The three-dimensional structure of both compressible and incompressible
components of turbulence is investigated at proton characteristic scales in
the solar wind. Measurements of the three-dimensional structure are typically
difficult, since the majority of measurements are performed by a single
spacecraft. However, the Cluster mission consisting of four spacecraft in a
tetrahedral formation allows for a fully three-dimensional investigation of
turbulence. Incompressible turbulence is investigated by using the three
vector components of the magnetic field. Meanwhile compressible turbulence is
investigated by considering the magnitude of the magnetic field as a proxy
for the compressible fluctuations and electron density data deduced from
spacecraft potential. Application of the multi-point signal resonator
technique to intervals of fast and slow wind shows that both compressible and
incompressible turbulence are anisotropic with respect to the mean magnetic
field direction

The solar wind is a collisionless, magnetised plasma originating from the Sun
and is often observed to be in a state of fully developed turbulence

To study the anisotropy, correlation lengths in the directions parallel and
perpendicular to the magnetic field

The solar wind plasma is also weakly compressible (e.g.

Several studies of the density power spectrum have highlighted the importance
of plasma

Several different approaches can be taken to investigate the spatial
structure of the plasma. Often, much of the information gained about solar
wind turbulence comes from single spacecraft measurements. Should the
fluctuations evolve slowly compared to the time it takes them to advect over
the spacecraft, spatial information can be gained along the sampling
direction, by assuming Taylor's hypothesis

In the study of

Given modern computer processing power, an alternative to using Taylor's
hypothesis is to investigate the spatial structure of plasma through direct
numerical simulations. Several different schemes have been employed including
magnetohydrodynamic (MHD) at large scales (e.g.

A final possibility to overcome the difficulties and ambiguities associated
with studying turbulence with a single spacecraft are to use multi-spacecraft
missions such as the Cluster mission

The multi-point signal resonator technique (MSR;

An interesting result from the aforementioned works of

In this paper we present a new study of the three-dimensional structure of
the turbulence in the solar wind, and this will be investigated using the MSR
technique

Table of the mean plasma and spacecraft parameters in the interval
organised from low to high

Multi-point data from the solar wind are provided from the Cluster mission

In total four intervals are analysed, where we have magnetic field and
spacecraft potential data available for all intervals. These intervals were
chosen as there is no connection to the foreshock and the Cluster spacecraft
configuration has inter-spacecraft distances of

As the MSR technique is a global technique (based on Fourier analysis), we use the mean magnetic field in the interval to define the parallel direction, whereas the perpendicular direction is defined as the mean over all angles which are perpendicular to the mean direction. It is important to note that anisotropy has been measured to be related to the local magnetic field direction, therefore intervals are selected so that there are no large changes in the magnetic field direction. The times are selected to be sufficiently short so that the local magnetic field at proton scales and the global magnetic field are approximately similar while having enough data points for sufficient averaging required for the MSR technique. Should a much longer time interval be used, then the global and local magnetic field directions might differ significantly and results may not be representative.

To estimate

The main advantage of the MSR technique is that it does not require assuming
Taylor's hypothesis, but does assume that the fluctuations can be described
as a superposition of incoherent plane waves (random phases), and that the
signal can be decomposed into separate signal and noise components

Typically when applying the MSR technique to magnetic field data, 12 time
series have been used as inputs: three components of the magnetic field at
four spacecraft. Additionally for the vector magnetic field data, the
solution can be constrained by using the divergence-free nature of the
magnetic field. In this study we will investigate the three-dimensional
incompressible turbulence by using the vector magnetic field (12 time series)
which we use as a proxy for the incompressible fluctuations in the solar
wind. The compressible component of the turbulence will be investigated by using the magnitude of
the magnetic field and the electron number density which both take an input
of four time series each. The technical details of the method and its
application to a scalar time series are discussed in detail and tested for
simulated and real data in

One important issue is that the presence of a constraining condition has the
effect of eliminating any mathematical solutions which do not satisfy the
condition. Conversely the absence of a constraining condition makes the
possibility of spatial aliasing occur more likely when two wave vectors
cannot be differentiated between each other, e.g.

To display the results of the four-dimensional

The total magnetic field which is dominated by incompressible fluctuations
is given in Fig. 1c, f, i, l. For the total magnetic fluctuations,
the divergence-free condition can also be used to improve the solution.
The result for the compressible components relating to electron density and
magnetic field magnitude are given in Fig.

