Observations of discrete harmonics emerging from equatorial noise

A number of modes of oscillations of particles and fields can exist in space plasmas. Since the early 1970s, space missions have observed noise-like plasma waves near the geomagnetic equator known as ‘equatorial noise'. Several theories were suggested, but clear observational evidence supported by realistic modelling has not been provided. Here we report on observations by the Cluster mission that clearly show the highly structured and periodic pattern of these waves. Very narrow-banded emissions at frequencies corresponding to exact multiples of the proton gyrofrequency (frequency of gyration around the field line) from the 17th up to the 30th harmonic are observed, indicating that these waves are generated by the proton distributions. Simultaneously with these coherent periodic structures in waves, the Cluster spacecraft observes ‘ring' distributions of protons in velocity space that provide the free energy for the waves. Calculated wave growth based on ion distributions shows a very similar pattern to the observations.

Distribution of phase space density for quasi-perpendicular ions. The Alfvén speed (the characteristic speed at which low-frequency waves propagate within a plasma), is indicated by the blue line. The peaks in phase space density near the Alfvén speed were suggested to be responsible for the generation of magnetosonic waves. Such peaked distributions are usually called 'ring' distributions and contain free energy for the excitation of waves. The location of the ring is indicated by the white line. c) A line plot of the ring distribution for the phase space density of protons gyrating near the equatorial plane (particles bouncing very near the equator). Y-axis is the density in phase space, and X-axis is the velocity of particles.  Figure 1 shows that, at harmonic frequencies of the proton gyrofrequency, there is a very high coherence between the two signals measured on satellites C3 and C4 indicating that the same wave structure is observed from two vantage points. Coherence analysis is a statistical method to examine the relation between the two signals. It is defined as C xy (f) = |G xy (f)| 2 /G xx (f)G yy (f), where G xy is the cross spectral density, and G xx and G yy are the auto spectral densities, resulting from the frequency decomposition of the input signals using methods such as the Fourier or Wavelet transform. A value of zero would indicate that the two signals are not related, while a value of one indicates that the signals are related by an idealized linear transfer function.

Supplementary Note 2
The first study that observed bands and structures within the magnetosonic noise was the study based on IMP6 and Hawkeye 1 data 1 . This pioneering study was the first to question the application of term noise to these emissions, and the study showed bands in the data. However, these measurements were not taken in the source region, the lines were not very clear nor did they match the gyrofrequency, and the ring was assumed to be a delta function. The study 1 clearly stated all the above mentioned shortcomings, and in conclusion, this study described three potential explanations for the observed bands. For detailed technical difficulties associated with the analysis of the data in the study 1 , please see study 2 . Another study 2 used GEOS 1 and 2 spacecraft observations. However, the instrument was only capable of observing waves up to 10 Hz, and most of the presented observations were not made exactly in the source region. Using an idealized model of a delta function ring-type distribution, the authors calculated the wave growth rates. The calculated growth rates showed clear periodic structures maximized around the 10 th harmonic, which is higher than that observed by the GEOS. The wave growth calculations were calculated with an assumption of wave vector orthogonal to the magnetic field (k ǁ =0). k denotes a wave vector, and the subscript symbol, ǁ, denotes the component parallel to the magnetic field.
The authors discussed a possibility that the instability is not produced by the ring but instead was produced by the loss cone distribution.
In the current study, we did not assume k ǁ =0 and instead we used the wave normal angle In summary, there are four major differences between the previous studies and the current study that allowed us to definitively resolve this long-standing question of the origin of these waves: 1) We did not use a delta function distribution. Instead we use a more realistic Gaussian ring distribution to represent the observed distribution. We did not assume k ǁ =0, 2) compared with GEOS observations, Cluster observations covered the entire frequency band, which is required to provide a definite answer to the generation mechanism.
3) The emission is in the source region because the harmonic spacing is equal to the local proton gyro frequency and the fact that changes in the emission frequency mirror the changes in the gyrofrequency. 4) Another highlight of the paper is the comparison of the growth rate based on the observed ion distribution with the observed waves over the entire frequency range and with high-frequency resolution.
A very recent study 3 searched for the harmonics of the gyrofrequency using Polar data.
They analyzed the 2 kHz mode receiver data, which has a maximum resolution of 2.2 Hz. They also looked at the low-frequency wave form receiver data (0 to 25 Hz data with resolution of 0.4 Hz) and could not find the harmonic structure. One possibility is that Polar was not in the source region, and the observed waves have propagated from their source region to the point of observation, which blurs the harmonic structures.
A similar explanation has been also proposed by one study 4 which showed multipoint measurements recorded by the Cluster spacecraft mission. They observed spectral structures, which did not match the harmonics of the local proton cyclotron frequency. Analysis of polarization properties of these waves implied that their group velocity has a radial component.

Supplementary Note 3
The ESA Cluster mission 5 consists of four identically instrumented spacecraft. During its lifetime, the inter-satellite separation has been varied from less than a few hundred km to over 20,000 km in order to explore the processes occurring within the magnetosphere at different spatial scales.

Supplementary Note 5
Supplementary Figure 5 shows the full distribution of the ion phase space density. The considered ring is the dominant feature near the peak Alfven speed. Other areas of high phase space density and other peaks in phase space density can be seen in Supplementary Figure 6. In the main part of the manuscript, we focused on the ring closest to the Alfven speed and high pitch angles, which are dominant for the excitation of the waves. Only ring distributions close to the Alfven speed can excite waves. In the main manuscript, we also focus on the particles close to 90° pitch angle, as there are more particles at high pitch angles than at small pitch angles.
Supplementary Figure 6 shows that the distributions were taken at 18:48:30.687. At this time, the background, due to penetrating particles from the outer radiation belt, is higher, but the ring distribution is still visible. At 18:54:33, the background was already lower, which explains the choice of the time 18:54:33 for the analysis of particle data.