D, MAPPING OF AUTOCQRREUTION LAGS TO THE: SOLAR BURFACE
IV: THE RELATIONSHIP OF DRVID OBSERVATIONS TO THE SOLAR-WIND TURBULENCE
DRVID measurements a r e of i n t e r e s t only i n s o f a r a s they can b e r e l a t e d t o t h e physical s t a t e of t h e s o l a r wind. A s was discussed i n Chapter I i t s f l u c t u a t i o n s a r e p a r t i c u l a r l y i n t e r e s t i n g because they a r e probably r e l a t e d t o t h e waves which heat t h e corona and d r i v e the s o l a r wind. One i s i n t e r - e s t e d i n t h e l o c a t i o n and e x t e n t of t h e region where t h e
heating occurs. One a l s o wish@@ t o characterl'.ze t h e f l u c t u - a t i o n s o r waves which cause t h e h e a t i n g by wavelength o r f r c - quency. These goals a r e both met by studying t h e power spectrum
( c o n t r i b u t i o n t o t h e rms d e n s i t y f l u c t u a t i o n by waves of a given frequency) of t h e solar-wind f l u c t u a t i o n s a s a f i n e t i o n of d i ~ t a n c ~ e from t h e sun.
I n t h e f i r s t s e c t i o n of t h i s chapter, t h e r e l a t i o n s h i p between t h e comoving m v e n w b e r spectrum 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 and t h e temporal. power spectrum of DRVID d a t a i s deduced. A s l - i g h t l y d i f f e r e n t v e r s i o n of t h i s s e c t i o n has been published i n Callahan (1974). The development here i s formal. The f i r s t s e c t i o n of Chapter V contains a q u a l i - t a t i v e ( i .e
. ,
without power s p e c t r a ) d i s c u s s i o n of the s c a l es i z e s and amplitudes of solar-wind d e n s i t y changes.
49
The e f f e c t s on t h e power spectrum of DRVID d a t a of d i f - f e r e n t density and v e l o c i t y d i s t r i b u t i o n s as functions of h e l i o - c e n t r i c distance a r e discussed i n t h e second section. It i s found t h a t the DRVID spectra a r e q u i t e s e n s i t i v e t o t h e velocity and density near the sun. This makes DRVID data a very useful
probe of t h i s region where it is l i k e l y t h a t much of t h e heating of t h e s o l a r wind occurs. The DRVID r e s u l t s f o r t h e r a d i a l d i s t r i b u t i o n of the solar-wind f l u c t u a t i o n s a r e given i n Chapter V. Their junplications f o r t h e o r i e s o f t h e s o h r wind a r e discussed i n Chapter V I I .
The r e l a t i o n s h i p between t h e spectrum of DRVUD data and t h e spectrum measured by a stationary spacecraft i s derived i n t h e t h i r d s e c t i o n .
I n
Chapter V t h i s r e h t i o n s h i p i s used t o compare DRVU) spectra t o data taken near 1 a , u . and thusf i n d the o v e r a l l r a d i a l dependence of t h e sohr-wind f l u c t w t i o m , For t h e reader more i n t e r e s t e d i n t h e observational f i n d i w s of t h i s study than i n formal mathematical development2 the tws key r e s u l t s of t h i s chapter w i l l be summarized here. This sunrmary
all
a l s o make c l e a r the goals of t h i s chapter f o r t h e reader who pursues it i n d e t a i l .
F i r s t , i f t h e wave number spectrum i n the r e ~ t - f r a m e of t h e s o l a r wind ("comoving") i s a power-law (g(k)
( 5 ia) ,
and ifrn
w
amplitude of the density f l u c t u a t i o n s depends on h e l i o c e n t r i c disr- tance as An(r) ac: r -'+y
,
then the temporal power spectrum of DRVmd a t a , denoted cnt ( v ) , i s [see Eqs. (4.19) and (4.21)]
c (v) [1 + e-m cos
~~1
n t
- -
Iq3Yry vP l
where q i s t h e distance t o t h e r a y p a t h ' s point of c l o s e s t approach t o t h e sun, and a and 1 a r e constants which depend on t h e exact geom- e t r y . The cosine term i n t h e numerator causes sharp minima i n t h e
s p e c t r a a t frequencies which a r e odd-integral m l t i p l e s of 3 x 10 -4 Hz f o r t h e geometry t h a t ob-tained during t h e observations r e - ported here. D i f f e r e n t v e l o c i t y and d e n s i t y d i s t r i b u t i o n s along t h e line-of-sight reduce, b u t do not eliminate, t h e minima. The minima occur because of the round t r i p manner i n which the DRVID measurements a r e made. From observations one wishes t o determine
p and y. The observed s p e c t r a w i l l a l s o be checked c a r e f u l l y f o r t h e predicted minima.
The second important r e s u l t r e l a t e s t h e power s s e t r u m of DRVID d a t a t o t h e power spectrum deduced from density measurements made aboard a spacecraft a t r e s t (approximately) with r e s p e c t t o t h e sun.
