Characteristics of the Near-Sun Solar Wind Turbulence from
Spacecraft Radio Frequency Fluctuations
A.I. Efimov, N.A. Armand, L.N. Samoznaev , M.K. Bird† , I.V. Chashei and P.
Edenhofer, D. Plettemeier, R. Wohlmuth‡
Inst. for Radio Engineering & Electronics, Russian Academy of Science, Moscow, 101999, Russia
†
Radioastronomisches Institut, Universität Bonn, 53121 Bonn, Germany
Lebedev Physical Institute, Russian Academy of Science, Moscow, 117924, Russia
‡
Institut für HF-Technik, Universität Bochum, 44780 Bochum, Germany
Abstract.
Frequency fluctuation temporal spectra measured during spacecraft radio occultation experiments are shown to have a
maximum at a well defined temporal frequency if the density turbulence power spectrum of the solar wind has a power
law form in the wavenumber range between the outer and inner turbulence scales. The frequency of this maximum and
the associated maximal spectral power of the frequency fluctuations both depend on the density variance, the solar wind
convection speed, the turbulence outer scale, and the power index of the density turbulence spectrum in the region of the
propagation medium near the line-of-sight proximate point. If the solar wind speed can be estimated from simultaneous
frequency fluctuation measurements at widely-spaced ground stations, then estimates can be derived for both the density
turbulence outer scale and density variance from the measured values of the power exponent of the frequency fluctuation
temporal spectra. Results of coronal radio sounding experiments in 1997 with the Galileo spacecraft were analysed using the
above method. Distinct differences were found between the temporal frequency fluctuation spectra observed at large and at
small heliocentric distances. Values of the fractional density variance in the solar wind at low heliolatitude during a period of
low solar activity are presented for the range of heliocentric distances between 7 R and 31 R .
INTRODUCTION
Investigations of the spatial and temporal (including solar cyclic) variations of the circumsolar plasma turbulence are of great importance for the physics of the solar wind. Coronal radio sounding experiments (Yakovlev
et al., 1980; Bird, 1982) are valuable tools for studying
the solar wind turbulence. One modification of the radio
occultation method is based on measurements of the radio frequency fluctuations. This observable has some advantages for investigation of density variance compared
with others (for instance, amplitude fluctuation measurements), because the large-scale regime of the spatial density turbulence spectrum, where the energy density is
greatest, produces most of the fluctuations of the radio
signal frequency (Wohlmuth et al., 2001).
In order to determine density turbulence spectra and
obtain estimates of the solar wind density fluctuation
variance we study here frequency fluctuation measurements of the Galileo radio signals at the carrier frequency
f = 2.295 GHz during the spacecraft’s solar occultation
in January-February 1997. Our results are thus applicable to a period of very low solar activity level and to low
heliolatitudes. Based on these estimates and published
data on the average electron density, we derive the radial
dependence of the fractional density variance over the
specific range of heliocentric distances which includes
the region of solar wind acceleration. We concentrate
here particularly on the fractional density fluctuations,
because this parameter is of crucial importance for understanding the turbulence regimes and evolution in the
solar wind. We also compare frequency fluctuation spectra observed in the developed solar wind with those applicable to the acceleration region.
THEORETICAL RELATIONS
In this section we consider the relation of the frequency
fluctuation temporal spectrum measured in radio occultation experiments to the basic spatial spectrum of solar
wind turbulence and to the solar wind speed. We assume
that the spatial density fluctuation spectrum in the solar
wind is a power law, nearly isotropic, and can be represented by the following form (Armand et al., 1987):
ΦN (q; r) = C2 (r)
exp( q2 =q2m )
(q2 + q2o ) p=2
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
469
(1)
where r is the heliocentric distance, q is the wavenumber,
Lo = 2π =qo and Lm = 2π =qm (Lo >> Lm ) are the turbulence outer and inner scales, respectively, and p is the 3D
power exponent. If the spectrum of Eq. (1) is normalized
to the density fluctuation variance σ N2 (r) then the temporal power spectrum of frequency fluctuations G f (ν ) is
defined by the relation:
G f (ν ) = Bπ
1
(λ re )
2
(νo2 + ν 2)
σN2 (R)Le vc (R)νoα ν 2
(α +2)=2
exp( ν 2 =νm2 )
(2)
where α = α f = p 3 is the power exponent of the
temporal frequency fluctuations spectrum, λ is the radio
wavelength, re = 2:82 10 13 cm is the classical electron radius, R is the solar offset distance of the radio ray
path, Le R is the effective thickness of the slab near the
solar proximate point along the ray path which contains
the radio frequency modulating turbulence, v c is the solar wind convection speed, ν o = vc =Lo and νm = vc =Lm
are the frequencies corresponding to the outer and inner
scales, and the constant B is equal to
B=
α
ln(2qm =q0 )
1
for 0 < α 1
for α = 0
(3)
In contrast to phase or amplitude fluctuation spectra,
the frequency fluctuation spectrum in Eq. (2) exhibits a
maximum at the fluctuation frequency ν max given by
νmax =
for 0 < α 1
for α = 0
(2=α )1=2 νo
(νo νm )1=2
(4)
Numerical values of ν max from Eq. (4) are expected
to be of order of 0.1 mHz in the regions of the developed solar wind r > 20 R with α 2=3 and v c = const
(Wohlmuth et al., 2001). In the solar wind acceleration region (r < 10 R ), however, for which spectra
with α 0 were reported (Woo and Armstrong, 1979),
the values of νmax may be considerably higher, up to
νmax > 10 mHz. In such cases the spectral maximum is
weakly pronounced and it is difficult to use the lower expression of Eq. (4) for estimates of ν o or νm .
