- Email: [email protected]
Turbulence in planetary occultations.
ICARUS 37, 322--335 (1979)
Turbulence in Planetary Occultations IV. Power Spectra of Phase and Intensity Fluctuations BJARNE
S. H A U G S T A D 1
Center for Radar Astronomy, Stanford University, Stanford, California 9~305 Received August 8, 1977; revised March 27, 1978 Power spectra of phase and intensity scintillations during occultation by turbulent planetary atmospheres are significantly affected by the inhomogeneous background upon which the turbulence is superimposed. Such coupling is particularly pronounced in the intensity, where there is also a marked difference in spectral shape between a central and a grazing .occultation. While the former has its structural features smoothed by coupling to the inhomogeneous background, such features are enhanced in the latter. Indeed, the latter power spectrum peaks around the characteristic frequency that is determined by the size of the free-space Fresnel zone and the ray velocity in the atmosphere ; at higher frequencies strong fringes develop in the power spectrum. A confrontation between the theoretical scintillation spectra computed here and those calculated from the Mariner 5 Venus mission by R. Woo, A. Ishimaru, and W. B. Kendall (1974, J. Atmos. Sci. 31, 1698-1706) is inconclusive, mainly because of insufficient statistical resolution. Phase and/or intensity power spectra computed from occultation data may be used to deduce characteristics of the turbulence and to distinguish turbulence from other perturbations in the refractive index. Such determinations are facilitated if observations are made at two or more frequencies (radio occultation) or in two or more colors (stellar occultation). 1. INTRODUCTION I n t e r e s t in w a v e p r o p a g a t i o n t h r o u g h r a n d o m l y i n h o m o g e n e o u s m e d i a s t e m s , in p a r t , f r o m t h e c o n t a m i n a t i n g effect s u c h refractive index inhomogeneities may have on e l e c t r o m a g n e t i c w a v e p r o p a g a t i o n a t microwave and optical frequencies. Over t h e p a s t d e c a d e o r so, i n t e r e s t h a s also f o c u s e d on t h e r e v e r s e p r o b l e m : h o w t h e s t a t i s t i c s of r a n d o m i n h o m o g e n e i t i e s m a y b e s t u d i e d f r o m t h e i r effects on t h e w a v e . A l t h o u g h e x t e n s i v e s t u d i e s of t h i s t y p e h a v e b e e n c o n d u c t e d in t h e E a r t h ' s t u r bulent boundary layer, similar techniques h a v e o n l y r e c e n t l y b e e n u s e d to d e d u c e t h e c h a r a c t e r i s t i c s of t u r b u l e n c e in p l a n e On leave from the Norwegian Defence Research Establishment, P.O. Box 25, N-2007 Kjeller, Norway. 322
0019-1035/79/010322-14502.00/0 Copyright O1979 b y Academic Press, Inc. All r i g h ~ of reproduction in any form r~erved.
tory atmospheres. Assuming the turbulence t o b e c o n f i n e d in a s p h e r i c a l l a y e r , W o o a n d I s h i m a r u (1974) e s t a b l i s h e d t h e p h a s e a n d l o g - a m p l i t u d e p o w e r s p e c t r a as seen from a spacecraft partially occulted by a planetary atmosphere. The theoretical spectra were then used to interpret observat i o n s of s i g n a l c h a r a c t e r i s t i c s a s s u m e d t o b e c a u s e d b y t u r b u l e n c e in t h e u p p e r a t m o s p h e r e of Venus, as o b t a i n e d b y t h e M a r i n e r 5 a n d 10 r a d i o e x p e r i m e n t s ( W o o et al., 1974; W o o , 1975). H o w e v e r , t h e a b o v e a n a l y s e s d i d n o t i n c l u d e effects d u e to coupling between the turbulence and the inhomogeneous background represented b y t h e q u i e s c e n t or a v e r a g e a t m o s p h e r e . 2 The effects of an inhomogeneous background on the fluctuations in signal intensity were later studied theoretically by these authors (see Woo et al., 1975; Ishimaru, 1977).
TURBULENCE IN OCCULTATIONS B u t we have shown in a previous paper (Haugstad, 1978c; referred to as P a r t I I I ) t h a t coupling to the inhomogeneous background affects the magnitude of the intensity fluctuations, and it is thus r a t h e r to be expected t h a t the power spectrum of this quantity, and also t h a t of the fluctuations in phase of a radio wave, is significantly affected as well. On the basis of the t h e o r y developed in Parts I (Haugstad, 1978a) and I I I we establish here the form of the phase and intensity power spectra t h a t accounts for coupling to the inhomogeneous background. In Section 2 we give the analytical form of these power spectra, and we proceed in Section 3 to present the results of a numerical evaluation of a selection of power spectra. In Section 4 we compare the theoretical results with observational evidence relating to turbulence in planet a r y atmospheres, and in Section 5 we discuss how scintillation spectra m a y be used as a diagnostic tool for probing the t u r b u l e n t state of planetary atmospheres, and how effects of turbulence on the signal m a y be distinguished from other, nonr a n d o m types of refractive index anomalies. When we refer to any specific equation or section in P a r t s I or III, the appropriate equation or section n u m b e r is preceded b y the R o m a n numeral I or III, respectively. 2. ANALYTICAL POWER SPECTRA Reference is made to the coordinate system defined in Fig. 1. The x axis is along the incoming ray a s y m p t o t e for a
ray from the E a r t h to spacecraft through a spherically symmetric atmosphere. The z axis is in the plane of the curved ray in the direction of bending, and the y axis completes a right-hand system of orthogonal axes. T h e proximal point of the ray is at x = x0, so t h a t L - - x0 = D is approximately the spacecraft-limb distance for a spacecraft located at x = L. T h e E a r t h replaces the spacecraft in Fig. 1 for a stellar occultation observed from the Earth, so t h a t D is the E a r t h - p l a n e t distance in this case. The atmospheric refractivity is the sum of a quiescent component vq, representing the average atmosphere, and a component vt due to turbulence. Let kS1 be the part of the total atmospheric phase perturbation t h a t is linear in ~t; k is the free-space radio wavenumber. The phase p a t h correlation function in the plane x = L is
-- g~(yo, zo)Bsl(O, Ay, Az). (2.1) Here Bsl is the normalized correlation function, and g(yo, zo) represents the slow variation of the rms fluctuations in refractivity with height. For the turbulence model assumed in P a r t I, g is spherically symmetric and varies exponentially with a constant scale height Ht in the radial direction. Thus representation (2.1) is valid only for [hy I and [Az I less t h a n (RHt) 1/~ and Ht, respectively. Assume t h a t turbulent inhomogeneities move across the line of sight at constant speeds v*u and v*z in the y and z directions,
Xo °
323
-~-.-.-.-.-.-.-.-~- ~ ~ ~ ~ ~ J ~ a q
L I
x
SiC
T
FIG. 1. Simplified occultation geometry. The spacecraft has velocity components vx, v~, v,, and a is the impact parameter for a ray deflected through an angle aQ.
324
BJARNE S. HAUGSTAD
respectively, so t h a t hy = v*~r and 5z = v*~r. In terms of the three-dimensional spatial spectrum ts,(u~, u~, u~) of S1, (2.1) then becomes
Xexp{i[u~v*~ -k u~v*~]r}du~du~du,.
(2.2)
The Fourier transform of (2.2) yields for the power spectrum of the phase scintillations (except for the factor ks)
An equation analogous to (2.4) is readily derived for the fluctuations in signal intensity. Let -71 be the first-order fluctuations in signal intensity when normalized by ¢. Using expression (III.2.15) for ~il(Uu, u~) we obtain
W2(oJ)-=167r2kS(RHt)1/2(~t2)/f ¢~(u) --
ao
Xsin 2 [ (D/2k) (u~2 -b tu~2) ]~[u~v*~
-~u,v*, - ~]du~du~.
(2.5)
Equations (2.4) and (2.5) both reduce to the respective R y t o v results for these quantities in the limit ¢ -~ 1. X~[uyv*y -F u,v*, ~]du~du~du,, (2.3) While (2.5) assumes a point source, the corresponding result for stellar occultawhere ~( ) is the delta function. In view tions m a y be obtained by substituting for of the previous restriction on hy and hz ~bi~ the form of this function t h a t includes it follows t h a t (2.3) is limited to frequen- averaging over the source. For the case cies o~ > v*/Ht, where v* -= (v*2~ + v*~) 1/a. in which the radius of the star as projected According to (2.3), any pair of spatial onto the occulting atmosphere exceeds the frequencies u~, u~ satisfying uvv*~ + u~v*~ radius of the principal Fresnel zone, = ~0 contributes to the same temporal ¢21(u~, u,) is given implicitly by (III.2.22). frequency o~, with the different contribu- For the scintillation power spectrum we tions weighted by Cs~ (u~, u~, u~). Realizing thus obtain t h a t 2~(S1 ~) is just the integral of ¢sl over all u~, u~, u,, we find, upon combining (I.7.1c) ~ with (2.3),
f
"~f
WI(~) = 2 7 r / / / ~ S l ( U , , -
u~, u~)
-
--¢o
X (u~2 ~- 4)u,2)~[2Jl(a,u*)/a,u*] 2
X~Eu~v*~ q- u,v** ×cos ~ (u,~D/2k)~[u,,v*~
+ u.v*. -
Xdu~du,.
