Four-dimensional energy spectra and a diagram for dispersion relations are determined for the first time in a magnetic reconnection region in the magnetotail using data from four-spacecraft measurements by the Cluster mission on a spatial scale of 200 km, about 0.1 ion inertial lengths. The energy spectra are anisotropic with an extension in the perpendicular direction and axially asymmetric with respect to the mean magnetic field. The dispersion diagram in the plasma rest frame is in reasonably good agreement with the ion Bernstein waves at the second and higher harmonics of the proton gyrofrequency. Perpendicular-propagating ion Bernstein waves likely exist in an outflow region of magnetic reconnection, which may contribute to bifurcation of the current sheet in the outflow region.

Magnetic reconnection in collisionless plasmas, as realized by auroral
substorms in the Earth magnetosphere and solar eruptions (flares and coronal
mass ejections), often exhibits wave activity. Two competing ideas exist on
the role of waves in reconnection. First, waves can trigger magnetic
reconnection through the anomalous resistivity caused by enhanced
electrostatic fluctuations such as the Buneman instability, the ion acoustic
mode, and the lower hybrid drift instability and through pitch angle
scattering by electromagnetic fluctuations such as kinetic Alfvén waves and
whistler waves (see review in

Here we present for the first time evidence for ion Bernstein waves in a
magnetic reconnection region in the Earth magnetotail. The analysis makes
extensive use of four-spacecraft measurements performed in situ by the
Cluster mission

On 24 August 2003, 18:30–18:50 UT, Cluster encountered a thin current sheet
in the Earth magnetotail from the Northern to the Southern Hemisphere at a
distance of about 17 Earth radii (Fig.

Time series plot of the magnetic field, the ion bulk velocity, the
ion number density, and the ion temperature measured by Cluster-3 spacecraft.
Data are obtained by fluxgate magnetometers

Four-point fluxgate magnetometer data are used to evaluate the energy spectra
for the fluctuations in the two intervals directly in the Fourier domain
spanning the spacecraft-frame frequencies

To obtain the spectra, the time series data are transformed into the
frequency domain at each spacecraft using the fast Fourier transform, and the
set of the frequency spectra is then projected onto the three-dimensional
wavevector domain using the wave telescope or minimum variance operator
(which is equivalent to the maximum likelihood method assuming a Gaussian
likelihood function)

It is worth mentioning that the projection method used here only assumes that the fluctuating field represents a set of plane waves and isotropic noise, and the spectra are determined by minimizing the isotropic noise. Therefore, the spectra are obtained without being biased from Taylor's frozen-in flow hypothesis, specific wave modes, or wavevector anisotropies.

The four-dimensional spectra are displayed for the fluctuations in
intervals 1 (Fig.

Four-dimensional magnetic energy spectrum on Interval 1 projected (by combining the frequency integration and averaging over the wavevector components) onto three planes spanning the wavevector components and the spacecraft-frame frequencies (top three panels) and three planes spanning the wavevector components. The frequencies and the wavevector components are normalized to the proton gyrofrequency and the proton inertial length, respectively.

Projection of the energy spectrum on Interval 2 (format of panels the same as in Fig.

Distribution of the wave frequencies in the plasma rest frame and
the wavevector magnitudes for the local peaks of the four-dimensional energy
spectra for different groups of wavevector angles from the mean magnetic
field. Left panel, top to bottom: dispersion relations from the linear Vlasov
theory are shown in gray for the whistler, kinetic slow, and the ion
cyclotron modes for ion beta

The wave modes are studied by comparing the diagram of dispersion relation
between the measurement and the linear Vlasov theory. The measured dispersion
diagrams are obtained by selecting the local peaks in the energy spectra and
plotting the frequencies in the rest frame of the plasma (co-moving with the
bulk flow),

The observationally determined dispersion diagram shows a rather poor
agreement with the theoretical modes in the oblique propagation directions
(up to

The measurement of the four-dimensional energy spectra and the dispersion diagram provides new insight into the waves in reconnecting magnetic fields on spatial scales smaller than the ion inertial length. The energy spectra are anisotropic and axially asymmetric in the wavevector domain, showing an extension in the perpendicular direction to the mean magnetic field. The dispersion diagram shows an indication of the ion Bernstein waves though the measured frequencies are somewhat scattered or deviate from the theoretical ones.

In our measurements, the ions can be in resonance with the Bernstein waves at the second harmonic and above because the ratio of the gyroradius to the inertial length is about 0.5. The wave is strongly damped at the fundamental mode, and this fact agrees with the dispersion diagram obtained from the Cluster data. Judging from the existence of an ion burst flow during the analyzed intervals, we conclude that the Bernstein waves likely exist in an ion outflow region.

Ion Bernstein waves may cause a bifurcation of the current sheet in the
reconnection outflow region through the Landau–Ginzburg-type transition

Y. Narita thanks R. A. Treumann for helpful discussions and suggestions
during the preparation of the manuscript. The work by K.-H. Glassmeier is
financially supported by the German Bundesministerium für Wirtschaft und
Energie and the Deutsches Zentrum für Luft- und Raumfahrt under contract 50
OC 1402. The work by U. Motschmann and H. Comişel is supported by
Collaborative Research Center 963,