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Characterize the relation between diffusion-induced streaming flux and Dμμ

Characterize and derive the relation between diffusion-induced streaming flux in pitch-angle space and the pitch-angle diffusion coefficient so that the total flux can be decomposed into streaming and diffusive components and Dμμ can be solved near the source in the stationary method.

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Background

In discussing the stationary solution method, the authors note that near the source the total flux consists of both streaming and diffusive contributions. They point out that translating the Fokker–Planck description to a streaming flux is unclear, and that without a known relationship between any diffusion-induced streaming term and Dμμ, one cannot solve for Dμμ from total flux in that regime.

Resolving this relation would improve applicability of the stationary method close to sources and clarify how to separate streaming and diffusive contributions.

References

Whether or not this can be directly translated to the streaming flux is unclear because it follows from the Fokker-Planck equation (see, e.g., vandenBerg2023) that \begin{align} \frac{\partial f}{\partial t} & = - \frac{\partial}{\partial \mu} \left[ \left\langle \frac{\Delta \mu}{\Delta t} \right\rangle f \right] + \frac{\partial2}{\partial2 \mu} \left[ \left\langle \frac{(\Delta \mu)2}{2\Delta t} \right\rangle f \right] \nonumber \ & = \frac{\partial}{\partial \mu} \left[ \left\langle \frac{(\Delta \mu)2}{2 \Delta t} \right\rangle \frac{\partial f}{\partial \mu} \right] \nonumber \ & = - \frac{\partial \dot{\cal N}{\partial \mu}, \end{align} implying a total flux of $\dot{\cal N} = - \langle (\Delta \mu)2 / 2 \Delta t \rangle$, which is again just Fick's law. Hence, if it is unknown exactly how such a ‘diffusion-induced streaming’ flux is related to the diffusion coefficient, then $\dot{\cal N} = \dot{\cal N}{\rm stream} - D{\mu\mu} ({\rm d}F / {\rm d}\mu)$ cannot be solved for the diffusion coefficient.

— An Evaluation of Different Numerical Methods to Calculate the Pitch-angle Diffusion Coefficient from Full-orbit Simulations: disentangling a rope of sand (2410.21001 - Berg et al., 28 Oct 2024) in Section 4.4 (Best Practices for M5)