-
-
o~]du~du,, (2.6)
,,,] (2.4)
Here ~(u) = ~F(u~ 2 + u~2)1/2] is the normalized power spectrum of the stochastic part of the turbulent refractivity (see Section 1.6), u , = (u~2+ ~bu~)~/2, and ~b is fractional intensity t h a t would be observed in a nonturbulent atmosphere if limb-curvature effects were negligible. SEquation (I.7.1c) relates 31 to the Fourier transform of yr. ¢si is then the integrand of the quantity ($12) formed from this equation.
where J1 is a Bessel function of order one, u * - - ( u ~ 2 q- ~b2u,2)1/2, and a, is the projected stellar radius before refraction. The factor E2Jl(a.u*)/a,u*] 2 in (2.6) represents the effect on the power spectrum from averaging the signal intensity over an elliptic region of semiminor axis af~b1/2 and semimajor axis af. We note, finally, t h a t in all of Eqs. (2.4) to (2.6), limb-curvature effects have been neglected. However, these equations are readily generalized to include such effects whenever they m a y be important.
TURBULENCE IN OCCULTATIONS 3. NUMERICAL EVALUATION The analytical power spectra established in the previous section are too general for conclusions to be reached about their form. We therefore seek a numerical evaluation of these spectra for a turbulence power spectrum of the form (see Section 1.6) Los +(u)
P (p/2)
-
325 exp( -loSu 2) X [1 + Lo*u2]r 12"
(3.1)
Here lo and L0 are the inner and outer scales, respectively, and p = 11/3 for Kolmogorov turbulence. T h e integral in (2.4) is evaluated b y first transforming it to polar coordinates and performing the integration over the angular variable. After a suitable change of integration variable in the remaining integral, we find
cos 2 { (w~/wo2)xS[cos2 (0~ + 5) + ¢ sin s (0, + 5)]}
Wl((2) ~-- /¢'lOJ~i=1fl ~x
[1 + (w2/wo2)X2"]vn(x2- 1) 1'2
In (3.2) we have defined 01 = cos -1 (1/x), 02 = - - 0 1 , wo = v*/Lo, w~ = 21r1/Sv*/af, 5 = tan -1 (v*~/v%), and K1 = 41r2A(RHt) 1/2 (vt2)/v *, where A is the normalization factor in (3.1). Thus the phase power spectrum is controlled b y the three parameters w0, we, and ~b, where the former f~x W2(~O)
KS(M,=1 ~" )x
dx.
(3.2)
two quantities are the characteristic frequencies associated with the ray speed v* in the atmosphere and the outer scale and the free-space Fresnel zone radius, respectively. For the point source intensity power spectrum (2.5) we similarly obtain
sin s { (~2/woS)xS[eos2 (0, + 5) + ¢ sin s (0~ + 5)]}
[1 + (w2/wo2)XS]~'12(xs -- 1) 1/2
dx,
(3.3)
where x2 = 16r2Ak2(RHt)l/2(vtS)/v * and w0, we, 01, and 0s are defined as before. For stellar occultations we obtain from (2.6)
W3(w)
K3O~3
f~
~=1/-"J~
xSl-eoss (0~ + 5) + ~ sin 2 (0, + 5)] 2
(x 2 -- 1)In(1 + xS/wo2)V/2['cos 2 (Oi + ~) + ~b2 sin s (& + 5)] XJ~S{ (w/wo)Ecos2 (0~ + 5) + ~2 sin s (0~ + 5)]ln}dx.
We have here defined m = 167r2AD2(RHt) 1/s (vtS)/a.:v .4, ~ = v*/as, and ~o, 01, and Os are defined as before. Figures 2, 3 , and 4 show phase power spectra evaluated numerically for different values of the parameters ~b and v%/v*=. In the computations we have assumed Lo = 3af, so t h a t w~/w0 = 3. In Fig. 2 we have first assumed v% = 0, corresponding
(3.4)
to a strictly central occultation. The curve for ¢ = 1 corresponds exactly to the power spectrum calculated b y the R y t o v method. Reducing ~ below unity is seen to reduce the amplitude of the periodic fringes t h a t appear for o~ ~ o~o,b u t it leaves the power spectrum otherwise unchanged. For all values of ~b, a plateau is reached as o~/w0--o0; since L0 ¢, w0-1, the phase
326
BJARNE S. HAUGSTAD
v;/v;
p l i t u d e of t h e f r i n g e s in t h e p o w e r s p e c t r u m for o~ ~> o~c; h o w e v e r , a t @ ~ 0.1 t h e s p e c t r a l f o r m is v i r t u a l l y i n d e p e n d e n t of a n y f u r t h e r d e c r e a s e in ¢. F i g u r e 4 s h o w s h o w t h e c h a n g e f r o m one s p e c t r a l s h a p e t o a n o t h e r p r o c e e d s a s a f u n c t i o n of t h e
:o
PHASE SPECTRUM
,= 0 . 3 3
PHASE SPECTRUM
W(~) iO °
i
I0-'
i
i
L
I
i
I0 °
i
i
~
J
i
i
i
J
I0' OJ/OJ0
LO.b!i!2
FIO. 2. Phase fluctuation power spectrum for a central occultation, as a function of relative average intensity ~band assuming ~/~o0 = 3. spectrum may thus be used to estimate t h e o u t e r scale of t h e t u r b u l e n c e . T h e v a r i a t i o n in s p e c t r a l s h a p e w i t h @ for a g r a z i n g o c c u l t a t i o n (v*~ = 0) is s h o w n in F i g . 3. C o u p l i n g t o t h e i n h o m o g e n e o u s b a c k g r o u n d is seen t o i n c r e a s e t h e a m -
I0 -I
I
I I I I0[ o I
I J II 0[ I
L
I I
,I)/(.L~0
FIG. 3. Phase fluctuation power spectrum for a grazing occultation, as a function of relative average intensity 4~and assuming coc/~o. = 3.
TURBULENCE IN OCCULTATIONS ratio v*~/v*=, assuming t h a t ~b has the constant value 0.1. While coupling between the turbulence and the average atmosphere did not change the gross shape of the phase power spectrum, the opposite is true for the intensity spectrum. Figure 5 shows the
327
v;/v~ = 0
¢=i
INTENSITY SPECTRUM
%(o J) v;/~ = o
~ =o i
V;/V; =4
~
PHASE SPECTRUM
w(~)
v;/vi: s
iO°1 -
v;/vi : 30 iO-I
,
I0 -I
,
,
,
I
.
I0 o
.
.
.
_
I
I0 I
~/~C
Fro. 5. R a d i o o c c u l t a t i o n i n t e n s i t y p o w e r spect r u m for a c e n t r a l occultation, as a f u n c t i o n of relative a v e r a g e i n t e n s i t y q~ a n d a s s u m i n g 0~c/~0 = 3.
k
w(oJ) I0° ~-
I0-*
I0 °
I0 I oJlo~o
FIG. 4. Phase fluctuation power spectrum for different occultation geometries, for a constant
relative average intensity ~ O,o/~o = 3.
=
0.1 and assuming
intensity power spectrum as a function of ¢ for a strictly central occultation. For ~b = 1, the spectrum has the familiar form predicted b y the R y t o v m e t h o d for a homogeneous background. However, with even a small reduction in average signal intensity, the " k n e e " in the spectrum at ¢0 = o0~ has become markedly gradual, and the fringes at higher frequencies have virtually disappeared. B y ~ ~---0.1, the shape has reached a final value as far as variation with this parameter is concerned.
328
BJARNE w2(~ ~b
~
•i.0
S. H A U G S T A D
:
=0.,
,,;/v;:
~o
I N T E N S I T Y SPECTRUM
:/v~=4
"
t
'°iI
I 0°~- Wlcu)
w(~)
to-,L_ i
I0"
,
i
,
I
i
I0 °
,
,
L 1
I0' (M/(.Uc
FIG. 6. Radio occultation intensity power speet r u m for a grazing occultation, as a f u n c t i o n of relative a v e r a g e i n t e n s i t y ¢ a n d a s s u m i n g ~ / ~ o = 3.
IO-'
10 0
I0'
¢o/¢~c Fro. 7. R a d i o o c c u l t a t i o n i n t e n s i t y power spect r u m for different o c c u l t a t i o n geometries, for a cons t a n t relative a v e r a g e i n t e n s i t y ¢ = 0.1 a n d ass u m i n g ~e/~o = 3.
