Nonlinear evolution of MHD instability in LHD

Direct numerical simulations of fully three-dimensional, compressible magnetohydrodynamic (MHD) equations are carried out in order to clarify nonlinear saturation mechanism of pressure-driven instabilities in the Large Helical Device (LHD). A special attention is paid to importance of the toroidal flows and compressibility, which are often discarded in the linear analysis and/or reduced MHD simulations. We refer to these effects as full-MHD effects. In recent LHD experiments under inward-shifted configurations, plasma is confined relatively well even though it passes through Mercier unstable region. Inspired by the experimental results, we aim to inspect influences of the full-MHD effects to the stability of the system. We solve the equations of continuity, momentum, pressure and magnetic field numerically by the 6th-order compact finite scheme and 4th-order Runge-Kutta-Gill scheme in our DNS code MINOS. Starting from the initial condition with β0 = 4% equilibrium which was calculated by the HINT code, the plasma kinetic energy grows exponentially. The growth is studied by decomposing the field quantities into Fourier modes in the Boozer coordinate. Having the peaked initial pressure profile, the initial equilibrium has very unstable nature. The poloidal (m) and toroidal (n) Fourier mode m/n = 2/1 is predominant over the other modes. It is shown that the toroidal flow contributes as much as the poloidal flow components to the growth of the kinetic energy. Though the poloidal components of the velocity field grows earlier than the toroidal component,the latter exceeds the former when the growth is saturated because of the nonlinearity of the MHD equations. By inspecting detailed views of the toroidal flow generation, it is shown that the toroidal flow generation contributes to reduce the impact of the instability to the confinement by distributing the energy obtained from the pressure gradient not only to the poloidal direction but also to the toroidal direction. An influence of the compressibility is explored by studying the right-hand-side terms of the budget equation of the kinetic energy. It is well known that the divergence of the displacement vector ξ contributes to suppress the linear growth, because the term is positive definite in the energy-principle-formulation of the linear analysis. It is shown that the compressible term in the kinetic energy budget equation has 50% - 80% magnitude of that of the driving term in our simulations. It indicates that the linear growth rate is suppressed to 1/2 to 1/5 of that value estimated under the incompressible assumption. Time evolutions of the flow field, pressure and magnetic field are observed closely by making use of visualization technique. Generation of the toroidal flow, compressibility will be discussed in the context of vortex generation associated with the growth of the pressure-driven unstable Fourier modes. Recent numerical results starting from a stabler equilibrium will also be reported. (author)