On the Outer Scale of Turbulence in the Solar Wind
I.V. Chashei , M.K. Bird† and A.I. Efimov
Lebedev Physical Institute, Russian Academy of Science, Moscow, 117924, Russia
†
Radioastronomisches Institut, Universität Bonn, 53121 Bonn, Germany
Inst. for Radio Engineering & Electronics, Russian Academy of Science, Moscow, 101999, Russia
Abstract.
The outer scale of turbulence in the solar wind has been estimated from frequency fluctuation data recorded during radio
sounding experiments using the Galileo and Ulysses spacecraft carrier signals at 2295 MHz. The outer scale was observed to
increase approximately linearly with increasing heliocentric distance in the range between 7 and 80 R . A model is presented
here for the formation and evolution of the turbulence outer scale. The model is based on the assumption that the density
fluctuations are caused by nonlinear interactions of Alfvén waves and that the rates of linear and nonlinear processes are nearly
equal in the spectral range near the inverse outer scale. The dependence of the turbulence outer scale on heliocentric distance
and local plasma parameters is investigated for three possible nonlinear wave coupling mechanisms: strong interactions
(Kolmogorov turbulence), three-wave interactions (Kraichnan turbulence), and four-wave interactions. Comparisons of the
model with the observations indicate that the three-wave decay processes with participation of both Alfvén and magnetosonic
waves are the main type of nonlinear interactions in the inertial range of the turbulence power spectra.
INTRODUCTION
The properties of the solar wind’s outer scale of turbulence, particularly its dependence on heliocentric distance and local plasma parameters, have not been studied
sufficiently, especially for the inner solar wind. Moreover, the outer scale of turbulence, in addition to its
significance for the problem of solar wind formation
and evolution, is a crucially important parameter for the
physics of turbulence itself. A model of the formation
and evolution of the density turbulence outer scale in the
inner solar wind is developed here and compared with
observational data obtained during radio occultation experiments with the Galileo and Ulysses spacecraft.
OBSERVATIONAL RESULTS
The density turbulence outer scale makes itself apparent,
for example, in long, uninterrupted intervals of frequency
fluctuation measurements. The specific data discussed
here were obtained in 1995/1996 using the Galileo and
Ulysses spacecraft carrier signals at the frequency 2295
MHz. Temporal power spectra were calculated for frequency fluctuations recorded by ground-based tracking
stations during solar conjunction for solar ray path proximate points in the range of heliocentric distances between 7 and 80 R (R = solar radius). Frequency fluctuation power spectra, an example of which is shown in
Fig. 1, generally display a clearly defined peak at temporal frequencies ν max ' 0.1 mHz and fall off beyond this
frequency with an index α f (inertial range). These peaks
have been interpreted as a manifestation of the density
turbulence outer scale in the outward-flowing solar wind
plasma. The peak associated with the turbulence outer
scale is much more pronounced in the frequency fluctuation spectra than, for example, in quasi 2D-spectra
for the phase fluctuations or the quasi 1D-spectra for in
situ measurements, because the instantaneous frequency
is equal to the time derivative of the radio wave phase
(Armand et al., 1987).
Values of the density turbulenceqouter scale L o were
estimated from the relation L o = 2=α f vc =νmax , with
the solar wind convection velocity v c determined from a
cross-correlation analysis of frequency fluctuations measured simultaneously at two widely-spaced ground stations (Bird et al., 2002). As illustrated in Fig. 2, it was
found that the turbulence outer scale L o (R) increases
with increasing heliocentric distance R of the radio ray
path proximate point.
Values with error bars in Fig. 2 were derived using
simultaneous two-station correlation determinations of
solar wind convection velocity. Values without error bars
use vc from a theoretical model. Solid (open) data points
are for occultation ingress (egress). A linear regression
best fit to the entire data set, shown by the thick solid line
in Fig. 2, yields a radial dependence L o (R) = a[R=R ]m ,
with a ' 0.23 R , and m ' 0:82. Preliminary analysis
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
445
ear Alfvén wave interactions. The propagation of Alfvén
waves can be described by the following equation (Tu
and Marsch, 1995):
∇
FIGURE 1. Example of a frequency fluctuation power spectrum. The peak at the fluctuating frequency νmax ' 0.1 mHz is
associated with the density turbulence outer scale. The spectrum decreases as a power-law ν α f for ν > νmax (adapted
from Bird et al., 2002).