TURBULENCE IN OCCULTATIONS For a strictly grazing occultation we find a somewhat different sequence of spectral shapes, as detailed in Fig. 6. Decreasing ~b below unity has the effect of narrowing W,(~) around ~ , and the fringe amplitude is seen to increase monotonically with further decrease in ~b. Figure 7 shows how W2(~) changes with v*v/v*z forc onstant ~b = 0.1. We conclude from these figures t h a t the form of the intensity power spectrum depends strongly on coupling to the inhomogeneous background and also on the occultation geometry. T h e strong a s y m m e t r y with respect to the value of v*Jv*~ is of course tied to the a s y m m e t r y of the Fresnel zone itself when ~b < 1. Stellar occultation power spectra follow the same general dependence on v*Jv*z and ~ as described above for the radio intensity. We show here only the dependence of W3(w) on ~ for a grazing occultation (Fig. 8). E x c e p t for the larger fringe period, these curves display the same general dependence on ~ and v*~/v*z as do the corresponding radio occultation power spectra. As might be expected, the intensity power spectra reported here for ~ < 1 have features in common with the theoretical spectra calculated b y Young (1969) for rectangular telescope apertures. I t should be stressed t h a t for a nonrotating planet v'y, v*~ coincide with the ray velocity components in the atmosphere, which in turn relate to the spacecraft velocity components vy, v~ as v*~ = v~ and v*~ = ~bv~, provided focusing of the wave front due to limb curvature m a y be neglected. For a rotating planet, v*y should be added to the linear horizontal velocity of the atmosphere relative to the coordinate system. Since ¢ m a y attain values of 0.01 or less, it follows t h a t v*~/v*~, and hence the shape of the intensity power spectrum will change markedly during the occultation. T o a lesser extent the phase power spectrum should also change.
329
-
/
°!I
iOI.l: i
I0-t
,
i
I
i
i01
0° co/~ c
Fro. 8. Stellar occultation intensity power spectrum for a grazing occultation, as a function of relative average intensity ~ and assuming ~/~0 = 3. 4. RELATION TO OBSERVATIONS Experimental intensity scintillation spectra from the Mariner 5 Venus mission have been computed b y Woo et al. (1974), and from the Mariner 10 Venus mission b y Woo (1975). The published spectra from the latter mission lack the low-
330
BJARNE S. HAUGSTAD
10 o
!
i
o•
IO-L
••
10-2-
°
-
•
• 10-3
~
LEVEL
1
f
..
10 0
Hz
FIG. 9. The dotted curve is the intensity power spectrum corresponding to region A of the Mariner 5 exit occultation by Venus, as defined and computed by Woo et al. (1974). Superimposed is the theoretical power spectrum (solid curve) based on the actual occultation parameters. frequency part essential to a comparison with the theoretical spectra derived here, and thus no comparison will be a t t e m p t e d for this case. Figures 3 to 6 of Woo et al. (1974) show experimental intensity scintillation power spectra for different epochs, termed A through D, of the Mariner 5 exit occultation. (The unsmoothed versions of the power spectra for regions A and C are also shown in Figs. 9 and 10.) T h e computational procedure for obtaining these spectra are described in detail in the paper b y Woo et al. and will no~ be iterated here. However, a few comments about the spectra seem to be in order. By inspection of Figs. 4 and 6 of Woo et al., it is seen t h a t the signal-to-noise ratio is probably too marginal for these spectra to be convincing. This contention is supported b y the fact t h a t the " k n e e " in the spectrum in Fig. 4 occurs at a
significantly higher frequency t h a n t h a t of the immediately preceding epoch (i.e., region A), even though the ray speed in the atmosphere decreases slightly from region A to B. Since the theoretical angular cut-off frequency is ~o ]y*l/af, it thus follows t h a t either the spectral features of Fig. 4 do not correspond to well-developed turbulence or t h e y are altogether spurious. In view of the low signal-to-noise ratio, the latter possibility is preferred here. Similar comments apply to the power spectrum in Fig. 6, corresponding to region D of the Mariner 5 exit occultation. But the power spectrum corresponding to region A (Fig. 3) is also a bit problematic. Indeed, the very regular course of this spectrum, b o t h its steady increase with frequency toward the " k n e e " and the regular fringes at higher frequencies, is difficult to reconcile with the large statistical uncertainty in the individual =
TURBULENCE IN OCCULTATIONS power estimates, as indicated b y the error bar in the unsmoothed spectrum. B y contrast, the scatter of points in the power spectrum corresponding to region C appears to be more consistent with the statistical significance of these estimates. The parameters necessary for computing the theoretical counterparts of the spectra of regions A and C of Woo et al. m a y be obtained from Fjeldbo et al. (1971). T h u s aft1.3 km, Ivl ~-~7.5 km sec -1, and inspection of Fig. 5 of Fjeldbo et al. shows t h a t v~, the component of the spacecraft velocity parallel to the planetary limb, is --~0.4 km sec -1 in b o t h regions. Since limb-curvature effects are negligible here, it follows t h a t v*~ ~--v~ and v*~--~ ~bv~. The value of ¢ is found from Fig. 1 of Woo et al. (1974) to be ---~0.05 for region A and ~-~0.01 for region C. The theoretical power spectra corresponding to these values are shown in Figs. 9 and 10, along
331
with the experimental spectra of Woo et al. I t is clear from Fig. 9 t h a t the positions of the "knees" agree quite well, as do the average slope of the spectra for w > we. However, b o t h the m a r k e d decrease in power at frequencies ~ < ¢0c and the sinusoidal fringes at higher frequencies are not reproduced in the theoretical spectra. T o the extent t h a t the experimental spectrum can be trusted (cf. previous comment), this discrepancy is remarkable because the present t h e o r y does indeed predict power spectra of this particular shape (see Figs. 6 and 7), b u t not for the parameters pertinent to region A of the Mariner 5 exit occultation. T h e comparison in Fig. 10 is somewhat difficult in view of the substantial scatter of the experimental points. T h e r e is an indication, however, t h a t the large fringes in the theoretical power spectrum are reproduced in this experiment, and there
100
1 0 -1 _
NOISE LEVEL 10-2_
4P•
•
*: •
•
.
•
. *..,.
.';. :.'.... ::
.
..':-
g° °
10-3 lO-f
O °
Hz
FIG. 10. The dotted curve is the intensity power spectrum corresponding to region C of the Mariner 5 exit occultation by Venus, as defined and computed by Woo et al. (1974). Superimposed is the theoretical power spectrum {solid curve) based on the actual occultation parameters.
332
BJARNE S. HAUGSTAD
m a y be a decrease in power for co < coc as well. In conclusion, the experimental power spectra for the Mariner 5 mission do not appear to be of sufficient quality to provide a satisfactory test on the theory outlined here. T o this end one would like to have spectra of higher statistical significance, as well as spectra covering a greater range in the parameters [v~/v~[ and ¢. In a partial response to this need, in a forthcoming paper we will compare experimental power spectra from the Mariner 10 Venus mission with the theory. 5. DIAGNOSIS OF SMALL-SCALE STRUCTURES We have shown in Part II (Haugstad, 1978b) and I I I t h a t atmospheric profiles derived from either Doppler or intensity measurements are virtually unaffected b y the presence of turbulence in the occulting atmosphere if the weak scattering criterion is satisfied. However, the effects of turbulence and other refractive index irregularities m a y be sufficiently large to yield observable effects in the received signal. We now discuss how such irregularities m a y be identified and studied from an examination of the phase and intensity power spectra of the received signal. Our discussion is rather general and is offered more to provide a guideline than to provide a tight-knit scheme for estimating the strength and character of small-scale atmospheric structures. A general discussion m a y be important, however, for realizing the full potential of the occultation technique to provide information on small-scale atmospheric structures in addition to the gross features commonly studied in experiments of this kind. Once this potential is realized, occultation experiments m a y be planned to take fullest possible advantage of their intrinsic capabilities.
