We have used the local-$δ{f}$ gyrokinetic code GS2 to perform studies of the effect of flux-surface shaping on two highly-shaped, low- and high-$β$ JT-60SA-relevant equilibria, including a successful benchmark with the GKV code. We find a novel destabilization of electrostatic fluctuations with increased elongation for plasma with a strongly peaked pressure profile. We explain the results as a competition between the local magnetic shear and finite-Larmor-radius (FLR) stabilization. Electromagnetic studies indicate that kinetic ballooning modes are stabilized by increased shaping due to an increased sensitivity to FLR effects, relative to the ion-temperature-gradient instability. Nevertheless, at high enough $β$, increased elongation degrades the local magnetic shear stabilization that enables access to the region of ballooning second-stability.
High-power-density tokamaks offer a potential solution to design cost-effective fusion devices. One way to achieve high power density is to operate at a high $β$ value (the ratio of thermal to magnetic pressure), i.e., $β\sim 1$. However, a $β\sim 1$ state may be unstable to various pressure- and current-driven instabilities or have unfavorable microstability properties. To explore these possibilities, we generate $β\sim 1$ equilibria and investigate their stability. Initially, we study an analytical technique that was used in the past to generate $β\sim 1$ equilibria and outline its limitatio…
Shaping of magnetic flux surfaces is found to have a strong impact on turbulence and transport in tokamak edge plasmas. A series of axisymmetric equilibria with varying elongation and triangularity, and a divertor configuration are implemented into a computational gyrofluid turbulence model. The mechanisms of shaping effects on turbulence and flows are identified. Transport is mainly reduced by local magnetic shearing and an enhancement of zonal shear flows induced by elongation and X-point shaping.
A poloidal tilting symmetry of the local nonlinear $δf$ gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equ…
Using analytic calculations, the effects of the edge flux surface shape and the toroidal current profile on the penetration of flux surface shaping are investigated in a tokamak. It is shown that the penetration of shaping is determined by the poloidal variation of the poloidal magnetic field on the surface. This fact is used to investigate how different flux surface shapes penetrate from the edge. Then, a technique to separate the effects of magnetic pressure and tension in the Grad-Shafranov equation is presented and used to calculate radial profiles of strong elongation for nearly constant …
Introducing up-down asymmetry into the tokamak magnetic equilibria appears to be a feasible method to drive fast intrinsic toroidal rotation in future large devices. In this paper we investigate how the intrinsic momentum transport generated by up-down asymmetric shaping scales with the mode number of the shaping effects. Making use the gyrokinetic tilting symmetry (Ball et al (2016) Plasma Phys. Control. Fusion 58 045023), we study the effect of envelopes created by the beating of different high-order shaping effects. This reveals that the presence of an envelope can change the scaling of the…
Breaking the up-down symmetry of tokamaks removes a constraint limiting intrinsic momentum transport, and hence toroidal rotation, to be small. Using gyrokinetic theory, we study the effect of different up-down asymmetric flux surface shapes on the turbulent transport of momentum. This is done by perturbatively expanding the gyrokinetic equation in large flux surface shaping mode number. It is found that the momentum flux generated by shaping that lacks mirror symmetry (which is necessarily up-down asymmetric) has a power law scaling with the shaping mode number. However, the momentum flux gen…
In this work, the role which three-dimensional shaping plays in the generation of rotational transform in toroidal magnetically confinement devices is explored. The susceptance matrix as defined by Strand & Houlberg (2001) is presented and compared to simulations of three dimensional MHD equilibria. The dependence of the edge rotational transform on axisymmetric shaping is briefly explored. It is then shown how simple three dimensional shaping can drive rotational transform in a simply shaped stellarator. It is found that modes with high poloidal mode number drive edge transform with little ef…
It is shown that small externally applied magnetic perturbations can significantly alter important geometric properties of magnetic flux surfaces in tokamaks. Through 3D shaping, experimentally relevant perturbation levels are large enough to influence turbulent transport and MHD stability in the pedestal region. It is shown that the dominant pitch-resonant flux surface deformations are primarily induced by non-resonant 3D fields, particularly in the presence of significant axisymmetric shaping. The spectral content of the applied 3D field can be used to control these effects.
Characterizing Flux-Surface Shapes in Tokamaks and Quasi-Symmetric Stellarators
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Max Planck Institute for Plasma Physics（马普等离子体物理研究所）；Dutch Institute for Fundamental Research（荷兰基础研究所）；Eindhoven University of Technology（埃因霍温理工大学）；Ruhr-Universität Bochum（波鸿鲁尔大学）
Bulk toroidal rotation has proven capable of stabilising both dangerous MHD modes and turbulence. In this thesis, we explore a method to drive rotation in large tokamaks: up-down asymmetry in the magnetic equilibrium. We seek to maximise this rotation by finding optimal up-down asymmetric flux surface shapes. First, we use the ideal MHD model to show that low order external shaping (e.g. elongation) is best for creating up-down asymmetric flux surfaces throughout the device. Then, we calculate realistic up-down asymmetric equilibria for input into nonlinear gyrokinetic turbulence analysis. Ana…
A local magnetic equilibrium model is presented, with finite aspect ratio and up-down asymmetrically shaped cross section, that depends on eight free parameters. In contrast with other local equilibria, which provide simple magnetic-surface parametrizations at the cost of complex poloidal-field flux descriptions, the proposed model is intentionally built to afford analytically tractable magnetic-field components. Therefore, it is particularly suitable for analytical assessments of equilibrium-shaping effects on a variety of tokamak-plasma phenomena.
