83
Columnar c o n t e n t v a r i a t i o n s l i k e t h a t i n F i g . 5.2 r e p r e s e n t a b o u t a 20'11, change i n t h e t o t a l columnar c o n t e n t . The f a c t t h a t t h e t o t a l columnar c o n t e n t v a r i e s o n l y about 2@when l o c a l d e n s i t y changes exceed
loo$!
g i v e s i n d i r e c t evidence of t h e s c a l e s i z e of t h e solar-wind t u r b u l e n c e . I f t h e r a y p a t h i s viewed as a number of c e l l s whose d e n s i t i e s vary randomly by a f a c t o r o f about 2 , t h e number o f c e l l s r e q u i r e d t o keep t h e v a r i a t i o n of t h e t o t a l colum- n a r c o n t e n t t o 2 @ i s given by AI/I- l/hc,
o r Nc'
25. Forr a y p a t h s which pass f a i r l y near t h e sun, t h e part of t h e p a t h n e a r e s t t h e sun makes t h e dominant c o n t r i b u t i o n t o t h e columnar c o n t e n t . Thus, t h e s c a l e s i z e of t h e t u r b u l e n c e , viewed simply as a number of e q u a l l y s i z e d c l o u d s , i s -- 0 . 5 a.u./25 = 3 x 1 0 6 km.
84
( 1 ) The d a t a averaged, d e s i g n a t e d by l i n e number from Table 2.1;
( 2 ) The number o f i n d i v i d u a l s p e c t r a averaged;
( 3 ) The parameters 5, D , (P
-
1 ) of t h e model spectrum f i t t o t h e d a t a ,- 4
( 4 ) The value o f t h e averaged spectrum a t v = 3 x 10 Hz, pA(v3);
3
-4
(5) The v a l u e of t h e model spectrum a t v = 3 x 10 Hz, P (v ); and
3 F 3
( 6 ) The average value of q i n u n i t s of 0 . 1 a . u .
The amplitudes of t h e s p e c t r a were measured a t 3 x 10
-4
Hz, i n s p i t e of t h e t h e o r e t i c a l minima, because t h e s p e c t r a s u f f e r e d t h e l e a s t c o r r u p t i o n from n o i s e and t h e l i m i t e d l e n g t h o f t h e d a t a r e c o r d s at t h a t frequency. A t h i g h e r f r e q u e n c i e s t h e i n - a c c u r a c i e s i n t h e s u b t r a c t i o n of t h e n o i s e l e v e l become important, while a t lower f r e q u e n c i e s t h e amplitudes cannot be r e l i a b l y e s t i m a t e d from d a t a r e c o r d s a s s h o r t a s t h o s e used.T a b l e 5 . 1 . Tine o v e r a l l a v e r a g e s p e c t r a o f t h e 14469 and MM7L DRVID d a t a a r e shown on t h e t o p two l i n e s . The e r r o r s i n t h e s e s p e c t r a a r e e s t i m a t e d by d i v i d i n g e a c h m i s s i o n ' s d a t a i n t o two g r o u p s by t a k i n g e v e r y o t h e r d a t a s e t , and t h e n a v e r a g i n g t h e s e g r o u p i n g s . T h i s p r o c e d u r e e s t i m a t e s b o t h t h e q u a l i t y o f t h e o r i g i n a l d a t a and t h e r e l i a b i l i t y of t h e f i t t i n g p r o c e s s u s e d t o g e t t h e m d e l p a r a m e t e r s i n c g l u a n 3. The columns a r e more f ' u l l y d e s c r i b e d i n t h e t e x t .
