Multi-satellite MMS analysis of electron holes in the Earth's magnetotail: origin, properties, velocity gap and transverse instability

We present a statistical analysis of more than 2400 electrostatic solitary waves interpreted as electron holes (EH) measured aboard at least three MMS spacecraft in the Earth's magnetotail. The velocities of EHs are estimated using the multi-spacecraft interferometry. The EH velocities in the plasma rest frame are in the range from just a few km/s, that is much smaller than ion thermal velocity V_Ti, up to 20,000 km/s, which is comparable to electron thermal velocity V_Te. We argue that fast EHs with velocities larger than about 0.1;V_Te are produced by bump-on-tail instabilities, while slow EHs with velocities below about 0.05;V_Te can be produced by warm bi-stream and, probably, Buneman type instabilities. We show that typically fast and slow EHs do not co-exist, indicating that the instabilities producing EHs of different types operate independently. We have identified a gap in the distribution of EH velocities between V_Ti and 2V_Ti, which is considered to be the evidence for self-acceleration [Zhou and Hutchinson, 2018] or ion Landau damping of EHs. Parallel spatial scales and amplitudes of EHs are typically between λ _D and 10 λ _D and between 10^-3;T_e and 0.1;T_e, respectively. We show that electrostatic potential amplitudes of EHs are below the threshold of the transverse instability and highly likely restricted by the nonlinear saturation criterion of electron streaming instabilities seeding electron hole formation: eΦ _0&l∼ m_{e}\varpi^{2d{||^2}}, where \varpi=min (γ ,1.5;ω _ce), where γ is the increment of instabilities seeding EH formation, while ω _ce is electron cyclotron frequency. The implications of the presented results are discussed.