I f the functions above a r e used f o r the wave number spectrum and t h e r a d i a l f a l l - o f f of t h e density f l u c t u a t i o n s t h e r e l a t i o n s h i p i s found t o be[ see Eq. (4.28)]
1 + em@' eos ~v 2
where t h e s p a c e c r a f t observed spectrum, denoted F~~ ( v ) , i s measured a t R (a.u.) from t h e sun, and Y i s t h e solar-wind v e l o c i t y . This
0 0
r e s u l t will be used t o determine t h e l a r g e - s c a l e
,
i , e. ,
between approximately 0.15 and l a . u . , r a d i a l Tall-off af t h e d e n s i t y f l u c t u a t i o n s . The f a l l - o f f w i l l be c h a r a c t e r i z e d by t h e value ofy
which gives t h e b e s t agreement between t h e observed values ofc n t ( v ) , q and Fnt(v), R o e
A . INTlERPRETATION OF DRVID IvlEASUREMENTS OF THE SOLAR-WIND TURBULENCE
Direct s p a c e c r a f t measurements near t h e e a r t h have shown t h a t t h e s o l a r wind i s t u r b u l e n t . Radio s c i n t i l l a t i o n obser- vations of regions nearer t h e sun a l s o show t h a t t h e s o l a r wind is t u r b u l e n t . Many discussions of t h e r e l a t i o n s h i p of
t h e scintill.€I.tion s p e c t r a t o t h e solar-wind turbulence
spectrum have appeared i n t h e X i t e r a t u r e * (see, f o r example, S a l p e t e r (1967), Young (1971), J o k i p i i and Lee (1972). )
Here t h e r e l a t i o n s h i p between t h e s p e c t r a of DRVTD measurements of t h e columnar content change and t h e solar-wind turbulence spectrum i s derived.
After d a t a reduction ( s e e Chapters 11 and 111) a time s e r i e s of DRVID measurements gives
where h ( r ) i s t h e f l u c t u a t i o n i n the e l e c t r o n content a t r ,
w
s measures d i s t a n c e along t h e r a y p a t h , and A.l ( t ) i s t h e change i n t h e e l e c t r o n colummr content. Observations of
A I ( ~ ) near a s p a c e c r a f t ' s superior conjunction can g i v e e s t i - mates of both t h e solar-wind turbulence spectrum near t h e sun and t h e r a d i a l dependence of An(r).
w
53
We wish t o i n v e s t i g a t e t h e r e l a t i o n s h i p between t h e t h r e e - dimensional wave number spectrum of An(r) t h a t would b e observed
,"
moving with the s o l a r wind ("comoving" ) and t h e frequency spectrum of A I ( ~ ) defined i n Eq.
(4.1).
Let t h e frame of t h e s o l a r wind s f move with v e l o c i t y v with r e s p e c t t o our framew
s. I n s ' , An(r') has a n a u t o c o r r e l a t i o n f u n c t i o n given by
,"
where t h e angular b r a c k e t s denote a s p t i a l average*
We e x p l i c i t l y show t h e n o n s t a t i o m r y nature of An($') caused by t h e v a r i a t i o n of t h e solar-wind d e n s i t y w i t h h e l l o -
c e n t r i c r a d i u s . Such v a r i a t i o n i s assumed t o b e on a ~ c a ~ ~
f a r l a r g e r than t h e turbul-ence. It i s assumed that t h e d e n s i t y a u t o c o r r e l a t i o n f u n c t i o n i s separable i n t o a function depending only on t h e s e p a r a t i o n of t h e p o i n t s involved m u l t i p l i e d by a n amplitude f u n c t i o n depending only on h e l i o c e n t r i c r a d i u s .
For t h i s nonstationary c a s e , we will d e f i n e t h e Fourier transform of GW (R)
,
enr (k ),
t o be t h e power spectrum ofC*
-
An(:'). Thus t h e power spectrum of An(r
- '
) i s given by(4
enr(k)-.
= G ~ , ( R )-
exp (-ik.~)- ."
d'~ 3 -mThe coordinate system s i s shown i n F i g . 4.1. The z-axis i s along t h e earth-spacecraft r a y path. The p r o j e c t i o n of v
.-"
i n t o the x - z plane i s shown. If s and s ' coincide a t t = 0,
g i v e s t h e coordinate transformation between t h e two frames.
Eq. (4.1) may be w r i t t e n e x p l i c i t l y i n frame is a s
('c
.4 >
where t i s measured a t t h e earth-based r e c e i v e r ,
It i s assumed that t h e s p a c e c r a f t i s a t a f i x e d d i s t a n c e L during t h e observation. P r o p g a t i o n t o and from t h e spacecraft is included. It i s a l s o assumed t h a t t h e s i g n a l propagates along a s i n g l e , s t r a i g h t r a y path. The e f f e c t s of multipathiny o r s c i n t i l l a t i o n must be small f o r t h e DRVZD technique t o work.
The curvature introduced by t h e varying refractive! index can
/
be shown t o be n e g l i g i b l e i n t h e s o l a r wind.
56
By using Eq. (4.4) one can rewrite Eq. (4.5) i n t h e r e s t frame of t h e s o l a r wind a s
It i s straightforward t o write t h e a u t o c o r r e l a t i o n of n ( t ) i n terms of G ~ ~ ( R ) .
-
The r e s u l t i swhere R
-
= (0, 0 , A), and t h e angular brackets sub-t denote a t i m e average.57
It i s assumed that v
...
does not change i n a c o r r e l a t i o n s c a l e , i. e . , v i s independent of A . Since v i s the bulk velocity of theU ..c
s o l a r wind, t h e approximation should be very good. For the complex Fourier transforms which follow only t h e t h i r d and f o u r t h t e r m of Eq. (4.7) will be used because cnt(') i s a r e a l , symmetric func- t i o n . One may recover c o r r e c t r e s u l t s by taking twice t h e r e a l p r t of any expression.
To reduce Eq. (4.7) t o t h e desired r e l a t i o n s h i p between t h e observed and t h e comoving power spectrum, one takes the Fourier transform of Eq.
(4.7)
and writes ~ ~ ~ ( 5 ) using Eq4 (4.3b)I
where
g
= R C1- -
(L/c)], M <b = R CI- ;
[r-
( 4 + 2 ( s- L)/c)]*
Rearrangement of
(4.8)
makes t h e important feature8 of the r e s u l t c l e a r e rIJ
d3k
... -.
c (v) = b ( r )ds
-
n t
-
( r n ? CL-S 03
58
The last two i n t e g r a l s express t h e convection of t h e s o l a r wind.
