~ = 0 ° ( t h e l o n g i t u d e of m a x i m u m e l e c t r o j e t c u r r e n t s ) , w h i l e t h e m a x i m u m E e occurs t o t h e w e s t of ~ ~ 0 °. T h e p o s i t i o n s o f t h e m a x i m a a r e n e a r l y i n d e p e n d e n t of n w h e n h i s small. F o r n < 4, E ~ i n t h e ~ ' - r e g i o n m a x i m i z e s (or m i n i m i z e s , or goes t h r o u g h zero) a b o u t 6 ° w e s t of, or a b o u t 24 r a i n earlier i n local t i m e t h a n , E= a n d J e a t 102 kin. T h i s p h a s e difference m a y b e of s o m e m i n o r i m p o r t a n c e w h e n a t t e m p t s are m a d e t o relate F-region vertical ionization drifts with E - r e g i o n h o r i z o n t a l e l e c t r o n drifts. TRANSMISSION OF ELECTRI0 FIELDS TO THE EQUATOR
We have seen in the previous section how electric fields g e n e r a t e d o u t s i d e of t h e e q u a t o r i a l region become increasingly attenuated within the e q u a t o r i a l r e g i o n as t h e l o n g i t u d i n a l v a r i a t i o n
284
J. GAGNEPAIN, M. CI~OCIIET and A. D. RICHMOND
400
400
I / /I
200
-5
~
0
64
5
-r Lu
10
1364~!
.40
15
{
-5
200
O
5
LONGITUDE [ degrees }
Fig. 3. Longitude of the maximum of the E= (left) and E~ (right) components of the electric field versus height for wavenurnbers n = 1, 2 . . . . , 64 (E= is chosen to be maximum at longitude ~0 = 0 at 102 krn).
becomes stronger. To illustrate the r a t e o f a t t e n u a tion a t higher latitudes, we h a v e p l o t t e d in Fig. 4, a s a f u n c t i o n of t h e l a t i t u d e 2 (90 ° -- 0"), t h e m a g n i t u d e o f t h e e l e c t r i c f i e l d , Mod(E)=(lea*12+[e¢*12)U2, where %* a n d %* are c o m p u t e d f r o m (14) a n d (15), being m a t c h e d to t h e e q u a t o r i a l solutions at 0* = 80 °. I t m a y be n o t e d t h a t t h e rates of a t t e n u a t i o n of IE=*I a n d of [e¢*[ t a k e n i n d i v i d u a l l y are a b o u t t h e s a m e as t h a t illustrated for Mod (E). To i n t e r p r e t Fig. 4, it is i m p o r t a n t to keep in m i n d two facts. First, in i n t e g r a t i n g (3) a n d (5) o u t to a g i v e n latitude, we h a v e i m p l i c i t l y a s s u m e d t h a t no sources of electric fields or currents are p r e s e n t at l a t i t u d e s below t h e l a t i t u d e in question. Second, all of our calculated values h a v e been a r b i t r a r i l y n o r m a l i z e d such t h a t E a ----20 m V . m -1 on t h e g e o m a g n e t i c field line p e a k i n g a t 102 k m at ~ = 0 °. A n y o t h e r n o r m a l ization for a n y p a r t i c u l a r v a l u e of n w o u l d cause t h e associated c u r v e in Fig. 4 to be shifted u p w a r d or d o w n w a r d on t h e l o g a r i t h m i c scale, Table 1 expresses in two different w a y s t h e efficiency w i t h which electric fields are t r a n s m i t t e d to t h e E q u a t o r . The 'transmission coefficient b e t w e e n 30 ° l a t i t u d e and t h e E q u a t o r ' is d e t e r m i n e d as follows. L e t us define the transmission coeffieient for t h e w a v e n u m b e r n = 0 as 1. T h e n t h e coefficient f o r a n y o t h e r w a v e n u m b e r is t h e r a t i o o f
lea(102 km)] for t h a t w a v e n u m b e r to [e=(102 km)] for n = 0 w h e n the values of Mod(E) for these two w a v e n u m b e r s are set e q u a l at t h e l a t i t u d e in
question (30°). I n o t h e r words, the transmission coefficient b e t w e e n 30 ° a n d t h e E q u a t o r is t h e ratio of t h e lYIod(E) values shown in Fig. 4 a t 30 ° l a t i t u d e for n = 0 a n d for t h e w a v e n u m b e r in question. I t is seen t h a t these coefficients r a p i d l y decrease w i t h increasing n so t h a t it becomes increasingly difficult to t r a n s m i t an electric field f r o m 30 ° to t h e electrojet for large w a v e n m n b e r s .
_
32
r i ~ i I 10 .7 [_
'E .~>
~'?' o :~ 10 .2
8
4
i'/ / / ,/ '
/ /
,
~
//
i t i~
/ / / / //
/
/
;_2 /
/ / /
//
/
/
/
/
/ , E i
103
16
0
/.~1
/
/
// / '
20
/
// / ~ • ~
40
,. .~_C_.O J
60
~
J
80
k[degrees) Fig. 4. Variations of the quantity Mod(E) = ([e=*l2 + [e¢*12)~/~withlatitudeforwavenumbersn = 0 . . . . . 32.
