Properties of solar wind turbulence near the Sun as deduced from coronal radio sounding experiments

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1 Advances in Space Research 36 (2005) Properties of solar wind turbulence near the Sun as deduced from coronal radio sounding experiments I.V. Chashei a, A.I. Efimov b, L.N. Samoznaev b, D. Plettemeier c, M.K. Bird d, * a Lebedev Physical Institute, Russian Academy of Science, Moscow, Russia b Institute of Radio Engineering and Electronics, Russian Academy of Science, Moscow, Russia c Technische Universität Dresden, Elektrotechnisches Institut, Dresden, Germany d Radioastronomisches Institut, Universität Bonn, Auf dem Hügel 71, Bonn, Germany Received 29 July 2004; received in revised form 6 May 2005; accepted 9 May 2005 Abstract The characteristics of plasma turbulence in the inner solar wind, as deduced from radio frequency fluctuation measurements recorded during solar conjunction, are reviewed. Special emphasis is placed on results from the radio occultation experiments performed with the Galileo and Ulysses spacecraft in the interval Estimates of the power spectral index are obtained over the range of heliocentric distances 5R x < R < 80R x and the radial evolution of this parameter is discussed. Low-heliolatitude data from Galileo are compared with high-heliolatitude data from Ulysses during the period of solar activity minimum. Unusual power spectra with a sharp break at high fluctuation frequencies are detected for the first time in the solar wind acceleration region. The observations are interpreted within the framework of a theoretical model based on local generation of density fluctuations via nonlinear interactions of Alfvén waves propagating away from the Sun. Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Solar corona; Radio sounding; Solar wind turbulence 1. Introduction * Corresponding author. Tel.: ; fax: addresses: chashey@prao.psn.ru (I.V. Chashei), efimov@ ms.ire.rssi.ru (A.I. Efimov), plettemeier@eti.et.tu-dresden.de (D. Plettemeier), mbird@astro.uni-bonn.de (M.K. Bird). Many years of solar wind plasma investigations by radio occultation (Yakovlev et al., 1980; Bird, 1982; Bird and Edenhofer, 1990; Wohlmuth et al., 2001; Chashei et al., 2003) and in situ methods have shown that turbulence is a permanent property of the interplanetary plasma at all heliocentric distances and heliolatitudes. Fluctuations of the electron density, magnetic field, velocity, etc. are characterized by spatial and temporal power spectra covering many decades in the wavenumber or temporal frequency domains, respectively. Turbulence evolution and its interaction with the background plasma is of great importance for solar wind physics, especially near the Sun where the plasma flow accelerates to supersonic and superalfvenic velocities. One modification of the radio occultation method is based on measurements of the radio frequency fluctuations. This observable has some advantages for investigation of solar wind density fluctuations compared with others (for instance, amplitude fluctuation measurements), because the large-scale regime of spatial density turbulence spectrum, where the turbulence energy density is greatest, produces most of the fluctuations of the radio signal frequency. Radio occultation experiments were carried out beginning from 1991 using coherent radio signals of the Ulysses spacecraft and in the period with the Galileo spacecraft. A large volume of observational /$30 Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi: /j.asr

2 I.V. Chashei et al. / Advances in Space Research 36 (2005) data was obtained during these experiments at heliocentric distances 5 80R x, which includes the outer region of plasma acceleration. The data were obtained at low heliolatitudes (Galileo and Ulysses) as well as at high heliolatitudes (Ulysses). This paper briefly reviews some new results on turbulence spectra and regimes deduced from the Ulysses and Galileo coronal radio sounding experiments for the period with special interest on the period of solar activity minimum in Observations and data processing Coronal sounding experiments were conducted using the radio signals of the spacecraft Ulysses and Galileo during their solar conjunctions. Ulysses used phase coherent dual-frequency downlinks at the carrier frequencies GHz (S-band) and GHz (X-band). Galileo was constrained to single frequency transmission at the downlink frequency GHz (S-band), but this signal was generated on board by an ultra-stable oscillator (USO), resulting in a sensitivity nearly equivalent to the Ulysses radio science experiment. The geometry of the experiment is presented schematically in Fig. 1. The radio signals were recorded at the three 70-m tracking stations Canberra (DSS 43), Madrid (DSS 63) and Goldstone (DSS 14) of the NASA Deep Space Network (DSN). The signal frequency was recorded at a nominal sampling rate of 1 s 1. Doppler residual time series were calculated by subtracting a slowly-varying component from the raw data to compensate for the spacecraft motion relative to the observer. Some examples of Doppler residual records are presented in Fig. 2. Temporal power spectra of the frequency fluctuations were calculated using a standard FFT algorithm for all observational intervals with a duration longer than samples. Spectral indices a f were determined for the spectral ranges where the power spectra can be represented as power law. Some typical frequency fluctuation spectra are illustrated in Fig. 3. Simultaneous Doppler recordings from two widelyseparated receivers, which provide the capability for estimating the solar wind speed, occur during the frequent observation overlap intervals between DSN stations. For these overlap periods, the solar wind speed was found as the ratio of the calculated 2-station ray path radial separation to the time lag of maximum frequency cross-correlation between the two stations. The typical duration of the overlap intervals is about 2 h, and the typical ray path radial separation on these occasions is several thousand kilometers. An example of a Fig. 2. Some examples of Doppler residual recordings (from Samoznaev et al., 2003). Fig. 1. Geometry for coronal radio sounding experiments (from Efimov et al., 2002). Fig. 3. Typical temporal spectra (power law) of frequency fluctuation data (from Samoznaev et al., 2003).

