Eigenfunctions of the magnetospheric radialdiffusion operator
Abstract
The magnetospheric radialdiffusion operator Λ ≡  L^{2}(∂/∂L)[(D_{LL}/L^{2})(∂/∂L)] for radiationbelt and ringcurrent particles has eigenfunctions g_{n}(L), and these can be expressed in closed form (as can the corresponding eigenvalues λ_{n}) if the diffusion coefficient D_{LL} is exactly proportional to L^{β} for some fixed β > 3. More specifically, the eigenfunctions are expressible in terms of a linear combination of ordinary Bessel functions of the first and second kinds, denoted J_{v}(θ_{n}) and Y_{v}(θ_{n}), respectively. The order v of the Bessel functions is given by v = (β  3)(β  2), and the argument θ_{n} is proportional to L^{(2β)/2}. The value of the proportionality constant L^{(β2)/2}θ_{n} and the relative weights of J_{v}(θ_{n}) and Y_{v}(θ_{n}) in the linear combination are to be chosen so as to make g_{n}(L_{0}) = g_{n}(L_{1}) = 0, where L_{0} and L_{1} are the labels of the innermost and outermost drift shells, respectively. Thus, the solution bar f(L, t) of the radialdiffusion equation ∂bar f/∂t =  Λbar f for the driftaveraged phasespace density bar f(L, t) is expressible as the sum of a quasistatic solution bar f_{∞}(L, t) = 0 and a timedependent superposition of the eigenfunctions of the radialdiffusion operator Λ. The quasistatic solution instantaneously satisfies the equation Λbar f_{∞}(L, t) = 0 and the boundary conditions that the magnetosphere imposes (at L = L_{0} and at L = L_{1}) on bar f(L, t) itself. Eigenfunctions and eigenvalues are obtained here for β = 10 with L_{1} = 8L_{0} = 9 and are employed in several idealized but illustrative applications of the eigenfunction method to situations in which the value of bar f(L_{1}, t) is made to vary with time while the value of bar f(L_{0}, t) is made to vanish.
 Publication:

Physica Scripta
 Pub Date:
 April 1988
 DOI:
 10.1088/00318949/37/4/023
 Bibcode:
 1988PhyS...37..632S
 Keywords:

 Atmospheric Diffusion;
 Earth Magnetosphere;
 Eigenvectors;
 Geomagnetism;
 Operators (Mathematics);
 Bessel Functions;
 Plasma Drift;
 Radiation Belts;
 Ring Currents;
 Geophysics