Theory of longitudinal gradients in the equatorial electrojet
285
Table 1. Attenuation parameters of ionospheric electric fields for wavenumbers n ~ 0 through 32 Transmission coefficient between 30 ° latitude and t h e Equator Limiting latitude for a 1 mV.m -1 source
0
1
2
4
8
16
32
1
0.93
0.73
0.33
0.073
0.0063
0.000068
630
50 °
38 °
27 °
18 °
12 °
T h e transmission coefficients for o t h e r l a t i t u d e s can be d e t e r m i n e d analogously, The 'limiting l a t i t u d e for a 1 m V . m -1 source' in T a b l e 1 is defined as t h e highest l a t i t u d e a t which a source (e.g., ionospheric winds) p r o d u c i n g an electric field of l~od(E) ~ 1 m V . m -1 can lie, a n d still cause a n Ea(102 km) a t t h e E q u a t o r of a t least 2 m V . m -1, or 10% of t h e t y p i c a l electrojet field, I n o t h e r words, this l a t i t u d e is t h a t at w h i c h a g i v e n c u r v e in Fig. 4 crosses t h e Mod(E) = 10 m V . m -1 level. A 1 m V . m -1 source lying a t a h i g h e r l a t i t u d e w o u l d produce a r e l a t i v e l y w e a k field a t t h e E q u a t o r , a n d wolfld h a v e a r e l a t i v e l y insignific a n t influence on t h e e l e c t r o j e t . T a b l e 1 shows t h a t t h e higher t h e w a v e n u m b e r of the source, t h e closer it m u s t lie to t h e E q u a t o r in order to h a v e a significant influence on t h e electrojet. A source m u c h s t r o n g e r t h a n 1 m V . m -1 is unlikely to occur in t h e ionosphere unless it is associated w i t h highl a t i t u d e m a g n e t o s p h e r e - i o n o s p h e r e interactions, which m a y t y p i c a l l y p r o d u c e a field of 50 m V . m -1. R e f e r e n c e to Fig. 4 shows t h a t only w a v e n u m b e r s less t h a n or e q u a l to 2 can produce significant effects a t t h e E q u a t o r for a 50 m V . m -1 source located a b o v e 60 ° latitude. T h e e q u a t o r i a l electric field due to h i g h - l a t i t u d e sources is therefore of necessity v e r y b r o a d in longitudinal e x t e n t , a n d can h a v e small-scale features o n l y where these are c r e a t e d locally b y c o n d u c t i v i t y inhomogeneities. DISCUSSION AND CONCLUSIONS T h e purpose of this w o r k has b e e n to p o i n t o u t t h e close r e l a t i o n b e t w e e n longitudinal g r a d i e n t s a n d h e i g h t - l a t i t u d e v a r i a t i o n s of e q u a t o r i a l electric fields a n d currents. I n addition, b y e x t e n d i n g our calculations to higher latitudes, we h a v e shown q u a n t i t a t i v e l y h o w electric fields d e c a y a w a y f r o m t h e i r source in t h e ionosphere in general, a n d specifically h o w t h e y are a t t e n u a t e d b e t w e e n t h e i r source a n d t h e E q u s t o r . Because wc h a v e ignored longitudinal v a r i a t i o n s o f c o n d u c t i v i t y , o t h e r t h a n to p o i n t to t h e m as p o t e n t i a l sources of electric field gradients, our results b e c o m e i n v a l i d w h e n t h e c o n d u c t i v i t y
g r a d i e n t in l o n g i t u d e is c o m p a r a b l e to or g r e a t e r t h a n t h e electric field gradient. I n particular, our results c a n n o t be applied to sunrise or sunset p h e n o m e n a . On t h e o t h e r hand, t h e f a c t t h a t we h a v e chosen n o o n t i m e c o n d u c t i v i t y conditions does n o t render our electric field results useless for o t h e r t i m e s of t h e day. N o t e t h a t t h e solution of (5) does n o t d e p e n d on t h e absolute values of t h e conductivities, b u t only on t h e i r r e l a t i v e v a r i a t i o n s w i t h a l t i t u d e a n d latitude. Since these r e l a t i v e v a r i a t i o n s p r o b a b l y do n o t change n e a r l y as m u c h in local t i m e as do the absolute values, our electric field results should be r e a s o n a b l y applicable a t n i g h t as well as during t h e day. I n addition, longitudinal v a r i a t i o n s of t h e c o n d u c t i v i t y due to v a r i a t i o n s of the m a g n e t i c field s t r e n g t h should n o t s t r o n g l y affect our results. The p r i m a r y effect of changing t h e c o n d u c t i v i t y will be to change t h e ratio of current i n t e n s i t y to electric field strength. Our results show t h a t alterations of t h e v e r t i c a l structures of t h e electric fields a n d currents f r o m t h e infinite-electrojet (n = 0) m o d e l are r e l a t i v e l y small for large-scale longitudinal g r a d i e n t s of t h e electrojet. T h e alterations m a y b e c o m e quite i m p o r t a n t , however, for small-scale gradients, such as these s o m e t i m e s observed b y r a d a r in S o u t h A m e r i c a (BALsLEY, 1970) a n d in Africa (results to be published). T h e sources of longitudinal v a r i a t i o n s in t h e electrojet m a y be c o n d u c t i v i t y gradients, Jonespheric winds, or lfigh-latitude m a g n e t o s p h e r e ionosphere interactions. Small-scale electrojet v a r i a t i o n s can be p r o d u c e d o n l y b y sources close to t h e E q u a t o r , since t h e electric fields of small-scale h i g h e r - l a t i t u d e sources are strongly a t t e n u a t e d b e t w e e n t h e source a n d t h e E q u a t o r .
Acknowledgements--Thiswork was supported partially by NATO through a fellowship to one of us (A. Richmend) and through the Grant No. 684 for multinational cooperation. Computers of I.P.G.P. and I.N.A.G. were used for numerical analysis while I.P.G.P., C.N.R.S. and I.N.A.G. supported the related experiments in Africa.
ft. GAGNEPAIN, M. CROCHET and A. D. RICHMOND
286
REFERENCES
BALSLEY B. GOUIN P. a n d MAYAtrD P . N . Y~YLOV A. L., SOBOImV~_T. N., FISHCHUK D . L . ,
1970 1967 1973
J. geophys. Re*. 75, 4291. Annla. Ggophys. 28, 41. Geomagn. Aeron. 13, 400.
1973 1969 1967
J. atmoa, terr. Phys. 36, 1083. J. geophys. Re*. 74, 4025. d. geophys. Re*. 72, 5799.
TSEDILINA YE. and SHCIIERBA2~OV V. P.
RICHMONI) A . D . SUGIURA M. a n d PoRes D . J . I~lq'TIEDT J.
Reference is also made to the following unpublished material: RICHMOND A . D .
