Numerical Simulation Of Line-Tied Instability On Magnetic Confinement Plasma In Symmetrical Cylindrical Geometry

There are many macro instabilities in astrophysical and laboratory plasmas, some of . them can release enormous energy within very short time and change the plasma state. Kink instability is a familiar instability which driven by current in all macro instabilities and can be studied with MHD model.Kink instability is used to explain the energy release in the solar flare. In laboratory plasmas, kink instability is a crucial component for sawtooth oscillations in tokamak.Kink instability is strongly depends on the system geometry and boundary conditions, and corresponding theory has been quite successful in predicting the behavior of toroidal plasma. It has not been thoroughly tested in cylindrical geometries for which the boundary conditions are line-tied at end plates.For further study of line-tied kink instability, two experiments have been made on cylindrical devices. The first investigates the evolution of line-tied kink modes at both ends of the machine at the University of Wisconsin, and the second is on the reconnection scaling experiment device line-tied only at one end plate at the Los Alamos National Laboratory.Evstatiev et al introduced a new method for analyzing line-tied kink modes in cylindrical geometry and simulated the experiments mentioned above. However, the effect of plasma pressure for kink instability is omitted in the paper of Evstatiev. In fact, Plasma pressure cannot be neglected any more if it is large enough, because many kinds of instabilities, such as ballooning mode and interchange mode are driven by its gradient.In this paper, the effect of plasma pressure for line-tied kink instability in cylindrical geometry is studied with the method introduced by evstatiev. Growth rate and complex spectra are analyzed for zero plasma pressure, uniform plasma pressure and heterogeneous plasma pressure. Effects of viscosity and resistivity in unideal MHD are analyzed for kink instability, tearing mode is studied too. The results are as follows:1. Curves of growth rate are nested and intersect the common point of k = -1,γ= 0 when plasma pressure is zero, which is a specific case for the magnetic components in this paper. That is to say, the point of k = -1,γ= 0 is a singularity, so eigenfunction must be zero for the balance of equation. 2. Growth rate decreases for the uniform plasma pressure P₀ = 0.001 and P₀ = 0.01. Scale of axial wave number k is not change for different uniform plasma pressure. Corresponding growth rate curves are intersecte the common point, which means the scale of instability is not extend when plasma pressure is uniform.3. Compared to zero plasma pressure, growth rate curves for heterogeneous plasma pressure are not intersect common point. The scale of axial wave number k and maximum value of growth rate are extend, the difference is distinct even plasma beta is 2% , which is caused by the gradient of heterogeneous plasma pressure.4. Maximum value of growth rate and 2-D eigenfunction decrease for the viscosity. It is because the viscosity can reduce velocity shear so MHD system is more stable, the effect is distinct for short wave mode. The stable effect for growth rate decreases with the decreases of viscosity and can be neglected when viscosity less than 10"5. Effect of resistivity for MHD system is that it will cause tearing mode, resistivity can be neglected if it is less than 10^-6.5. Enrgy principle is used to analyze MHD instability in the last chapter. Paths of steepest descent are used for the variational principle of potential energy in plasma and vacuum region, instability in D-shaped Tokamak is analyzed.