(
3
~
vc +~va )Wk
2
Wk =
of the Ulysses radio occultation data yields very similar
results with a ' 0.25 R , and m ' 0:88 (Wohlmuth et
al., 2001).
Ck γ
Ckoα γ k
Fk = ε k va
2
We now consider a possible formation mechanism for the
turbulence outer scale in the outward flowing nonuniform solar wind plasma. We assume that: (a) the main
energy-containing disturbances are Alfvén waves, generated initially at the coronal base and propagating away
from the Sun, and (b) density fluctuations are associated with magnetosonic waves excited locally by nonlin-
446
∂ Fk
∂k
(1)
α
for k < ko
for k > ko
(2)
where C is the structure constant, γ is an initial (shallow)
power exponent, and the power exponent α (α > γ ) is
defined by the k-dependence of the cascading function
Fk (see Fig. 3). The exponent α is related to the index α f
of the frequency fluctuation spectrum (previous section)
by α = α f + 1.
Because the spectral range k k o serves as an energy
source for the spectral energy flux and for the turbulence
at k > ko , the value of k o shifts to lower k with increasing
distance from the Sun. The cascading function Fk can be
specified in the following form:
FORMATION MECHANISM OF THE
TURBULENCE OUTER SCALE
1
~
vc ∇Wk =
2
where Wk is the spectral density of the 1D Alfvén wave
power spectrum, k is the wavenumber, and ~v c and ~va are
the solar wind convection speed and Alfvén speed, respectively. The left-hand side of Eq. (1) basically corresponds to a WKB approach, and the right-hand side describes the various nonlinear cascading processes, with
Fk being a cascading function. Following the work of
Chashei and Shishov (1981), we define the inverse turbulence outer scale k o = 2π =Lo as that wavenumber at
which the typical time scales of linear t l = R=(vc + va )
and nonlinear t nl = Fk =k processes are approximately
equal. Those waves with k < k o propagate in the linear
regime, and the wave energy is pumped to higher k by
cascading processes to form an inertial scaling spectrum
with Fk = const in the range k > k o , dominated by nonlinear processes. Magnetosonic waves (and consequently
density fluctuations) are generated locally by nonlinear
wave interactions in the spectral range k > k o . The power
spectra of the Alfvén waves (magnetic field turbulence)
and the magnetosonic waves (density turbulence) have a
similar shape at k > ko (Chashei and Shishov, 1985). In
the model considered here, we represent the turbulence
power spectrum by a broken power law:
FIGURE 2. Radial dependence of the density turbulence
outer scale Lo (R) deduced from Galileo measurements during
the solar conjunction in 1995/96 (adapted from Bird et al.,
2002).
4π kWk
B2
n
Wk
(3)
where ε = const 0:1 (Tu and Marsch, 1995), and B
is the local ambient magnetic field strength. The values
of the power exponent n in Eq. (3) and the corresponding
values of the inertial spectral range power exponents α in
Eq. (2), as obtained from dimensional analysis, are equal
to:
n = n1 = 1=2; α = α1 = 5=3
n = n2 = 1;
α = α2 = 3=2
(4)
n = n3 = 2;
α = α3 = 4=3
mately constant at a fixed distance R. We then have:
Lo =
with
2π
ko
vcµ
(7)
µ1 = (1 2γ )=2(3 γ )
µ2 = 2(1 γ )=(2 γ )
µ3 = (5 4γ )=2(3 2γ )
(8)
COMPARISON WITH OBSERVATIONS
FIGURE 3. Turbulence power spectrum given by Eq. (2).
The intrinsic spectrum for k < ko is characterized by a spectral
index γ. The spectrum steepens in the inertial range (k > ko ),
starting from the wavenumber corresponding to the outer scale
and extending up to a dissipation scale indicated here by km .
The index n 1 = 1=2 corresponds to the Kolmogorov
spectrum with strong turbulence (Tu and Marsch, 1995).
