Theory of tokamak equilibria with central current density reversal

It is found that, with a model current profile, the Grad-Shafranov equation can be reduced to the Helmholtz equation, which can describe a variety of equilibrium configurations. With the eigenvalue problem solved in the toroidal coordinate system, an analytical solution to the Grad-Shafranov equation is found. It is demonstrated that current reversal equilibrium configurations exist with finite radial gradient of plasma pressure and continuous current density, and that current density reversal is accompanied by pressure gradient reversal.     

非均匀纵向场对等离子体平衡的影响

讨论了纵向不均匀磁场导致磁面与等离子的畸变。对于锐边界情形和具有体电流分布情形求解了平衡方程，特别是给出了磁面变形依赖于各参量变化的表达式。     

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

The reconstruction of the equilibrium of a plasma in a Tokamak is a free boundary problem described by the Grad–Shafranov equation in axisymmetric configuration. The right-hand side of this equation is a nonlinear source, which represents the toroidal component of the plasma current density. This paper deals with the identification of this nonlinearity source from experimental measurements in real time. The proposed method is based on a fixed point algorithm, a finite element resolution, a reduced basis method and a least-square optimization formulation. This is implemented in a software called Equinox with which several numerical experiments are conducted to explore the identification problem. It is shown that the identification of the profile of the averaged current density and of the safety factor as a function of the poloidal flux is very robust.     

Theoretical Optimization of Stellarators

Net current free toroidal ("stellarator") confinement is studied with a combination of several methods: a complete set of analytical vacuum fields for finding favorable vacuum field configurations; three-dimensional MHD codes for finite-?, equilibrium computations; the expansion of a general toroidal equilibrium around its magnetic axis as guideline for the computational search in configurational space and for finite-?, MHD stability; Monte Carlo simulations for particle containment; continuous modular coil systems generating the configurations considered. Results are: vacuum field configurations with sizeable Q = 0, 1, 2, 3 helical fields, substantial twist number (? 1/2), significant reduction of the parallel current density, and vacuum magnetic well exist for a toroidal aspect ratio of 15-20 and can be generated by modular coils whose excursions from meridional planes are small compared to the toroidal period length. In these configurations, the finite-? toroidal shift is strongly reduced, so that a larger ? value (factor 2-4) than in the equivalent Q = 2 stellarator can be achieved. Stability calculations do not exclude the possibility of stable equilibria of this kind with (?) ? 0.05-0.1; transport calculations without electrical field show improvement-as compared to the Q = 2 stellarator-in the collisional and plateau regimes.     

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采用流体模型理论推导了等熵平衡条件下环向转动托卡马克等离子体中带状流的色散关系。从理论上分析了环向转动对测地声模、低频带状流和声波的频率、压力和密度扰动量的影响。结果表明,环向转动对低频带状流的频率没有影响,但会使测地声模的频率逐渐增大。此外,存在环向转动时,低频带状流会具有驻波形式的压力和密度扰动量,且测地声模和声波可以沿着极向传播。而且还发现,等熵平衡可以看成是等温平衡的一种特殊情况。     

托卡马克等熵平衡条件下等离子体环向转动对带状流的影响

采用流体模型理论推导了等熵平衡条件下环向转动托卡马克等离子体中带状流的色散关系。从理论上分析了环向转动对测地声模、低频带状流和声波的频率、压力和密度扰动量的影响。结果表明,环向转动对低频带状流的频率没有影响,但会使测地声模的频率逐渐增大。此外,存在环向转动时,低频带状流会具有驻波形式的压力和密度扰动量,且测地声模和声波可以沿着极向传播。而且还发现,等熵平衡可以看成是等温平衡的一种特殊情况。     

Spontaneous formation of closed-field torus equilibrium via current jump observed in an electron-cyclotron-heated plasma

Spontaneous current jump resulting in the formation of closed field equilibrium has been observed in electron-cyclotron-heated toroidal plasmas under steady external fields composed of a toroidal field and a relatively weak vertical field in the low aspect ratio torus experiment device. This bridges the gap between the open field equilibrium maintained by a pressure-driven current in the external field and the closed field equilibrium at a larger current. Experimental results and theoretical analyses suggest a current jump model that is based on the asymmetric electron confinement along the field line appearing upon simultaneous transitions of field topology and equilibrium.     