' i g u r e 5.3. O v e r a l l average spectrum of a 6 9 data. E r r o r b a r s show var-l i t i o n s found when a l t e r n a t e d a t a sets a r e averaged ( s e e Table 5.1 ). The M ' s g i v e t h e s p e c t r a l v a l u e s a f t e r t h e minimum o f t h e average v a l u e s ( A ' S ) i s s u b t r a ~ t e d from a l l p o i n t s . "AVG IMPACT"^^ t h e average d i s t a n c e of c l o s e s t
approach i n t e n t h s of a.u. b
87
AVERAGE D POUE R S P E C T R U M OF D A T A VS F RE OUENCY
F i g u r e
F R E B U E N C Y [ HERTZI
ClCI? C I L C L I B ? P I C # rtuw A V M A V 6 I W A C r 1.)@01911401 ?.11011111*8? D.bOl)W00*81 I ~ B O ~ ( W ) ( 1 ~ * 0 ~
5 . 4 O v e r a l l average spectrum of MM.In data. Averaging more d a t a g i v e s s m a l l e r e r r o r b a r s and a b e t t e r e s t i m a t e of t h e n o i s e t h a n f o r t h e m69 d a t a . Even with t h e n o i s e s b t r a c t i o n t h e spectrum f l a t t e n s somewhat between 7 and 9x10-' Hz. Other c o m e n t s of Figure 5.2. a l s o apply t o t h i s figure.
88
The parameters f o r t h e o v e r a l l average s p e c t r a a r e on t h e t o p two l i n e s of Table 5.1. The parameter of most i n t e r e s t i s t h e s p e c t r a l index. I t s value i s 2.85 f o r the MM69 data, 2.95 f o r the MM7l d a t a . The e r r o r i n these s l o p e s , based on t h e p o s s i b l e f i t s t o t h e
s p e c t r a , i s approximately
-
+ 0.2. The e r r o r may a l s o be estimated by forming average spectra with a l t e r n a t e data s e t s f r m Table 2 . 1 and comparing t h e r e s u l t s t o Figures 5.3 and 5.4. The next fourl i n e s of Table 5.1 give t h e parameters f o r t h e s e s p e c t r a . Again t h e e r r o r i n t h e s l o p e s i s about
-
+ 0.2Figures 5.3 and 5.4 show t h e
JN
improvement i n s i g n a l t o n o i s e r a t i o expected a f t e r averaging. I n Figure 5.4 t h e noise l e v e l a f t e r s u b t r a c t i o n i s a f a c t o r of 6 ( N = 33) l e s s than before. Since t h e spectrum has a slope of about 3 , t h i s g i v e s a n extension of t h e frequency range by a f a c t o r of 2 from- 4
x 10-4
Hz t o -- 7 x Hz. The spectrum i n the region 8 x t o 1 . 3 xl o 3
Hz has i t s slope reduced s l i g h t l y by the n o i s e .For t h e MI469 data ( ~ i ~ u r e 5.3 ) t h e amplitude of t h e lowest frequency p o i n t i s reduced because of t h e shortness of t h e
o r i g i n a l d a t a records. The d a t a were only s u f f i c i e n t t o provide a r e l i a b l e spectrum above -- 3 x 10
-4
Hz. However, t o o b t a i nincreased r e s o l u t i o n a t higher frequencies, t h e point a t 1.5 x 10
-4
Hz was computed. The s i g n a l t o noise r a t i o was improved by about a f a c t o r of
4
by t h e noise removal.89
The average spectrum of t h e W1 d a t a shows Lj.ttle o r no e f f e c t which could be a t t r i b u t e d t o 'a long wavelength l i m i t t o t h e s o l a r wind turbulence. The numerical i n t e g r a t i o n s of Eq. (4.16) t h e n suggest t h a t A,,> 1.2 x 10 7 krn. I n t h i s case t h e use of
Eq. (4.19) which ignores t h e long wavelength c u t o f f i s completely j u s t i f i e d .
A l i n e of slope 2.9 provides a good f i t t o both averaged s p e c t r a . It implies a comoving s p e c t r a l index $ = 3.9
-
+ 0 . 2 .IW r e p o r t a s p e c t r a l index f o r proton d e n s i t y f l u c t u a t i o n s
which g i v e s B = 3.3
-
+ 0.3. However, j3 = 3.5 provides a good f i t t o t h e i r d a t a and values up t o $ = 3.8 a r e p o s s i b l e . Because of t h e small 10 cm) Debye length i n t h e s o l a r wind, one 3would expect t h e e l e c t r o n s and protons t o have t h e same turbulence spectrum. Thus, over a considerable range of h e l i o c e n t r i c d i s - t a n c e s and more than one-half a s o l a r c y c l e t h e s p e c t r a l index of t h e s o l a r wind turbulence appears t o vary h a r d l y a t a l l .
