Solar Wind Temperature Anisotropies
J. C. Kasper1, A. J. Lazarus1, S. P. Gary2, A. Szabo3
1
MIT/CSR,77 Massachusetts Avenue, Cambridge, USA, 2LANL/NIS-1, M.S. D466, Los Alamos, USA,
3
NASA/GSFC, Code 696, Greenbelt, USA
Abstract. Solar wind proton and alpha spectra from the Faraday Cup portion of the SWE experiment on the Wind
spacecraft have been analyzed to determine the temperature anisotropy of each species, under the assumption of
convected, bi-Maxwellian distributions. From the start of the mission in late 1994 to date we have collected over 2
million measurements of the anisotropies in the solar wind. This dataset is sufficiently large to conduct a statistical
study, comparing the observed temperature anisotropies to various limits imposed by instabilities. Specifically we will
discuss the effects of the firehose, mirror, and cyclotron instabilities. With a limit to the proton temperature anisotropy
established, we examine several cases where this limit is approached or exceeded and comment on magnetic field
activity and alpha parameters during these intervals. In the large plasma beta regime we illustrate evidence of a
transition from the cyclotron to the mirror instability as the dominant limit.
INTRODUCTION
The first-order departure from a simple Maxwellian
velocity distribution function (VDF) of a particle
species in the solar wind is due to the existence of a
temperature anisotropy. Generally this anisotropy is
well described by a convected, field-aligned, biMaxwellian VDF, with two temperatures, T⊥ and T||
perpendicular and parallel to the ambient magnetic
field Bo. The solar wind is a collisionless plasma, and
one might expect it to satisfy the double adiabatic
equations of state,
2
d æ T⊥ ö
d æç T|| B ö
ç ÷ = 0; ç 2 = 0
dt è B
dt è n
(1)
In this case an initially isotropic parcel of solar
wind would be expected to develop a temperature
anisotropy, R,
R = T⊥ T|| − 1
(2)
which might vary by orders of magnitude. While a
radial evolution of the proton anisotropy has been
observed [1], generally |R| ≤ unity.
FIGURE 1. Solar wind observations by the Wind spacecraft
on April 30, 1997. The alpha particle abundance is very low
and may be neglected. Note that the predicted value of T|| is
very high and off the scale of the plot for 0500-1500 UT.
Consider the solar wind observations on April 30,
1997 by the Wind spacecraft which are summarized in
Figure 1. There was relatively little activity in the first
15 hours of the day; in general the density was
increasing while the magnetic field strength decreased.
The observed parallel and perpendicular temperatures
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
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a range of values. For R<0 the firehose instability [5]
provides the lower limit, with
are shown along with predictions using (1). As the
solar wind evolves, T⊥ agrees with the adiabatic
predictions, while T|| decreases and dramatically
disagrees with that calculation.
R f = − S p ( β || p )
The permissible range of R is constrained by the
effects of kinetic micro-instabilities driven by the
temperature anisotropy. In a collisionless magnetized
plasma with R>0 mirror [2] and cyclotron [3]
instabilities may arise. These growing modes have
real frequency ωr with ωr~Ωp for the cyclotron
instability and 0<ωr<Ωp for the cyclotron instability,
where Ωp is the proton cyclotron frequency. Electrons
are non-resonant with these modes and to first order
the alpha particles may be neglected, so we may study
them with proton and magnetic field observations. For
a given fixed value of the dimensionless growth rate
γ/Ωp of these instabilities, linear theory and hybrid
simulations [4] have shown that the maximum values
Rm allowed by the mirror, and Rc by the cyclotron
instability take the form,
Rm ,c = S p ( β || p )
αp
αp
(4)
The resulting electromagnetic fluctuations scatter
the protons and drive the proton VDF to isotropy. The
final panel in Figure 1 shows the measured value of R
along with the theoretical upper bounds due to the
cyclotron (purple) and mirror (blue) instabilities using
(3), and the lower bound due to the firehose (red)
instability calculated with (4). It is clearly seen that
these instabilities limit the anisotropy accessible to the
solar wind protons, especially in the region with
β||p>20. The effects of the mirror and cyclotron
instabilities have been studied extensively in the
magnetosheath [6], but the only previous study of the
firehose instability is based on several hours of Vela 4
data [7]. We have performed a bi-Maxwellian analysis
of solar wind ions observed by the Solar Wind
Experiment (SWE) Faraday Cup (FC) instruments on
the Wind spacecraft [8]. A typical FC spectrum
contains 300 measurements of the proton VDF. By
comparing these observations with a model
distribution function using a non-linear fitting routine
we are able to extract the best-fit parameters, their
uncertainties, and χ2 per degree of freedom (dof).
These data, with the addition of magnetic field
measurements by the MFI experiment, have been used
to produce the first statistical study of the firehose
instability in the solar wind. The results of that study
and a complete description of the instrument analysis
techniques may be found in [9]. In this paper we
outline two separate approaches for understanding the
effects of these instabilities on the solar wind: by
statistical studies of a large sample of ion spectra over
a range of solar wind parameters and by examining the
detailed properties of specific events.
(3)
STATISTICAL METHOD
For the statistical study we look at the values of R
which the protons have access to as a function of
plasma beta. Based on (3) and (4) we expect to see
signatures of a bounds on R as a function of beta.