磁螺度(helicity)注入电流驱动实验的平衡反演

磁螺度注入是一种国际上正处于探索性研究阶段的极富吸收力的电汉驱动方案,尤其对于球环堆芯更具有至和关重要的意义。对磁螺度注入电流驱动实验进行平衡反演工作,将实验数据进行理论分析研究,拟合所有的测量数据以确立实际的等离子体位形,平衡的磁拓扑结构以及环向电流的空间分布,从而确定闭合橛面区的驱动电流大小以及等离子体的性质,计算结果表明等离子体具有托卡马克型ｑ分布,环向电汉分布呈中空形,磁面结构为有三角形变     

托卡马克等离子体不同平衡位形下的电子回旋波电流驱动

在给定的等离子体总电流和中心电流密度条件下,数值求解平衡方程,求出不同拉长比和三角形变因子的托卡马克等离子体温度、密度、磁场分布,然后通过求解波迹方程和Fokker-Planck方程,分别计算这些位形下的电子回旋波波迹和电流驱动.结果表明：电子回旋波X模从顶部发射时,随着拉长比的增大,波迹会向弱场侧偏移.电子回旋波X模从弱场侧发射时,电子回旋波在等离子体中传播沉积的功率份额随着拉长比的增大而增加,驱动电流位置随着三角形变因子的增大向等离子体中心移动.驱动电流位置随环向和极向发射角的减小向中心移动,对应的电流密度峰值也变大.     

An improved “peeling” stability criterion

The stability of a general toroidal magnetohydrostatic equilibrium configuration surrounded by a vacuccum is re-investigated. The “peeling” criterion given perviously has been improved. In the case of axisymmetry with vanishing poloidal current the new form of the criterion requires vanishing of the toroidal current at the edge of the plasma.     

托卡马克高场侧低杂波电流驱动模拟研究

利用GENRAY/CQL3D程序研究了EAST装置双零位形下高场侧和低场侧发射低杂波的电流驱动情况.模拟发现,电子密度较小时,高场侧低杂波电流驱动效果不如低场侧.随着电子密度的增加,高场侧低杂波电流驱动的优势逐渐显现,从高场侧发射的低杂波可以将能量沉积在更加靠近等离子体中心的位置.提高环向磁场强度有利于低杂波在高密度条...     

Geodesic acoustic modes and zonal flows in toroidally rotating tokamak plasmas

The effect of equilibrium toroidal rotation on the rotational eigen-modes in large aspect ratio tokamak is studied. The case of equilibrium with uniform plasma density on magnetic surfaces is considered. It is shown that the toroidal rotation results in a frequency up-shift of ordinary Geodesic Acoustic Modes. A new unstable low frequency branch of the continuum modes is found. This mode appears as a consequence of the non-uniform plasma pressure created by the centrifugal force on the magnetic surfaces. This mode represents a linear eigen-mode counterpart of Zonal Flow modes. It is shown that the growth rate of such a mode increases with the increase of the angular velocity of toroidal rotation.     

托卡马克等离子体环向电流的锯齿效应

本文从动力论出发，分析了在托上瞳马克欧姆感应驱动和低杂波驱动等离子体的组合放电中，锯齿崩塌时，循环电子和俘获电子对环向电流及其密度产生的调制作用。     

环径比为2的托卡马克反应堆中氘氚燃烧等离子体特性

本文对环径比为2（A=2）的托卡马克反应堆中氘氚燃烧等离子体的特性，利用一类简单的平衡位形，即Solov’ev位形，进行了研究。作为基准，采用了Troyon比压极限和Greenwald密度极限，以及 ITER的H-模约束定标关系。我们发现，除了增大拉长比外，大的三角变形也有利于提高比压值及获得高的聚变功率输出。与 ITER设计相比，对相同的总等离子体电流而言，A=2的反应堆因边缘安全因子很大，具有可以避免破裂不稳定性的优点。由于减小环径比时环内侧的可利用空间相应减小，这类反应堆的一个主要缺点是环向磁场强度也会减小，从而降低聚变堆品质。实际研究工作中要对这些方面做出某种折衷。另外，虽然由于我们采用了简单的位形，我们不能对自举电流的准直性进行优化，但是通过本研究仍然可以对中等环径比反应堆系统（目前是研究工作的空白）的特点获得了解。     

A Numerical Study on Current Drive by Electron Cyclotron Waves in a Toroidal Plasma Device

A survey of the main features of ECR current driving in toroidal plasmas is presented. Numerical calculations are performed in order to obtain the spatial profiles of the ECT energy deposition and of the correlated current density as a function of temperature and of the direction of the wave vector with respect to the toroidal magnetic field. The results indicate a very strong dependence of the ECR absorption profiles and of the driven current densites on the direction of launching of the wave.     