The averaged DRVID s p e c t r a (cnt ( v ) ) can be compared t o the spectrum of IN ( F , ~ ( V ) ) by using Eq. ( 4 . ~ 8 ) ~ where a f a c t o r of 2 has been introduced t o account f o r t h e one-sided spectrum output by t h e d a t a processing programs.
R,/ a.u. 3 + 2 ~ 1 + e-d cos dp ( 4 0 0 s 1
90
Since the s p e c t r a l indices a r e very similar t h e canparison gives t h e index y which measures t h e r a t e of f a l l - o f f of t h e density f l u c t u a t i o n s between t h e point of the ray p a t h ' s c l o s e s t approach t o t h e sun and 1 a.u. where t h e proton spectrum was measured.
Comparisons were made a t 3 x 10
-4
Hz and 6 x Hz f o r both t h e Mh69 and t h e MM71 d a t a . The e f f e c t of t h e t h e o r e t i c a l m i a t 3 x Hz was ignored; t h e term i n braces was s e t t o 1 a t3 x Hz and t o l / 2 a t 6 x Hz. Table 5.2 gives t h e r e s u l t s of t h e comparisons and t h e i r e r r o r s based on t h e range of the a l t e r n a t e spectra. The average value of y i s =
0.38
-
+ 0.11. The quoted e r r o r i s t h e standard deviation of thevalues from
7.
The small value of y f o r t h e W 1 data a t 3 x 10 -4 Hz may r e s u l t from ignoring t h e predicted minima t h e r e ; using pF(v3)= 1.2 x
lo3'
gives y = 0.28.The comparisons between t h e DRVID spectra and t h e spectrum of show that, on t h e average, t h e amplitude of solar-wind density f l u c t u a t i o n s fa 11s o f f considerably f a s t e r than re2 between 0.15 and 1 a .u. The determination of y i s based on t h e assumption t h a t the solar-wind density f l u c t u a t i o n s have a uniform power - l a w dependence on h e l i o c e n t r i c distance between t h e point of c l o s e s t approach t o the sun of the W r i n e r line-of-sight and 1 a . u . If t h i s assumption i s not correct and t h e r e i s a region near t h e sun where the density f l u c t u a t i o n s f a l l off more slowly than r-2 (see t h e next s e c t i o n ) , t h e e f f e c t i v e value of y outside t h i s region my be reduced f r m t h e values given above. That i s , from the outer e d ~ e of the region of enhanced turbulence t o 1 a.u. t h e fluctua+tions may d e c l i n e as r -2
,
but i n camparing the i n t e r i o r of the region t o-2.38 1 a.u. one f i n d s h n ( r ) a r
91-
Table 5.2
Summary of the comparison of DRVU) s p e c t r a l amplitudes t o t h e spectrum of I n t r i l i g a t o r and Wolfe (1970) t o find t h e large-scale r a d i a l dependence of the solar-win turbulence
assumed t o behave a s k ( r ) a
r-(2+J.
Ro = 0.95 a.u.Average y =
0.38 -
+ 0.1192
The comparisons a l s o ignore any short-or long-term time v a r i - a t i o n s i n the amplitude of the turbulence. The short-term e f f e c t s
should not be t o o important because of the long spans of data averaged. The data of IW cover t h r e e weeks. The DRVID spectra each cover a t l e a s t s i x weeks and agree well over a separation of two years. The p o s s i b i l i t y of long-term v a r i a t i o n s cannot be r u l e d out. The DRVID data were obtained near the maximum of the current sunspot cycle, while t h e data of IW were obtained (1965 December, 1966 ~ a n u a r y ) near t h e minimum. Data presented i n the next section shows t h a t short-term v a r i a t i o n s i n the amplitude of
the turbulence spectra c o r r e l a t e with sunspot a c t i v i t y . The d a t a i n Table 5.2 show that t h e Mb69 spectrum taken nearer the sunspot maximum give l a r g e r values of y than the M M 7 l spectrum. If these c o r r e l a t i o n s between sunspot number and s p e c t r a l amplitude hold throughout the s o l a r cycle, t h e value of y corrected f o r t h i s e f f e c t may not d i f f e r s i g n i f i c a n t l y f r m zero. Thus, t h e r e i s e i t h e r strong o v e r a l l r a d i a l v a r i a t i o n O r strong long-term time v a r i a t i o n i n the solar-wind t u r u u l ence
.
