### Licentiate thesis 2001-004

# Numerical Simulation of Kinetic Effects in Ionospheric Plasma

### Bengt Eliasson

April 2001

**Abstract:**In this thesis, we study numerically the one-dimensional Vlasov equation for a plasma consisting of electrons and infinitely heavy ions. This partial differential equation describes the evolution of the distribution function of particles in the two-dimensional phase space *(x,v)*. The Vlasov equation describes, in statistical mechanics terms, the collective dynamics of particles interacting with long-range forces, but neglects the short-range "collisional" forces. A space plasma consists of electrically charged particles, and therefore the most important long-range forces acting on a plasma are the Lorentz forces created by electromagnetic fields. What makes the numerical solution of the Vlasov equation to a challenging task is firstly that the fully three-dimensional problem leads to a partial differential equation in the six-dimensional phase space, plus time, making it even hard to store a discretized solution in the computer's memory. Secondly, the Vlasov equation has a tendency of structuring in velocity space (due to free streaming terms), in which steep gradients are created and problems of calculating the *v* (velocity) derivative of the function accurately increase with time. The method used in this thesis is based on the technique of Fourier transforming the Vlasov equation in velocity space and then solving the resulting equation. We have developed a method where the small-scale information in velocity space is removed through an outgoing wave boundary condition in the Fourier transformed velocity space. The position of the boundary in the Fourier transformed variable determines the amount of small-scale information saved in velocity space. The numerical method is used to investigate a phenomenon of tunnelling of information through an ionospheric layer, discovered in experiments, and to assess the accuracy of approximate analytic formulae describing plasma wave dispersion. The numerical results are compared with theoretical predictions, and further physical experiments are proposed.

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