Estimators
The wavelet cross-correlation is given in Eq. (), where
ñe and B̃ are the complex wavelet coefficients for the
electron density and the magnitude of the magnetic field respectively, while
the Re denotes the real part and the asterisk denotes the complex conjugate.
The value of CC varies between -1 (full anti-correlation) and +1 (full
correlation). In order to compare the magnetic field data with the electron
density data in this way, the magnetic field data are resampled at 5 Hz.
CCn,B(t,f)=Reñe(t,f)B̃*(t,f)|ñe(t,f)||B̃(t,f)|
As a further diagnostic tool for the magnetic fluctuations, we also analyse
the magnetic helicity. The magnetic helicity is the spatial rotation sense of
the magnetic fluctuation about the wave vector. We use the definition in Eq. (), where the tilde denotes the wavelet
coefficients of the GSE components of the magnetic field in GSE and Im
denotes imaginary parts.
σm(t,f)=ImB̃y*(t,f)B̃z(t,f)|B̃y(t,f)|2+|B̃z(t,f)|2
It is important to note that this definition assumes that the wave vector
k is along -xGSE (approximately along the flow) and also
implicitly assumes Taylor's hypothesis, i.e. that there is no temporal change.
Thus, the magnetic helicity varies between -1 and +1 and for this definition
can be regarded as being the same as polarisation (temporal rotation sense of
the fluctuations with respect to the magnetic field). Thus, positive
(negative) helicity is indicative of a polarisation sense in the right- (left-)
handed direction in the direction of electron (proton) gyration. However, an
important caveat of this method is that it does not contain information about
the propagation direction of the fluctuations. As such a right-hand polarised
wave can give the appearance of a left-handed wave if the propagation
direction is reversed .
Results
Figure a shows the cross-correlation spectra, where the white line
denotes the cone of influence associated with the wavelet transform
; below this line the spectrum is unreliable due to edge
effects. The spectrum is dominated by anti-correlated fluctuations throughout
with small regions of sporadic positive correlation peppering the spectrum.
This suggests that IBWs are not present or are only present for very brief
times.
Figure b shows the magnetic helicity spectrum. The value of the
helicity shows a mix of different values on fluid scales f∈[0.01,0.2]
and a general increase on kinetic scales f∈[0.4,2] towards a value
of σm∼0.5 near 1 Hz. At spacecraft frame frequencies of
fsc≥1.0 Hz and fsc≥2 Hz, instrumental noise becomes
significant in the spacecraft potential and in the magnetic field data
respectively. Another important issue is that of aliasing, which affects high
frequencies and can artificially reduce the value of the helicity (e.g.
). However, in previous studies of the helicity, the sampling
rates of magnetic field instruments is much lower than for Cluster, and the
data presented here are not as strongly affected. Aliasing would be expected
to significantly affect frequencies above 5.5 Hz in the helicity measurement
and 1.25 Hz for the cross-correlation; however, we are already near the
instrument noise levels.
(a) Wavelet cross-correlation of the electron density and magnetic
field strength. (b) Magnetic helicity. The white dashed line denotes the cone
of influence region where the spectrum is unreliable due to edge effects.
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The negative cross-correlation signature and negative helicity on fluid
scales is likely to be due to MHD slow waves or pressure-balanced structures . However, there are regions on fluid
scales which have positive helicity and negative cross-correlation which
could be due to a fluid-scale Alfvén wave. On ion kinetic scales, more
regions of positive cross-correlation appear, but they are still less numerous than
the negatively correlated regions, while the magnetic helicity increases and
is predominantly above 0.
The mean value at each frequency for the cross-correlation and the helicity
is given in Fig. a. The cross-correlation remains negative
throughout, with a minimum near 0.5 Hz, and then remains at a similar value
until Taylor-shifted proton scales, and then it increases before being
influenced by noise near 2 Hz. The magnetic helicity is close to 0 on fluid
scales and then increases at 0.5 Hz, reaching a maximum plateau at the Taylor-shifted scales (denoted by the green and purple vertical line) before
decreasing. Analysis of the helicity from the search coil magnetometer (SCM)
is shown as the dotted red line in Fig. a. The
SCM is more sensitive than FGM and is sampled at 25 Hz. The data from
SCM show that the helicity is indeed lower at 2<fsc<5.5 Hz, and due
to the higher sampling rate and sensitivity, this is not an artefact of aliasing
or noise.
To understand the relative importance of the competing wave modes, we use the
thresholds in Table to find the relative occurrence rates.
These thresholds are motivated by what is expected for each wave mode from
linear theory (e.g. ) and assume anti-sunward propagation.
Table of the thresholds used to identify different wave modes for fluid (fl) and kinetic (ki) scales.
Branch
σm (fl)
CC (fl)
σm (ki)
CC (ki)
Alfén (AW)
>0, <0.5
<-0.5
>0.5
<-0.5
Slow (SW)
<0, <-0.5
<-0.5
<-0.5
<-0.5
Fast (FW)
>-0.05, <0.05
>0.5
>-0.3, <-0.05
>0.5
Figure b shows the ratio r of data points which reach the
thresholds outlined in Table to the total number of data points.
On large scales the plasma is dominated by MHD Alfvén waves and slow waves.
However, as we approach proton scales, these both decrease and this method
clearly identifies that the dominant type of fluctuation on proton scales is
the KAW, with very few points at proton kinetic scales reaching the
thresholds for KSWs or IBWs. At the highest frequencies there is a small
increase in both of these wave modes; however, it is unclear whether this is
true or related to the instrumental noise. It should be noted that a
sunward-propagating KSW would in this case produce the same signature as the
KAW; therefore, in interpreting the data in and
Fig. 1c and d this mixture of two waves may explain the scatter seen.
Moreover, even though some points in Fig. 1c and d, may agree better with the
curves for IBWs as there is no significant signature in the
cross-correlation, this is most likely due to wave–wave interactions between
individual KAW packets or KAWs and KSWs . It is
also interesting to note that while the KAW dominates both other wave modes,
the majority of the fluctuations do not fall into the thresholds in
Table .
(a) Mean values at each spacecraft frame frequency for the
helicity (red) and the wavelet cross-correlation (blue). Light shaded areas
denote 1 standard deviation, and the dashed red line denotes the helicity
from the search coil magnetometer. (b) Ratio of data points meeting
the thresholds to the total number of points in the interval, which are
defined in Table . The solid lines denote the ratio of points that
meet the kinetic thresholds. Lighter dotted lines denote the ratio for
the fluid thresholds.
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