Reduced two-dimensional spectra which have been integrated in
frequency and integrated azimuthally as described in Eq. (

It is clear for the incompressible components of all intervals that there is
an anisotropy in the direction perpendicular to the magnetic field direction,
in agreement with the vast amount of literature at MHD scales to proton
kinetic scales (e.g.

One interesting feature in the incompressible component of the fast wind
intervals is that there is a distinct enhancement along the magnetic field
direction up to around

Another interesting feature is that this parallel component shows no
associated counterpart in either compressible component suggesting that these
parallel wave vectors are not very compressive, ruling out that the
components at these scales are parallel magnetosonic waves which have higher
compressibility

To quantify the degree of anisotropy in both cases for the compressible and
incompressible components, we use the anisotropy index

The range of physical scales are limited by the spacecraft separation (which
was roughly 200 km in 2004), but are also limited by the plasma parameters
giving slightly different ranges (in normalised units) for the spectra in
Fig.

Generally all of the incompressible points follow the empirical law obtained
in

In the two intervals of slow wind the compressible magnetic field is the
least anisotropic, with one interval showing the incompressible fluctuations
(

There is also a difference between the two compressible components (density
and magnitude of the magnetic field). This suggests that a single wave mode
cannot be used to describe the fluctuations present in the solar wind. For
example, should the turbulence only contain slow waves, both compressible
components would be expected to have similar shapes and anisotropies by
virtue of the strong anti-correlation of both compressible
fluctuations. KAWs also exhibit similar anti-correlations; however, the fluctuations have
smaller amplitudes. For the fast wind, where ICWs are seen,
there is little to no correlation and any compressible fluctuations are
likely to be too small to be measured effectively

The results suggest that both the value of the plasma

Recent studies have found a tendency towards isotropy at sub-ion scales

1-D integrations of the two-dimensional spectra presented in
Fig.

In Fig.

The perpendicular magnetic field at inertial range scales with a Kolmogorov
power law of

For the density spectra, similar power laws are often observed to have a
Kolmogorov-like inertial range followed by a flattening in the spectra
between inertial and kinetic ranges, which is sensitive to the value of the
plasma

The total magnetic fluctuations perpendicular slopes show similar scalings at
inertial range and kinetic range scales in line with previous studies of the
trace magnetic fluctuations

The orange and green arrows denote the

The spectra for all components are shown to be steeper in the parallel
direction and the break scale for the different directions is at a different
location with the parallel break being at larger scale than the perpendicular
direction. A rough agreement is found for some intervals with the spectral
indices expected for critical balance at fluid scales for the incompressible
components (e.g. Fig.

In this paper we have sought to investigate the morphology and the shape of
incompressible and compressible turbulence in the solar wind with multi-point
measurements as well as the one-dimensional spectra as a function of
wave number. It is clear that density and compressible magnetic field
fluctuations display the same power anisotropy as the incompressible magnetic
field with the power being elongated in the perpendicular direction. However,
in accordance with previous studies we see that the value of plasma

The anisotropy is also shown to be different in the fast and slow wind, where
extensions in power are seen in the

The multiple spacecraft of Cluster also allow a simultaneous measurement of the spectral index in the parallel and perpendicular directions, and revealed that the spectral index is steeper in the parallel direction for the total magnetic field, the magnitude of the magnetic field and the density in some cases; the spectra of the vector magnetic field are approximately consistent with the predictions of a critically balanced cascade at fluid scales. However, at kinetic scales the agreement is weaker.

To conclude, we have shown that the anisotropy is a function of the plasma

All Cluster data are obtained from the ESA Cluster Science
Archive:

OWR contributed to the analysis of the data, the writing and co-ordination; YN and CPE contributed to the interpretation of the data and general improvements to the manuscript.

The authors declare that they have no conflict of interest.

All Cluster data are obtained from the ESA Cluster Science Archive. We thank the FGM, CIS, WHISPER, EFW and PEACE instrument teams and the ESA Cluster Science Archive. Owen W. Roberts is funded by a ESA Science Fellowship. The topical editor, Minna Palmroth, thanks two anonymous referees for help in evaluating this paper.