Physics of Space Storms pp 141-162 | Cite as

# Vlasov Theory

Chapter

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## Abstract

The cold plasma approximation of Chap. 4 was based on the assumption that the phase velocities of the waves are much larger than the thermal velocities of the particle populations. This is essentially the same as approximating the particle distribution functions by delta functions, although taking the limit may be tricky and not necessarily mathematically rigorous. The approximation is evidently not valid at resonances and many aspects of wave–particle interactions are lost. In this chapter we introduce the thermal (or kinetic) effects starting from the Vlasov equation

$${\frac{{\partial f_{\alpha}}}{\partial t}}+{{\bf v}}\cdot{\frac{\partial f_{\alpha}}{\partial {\bf r}}}+{\frac{q_{\alpha}}{m_{\alpha}}({\bf E}+{\bf v}\times{\bf B})}\cdot{\frac{\partial f_{\alpha}}{\partial {\bf v}}}=0.$$

(5.1)

## Keywords

Dispersion Equation Vlasov Equation Langmuir Wave Whistler Mode Ballistic Term
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© Springer-Verlag Berlin Heidelberg 2011