Substituting νmax from Eq. (4) into Eq. (2) leads to the
following relation for the maximum spectral density
G f (νmax ) =
with
(
B1 =
B1
2 2
(λ re ) σN (R)Le vc
π
(α +2)=2
2 αα+2
[ln(2qm =qo )]
1
for 0 < α 1
for α = 0
(5)
(6)
Based on frequency fluctuation observations of power
index α , solar wind speed v c and maximum spectral
power G f (νmax ), Eqs. (5) and (6) are used below for
estimating the density variance σ N2 (R).
470
FIGURE 1. A series of frequency fluctuation temporal spectra. The start and stop times are labeled for each spectrum. Dotted lines show fitted theoretical spectra from Eq. (2) normalized
to the maximum spectral density given by Eq. (5).
OBSERVATIONAL DATA
Long observation records of frequency fluctuations at
high sampling rate (1 s 1 ) were processed according to
the above presented method. The Galileo S-band radiometric signal parameters were recorded with the three
large antennas of the global NASA Deep Space Network (DSN): Goldstone (DSS 14), Canberra (DSS 43)
and Madrid (DSS 63). The Doppler residual time series
was calculated after subtracting a slowly varying component from the raw data to compensate for the spacecraft
motion relative to the observer known from navigation
data. Two continuous frequency fluctuation records are
described in this section. The first one is typical for comparatively large heliocentric distances (' 30 R ); the
other for the region closer to the Sun ' 7 R .
A series of successive temporal frequency fluctuation
power spectra from the first record is presented in Fig. 1.
The spectral parameters α and ν max were estimated from
the normalized spectra (dotted lines in Fig. 1).
Simultaneous observations, and a corresponding capability for estimating the solar wind speed, occur during
the overlap time intervals between DSN stations. In these
FIGURE 3.
FIGURE 2. Temporal frequency fluctuation spectra for data
recorded at smaller solar offset distances.
cases 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 fluctuation cross correlation
between the two stations.
Spectral parameters for the Galileo 1997 observations
are summarized in Table 1. The Spectra #1–4 were obtained at a larger solar offset distance and have generally
different characteristics from those recorded at smaller
solar distances (Spectra #5–8). Estimates of outer scale
were obtained using Eq. (4) for observed values of spectral index α and solar wind velocity v c . The RMS values
of density fluctuations in the solar wind σ N were found
from Eq. (5). Results from Viking ranging measurements
(Muhleman and Anderson, 1981), which are applicable
to the solar minimum period, were used for computing
the mean electron density Ne in Table 1.
Three temporal frequency fluctuation spectra corresponding to the Doppler residuals recorded at solar ray
path offsets much closer to the Sun are shown in Fig. 2.
These spectra, in strong contrast to those shown in Fig. 1,
display a very flat low frequency part (power exponent
α 0) and a very sharp spectral break (power exponent
> 2) at frequencies ν 0:02 Hz.
Because the spectra of Fig. 2 have a flat low frequency
part and a sharp high frequency break, we cannot determine the frequency ν max and corresponding turbulence
outer scale Lo . Furthermore, the frequency fluctuation
variance for spectra of this type is dependent only on the
nearly constant spectral density at low frequencies G f l f
and the break frequency ν b .
The values of G f l f and νb needed for the estimates
of σ f and σN , as well as the minimal frequency in the
temporal spectra νmin (defined by the length of the data
record), are presented in Table 1 (Spectra #5–8). The
same method described in the previous section was used
471
Fractional density fluctuations σN =Ne versus R.
to calculate σN from the frequency fluctuation spectra.
The only difference is that the ratio ν b =νmin was substituted in Eqs. (5),(6) instead of q m =q0 . It should be noted
that the variances σ f and σN for spectra #5–8 of Table 1
are defined mainly by the spectral density in the spectral
range near the break frequency ν b .