1972
A. D. R i c h m o n d is grateful to Dr, A. R. J x I ~ for kindly pointing out a sign error in ~he paper of RIC~MeND (1973). The corrected forms of equations (11) a n d (12) of t h a t paper should read:
H~(~,~) =
,1 , [8 d~ (c¢,fl') h9 (c¢,fl') h~(cc,fl') dfl', (11)
Air Force Cambridge Research Laboratories Rept. 72-0668.
a-~ Lh¢Lhq~(~o
(h~Hg)
[
] + ~ " h ~ ~i a~ (hgH~)] ] [( l
O h~ ~ E~ = 0-~
O -- ~
~ v~/B0
h~ v~0 +
. -J (12) The only result of RICHMOI~D (1973) affected by this error is the sign of H~ shown in his Fig. 5. The directions of current flow stated in the figure caption are correct. ol
/
Theory o~ longitudinal gradients in the equatorial electrojet J. GAGI,IEPAII,~ a n d M. C R O C H E T
Institut de Physique du Globe, Universit6 Pierre et Marie Curie, 75230 Paris Cedex 05, France and A. I). RIC~OZ~D* Laboratoire de Physique de l'Exosphbrc, Universitd Pierre et Marie Curie, 75230 Paris Cedex 05, France (Received 1 M a y 1975; in revised form 8 J u l y 1975)
Abstract--Longitudinal gradients in the equatorial electrojet are observed with scales ranging from several thousands of kilometers on down. Since previous theoretical models of the electrojet electric fields and currents have assumed that no longitudinal gradients exist, we examine theoretically the consequences of allowing for them. The vertical structures of the electric fields and currents are altered in predictable ways, the alterations being increasingly important as the electric field and current gradients increase. In contrast to the infinite-eleetrojet model, reversals of the electric field and current directions with height become possible when longitudinal gradients are present. We show how electric fields produced by hypothetical generating sources located at various latitudes are attenuated between the source and the Equator. For electric fields with small longitudinal scale lengths, the source must lie close to the magnetic equator in order to cause noticeable effects in the electrojet.
INTRODUCTION The e q u a t o r i a l electrojet is generally i m a g i n e d to be a r i b b o n of current a few h u n d r e d kilometers wide, flowing along t h e E a r t h ' s m a g n e t i c e q u a t o r a t a n a l t i t u d e of a b o u t 100 kin. Theoretical models w h i c h h a v e been d e v e l o p e d up to n o w (e.g. UNTIEDT, 1967; SU•IURA a n d PORES, 1969; KI~YLOV et al., 1973; RICHMOND, 1973) h a v e a s s u m e d for simplicity t h a t t h e longitudinal e x t e n t of t h e electrojet is so great in c o m p a r i s o n w i t h its w i d t h , t h a t longitudinal g r a d i e n t s of t h e electric field, c o n d u c t i v i t y a n d c u r r e n t can be c o m p l e t e l y n e g l e c t e d w h e n only local features are being considered. These models h a v e b e e n fairly successful in r e p r o d u c i n g several o b s e r v e d features of t h e eleetrojet, a n d t h e y p r o v i d e a useful s t r u c t u r e for organizing observations a n d for r e l a t i n g v a r i o u s physical p h e n o m e n a in t h e e q u a t o r i a l ionosphere, L o n g i t u d i n a l g r a d i e n t s of t h e eleetrojet, however, n e e d n o t always be so insignificant as these models assume. To be sure, t h e giobal-scale g r a d i e n t s associated w i t h t h e diurnal changes of t h e conduct i v i t y a n d t h e electric field were e s t i m a t e d b y I~ICHMOND (1973) tO cause errors o n l y on t h e order of 10~o in t h e features of his model. T h e r e exist variations of smaller scale, however, w h i c h can be * Present address: High Altitude Boulder, Colorado 80303, U.S.A.
Observatory,
e x p e c t e d to cause errors of correspondingly greater m a g n i t u d e . F o r example, t h e ' c o u n t e r electrojet', a d a y t i m e reversal in t h e direction of current flow, is u s u a l l y o b s e r v e d to be l i m i t e d in longitudinal e x t e n t to a few tens of degrees (Gouz~ a n d MAYAUD, 1967). On a n e v e n smaller scale, r a d a r measurem e n t s of t h e electron speed w i t h i n t h e electrojet h a v e r e v e a l e d significant longitudinal v a r i a t i o n s o v e r a distance of a few h u n d r e d kilometers (BALsLEY, 1970). T h e r e also exist v e r y i m p o r t a n t v a r i a t i o n s of t h e electric field a t sunrise a n d sunset ( B ~ s L E Y , 1970). T h e p h y s i c a l sources of these smaller scale v a r i a t i o n s in t h e electric field a n d cLurrent of t h e electrojet can be classified, for t h e purposes of our present discussion, as being either 'local' or ' r e m o t e ' . One local source w o u l d be t h e presence of eonduct i v i t y v a r i a t i o n s w i t h longitude, such as a t sunrise a n d sunset, which cause a disruption of the largescale current flow a n d t h e creation of smaller-scale electric fields. A second local source w o u l d be t h e presence of l o n g i t u d i n a l l y v a r y i n g local n e u t r a l winds, which can drive electric currents b y action of b o t h t h e d y n a m o electric field, V × B (where V is t h e w i n d v e l o c i t y a n d B t h e g e o m a g n e t i c field), a n d t h e electrostatic field, E, w h i c h in general exists in association w i t h t h e d y n a m o field due to a c c u m u l a t i o n of space charge, l~emote sources, t h a t is, sources existing outside of t h e e q u a t o r i a l 279
