1. Solar Wind Turbulence
Why study turbulence.
Turbulence studies are important because studies of fluid and magnetofluid flows have shown
that turbulent dissipation is known to be an effective mechanism for the heating of fluids and for
the transfer of momentum and energy. Coleman [1968] was one of the first to demonstrate the
presence of turbulent fluctuations within the solar wind.
Since Coleman [1968] first study on solar wind turbulence many other studies have employed
a large array of tools to classify the type of solar wind turbulence (I.e., Kolmogorov [1941],
Kraichnan [1965], or multifractal [Burlaga 1991a;b;c;1993]), characterize the strength of the
turbulence [Lui, 2001], and identify the source [Matthaeus et al., 1986; 1990]. The most common
means employed to identify the type of turbulence is to determine the spectral index from power
spectral density plots within the inertial range of the turbulent cascade. Both Kolomorogov
[1941] and Kraichnan [1965] derived specific spectral index values for homogenous neutral
fluids (α = -5/3) and magnetohydrodynamic plasma turbulence (α = -3/2), respectively, under the
assumption of a constant energy transfer rate. Spectral indices that vary from these could be
associated with intermittent turbulence (i.e., none constant energy transfer rate). This type of
turbulence can be best identified with mutifractal methods. Multifractal studies of plasma
turbulence often rely on the relationship:
s ( p)
S jp     j
(1)
that holds for fully developed turbulence. Here S is the normalized times series data averaged to
some resolution τ (=2nΔt) with Δt the data resolution, j refers to a vector component, p is some
moment of the distribution, and s(p) is the scaling exponential function [Anselmet et al., 1984;
Castaing et al., 1990; Burlaga, 1991a]. The scaling exponential function is expected to be linear
in flow and magnetic field fluctuations for Kolomogorov and Kraichnan turbulence if the energy
transfer rate is constant. Burlaga [1991a,b,c; 1993] showed that the scaling exponentials in SW
flow and magnetic field data are nonlinear suggesting that the energy transfer from the driving
scale to dissipation scales is variable (i.e., intermittent turbulence). The strength of the
intermittent turbulence is measured with the intermittent coefficient, which has values between 0
and 1 where a value of zero indicates the energy transfer rate is constant and a value of 1
indicates the energy transfer rate is highly variable [Antonia et al., 1982; Consolini et al., 1996].
In the SW Burlaga [1991a; 1993] determined values between 0.19 and 0.28.
Nearly all these previous studies rely on single spacecraft measurements and the assumption
that the observed fluctuations are frozen into the plasma flow (i.e., the fluctuations evolve slowly
with respect to the time it takes for the plasma to travel pass the instrument). In order to make
statements about the turbulent fluctuations across space the mean plasma speed is used to convert
from time to space (Δx = Δt·v). It has only been within the last 5 years that multi spacecraft
measurements have been utilized to examine turbulent fluctuations across space without the
assumption of frozen in flow. The work of Mattheaus et al. [2005] and Weygand et al. [2007a]
have examined two point cross correlation function obtained Cluster and other solar wind
spacecraft data from many different intervals in the solar wind to determine the magnetic
correlative and Taylor microscale from simultaneous multiple point measurements. These
studies had intervals with spacecraft separation as small as 150 km, which is much smaller that
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the Taylor micro scale of about 2400 km, and as large as 2.3x106 km, which is much larger than
the correlative scale at about 1.3x106 km. See Figure 1. The only spacecraft separation that they
were unable to obtain 2 point cross correlation values from was between about 10,000 km and
110,000 km. This lack of data was due to the little or no spacecraft separations at these distances.
Measurements within this range of separates would help improve our understanding of the
turbulent correlative scale and Taylor microscale value. With reliable correlative and Taylor scale
values we can calculate the effective magnetic Reynolds number in the solar wind. The effective
magnetic Reynolds number would be useful in magnetohydrodynamic modeling of the solar
wind and may provide constraints on kinetic theories of dissipation in space plasmas.
While relatively reliable correlative and Taylor scales have been determined from many
different solar wind intervals, what has not yet been done is an examination of the variation of
these values with respect to the angle they make to the mean magnetic field direction using
multiple spacecraft. The work of Matthaeus et al. [1986; 1990] using a single spacecraft suggests
that the correlative scale varies with respect to the mean magnetic field direction. Additional two
point cross correlation measurements with in the range of 10,000 to 110,000 km combined with
the already existing database of separations could be used to provide the first multi spacecraft
study of the variation of the correlative scale and Taylor scale with their angle with respect to the
magnetic field direction. Preliminary work by Weygand et al. [2007b] and [Osman et al., 2007]
suggests that the correlative scale varies significantly, but the Taylor scale remained relatively
constant. The work of Osman et al. is different from Weygand in the fact that they only look at
scale smaller than 40,000 km. Both studies, however, lacked measurements within the 40,000 to
110,000 km range and lacked cross correlation measurements for separation along the mean
magnetic field direction. If that preliminary study is correct, then the implication is that the
effective magnetic Reynolds number varies with respect to the mean magnetic field direction.
Outstanding questions:
1) Does the correlation scale vary with its angle to the magnetic field direction?
2) Does the correlation and Taylor scale vary with solar wind speed?
