Geodesic Acoustic Mode in Toroidally Axisymmetric Plasmas with Non-Circular Cross Sections

The geodesic acoustic mode （GAM）, first predicted by Winsor, Johnson and Dawson in an attempt to explain some experimentally observed low frequency oscillations in stellarators, is a special electrostatic fluid mode with low mode number and coupled with the so-called geodesic curvature of a toroidally confined plasma. The recent on this mode are due to the close relevance of the stabi- lization of drift turbulence by both the zonal flow and the GAM. Previously, the GAM was illustrated in very simple geometries,     

Simplified Forms of Generalized MHD Equations in a Cylindrical Plasma

The simplified forms of generalized magnetohydrodynamic equations have been derived. The K. Appert theory and Hain-Lüst equation are two special cases of our results when p→0 and ω/ω_ci→0. It is shown that the process of taking any limits (p→0 or ω/ω_ci→0) will result in a singularity at the Alfvén resonant layer.     

Three-Wave Coupling Coefficients for Non-Uniform MHD Plasmas

Resonant three-wave interaction in a spatially non-uniform MHD plasma is considered, and the coupling coefficients are derived.     

A Roe Scheme for Ideal MHD Equations on 2D Adaptively Refined Triangular Grids

In this paper we present a second order finite volume method for the resolution of the bidimensional ideal MHD equations on adaptively refined triangular meshes. Our numerical flux function is based on a multidimensional extension of the Roe scheme proposed by Cargo and Gallice for the 1D MHD system. If the mesh is only composed of triangles, our scheme is proved to be weakly consistent with the condition …B=0. This property fails on a cartesian grid. The efficiency of our refinement procedure is shown on 2D MHD shock capturing simulations. Numerical results are compared in case of the interaction of a supersonic plasma with a cylinder on the adapted grid and several non-refined grids. We also present a mass loading simulation which corresponds to a 2D version of the interaction between the solar wind and a comet.     

李变换方法在超环面光栅成像理论中的应用

应用李变换方法研究了超环面光栅的成像及其像差,介绍了李变换基本原理.光栅成像过程可分成五个部分,分别对应五个李变换;其中最重要的变换就是超环面光栅的衍射李变换.利用这五个李变换推导出了超环面光栅的成像公式,并应用光线追迹,对研究结果进行了验证,说明了李变换的方法能准确地描述超环面光栅的成像.该方法可以处理平面光源,能计算远离子午焦平面处的像差.李变换方法导出的成像公式描述了像平面坐标与物平面坐标、方向余弦之间的函数关系,体现了物空间变量与像空间对应变量之间映射关系.     

Stability of Tokamak Plasmas with Internal Transport Barriers against High Ideal Magnetohydrodynamic Ballooning Mode

High n （ the toroidal mode number） ballooning mode analysis is generally thought to be the most fundamental tool for the study of the magnetohydrodynamic （MHD） stability in magnetically confinement toroidal fusion devices. Very recently, new interest on ballooning mode study is put on the internal transport barrier （ITB） phenomena, an improved confinement structure characterized by a central negative shear region and a much smaller energy transport region near the minimum of q.     

Relation Between Shape Equations for Axisymmetric vesicles

In deriving the shape equation for axisymmetric vesicles from the Helfrich free energy the variation must be taken with respect to the contour. It is pointed out that the widely used Helfiich shape equation and the other one given by Peterson were derived by improper variations, hence both equations need to be examined. The validity of the two equations is discussed based on analysis of their relation revealed by two derived first integrals. A newly found exact solution for the general shape equation is studied and the relation of this exact solution of spherical topology with that of the Helfrich shape equation is discussed.     

Hyperbolic Divergence Cleaning for the MHD Equations

In simulations of magnetohydrodynamic (MHD) processes the violation of the divergence constraint causes severe stability problems. In this paper we develop and test a new approach to the stabilization of numerical schemes. Our technique can be easily implemented in any existing code since there is no need to modify the solver for the MHD equations. It is based on a modified system in which the divergence constraint is coupled with the conservation laws by introducing a generalized Lagrange multiplier. We suggest a formulation in which the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same time. This corrected system is hyperbolic and the density, momentum, magnetic induction, and total energy density are still conserved. In comparison to results obtained without correction or with the standard “divergence source terms,” our approach seems to yield more robust schemes with significantly smaller divergence errors.     

A Triangulated Vortex Method for the Axisymmetric Euler Equations

A new method is presented for the prediction of unsteady axisymmetric inviscid flows. By combining a triangulated vortex approach with a novel evaluation technique for the Biot–Savart integrals, a Lagrangian vortex method is developed which eliminates the singularities usually present in axisymmetric methods, without recourse to normalizations or other approximations. Furthermore, the computational effort scales as the number of control points N and, in the large N limit, depends only on the order of quadrature employed. The accuracy and computational effort are assessed by comparison with the velocity field of a Gaussian core vortex ring and the use of the technique is illustrated by computation of the motion of Norbury rings and of vortex ring pairing.     

