Outer scale of solar wind turbulence deduced from two-way coronal radio sounding experiments

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1 Advances in Space Research 33 (2004) Outer scale of solar wind turbulence deduced from two-way coronal radio sounding experiments A.I. Efimov a, M.K. Bird b, *, I.V. Chashei c, L.N. Samoznaev a a Institute of Radio Engineering and Electronics, Russian Academy of Science, Moscow, Russia b Radioastronomisches Institute, Universit at Bonn, Bonn, Germany c Lebedev Physics Institute, Russian Academy of Science, Moscow, Russia Abstract Results of two-way mode coronal radio sounding experiments using the Helios-1,-2, Pioneer-10,-11, and Viking-1 spacecraft are presented. These observational data apply to the low-latitude solar wind during solar activity minimum. Enhancements of the Doppler scintillation variance were detected for spacecraft located closer to the effective scattering layer with respect to the variance measured for a more distant spacecraft at the same time and solar elongation. These enhancements can be explained by the assumption that the radio signal passes twice through the same solar wind density irregularities for close spacecraft but through uncorrelated irregularities when the heliocentric distance becomes too large. Estimates of the outer scale of solar wind turbulence at different heliocentric distances are found from a comparison of Doppler scintillations from near and distant spacecraft. Ó 2003 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Solar wind; Corona; Turbulence; Radio sounding 1. Introduction Coherent radio wave propagation in the solar corona has been investigated now for over 37 years, beginning with the first radio occultation experiment on the Mariner 4 spacecraft in 1966 (Goldstein, 1967). Various spacecraft signal parameters, including intensity scintillations, frequency (Doppler) fluctuations, Faraday rotation variations and spectral line broadening, have been studied. Two basic radio wave propagation configuration modes have been used for these experiments. In the first configuration mode one way propagation the radio waves are transmitted from a spacecraft situated behind the Sun and received by a ground station after a single traversal of the solar corona. In the second (twoway) mode, the radio waves are transmitted from a ground station, received and phase coherently retransmitted by a spacecraft, and finally received by the same or another ground station after a double transit of the solar corona. It has usually been assumed that radio * Corresponding author. Tel.: ; fax: address: mbird@astro.uni-bonn.de (M.K. Bird). wave fluctuations in the solar corona depend only on the distance of the closest approach of the ray path to the Sun (solar offset point), the heliolatitude and the solar activity. Analyses of two-way propagation data, however, have revealed an additional dependence on the distance of the spacecraft from the scattering solar corona, which can be well simulated by a phase screen of limited thickness centered about the solar offset point (Efimov et al., 1984, 1989). In the following we analyze two-way mode data associated with low heliolatitudes during solar minimum, for which the solar wind is slow and emanates predominantly from coronal regions near the equatorial streamer belt. An estimate for the outer scale of the electron density fluctuation spectrum, an important turbulence parameter that marks the lower boundary of the turbulent energy flux to smaller scales, can be obtained from these data. 2. Theoretical analysis Effects of two-way propagation have been previously investigated for such inhomogeneous media as the EarthÕs ionosphere and troposphere, and for turbulent air flow in the laboratory (Kravtzov and Saichev, 1982) /$30 Ó 2003 COSPAR. Published by Elsevier Ltd. All rights reserved. doi: /s (03)

2 702 A.I. Efimov et al. / Advances in Space Research 33 (2004) It was shown in this latter work that, whenever the wave reflector is situated close to the inhomogeneous medium, then the waves propagate through the same inhomogeneities in the direction from reflector to receiver as they did from transmitter to reflector (correlated inhomogeneities). In this case, phase and amplitude fluctuations will exceed the fluctuations occurring in the medium for wave propagation with uncorrelated inhomogeneities. The variance of the frequency fluctuations for a radio wave passing only once through the solar corona is given by (Efimov et al., 2003): r 2 f 1 ¼ ar2 e k2 r 2 N ðrþl e pð1 aþ v c m a o m1 a u m 1 a l ; ð1þ where r e ¼ 2: cm is the classical electron radius, k is the radio wavelength, r N is the RMS electron density fluctuation, m o ¼ v c =L o is the fluctuation frequency corresponding to the outer turbulence scale L o, ðm l ; m u Þ are the fluctuation frequencies corresponding to the lower and upper bounds of the observed spectral range, a is the spectral index of the temporal frequency fluctuation spectrum (p ¼ a þ 3 is the corresponding index of the spatial spectrum of electron density irregularities); v c is the convective velocity of the plasma irregularities across the ray path, and L e is the effective thickness of the scattering layer (L e R), where R is the solar proximate point along the radio ray path (solar offset). When the radio waves propagate twice (uplink + downlink) through the scattering layer of the solar corona, the variance of the frequency fluctuations is determined by the expression r 2 f 2 ¼ br2 f 1 ; ð2þ where b ¼ 2 for the case of uncorrelated irregularities (simply a doubling of the effective scattering layer with uncorrelated phase fluctuations) and b ¼ 4 for the case of completely correlated irregularities, when the phase fluctuations are coherently summed (e.g., Kravtzov and Saichev, 1982). A similar double propagation effect holds for spectral broadening. The equivalent bandwidth of the spectral line for a single transit of the solar corona is determined by the expression (Armand et al., 1987) 1 Df 1 ¼ 2C½ðp 1Þ=ðp 2ÞŠ pðp 3Þv p 2 1=ðp 2Þ c C½p 2Šðp 2Þsin½ðp 2Þp=2Š r2 N qp 3 o r 2 e k2 L e ; ð3þ where q o ¼ 2p=L o is the inverse turbulence outer scale. Combining Eqs. (1) and (3), one obtains 1 Df 1 ¼ 2C½ðp 1Þ=ðp 2ÞŠ ( ) 1=ðp 2Þ p p 1 2 p 3 ð4 pþ r 2 f 1 ; C½p 2Šðp 2Þ sin½ðp 2Þp=2Š m 4 p u where we have simplified m 1 a u m 1 a l ð4þ m 4 p u, an adequate approximation for p < 3:8 (Efimov et al., 2003). As evident in Eq. (4), spectral line broadening depends strongly on the spectral index p. A similar expression may be derived for two-way line broadening Df 2 by replacing r f 1 in Eq. (4) with r f 2. Let us define a parameter c for comparison of oneway and two-way spectral broadening: cðbþ ¼ Df 2 Df 1 ¼ b 1=ðp 2Þ ; ð5þ where c is the bandwidth enhancement for two-way propagation, which ranges between the cases of either uncorrelated irregularities c 2 ¼ cðb ¼ 2Þ or completely correlated irregularities c 4 ¼ cðb ¼ 4Þ. Table 1 gives values of the factors c for both of these cases at various values of the spectral index p. From Table 1 it is seen that the bandwidth enhancement for two-way propagation increases for smaller spectral index p. Furthermore, the enhancement with correlated irregularities is larger and increases more quickly toward smaller index p than with uncorrelated irregularities. At this point we must formulate a quantitative definition for correlated irregularities. The irregularities are said to be correlated if the radio propagation time over the distance L from the scattering layer up to the reflector (spacecraft transponder) and back again to the scattering layer (again L) is less than the time required for total replacement of the irregularities. At greater heliocentric distances L the irregularities become uncorrelated. The transition from correlated to uncorrelated irregularities occurs at L L 1, where L 1 is defined by equating the total propagation time to the convection time across the ray path of those large-scale irregularities with dimensions close to the outer turbulence scale L o. This condition may thus be expressed as: 2L 1 c L o v c ; ð6þ Table 1 Spectral bandwidth enhancement factor c p c c

3 where v c is the convection speed and c is the velocity of light. Knowing v c and estimating L 1 from measurements of the two-way propagation enhancement factors b, c, it is possible to place bounds on the outer turbulence scale L o from Eq. (6). A.I. Efimov et al. / Advances in Space Research 33 (2004) Experimental data 3.1. Frequency fluctuation measurements Fig. 1 presents a schematic of the geometry for the coronal sounding experiments in 1975 and 1976 with the spacecraft Helios-1, Helios-2, Pioneer-10, Pioneer-11 and Viking. As indicated in panel (a) of Fig. 1, the coronal west limb was sounded with signals from the spacecraft Helios-1 (distance from spacecraft to scattering layer L varied over the range 0:55 AU < L < 0:90 AU), Pioneer-10 (L ¼ 6:95 AU) and Pioneer-11 (L ¼ 4:45 AU). All of these measurements were carried out in the same time interval, 1975 DOY (Berman and Rockwell, 1975). Fig. 2 shows the two-way RMS frequency fluctuation r f 2 for each