(i) Turbulence Apparently the most direct way to establish the existence of turbulence would be to compare experimentally obtained power spectra of phase a n d / o r intensity with their theoretical counterparts. In making such a comparison it is very imp o r t a n t to realize t h a t theoretical spectra correspond to the idealized situation of averaging over a very large n u m b e r of realizations of the turbulence. However, in a given experimental situation only one such distribution of scale sizes is realized, and the specific power spectra calculated from this piece of the time series may depart very markedly from the theoretical average shape. This was clearly demonstrated by the a t t e m p t e d comparison in the previous section. D e p a r t u r e from the average shape is likely to be particularly pronounced in the upper part of the occultation because the ray moves rapidly in the vertical direction here, so t h a t the time the ray spends within about a scale height Ht (over which meaningful determinations of the power spectra can be made) is correspondingly low. Deeper in the atmosphere, where v*~ is small and the ray is moving essentially in the horizontal direction, the time the ray stays within each scale height is greatly increased, and power spectra corresponding to essentially the same rms refractivity, but corresponding to different spatial realizations of the turbulence, m a y be averaged to yield estimates t h a t more closely resemble the theoretical spectra. If turbulence is detected, natural quantities to estimate are the rms refractivity and its variation with height, and perhaps also the outer scale of the turbulence. The latter q u a n t i t y m a y be obtained from the phase power spectrum, since the " k n e e " in this spectrum is determined b y L0 and the ray speed in the atmosphere, which is known. For stellar occultations there is no phase information, and L0 must here be
TURBULENCE IN OCCULTATIONS determined from other considerations. A rough estimate would be Hq, since it is difficult to maintain an isotropic input of energy over vertical distances much greater t h a n a scale height. High in the atmosphere it is also hard to visualize processes b y which L0 could be limited to values much smaller t h a n Hq. Once L0 has been measured or estimated, the rms turbulent refractivity m a y be obtained from the equation expressing the scintillation index in terms of the rms refractivity, @t~)~/2. In view of (III.4.1) and (III.4.4), we have (assuming ~ < 1) (~ "~ ~(p) (Lo/a) (3-p)/2
X [(RHt)~/4D/a3/2-]@t2) 1/2, (5.1) where a is the scintillation index, ~(p) is a function of the slope p a r a m e t e r p t h a t is different for radio [-~(11/3) "~ 5.7J and stellar [~(11/3) -'~ 1.9-] occultations, a = a~ for radio occultations, and for stellar occultations a = a., provided as is somewhat larger t h a n the principal Fresnel zone in this case. According to (5.1), a ~ Lol/3Ht 1/4 (assuming p = 11/3), which implies t h a t the rms refractivity m a y be determined quite accurately from an evaluation of a even though L0 and H t a r e not known with a high precision. Thus a depends essentially on @t2)1/2 alone, so t h a t Ht m a y be accurately determined knowing the variation of ~ with height. If this variation differs appreciably from t h a t of an exponential, the actual course of @t2)1/2 with height m a y be used to construct a refined alternative to the exponential model assumed throughout, and which in turn m a y result in an improved formula of type (5.1) and thus a more significant estimate of (pt2) 112 itself. The above procedure illustrates how to determine relevant turbulence characteristics given t h a t turbulence has been positively identified. We have suggested t h a t this could be done b y comparing the data with the theoretical power spectra, b u t for reasons outlined above experi-
333
mental spectra m a y deviate markedly from their theoretical counterparts and yet be caused b y turbulence. As an alternative, or complementary, approach, one could look for any wavelength dependences if measurements are done at more t h a n one wavelength, or for stellar occultations in two or more colors. For radio occultations, assuming Kolmogorov turbulence, the scintillation index is proportional to ~-7/~, and a should scale with wavelength according to this relationship if the intensity scintillations are caused b y well-developed turbulence. However, the most reliable check on turbulence using the wavelength dependence would probably be to identify the " k n e e " in the radio intensity power spectrum at ~ = we, and see if it scales with wavelength as k 1/2. Within certain limits, this scaling m a y be shown to be independent of the detailed shape of the turbulence power spectrum. For optical occultations the scintillations do not depend on wavelength when as > af, and all spectral features should be independent of h if the scintillations are caused by turbulence alone. (ii) Layers
Localized layers might produce features in the i n t e n s i t y - t i m e record t h a t resemble turbulence. However, if over some interval of height individual structures have a multiple structure with roughly constant separation in altitude, as might be caused b y large-scale coherent waves propagating from deeper regions of the atmosphere, then in the power spectrum the resultant intensity fluctuations would be strongly confined in frequency, whereas turbulence would be identified b y the presence of a continuum of scale sizes. Furthermore, should such structures extend over several scale heights, the frequency at which two adjacent layers are traversed b y the ray will decrease markedly as the vertical ray velocity in the atmosphere is slowed b y
334
BJARNE S. HAUGSTAD
differential refraction, and the structure in W(~) that is caused by layers should shift to progressively lower frequencies as the ray probes increasingly deeper regions of the atmosphere. If the simultaneous phase spectrum is available, layered structure there would also have the appearance of sharp, isolated structures corresponding to the rate at which adjacent layers are traversed. Intensity scintillations caused by layered structures should have no dependence on wavelength, unless the vertical scale over which the density variation occurs is comparable to or smaller than the principal Fresnel zone.
(iii) Absorptive structures Variations in signal intensity resembling those produced by turbulence may be caused by "clouds" of absorbing material in the atmosphere; the clouds may have a distribution of scale sizes. If the refractive anomaly associated with such clouds is sufficiently small, they may be distinguished by the wavelength dependence, which will be typical of the absorbing mechanism and is the same for all scale sizes. The simultaneous phase spectrum will contain no structure corresponding to those found in the intensity power spectrum, again provided that the refractivity perturbations associated with the absorber are sufficiently small, and provided also that the intensity perturbations are small. In a practical context, small-scale structures in the intensity-time record may not be caused by any one of the possibilities considered above, but rather by a combination of several scintillation sources. With the limited amount of data available, reliable identification of these sources may have to be based on information additional to that provided by the power spectra. Meteorological arguments, for instance, may be advanced to argue for or against a layering hypothesis, or whether or not absorbing material is likely to
exist at the atmospheric level being probed. Together with considerations of this type, experimentally obtained power spectra of phase and/or intensity may provide an important new source of information about the small-scale structures of planetary atmospheres, as originally suggested by Woo et al. (1973). 6. CONCLUDING REMARKS We have shown that coupling to the inhomogeneous background significantly alters the shape of the intensity power spectrum. While the form of this spectrum is the same for all occultation geometries when the background is homogeneous, we now find a marked asymmetry with respect to the ratio v*~/v*~ when the average intensity is reduced by about a factor of 2 or more. To a lesser extent, the phase power spectrum also becomes asymmetric with respect to v*y/v*~. Although the results presented here neglect the effect of limb curvature, the necessary accommodations to incorporate such effects are straightforward. It is believed that experimentally determined phase and intensity power spectra, when compared with their theoretical counterparts, may be analyzed for turbulence and other smallscale structures along the general guidelines given in Section 5. This technique could be developed beyond the point considered here, for example, by assuming specific models for either layers or absorptive structures, and then evaluating the quantitative effects on phase and/or intensity as a function of the parameters of these models. A natural extension of the results derived here, and also of those given in Parts I, II, and III, would be to develop analogous results for the strong scintillation case. Steps in this direction have already been taken by Ishimaru (1977) for the scintillations in intensity. Ultimately, however, a complete theory should be developed that also covers the phase
TURBULENCE IN OCCULTATIONS a n d b e n d i n g a n g l e of t h e wave. F r o m s u c h a d e s c r i p t i o n one m i g h t assess t h e effects of t u r b u l e n c e o n a t m o s p h e r i c profiles d e r i v e d f r o m e i t h e r D o p p l e r or intensity measurements under the most general conditions. Such a theory would n a t u r a l l y p r o v i d e t h e p o w e r s p e c t r a for these q u a n t i t i e s as well, a n d c o n s i d e r a t i o n s a n a l o g o u s to t h o s e p r e s e n t e d here (which are l i m i t e d to w e a k s c a t t e r i n g conditions) could be made. ACKNOWLEDGMENTS The author is very much indebted to Professors V. R. Eshlcman and G. L. Tyler, whose continued interest, encouragement, and constructive criticism have been of great help throughout the preparation of this work. Discussions with Drs. T. A. Croft, E. C. Ha, R. W. Lee, and Prof. A. T. Waterman, Jr., are also appreciated. This work has been supported in part by the Royal Norwegian Council for Scientific and Industrial Research and by the Planetary Atmospheres Program of the National Aeronautics and Space Administration under Grant No. NGL 05-020-014. REFERENCES FJELDBO, G., KLIORE, A., AND ESHLEMAN, V. R. (1971). The neutral atmosphere of Venus as studied with the Mariner V radio occultation experiments. Astron. J. 76, 123-140. HAUGSTAD,B. S. (1978a). Turbulence in planetary occultations, I. A theoretical formulation. Icarus 35, 121-138. HAUGSTAD,B. S. (1978b). Turbulence in planetary
335
occultations, II. Effects on atmospheric profiles derived from Doppler measurements. Icarus 35, 41(}-421. HAUGSTAD, B. S. (1978c). Turbulence in planetary occultations, III. Effects on atmospheric profiles derived from intensity measurements. Icarus, 35, 422-435. HAUGSTAD, B. S. (1978d). Effects of the inhomogcneous background on radiation propagating through turbulent planetary atmospheres. Radio Sci., 13, 435-440. ISHIMARU, A. (1977). Effects of ray-bending on wave fluctuations. Paper presented at Annual Meeting of URSI, Stanford University, Calif. YOUNG, A. T. (1969). Photometric error analysis. VIII. The temporal power spectrum of scintillations. J. Appl. Opt. 8, 869-885. Woo, R,, AND ISHIMARU,A. (1973). Remote sensing of the turbulence characteristics of a planetary atmosphere by radio occultation of a space probe. Radio Sci. 8, 103-108. Woo, R., AND ISHIMARU, A. (1974). Effects of turbulence in a planetary atmosphere on radio occultation. I E E E Trans. AP-22, 566-573. Woo, R., ISHIMARU, A., AND KENDALL, W. B. (1974). Observations of small-scale turbulence in the atmosphere of Venus by Mariner 5. J. Atmos. Sci. 31, 1698-1706. Woo, R. (1975). Observations of turbulence in the atmosphere of Venus using Mariner 10 radio occultation measurements. J. Atmos. Sci. 32, 1084-1090. Woo, R., YANG, F. C., AND ISHIMARU, A. (1975). Wave propagation in a random medium with inhomogeneous background. Proceedings, 1975 Annual Meeting of URSI, University of Colorado, Boulder.
