Analytic Expressions for Radial Diffusion

I briefly review, compare and contrast two theoretical works that have significantly influenced radial diffusion research thus far, namely, the works of Fälthammar (1965, https://doi.org/10.1029/JZ070i011p02503) and the works of Fei et al. (2006, https://doi.org/10.1029/2005JA011211). Leveraging Fälthammar's model for magnetic field disturbances, I demonstrate that Fei et al's formulas are incorrect: they underestimate radial diffusion by a factor two in the presence of magnetic field disturbances. This underestimation comes from the erroneous assumption that radial displacements driven by magnetic field disturbances are statistically independent from radial displacements driven by induced electric fields while in fact both displacements are proportional to each other. Fei et al.'s approach is similar to Fälthammar's approach in that they both analyze radial diffusion by pieces, depending on the nature of the driver. Yet, the Fokker-Planck equation requires only one radial diffusion coefficient to characterize statistically a trapped radiation belt population cross drift shell motion. Thus, it is worth questioning the practice that consists of defining the coefficient as a sum of independent contributions. In addition, both theoretical models rely on the assumption that the background magnetic field is primarily dipolar, leading to flawed estimates. To overcome these limitations and to improve radial diffusion quantification, I use a general formulation for the variation of the third adiabatic invariant (1) to describe how to compute a radial diffusion coefficient in the most general way and (2) to highlight the assumptions that need to be questioned.