3 1456 I.V. Chashei et al. / Advances in Space Research 36 (2005) Fig. 4. Example of a frequency fluctuation cross-correlation function (adapted from Armand et al., 2003). frequency fluctuation cross-correlation function is presented in Fig Main theoretical relations In this section, we recall the theoretically derived expressions relating the observed statistical characteristics of the frequency fluctuations to the physical characteristics of the solar wind, namely its speed and its spatial turbulence power spectrum. We assume that the spatial density fluctuation spectrum of the solar wind plasma is a nearly isotropic power law in the spectral range between the inverse outer scale q 0 =2p/L 0 and the inverse inner scale q m =2p/L m, q m q 0, U N ðq; rþ ¼C N ðrþðq 2 þ q 2 0 Þ p=2 ; ð1þ where r is heliocentric distance, q is the spatial wavenumber, and C N is the structure constant. In this case the temporal frequency fluctuation spectrum G f (m) in the frequency range m m m = q m V eff has the form (Efimov et al., 2003) G f ðmþ ¼Ar 2 p 2 NðRÞRV eff m 2 ðm 2 0 þ m2 Þ ðp 1Þ=2 ; where A is a function of p and k (radio wavelength), m is the fluctuation frequency, R is the heliocentric distance of the line-of-sight proximate point (see Fig. 1), V eff is the effective velocity of the irregularities relative to the line of sight, and r 2 N is the electron density fluctuation variance, where r 2 N ðrþ ¼R U N ðq; r ¼ RÞd 3 q. Eq. (2) is obtained under the assumption that the effective thickness of the medium containing the random inhomogeneities is K eff. R. If the density inhomogeneities are convected with one and the same velocity and do not change during their motion (frozen-in hypothesis), then the velocity V eff ¼j~V sw ~V rp j, where ~V sw is the solar wind velocity and ~V rp is the ray path velocity relative to the Sun. Possible deviations from the frozen-in condition arising from the velocity spread along the line-of-sight or from temporal evolution of the inhomogeneities can be described by a velocity distribution function u(v). In this case the average value hv p 2 i¼ ð2þ R V p 2 uðv ÞdV should be used in Eq. (2) instead of V p 2 eff. As indicated in Eq. (2), the power exponent of the frequency fluctuation temporal spectrum a f is related very simply to the power exponent of the 3D density turbulence spectrum p, a f ¼ p 3. ð3þ The spectrum G f (m) has a local maximum at the frequency m = m max. [2/(p 3)] 1/2 m 0 (Efimov et al., 2003), at which point the spectral density reaches a value given by G f ðm max Þ¼A 1 r 2 N ðrþrv eff; ð4þ where A 1 =2A[(p 3)/(p 1)] (p 1)/2, and V eff = (ÆV p 2 æ) 1/(p 2). If the individual observational records are long enough to enable detection of the lowfrequency spectral maximum at m = m max, and if the velocity V eff can be estimated simultaneously, then Eq. (4) can be used to estimate the density variance r 2 N ðrþ. Estimates of the velocity of the inhomogeneities can be obtained, for example, from simultaneous measurements of the frequency fluctuations at widely-spaced ground based stations. The temporal cross-correlation function of frequency fluctuations K(q, s) is related to the corresponding cross-correlation function of phase fluctuation by the equation Kðq; sþ ¼ o2 K U ðq; sþ os 2 ; ð5þ where q(x,y) is projection of the baseline vector onto the plane perpendicular to the line-of-sight, x is its radial component, and s is the time lag. The function K(q,s) reaches its maximum at some lag s max, from which the apparent pattern speed V ap = x/s max can be determined. This parameter is defined by the third and fourth moments of the velocity distribution (Chashei et al., 2000), V ap ¼hV 4 i=hv 3 i ð6þ and is equal to the convection speed if the velocity spread can be neglected. The difference between the parameters V eff and V ap is not very strong under typical solar wind conditions (Chashei et al., 2000), and we assume below that V eff V ap. 4. Radial evolution of power law turbulence spectra Typical frequency fluctuation power spectra assume the form of a power law over the central range of temporal frequencies. These spectra have a smooth local maximum at some frequency m max and a power law spectral interval for m > m max that extends up to frequencies where the measured fluctuation power drops to the noise level. Parameters available for study of their radial evolution are the frequency of spectral maximum m max, the maximum spectral density G f (m max ), and the power law index a f at m > m max.