Turbulence in Planetary Occultations IV. Power Spectra of Phase and Intensity Fluctuations BJARNE
S. H A U G S T A D 1
Center for Radar Astronomy, Stanford University, Stanford, California 9~305 Received August 8, 1977; revised March 27, 1978 Power spectra of phase and intensity scintillations during occultation by turbulent planetary atmospheres are significantly affected by the inhomogeneous background upon which the turbulence is superimposed. Such coupling is particularly pronounced in the intensity, where there is also a marked difference in spectral shape between a central and a grazing .occultation. While the former has its structural features smoothed by coupling to the inhomogeneous background, such features are enhanced in the latter. Indeed, the latter power spectrum peaks around the characteristic frequency that is determined by the size of the free-space Fresnel zone and the ray velocity in the atmosphere ; at higher frequencies strong fringes develop in the power spectrum. A confrontation between the theoretical scintillation spectra computed here and those calculated from the Mariner 5 Venus mission by R. Woo, A. Ishimaru, and W. B. Kendall (1974, J. Atmos. Sci. 31, 1698-1706) is inconclusive, mainly because of insufficient statistical resolution. Phase and/or intensity power spectra computed from occultation data may be used to deduce characteristics of the turbulence and to distinguish turbulence from other perturbations in the refractive index. Such determinations are facilitated if observations are made at two or more frequencies (radio occultation) or in two or more colors (stellar occultation). 1. INTRODUCTION I n t e r e s t in w a v e p r o p a g a t i o n t h r o u g h r a n d o m l y i n h o m o g e n e o u s m e d i a s t e m s , in p a r t , f r o m t h e c o n t a m i n a t i n g effect s u c h refractive index inhomogeneities may have on e l e c t r o m a g n e t i c w a v e p r o p a g a t i o n a t microwave and optical frequencies. Over t h e p a s t d e c a d e o r so, i n t e r e s t h a s also f o c u s e d on t h e r e v e r s e p r o b l e m : h o w t h e s t a t i s t i c s of r a n d o m i n h o m o g e n e i t i e s m a y b e s t u d i e d f r o m t h e i r effects on t h e w a v e . A l t h o u g h e x t e n s i v e s t u d i e s of t h i s t y p e h a v e b e e n c o n d u c t e d in t h e E a r t h ' s t u r bulent boundary layer, similar techniques h a v e o n l y r e c e n t l y b e e n u s e d to d e d u c e t h e c h a r a c t e r i s t i c s of t u r b u l e n c e in p l a n e On leave from the Norwegian Defence Research Establishment, P.O. Box 25, N-2007 Kjeller, Norway. 322
0019-1035/79/010322-14502.00/0 Copyright O1979 b y Academic Press, Inc. All r i g h ~ of reproduction in any form r~erved.
tory atmospheres. Assuming the turbulence t o b e c o n f i n e d in a s p h e r i c a l l a y e r , W o o a n d I s h i m a r u (1974) e s t a b l i s h e d t h e p h a s e a n d l o g - a m p l i t u d e p o w e r s p e c t r a as seen from a spacecraft partially occulted by a planetary atmosphere. The theoretical spectra were then used to interpret observat i o n s of s i g n a l c h a r a c t e r i s t i c s a s s u m e d t o b e c a u s e d b y t u r b u l e n c e in t h e u p p e r a t m o s p h e r e of Venus, as o b t a i n e d b y t h e M a r i n e r 5 a n d 10 r a d i o e x p e r i m e n t s ( W o o et al., 1974; W o o , 1975). H o w e v e r , t h e a b o v e a n a l y s e s d i d n o t i n c l u d e effects d u e to coupling between the turbulence and the inhomogeneous background represented b y t h e q u i e s c e n t or a v e r a g e a t m o s p h e r e . 2 The effects of an inhomogeneous background on the fluctuations in signal intensity were later studied theoretically by these authors (see Woo et al., 1975; Ishimaru, 1977).
TURBULENCE IN OCCULTATIONS B u t we have shown in a previous paper (Haugstad, 1978c; referred to as P a r t I I I ) t h a t coupling to the inhomogeneous background affects the magnitude of the intensity fluctuations, and it is thus r a t h e r to be expected t h a t the power spectrum of this quantity, and also t h a t of the fluctuations in phase of a radio wave, is significantly affected as well. On the basis of the t h e o r y developed in Parts I (Haugstad, 1978a) and I I I we establish here the form of the phase and intensity power spectra t h a t accounts for coupling to the inhomogeneous background. In Section 2 we give the analytical form of these power spectra, and we proceed in Section 3 to present the results of a numerical evaluation of a selection of power spectra. In Section 4 we compare the theoretical results with observational evidence relating to turbulence in planet a r y atmospheres, and in Section 5 we discuss how scintillation spectra m a y be used as a diagnostic tool for probing the t u r b u l e n t state of planetary atmospheres, and how effects of turbulence on the signal m a y be distinguished from other, nonr a n d o m types of refractive index anomalies. When we refer to any specific equation or section in P a r t s I or III, the appropriate equation or section n u m b e r is preceded b y the R o m a n numeral I or III, respectively. 2. ANALYTICAL POWER SPECTRA Reference is made to the coordinate system defined in Fig. 1. The x axis is along the incoming ray a s y m p t o t e for a
ray from the E a r t h to spacecraft through a spherically symmetric atmosphere. The z axis is in the plane of the curved ray in the direction of bending, and the y axis completes a right-hand system of orthogonal axes. T h e proximal point of the ray is at x = x0, so t h a t L - - x0 = D is approximately the spacecraft-limb distance for a spacecraft located at x = L. T h e E a r t h replaces the spacecraft in Fig. 1 for a stellar occultation observed from the Earth, so t h a t D is the E a r t h - p l a n e t distance in this case. The atmospheric refractivity is the sum of a quiescent component vq, representing the average atmosphere, and a component vt due to turbulence. Let kS1 be the part of the total atmospheric phase perturbation t h a t is linear in ~t; k is the free-space radio wavenumber. The phase p a t h correlation function in the plane x = L is
Xo °
323
-~-.-.-.-.-.-.-.-~- ~ ~ ~ ~ ~ J ~ a q
L I
x
SiC
T
FIG. 1. Simplified occultation geometry. The spacecraft has velocity components vx, v~, v,, and a is the impact parameter for a ray deflected through an angle aQ.
324
BJARNE S. HAUGSTAD
respectively, so t h a t hy = v*~r and 5z = v*~r. In terms of the three-dimensional spatial spectrum ts,(u~, u~, u~) of S1, (2.1) then becomes
Xexp{i[u~v*~ -k u~v*~]r}du~du~du,.
(2.2)
The Fourier transform of (2.2) yields for the power spectrum of the phase scintillations (except for the factor ks)
An equation analogous to (2.4) is readily derived for the fluctuations in signal intensity. Let -71 be the first-order fluctuations in signal intensity when normalized by ¢. Using expression (III.2.15) for ~il(Uu, u~) we obtain
W2(oJ)-=167r2kS(RHt)1/2(~t2)/f ¢~(u) --
ao
Xsin 2 [ (D/2k) (u~2 -b tu~2) ]~[u~v*~
-~u,v*, - ~]du~du~.
(2.5)
Equations (2.4) and (2.5) both reduce to the respective R y t o v results for these quantities in the limit ¢ -~ 1. X~[uyv*y -F u,v*, ~]du~du~du,, (2.3) While (2.5) assumes a point source, the corresponding result for stellar occultawhere ~( ) is the delta function. In view tions m a y be obtained by substituting for of the previous restriction on hy and hz ~bi~ the form of this function t h a t includes it follows t h a t (2.3) is limited to frequen- averaging over the source. For the case cies o~ > v*/Ht, where v* -= (v*2~ + v*~) 1/a. in which the radius of the star as projected According to (2.3), any pair of spatial onto the occulting atmosphere exceeds the frequencies u~, u~ satisfying uvv*~ + u~v*~ radius of the principal Fresnel zone, = ~0 contributes to the same temporal ¢21(u~, u,) is given implicitly by (III.2.22). frequency o~, with the different contribu- For the scintillation power spectrum we tions weighted by Cs~ (u~, u~, u~). Realizing thus obtain t h a t 2~(S1 ~) is just the integral of ¢sl over all u~, u~, u,, we find, upon combining (I.7.1c) ~ with (2.3),
f
"~f
WI(~) = 2 7 r / / / ~ S l ( U , , -
u~, u~)
-
--¢o
X (u~2 ~- 4)u,2)~[2Jl(a,u*)/a,u*] 2
X~Eu~v*~ q- u,v** ×cos ~ (u,~D/2k)~[u,,v*~
+ u.v*. -
Xdu~du,.