5
3C
.
RADIAL VARIATION OF THE DRVID SPECTRA NEAR THE SUNThe measurement of the r a d i a l dependence of t h e solar-wind d e n s i t y f l u c t u a t i o n s near t h e sun i s extremely important f o r t h e o r i e s of s o l a r wind heating. DRVID spectra a r e a
u s e f u l probe of t h e s o l a r wind near t h e sun. To m k e c l e a r any systematic changes of t h e s p e c t r a with q by smoothing out the l a r g e day t o day v a r i a t i o n s i n t h e amplitudes, s e v e r a l spectra with s i m i l a r values of q were averaged together. The data were grouped s o t h a t approximately one-quarter of t h e spectra were i n each averaged spectrum. During t h e MM'i'lmission data were ob- t a i n e d before and a f t e r the s p a c e c r a f t ' s superior conjunction s o the data were grouped by both time period and q.
The r e s u l t s of averaging t h e grouped s p e c t r a a r e given i n Table 5.3; i t s format i s the same as Table 5.1. The amplitudes of t h e averaged s p e c t r a a t 3 x 10
-4
Hz a r e p l o t t e d a g a i n s t q i n Figure 5.5a. The o v e r a l l averaged amplitudes a r e a l s o shown with t h e i r e r r o r s (based on a l t e r n a t e data s e t s ) t o i n d i c a t et h e range of v a r i a t i o n expected. Figure 5.5a shows t h a t generally t h e
~ ~
spectra f i t i n r a t h e r well with the MWl data. However,6 9
t h e m69 data may f a l l off somewhat more rapidly with q. Figure 5.5b shows that t h e W l post-conjunction amplitudes a r e a l l s u b s t a n t i - a l l y l e s s than t h e preconjunction amplitudes, but t h e r a d i a l de- pendence i s very s i m i l a r .
Table 5.3
Groupings of DRVU) data used to investigate variation of the spectra near the sun.
F i g u r e 5 . 5 a . Averaged s p e c t r a l a m p l i t u d e s p l o t t e d a g a i n s t d i s t a n c e of c l o s e s t approach. The d a t a were d i v i d e d i n t o f o u r groups based on t h e d i s - t a n c e of c l o s e s t approach:A, MM69;Q, MM71.
The o v e r a l l average a m p l i t u d e s , w i t h e r r o r b a r s r e p r e s e n t i n g t h e v a r i a t i o n s between a l t e r n a t e s p e c t r a (Table 5 . 1 ) , a r e g i v e n w i t h t h e f i l l e d symbols. The l i n e s have
s l o p e s : s 0 l i d ~ 2 . 0 ; dashed,3.0; dash-dot,4.0.
These s l o p e s correspond t o b n ( r ) * r - ' w i t h 8= 1.5, 2.Q, 2.5, r e s p e c t i v e l y .
F i g u r e 5.5b. MM71 averaged s p e c t r a l a m p l i t u d e s p l o t t e d a g a i n s t d i s t a n c e of c l o s e s t approach. The d a t a were f i r s t d i v i d e d i n t o f o u r g r o u p s based o n l y on d i s t a n c e of c l o s e s t approach
(D). The d a t a w i t h i n t h e s e g r o u p s were s e p a r a t e d i n t o p r e c o n j u n c t i o n (0) and p o s t
-
c o n j u n c t i o n (a) s e t s . The group w i t h t h e l a r g e s t q c o n t a i n s n o p r e c o n j u n c t i o n d a t a . Note t h a t t h e s l o p e s a r e s i m i l a r a l t h o u g h
t h e a c t u a l a m p l i t u d e changed. The l i n e h a s s l o p e 2.0 i m p l y i n g t h a t b n ( r ) 4 r ' l a 5 .
