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

• Published in: Journal of geophysical research. Space physics 2017-06, Vol.122 (6), p.5979-6006
• Main Authors: Loridan, Vivien, Ripoll, Jean‐François, Vuyst, Florian
• Format: Article
• Language: English
• Subjects: analytical solution; Computer Science; Computer simulation; Continuity (mathematics); convergence criteria; Diffusion coefficient; Exact solutions; Formulas (mathematics); Jupiter; Magnetosphere; Magnetospheres; Modeling and Simulation; nonuniform electron lifetime; Numerical simulations; Power law; radial diffusion; reduced Fokker‐Planck equation; Saturn; Sturm‐Liouville problem with interfaces; Transportation models; Well posed problems
• ISSN: 2169-9380; 2169-9402
• DOI: 10.1002/2017JA023868

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. Key Points Analytical solution found for the radial diffusion equation with a nonuniform loss term (solving a Sturm‐Liouville problem with interfaces) Verification of the method with full agreement between numerical and analytical solutions for various nonuniform loss terms Estimations of the required number of modes for convergence (usually large) and of the time to reach equilibrium