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We present FARR (Finite-difference time-domain ARRay), an open source, high-performance, finite-difference time-domain (FDTD) code. FARR is specifically designed for modeling radio wave propagation in collisional, magnetized plasmas like those found in the Earth’s ionosphere. The FDTD method directly solves Maxwell’s equations and captures all features of electromagnetic propagation, including the effects of polarization and finite-bandwidth wave packets. By solving for all vector field quantities, the code can work in regimes where geometric optics is not applicable. FARR is able to model the complex interaction of electromagnetic waves with multi-scale ionospheric irregularities, capturing the effects of scintillation caused by both refractive and diffractive processes. In this paper, we provide a thorough description of the design and features of FARR. We also highlight specific use cases for future work, including coupling to external models for ionospheric densities, quantifying HF/VHF scintillation, and simulating radar backscatter. The code is validated by comparing the simulated wave amplitudes in a slowly changing, magnetized plasma to the predicted amplitudes using the WKB approximation. This test shows good agreement between FARR and the cold plasma dispersion relations for O, X, R, and L modes, while also highlighting key differences from working in the time-domain. Finally, we conclude by comparing the propagation path of an HF pulse reflecting from the bottomside ionosphere. This path compares well to ray tracing simulations, and demonstrates the code’s ability to address realistic ionospheric propagation problems. 

Abstract Incoherent scatter radars (ISR) estimate the electron and ion temperatures in the ionosphere by fitting measured spectra of ion‐acoustic waves to forward models. For radars looking at aspect angles within 5° off perpendicular to the Earth's magnetic field, the magnetic field constrains electron movement and Coulomb collisions add an additional source of damping that narrows the spectra. Fitting the collisionally narrowed spectra to collisionless forward models leads to errors or underestimates of the plasma temperatures. This paper presents the first fully kinetic particle‐in‐cell (PIC) simulations of ISR spectra with collisional damping by velocity‐dependent electron‐electron and electron‐ion collisions. For aspect angles between 0.5° and 2° off perpendicular, the damping effects of electron‐ion and electron‐electron collisions in the PIC simulations are the same and the resulting spectra are narrower than what current theories and models predict. For aspect angles larger than 3° away from perpendicular, the simulations with electron‐ion collisions match collisionless ISR theory well, but spectra with electron‐electron collisions are narrower than theory predicts at aspect angles as large as 5° away from perpendicular. At aspect angles less than 5° the PIC simulations produce narrower spectra than previous simulations using single‐particle displacement statistics that include both electron‐ion and electron‐electron collisions. The narrowing of spectra by electron‐electron collisions in the PIC code between 3° and 5° away from perpendicular is currently neglected when fitting measured spectra from the Jicamarca and Millstone Hill radars, leading to underestimates of electron temperatures by as much as 25% at small aspect angles. 

Abstract 150 kilometer echoes are strong, coherent echoes observed by equatorial radars looking close to perpendicular to Earth's magnetic field. Observations over a day show a distinct necklace pattern with echoes descending from 170 km at sunrise to 130 km at noon, before rising again and disappearing overnight. This paper shows that the upper hybrid instability will convert photoelectron energy into plasma wave energy through inverse Landau damping. Using parameters from a WACCM‐X simulation, the upper hybrid wave growth rates over a day show a nearly identical necklace pattern, with bands of positive growth rate following contours of the plasma frequency. Small gaps in altitude with no echoes are explained by thermal electrons Landau damping the instability where the upper hybrid frequency is a multiple of the gyrofrequency. This theory provides a mechanism that likely plays a crucial role in solving a long‐standing mystery on the origin of 150‐km echoes. 

Abstract The Millstone Hill incoherent scatter (IS) radar is used to measure spectra close to perpendicular to the Earth's magnetic field, and the data are fit to three different forward models to estimate ionospheric temperatures. IS spectra measured close to perpendicular to the magnetic field are heavily influenced by Coulomb collisions, and the temperature estimates are sensitive to the collision operator used in the forward model. The standard theoretical model for IS radar spectra treats Coulomb collisions as a velocity independent Brownian motion process. This gives estimates ofT_e/T_i < 1 when fitting the measured spectra for aspect angles up to 3.6°, which is a physically unrealistic result. The numerical forward model from Milla and Kudeki (2011,https://doi.org/10.1109/TGRS.2010.2057253) incorporates single‐particle simulations of velocity‐dependent Coulomb collisions into a linear framework, and when applied to the Millstone data, it predicts the sameT_e/T_iratios as the Brownian theory. The new approach is a nonlinear particle‐in‐cell (PIC) code that includes velocity‐dependent Coulomb collisions which produce significantly more collisional and nonlinear Landau damping of the measured ion‐acoustic wave than the other forward models. When applied to the radar data, the increased damping in the PIC simulations will result in more physically realistic estimates ofT_e/T_i. This new approach has the greatest impact for the largest measured ionospheric densities and the lowest radar frequencies. The new approach should enable IS radars to obtain accurate measurements of plasma temperatures at times and locations where they currently cannot.