spacecraft as a function of the solar offset distance R. Helios-1 was approaching the Sun (occultation ingress); the Pioneer-10 and Pioneer-11 spacecraft were receding from the Sun (occultation egress). It is evident that the level of the frequency fluctuations was substantially higher for the case of minimal distance L (for Helios-1) between spacecraft and scattering layer. On two occasions, marked by open circles in Fig. 2, the solar offsets of Helios-1 were virtually the same as those Fig. 2. Mean frequency fluctuation hr f 2 i vs. solar offset for coronal sounding experiments in 1975 (Helios-1: circles; Pioneer-10: triangles; Pioneer-11: squares). The arrows indicate whether the ray path from the spacecraft was approaching or receding from the Sun. of Pioneer-10 ðr 16:2 R Þ and Pioneer-11 (R 26:0 R ). In both cases the two-way scintillation level of Helios-1 distinctly exceeded that of the more distant spacecraft. A similar result was obtained in 1976 (Fig. 3), for which the geometrical conditions are sketched in Fig. 1, panel (b). The spacecraft Helios-2 (0:38 AU < L < 0:63 AU) and Pioneer-10 (9:7 AU < L < 9:8 AU) were observed during the same period, 1976 DOY (Berman et al., 1976). Again, the mean level of frequency fluctuations is higher for the spacecraft closer to the Sun (Helios-2) than that measured for large distance (Pioneer-10). The two oppositely moving spacecraft Helios-2 and Pioneer-10 were located at the same solar Fig. 1. Schematic geometry of two-way mode coronal sounding experiments in 1975 (a) and 1976 (b).

4 704 A.I. Efimov et al. / Advances in Space Research 33 (2004) Observations of the Viking spacecraft (L ¼ 1:52 AU), carried out in the interval DOY (Berman and Wackley, 1977), have been added to the 1976 data. The data are grouped together from top to bottom according to the solar offset distance R ¼ 10, 15 and 20 R. In all cases the mean frequency fluctuation hr f 2 i is a decreasing function of L Spectral line broadening Fig. 3. Same as Fig. 2 for coronal sounding experiments in 1976 (Helios-2: circles; Pioneer-10: triangles). offset on DOY 122 (circled in Fig. 3). The fluctuations have an enhanced value on this particular day, but a persistent difference between the level of r f 2 from relatively near and relatively distant spacecraft remains substantial at virtually all solar offsets. The pronounced dip in the Helios-2 data near 22 R is likely due to a locally strong decrease in the solar wind turbulence or velocity on this day. Fig. 4 presents the same quantity as Figs. 2 and 3, but now as a function of the distance of the spacecraft from the Sun for 1975 (upper panel) and 1976 (lower panel). Two-way spectral line broadening measurements of the Mariner 4 spacecraft were found to be larger than those obtained in one-way mode by a factor 3.5 (Hollweg and Harrington, 1968). As shown in Eq. (5) and Table 1, it is now apparent that this unexpectedly large increase from one-way to two-way propagation can be explained by an enhancement of the fluctuations imposed by a propagation medium with correlated irregularities. This particular enhancement factor occurs when the spectral index p ¼ 3:1. Upon additional processing of the initial Mariner-4 spectral broadening data (Goldstein, 1967), we computed values of Df 1, Df 2 and c, which is presented in Fig. 5 (circles). The distance between spacecraft and scattering layer was L ¼ 1:18 AU during these measurements. The long-dash lines in Fig. 5 show the dependence of the two-way spectral broadening enhancement factor c from Eq. (5). These are based on observations of the spectral index pðrþ (Woo and Armstrong, 1979), assuming correlated irregularities (b ¼ 4, upper line a) and uncorrelated irregularities (b ¼ 2, lower line b). The average value of the enhancement coefficient at heliocentric distances R between 3 and 5 R is equal to c ¼ 3:92 1:05. This would imply a spectral index 2:9 < p < 3:3 from Eq. (5) in the Fig. 4. Mean frequency fluctuation hr f 2 i vs. L for three different solar offsets: 10, 15 and 20 R. The ranges of L for the three spacecraft during the observations are indicated by the vertical dashed lines. Upper panel: 1975; lower panel: Fig. 5. The dependence of the spectral bandwidth enhancement ratio c between two-way and one-way propagation as a function of solar offset R. Data are presented from Mariner-4 (1966): circles, Venera-15/ 16 (1984): triangles, and Pioneer-10 (1973): squares. The long-dash lines are theoretical curves for completely correlated plasma irregularities (curve a) and for uncorrelated irregularities (curve b).

