Powerful ground-based very low frequency (VLF) transmitters, which operate at tens of kHz with power ranging from several 100 s to 1,000 kW (Volakis, 2007;Watt, 1967), have been utilized for maintaining long distance communication with submarines for decades dating back to the era before World War II (Sterling, 2007). The long-distance communication is realized by radio wave propagation in the ionosphere-Earth waveguide (Wait, 1957). A fraction of the radio wave power can leak through the ionosphere into the magnetosphere (Maeda & Oya, 1963), where VLF signals propagate in the whistler-mode (Helliwell, 1965;Leiphart et al., 1962). There are two modes of propagation for VLF transmitter signals in the inner magnetosphere: ducted and nonducted. Ducted waves propagate inside a density enhancement or depletion known as a duct, with wave energy and wave normal direction confined nearly along the ambient field lines (

We propose a new method to construct a controllable and quantifiable loss cone distribution. Then, we derive a linear growth rate formula of the electrostatic mode for a realistic and arbitrary distribution function. Such formula is used to perform a parametric study of instability analysis of the electron cyclotron harmonic (ECH) wave in a plasma consisting of electron loss cone distribution and isotropic cold electron distribution. We find (1) the peak linear growth rate and the corresponding wave frequency increase with the loss cone size. The wave frequency of peak growth rate is about 1.5 and 2.5 times electron cyclotron frequency when loss cone is about 4-6 ∘ wide. The wave normal angle corresponding to the growth rate peak decreases with the loss cone size. (2) Increasing hot electron temperature anisotropy decreases the growth rate but hardly changes the wave frequency and wave normal angle corresponding to the growth rate peak. (3) Increasing hot electron parallel temperature tends to increase both the peak growth rate and wave normal angle but only change the corresponding wave frequency slightly. (4) The peak growth rate and its corresponding wave frequency increase with pe ∕|Ω e | for wavebands above or passing upper hybrid resonance frequency UHR and almost remain unchanged for wavebands below UHR . (5) Increasing cold electron temperature tends to decrease wave frequency and increase wave normal angle and peak growth rate for wavebands below UHR . The impact of our work on ECH wave generation and the significance of ECH waves on diffuse aurora are also discussed.

In this study using Van Allen Probe wave observations we investigate the statistical properties of exohiss waves, which are structureless whistler mode waves observed outside the plasmapause. The exohiss waves are identified based on the cold electron number density, frequency distribution, ellipticity, and wave normal angle. The statistical analysis on exohiss wave properties shows that exohiss waves prefer to occur over 3 < L < 6 from dawnside to noon and duskside during geomagnetic quiet times, and their wave power is larger at smaller values of L shell. The frequency of exohiss mainly ranges from 200 to 300 Hz and is almost independent of L shell. The median values of ellipticity and electromagnetic planarity of exohiss waves are 0.91 and 0.54, respectively. Furthermore, the equatorward Poynting flux is comparable to the poleward Poynting flux at the equator and it becomes dominant at the absolute value of the magnetic latitude | | ∼20 •. Our observation results reveal the statistical features of exohiss waves for the first time and support the exohiss formation mechanism that exohiss originates from plasmaspheric hiss leakage. The results demonstrate exohiss as an important energy dissipation route for plasmaspheric hiss and significantly improve the understanding of plasmaspheric hiss evolution in the radiation belt region.

The very‐low frequency (VLF) and low frequency (LF) waves from ground transmitters propagate in the ionospheric waveguide, and a portion of their power leaks to the Earth's inner radiation belt and slot region where it can cause electron precipitation loss. Using Van Allen Probes observations, we perform a survey of the VLF and LF transmitter waves at frequencies from 14 to 200 kHz. The statistical electric and magnetic wave amplitudes and frequency spectra are obtained at 1 < L < 3. Based on a recent study on the propagation of VLF transmitter waves, we divide the total wave power into ducted and unducted portions, and model the wave normal angle of unducted waves with dependences on L shell, magnetic latitude, and wave frequency. At lower frequencies, the unducted waves are launched along the vertical direction and the wave normal angle increases during the propagation until reaching the Gendrin angle; at higher frequencies, the normal angle of unducted waves follows the variation of Gendrin angle. We calculate the bounce‐averaged pitch angle and momentum diffusion coefficients of electrons due to ducted and unducted VLF and LF waves. Unducted and ducted waves cause efficient pitch angle scattering at L = 1.5 and 2.5, respectively. Although the wave power from ground transmitters at frequencies higher than 30 kHz is low, these waves can cause the pitch angle scattering of lower energy (2–200 keV at L = 1.5) electrons, which cannot resonate with the VLF transmitter waves at frequencies below 30 kHz, lightning generated whistlers, or plasmaspheric hiss.

The plasmasphere is a vast torus-shaped region in the inner magnetosphere filled with dense (  2 6 3 10 10 #/ cm ) and cold (less than 10 eV) ions and electrons. Controlled by the co-rotation and convection electric fields, plasmasphere erosion and refilling occur regularly due to the increase and decrease in geomagnetic activities, respectively (Carpenter & Anderson, 1992;Darrouzet et al., 2013;Grebowsky, 1970). The outer boundary of the plasmasphere, called plasmapause, is a sharp plasma density boundary that separates the closed and open drift paths for cold plasma. Due to the dramatic variation of the plasma properties across the plasmapause (Liu, Chen, & Xia, 2020), the plasmapause plays a crucial role in determining energetic and relativistic particle distributions (Lorentzen et al., 2001) and controlling the growth and propagation of various electrostatic and electromagnetic emissions in the inner magnetosphere (Liu et al., 2015, and reference therein). For example, as shown by satellite observations and numerical simulations, electron cyclotron harmonic (ECH) waves (e.g.,