The sum i n t h e braces e n t e r s because of t h e round t r i p nature of W I D d a t a . The change from t h e time t o t h e frequency do- mains r e q u i r e s t h e k - i n t e g r a l . F i n a l l y , t h e s - i n t e g r a l s m s
up t h e c o n t r i b u t i o n s t o t h e observed spectrum along t h e r a y path.
A g r e a t s i m p l i f i c a t i o n occurs i n E q ,
(4.9)
i f one assumes t h a t t h e c o r r e l a t i o n s c a l e i s much smaller t h a n t h e o t h e r s c a l e s i n t h e problem. For t h e s o l a r wind more t h a n a few degrees from t h e sun such an approximation i s reasonable s i n c e t h e c o r r e l a t i o n s c a l e i s L.. 1-06
km while t h e d i s t a n c e s c a l e i s2 lo7
km. This assumption allows t h e l i m i t s on t h e 4 - i n t e g r a l t o be moved t o i n f i n i t y . Each of t h e l a s t two i n t e g r a l s t h e n r e s u l t s i n a &-function so t h a t one o b t a i n sThe S-functions allow two o f t h e k - i n t e g r a l s t o be done a t once.' The r e s u l t i s t h e most transparent i f one does the k and
X
kZ i n t e g r a l s t o g e t
cL b ( r )ds
Y
dk
c n t (v) =
--
J [ l + e - i 4 w (s-L)/C].
3'
0 X- a 2
Under the assumptions that v does not change i n a c o r r e l a t i o n
-
s c a l e and that t h e c o r r e l a t i o n s c a l e i s much l e s s than t h e distance s c a l e , Eq. (4.11) i s t h e general r e l a t i o n between t h e three-dimensional wave number spectrum of the s o l a r wind and t h e temporal spectrum observed i n BRVID d a t a . Eliminating t h e second term i n square brackets gives t h e r e s u l t f o r one-way
spacecraft d a t a . The assumptions used i n t h i s a n a l y s i s remove t h e solar-wind v e l o c i t y from t h e terms i n Eq. (4.11) which directly' involve v . Thus, only t h e bulk properties af t h e v e l o c i t y are important t o the predicted spectrum.
To evaluate Eq. (4.11) it i s necessary t o assume s p e c i f i c forms f o r %(k), b ( r ) , and vx. l4m-1~ spacecraPe observations have
II CI
shown that solar-wind magnetic f i e l d f l u c t u a t i o n s can be rspreaenked by a power-law spectrum, f (v) or v
- ',
over a Large range of frequencies,10')
<
v < ~ o - ~ H z . The suggestion t h a t the charged p a r t i c l e f luctu-(r re
a t i o n s would ham t h e same spectra was confirmed by I n - t r i l f g a t o r and
Wolfe (1970). Using i n s i t u spacecraft measurements they found a power l a w index of 1.3
-
+ 0.3, C r (1970) hras shown t h a t an in-dex derived from i n s i t u observations i s r e l a t e d t o t h e camoving index by
60
The schematic s o l a r wind spectrum of J o k i p i i a d Coleman (1968) suggests a long wavelength f l a t t e n i n g t o t h e spectrum.
A f l a t t e n i n g must occur i n a power-law spec$rum i f t h e t o t a l power i s t o be f i n i t e . Thus f o r a model spec-brum we choose
- ~ / 2 (k) = (kg + a:k: + a2k2 + a2k2)
g n r
-
Y Y z zwhere t h e a ' s allow f o r a n a n i s o t r o p i c spectrum. One expects f3 t o be between 3 and
4.
I f Eq. ('1.13) i s used i n Eq, (bell) t h e k - i n t e g r a l can Y
be done simply only i f $ =
4.
However,
observations show t h a t t h e s o l a r wind v e l o c i t y i s e s s e n t i a l l y r a d i a l so v may be takenM
t o be i n t h e x-z plane. Since t h e s o l a r wind v e l o c i t y i s a l - ways much l e s s than t h a t of l i g h t , vZ may be neglected i n t h e expression f o r k
.
With t h e s e approximations E q . (4.11.) r e -X
duces t o
6 1
To f i n i s h the evaluation one m u s t s p e c i f y t h e functions b ( r ) , vx, a x 2 a y, &
.,
and k along the ray path. Of course,sw 0
observations a r e not n e a r l y complete enough t o determine s o many f u n c t i o n s , s o l e t us s e t ax, ay, aZ and k t o constants. For
0
b ( r ) t h e obvious choice i s a power-law of h e l i o c e n t r i c d i s -
. - - 3
t a n c e ; from observation one can attempt t o determine t h e ex- ponent. The s o l a r wind v e l o c i t y i s taken t o be constant and r a d i a l so t h a t v ( s ) i s j u s t t h e p r o j e c t i o n of v onto the
X M
x - a x i s , Except very near t h e sun (< 10R ) t h i s should be a
-
0good approximation f o r v ( ~ k e r s and L i t t l e , 1971, See t h e
w
next s e c t i o n , ).
The i n t e g r a l i s more e a s i l y w r i t t e n i n terms of (see F i g . I+. 1)
where D i s a s c a l e f a c t o r f o r t h e solar-wind d e n s i t y . S u b s t i t u t i o n of E q s , (4,15) i n t o Eq. (4.14) and some a l g e b r a i c manipulations y i e l d
where
-
- A cos e -- L - A c o s e
g '1.12 q
and
I n Eq. (4.16) we can see the form of c (v ) s i n c e t h e n t
i n t e g r a l i s of t h e order of 1 except f o r c e r t a i n frequencies where t h e numerator vanishes. Except f o r t h i s f a c t o r t h e ob-
served spectrum i s one power less s t e e p than t h e comoving spectrwn, and i t s amplitude depends on t h e d i s t a n c e of c l o s e s t approach q t o t h e (3+2y) power. Frm Counsel-man and Rankf n (:1972 ) we expect t h a t
o
<..,
y <".