Statistical Equations for Particle Distribution Functions in Collisionless Quark-Gluon Plasmas

A system of statistical equations has been derived to analyze the behavior of nonequilibrium collisionless quark-gluon plasmas in strong gravitational fields at superhigh temperatures (T < T _KP =1/8·(m _t c ₂/k) = 2.5·10¹⁴ K with mt being the mass of t-quark) and densities (10²⁴ g/cm³). It has been shown that in a system where perturbations appear which are accompanied by an increase in the temperature over Tcr, an explosion occurs. This explosion is characterized by an anomalous prevalence of high particle velocities and leads to the formation of flows of high-energy hadrons.     

Unified System of Differential Equations for Plasmas and Boundary Layers

Starting from the Boltzmann equation the moment equations up to the third order for a non-quasineutral plasma are derived. It is shown that for a non-quasineutral plasma, in the distinction from the second order set of moment equations, the systems of the first order and of the third order can be closed in a physically reasonable manner by a relatively simple method when near the walls the ion gas has a high drift velocity caused by an electric field. From this follows that in non-quasineutral plasmas the ion pressure must be neglected in the first order set of equations consisting of the equations of continuity and of momentum transfer. Terms of the first order in the ion pressure related to the electron pressure are taken into consideration for the third order system. In this way one obtains a unified set of differential equations to treat both the quasineutral plasma core and the space charge sheath at the walls jointly taking into account a non-vanishing ion temperature. This set of equations is applied to the positive column in the free-fall regime.     

On the Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

In suitable coordinates Einstein's field equations for a rigidly rotating perfect fluid in equilibrium can be written as a semilinear system of purely elliptic partial differential equations of second order. Therefore, the formulation of a boundary value problem is appropriate in this situation. It is shown that the Dirichlet problem for the vacuum region outside a ball, and for a ball inside the matter region, has a unique regular solution if the boundary data are in a characteristic way limited by the “diameter” of the ball. This restriction seems to be closely connected with stability limits for rotating stars. Furthermore, the used mathematical methods are directly related to a numerical solution technique for such physical systems. Received: 30 November 1995/ Accepted: 15 April 1997     

Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B ³ _s with s greater than 1/3. B ³ _s consists of functions that are Lip(s) (i.e., H?lder continuous with exponent s) measured in the L ^p norm. Here this result is applied to a velocity field that is Lip(α₀) except on a set of co-dimension on which it is Lip($agr;₁), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if . Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics. Received: 21 March 1995 / Accepted: 6 August 1996     

A Modular Grad-Div Stabilization Method for Time-Dependent Thermally Coupled MHD Equations

In this paper, we consider a fully discrete modular grad-div stabilization algorithm for time-dependent thermally coupled magnetohydrodynamic (MHD) equations. The main idea of the proposed algorithm is to add an extra minimally intrusive module to penalize the divergence errors of velocity and improve the computational efficiency for increasing values of the Reynolds number and grad-div stabilization parameters. In addition, we provide the unconditional stability and optimal convergence analysis of this algorithm. Finally, several numerical experiments are performed and further indicated these advantages over the algorithm without grad-div stabilization.     

A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics

We present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio–Wu Riemann problems, the two-dimensional Kelvin–Helmholtz instability problems, and the two-dimensional Orszag–Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points.     

A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations

Based on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.     

一种基于新型斜率限制器的理想磁流体方程的高分辨率熵相容格式

针对磁流体动力学方程, 通过分析数据重建所需的条件, 构造一种基于MUSCL(Monotone Upstream-Centred Scheme for Conservation Laws)型重建方法的斜率限制器, 获得了一种求解理想磁流体动力学方程的高分辨率熵相容格式。该格式在解的光滑区域具有高精度; 在解的间断区域可以合理地控制耗散, 可有效避免非物理现象的产生。采用熵稳定格式、熵相容格式和新的高分辨率熵相容格式对一维、二维理想磁流体动力学方程进行数值模拟。结果表明: 新格式能准确地捕捉解的结构, 且具有无振荡、高分辨、鲁棒等特性。     

Boundary Value Problems for the Stationary Axisymmetric Einstein Equations: A Disk Rotating Around a Black Hole

We solve a class of boundary value problems for the stationary axisymmetric Einstein equations involving a disk rotating around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.     

Error Analysis of a PFEM Based on the Euler Semi-Implicit Scheme for the Unsteady MHD Equations

In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint “∇·u=0”, which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter ϵ, the time-step size τ, and the mesh size h. Finally, two numerical tests show that our scheme is effective.