280
J. GAG~EPAI~r,M. CROCHET and A. D. RICHMOND
region itself, can affect the electrojet only b y means of the electric field they create, which can be transferred from the source region through the conducting ionosphere to the electrojet region in association with current flow between the source a n d eleetrojet regions. Possible remote sources are (1) ionospheric winds existing outside the equatorial region, (2) solar wind-magnetosphere interactions, which transfer electric fields to the ionosphere at high latitudes, and (3) conductivityinhomogeneities at non-equatorial latitudes. The present study has two goals. First, we wish to estabhsh quantitatively how the presence of longitudinal gradients in the electrojet electric field and current affect the vertical and latitudinal structure of eleetrojet features. Second, we wish to examine the 'transmissivity' of the ionosphere to electric fields produced by remote sources; i.e. we wish to be able to tell how remote from the E q u a t o r a source can be and still have a significant influence on the electrojet. We shall not examine how the electric fields are actually produced b y hypothesized sources. Thus our mathematical t r e a t m e n t neglects a n y longitudinal variation of the ionospheric conductivity, and also neglects the presence of winds within the latitudinal region under examination. For this reason, our results cannot be used to examine sunrise-sunset effects, nor can they be applied to effects due to winds existing within the electrojet region itself. I n spite of these limitations, we believe t h a t our results m a y be applicable to m a n y types of longitudinal variations in the electrojet, MATHEMATICAL DEVELOPMENT Following RICHMOND (1973), we assume the geomagnetic field to be dipolar, and use the dipolar coordinates a, ~, fl defined b y = r/sin ~ 0, = ~0, fl
=
cos
O/r~
= si.~ o/(~ + 3
h e = r sin
O,
h~
-I-
=
r8/(1
cos ~
=
e~((~zE ~
--
asE~)
+ e~(%E~ + ~IE~) - eB%E p,
(1)
1 FO(h~,hpJc,)+ -O(hph=J~) h~h~h# L ~ Oq~ ~(hahej#).~
V. ~
-F ~-fl V × E =
| J
= 0,
(2)
a(h~E~)I"~ ~-# .~
e~ ~ ~l.a(,~,~#) ~
h~h# L °T -F - e~ -
vO(haEa) a(hflE#)~ h#haL" ~-fl ~c¢ J
-F e# F~(h~E~) ~(h~E~)~ l~hq,L. ~ -~ J =
0,
(3)
where J is the current density, ~0, ~1, °2 are the parallel, Pedersen, Hall eonductivities, and ea, e~, e# are u n i t vectors. We follow a suggestion b y KRYr,OVet al. (1973) a n d RICHMOND (1973), who showed t h a t a reasonable solution for the electric fields a n d currents can be obtained b y assuming ~0 to be infinite, so that E~ = 0. (4) Then the a a n d ~ components of (3) tell us that (h~Ea) and (h~Ee) are constant along a magnetic line of force, i.e. these quantities are functions of only ~ a n d ~. The fl-component of (3) provides us with one relation between (h~Ea) and (h~E~). The second equation relating these two quantities is obtained b y using (1) in (2) a n d b y intregrating hahq~hpV . ~ with respect to fl along a line of force from one end, at the southern base of the conducting ionosphere, to the other end, a t the northern base. Nothing t h a t 3"# = 0 at each end, we then have :
0 = ZF ( fP'h_..T.~a~hp dfl ) (haEa)
where r, 0, ~ are the usual spherical coordinates. The associated metric coefficients are:
4
instabilities, we have the following set of equations to solve:
a=L\J~, ha
-- ||//*P~~h#r dfl
~#,
0)'~,
a F / I@'~
-F - - I I I
a~L\J#,
3 COS2 0) 1/~. -I-
Neglecting ionospheric winds and possible plasma
(bee ~) \
azhp dfl) (haEa)
( f flzha~ 61h~dfl)(hq)Eq~). \,]p~ h~
(5)
Theory of longitudinal gradients in the equatorial eleetrojet W e eliminate t h e ~ - d e r i v a t i v e s in (3) a n d (5) b y a s s u m i n g t h a t E~ a n d E~ v a r y in l o n g i t u d e as Ea, ~
=
Re
(ea.~ein~)
(6)
where R e denotes t h e real p a r t , i i s , v / - - ~ a n d n is the longitudinal wavenumber; t h e n a]a~ is r e p l a c e d b y in. I t is a s s u m e d t h a t t h e c o n d u c t i v i t ies are c o n s t a n t w i t h longitude. T w o b o u n d a r y conditions are required, as we n o w h a v e t w o firstorder o r d i n a r y differential e q u a t i o n s for (haea) a n d (h~e~) w i t h respect to ~. One c o n d i t i o n is t h a t t h e v e r t i c a l c u r r e n t d e n s i t y v a n i s h a t t h e base of t h e c o n d u c t i n g region (which we t a k e t o be at an a l t i t u d e of 80 kin). S e t t i n g J a = 0 a t 80 kin a t t h e E q u a t o r gives ale a -- ~2e~ = 0
at
fl = 0, ct =
(R E +
80km)
(7) w h c r o R E i S t h e radius of t h e E a r t h . F o r t h e second condition, we choose: e~ = 2 O m V . m
-1
at
fi = 0 , c ¢ = ( R E + 102 k m ) .
281
(11) to step u p w a r d to find s~ a n d % at each sueeessive l e v e l ; finally we n o r m a l i z e our results to t h e c o n d i t i o n (8). The c o n d u c t i v i t i e s al, o 2 a n d t h e l i n e - i n t e g r a t e d c o n d u c t i v i t i e s (9) are o b t a i n e d f r o m I:~ICB-~OND'S (1972, 1973) m o d e l w i t h t h e m a g n e t i c field s t r e n g t h B 0 = 3.21 × 10 -5 T, t h e solar flux Fxo. 7 = 90 × 10 -22 W . m -~ . Hz-~ a n d t h e zenith angle X = 0 °. Outside t h e e q u a t o r i a l region, we define a new v a r i a b l e 0* as t h e c o l a t i t u d e w h e r e a g i v e n ]ine of force intersects t h e 130 k m a l t i t u d e level, i.e. 0* ---- sin -1
{r*/~) 1/2.