97
The data of Figure 5.5& show that while t h e &9 spectra f i t i n f a i r l y well w i t h those of M1, they may f a l l off sumewhat more r a p i d l y with q . Because t h e range i n q i s only a f a c t o r 2 and t h e point a t q = 0.24 a . u . has a low s i g n a l t o noise r a t i o , i t i s d i f f i c u l t t o r e l i a b l y estimate this f a l l o f f . However, the
-3.0
amplitudes roughly decrease a s q t o q-4.0. This implies that ( s e e Eq. 4.19) hn decreases a s r -2'0 t o r -2 * 5 i n the region
0.12 <
-
r <-
0.24 a . u . This f a l l - o f f may be compared t o t h e obser- vatiuns of Counselman and Rankin (1972) (hereafter c a l l e d C R ) who found t h a t t h e e l e c t r o n d e n s i t y i n t h e range 5 <-
r <-
20 Rg-2.9w.2 during 1969 and 1970 June decreased a s r
-
ments were made near t h e sun's south pole while t h e DRVID ob- servations were taken near the north pole, but they s t a t e t h a t t h e corona had a symmetric appearance during t h i s period. The
MM69
R e l a t i v i t y Experiment ( ~ n d e r son e t a l .,
1972), which used range data associated with the MM69 DRVID measurements, found that f o r a l a r g e range of s o l a r distances (0.03-
< r <-
0.30 a.u.)-2'0X)*25. Thus, t h e r e i s t h e solar-wind density declined a s r
-
considerable s c a t t e r i n the reported exponents
,
although same of the apparent disagreement may be due t o the d i f f e r e n t r a d i a l regimes sampled. I n t h e r a d i a l regime r 10 R t h e density i s0
known t o have a s i g n i f i c a n t component which has a r a d i a l dependence r -6 ( ~ n d e r s o n e t a l . , 1972) and thus increases r a p i d l y near the
sun. Much of t h e data of CR i s i n t h i s region so t h e i r r e s u l t
m y not apply t o t h e large-scale v a r i a t i o n of the solar-wind density.
Anderson e t a l . (1972) have l i t t l e data i n s i d e r = 10 R 0'
98
The amplitudes of t h e MM71 grouped spectra a r e p l o t t e d a g a i n s t q i n Figure 5.w. The data a r e a l s o separated i n t o pre- and post- conjunction groups. The l i n e drawn through t h e data has slope
-2.0. It i s a good f i t t o the data, and t h e v a r i a t i o n i n t h e slope i s only
-
+ 0.2. These observations imply t h a t An(r) f a l l s off a s r-1*5+0*2 f o r 0.07-
< r <-
0.22 a.u. This decline i s much l e s s than t h a t suggested f o r t h e ~ ~ data. The MM71Relativity 6 9Experiment ( ~ n d e r s o n and Lau, 1973) again found t h a t the Large-scale -2 *Ow l.
CR r e p o r t v a r i a t i o n o f t h e s o l a r - w i n d d e n s i t y i s r
-
preliminary r e s u l t s of t h e i r experiment conducted i n 1971 June.
-1.4m.1
They f i n d t h a t the density declined as r
-
f o r 5 Ro5
rC
-
20 Ra, i f t h e corona i s assumed t o be s p h e r i c a l l y symmetric.However, they point out t h a t a s s o l a r a c t i v i t y declines t h e corona develops a strong enhancement near t h e equator, thus d i s t o r t i n g observations a t the poles. They obtain a good f i t t o t h e i r ob- servations i f the density v a r i e s from equator t o pole as cos 2 1, where
x
i s heliographic l a t i t u d e , and t h e density decreases with-2.4-1-0.4 r a d i u s a s r
-
Solar a c t i v i t y i n 1971 June when CR made t h e i r observations was similar t o t h a t i n late-1972 when t h e MM71 DRVU) data were obtained. I n Figure 5.6 t h e amplitudes of the MM71 grouped
spectra multiplied by 93*5 a r e p l o t t e d against tke -la%itQde of the point of c l o s e s t approach. The q3*5 f a c t o r i s t o account f o r t h e steep r a d i a l f a l l - o f f found i n t h e MI69 data and would
give a r a d i a l density dependence of An a r - 2 0 2 similar t o the se- cond value of CR. The scaled data show a very strong l a t i t u d e
1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0
HELIOGRAPHIC LATITUDC:
Figure 5.6. Scaled s p e c t r a l amplitudes of EWl d a t a p l o t t e d
a g a i n s t h e l i o g r a p h i c i a t i t u d e o f t h e p o i n t o f c l o s e s t approach. The s p e c t r a l amplitudes have been m u l t i - p l i e d by q 3 . 5 ( q i n u n i t s of 0.1 a . u . ) , t h e r a d i a l v a r i a t i o n t:hnt would o b t a i n i f n ( r ) r-2.25--
approximately t h e r e s u l t f o r t h e ~ ~ d a t a . The 6 9
f i g u r e t e s t s t h e h y p o t h e s i s t h a t t h e slow r a d i a l f a i l - o f f of t h e W1 s p e c t r a i s caused by a l a t i - tude-dependent e f r e c t
.