5 A.I. Efimov et al. / Advances in Space Research 33 (2004) case of completely correlated irregularities, or an even smaller p (more probable) if the irregularities are only partially correlated. Fig. 5 also presents results obtained with the decimeter (k ¼ 32 cm) signals of the Venera-15 and Venera- 16 spacecraft in 1984 (triangles), located at a distance L ¼ 0:72 AU behind the Sun, for solar offsets ranging from 7 to 20 R. These two-way data are consistent with the presence of completely correlated irregularities and a spectral index p ¼ 3:3 0:3. The remaining data set in Fig. 5 was obtained from an analysis of S-band (k ¼ 13 cm) spectral line broadening using the Pioneer-10 spacecraft in 1973 (Cannon, 1976). The distance between spacecraft and scattering layer was L 3:45 AU. The Pioneer-10 data (squares) appear to correspond to the case of a medium with partially correlated irregularities (2 < b < 4). 4. Outer scale of coronal turbulence The data of Fig. 4 correspond to a deep minimum of solar activity. Combining all data of 1975 and 1976 we can obtain a statistically more representative dependence of the mean frequency fluctuation on the distance L. Table 2 gives the resulting values of hr f 2 i grouped at similar distances L for three solar offsets R ¼ 10, 15 and 20 R. These results provide information on the characteristics of two-way radio wave propagation in the solar corona. For L 0:5 AU it may be assumed that a geometrical configuration with correlated irregularities hrf 2 ð0:5 AUÞi ¼ 2hrf 1 iðb ¼ 4Þ is attained. Using the dependence hrf 2 ðlþi indicated in Table 2, one can then determine the distance L 1 at which intensity ofpthe frequency fluctuations drops to hr f 2 ðl 1 Þi ¼ ffiffiffi 2 hrf 1 i ðb ¼ 2Þ, corresponding to the case of radio wave propagation in a medium with uncorrelated irregularities. These distances are L 1 ¼ 1:4 AU, 1.15 AU and 3.1 AU for the solar offsets R ¼ 10, 15 and 20 R, respectively. Table 2 Mean frequency fluctuation hr f 2 i at different distances L [AU] R ¼ 10 R R ¼ 15 R R ¼ 20 R Table 3 Estimates of the outer turbulence scale R ½R Š L 1 [AU] v c [km s l ] L o ½R Š 3 5 > > It follows from the spectral line broadening data (Fig. 5) that partially correlated irregularities remain at least out to L < 1:18 AU for solar offsets 3R < R < 5R as well as out to L ¼ 0:72 AU for solar offsets 7 R < R < 20 R. At the same time L ¼ 3:45 AU is an upper bound for partially correlated irregularities. In order to estimate the outer turbulence scale L o from Eq. (6), it is necessary to know the typical convection velocity of the irregularities v c. We use here as a model the velocity measurements of convected irregularities observed in the slow solar wind with the LASCO coronograph on the SOHO spacecraft during the solar activity minimum in 1996 (Sheeley et al., 1997). The parameters used for the calculation of L o are given in Table 3. Estimates of the outer turbulence scale are thus of the order of the solar radius and seem to display a tendency for an increase with increasing heliocentric distance. 5. Conclusions Radio occultation experiments carried out in twoway mode demonstrate that an enhancement of fluctuation effects by correlated irregularities occurs for radio wave propagation through the solar corona when the spacecraft is located close enough to the Sun (L 1 K 1 AU), compared with more distant spacecraft (L 1 J 5 AU). This effect is explained by the presence of an outer turbulence scale, which determines the replacement time of the irregularities along the ray path. If this replacement time is larger than the round trip light time between scattering layer and spacecraft, then the radio wave encounters the same irregularities twice (correlated irregularities on uplink and downlink). Estimates of the outer turbulence scale have a magnitude close to a solar radius and tend to increase with increasing heliocentric distance. The estimates are generally lower than the more accurate values of L o obtained from an analysis of frequency fluctuation measurements (Bird et al., 2002; Efimov et al., 2002). The two-way propagation effect reported here provides additional evidence of the existence of the outer scale, an important, previously unrecognized characteristic of the solar wind plasma. Acknowledgements The present work is supported by the Russian Foundation of Basic Research (RFBR), Grants , and , and the Russian Ministry of Industry, Technology and Science. This paper presents results of research partly funded by the Deutsche Forschungsgemeinschaft (DFG) under a cooperative program between the DFG and RFBR.

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