I n t h e following we d e n o t e w i t h a star t h e v a l u e of a q u a n t i t y e v a l u a t e d a t t h e coordinates r = r*, 0 = 0". A s s u m i n g t h a t
;
~2(h*/hct)±1~lha dfl =(h~*/ha*)± I
(8)
This choice has b e e n m a d e in order a p p r o x i m a t e l y to normalize all of our c o m p u t a t i o n s such t h a t t h e m a x i m u m h e i g h t - i n t e g r a t e d eleetrojet c u r r e n t d e n s i t y (and hence, a p p r o x i m a t e l y , t h e m a x i m u m g r o u n d - l e v e l m a g n e t i c effect) occurs a t ~ = 0 a n d has t h e s a m e v a l u e for a n y n. As t h e m a x i m u m e a s t - w e s t electrojet c u r r e n t d e n s i t y occurs approxi m a t e l y a t 102 k m a n d is a l m o s t e n t i r e l y d e p e n d e n t on E a (rather t h a n Ee) a t this altitude, t h e normali z a t i o n expressed b y (8) t u r n s o u t to be a good a p p r o x i m a t i o n to a n o r m a l i z a t i o n in t e r m s of g r o u n d - l e v e l m a g n e t i c effect, T e solve (3) a n d (5) numerically, we use a p p r o x i m a t e forms of these equations. I n t h e e q u a t o r i a l region, defined as
(12)
alh~dfl
(13)
d~ a n d l e t t i n g I be t h e inclination of t h e geomag,letie field below t h e horizontal, we o b t a i n t h e following e q u a t i o n s f r o m (3) a n d (5): d [sin 0*ee*] = --in sin l*¢a*, dO* d [sin O*(Exe * dO* -- Zse~*)] =
in
sin I*[Z2e * + Z x e * ]
(14)
(15)
w h i c h are solved in 1° i n c r e m e n t s of 0* b e t w e e n 80 ° a n d 20 ° colatitudo. T h e c e n d u c t i v i t i e s are comp u t e d b y lotting B a n d X v a r y w i t h c o l a t i t u d e as B = B0[1 + 3 c o s 2 0"] 1/2,
(R E + 8 0 k i n ) < ~ < (R E + 4 0 0 k i n ) , we assume t h e m e t r i c coefficients to be constant, w i t h h a = 1, h~ = r* ~ (R E + 1 3 0 k m ) . D e n o t i n g
Z = -~r -- 0". 2
f~x ~ L 2 h ~ dfl ~ Ex. 2
The line integrals are a p p r o x i m a t e d b y (9)
we t h e n h a v e _ (d/d~)[Xxe ~
de~/d~ = = E2s~]
(in/r*)ea,
--(in/r*)[E~e~ +
(10) Xle,] (11)
w h i c h are solved in 2 k m i n c r e m e n t s of ~. I n practice, we s t a r t w i t h an a r b i t r a r y v a l u e of s , a t 80 k m a n d find ea f r o m (7) ; we t h e n use (10) a n d
;~
~1,2h~ d~ =
2 fff s i n 1"
~1,~ dr
o
where r 0 = (R E + 8 0 k i n ) . T h e b o u n d a r y conditions are t h a t t h e longitudinal electric field and t h e m e r i d i o n a l c u r r e n t (E~s~* -- E2e** ) ~*be c o n t i n u o u s w i t h t h e values f o u n d f r o m t h e equatorial solution for t h e line of force passing t h r o u g h 130 k m a l t i t u d e at 10 ° l a t i t u d e (80 ° colatitude),
J. GAGNEPAIN,~VI. CROCHET and A. D. RICHMOND
282
i.e. t h e field line p e a k i n g a t 330 k m at t h e E q u a t o r . T h e electric field c o m p o n e n t %* is n o t e x a c t l y c o n t i n u o u s because of slight discontinuities in t h e c o n d u c t i v i t i e s ZL2. VERTICAL STRUCTURE OF ELECTRIC FIELDS AND CURRENTS
F i g u r e s 1 a n d 2 show t h e h e i g h t s t r u c t u r e s of t h e electric field, c o m p u t e d f r o m (7), (8), (10), a n d (11), a n d of t h e e a s t w a r d c u r r e n t density, c o m p u t e d f r o m (1), a t t h e E q u a t o r , for f o u r different l o n g b t u d i n a l phases of t h e sinusoidal v a r i a t i o n ( n ~ 0 - - - 4 5 °, 0 °, 45 °, and 90°) a n d for longitudinal w a v e n u m b e r s (wavelengths) of 0 (oo km), 1 (40,000 kin), 2 (20,000 kin), 4 (10,000 km), 8 (5000 km), 16 (2500 km), 32 (1250 kin), a n d 64 (625 km). N o t e t h a t t h e scale for E e in Fig. 1 is e x p a n d e d b y a factor of 10 o v e r t h a t for Ea. F o r l o n g i t u d i n a l phases of n~0 = 135 °, 180 °, 225 °, a n d
_@o
270 °, t h e v e r t i c a l structures of the electric field a n d c u r r e n t are identical to those shown in Figs. 1 a n d 2 for nW = 45 °, 0 °, 45 ° a n d 90 °, respectively, e x c e p t t h a t t h e signs are reversed. R e c a l l t h a t nW = 0 ° represents a p p r o x i m a t e l y t h e m a x i m u m of t h e h e i g h t - i n t e g r a t e d e a s t w a r d eleetrojet c u r r e n t density. T h e phase nq0 = --45 ° is w e s t w a r d of t h e m a x i m u m , a n d hence represents an eleetrojet c u r r e n t whose s t r e n g t h is increasing t o w a r d t h e east. Conversely, n~0 ~ ÷ 4 5 ° represents a c u r r e n t w h i c h is decreasing t o w a r d t h e east. T h e phase n ~ = 90 ° represents a t r a n s i t i o n f r o m an e a s t w a r d to a w e s t w a r d clectrojet. The ease n = 0 is identieM to RIOHSIOND'S (1973) % = CO m o d e l ; for longitudinal phases of --45 °, 45 °, a n d 90 °, we h a v e simply t a k e n the ~0 -- 0 ° solutions a n d multiplied t h e m b y cos( 45°), cos(45°), a n d cos(90°), respectively. (It should be n o t e d t h a t n -- 0 represents an u n p h y s i c a l s i t u a t i o n in t h e sense t h a t
,oo
,r
J o
16
~ :
_/o
1
4__
~ 6 4 /
/
~-
/
8
-
--
/'
/
,/ 100 16
0
1
2
I o,,I
l
0
/
!'
,,
,
o,t2_
I
. "
/" 32
i,l,: .'
0
/
/
.~/_
/ /
32
l
1 O0 L "~-~-
1
2
2
-1
O
" i; L__o
1
s._
2
_
0
t
/
1 .. ~
32
~
!