4 I.V. Chashei et al. / Advances in Space Research 36 (2005) The radial dependence of the power law index a f for low heliolatitudes (u < 10 ), derived from Galileo and Ulysses data in the years , is presented in Fig. 5. Considering results from both spacecraft to increase the data volume, average values of a f and its standard deviation are presented for 10R x bins in the range 30 80R x,5r x bins in the range 10 30R x, and 2.5R x bins in the range 5 10R x. Circles and squares denote measurements on the west and east solar limbs, respectively. Data averages below 30R x (triangles) were taken from both solar limbs. The power exponent a f clearly increases with heliocentric distance from values a f. 0.2 up to a f at R [ 20R x and remains approximately constant for R > 20R x. The observed value of a f at R > 20 R x is consistent with the expected value a f = 2/3 for Kolmogorov turbulence spectra. The 3D power index of the density turbulence power spectra p = a f + 3 must follow the same radial dependence. A similar trend was found earlier by Woo and Armstrong (1979) from spectral analysis of Viking radio signal phase fluctuations. The Ulysses data of 1995 are well suited for investigating the heliolatitude dependence of the spectral index a f. As shown in Fig. 6, a f. 2/3 in the range of heliolatitude 50 < u <0, but has a tendency to decrease when the line of sight approaches the south polar region. Fig. 5. Radial dependence of power law index a f at low heliolatitudes (from Samoznaev et al., 2003). These results agree with the spectral changes found in phase fluctuation spectra by Pätzold et al. (1996). The data of Fig. 6 most likely reflect a real heliolatitude dependence. The variation in solar offset distance during this measurement interval was rather moderate (22R x < R < 32R x ). Whereas a f. 2/3 within this distance range at low heliolatitudes (Fig. 5), Fig. 6 shows that the spectral index of the frequency fluctuation spectrum is substantially less than 2/3 at heliolatitudes juj > 65. We thus conclude that the transition from flat to steeper turbulence spectra occurs at greater heliocentric distances in high latitude regions (Fig. 6) than at low latitudes (Fig. 5). 5. Spectral features observed at small heliocentric distances Roughly 30% of the Galileo frequency fluctuation spectra recorded at small heliocentric distances were found to have a sharp spectral break. As a rule, the spectral breaks are observed inside 10R x at relatively low levels of RMS frequency fluctuations. Fig. 7 presents contrasting examples of spectra that are best fit by (a) a single power law, or (b) a double power law (with break frequency m b Hz). Fig. 7(a), from Ulysses measurements on August 1991, was recorded during very high solar activity (Wolf number W = 212) when the radio ray path proximate point was at a heliolatitude.40. Fig. 7(b), on the other hand, is a Galileo spectrum from data on January 1997, during very low solar activity (W < 9) and at a heliolatitude.20. Whereas the heliocentric distances were nearly identical (R. 7R x ), the levels of the RMS frequency fluctuations were very different: (a) r f Hz, (b) r f Hz. The value of the frequency m max is less than 10 3 Hz for the temporal spectrum presented in Fig. 7(a). The break frequency in Fig. 7(b) is much higher ( Hz). The power expo- Fig. 6. Heliolatitude dependence of the power law index a f (from Samoznaev et al., 2003). Fig. 7. Contrasting power spectra: (a) single power law and (b) double power law with break frequency (from Samoznaev et al., 2003).