The s t r o n g l a t i t u d e f u n c t i o n r e q u i r e d t o match t h e s c a l e d d a t a s u g g e s t s t h a t t h e slow f a l l - o f f i s not due t o a l a t i t u d i n a l v a r i a t i o n o f t h e t u r b u l e n c e . I),
MM71 o v e r a l l a v e r a g e .a, MM'j'1 d a t a g r ~ u p e d i n t o q u a r t e r s by q . 0, Mi,
p r e c o n j u n c t i o n d a t a from each q u a r t e r . @, MMi'l, p o s t c o n j u n c t i o n d a t a from each q u a r t e r .
X, l / s i n ( i a t i t u d e ) , s c a l e d t o f i t t h e m's.
100
dependence--approximately I/ s i n A . A cos2
X
dependence cannot account f o r t h e d a t a i n Figure !j.G. The r a d i a l f a l l - o f f i sl e s s s t e e p i n t h e MIvi71 d a t a than i n t h e m6g
d a t a . However, a determination of t h e l a t i t u d e v a r i a t i o n i s very d i f f i c u l t because of t h e s t r o n g c o r r e l a t i o n between t h e
d i s t a n c e of c l o s e s t approach and t h e l a t i t u d e of c l o s e s t approach.
We may summarize these observations a s follows: F i r s t , t h e l a r g e - s c a l e (0.03 <
-
r <-
0.30 a . u . ) decrease of the solar-wind-2.0+0.2
d e n s i t y i s r
-
( ~ n d e r s o n e t a l . , 1972. Andersonand U u , 1973).This r e s u l t seems t o be independent of t h e s o l a r cycle, and i t s determination i s not s i g n i f i c a n t l y influenced by d e n s i t y v a r i a - t i o n s with l a t i t u d e o r i n t h e inner corona ( r 5 10
R ~ ) .
Second, t h e decrease of both t h e d e n s i t y and f l u c t u a t i o n s with r a d i u s may depend on the s o l a r cycle. I t appears t h a t near s o l a r maximum t h e d e c l i n e i s more r a p i d (An -- r -2.25,
n-
r'2'9) than l a t e r i n t h e c y c l e (An n-
.-Im5). However, a s s o l a r a c t i v i t y d e c l i n e s heliographic l a t i t u d e dependence, which i s d i f f i c u l t t o s e m r a t efrom the r a d i a l dependence with t h e r a d i o techniques employed, may come i n t o play and account f o r some of t h e apparent lessening
of t h e r a d i a l f a l l - o f f . Third, i n t h e region near t h e sun
(0.07 <
-
r-
< 0.20 a . u . ) t h e d e n s i t y f l u c t u a t i o n s d e c l i n e more slowly t h a n t h e y do on t h e average ( s e c t i o n B) between -- 0.15 a.u. and 1 a . u . The implications of t h i s r e s u l t w i l l b e discussed i n Chapter VII.
101-
Other r a d i a l v a r i a t i o n s of t h e DRVID s p e c t r a were a l s o i n v e s t i g a t e d . The s p e c t r a l i n d i c e s of t h e grouped s p e c t r a were p l o t t e d a g a i n s t q. The data show no v a r i a t i o n with q. The f a c t t h a t t h e s p e c t r a l index i s independent of q i s i n t e r e s t i n g i n
i t s e l f . It i s a l s o important because i t shows t h a t t h e model s p e c t r a f i t t o t h e d a t a were not influenced by t h e noise i n t h e high frequency regime.