The cases n2 = 1 and n3 = 2 apply to the weak threewave (Kraichnan spectrum) and weak four-wave interactions, respectively (Chashei and Shishov, 1977).
OUTER SCALE DEPENDENCE ON
HELIOCENTRIC DISTANCE AND
LOCAL PLASMA PARAMETERS
Assuming that the typical rates of linear and nonlinear
processes are equal, we find the following radial dependence for the turbulence outer scale in the developed solar wind (where v c = const, B R 2 , va R 1 , and, as
follows from the WKB approach, C R 3 ):
Lo =
with
2π
ko
Rm
m1 = 1=(3
m2 = 1=(2
m3 = 1=(3
γ)
γ)
2γ )
(5)
We now compare the radial dependence of the turbulence
outer scale predicted by the models with the Galileo
observations presented in Fig. 2. It is found that γ 1 =
7=4, γ2 = 3=4, and γ3 = 1=4. As known from in situ
measurements of magnetic field fluctuations near R = 1
AU (Matthaeus and Goldstein, 1986), as well as from
Faraday rotation fluctuations near the Sun (Chashei et
al., 2000), that the low frequency exponent of the power
spectra is γobs 1 (so-called “flicker” spectrum).
A comparison of the above estimates γ 1;2;3 with γobs =
1 shows convincingly that the model with n = n 2 = 1
agrees best with the observed radial dependence of the
turbulence outer scale. The estimate γ 1 = 7=4, in fact,
would mean that the turbulence spectrum does not even
have an outer scale, because γ 1 = 7=4 > α1 = 5=3. Finally, it is noted that the value γ 3 = 1=4 is reasonable,
in principle, but differs rather strongly from the observational value γobs = 1.
Within the framework of our model with n = n 2 = 1,
we can use the numerical values of the turbulence outer
scale Lo to estimate the fractional turbulence level and its
dependence on heliocentric distance:
4π C
B2
03
:
R
50R
(9)
where the right-hand side of Eq. (9) is in reasonably
good agreement with the in situ measurements of the
Helios spacecraft (Tu and Marsch, 1995) over the radial
distances 0.3 AU < R < 1.0 AU, as well with the radial dependence of C expected from the WKB approach
(Belcher and Davis, 1971).
(6)
where the indicies 1,2,3 are related to the corresponding
cascading functions given by Eqs. (3) and (4). Similarly,
we can find the dependence of L o on local plasma parameters. This can be reduced to the dependence of L o
on the solar wind speed v c if we assume that the solar
wind acceleration is associated mainly with the Alfvén
waves and the solar wind mass flux density is approxi-
447
CONCLUSIONS
The density turbulence outer scale deduced from the
Galileo and Ulysses radio occultation data increases approximately linearly with increasing heliocentric distance.
Comparison of the observed radial dependence of the
turbulence outer scale with different versions of the theoretical model for turbulence evolution shows that the
Kraichnan type model with n = 1 appears to yield better
agreement than the models with n = 1/2 or n = 2. The
physical consequences of this fact, as well as comparisons with other observational data, should be considered
in more detail in future work.
As follows from the estimate given in Eq. (9) for
the Galileo data, the fractional level of turbulence is
moderate for the developed low-latitude solar wind in the
inner heliosphere, at least for the period of solar activity
minimum.
Further studies of the turbulent cascading mechanism
could include a more detailed analysis of the parametric
dependence of the turbulence outer scale on the solar
wind speed. Indeed, the dependence of the outer scale
on solar wind speed, L o (vc ) in Eqs. (7) and (8), differs for various cascading mechanisms. For example,
Lo decreases with increasing solar wind speed v c for
the Kolmogorov turbulence with n = 1=2, is almost
independent of v c for the Kraichnan turbulence with
n = 1, and increases with increasing v c for the four-wave
interactions with n = 2 (or for higher-order interactions
with n > 2).
ACKNOWLEDGMENTS
This work presents results of a bi-national Research
Project partially funded by the Deutsche Forschungsgemeinschaft (DFG) and the Russian Foundation for Basic
Research (RFBR), Grant 00-02-04022. Additional support from the RFBR, Grant 00-02-17845, is acknowledged.
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