5 1458 I.V. Chashei et al. / Advances in Space Research 36 (2005) Fig. 8. Radial dependence of the break frequency m b (from Samoznaev et al., 2003). nent for the spectrum (a) and the low-frequency part of spectrum (b) are approximately equal, and close to the values presented in Fig. 5 for the corresponding heliocentric distance. Spectra with obvious break frequencies are not observed at high fluctuation levels or in regions outside 10R x. Additional data would be required to determine just what environmental conditions (heliolatitude, solar activity level, others?) are necessary for producing the unusual spectra with a break frequency. Fig. 8 presents all observed values of the spectral break frequency m b as a function of heliocentric distance. The data for Fig. 8 were obtained during the period January 1997 when solar activity was very low (sunspot numbers from 0 to 17). The heliolatitudes of the sounded regions were low, varying from 16.9 to 23.2 (ingress-east limb) and from 19.3 to 13.2 (egress-west limb). The solar wind speed was not atypical for these distances during this period, increasing from 78 km s 1 at 6.3R x up to 103 km s 1 at 9.06R x, i.e., characteristic for slow wind. Fig. 8 indicates that the value of m b ranges from 0.01 to 0.03 Hz. There is a slight tendency for an increase in m b with solar offset distance R, but the data statistics are insufficient for drawing a more definite conclusion about this dependence. 6. Fractional density fluctuations in the inner solar wind We now use the frequency fluctuation data to derive the fractional density fluctuations over the range of heliocentric distances from 7R x up to 31R x. The RMS values of density fluctuations in the solar wind r N were calculated from the measured values of maximum spectral density G f (m max ), the power exponent a f, and the apparent velocity of the inhomogeneities V ap, using Eq. (4). The estimates of r N are valid for the period of low solar activity between 16 January and 30 January 1997 from Galileo data. Typical values of r N range from 10 2 to 10 3 cm 3. Results from Viking ranging measurements (Muhleman and Anderson, 1981), which are applicable to the epoch of solar minimum, were used for computing the mean electron density N e. The values of fractional Fig. 9. Fractional density fluctuations r N /N e vs. R (from Efimov et al., 2003). density fluctuations r N /N e, valid here for the slow solar wind, are shown in Fig. 9. The fractional density fluctuations are typically of the order of at low heliolatitutes and low solar activity levels. On the average, these estimates are in good agreement with the results of Woo et al. (1995) determined from a study of Ulysses dual-frequency ranging data, which are applicable to temporal frequencies in the range Hz < m < Hz. The data of Fig. 9 display a slight tendency for an increase in fractional density fluctuations r N /N e with heliocentric distance. Woo et al. (1995) found the same trend for fast solar wind streams. The data presented in Fig. 9 extend their conclusion to the slow solar wind. 7. Conclusion The following conclusions may be drawn from the above results: As a rule, spatial turbulence power spectra in the inner solar wind take the form of a power law over a wide range of inhomogeneity scales. The spectral regime of solar wind turbulence changes at a heliocentric distance near 20R x : power spectra are comparatively flat for R [ 20R x ; power spectra become steeper for R > 20R x. During the period of minimum solar activity the change of turbulence regime occurs at greater heliocentric distances for (fast) high-latitude solar wind than for (slow) low-latitude solar wind. The data of Fig. 9 display a slight tendency for an increase in fractional density fluctuation with increasing heliocentric distance inside 25R x, but more data are needed to confirm this preliminary conclusion. Provided the absolute level of turbulence is low, the power spectra in the range inside 10R x may have a high frequency break. These conclusions may be reasonably interpreted within the framework of the turbulence model developed by Chashei and Shishov (1983). According to this

