The Geometry of Turbulent Magnetic Fluctuations at High
Heliograpahic Latitudes
Charles W. Smith
Bartol Research Institute, University of Delaware, Newark, Delaware
Abstract. Although the orientation of wave vectors for interplanetary magnetic fluctuations is often discussed or assumed
when modelling the solar wind turbulence, there were few if any direct measurements of the wave vector orientation until very
recently. Indirect inferences abound. For instance, the transverse nature of magnetic fluctuations has often been used to infer
a wave vector parallel to the mean magnetic field, but transverse fluctuations do not necessarily lead to parallel wave vectors
– the fluctuations of two-dimensional turbulence are fully tranverse, but their wave vectors are also transverse to the mean
field. Bieber et al. [3] offer the first single-point algorithm for separating field-aligned wave vectors from perpendicular wave
vectors using the component spectra. This method has since been used by others [5, 6, 8, 9] and is here applied to Ulysses
measurements in a comparison of high latitude turbulence with the results of previous studies of near-ecliptic observations.
INTRODUCTION
Until now, this method has not been applied to the
Ulysses observations at high heliographic latitude.
Although the anisotropy of interplanetary magnetic field
(IMF) fluctuations has been recognized for some time
[1], little is known of the orientation of the associated
wave vectors. Under limited conditions and assumptions
the minimum variance direction corresponds to the direction of the wave vector (nonlinear or eliptically polarized Alfvén and fast mode waves). However, under
many more general assumptions (multiple waves projecting onto the same spacecraft frame frequency and 2dimensional (2-D) turbulence where the wave vector is
normal to the mean field direction B 0 , but the minumum
variance direction is parallel to B 0 ) the wave vector and
minimum variance direction are unrelated.
Two clear conclusions of the observed correlation
function for IMF fluctuations at 1 AU [7] are: (1) the angular width of the correlation function is sufficient to permit multiple wave vectors to project onto the same spacecraft frequency and (2) wave vectors at large angles to B 0
with kk =k? 0 are present. Therefore, it is suspect that
minimum variance directions are an inadequate prescription by which to determine the orientation of the wave
vector.
The relative importance of the large k ? oscillations
during energetic solar particle events was made clear
by Bieber et al. [1996] who demonstrated that 85%
of the energy in the observed IMF fluctuations at 1 AU
resided in the 2-D component. Previously, Bieber et al.
[2] demonstrated that nearly this same 20/80 ratio of slab
(1-D) to 2-D energy could account for the observed mean
free path of solar energetic particles.
TECHNIQUE
We adopt the technique described by Bieber et al. [3].
The observed spectrum of IMF fluctuations is assumed
to be an admixture of transverse fluctuations, some of
which have wavevectors parallel to the mean magnetic
field (k k B0 ) and some having k ? B 0 . This is a simplification of the “Maltese Cross” correlation function for
IMF fluctuations at 1 AU [7]. Figure 1 illustrates the two
conditions.
The IMF data is rotated into the same mean field
coordinate system used by Belcher and Davis [1]. In this
coordinate system one component perpendicular to B 0 is
also perpendicular to the wind velocity V SW and one is
not. We denote the “Y” component as that component
which is perpendicular to both B 0 and VSW while the
FIGURE 1. Illustration of the two k orientation for slab (1D) and 2-D geometries. Both have IMF fluctuations δ b ? B0 .
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
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413
FIGURE 2. The %Slab, power law index q, and IMF fluctuation anisotropy are plotted for the analyzed intervals. Square
symbols represent averages over the time intervals represented by horizontal lines. Vertical lines give the variance of the underlying
subset.
“X” component is ? B 0 but has a nonzero projection onto
VSW . It can then be shown [3]:
PY
PX
where
and
=
ks1
q + r0
ks1
q + r0
2q
1+q
2
1+q
k21
q
k21
q
is the fraction of energy contained in the slab (1-D)
component.
From this it follows that PY =PX = 1 if r = 1 and the
spectrum is entire composed of k k B 0 wave vectors.
If PY =PX = q, then r = 0 and the spectrum is entirely
composed of 2-D fluctuations. Also, 1 < PY =PX < q for
all ΘBV . Belcher and Davis [1] found that PY =PX 5 : 4.
This ratio implies that r = 0:57 for Θ BV = 45Æ and q =
5=3 indicating that the interval studied by Belcher and
Davis contains an approximately equal admixture of slab
(1-D) waves and 2-D turbulence.
The geometry analysis assumes that the spectrum in
the rest frame of the plasma is axisymmetric and invariant to rotation about the mean magnetic field. One test
of this assumption is that both transverse components
should have the same power law index. Sometimes they
do and often they do not. Intervals when the two power
law indices differ substantially are excluded from this
analysis.