1/2.63
A s Eq. (4.16) s t a n d s it must be i n t e g r a t e d numerically.
The parameter of p r i n c i p a l i n t e r e s t i s k I n t e g r a t i o n s have 0 '
been c a r r i e d out a s functions of c , f3, y, and ko with a = a =
x Y
a z = 1, L = 2.6 a.u., and lvo)/c] 2 neglected. F i g . 4.2 shows t h e r e s u l t s of t h e i n t e g r a t i o n s f o r t h r e e values of c and two values of k with $ = 3.5, Y = 0.25. For frequencies v
>
2 x 10 -4 Hz0
t h e e f f e c t of changing k i s seen t o be n e g l i g i b l e . I n t e s a - 0
t i o n s with smaller values of k 0 ( i .e
. ,
to = 2rr/k0>
8 x 1.06
h)show t h a t l i t t l e a d d i t i o n a l change occurs as 1 -. a~ f a r v
>
2.Q
U
x 10
-4
Hz, I n t h i s frequency range t h e envelope of t h e s p e c t r aThe deep minima i n t h e s p e c t r a occur a t odd-integral m u l t i p l e s of t h e frequency corresponding, t o twice t h e round
t r i p l i g h t t r a v e l time f r a n t h e r a y px-bhPs p o i n t of c l o s e s t approach t o t h e sun t o t h e s p a c e c r a f i . The minima occur because t h e round
t r i p nature of t h e measurement i n s u r e s t h a t fluctuations a t t h e s e frequencies (v = kv /2n) w i l l be observed by t h e up and
X
down l e g s with a phase d i f f e r e n c e of n. The v a r i a t i o n of v e l o c i t y along t h e r a y path i s slow enough s o t h a t t h e r a t h e r s t e e p spectrum
( e -
3-4) and d e n s i t y ft&ll-off (a w4f2y) give a sharp resonance. The numerical i n t e g r a t i o n s show t h e expected r e s u l t s a s parameters a r e changed: The minima a r e deeper f o r l a r g e r values of B and y, and a r e l e s s deep f o r l a r g e r values of c . I n t h e f i r s t two cases t h e range of k and g, r e s p e c t i v e l y , which c o n t r i b u t e s i g n i f i c a n t l y t o t h e i n t e g r a l are reduced; t h u s65
t h e resonance i s sharpened. A s c i s increased, t h e r a y path no longer goes near t h e sun s o t h a t n o ' p r t i c u l a r value of /f
dominates t h e i n t e g r a l .
Examination of Eqs
.
(4.8) and (4.9) show t h a t t h e deep minima i n t h e t h e o r e t i c a l power s p e c t r a occur because of t h e assumption t h a t the s o l a r wind v e l o c i t y does not change i n a d e n s i t y c o r r e l a t i o n length. Ekcept very near t h e sun (< 10R )0
o r when t h e s o l a r wind i s very a c t i v e t h i s should be a v a l i d assumption, and Eq. (4.16) should r e p r e s e n t t h e observed spectrum f a i r l y well. DRVID d a t a taken near t h e sun ( 1 % 5 q
<
p ) w i l l0 0
be c a r e f u l l y examined f o r t h e deep minima found i n Eq. (4.16).
The numerical i n t e g r a t i o n s of Eq. (4.16) suggest t h a t f o r r e a l i s t i c values of k and v
>
1 x 10-4
Hz t h e term0 .a.
[(v0k,)/(2nv)] 2 (C 0.2) can be ignored without d i s a s t r o u s con- sequences. I f t h a t term i s ignored ( a s w e l l a s ( v d c ) 2 ),
Eq. (4.16) reduces t o t h e r e l a t i v e l y simple form
c ( v ) =
n t
a / ,
ar
(812) (2mrax)'-
1X
Y
66
For s e v e r a l plausible p a i r s of values of $ and y t h e i n t e g r a l reduces t o a s i n g l e case. I f one chooses
(a) i3 = 3.0, y = 0.5, or (b) $ = 3.5, y = 0.25, o r
( c ) @ =
4.0,
y = 0t h e power i n t h e denominator i s 3. These values of t h e ex- pognents span the expected range, so we w i l l e v a l w t e t h i s case i n more d e t a i l . O f course, any combination of y and fl t h a t gives
1 + y -1- @/2
>
0 allows the i n t e g r a li n
Eq. (4.17) t o be done by t h e method given i n Appendix A .U s i n g the r e s u l t of Appndix A with y + 8/2 = 2 the observed spectrum i s
67 where p = (A cos 6 ) / 9 ,
1 = (L
-
A cos c ) / q , q = A s i n s, and$
= ( ~ I W ~ ~ J , ~ ) / C . The constant i n Eq. (4.19) i s a p p r o p r i a t e f o r i n t e r p r e t i n g DRVID d a t a a s one-way columnar content measure- ments when t h e power spectrum i s defined a s i n Eq. (4.3a).The s p e c t r a l amplitudes obtained from Eq. (4.19) w i l l be one- h a l f a s l a r g e as those output by t h e computer r o u t i n e s described i n Chapter 111.
Eq. (4.19) has been evaluated as a function of frequency f o r s e v e r a l values of' E i n t h e t h r e e cases of Eq, (4.18). The r e s u l t s of the evaluation f o r B = 3.5, Y = 0.27 and t h r e e values of 6. a r e shown i n Figure 4.3. Comparison of Figures
4
.%and 4.3 shows e x c e l l e n t agreement except a t t h e lowest f r e - quencies. The deep minima s t i l l occur i n t h e s p e c t r a a t t h e
same frequencies. The c o r r e c t i o n terms ( ~ q . A - 5 ) were n e g l i g i b l e , even a t t h e minima. Thus Eq.