0
1
3oo /
!(~
0
1 2 E ~ ( l O - 2 v m -1 )
,oo ~
8
_
- 64
I
2
_
-1
64
0
1
-2
-I
E~( 10 - 3 v . r n - l l
Fig. 1. Height profiles of the E= (upper) and E~ (lower) components of ~he electric field for wavenumbcrs n = 0, 1, 2 . . . . . 64. For each value o f n, the profiles are drawn at four longitudes: from left to right, n~0 = --45 °, 0 °, 45 °, 90 °.
0
o
Theory of longitudinal gradients in the equatorial electroje~
283
200 !
"•
150
F-,
1 6 ~
8
4,2,1,0
32__ ~ 644 . ,'~ f / ",'1
"r
16 B ¢
100
64 32 16 8 4~2,1,o
64 32
16,84,2fl,o
50~_
0
5
10
I
0
!;
5 10 JJ/( 10 -6 arnp.m -2 |
15
_
5
10
-5
0
5
Fig. 2. I-teight profiles of the eastward current d e n s i t y J ~ for wavenumbers n = 0, 1 . . . . . 64. For each n value, the profiles are drawn a t four longitudes: from left to right, ng0 ~ --45 °, 0 °, 45 °, 90 °. t h e l o n g i t u d i n a l i n t e g r a l of E e a r o u n d t h e E a r t h is n o t zero, as r e q u i r e d b y t h e c o n d i t i o n V × E = 0). W h e n n is small, t h e electric field a n d c u r r e n t do n o t differ g r e a t l y f r o m t h e case n = 0. O u r r e s u l t s c o n f i r m I~ICHMONDS'S (1973) e s t i m a t e o f t h e influence of w e a k l o n g i t u d i n a l g r a d i e n t s o n t h e r a t i o .E=(lO2km)/E,p(350km): f e r n = l a n d ~ = ~=45 °, c o r r e s p o n d i n g t o L E = =~6490 k m i n R i c h m o n d ' s t r e a t m e n t , we find E=(102 k m ) / E e ( 3 5 0 kin) t o b e a b o u t 10~o s m a l l e r (~0 = --45 °) or 1 1 % l a r g e r (~ = 45 °) t h a n for n = 0. U p t o a b o u t n = 4, t h e d e p a r t u r e s of E=, E e , a n d J e f r o m t h e n = 0 s o l u t i o n s a r e a p p r o x i m a t e l y l i n e a r w i t h n. F i g u r e 1 s h o w s t h a t , for l a r g e n, t h e m a g n i t u d e of t h e electric field t e n d s t o g r o w r a p i d l y w i t h a l t i t u d e . C o n c e p t u a l l y , i t is p r o b a b l y m o r e u s e f u l t o c o n s i d e r t h i s t e n d e n c y as o n e of s t r o n g a t t e n u a tion with decreasing altitude. We have assumed i m p l i c i t l y t h a t t h e m e c h a n i s m r e s p o n s i b l e for g e n e r a t i n g t h e electric fields a n d c u r r e n t s is l o c a t e d o u t s i d e of t h e e q u a t o r i a l region, b e y o n d t h e field line w h i c h p e a k s a t 400 kin. T h e m o r e r a p i d t h e l o n g i t u d i n a l v a r i a t i o n s of t h e s e c u r r e n t s a n d fields, t h e m o r e difficult i t is for t h e m t o p e n e t r a t e d e e p l y i n t o t h e e q u a t o r i a l region. I n t h e n e x t s e c t i o n we e x t e n d o u r t r e a t m e n t of t h i s p h e n o m e n o n t o still l a r g e r v a l u e s o f ~. T h e i n t r o d u c t i o n of l o n g i t u d i n a l g r a d i e n t s c a u s e s c e r t a i n q u a l i t a t i v e differences o f t h e electric field a n d c u r r e n t profiles f r o m t h e n ~ 0 case. T h e e a s t w a r d electric field E ~ is n o l o n g e r c o n s t a n t w i t h a l t i t u d e , a n d r e v e r s a l s o f t h e electric field a n d c u r r e n t d i r e c t i o n s w i t h h e i g h t b e c o m e possible, For J~, the reversed current (with respect to the c u r r e n t d i r e c t i o n a t 102 k m ) occurs b d o ~ t h e m a i n
current layer when the eleetrojet strength increases t o w a r d t h e e a s t (e.g. n T = --45°), a n d i t occurs above t h e m a i n c u r r e n t l a y e r w h e n t h e e l e c t r o j e t s t r e n g t h d e c r e a s e s t o w a r d t h e e a s t (e.g. ng0 = 45°). T h e s e r e v e r s a l s are a c c o m p a n i e d b y a s l i g h t u p w a r d or d o w n w a r d d i s p l a c e m e n t of t h e m a i n c u r r e n t l a y e r itself. I n p r a c t i c e , i n o r d e r to o b s e r v e a r e v e r s a l of t h e electric field or c u r r e n t w i t h a l t i t u d e , it would probably be necessary that the longitudinal variation be very strong, corresponding to a w a v e l e n g t h less t h a n 2000 k m , a n d t h a t t h e obscrr a t i o n b e m a d e n e a r a n e a s t - t o - w e s t or w e s t - t o e a s t e l e c t r o j e t t r a n s i t i o n a t n ~ = -~90 °. F i g u r e 3 s h o w s h o w t h e l o n g i t u d i n a l p o s i t i o n s of t h e m a x i m u m (or m i n i m u m , or zero) v a l u e s of Ea a n d E e v a r y w i t h a l t i t u d e . I n t h e i o n o s p h e r i c _F-region, t h e m a x i m u m E~ occurs t o t h e e a s t of ~ = 0 ° ( t h e l o n g i t u d e of m a x i m u m e l e c t r o j e t c u r r e n t s ) , w h i l e t h e m a x i m u m E e occurs t o t h e w e s t of ~ ~ 0 °. T h e p o s i t i o n s o f t h e m a x i m a a r e n e a r l y i n d e p e n d e n t of n w h e n h i s small. F o r n < 4, E ~ i n t h e ~ ' - r e g i o n m a x i m i z e s (or m i n i m i z e s , or goes t h r o u g h zero) a b o u t 6 ° w e s t of, or a b o u t 24 r a i n earlier i n local t i m e t h a n , E= a n d J e a t 102 kin. T h i s p h a s e difference m a y b e of s o m e m i n o r i m p o r t a n c e w h e n a t t e m p t s are m a d e t o relate F-region vertical ionization drifts with E - r e g i o n h o r i z o n t a l e l e c t r o n drifts. TRANSMISSION OF ELECTRI0 FIELDS TO THE EQUATOR
We have seen in the previous section how electric fields g e n e r a t e d o u t s i d e of t h e e q u a t o r i a l region become increasingly attenuated within the e q u a t o r i a l r e g i o n as t h e l o n g i t u d i n a l v a r i a t i o n
284
J. GAGNEPAIN, M. CI~OCIIET and A. D. RICHMOND
400
400
I / /I
200
-5
~
0
64
5
-r Lu
10
1364~!