-
-
o~]du~du,, (2.6)
,,,] (2.4)
Here ~(u) = ~F(u~ 2 + u~2)1/2] is the normalized power spectrum of the stochastic part of the turbulent refractivity (see Section 1.6), u , = (u~2+ ~bu~)~/2, and ~b is fractional intensity t h a t would be observed in a nonturbulent atmosphere if limb-curvature effects were negligible. SEquation (I.7.1c) relates 31 to the Fourier transform of yr. ¢si is then the integrand of the quantity ($12) formed from this equation.
where J1 is a Bessel function of order one, u * - - ( u ~ 2 q- ~b2u,2)1/2, and a, is the projected stellar radius before refraction. The factor E2Jl(a.u*)/a,u*] 2 in (2.6) represents the effect on the power spectrum from averaging the signal intensity over an elliptic region of semiminor axis af~b1/2 and semimajor axis af. We note, finally, t h a t in all of Eqs. (2.4) to (2.6), limb-curvature effects have been neglected. However, these equations are readily generalized to include such effects whenever they m a y be important.
TURBULENCE IN OCCULTATIONS 3. NUMERICAL EVALUATION The analytical power spectra established in the previous section are too general for conclusions to be reached about their form. We therefore seek a numerical evaluation of these spectra for a turbulence power spectrum of the form (see Section 1.6) Los +(u)
P (p/2)
-
325 exp( -loSu 2) X [1 + Lo*u2]r 12"
(3.1)
Here lo and L0 are the inner and outer scales, respectively, and p = 11/3 for Kolmogorov turbulence. T h e integral in (2.4) is evaluated b y first transforming it to polar coordinates and performing the integration over the angular variable. After a suitable change of integration variable in the remaining integral, we find
cos 2 { (w~/wo2)xS[cos2 (0~ + 5) + ¢ sin s (0, + 5)]}
Wl((2) ~-- /¢'lOJ~i=1fl ~x
[1 + (w2/wo2)X2"]vn(x2- 1) 1'2
In (3.2) we have defined 01 = cos -1 (1/x), 02 = - - 0 1 , wo = v*/Lo, w~ = 21r1/Sv*/af, 5 = tan -1 (v*~/v%), and K1 = 41r2A(RHt) 1/2 (vt2)/v *, where A is the normalization factor in (3.1). Thus the phase power spectrum is controlled b y the three parameters w0, we, and ~b, where the former f~x W2(~O)
KS(M,=1 ~" )x
dx.
(3.2)
two quantities are the characteristic frequencies associated with the ray speed v* in the atmosphere and the outer scale and the free-space Fresnel zone radius, respectively. For the point source intensity power spectrum (2.5) we similarly obtain
sin s { (~2/woS)xS[eos2 (0, + 5) + ¢ sin s (0~ + 5)]}
[1 + (w2/wo2)XS]~'12(xs -- 1) 1/2
dx,
(3.3)
where x2 = 16r2Ak2(RHt)l/2(vtS)/v * and w0, we, 01, and 0s are defined as before. For stellar occultations we obtain from (2.6)
W3(w)
K3O~3
f~
~=1/-"J~
xSl-eoss (0~ + 5) + ~ sin 2 (0, + 5)] 2
(x 2 -- 1)In(1 + xS/wo2)V/2['cos 2 (Oi + ~) + ~b2 sin s (& + 5)] XJ~S{ (w/wo)Ecos2 (0~ + 5) + ~2 sin s (0~ + 5)]ln}dx.
We have here defined m = 167r2AD2(RHt) 1/s (vtS)/a.:v .4, ~ = v*/as, and ~o, 01, and Os are defined as before. Figures 2, 3 , and 4 show phase power spectra evaluated numerically for different values of the parameters ~b and v%/v*=. In the computations we have assumed Lo = 3af, so t h a t w~/w0 = 3. In Fig. 2 we have first assumed v% = 0, corresponding
(3.4)
to a strictly central occultation. The curve for ¢ = 1 corresponds exactly to the power spectrum calculated b y the R y t o v method. Reducing ~ below unity is seen to reduce the amplitude of the periodic fringes t h a t appear for o~ ~ o~o,b u t it leaves the power spectrum otherwise unchanged. For all values of ~b, a plateau is reached as o~/w0--o0; since L0 ¢, w0-1, the phase
326
BJARNE S. HAUGSTAD
v;/v;
p l i t u d e of t h e f r i n g e s in t h e p o w e r s p e c t r u m for o~ ~> o~c; h o w e v e r , a t @ ~ 0.1 t h e s p e c t r a l f o r m is v i r t u a l l y i n d e p e n d e n t of a n y f u r t h e r d e c r e a s e in ¢. F i g u r e 4 s h o w s h o w t h e c h a n g e f r o m one s p e c t r a l s h a p e t o a n o t h e r p r o c e e d s a s a f u n c t i o n of t h e
:o
PHASE SPECTRUM
,= 0 . 3 3
PHASE SPECTRUM
W(~) iO °
i
I0-'
i
i
L
I
i
I0 °
i
i
~
J
i
i
i
J
I0' OJ/OJ0
LO.b!i!2
FIO. 2. Phase fluctuation power spectrum for a central occultation, as a function of relative average intensity ~band assuming ~/~o0 = 3. spectrum may thus be used to estimate t h e o u t e r scale of t h e t u r b u l e n c e . T h e v a r i a t i o n in s p e c t r a l s h a p e w i t h @ for a g r a z i n g o c c u l t a t i o n (v*~ = 0) is s h o w n in F i g . 3. C o u p l i n g t o t h e i n h o m o g e n e o u s b a c k g r o u n d is seen t o i n c r e a s e t h e a m -
I0 -I
I
I I I I0[ o I
I J II 0[ I
L
I I
,I)/(.L~0
FIG. 3. Phase fluctuation power spectrum for a grazing occultation, as a function of relative average intensity 4~and assuming coc/~o. = 3.
TURBULENCE IN OCCULTATIONS ratio v*~/v*=, assuming t h a t ~b has the constant value 0.1. While coupling between the turbulence and the average atmosphere did not change the gross shape of the phase power spectrum, the opposite is true for the intensity spectrum. Figure 5 shows the
327
v;/v~ = 0
¢=i
INTENSITY SPECTRUM
%(o J) v;/~ = o
~ =o i
V;/V; =4
~
PHASE SPECTRUM
w(~)
v;/vi: s
iO°1 -
v;/vi : 30 iO-I
,
I0 -I
,
,
,
I
.
I0 o
.
.
.
_
I
I0 I
~/~C
Fro. 5. R a d i o o c c u l t a t i o n i n t e n s i t y p o w e r spect r u m for a c e n t r a l occultation, as a f u n c t i o n of relative a v e r a g e i n t e n s i t y q~ a n d a s s u m i n g 0~c/~0 = 3.
k
w(oJ) I0° ~-
I0-*
I0 °
I0 I oJlo~o
FIG. 4. Phase fluctuation power spectrum for different occultation geometries, for a constant
relative average intensity ~ O,o/~o = 3.
=
0.1 and assuming
intensity power spectrum as a function of ¢ for a strictly central occultation. For ~b = 1, the spectrum has the familiar form predicted b y the R y t o v m e t h o d for a homogeneous background. However, with even a small reduction in average signal intensity, the " k n e e " in the spectrum at ¢0 = o0~ has become markedly gradual, and the fringes at higher frequencies have virtually disappeared. B y ~ ~---0.1, the shape has reached a final value as far as variation with this parameter is concerned.
328
BJARNE w2(~ ~b
~
•i.0
S. H A U G S T A D
:
=0.,
,,;/v;:
~o
I N T E N S I T Y SPECTRUM
:/v~=4
"
t
'°iI
I 0°~- Wlcu)
w(~)
to-,L_ i
I0"
,
i
,
I
i
I0 °
,
,
L 1
I0' (M/(.Uc
FIG. 6. Radio occultation intensity power speet r u m for a grazing occultation, as a f u n c t i o n of relative a v e r a g e i n t e n s i t y ¢ a n d a s s u m i n g ~ / ~ o = 3.
IO-'
10 0
I0'
¢o/¢~c Fro. 7. R a d i o o c c u l t a t i o n i n t e n s i t y power spect r u m for different o c c u l t a t i o n geometries, for a cons t a n t relative a v e r a g e i n t e n s i t y ¢ = 0.1 a n d ass u m i n g ~e/~o = 3.