(4.19)
i s a good r e p r e s e n t a t i o n of t h e temporal spectrum of two-way spacecraft observations f a r f'requencies v21.5
x LO-4
Hz.I n Chapter V DRVID measurements made near t h e s u p e r i o r conjunctions of Mariners 6, 7, and 9 w i l l be presented and analyzed within t h e framework developed here. The observations w i l l a l l o w t h e determination of t h e solar-wind-density s p e c t r a l index f! near t h e sun. A search f o r sharp minima i n t h e s p e c t r a w i l l shed l i g h t on t h e v e l o c i t y and d e n s i t y d i s t r i b u t i o n s near t h e sun. A s can be seen from Eq. (4.16) measurements of t h e spcctrwn a t d i f f e r e n t values of c wi 11 a 1 l . o ~ t h e d e t e m j nat ion of3 Y whi ctl
.d a, u E P I C -.
m e , u
69
measures t h e f a l l ~ o f l f of d e n s i t y f l u c t u a t i o n s w i t h h e l i o c e n t r i c d i s t a n c e . Comparison of measurements near t h e sun with t h o s e a t 1 a . u . w i l l g i v e a very s e n s i t i v e determination of t h e value
of y which c h a r a c t e r i z e s t h e l a r g e s c a l e behavior of t h e s o l a r wind.
B e PREDICTm SPETRA FOR DIFFERENT VELOCITY AND DENSITY MODELS The r e s u l t s of the f i r s t s e c t i o n a r e f o r t h e simplest
v e l o c i t y and d e n s i t y models t h a t might d e s c r i b e t h e s o l a r wind.
The f u n c t i o n s a r e smooth f u n c t i o n s of r and s , and t h u s do not a l l o w f o r observed s t r u c t u r e s such as high speed streams
( ~ e u ~ e b a u e r and Snyder, 1366) o r d e n s i t y enhancements a s s o c i a t e d with magnet,ic s e c t o r s ( ~ i l c o x and Ness, l % > ) . It i s a l s o of i n t e r e s t t o f i n d out what s o r t of v a r i a t i o n s of h ( r ) a n d
-
vxw i l l remove t h e deep minima i n t h e s p e c t r a p r e d i c t e d by Eq. (4.19).
The a n a l y s i s h e r e w i l l d e a l only with l a r c e s c a l e d i s t u r b a n c e s and w i l l n o t a t t e m p t t o modif'y t h e assumption t h a t t h e v e l o c i t y does not change s i g n i f i c a n t l y i n a c o r r e l a t i o n s c a l e o r t h a t t h e d i s t a n c e s c a l e i s much l a r g e r than t h e c o r r e l a t i o n s c a l e .
To f a c i l i t a t e t h e i n v e s t i g a t i o n of v e l o c i t y and d e n s i t y models Eq. (4.14) i s r e c a s t i n terms of p and s i m p l i f i e d . Thc
2 2
terms involvi ng kg and vx/c
,
which were shown t o be negli g i b Le,
a r e discarded. The a ' s a r e s e t t o 1. A f t e r t h e s e s i m p l i f i c a t i o n s Eq.
(4.1.4)
becomeswhere b ( u ) and vx(,*) a r e t o be s p e c i f i e d .
Before examining s p e c i f i c functions f o r b ( y ) and vX(y) l e t us r e c a l l t h e r e s u l t s of t h e f i r s t s e c t i o n . The s p e c t r a l m i n i m were deeper f o r Larger values of p and y, and f o r r a y wths nearer t h e s u n . The region near t h e sun i s c r i t i c a l f o r t h e resonance e f f e c t which produces t h e minima. Since i n any sen- s i b l e model b ( p ) and vx(p) must vary i n v e r s e l y with p, Eq. (4.20)
shows t h a t t h i s region w i l l be l e s s important i f b ( p ) and vX(p) vary slowly with y. P h y s i c a l l y , t h i s detunes t h e resonance by making a whole region near t h e sun equivalent. Mathematically,
it changes t h e phase of t h e second term i n Eq. (4.20). However, t h e l a r g e s c a l e v a r i a t i o n of b ( y ) and v , ( ~ ) i s f i x e d by s o l a r wind observations near t h e e a r t h . Thus, any changes i n vX(p) and
b ( y ) must occur i n f a i r l y small (-- 10 7 km) s t r u c t u r e s o r i n n region near the sun. The l a t t e r case i s p a r t i c u l a r l y i n t e r e s t i n g because of t h e observations by Ekers and L i t t l e (1971) of Large t u r b u l e n t ( i . e . , non-radial) v e l o c i t i e s i n t h e region from 6
71.
The e f f e c t of s e v e r a l k i n d s of v e l o c i t y and d e n s i t y d i s t r i - b u t i o n s on t h e observed s p e c t r a o f DRVID d a t a w i l l be d i s c u s s e d . Three c a s e s a r e easily a n a l y z e d u s i n g Eq. (4.20) :
A l o c a l i z e d d e n s i t y enhancement ("stream" ) of v a r i a b l e width.
A high v e l o c i t y stream.
