The analytical solution of the transient radial diffusion equation with a nonuniform loss term

Abstract

Many works have been done during the past 40 years to perform the analytical solution of the radial diffusion equation that models the transport and loss of electrons in the magnetosphere, considering a diffusion coefficient proportional to a power law in L shell and a constant loss term. In this paper, we propose an original analytical method to address this challenge with a nonuniform loss term. The strategy is to match any L‐dependent electron losses with a piecewise constant function on M subintervals, i.e., dealing with a constant lifetime on each subinterval. Applying an eigenfunction expansion method, the eigenvalue problem becomes presently a Sturm‐Liouville problem with M interfaces. Assuming the continuity of both the distribution function and its first spatial derivatives, we are able to deal with a well‐posed problem and to find the full analytical solution. We further show an excellent agreement between both the analytical solutions and the solutions obtained directly from numerical simulations for different loss terms of various shapes and with a diffusion coefficient DLL∼L6. We also give two expressions for the required number of eigenmodes N to get an accurate snapshot of the analytical solution, highlighting that N is proportional to 1/t0, where t0 is a time of interest, and that N increases with the diffusion power. Finally, the equilibrium time, defined as the time to nearly reach the steady solution, is estimated by a closed‐form expression and discussed. Applications to Earth and also Jupiter and Saturn are discussed.

Particle Dynamics in the Earth's Radiation Belts: Review of Current Research and Open Questions

“…The 1‐D Fokker‐Planck formulation has been commonly used since the 1970s for Earth's (and other planets) radiation belts (e.g., Spjeldvik & Thorne, , ; Spjeldvik & Lyons, ; Brautigam & Albert, 2000; Shprits et al, ; Shprits, Thorne, Horne, et al, ; Tu et al, 2009; Ozeke et al, ; Li, Millan, et al, ; Ripoll, Loridan, et al, , Ripoll, Reeves, et al, ; Schiller et al, ; Loridan et al, ). There exist tractable analytical solutions of this equation according to the form of the diffusion coefficient and/or the lifetime model, for the steady problem (Haerendel, ; Hood, ; Jentsch, ; Thomsen et al, , ) and for the general (unsteady) problem (Loridan et al, ; Schulz, ; Schulz & Newman, ; Walt, ). Tridimensional full Fokker‐Planck codes only became readily available and operational in a common manner in the years 2005–2010 (e.g., Albert et al, ; Subbotin & Shprits, ; Varotsou et al, , ).…”

Observations and Fokker‐Planck Simulations of the L‐Shell, Energy, and Pitch Angle Structure of Earth's Electron Radiation Belts During Quiet Times

“…The intrinsic conversion between magnetic invariants and physical variables made in equation in order to define τ ( E , L ) at fixed ( μ , K ) is also performed with either a dipole or the T89 magnetic field models. The 1D‐RFP code we use here has been used successfully numerous times (Ripoll, Loridan et al, ; Ripoll, Reeves et al, ; Ripoll et al, ), and it reproduces analytical solutions (Loridan et al, ). A novel aspect of this work is to solve equation with respect to all K (and not only K = 0 since τ is independent of K ).…”

“…Here daily variation in the flux snapshots of Figure (and Figures and ) shows equilibrium is not reached, that is, ∂f / ∂t ≠ 0. Besides, daily variations of the diffusion coefficients and boundary conditions do not allow equilibrium to exist, or, in other words, each set of Dα 0 α 0 and boundary condition leads to a different equilibrium solution (Loridan et al, ; Ripoll, Reeves et al, ). Running the simulations to longer times (in freezing the parameters) to establish when equilibrium is reached showed that great variations exist in the outer belt until at least 25 days and that equilibrium is reached after ~55 days in the outer belt ( L > 4, E < 4 MeV).…”

On the Use of Different Magnetic Field Models for Simulating the Dynamics of the Outer Radiation Belt Electrons During the October 1990 Storm

“…Such L-shell profile can be obtained by solving for the steady state of the reduced Fokker-Planck equation with no loss term (radial diffusion only) for D LL ∼ L n+3 (e.g., Loridan et al, 2017). Such L-shell profile can be obtained by solving for the steady state of the reduced Fokker-Planck equation with no loss term (radial diffusion only) for D LL ∼ L n+3 (e.g., Loridan et al, 2017).…”

“…for which L 0 and L M are respectively the inner and outer boundaries in L-shell, and n > 0 is the power of the radial diffusion coefficient. Such L-shell profile can be obtained by solving for the steady state of the reduced Fokker-Planck equation with no loss term (radial diffusion only) for D LL ∼ L n+3 (e.g., Loridan et al, 2017). The second case is the strong diffusion (SD) limit of case 1 with n ≫ 1 so that v(L) = v SD (L) ∼ (L M ∕L) 3 .…”