3) Does the strength of the plasma sheet turbulence directly proportional to the
strength of the turbulence with in the solar wind?
4) Ask Mel Goldstein. I have not heard back from him for about a week now.
2. Plasma Sheet turbulence
One of the primary goals of current investigations in space physics is to understand and
determine how electromagnetic energy stored in the magnetotail is transferred to plasma energy.
Wave-particle interactions and reconnection are surely relevant to energy transfer, but recently it
has been proposed that turbulent flows in the plasma sheet (PS) dynamics play a dominant role in
the process [Borovsky et al., 1997; 2003, Angelopoulos et al ., 1999; Chang, 1999; and Klimas,
2000]. From studies of fluid and magnetofluid flows, turbulent dissipation is known to be an
effective mechanism for the heating of fluids and for the transfer of momentum and energy.
However, measurements to characterize spatial and temporal variations independently and
ascertain the role of PS turbulence in energy transport remain incomplete. In particular, the
possible flow of turbulence in the PS plasma and in mediating energy exchange between lobe
plasma and ionosphere plasma has not been well studied.
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While a number of studies have been done to demonstrate the presence of turbulence and
characterize those turbulent fluctuations [Angelopoulos et al.,1999; Lui et al. 1998; 2001; 2002;
Borovsky et al. 1997; 2003; Vörös et al., 2003; 2004; 2005; 2006; Weygand et al., 2005; 2006;
2007a] many of these studies are limited to single spacecraft observations. Many of the same
tools for identifying and characterizing the turbulent fluctations are used even when the Taylor
hypothesis is no longer applicable within the plasma sheet. Ideally, when frozen in fluctuation no
longer apply one should use two point measurements over a range of spatial scales Angelopoulos
et al.,1999; Vörös et al., 2005; Weygand et al., 2005; 2006; 2007a]. The tools for identifying and
characterizing turbulence with in the plasma sheet are nearly the same as those used in the solar
wind and consist of calculating the Reynolds number, examining probability distribution
functions and their scaling features, multifractal analysis, wavelet analysis, power density
spectra, and calculating the local intermittence measure. Despite ambiguous results in some
studies the evidence suggests that the plasma sheet frequently display turbulent properties. The
study of Weygand et al. [2007] has taken plasma sheet turbulence studies a step further, but using
multi spacecraft measurements to determine both the correlative and Taylor scale for turbulence
with in the plasma sheet. Figure 2 shows the correlation function derived from a range of
spacecraft separations and used to obtain the two scales. While the data are adequate for
calculating the correlative scale, the figure shows that correlations coefficients of zero have not
yet been observed. Spacecraft separation larger than 10,000 km would be critical in identifying
the far end of the correlation function. With these two scale Weygand et al. obtained the effective
magnetic Reynolds number with in the plasma sheet at about 20 RE down the tail. This value is
critical for MHD magnetospheric models and studies on particle scattering within the plasma
sheet.
While the evidence suggests that turbulence is a consistent feature with in the plasma sheet a
number of outstanding question still remain:
1) At what spatial separation is the end of the correlation scale? Two point cross
correlation measurements suggest that the correlation coefficient becomes
zero by about 3 RE [Weygand et al., 2007a]. Measurements to support this
hypothesis do not yet exist and can only be provide with spacecraft separation
greater than 3 RE.
2) Do turbulence characteristics vary with the radial distance down the plasma
sheet? While observations have been made at distance from about 10 to 30 RE
[Lui et al., 1998; Borovsky et al., 1997; Weygand et al. 2005; 2006], few
measurements have been made at distances greater than 30 Re.
3) Does the strength of the turbulence with in the plasma sheet vary with
geomagnetic activity? No one has yet examined the variation of the strength of
turbulent magnetic field fluctuations with geomagnetic activity mainly due to
a lack of measurements for a good statistical study.
4) What is the mechanism(s) that generate turbulent fluctuations in the magnetic
field? It has been suggested that velocity shears [Borovsky et al., 1997; 2003]
and or reconnection may be the generation mechanism, but this has not been
thoroughly investigated.
5) Two point spacecraft measurements can be used to calculate energy transfer
rates within the plasma sheet. With spacecraft position far down the tail and
some closer to the Earth we can examine the differences in the energy transfer
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rate at difference radial distance to variation of the energy transfer rate
associated with the turbulent cascade at different distance down the
magnetotail.
3. Magnetosheath turbulence
Similar to the solar wind and plasma sheet, fluctuations associated with magnetosheath
turbulence have been studied for years with many of the same techniques discussed above.
However, as far as we are aware no one has used multi spacecraft techniques to characterize the
magnetosheath turbulence. Because most spacecraft measurement occur close to the nose of the
magnetosphere, the time it takes spacecraft to cross through the magnetosheath is relatively short
and long stable intervals are difficult to obtain. Multi spacecraft measurements obtained further
down the tail where the sheath is potentially thicker provide an opportunity to use two point
correlations to determine the correlative scale of turbulence with in the sheath.
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Figure 1. From Weygand et al. [2007]. This figure demonstrates the range of spacecraft
separations available in the solar wind and shows the lack of available measurements between
about 20,000 km and 110,000 km.
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Figure 2. From Weygand et al. [2007]. This figure demonstrates the range of spacecraft
separations available in the plasma.
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