A v e l o c i t y everywhere p e r p e n d i c u l a r t o t h e r a y p a t h . I n a d d i t i o n a number of c a s e s s i m u l a t i n g a r e g i o n o f t u r b u l e n t v e l o c i t y o r a slow d e n s i t y f a l l - o f f near t h e sun were i n v e s t i - g a t e d w i t h t h e computer program used t o numerically i.nte~.:rate E q . (4.16). The r e s u l t s o b t a i n e d f o r t h e s e p e r t u r b e d c a s e s
may b e cclmpared t o a s l i g h t l y s i m p l i f i e d form o f E q . ( 4 . 1 9 ) , v i z .
where d = 4mrq/c, and
Y +
@/2 = 2 .A h t g h d e n s i t y stream of v a r i a b l e width can be r e p r e s e n t e d by a d e l t a f u n c t i o n as
where
R
i s t h e dimensionless width of t h e stream, A0- -
o
Lstreamk ,
measured a t I i s t h e square o f t h e o v e r d e n s i t y of t h e
0' S
stream, and IJ, i s t h e p o s i t i o n a t which t h e stream c r o s s c s t11e S
r a y p a t h . Such n stream adds a c o n s t a n t columnar c o n t e n t czt
72
along with a r a d i a l v e l o c i t y , one f i n d s f o r t h e a d d i t i o n t o t h e spectrum of DRVID d a t a
c
(4
Xnt stream
For reasonable values of I
R
of t h e same order a s D, t h e S 0'p e r t u r b a t i o n i s only important f o r y 0 .
he
range of 1 . ~ i s S0 <
-
)I <."
30, f o r r a y p a t h s near t h e s u n . ) The stream must i n t e r - s e c t t h e ray p a t h near t h e sun i n order t o have a s i g n i f i c a n t s p e c t r a l amplitude. The cosine term i s then i n phase with the,&i n Eq. (4.21) and no c a n c e l l a t i o n of t h e minima occurs. Note t h a t t h e spectrum of the stream has t h e same q dependence a s t h e unperturbed spectrum.
A high v e l o c i t y stream represented by v16()r-1L ) coupled 8
with t h e normal d e n s i t y d i s t r i b u t i o n w i l l c l e a r l y give a r e s u l t very s i m i l a r i n form and i n t e r p r e t a t i o n t o Eq. ( t 2 To study a v e l o c i t y everywhere normal t o the r a y path, i . e . , vX(p) = vO, t h e i n t e g r a l i n ~ ~ . ( 4 . 2 0 ) must be c a r r i e d out by contour i n t e g r a t i o n . The r e s u l t i s ( t o within t h e approximations used) i s e x a c t l y t h e same a s Eq,. (4.21).
From t h e s e t h r e e c a s e s i t i s 73 c l e a r t h a t n e i t h e r streams nor l a r g e s c a l e v e l o c i t y and d e n s i t y d i s t r i b u t i o n s t h a t p r i - marily a f f e c t regions f a r from the sun (p 2
-
2 ) w i l l a f f e c t t h es p e c t r a l minima p r e d i c t e d by Eq. (4.21). However, t h e analysi s suggests t h a t e v e n t s occurring near t h e sun may be very important i n determining t h e shape of t h e DRVID s p e c t r a .
Many numerical i n t e g r a t i o n s of Eq. (4.16) were made i n which t h e v e l o c i t y and/or d e n s i t y i n t h e region within 0.2 a . u . of t h e ray p a t h ' s c l o s e s t approach t o t h e sun were modified.
1'he sense of t h e modifications taas t o reduce or eliminate the r a d i a l dependence ( i . e . dependence on p ) of An and vx i n t h i s region. Reducing t h e p-dependence of vx introduces a non-radial v e l o c i t y component. Fig.
4.4
i s an example of t h e r e s u l t s . I n t h i s case both t h e v e l o c i t y and t h e d e n s i t y were independent of p i n the region near t h e sun. For a l l t h e modified densi- t i e s and v e l o c i t i e s t h e depth of t h e minima was reduced,p a r t i c u l a r l y a t t h e higher frequencies. However, no reasomb La d i s t r i b u t i o n of veloci.t,y and d e n s i t y cou1.d be found which r c - moved the minima completely. Fur t h e m o r e , none of t h e mod i l'i cd d i s t r i b u t i o n s a f f e c t e d t h e l a r g e s c a l e r a d i a l ( q ) dependence of t h e s p e c t r a .
This a n a l y s i s has not considered t h e problem of t h e velocity having a c o r r e l a t i o n s c a l e s i m i l a r t o t h a t of the d e n s i t y . No doubt t h i s w i l l a l s o tend t o reduce t h e s p e c t r a l minima b y snearing the resonance e f f e c t . Ifowever, when t h e nolar wind i t ;
.-
n z ) ~ . Q L > - P w r l -ao) I - u u 7 ' 1 -4lo-"
10-3 LO-?r
Frequency, ( H Y )
7 5
q u i e t , one might hope t h a t t h e v e l o c i t y would be f a i r l y
s t e a d y , and t h i s smearing would be minimized. The minima w i l l a l s o be reduced because near t h e sun t h e d i s t a n c e s c a l e i s only a few times t h e c o r r e l a t i o n s c a l e . This r e s u l t s i n some
broadening of t h e d e l t a f u n c t i o n f r m t h e a - i n t e g r a l i n Eq. (4.10).
The broadened d e l t a f u n c t i o n w i l l f i l l up t h e minima from nearby r e g i o n s o f t h e spectrum. However, a n a l y s i s shows t h a t t h e d e l t a f u n c t i o n approximation should b e adequate i n t h e regime probed by t h e DRVID measurements. The s p e c t r a w i l l be examined f o r t h e p r e d i c t e d minima a t moderate d i s t a n c e s from t h e s u n when t h e s o l a r wind i s q u i e t t o shed l i g h t on t h e v e l o c i t y and d e n s i t y d i s t r i b u t i o n s near t h e sun.
C
.
THE RELATION OF DRVIB SPECTRA TO--
I N SITU SPACECRAFT MEASUREMENTSMost o b s e r v a t i o n s of t h e low-frequency spectrum o f t h e solar-wind turbulence r e p o r t e d t o d a t e have been made by space-
c r a f t a t r e s t (v << v ) with r e s p e c t t o t h e sun. The spectrum found SW
f r ~ m such o b s e r v a t i o n s , F ( v ) , can e a s i l y be r e l a t e d t o t h e n t
comoving wavenumber spectrum of t h e s o l a r wind (Cronyn, 1970).