6 I.V. Chashei et al. / Advances in Space Research 36 (2005) model, the solar wind turbulence is driven by a random ensemble of interacting Alfvén and magnetosonic waves. The turbulence energy source is provided by low-frequency Alfvén waves propagating outward from the coronal base. Magnetosonic waves responsible for the density fluctuations are generated locally by nonlinear wave wave interactions. The magnetic field is strong in the solar wind acceleration region (R <20R x ) so that the Alfvén speed is larger than the speed of sound. The turbulence here is weak and the density fluctuations are dominated by slow magnetosonic waves. Accordingly, power spectra for the magnetic field and density fluctuations are comparatively flat, with a 3D power-law index p. 3.0 (i.e., corresponding to a f. 0, consistent with the frequency fluctuation measurements). Such spectra are formed as a result of the nonlinear conversion of Alfvén quanta to lower frequencies by conservation of their total number. It is very important that these power spectra are noncascading, i.e., the flux of turbulent energy to the small-scale spectral range is absent. The fractional level of turbulence increases with heliocentric distance because the energy density of the Alfvén waves decreases more slowly than the energy density (magnetic plus thermal) of the ambient plasma. Moreover, the ratio of the speed of sound to the Alfvén speed also increases. The combination of these two effects results in an increase of the fractional level of fast magnetosonic waves, thereby enhancing the cascading process and the corresponding transition to developed turbulence characterized by steeper spectra of the type Kolmogorov (p = 11/ 3, a f = 2/3) or Kraichnan (p = 7/2, a f = 1/2). This transition, as well as the increase in fractional turbulence level, is observed in the radio sounding measurements. The transition zone is located at R. 20R x, thereby coinciding with the radial evolution from the acceleration region to the region of fully developed solar wind. The more distant transition for the fast solar wind can be explained by the stronger magnetic field above the coronal holes and, as a consequence, the lower values of fractional turbulence level and ratio of sound to Alfvén speed. The spectra with a sharp spectral break observed at small heliocentric distances also can be considered as evidence favoring the absence of cascading processes. Indeed, the observed break frequency m b Hz is much lower than those plasma frequencies corresponding to the typical plasma scales (ion gyroradius or ion inertial scale), at which one can expect strong linear damping or changes in the wave dispersion. If cascading processes were important then, in contrast to the observations, the value of m b should decrease with heliocentric distance. The spectral break in the corona is most likely formed from the linear damping of Alfvén waves. If the fractional level of outward propagating waves is anomalously low, the nonlinear processes in the outer corona do not have enough time to eradicate this spectral feature. The above considerations show that important conclusions on the physics of the turbulent inner solar wind may be drawn from radio sounding data. An analysis of radio occultation results in combination with in situ measurements would be particularly beneficial. The envisioned Solar Orbiter (Fleck et al., 2000; Marsch et al., 2002), which would explore partially overlapping ranges of solar distance and heliographic latitude, will provide an excellent opportunity for comparison with radio sounding observations. Turbulence temporal spectra measured by in situ and radio sounding methods correspond to 1D and 2D spatial power spectra, respectively. A comparison of spectra derived from in situ density fluctuations and radio phase (or frequency) fluctuations thus provide valuable information about the anisotropy of density irregularities. Furthermore, conclusions concerning the physical origin of turbulence can be inferred from 1D/2D turbulence spectra and fractional fluctuation levels of the density or any other fluctuating plasma parameter with the same statistical characteristics such as magnetic field, bulk velocity, etc. In particular, such comparisons should provide a better understanding of the role of Alfvén waves in the local generation of density fluctuations. Acknowledgements The present work was supported by the Program Solar Wind of the Russian Academy of Sciences and by Grants and of the Russian Foundation of Basic Research (RFBR). This paper present results of research partly funded by the Deutsche Forschungsgemeinschaft (DFG) under a cooperative program between the DFG and RFBR. The authors wish to express their gratitude for the continued support of the Galileo Project, the Multi-Mission Radio Science Team and the NASA Deep Space Network. References Armand, N.A., Efimov, A.I., Samoznaev, L.N., et al. The spectra and cross correlation of radio frequency fluctuations observed during coronal plasma sounding with spacecraft signals. J. Commun. Technol. Electron. 48 (9), , Bird, M.K. Coronal investigations with occulted spacecraft signals. Space Sci. Rev. 33, , Bird, M.K., Edenhofer, P. Remote sensing observations of the solar corona. in: Schwenn, R., Marsch, E. (Eds.), Physics of the Inner Heliosphere, vol. 1. Springer-Verlag, Heidelberg, pp , Chashei, I.V., Shishov, V.I. Solar wind turbulence in the acceleration region. Sov. Astron. 27 (3), , Chashei, I.V., Efimov, A.I., Rudash, V.K., Bird, M.K. 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