.40
15
{
-5
200
O
5
LONGITUDE [ degrees }
Fig. 3. Longitude of the maximum of the E= (left) and E~ (right) components of the electric field versus height for wavenurnbers n = 1, 2 . . . . , 64 (E= is chosen to be maximum at longitude ~0 = 0 at 102 krn).
becomes stronger. To illustrate the r a t e o f a t t e n u a tion a t higher latitudes, we h a v e p l o t t e d in Fig. 4, a s a f u n c t i o n of t h e l a t i t u d e 2 (90 ° -- 0"), t h e m a g n i t u d e o f t h e e l e c t r i c f i e l d , Mod(E)=(lea*12+[e¢*12)U2, where %* a n d %* are c o m p u t e d f r o m (14) a n d (15), being m a t c h e d to t h e e q u a t o r i a l solutions at 0* = 80 °. I t m a y be n o t e d t h a t t h e rates of a t t e n u a t i o n of IE=*I a n d of [e¢*[ t a k e n i n d i v i d u a l l y are a b o u t t h e s a m e as t h a t illustrated for Mod (E). To i n t e r p r e t Fig. 4, it is i m p o r t a n t to keep in m i n d two facts. First, in i n t e g r a t i n g (3) a n d (5) o u t to a g i v e n latitude, we h a v e i m p l i c i t l y a s s u m e d t h a t no sources of electric fields or currents are p r e s e n t at l a t i t u d e s below t h e l a t i t u d e in question. Second, all of our calculated values h a v e been a r b i t r a r i l y n o r m a l i z e d such t h a t E a ----20 m V . m -1 on t h e g e o m a g n e t i c field line p e a k i n g a t 102 k m at ~ = 0 °. A n y o t h e r n o r m a l ization for a n y p a r t i c u l a r v a l u e of n w o u l d cause t h e associated c u r v e in Fig. 4 to be shifted u p w a r d or d o w n w a r d on t h e l o g a r i t h m i c scale, Table 1 expresses in two different w a y s t h e efficiency w i t h which electric fields are t r a n s m i t t e d to t h e E q u a t o r . The 'transmission coefficient b e t w e e n 30 ° l a t i t u d e and t h e E q u a t o r ' is d e t e r m i n e d as follows. L e t us define the transmission coeffieient for t h e w a v e n u m b e r n = 0 as 1. T h e n t h e coefficient f o r a n y o t h e r w a v e n u m b e r is t h e r a t i o o f
lea(102 km)] for t h a t w a v e n u m b e r to [e=(102 km)] for n = 0 w h e n the values of Mod(E) for these two w a v e n u m b e r s are set e q u a l at t h e l a t i t u d e in
question (30°). I n o t h e r words, the transmission coefficient b e t w e e n 30 ° a n d t h e E q u a t o r is t h e ratio of t h e lYIod(E) values shown in Fig. 4 a t 30 ° l a t i t u d e for n = 0 a n d for t h e w a v e n u m b e r in question. I t is seen t h a t these coefficients r a p i d l y decrease w i t h increasing n so t h a t it becomes increasingly difficult to t r a n s m i t an electric field f r o m 30 ° to t h e electrojet for large w a v e n m n b e r s .
_
32
r i ~ i I 10 .7 [_
'E .~>
~'?' o :~ 10 .2
8
4
i'/ / / ,/ '
/ /
,
~
//
i t i~
/ / / / //
/
/
;_2 /
/ / /
//
/
/
/
/
/ , E i
103
16
0
/.~1
/
/
// / '
20
/
// / ~ • ~
40
,. .~_C_.O J
60
~
J
80
k[degrees) Fig. 4. Variations of the quantity Mod(E) = ([e=*l2 + [e¢*12)~/~withlatitudeforwavenumbersn = 0 . . . . . 32.