I n t h e n o t a t i o n of S e c t i o n A t h e r e s u l t i s
where R i s t h e s p a c e c r a f t ' s d i s t a n c e from the sun. I f a power-
0
law spectrum of t h e form (4.13) i s used t h e i n t e g r a l s may be c a r r i e d out t o g i v e
where Eq. (4.15) has been used f o r b ( ~ ~ ) . AS pointed o u t by Cronyn (1970) t h e spectrum i s two powers l e s s s t e e p i n frequency t h a n t h e comoving spectrum. Comparison with t h e DRVID spectrum, Eq. (4.21), shows t h a t F ( v ) i s one power l e s s s t e e p i n f r e -
n t
quency
,
but one power more s t e e p i n h e l i o c e n t r b d i s tsnce.
I n order t o f a c i l i t a t e t h e comparison o f DRVID s p e c t r a t o t h o s e of other i n v e s t i g a t o r s it i s u s e f u l t o w r i t e t h e DRVU)
spectrum c (v ) i n terms of F , ~ ( v ) . The comoving amplitude D n t
can be eliminated between Eqs. (4.21) and (4.26) t o g i v e
+2
( 1 +
cad
cos dp2)2
Eq. (4.28) s t r i c t l y holds only f o r y+@/2. However, t h e only change caused by a v i o l a t i o n o f t h i s r e s t r i c t i o n i s t o s l i g h t l y modify t h e term i n braces, which i n any case i s of t h e order of
1. Eq. (4.28) w i l l be used i n Chapter V t o f i n d t h e index y which c h a r a c t e r i z e s t h e f a l l - o f f of t h e d e n s i t y f l u c t u a t i o n s with h e l i o c e n t r i c d i s t a n c e .
77
V : DRVm MEASUREMENTS OF THE SOLAR-WIND TURBULENCE
The physics, implementation, and gathering of DRVIH) d a t a have been discussed i n Chapter 11. Data reduction procedures were out- l i n e d i n Chapter 111. Chapter IV derived t h e r e l a t i o n s h i p between t h e spectrum of DRVID d a t a and t h e cmoving wavenumber spectrum of t h e s o l a r wind. I n t h i s chapter t h e s p e c t r a of t h e DRVID d a t a ob- t a i n e d during t h e m69 and MM'7'1 missions a r e presented.
I n t h e f i r s t s e c t i o n the DRVID d a t a records a r e discussed i n terms of t h e i r columnar content change and t h e s i z e of t h e d e n s i t y f l u c t u a t i o n s which c o n t r i b u t e t o t h e change. The data show t h a t t h e dominant s c a l e s i z e f o r d e n s i t y changes i s
-
1.5-3.0 x 10 6 km, i n agreement with t h e suggestion of J o k i p i i and Hollweg (1970). Local d e n s i t y changes exceeding 100$ a r e i n f e r r e d .The average s p e c t r a of t h e DRVID d a t a f o r t h e MI469 and W1 missions a r e presented. The s p e c t r a may be well represented by a power-law whose slope implies a cmoving s p e c t r a l index of
p = 3.9 1 0 . 2 f o r 1.0 x l f 4 d \, 5 i x Hz. The slope and ampli- tude of t h e s e s p e c t r a a r e compared t o t h e spectrum of proton d e n s i t y f l u c t u a t i o n s of I n t r i l i g a t o r and Wolfe (1970) ( h e r e a f t e r c a l l e d IW)
.
The D R V D s p e c t r a a r e somewhat s t e e p e r t h a n t h a t of I W whose data crave a comovinl; s p e c t r a l index of P = 3.3 + 0 . 3 . I n order f o r t h e
i> -
78
amplitudes of t h e DRVID s p e c t r a t o agree with t h a t of IN t h e rms solar-wind d e n s i t y f l u c t u a t i o n s must (1) d e c l i n e a s r -2.38-t-0.11
-
between r
"
0.15 and r = 1 a . u . , assuming no long-term time v a r i a - t i o n s ; o r ( 2 ) d e c l i n e a p p r o x i m t e l y a s r -2,
i f t h e turbulence am- p l i t u d e i s p r o p o r t i o n a l t o sunspot number throughout t h e s o l a r c y c l e.
The temporal behavior of t h e M M 7 1 d a t a was i n v e s t i g a t e d . The post-conjunction s p e c t r a have s y s t e m a t i c a l l y lower amplitudes t h a n t h e pre-conjunction s p e c t r a . This v a r i a t i o n c o r r e l a t e s with
l a r g e s c a l e i n d i c e s of s o l a r a c t i v i t y , v i z . Zurich sunspot number and 2.8 GHz r a d i o f l u x . No c o r r e l a t i o n with heliographic l a t i t u d e and longitude, or with McMath plage regions i s found.
To i n v e s t i g a t e t h e v a r i a t i o n with r a d i u s near t h e sun t h e DRVID s p e c t r a were grouped by t h e d i s t a n c e of t h e r a y p a t h ' s c l o s e s t approach t o t h e sun ( q ) and averaged. For t h e MM71 d a t a
~ ( r ) i s p r o p o r t i o n a l t o
r-1*5+0.2
i n t h e r e g i o n 0.07 5 r 5 0.22 a.u.The slope f o r t h e MM6p d a t a i s not w e l l determined but i s approxi- mately ,-2.0 t o -2.5.
The average s p e c t r a c o n t a i n only marginal evidence f o r t h e e x i s t e n c e of t h e minima found i n t h e a n a l y s i s of Chapter IV.
