Nonlinear Theory

Book chapter, 2012

We have up to now mainly studied linear and quasilinear phenomena (with the exception for Sect. 6.10.4 in Chap. 6). Although quasilinear equations, in combination with an estimate of the saturation level, can be used to derive transport coefficients, it is important to go beyond this description in order to understand its region of applicability [1–83]. In particular nonlinear cascade rules [18, 20, 25, 26, 29, 30, 55] are important for the interplay between sources and sinks in k-space and the resulting saturation level and correlation length. We will thus here consider some simple nonlinear systems for turbulence in magnetized plasmas. We will also make a kinetic derivation of the diffusion coefficient which involves the turbulent transport itself as a decorrelation mechanism [3–5, 7, 8]. As we have pointed out in Chap. 3, the parallel ion motion may often be ignored in drift and flute modes. This is possible if$$ \omega \, > > {{\hbox{k}}_{\parallel }}\,\,{{\hbox{c}}_{\rm{s}}} $$. For this case it is possible to derive a simple but still rather general nonlinear equation for the ion vorticity Ω = rot vi. We start from the fluid equation of motion for ions$$ \frac{{\partial {{{\mathbf{v}}}_i}}}{{\partial t}} + ({{{\mathbf{v}}}_i} \cdot \nabla ){{{\mathbf{v}}}_i} = \frac{e}{{{{m}_i}}}({\mathbf{E}} + {{{\mathbf{v}}}_i} \times {\mathbf{B}}) - \frac{1}{{{{m}_i}n}}\nabla {{P}_i} + {\mathbf{g}} $$