σN =NE IN THE INNER SOLAR WIND
We use now the frequency fluctuation data to derive the
fractional density fluctuations in the range of heliocentric
distances between 7 R and 31 R . The values of fractional density fluctuations σ N =Ne are given in Table 1
and shown in Fig. 3.
The fractional density fluctuations over the investigated heliocentric distance range, as shown in Fig. 3,
are typically of the order of 0.1–0.3 at low heliolatitudes and low solar activity levels. These estimates are
(on average) in good agreement with the results of Woo
et al. (1995) found from investigation of Ulysses dualfrequency ranging data, applicable to temporal frequencies in the range 6 10 6 Hz < ν < 8 10 4 Hz.
The data of Fig. 3 display a slight tendency for an increase in fractional density fluctuations σ N =Ne with increasing heliocentric distance. Such behavior would be
quite reasonable from a physical point of view. Indeed,
the typical temporal spectra observed at low solar activity close to the Sun (see, for instance, Fig. 2) correspond to spatial density fluctuations with an initial power
spectrum that slowly evolves as it is carried outward by
the solar wind flow. No cascading is expected for such
spectra. Switching on the cascading processes, with a
corresponding transition to developed Kolmogorov turbulence, would be possible at larger distances from the
Sun if the fractional turbulence level increases with increasing heliocentric distance as a result of smooth radial
gradients in the ambient plasma parameters. It should be
noted that Woo et al. (1995) found a radial increase of
TABLE 1.
Summary of derived spectral parameters: Galileo 1997
Spectrum
#1
#2
#3
#4
#5
#6
#7
#8
Date, Jan 1997
Start time [UT]
Stop time [UT]
R [R ]
σ f [mHz]
α
vc [km/s]
νmax [10 4 Hz]
G f (νmax ); [Hz2 /Hz]
Lo , [R ]
νmin [10 4 Hz]
G f l f [Hz2 /Hz]
νb [10 2 Hz]
σN [102 cm 3 ]
Ne [102 cm 3 ]
σN =Ne
27/28
17:12:01
11:24:15
25.5
73.4
0.677
24960
0.6
0.9
10.22.5
1.2
4.5
0.26
28/29
11:24:16
05:36:32
27.7
63.0
0.775
20829
0.7
0.72
6.81.0
1.1
3.8
0.29
29
05:36:33
23:48:48
29.8
51.6
0.667
25973
0.7
0.28
9.22.6
0.62
3.2
0.19
29/30
15:41:50
09:54:05
30.9
46.9
0.640
26289
0.65
0.27
10.23.5
0.56
3.0
0.19
16-17
23:35:51
00:44:07
7.0
699
0.0
145
1.0
10.0
2.0
1.4
10.5
0.13
21
21:07:56
22:16:12
7.9
548
0.0
110
1.0
8.0
2.0
1.3
8.0
0.17
22
06:48:11
07:56:27
9.1
579
0.0
120
1.0
8.0
2.0
1.2
6.1
0.20
24
03:38:14
10:14:21
14.6
149
0.446
160
3.0
1.4
8.0
0.44
2.1
0.21
σN =Ne for the fast solar wind streams. Although more
data would be needed to confirm this preliminary conclusion, we find evidence in this work that the same tendency may well hold for the slow solar wind.
support from the RFBR, Grant 00-02-17845, and from
the Russian Ministry of Industry, Technology and Science, is acknowledged.
REFERENCES
CONCLUSIONS
Continuous Galileo spacecraft signal frequency fluctuation records in the period of low solar activity at low heliolatitudes reveal a strong contrast between density turbulence spectra observed at large (R > 20 R ) and small
(R < 10 R ) solar offset distances.
Typical density turbulence power spectra have a pronounced outer scale and power-law decrease at high frequencies with a 3D power exponent p 3:6 3:7 far
from the Sun. Such spectra correspond to developed turbulence of the Kolmogorov type. In contrast, the power
exponent close to the Sun is typically p 3, or even less,
in the low frequency spectral range, and becomes very
large (p > 4) at frequencies above about 0.02 Hz (break
frequency). Spectra of this type are possible only in the
absence of nonlinear cascading processes.
The fractional density fluctuation level lies within the
range 0.1–0.3 for heliocentric distances between 7 R and 31 R . The slight increasing trend toward larger R
requires further investigations for confirmation.
ACKNOWLEDGMENTS
This work presents results of a bi-national research
project partially funded by the Deutsche Forschungsgemeinschaft (DFG) and by the Russian Foundation for
Basic Research (RFBR), Grant 00-02-04022. Additional
472
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