Initial Measurements of Plasma Potential in the Core of the MST Reversed Field Pinch with a Heavy Ion Beam Probe

Measurement of the plasma potential in the core of MST marks both the first interior potential measurements in an RFP, as well as the first measurements by a Heavy Ion Beam Probe (HIBP) in an RFP. The HIBP has operated with (20-110) keV sodium beams in plasmas with toroidal currents of (200-480) kA over a wide range of densities and magnetic equilibrium conditions. A positive plasma potential is measured in the core, consistent with the expectation of rapid electron transport by magnetic fluctuations and the formation of an outwardly directed ambipolar radial electric field. Comparison between the radial electric field and plasma flow is underway to determine the extent to which equilibrium flow is governed by E×B. Measurements of potential and density fluctuations are also in progress.Unlike HIBP applications in tokamak plasmas, the beam trajectories in MST (RFP) are both three-dimensional and temporally dynamic with magnetic equilibrium changes associated with sawteeth. This complication offers new opportunity for magnetic measurements via the Heavy Ion Beam Probe (HIBP). The ion orbit trajectories are included in a Grad-Shafranov toroidal equilibrium reconstruction, helping to measure the internal magnetic field and current profiles. Such reconstructions are essential to identifying the beam sample volume locations, and they are vital in MST's mission to suppress MHD tearing modes using current profile control techniques. Measurement of the electric field may be accomplished by combining single point measurements from multiple discharges, or by varying the injection angle of the beam during single discharges.The application of an HIBP on MST has posed challenges resulting in additional diagnostic advances. The requirement to keep ports small to avoid introducing magnetic field perturbations has led to the design and successful implementation of cross-over sweep systems. High levels of ultraviolet radiation are driving alternative methods of sweep plate operation. While, substantial levels of plasma flux into the HIBP diagnostic chambers has led to the use of magnetic plasma suppression.     

平衡电流位形对电阻壁模稳定性影响的数值模拟

数值研究平衡电流位形对电阻壁模式稳定性的影响.研究发现,对于不同的电流位形,当等离子体边缘处安全因子一定时,最不稳定的电阻壁模的环向模数和极向模数相同.在同一壁位置下,非均匀电流位形驱动的电阻壁模的线性增长率比均匀电流位驱动的电阻壁模的线性增长率大.等离子体速度流在不同的初始电流位形下对电阻壁模稳定性的影响不同.由于磁力线在壁上的挤压,经过线性演化后,电阻壁模进入非线性演化并达到饱和状态,非均匀电流位形下的扰动磁能比均匀电流位形下的扰动磁能饱和度低.     

Role of nonlinear coupling and density fluctuations in magnetic-fluctuation-induced particle transport

Three-wave nonlinear coupling among spatial Fourier modes of density and magnetic fluctuations is directly measured in a magnetically confined toroidal plasma. Density fluctuations are observed to gain (lose) energy from (to) either equilibrium or fluctuating fields depending on the mode number. Experiments indicate that nonlinear interactions alter the phase relation between density and magnetic fluctuations, leading to strong particle transport.     

Toroidal momentum pinch velocity due to the coriolis drift effect on small scale instabilities in a toroidal plasma

In this Letter, the influence of the "Coriolis drift" on small scale instabilities in toroidal plasmas is shown to generate a toroidal momentum pinch velocity. Such a pinch results because the Coriolis drift generates a coupling between the density and temperature perturbations on the one hand and the perturbed parallel flow velocity on the other. A simple fluid model is used to highlight the physics mechanism and gyro-kinetic calculations are performed to accurately assess the magnitude of the pinch. The derived pinch velocity leads to a radial gradient of the toroidal velocity profile even in the absence of a torque on the plasma and is predicted to generate a peaking of the toroidal velocity profile similar to the peaking of the density profile. Finally, the pinch also affects the interpretation of current experiments.     

Experimental Studies on the Sustainment of Splieromak Plasmas by an Inductive Drive of the Toroidal Current

Sustainment of spheromak plasmas produced in an external equilibrium field has been demonstrated with a center current transformer (ohmic heating (OH) coil) which is used to inductively drive the toroidal current of the plasma. The OH coil is covered by a cylindrical metal liner. It provides the stability against the tilt and shift motions of spheromaks at the expense of the simple connection of its geometry. Since the spheromak is characterized by the elimination of external toroidal fields in association with nonconservation of a toroidal flux during magnetic relaxation, the metal liner was made electrically disconnected from the main vacuum vessel (spheromak mode). In the experiments, existense of the dynamo effect, meaning automatic generation of toroidal flux similar to that of a reversed field pinch (RFP), is observed. Measured MHD activity consists of multihelicity helical modes with toroidal mode numbers N = 1-3. In order to investigate the difference between spheromaks and RFP's in the MHD activity during sustainment, experiments have also been made with the metal liner of the OH coil connected with the vessel (RFP mode). The dynamics of the MHD activities observed are compared with those obtained from the three-dimensional MHD simulations by Katayama and Katsurai [18], and their implication in the dynamo effect is discussed.