Transport Equation
AURORA solves the time-dependent, multi-stream electron transport equation in the Earth's ionosphere. The equation describes how precipitating auroral electrons propagate along the magnetic field, scatter off neutral atoms and molecules, lose energy through inelastic collisions, and produce secondary electrons through ionization.
The equation
\[\frac{\partial I_e}{\partial t} = -\mu \, v \frac{\partial I_e}{\partial z} - A \cdot I_e + B \cdot I_e + Q\]
| Term | Physical meaning |
|---|---|
| $-\mu v \, \partial I_e / \partial z$ | Field-aligned streaming |
| $-A \cdot I_e$ | Losses (collisions with neutrals + thermal electrons) |
| $+B \cdot I_e$ | Pitch-angle scattering (elastic + inelastic) |
| $+Q$ | Sources from energy cascading (ionization secondaries) |
The independent variables are altitude $z$, pitch-angle cosine $\mu$, time $t$, and energy $E$. The loss and source terms couple the solution across the energy grid because higher-energy electrons can degrade and produce lower-energy secondaries.
For how this equation is discretized and solved in AURORA, see Internals. The numerical implementation also includes a spatial diffusion term arising from the finite pitch-angle bin width, described in the Diffusion coefficient D section.
References
- Gustavsson, B. (2022). Time-Dependent Electron Transport I: Modelling of Supra-Thermal Electron Bursts Modulated at 5–10 Hz With Implications for Flickering Aurora. Journal of Geophysical Research: Space Physics, 127(6), e2019JA027608.
- Chapter 3 of Gavazzi, E. (2022). The effects of time-variation of electron fluxes from the auroral ionosphere on M-I coupling [Master thesis, UiT Norges arktiske universitet].