Self-consistent numerical simulations of free boundary instabilities, such as resistive wall modes (RWMs) and external kink modes, require special attention given their relevance on the path towards magnetic fusion based nuclear reactors. In this work, a new implementation and a verification benchmark are presented for the nonlinear three-dimensional (3D) full magneto-hydrodynamics (MHD) code SpeCyl, which self-consistently combines a free flow 3D treatment for the edge plasma velocity with double resistive wall thin shell magnetic boundary conditions, capable of capturing the essential features of common setups in experimental devices. Two coaxial resistive shells surround the plasma. Vacuum conditions are assumed between and outside the resistive shells, whose relative positions and time constants can be varied on purpose.
The implementation is extensively verified against Shafranov's analytic theory, valid for a circular cross section large aspect ratio tokamak and assuming a flat current profile. The correctness of newly implemented Boundary Conditions (BCs) is confirmed by the good agreement obtained for the linear growth rates of (m=2,n=1) RWM and external kink, over a wide range of wall positions, resistive shell time constants, and for different initial reference equilibria. The (m=2,n=1) RWM growth rates are also benchmarked against previous different RWMs modelling and successful validation studies for RFX-mod tokamak plasmas. The new BCs make SpeCyl a flexible and robust tool to investigate free boundary MHD modes in large aspect ratio magnetic confinement configurations, such as those achieved with the RFX-mod machine and expected in RFX-mod2.
In this work, a quasi-linear model for plasma flow response to the resonant magnetic perturbation (RMP) in a tokamak has been rigorously developed in the resistive-inertial (RI) and viscous-resistive (VR) regimes purely from the two-field reduced MHD model. Models for plasma response to RMP are commonly composed of equations for the resonant magnetic field response (i.e. the magnetic island) and the torque balance of plasma flow. However, in previous plasma response models, the magnetic island and the torque balance equations are often derived separately from reduced MHD and full MHD equations…
We present a scalable block preconditioning strategy for the trace system coming from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block (the upper block), for which a system algebraic multigrid (AMG) is used for the approx…
Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresolution analysis is applied which guarantees the precision of the adaptive computations, while reducing CPU time and memory requirements. Implementation issues of the open source code CARMEN-MHD are discussed. To illustrate its precision and efficiency…
An analytic scaling relation for the maximum tokamak elongation against n=0 MHD resistive wall modes
为了增加等离子体压力和能量约束以最大化聚变功率输出，高度伸长的等离子体是期望的。然而，存在对最大可实现伸长的限制，其由$n=0$ MHD模式驱动的垂直不稳定性设定。通过优化表征等离子体和壁的几个参数，可以增加该限制。我们研究的目的是探讨如何以及在多大程度上可以做到这一点。具体地说，我们扩展了许多早期的计算的$n=0$模式和数值确定的最大伸长率作为一个功能的无量纲参数描述（1）等离子体轮廓（$β_p$和$l_i$），（2）等离子体形状（$ε$和$δ$），（3）壁半径（$b/a$）和（4）最重要的是反馈系统的能力参数$γτ_w$的缩放关系。这些数值计算依赖于一个新的制定$n=0$磁流体理论，我们最近开发[Freidberg等。等人2015; Lee et. 2015]，将二维稳定性问题简化为一维问题。该方法包括了理想磁流体轴对称不稳定性的所有物理过程，同时显著减少了计算时间，因此在优化过程中可以探索许多参数。也许最有用的最终结果是对模拟的简单解析拟合，其给出最大伸长和相应的优化三角形度作为函数$κ(ε,β_p,l_i,b/a,γτ_w)$和$δ(ε,β_p,l_i,b/a,γτ_w)$。我们提出的比例关系包括最佳的三角形和有限的纵横比上的最大伸长率的影响，并可以用于确定优化的等离子体形状在当前的实验和未来的托卡马克设计。
Stationary Resistive MHD Solutions Near an Isolated Three-Dimensional Magnetic Null
Astronomical Institute, Czech Academy of Science（捷克科学院天文研究所）；Astronomical Institute, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院天文研究所）；Escuela de Ingeniería Civil, Universidad de Valparaíso（瓦尔帕莱索大学土木工程学院）
In this work we introduce a novel semi-implicit structure-preserving finite-volume/finite-difference scheme for the viscous and resistive equations of magnetohydrodynamics (MHD) based on an appropriate 3-split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field and a third subsystem involving the pressure-velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfven waves and the magneto-acoustic wav…
We investigate the nonlinear evolution of the $m = 1$ resistive internal kink mode in a two-dimensional (2D) configuration containing a central region of negative current density, also known as the "current hole" setup. The finite-element code FINMHD is used to solve a reduced set of incompressible Magnetohydrodynamic (MHD) equations with a current-vorticity formulation. First, the kink instability linearly develops in agreement with the general theory of resistive internal kink mode, and it subsequently leads to the formation of a current sheet. At relatively low Lundquist number $S$, a magne…
We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin-Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instabilit…