Figure 2 is a two-dimensional histogram of the 1.6
million observations with Vp<400 km/s, χ2/dof < 10,
and an uncertainty σR<0.3 collected by Wind from
1994-2001. The average uncertainty in R was 0.18, or
roughly the size of each bin in the histogram. The
most probable state of the protons is R=-0.3, β||p=0.8.
It is clear that for β||p>1 the protons are more strongly
FIGURE 2. Two-dimensional histogram of all solar wind
spectra with speeds less than 500 km/s as a function of
anisotropy and plasma beta.
where β||p is the parallel proton plasma beta,
β||p=8πnpkBT||p/Bo2, Sp is of order unity and determined
by the choice of γ/Ωp, and αp is roughly constant over
539
constrained to isotropy as β||p increases, in agreement
with the predictions of theory. To compare the
observed distribution of spectra with the predictions of
theory we employ two methods, both shown in Figure
3. In the top panel the number of spectra seen as a
function of R are plotted for several values of β||p. The
nature of the curves for R>0 are determined by the
competing effects of the mirror and cyclotron
instabilities, and observationally they are well
described by exponential curves. A fit to the β||p =0.1
interval is shown. In [9] the same analysis was applied
to the R<0 data to quantify the effect of the firehose
instability, in which it was assumed that for each
interval in β||p the observed limiting anisotropy Rf (β||p)
is given by the value of R at which the number of
spectra falls to 10% of its maximum value,. The best
fit of (4) to the measured values of Rf (β||p) was given
by (Sp, αp )=(1.21±0.26, 0.76±0.14), compared to
hybrid simulation results [5] of (1, 0.74) . The bottom
panel shows a version of the histogram in Figure 2
which has been normalized to remove the distribution
of solar wind spectra in β||p. The theoretical limits to R
are indicated. New features may be seen, such as the
clear competition between the expansion of the solar
wind and the effect of the firehose instability for R<0.
CLOSE STUDY OF SINGLE EVENTS
FIGURE 3. Two views of the histogram in Figure 2.
Upper: Number of spectra vs R for several intervals in β||p, fit
to β||p=0.1; Lower: The original histogram normalized to
illuminate the effect of the instabilities: Each column in β||p
is separated into the parts with R>0 and R<0, and each of
those parts is normalized to unity. Limits from instabilities
Rm(solid,R>0), Rc (dashed), Rf(solid,R<0).
FIGURE 4. Close inspection of three hours of observations
comparing anisotropy, limits, and magnetic fluctuations.
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We have shown that the properties of the cyclotron
and firehose instabilities may be probed with large
statistical studies of the allowed values of R as a
function of β||p, but there are limitations to this method.
It is clear from Figure 3 that the mirror instability is
not expected to contribute to a great extent until β||p
>5, so there are not enough measurements to perform a
similar statistical study; and instead we should focus
on individual periods with high β||p. Additionally,
while the variation of the slope of the curves in the top
panel of Figure 3 is likely related to the growth rate of
the instabilties as a function of β||p, a more direct
measurement of that growth may come from a study of
the electromagnetic fluctuations seen during individual
events [10].
3. An unanticipated lower limit to R at β||p<1,R<0, is
indicative of the existence of another instability.
4. Frequency and temporal dependence of magnetic
fluctuations during a high-β||p interval provides
evidence of the mirror instability.
More case studies of single events will permit a
quantitative probe of the mirror instability. We are
studying the effect of alpha particles. Work should be
done to relate the observed exponential fall-off shown
in Figure 3 to the growth rate of the instabilities and
possibly to the rate of cyclotron heating for R>0.
REFERENCES
We can sort through this large dataset of plasma
conditions to identify ideal periods for close
examination. The interval shown in Figure 1 is such
an example: The alpha abundance was extremely low,
the anisotropy was right at the theoretical limit, and β||p
was so high that the mirror instability contributed
equally to the enhanced field fluctuations which
scattered the proton VDF and prevented it from
becoming even more anisotropic. Figure 4 shows a
period in which R is near the limiting values Rc and
Rm. The second panel shows the RMS fluctuation of
the magnetic field about its ambient value as a
function of frequency and time. The red curve is Ωp.
The bottom panel shows the power in the frequency
range marked in the middle panel by the dotted line.
Note that for the duration of the interval there is an
enhancement of power near Ωp, but that between 0630
and 0730 large fluctuations are seen at ωr<Ωp, exactly
in agreement with the theoretical signiatures of those
instabilities discussed above.
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CONCLUSIONS
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We have used a large set of measurements by the SWE
experiment on the Wind spacecraft of the proton
temperature anisotropy in the solar wind to constrain
theoretical models of the mirror, cyclotron, and
firehose instabilities. Our results may be summarized
as follows:
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instrument for the Wind spacecraft, Space Sci. Rev., 71,
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observations of firehose constraint on solar wind proton
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10.1029/2002GL015128.
1. Statistically, the observed bounds on R are in
agreement with the limits imposed by the cyclotron
and firehose instabilities.
2. Solar wind with R<0 is pinned between the
competing effects of adiabatic expansion and the
firehose instability.
10. Anderson, B. J., S. A. Fuselier, S. P. Gary, and R. E.
Denton, Magnetic spectral signatures in the Earth’s
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Res., 99, 5877, 1994.
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