(1)
ks = 2πν =VSW cos(ΘBV )
(2)
k2 = 2πν =VSW sin(ΘBV ):
(3)
ΘBV is the angle between the B 0 and VSW , where the
wind velocity is assumed to be in the Radial R direction,
q is the power law spectral index, and ν is the spacecraft frame frequency. From this we can obtain:
r0 = [tan(Θ
1 q
BV )]
where
r=
PY =PX 1
q PY =PX
1
1 + r0
1+q
2
(4)
(5)
414
a slab percentage less than zero or greater than 100%.
This reflects statistical fluctuations within the ensemble
that can push the ratio PY =PX beyond the bounds of the
geometry analysis. Only the statistics of the distribution
are truly meaninful. The low-latitude %Slab results are
unusually high compared with the low-latitude, 1 AU
results of Bieber et al. [3] and Leamon et al. [5, 6],
but settle down to the expected value by mid-1993. As
the spacecraft climbs in latitude toward the south solar
pole the slab fraction of the spectrum climbs to a value
of 40%, a value that is in excess of the nominal
low latitude results at 1 AU. Temporarily setting aside
the low latitude results of this analysis, this means that
the high latitude wind at solar minimum is more wavelike than the low latitude wind, but is not exclusively
composed of parallel propagating waves. Both waves oof
the slab (1-D) component and turbulence as represented
by the 2-D component are present in approximately equal
proportion.
Figure 3 shows the probability distribution function
of %Slab values computed by this analysis for the year
1994 when Ulysses was closest to the south solar pole.
The poorness of the PDF is in part due to the excessive application of the axisymmetry assumption which
reduced the number of usable intervals. Nevertheless, a
peak near 50% is evident as opposed to a bi-modal distribution peaked at 0% and 100% that would be the case
if the two populations of slab (1-D) and 2-D geometries
were spatially isolated. This distribution, whose mean is
44%, indicates that the two populations of slab (1-D) and
2-D wave vectors coexist in the same plasma elements.
Figure 4 compares the observed variation of PY =PX
with the prediction based on the geometry analysis for
varying %Slab using the observed average value of the
power law index during this time q = 1:75 for the year
1994. Although the fit is not as good as seen by Bieber et
al. [3] at low latitudes, it is generally consistent with an
equal admixture of slab (1-D) and 2-D geometries.
FIGURE 3. Probability distribution function for %Slab as
computed from Ulysses data intervals during 1994.
ANALYSIS
We have examined 530 intervals of Ulysses data, each
12 hours to 2 days in duration with 1 minute data resolution obtained from the National Space Science Data
Center (NSSDC). The analysis begins with day 298 of
1990 and extends over the southern solar pole at the end
of 1994. The analysis applies an automated badpoint removal algorithm before rotating to mean field coordinates. A first-order difference filter is applied followed
by a Blackman-Tukey correlation function and spectral
analysis. The resulting spectra are post-darkened to correct for the first-order differencing filter and the geometry analysis is applied.
Figure 2 shows the results of the analysis. The geometry analysis neglects the parallel fluctuation component and assumes the fluctuations are all transverse to
the mean field. The bottom panel addresses this assumption by plotting the ratio of the perpendicular and parallel
B =E B . Although highly
components of the fluctuation E ?
k
variable at low latitudes, the magnetic power ratio settles down to a consistent value of 3 beyond 30 Æ latitude and slowly increases as the spacecraft approaches
the southern solar pole.
The power law index q is also critical to the analysis.
While again variable at low latitudes, it also settles into
a consistent value of 1:75 at southern latitudes below
30Æ and approaches 5=3 over the solar pole. The relatively high values of q are troubling and demonstrate the
leading reason why the results shown here are considered
preliminary. Low latitude values of q are normally found
to be 5=3 < q < 3=2.
The results of the %Slab= r 100 analysis is shown
in Figure 2. It must be noted that this analysis can yield
CONCLUSIONS
We have applied the technique and assumptions of
Bieber et al. [3] to Ulysses IMF measurements in an
attempt to obtain some insight into the relative orientation of wave vectors at high latitudes during solar minimum. This method assumes that the wave vectors are
either parellel or orthogonal to B 0 in keeping with a simplification of the observed correlation function for 1 AU
near-ecliptic observations. We find the IMF fluctuation
energy to be almost equally divided between the two assumed extremes.
It is sometimes assumed but never proven that the k k
population is waves while the k ? fluctuations constitute
415
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FIGURE 4. Ratio of the observed average PY =PX as a function of ΘBR compared with the prediction from the geometry
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Horizontal lines on the observed means represent the averaging
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7.
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equally distributed at high latitudes during solar minimum.
8.
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ACKNOWLEDGMENTS
This work was supported by NASA Sun Earth Connection Guest Investigator program grant NAG5-10911 to
the Bartol Research Institute.
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