The observational r e s u l t s on t h e spectrum of t h e solar-wind turbulence a r e summarized a t t h e end of t h i s _ . chapter. The impli- c a t i o n s of t h e s e r e s u l t s f o r t h e o r i e s of t h e s o l a r wind a r e d i s - cussed i n Chapter VII.
79 A.
P l o t s of t h e DRVID d a t a records r e v e a l much i n f o m a t i o n about t h e s c a l e of f l u c t u a t i o n s i n t h e solar-wind d e n s i t y . Callahan
(1973) discusses some of t h e W1 D R V D data q u a l i t a t i v e l y . The d a t a my be c h a r a c t e r i z e d by t h e change i n t h e columnar content during a day, t h e r a t e of t h e columnar content c b w e , and t h e o v e r a l l s k p e of t h e change.
The BRVTS) records show t h a t t h e c o l r content c b n g e i n a record (6-10 hours) i s highly v a r i a b l e f r o n t day t o day. Values of hI a r e r a t h e r uniformly spread i n t h e i n t e r v a l 1.8 t o 20 x 10 14
-2 -2
cn
,
with two examples approaching 30 x 10 on.
Theday t o day v a r i a t i o n s i n AI m s k any changes caused by t h e varying d i s t a n c e of t h e r a y p a t h f'ram t h e sun. Figure 5 . 1 shows t h e stan- dard d e v i a t i o n of each DRVlD record about i t s mean p l o t t e d a p i n s t q. There i s some i n d i c a t i o n t h a t t h e value i n c r e a s e s near t h e sun, b u t t h e s c a t t e r i s n e a r l y a s l a r g e as t h e e f f e c t .
The r a t e of chebnge of hl: can be found frm t h e p l o t s of t h e
-
1D R V m d a t a . Typical r a t e s a r e -- 4 x 10" sec
.
The maximum11 -2 -1
r a t e foundwas 2.5 x 10 cm sec
.
When AI r e v e r s e s i t s sense of v a r i a t i o n , t h e change r e q u i r e s l e s s than one hour. Thus,7 -2 -2
the second d e r i v a t i v e of AI is of t h e order of 2 x 1 0 cm sec
.
Two general types of D R V D records a r e found. F i r s t , t h e r e are "slopes" i n which t h e columnar content changes a t a r a t h e r uniform r a t e f o r s e v e r a l hours and then slowly t a k e s on a new sate. The d a t a k v e l i t t l e s m l l scale etructure that oxcaede
Figure 5 . 1 . The standard d e v i a t i o n o f the DRVID data records about t h e i r i n d i v i d u a l average v a l u e s p l o t t e d a g a i n s t the d i s t a n c e o f c l o s e s t approach. Note t h e l a r g e s c a t t e r . x,MM69. O,MM71 preconjunction.
@,MM71 p o s t c o n j u n c t i o n . The standard deviation values nay be converted t o one way olumnar content
1
E
(err.'*) by rn!~Lt,ipljring by e.39 x 10
.
81
t h e d a t a n o i s e . Second, t h e r e a r e "humps" i n which t h e columnar content r a t e r e v e r s e s i t s s i g n . The columnar content r a t e domi- n a t e s t h e d a t a noise. Figure 9.2 i s a p a r t i c u l a r l y s t r i k i n g
example of t h i s form. One has t h e impression of tw clouds of high d e n s i t y p l a s m c r o s s i n g the r a y path. I f t h e time s c a l e , -- 10 s e c ,
4
i s m u l t i p l i e d by t h e s o l a r wind v e l o c i t y normal t o t h e r a y path,
-
300 km s e c - l , t h e s c a l e s i z e i s-
3 x 106
)on. Many o t h e r data records have s t r u c t u r e s of a similar s c a l e .A few m i l l i o n kilometers i s o f t e n mentioned a s a t y p i c a l s i z e f o r the most important d e n s i t y f l u c t u a t i o n s i n t h e s o l a r wind. The BRVPB d a t a records support t h i s contention. I n a q u a l i t a t i v e a n a l y s i s t h e DRVID d a t a a r e s e n s i t i v e t o changes t a k i n g more than about 1/2 hour and l e s s than about 18 hours.
However, a l l important columnar content changes appear t o t a k e place on a time s c a l e g r e a t e r than m e hour and u s u a l l y l e s s than
5 hours. These times correspond t o s c a l e s i z e s of g r e a t e r than 1 m i l l i o n , b u t l e s s than 6 m i l l i o n kilometers, i f t h e v e l o c i t y normal t o t h e r a y path i s taken t o be 300 km s e c - l .
If one assumes t h a t t h e humps a r e caused by a s i n g l e , dense, roughly s p h e r i c a l cloud crossing t h e r a y path, t h e l o c a l d e n s i t y change can be i n f e r r e d . The thickness of t h e cloud i s
-
L,where L i s estimated a s above, and An = AI/L. For t h e d a t a of Figure 5.2 one g e t s An
-
2 xlo3,
which r e p r e s e n t s about a 20@change i n t h e l o c a l e l e c t r o n d e n s i t y (assumed t o vary as r 2 ) i f t h e region c r o s s e s t h e r a y m t h a t i t s p o i n t of c l o s e s t approach t o t h e sun. Of t h e 50 data records used i n this study, 12 show columnar content changes e q u a l l i n g o r exceeding that in Figure 5.2.
HOURS UTC
F i g u r e 5 . 2 . An example o f a DRVID d a t a r e c o r d showing a l a r g e columnar c o n t e n t change. The columnar c o n t e n t change can be viewed a s two c l o u d s c r o s s i n g t h e l i n e of s i g h t . The d a t a were o b t a i n e d w h i l e t r a c k i n g M a r i n e r 7 on 1970 May29. The s u n - e a r t h - p r o b e a n g l e was 6 4 and t h e d i s t a n c e t o t h e s p a c e - c r a f t w a s 2 . 5 a . u .
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
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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,
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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.