Multiscale behavior of the magnetosphere: Modeling with Levy flights and fractional kinetics

Multiscale phenomena are ubiquitous in nature and arise from the presence of a broad range of interacting space and time scales. In the solar wind-magnetosphere interaction, multiscale features coexist along with the global or coherent features, and have been studied extensively using nonlinear dynamical techniques. The detailed properties of these phenomena are studied using Levy flights and fractional kinetics. In the solar wind - magnetosphere coupling, a technique to analyze the Levy-type processes is appied to the time series data of the solar wind electric field and the auroral electrojet index. The probability distribution function of the flights show similarities and differences, and provides a new insight into the origin of the multiscale behavior through the different values of the scaling indices. In a complementary approach, the fractional kinetic equations, which uses fractional derivatives to represent the complexity, provide a suitable mathematical framework for the multiscale behavior. Unlike the usual diffusion equations, the solutions of these equations yield non-convergent moments, showing its multiscale features. The origin of multiscale behavior is studied by using a fractional kinetic equation and a diffusion equation with a space dependent diffusion coefficient. Numerical solutions of these equations are used to analyze the nature of the equations by computing the moments. The fractional kinetic equations yield solutions with large moments and are non-convergent. On the other hand the solutions of the diffusion equation have convergent moments similar to those of Gaussian distributions. These results lead to the conclusion that the fractional kinetic equations are suitable models of multiscale phenomena.