Trapped-particle instability and flute instability in Tandem mirror; scale separation closure in Alfven-wave turbulence

Two different topics are discussed: instability in Tandem mirrors and closure schemes in turbulence theories. In the tandem mirror study, an appropriate quadratic functional form is constructed taking into account equilibrium rotational effects, electron temperature gradients, ion Landau damping, and electron-collisional effects. The collisional effects, which are described by the Fokker-Planck equation, are evaluated by using the solution for the particle and energy lifetime of an electron in an ambipolar trap and a magnetic trap. With the quadratic form, a dispersion relation that covers both the trapped-particle modes and the flute modes is derived and discussed. The numerical studies utilizing parameters suitable to the TMX-U mirror (Livermore) are conducted. The second topic is to study the renormalized turbulence theory. After analyzing the process of the wave-wave incoherent interaction, it is argued that the incoherent source term for most turbulence systems lacks fine structure in the k-w space. Based on the concept of scale separation between the coherent response function and the incoherent source, a closure scheme for the renormalized turbulence equation is proposed. A model dealing with the shear-Alfven wave turbulence is numerically solved