MMS data analysis

The MMS electron data used in this study and shown in Fig.  5 are drawn from the level 2 science moments provided by the instrument team³¹ and held at the mission science data center (see the “Acknowledgements” section). The temperature moments are the direct integration of the full 3D velocity space sampled at 32 energies, 16 elevation and 32 polar angles every 30ms in burst mode for this interval. The moment algorithms remove or correct for the influence of the spacecraft potential, and the presence of photoelectrons and internally produced secondary electrons.

The electric field data used to diagnose plasma waves in this study (Fig. 5) are drawn from the level 2 science data provided by the FIELDS instrument team³⁶ and held at the mission science data center (see the “Acknowledgements” section). To derive the power spectra and cross-spectral analysis results shown in Fig. 5 the 8,192 sample per second 3-axis time series electric field data are first rotated into magnetic field-aligned coordinates. The DC-coupled magnetic field data used to perform this rotation are the four-spacecraft barycentric magnetic field data, sampled at 16 Hz. A windowed Fast Fourier Transform (FFT) is then applied to the magnetic field-aligned electric field data. The FFT used has 2048 points, 50% overlap between windows, and a Hanning window applied.

From the real and imaginary FFT outputs, A 3×3 cross-spectral matrix is determined for each electric field spectrogram time and frequency bin shown in Fig. 5. The power spectral matrix shown is the sum of the squares of the cross-spectral matrix diagonal elements. The cross-spectral matrix data are smoothed in time with a Hanning-based smoothing profile for the remaining cross-spectral analysis.

Wave properties are defined from the cross-spectral matrices using standard definitions. The wave normal angle is defined as the angle between the magnetic field direction and the minimum variance direction of the 3D wave field for a given time and frequency bin³⁷. Ellipticity is defined as the ratio of the smallest to largest eigen value of the cross-spectra matrix defined in the 2D plane transverse to the magnetic field direction. The sign indicates the handedness (+ for right-handed, − for left-handed) of the rotation of the electric field vector in this plane³⁷. Degree of polarization (Fig. 5e) is defined as in³⁸. When degree of polarization is near one, the wave polarization is close to circular. When it is near zero, the wave polarization is close to linear.