Development of a 3D Radiation Belt Model in Adiabatic Invariant Coordinates Using Stochastic Differential Equations
Abstract
Modeling the phase-space diffusion of radiation belt electrons interacting with electromagnetic plasma waves is a key step to understanding energization and transport of relativistic electrons in Earth's radiation belts. Existing models use finite-difference methods to solve a 3D Fokker-Planck equation (FPE) using the electron's equatorial pitch angle, momentum (or equivalently kinetic energy), and Roederer's L as the phase-space coordinates. Apart from their close connection with observations, equatorial pitch angle and momentum are not adiabatically invariant coordinates, and these models rely on the geometry of the Earth's magnetic field line, which is often idealized as a dipolar. The adiabatic invariants M, K and L can be chosen as phase-space coordinates, but the implementation of the corresponding boundary conditions becomes more complicated. A new mathematical approach, the stochastic differential equation (SDE) method, views the diffusion process as an ensemble of stochastic motions, and represents the solution of the FPE as the expectation of functionals of the stochastic paths. The SDE method has the advantages that it can easily deal with complicated boundaries and mixed diffusion, and is computationally more efficient than finite-difference methods when solutions are only required in limited regions of the phase-space. Based on these theories, we present here the development of a new bounce-averaged pitch-angle-energy diffusion model that works in the adiabatic invariant coordinates, and show its validity by obtaining solutions that agree well with previous methods. We also discuss the complication of boundary conditions that arises in adiabatic invariant coordinates, which had been largely overlooked in the literature previously. This new model can be easily coupled with a radial transport model, resulting in a 3D radiation belt model in adiabatic invariant coordinates. A further aim of this work is to integrate this model into an MHD-particle simulator to build a 4D (M, K, plus coordinates of the guiding field line) model that uses dynamic large-scale magnetospheric fields.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2012
- Bibcode:
- 2012AGUFMSM31C2352Z
- Keywords:
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- 2753 MAGNETOSPHERIC PHYSICS / Numerical modeling;
- 2774 MAGNETOSPHERIC PHYSICS / Radiation belts