Theory of longitudinal gradients in the equatorial electrojet
285
Table 1. Attenuation parameters of ionospheric electric fields for wavenumbers n ~ 0 through 32 Transmission coefficient between 30 ° latitude and t h e Equator Limiting latitude for a 1 mV.m -1 source
0
1
2
4
8
16
32
1
0.93
0.73
0.33
0.073
0.0063
0.000068
630
50 °
38 °
27 °
18 °
12 °
T h e transmission coefficients for o t h e r l a t i t u d e s can be d e t e r m i n e d analogously, The 'limiting l a t i t u d e for a 1 m V . m -1 source' in T a b l e 1 is defined as t h e highest l a t i t u d e a t which a source (e.g., ionospheric winds) p r o d u c i n g an electric field of l~od(E) ~ 1 m V . m -1 can lie, a n d still cause a n Ea(102 km) a t t h e E q u a t o r of a t least 2 m V . m -1, or 10% of t h e t y p i c a l electrojet field, I n o t h e r words, this l a t i t u d e is t h a t at w h i c h a g i v e n c u r v e in Fig. 4 crosses t h e Mod(E) = 10 m V . m -1 level. A 1 m V . m -1 source lying a t a h i g h e r l a t i t u d e w o u l d produce a r e l a t i v e l y w e a k field a t t h e E q u a t o r , a n d wolfld h a v e a r e l a t i v e l y insignific a n t influence on t h e e l e c t r o j e t . T a b l e 1 shows t h a t t h e higher t h e w a v e n u m b e r of the source, t h e closer it m u s t lie to t h e E q u a t o r in order to h a v e a significant influence on t h e electrojet. A source m u c h s t r o n g e r t h a n 1 m V . m -1 is unlikely to occur in t h e ionosphere unless it is associated w i t h highl a t i t u d e m a g n e t o s p h e r e - i o n o s p h e r e interactions, which m a y t y p i c a l l y p r o d u c e a field of 50 m V . m -1. R e f e r e n c e to Fig. 4 shows t h a t only w a v e n u m b e r s less t h a n or e q u a l to 2 can produce significant effects a t t h e E q u a t o r for a 50 m V . m -1 source located a b o v e 60 ° latitude. T h e e q u a t o r i a l electric field due to h i g h - l a t i t u d e sources is therefore of necessity v e r y b r o a d in longitudinal e x t e n t , a n d can h a v e small-scale features o n l y where these are c r e a t e d locally b y c o n d u c t i v i t y inhomogeneities. DISCUSSION AND CONCLUSIONS T h e purpose of this w o r k has b e e n to p o i n t o u t t h e close r e l a t i o n b e t w e e n longitudinal g r a d i e n t s a n d h e i g h t - l a t i t u d e v a r i a t i o n s of e q u a t o r i a l electric fields a n d currents. I n addition, b y e x t e n d i n g our calculations to higher latitudes, we h a v e shown q u a n t i t a t i v e l y h o w electric fields d e c a y a w a y f r o m t h e i r source in t h e ionosphere in general, a n d specifically h o w t h e y are a t t e n u a t e d b e t w e e n t h e i r source a n d t h e E q u s t o r . Because wc h a v e ignored longitudinal v a r i a t i o n s o f c o n d u c t i v i t y , o t h e r t h a n to p o i n t to t h e m as p o t e n t i a l sources of electric field gradients, our results b e c o m e i n v a l i d w h e n t h e c o n d u c t i v i t y
g r a d i e n t in l o n g i t u d e is c o m p a r a b l e to or g r e a t e r t h a n t h e electric field gradient. I n particular, our results c a n n o t be applied to sunrise or sunset p h e n o m e n a . On t h e o t h e r hand, t h e f a c t t h a t we h a v e chosen n o o n t i m e c o n d u c t i v i t y conditions does n o t render our electric field results useless for o t h e r t i m e s of t h e day. N o t e t h a t t h e solution of (5) does n o t d e p e n d on t h e absolute values of t h e conductivities, b u t only on t h e i r r e l a t i v e v a r i a t i o n s w i t h a l t i t u d e a n d latitude. Since these r e l a t i v e v a r i a t i o n s p r o b a b l y do n o t change n e a r l y as m u c h in local t i m e as do the absolute values, our electric field results should be r e a s o n a b l y applicable a t n i g h t as well as during t h e day. I n addition, longitudinal v a r i a t i o n s of t h e c o n d u c t i v i t y due to v a r i a t i o n s of the m a g n e t i c field s t r e n g t h should n o t s t r o n g l y affect our results. The p r i m a r y effect of changing t h e c o n d u c t i v i t y will be to change t h e ratio of current i n t e n s i t y to electric field strength. Our results show t h a t alterations of t h e v e r t i c a l structures of t h e electric fields a n d currents f r o m t h e infinite-electrojet (n = 0) m o d e l are r e l a t i v e l y small for large-scale longitudinal g r a d i e n t s of t h e electrojet. T h e alterations m a y b e c o m e quite i m p o r t a n t , however, for small-scale gradients, such as these s o m e t i m e s observed b y r a d a r in S o u t h A m e r i c a (BALsLEY, 1970) a n d in Africa (results to be published). T h e sources of longitudinal v a r i a t i o n s in t h e electrojet m a y be c o n d u c t i v i t y gradients, Jonespheric winds, or lfigh-latitude m a g n e t o s p h e r e ionosphere interactions. Small-scale electrojet v a r i a t i o n s can be p r o d u c e d o n l y b y sources close to t h e E q u a t o r , since t h e electric fields of small-scale h i g h e r - l a t i t u d e sources are strongly a t t e n u a t e d b e t w e e n t h e source a n d t h e E q u a t o r .
Acknowledgements--Thiswork was supported partially by NATO through a fellowship to one of us (A. Richmend) and through the Grant No. 684 for multinational cooperation. Computers of I.P.G.P. and I.N.A.G. were used for numerical analysis while I.P.G.P., C.N.R.S. and I.N.A.G. supported the related experiments in Africa.
ft. GAGNEPAIN, M. CROCHET and A. D. RICHMOND
286
REFERENCES
BALSLEY B. GOUIN P. a n d MAYAtrD P . N . Y~YLOV A. L., SOBOImV~_T. N., FISHCHUK D . L . ,
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J. geophys. Re*. 75, 4291. Annla. Ggophys. 28, 41. Geomagn. Aeron. 13, 400.
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J. atmoa, terr. Phys. 36, 1083. J. geophys. Re*. 74, 4025. d. geophys. Re*. 72, 5799.
TSEDILINA YE. and SHCIIERBA2~OV V. P.
RICHMONI) A . D . SUGIURA M. a n d PoRes D . J . I~lq'TIEDT J.
Reference is also made to the following unpublished material: RICHMOND A . D .
1972
A. D. R i c h m o n d is grateful to Dr, A. R. J x I ~ for kindly pointing out a sign error in ~he paper of RIC~MeND (1973). The corrected forms of equations (11) a n d (12) of t h a t paper should read:
H~(~,~) =
,1 , [8 d~ (c¢,fl') h9 (c¢,fl') h~(cc,fl') dfl', (11)
Air Force Cambridge Research Laboratories Rept. 72-0668.
a-~ Lh¢Lhq~(~o
(h~Hg)
[
] + ~ " h ~ ~i a~ (hgH~)] ] [( l
O h~ ~ E~ = 0-~
O -- ~
~ v~/B0
h~ v~0 +
. -J (12) The only result of RICHMOI~D (1973) affected by this error is the sign of H~ shown in his Fig. 5. The directions of current flow stated in the figure caption are correct. ol
/