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This article was submitted to Space Physics, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In our work, for the first time, we conducted a comprehensive multispacecraft wave study of flapping current sheet oscillations in Earth’s magnetotail. Measurements taken from the Magnetospheric Multiscale (MMS) mission were analyzed for two flapping events with different morphologies of oscillation behavior: stationary-like and kink-like type. A comparison of the results calculated by the methods of phase difference, wave surveyor, and Multipoint Signal Resonator technique was carried out. For the first time, using observations, it was found that the energy distribution of wavy magnetic field contains complex multi-branch dispersion dependencies on

Flapping motions are wave-like oscillations of the current sheet (CS), and they often accompany explosive activity released in the geomagnetic tail (

Flapping oscillations began to be studied more than half a century ago (

Currently, the question of what is the contribution of flapping oscillations to the energy budget of the geomagnetic tail and what type of instability is responsible for flapping motions in one or another case remains open. Among the possible triggers are considered external—solar wind, interplanetary magnetic field and internal—ion–ion kink instability (

Dispersion analysis of such waves is absolutely important, which will make it possible to consider the trigger instability present. The main dispersive features of flapping oscillations have already been considered in the works of

Turbulent or wave-like fluctuations of plasma parameters during measurements onboard a SC have mixed manifestations of spatial and temporal variations due to the dynamics of the environment itself. Therefore, the distribution of such plasma disturbances in terms of frequency and spatial range plays the most important role in identifying their spectral properties, including spectral laws, dispersion branches, and turbulence features (

First, we describe the phase difference method (

For two scalar time series

Phase

Having found

After that, the power spectrum is calculated as follows:

Alternatively,

A fast method for finding dispersion patterns is the wave surveyor technique (

Here,

Raising to the power of -1 means taking the inverse matrix or pseudo-inverse (for cases with S = 2,3).

Here,

The k-filtering method has become widely used in the geophysical field studies (

The specificity of other methods based on the search for eigenvalues consists of the formation of additional matrices, which contain a vector of columns of eigenvectors and diagonal elements formed from eigenvalues (

The power spectrum according to the MSR method is calculated as follows (

The trace of the power matrix

Here,

Later in the paper, we will consider all spectra and dispersion patterns depending on the usual frequency in Hz and not on the angular frequency:

Analysis the CS flapping oscillations in Earth’s magnetotail was performed using measurements from the four-spacecraft Magnetospheric Multiscale (MMS) mission. Magnetic field data have been given from fluxgate magnetometer (FGM) instruments (

Position of MMS spacecraft during two flapping events on 2020/08/26 (red triangle indicates the location of the first event, and green indicates the location of the second event) in the XZ and XY GSM planes. The color gradation indicates the magnetic field magnitude (carried out with the SpacePy package).

Flapping events 2020/08/26 (highlighted in gray) as observed by MMS.

For two flapping events, an analysis of the minimum variation (MVA) was applied to the magnetic field for the selected intervals, where a change in the sign of the

The first event shows the alternation of the κ parameter (+1, −1, +1...), which means a stationary type of flapping, that is, an up–down oscillation. For the second event

We calculated the phase difference spectra for each component of vector

P (k,f) spectrum of the magnetic field vector for

This makes it possible to correctly compare the results of this method with two others: the wave survey and the MSR method. The energy distribution was transferred to the plasma moving system using the Doppler correction (

The binning of frequencies is performed in a linear way, since it allows us to provide a Doppler shift for a specific value of

Let’s analyze the dispersion patterns constructed by the wave surveyor method (

Spectra obtained using the MSR method are constructed for a separate set of three frequencies. The latter corresponds to the maxima of the wavelet power for the corresponding intervals. A set with the dominant frequencies of 0.02, 0.05, and 0.5 Hz was selected for the first flapping and 0.02, 0.046, and 0.5 Hz for the second. For a simplified visualization of the obtained 3D spectra

MSR 3D power spectra of

We present, for the first time, a low-frequency multispacecraft wave analysis for two flapping oscillation events with high-speed plasma flows. It was found from magnetic polarity that two events of flapping oscillations differ in their type: the first event is characterized by up-and-down movements, and the second event demonstrates kink-type oscillations.

We found that the energy distribution of fluctuations of the magnetic field vector contains previously undetected complex multi-branch dispersion dependences on

The used methods complement each other, and their differences made it possible to assess the presence of non-linear wave packets when considering kink flapping and the asymmetry of the CS profile in the azimuthal direction (

Publicly available datasets were analyzed in this study. These data can be found at:

LK proposed to conduct a comparative analysis of the results of specific multispacecraft methods. EK and RA made valuable suggestions about data visualization and text readability.

This work was supported by grant No. 97742 of the Volkswagen Foundation (VW-Stiftung) and the Royal Society International Exchanges Scheme 2021 IES\R1\211177, BF/30-2021. The work of Elena Kronberg was supported by the German Research Foundation (DFG) under number KR 4375/2-1 within SPP “Dynamic Earth.”

The data used in this work are publicly available at the MMS Science Data Center website

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.