TURBULENCE IN OCCULTATIONS For a strictly grazing occultation we find a somewhat different sequence of spectral shapes, as detailed in Fig. 6. Decreasing ~b below unity has the effect of narrowing W,(~) around ~ , and the fringe amplitude is seen to increase monotonically with further decrease in ~b. Figure 7 shows how W2(~) changes with v*v/v*z forc onstant ~b = 0.1. We conclude from these figures t h a t the form of the intensity power spectrum depends strongly on coupling to the inhomogeneous background and also on the occultation geometry. T h e strong a s y m m e t r y with respect to the value of v*Jv*~ is of course tied to the a s y m m e t r y of the Fresnel zone itself when ~b < 1. Stellar occultation power spectra follow the same general dependence on v*Jv*z and ~ as described above for the radio intensity. We show here only the dependence of W3(w) on ~ for a grazing occultation (Fig. 8). E x c e p t for the larger fringe period, these curves display the same general dependence on ~ and v*~/v*z as do the corresponding radio occultation power spectra. As might be expected, the intensity power spectra reported here for ~ < 1 have features in common with the theoretical spectra calculated b y Young (1969) for rectangular telescope apertures. I t should be stressed t h a t for a nonrotating planet v'y, v*~ coincide with the ray velocity components in the atmosphere, which in turn relate to the spacecraft velocity components vy, v~ as v*~ = v~ and v*~ = ~bv~, provided focusing of the wave front due to limb curvature m a y be neglected. For a rotating planet, v*y should be added to the linear horizontal velocity of the atmosphere relative to the coordinate system. Since ¢ m a y attain values of 0.01 or less, it follows t h a t v*~/v*~, and hence the shape of the intensity power spectrum will change markedly during the occultation. T o a lesser extent the phase power spectrum should also change.
329
-
/
°!I
iOI.l: i
I0-t
,
i
I
i
i01
0° co/~ c
Fro. 8. Stellar occultation intensity power spectrum for a grazing occultation, as a function of relative average intensity ~ and assuming ~/~0 = 3. 4. RELATION TO OBSERVATIONS Experimental intensity scintillation spectra from the Mariner 5 Venus mission have been computed b y Woo et al. (1974), and from the Mariner 10 Venus mission b y Woo (1975). The published spectra from the latter mission lack the low-
330
BJARNE S. HAUGSTAD
10 o
!
i
o•
IO-L
••
10-2-
°
-
•
• 10-3
~
LEVEL
1
f
..
10 0
Hz
FIG. 9. The dotted curve is the intensity power spectrum corresponding to region A of the Mariner 5 exit occultation by Venus, as defined and computed by Woo et al. (1974). Superimposed is the theoretical power spectrum (solid curve) based on the actual occultation parameters. frequency part essential to a comparison with the theoretical spectra derived here, and thus no comparison will be a t t e m p t e d for this case. Figures 3 to 6 of Woo et al. (1974) show experimental intensity scintillation power spectra for different epochs, termed A through D, of the Mariner 5 exit occultation. (The unsmoothed versions of the power spectra for regions A and C are also shown in Figs. 9 and 10.) T h e computational procedure for obtaining these spectra are described in detail in the paper b y Woo et al. and will no~ be iterated here. However, a few comments about the spectra seem to be in order. By inspection of Figs. 4 and 6 of Woo et al., it is seen t h a t the signal-to-noise ratio is probably too marginal for these spectra to be convincing. This contention is supported b y the fact t h a t the " k n e e " in the spectrum in Fig. 4 occurs at a
significantly higher frequency t h a n t h a t of the immediately preceding epoch (i.e., region A), even though the ray speed in the atmosphere decreases slightly from region A to B. Since the theoretical angular cut-off frequency is ~o ]y*l/af, it thus follows t h a t either the spectral features of Fig. 4 do not correspond to well-developed turbulence or t h e y are altogether spurious. In view of the low signal-to-noise ratio, the latter possibility is preferred here. Similar comments apply to the power spectrum in Fig. 6, corresponding to region D of the Mariner 5 exit occultation. But the power spectrum corresponding to region A (Fig. 3) is also a bit problematic. Indeed, the very regular course of this spectrum, b o t h its steady increase with frequency toward the " k n e e " and the regular fringes at higher frequencies, is difficult to reconcile with the large statistical uncertainty in the individual =
TURBULENCE IN OCCULTATIONS power estimates, as indicated b y the error bar in the unsmoothed spectrum. B y contrast, the scatter of points in the power spectrum corresponding to region C appears to be more consistent with the statistical significance of these estimates. The parameters necessary for computing the theoretical counterparts of the spectra of regions A and C of Woo et al. m a y be obtained from Fjeldbo et al. (1971). T h u s aft1.3 km, Ivl ~-~7.5 km sec -1, and inspection of Fig. 5 of Fjeldbo et al. shows t h a t v~, the component of the spacecraft velocity parallel to the planetary limb, is --~0.4 km sec -1 in b o t h regions. Since limb-curvature effects are negligible here, it follows t h a t v*~ ~--v~ and v*~--~ ~bv~. The value of ¢ is found from Fig. 1 of Woo et al. (1974) to be ---~0.05 for region A and ~-~0.01 for region C. The theoretical power spectra corresponding to these values are shown in Figs. 9 and 10, along
331
with the experimental spectra of Woo et al. I t is clear from Fig. 9 t h a t the positions of the "knees" agree quite well, as do the average slope of the spectra for w > we. However, b o t h the m a r k e d decrease in power at frequencies ~ < ¢0c and the sinusoidal fringes at higher frequencies are not reproduced in the theoretical spectra. T o the extent t h a t the experimental spectrum can be trusted (cf. previous comment), this discrepancy is remarkable because the present t h e o r y does indeed predict power spectra of this particular shape (see Figs. 6 and 7), b u t not for the parameters pertinent to region A of the Mariner 5 exit occultation. T h e comparison in Fig. 10 is somewhat difficult in view of the substantial scatter of the experimental points. T h e r e is an indication, however, t h a t the large fringes in the theoretical power spectrum are reproduced in this experiment, and there
100
1 0 -1 _
NOISE LEVEL 10-2_
4P•
•
*: •
•
.
•
. *..,.
.';. :.'.... ::
.
..':-
g° °
10-3 lO-f
O °
Hz
FIG. 10. The dotted curve is the intensity power spectrum corresponding to region C of the Mariner 5 exit occultation by Venus, as defined and computed by Woo et al. (1974). Superimposed is the theoretical power spectrum {solid curve) based on the actual occultation parameters.
332
BJARNE S. HAUGSTAD
m a y be a decrease in power for co < coc as well. In conclusion, the experimental power spectra for the Mariner 5 mission do not appear to be of sufficient quality to provide a satisfactory test on the theory outlined here. T o this end one would like to have spectra of higher statistical significance, as well as spectra covering a greater range in the parameters [v~/v~[ and ¢. In a partial response to this need, in a forthcoming paper we will compare experimental power spectra from the Mariner 10 Venus mission with the theory. 5. DIAGNOSIS OF SMALL-SCALE STRUCTURES We have shown in Part II (Haugstad, 1978b) and I I I t h a t atmospheric profiles derived from either Doppler or intensity measurements are virtually unaffected b y the presence of turbulence in the occulting atmosphere if the weak scattering criterion is satisfied. However, the effects of turbulence and other refractive index irregularities m a y be sufficiently large to yield observable effects in the received signal. We now discuss how such irregularities m a y be identified and studied from an examination of the phase and intensity power spectra of the received signal. Our discussion is rather general and is offered more to provide a guideline than to provide a tight-knit scheme for estimating the strength and character of small-scale atmospheric structures. A general discussion m a y be important, however, for realizing the full potential of the occultation technique to provide information on small-scale atmospheric structures in addition to the gross features commonly studied in experiments of this kind. Once this potential is realized, occultation experiments m a y be planned to take fullest possible advantage of their intrinsic capabilities.
