Eigenfunctions of the magnetospheric radial-diffusion operator
Abstract
The magnetospheric radial-diffusion operator Λ ≡ - L2(∂/∂L)[(DLL/L2)(∂/∂L)] for radiation-belt and ring-current particles has eigenfunctions gn(L), and these can be expressed in closed form (as can the corresponding eigenvalues λn) if the diffusion coefficient DLL is exactly proportional to Lβ 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 Jv(θn) and Yv(θn), respectively. The order v of the Bessel functions is given by v = (β - 3)(β - 2), and the argument θn is proportional to L(2-β)/2. The value of the proportionality constant L(β-2)/2θn and the relative weights of Jv(θn) and Yv(θn) in the linear combination are to be chosen so as to make gn(L0) = gn(L1) = 0, where L0 and L1 are the labels of the innermost and outermost drift shells, respectively. Thus, the solution bar f(L, t) of the radial-diffusion equation ∂bar f/∂t = - Λbar f for the drift-averaged phase-space density bar f(L, t) is expressible as the sum of a quasi-static solution bar f∞(L, t) = 0 and a time-dependent superposition of the eigenfunctions of the radial-diffusion operator Λ. The quasi-static solution instantaneously satisfies the equation Λbar f∞(L, t) = 0 and the boundary conditions that the magnetosphere imposes (at L = L0 and at L = L1) on bar f(L, t) itself. Eigenfunctions and eigenvalues are obtained here for β = 10 with L1 = 8L0 = 9 and are employed in several idealized but illustrative applications of the eigenfunction method to situations in which the value of bar f(L1, t) is made to vary with time while the value of bar f(L0, t) is made to vanish.
- Publication:
-
Physica Scripta
- Pub Date:
- April 1988
- DOI:
- 10.1088/0031-8949/37/4/023
- Bibcode:
- 1988PhyS...37..632S
- Keywords:
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- Atmospheric Diffusion;
- Earth Magnetosphere;
- Eigenvectors;
- Geomagnetism;
- Operators (Mathematics);
- Bessel Functions;
- Plasma Drift;
- Radiation Belts;
- Ring Currents;
- Geophysics