(i) Turbulence Apparently the most direct way to establish the existence of turbulence would be to compare experimentally obtained power spectra of phase a n d / o r intensity with their theoretical counterparts. In making such a comparison it is very imp o r t a n t to realize t h a t theoretical spectra correspond to the idealized situation of averaging over a very large n u m b e r of realizations of the turbulence. However, in a given experimental situation only one such distribution of scale sizes is realized, and the specific power spectra calculated from this piece of the time series may depart very markedly from the theoretical average shape. This was clearly demonstrated by the a t t e m p t e d comparison in the previous section. D e p a r t u r e from the average shape is likely to be particularly pronounced in the upper part of the occultation because the ray moves rapidly in the vertical direction here, so t h a t the time the ray spends within about a scale height Ht (over which meaningful determinations of the power spectra can be made) is correspondingly low. Deeper in the atmosphere, where v*~ is small and the ray is moving essentially in the horizontal direction, the time the ray stays within each scale height is greatly increased, and power spectra corresponding to essentially the same rms refractivity, but corresponding to different spatial realizations of the turbulence, m a y be averaged to yield estimates t h a t more closely resemble the theoretical spectra. If turbulence is detected, natural quantities to estimate are the rms refractivity and its variation with height, and perhaps also the outer scale of the turbulence. The latter q u a n t i t y m a y be obtained from the phase power spectrum, since the " k n e e " in this spectrum is determined b y L0 and the ray speed in the atmosphere, which is known. For stellar occultations there is no phase information, and L0 must here be
TURBULENCE IN OCCULTATIONS determined from other considerations. A rough estimate would be Hq, since it is difficult to maintain an isotropic input of energy over vertical distances much greater t h a n a scale height. High in the atmosphere it is also hard to visualize processes b y which L0 could be limited to values much smaller t h a n Hq. Once L0 has been measured or estimated, the rms turbulent refractivity m a y be obtained from the equation expressing the scintillation index in terms of the rms refractivity, @t~)~/2. In view of (III.4.1) and (III.4.4), we have (assuming ~ < 1) (~ "~ ~(p) (Lo/a) (3-p)/2
X [(RHt)~/4D/a3/2-]@t2) 1/2, (5.1) where a is the scintillation index, ~(p) is a function of the slope p a r a m e t e r p t h a t is different for radio [-~(11/3) "~ 5.7J and stellar [~(11/3) -'~ 1.9-] occultations, a = a~ for radio occultations, and for stellar occultations a = a., provided as is somewhat larger t h a n the principal Fresnel zone in this case. According to (5.1), a ~ Lol/3Ht 1/4 (assuming p = 11/3), which implies t h a t the rms refractivity m a y be determined quite accurately from an evaluation of a even though L0 and H t a r e not known with a high precision. Thus a depends essentially on @t2)1/2 alone, so t h a t Ht m a y be accurately determined knowing the variation of ~ with height. If this variation differs appreciably from t h a t of an exponential, the actual course of @t2)1/2 with height m a y be used to construct a refined alternative to the exponential model assumed throughout, and which in turn m a y result in an improved formula of type (5.1) and thus a more significant estimate of (pt2) 112 itself. The above procedure illustrates how to determine relevant turbulence characteristics given t h a t turbulence has been positively identified. We have suggested t h a t this could be done b y comparing the data with the theoretical power spectra, b u t for reasons outlined above experi-
333
mental spectra m a y deviate markedly from their theoretical counterparts and yet be caused b y turbulence. As an alternative, or complementary, approach, one could look for any wavelength dependences if measurements are done at more t h a n one wavelength, or for stellar occultations in two or more colors. For radio occultations, assuming Kolmogorov turbulence, the scintillation index is proportional to ~-7/~, and a should scale with wavelength according to this relationship if the intensity scintillations are caused b y well-developed turbulence. However, the most reliable check on turbulence using the wavelength dependence would probably be to identify the " k n e e " in the radio intensity power spectrum at ~ = we, and see if it scales with wavelength as k 1/2. Within certain limits, this scaling m a y be shown to be independent of the detailed shape of the turbulence power spectrum. For optical occultations the scintillations do not depend on wavelength when as > af, and all spectral features should be independent of h if the scintillations are caused by turbulence alone. (ii) Layers
Localized layers might produce features in the i n t e n s i t y - t i m e record t h a t resemble turbulence. However, if over some interval of height individual structures have a multiple structure with roughly constant separation in altitude, as might be caused b y large-scale coherent waves propagating from deeper regions of the atmosphere, then in the power spectrum the resultant intensity fluctuations would be strongly confined in frequency, whereas turbulence would be identified b y the presence of a continuum of scale sizes. Furthermore, should such structures extend over several scale heights, the frequency at which two adjacent layers are traversed b y the ray will decrease markedly as the vertical ray velocity in the atmosphere is slowed b y
334
BJARNE S. HAUGSTAD
differential refraction, and the structure in W(~) that is caused by layers should shift to progressively lower frequencies as the ray probes increasingly deeper regions of the atmosphere. If the simultaneous phase spectrum is available, layered structure there would also have the appearance of sharp, isolated structures corresponding to the rate at which adjacent layers are traversed. Intensity scintillations caused by layered structures should have no dependence on wavelength, unless the vertical scale over which the density variation occurs is comparable to or smaller than the principal Fresnel zone.
(iii) Absorptive structures Variations in signal intensity resembling those produced by turbulence may be caused by "clouds" of absorbing material in the atmosphere; the clouds may have a distribution of scale sizes. If the refractive anomaly associated with such clouds is sufficiently small, they may be distinguished by the wavelength dependence, which will be typical of the absorbing mechanism and is the same for all scale sizes. The simultaneous phase spectrum will contain no structure corresponding to those found in the intensity power spectrum, again provided that the refractivity perturbations associated with the absorber are sufficiently small, and provided also that the intensity perturbations are small. In a practical context, small-scale structures in the intensity-time record may not be caused by any one of the possibilities considered above, but rather by a combination of several scintillation sources. With the limited amount of data available, reliable identification of these sources may have to be based on information additional to that provided by the power spectra. Meteorological arguments, for instance, may be advanced to argue for or against a layering hypothesis, or whether or not absorbing material is likely to
exist at the atmospheric level being probed. Together with considerations of this type, experimentally obtained power spectra of phase and/or intensity may provide an important new source of information about the small-scale structures of planetary atmospheres, as originally suggested by Woo et al. (1973). 6. CONCLUDING REMARKS We have shown that coupling to the inhomogeneous background significantly alters the shape of the intensity power spectrum. While the form of this spectrum is the same for all occultation geometries when the background is homogeneous, we now find a marked asymmetry with respect to the ratio v*~/v*~ when the average intensity is reduced by about a factor of 2 or more. To a lesser extent, the phase power spectrum also becomes asymmetric with respect to v*y/v*~. Although the results presented here neglect the effect of limb curvature, the necessary accommodations to incorporate such effects are straightforward. It is believed that experimentally determined phase and intensity power spectra, when compared with their theoretical counterparts, may be analyzed for turbulence and other smallscale structures along the general guidelines given in Section 5. This technique could be developed beyond the point considered here, for example, by assuming specific models for either layers or absorptive structures, and then evaluating the quantitative effects on phase and/or intensity as a function of the parameters of these models. A natural extension of the results derived here, and also of those given in Parts I, II, and III, would be to develop analogous results for the strong scintillation case. Steps in this direction have already been taken by Ishimaru (1977) for the scintillations in intensity. Ultimately, however, a complete theory should be developed that also covers the phase
TURBULENCE IN OCCULTATIONS a n d b e n d i n g a n g l e of t h e wave. F r o m s u c h a d e s c r i p t i o n one m i g h t assess t h e effects of t u r b u l e n c e o n a t m o s p h e r i c profiles d e r i v e d f r o m e i t h e r D o p p l e r or intensity measurements under the most general conditions. Such a theory would n a t u r a l l y p r o v i d e t h e p o w e r s p e c t r a for these q u a n t i t i e s as well, a n d c o n s i d e r a t i o n s a n a l o g o u s to t h o s e p r e s e n t e d here (which are l i m i t e d to w e a k s c a t t e r i n g conditions) could be made. ACKNOWLEDGMENTS The author is very much indebted to Professors V. R. Eshlcman and G. L. Tyler, whose continued interest, encouragement, and constructive criticism have been of great help throughout the preparation of this work. Discussions with Drs. T. A. Croft, E. C. Ha, R. W. Lee, and Prof. A. T. Waterman, Jr., are also appreciated. This work has been supported in part by the Royal Norwegian Council for Scientific and Industrial Research and by the Planetary Atmospheres Program of the National Aeronautics and Space Administration under Grant No. NGL 05-020-014. REFERENCES FJELDBO, G., KLIORE, A., AND ESHLEMAN, V. R. (1971). The neutral atmosphere of Venus as studied with the Mariner V radio occultation experiments. Astron. J. 76, 123-140. HAUGSTAD,B. S. (1978a). Turbulence in planetary occultations, I. A theoretical formulation. Icarus 35, 121-138. HAUGSTAD,B. S. (1978b). Turbulence in planetary
335
occultations, II. Effects on atmospheric profiles derived from Doppler measurements. Icarus 35, 41(}-421. HAUGSTAD, B. S. (1978c). Turbulence in planetary occultations, III. Effects on atmospheric profiles derived from intensity measurements. Icarus, 35, 422-435. HAUGSTAD, B. S. (1978d). Effects of the inhomogcneous background on radiation propagating through turbulent planetary atmospheres. Radio Sci., 13, 435-440. ISHIMARU, A. (1977). Effects of ray-bending on wave fluctuations. Paper presented at Annual Meeting of URSI, Stanford University, Calif. YOUNG, A. T. (1969). Photometric error analysis. VIII. The temporal power spectrum of scintillations. J. Appl. Opt. 8, 869-885. Woo, R,, AND ISHIMARU,A. (1973). Remote sensing of the turbulence characteristics of a planetary atmosphere by radio occultation of a space probe. Radio Sci. 8, 103-108. Woo, R., AND ISHIMARU, A. (1974). Effects of turbulence in a planetary atmosphere on radio occultation. I E E E Trans. AP-22, 566-573. Woo, R., ISHIMARU, A., AND KENDALL, W. B. (1974). Observations of small-scale turbulence in the atmosphere of Venus by Mariner 5. J. Atmos. Sci. 31, 1698-1706. Woo, R. (1975). Observations of turbulence in the atmosphere of Venus using Mariner 10 radio occultation measurements. J. Atmos. Sci. 32, 1084-1090. Woo, R., YANG, F. C., AND ISHIMARU, A. (1975). Wave propagation in a random medium with inhomogeneous background. Proceedings, 1975 Annual Meeting of URSI, University of Colorado, Boulder.