An Inverse Grad-Shafranov Neural Network Approach to Tokamak Magnetic Control
MIT Plasma Science and Fusion Center（麻省理工学院等离子体科学与聚变中心）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Google DeepMind（谷歌DeepMind）；Fusionality SA；Commonwealth Fusion Systems（联邦聚变系统）
Instability to high toroidal mode number ($n$) ballooning modes has been proposed as the primary gradient-limiting mechanism for tokamak equilibria with negative triangularity ($δ$) shaping, preventing access to strong H-mode regimes when $δ\ll0$. To understand how this mechanism extrapolates to reactor conditions, we model the infinite-$n$ ballooning stability as a function of internal profiles and equilibrium shape using a combination of the CHEASE and BALOO codes. While the critical $δ$ required for avoiding $2^\mathrm{nd}$ stability to high-$n$ modes is observed to depend in a complicated …
Plasma terminating disruptions in tokamaks may result in relativistic runaway electron beams with potentially serious consequences for future devices with large plasma currents. In this paper we investigate the effect of plasma elongation on the coupled dynamics of runaway generation and resistive diffusion of the electric field. We find that elongated plasmas are less likely to produce large runaway currents, partly due to the lower induced electric fields associated with larger plasmas, and partly due to direct shaping effects, which mainly lead to a reduction in the runaway avalanche gain.
High-order plasma shaping (mainly elongation and shift, as opposed to low-order toroidicity) is shown, under certain conditions, to open gaps in the coupled shear-Alfvén and acoustic continua at frequencies significantly above the values predicted by previous theories. Global eigenmodes in these gaps, which lie between those of geodesic acoustic modes (GAMs) and toroidicity-induced Alfvén eigenmodes (TAEs), are found unstable to hot-ion populations typical of tokamak operation, whilst their fundamental resonances with circulating particles are shown to take place at velocities near the geometr…
Breaking the up-down symmetry of the tokamak poloidal cross-section can significantly increase the spontaneous rotation due to turbulent momentum transport. In this work, we optimize the shape of flux surfaces with both tilted elongation and tilted triangularity in order to maximize this drive of intrinsic rotation. Nonlinear gyrokinetic simulations demonstrate that adding optimally-tilted triangularity can double the momentum transport of a tilted elliptical shape. This work indicates that tilting the elongation and triangularity in an ITER-like device can reduce the energy transport and driv…
Recent gyrokinetic stability calculations have revealed that the spherical tokamak is susceptible to tearing parity instabilities with length scales of a few ion Larmor radii perpendicular to the magnetic field lines. Here we investigate this 'micro-tearing' mode in greater detail to uncover its key characteristics, and compare it with existing theoretical models of the phenomenon. This has been accomplished using a full numerical solution of the linear gyrokinetic-Maxwell equations. Importantly, the instability is found to be driven by the free energy in the electron temperature gradient as d…
In this work, we demonstrate a method for constructing the Solov'ev equilibrium with any given 2D shape surrounded by a vacuum region using external poloidal field coils. The computational domain consists of two parts: the plasma region, where the solution is the same as the Solov'ev solution, and the vacuum region, where the magnetic field generated by external coils as well as plasma current is determined using the Green function method through a matching condition near the separatrix. However, the method is not limited to the Solov'ev equilibrium in particular. The accuracy, efficiency, and…
In this work, we modify the standard flux tube simulation domain to include arbitrary ion gyroradius-scale variation in the radial profile of the safety factor. To determine how to appropriately include such a modification, we add a strong ion gyroradius-scale source (inspired by electron cyclotron current drive) to the Fokker-Planck equation, then perform a multi-scale analysis that distinguishes the fast electrons driven by the source from the slow bulk thermal electrons. This allows us to systematically derive the needed changes to the gyrokinetic model. We find new terms that adjust the io…
Theoretical and numerical studies of wave-packet propagation are presented to analyze the time varying 2D mode structures of electrostatic fluctuations in tokamak plasmas, using general flux coordinates. Instead of solving the 2D wave equations directly, the solution of the initial value problem is used to obtain the 2D mode structure, following the propagation of wave-packets generated by a source and reconstructing the time varying field. As application, the 2D WKB method is applied to investigate the shaping effects (elongation and triangularity) of tokamak geometry on the lower hybrid wave…
Accurate plasma shape control is the basis of tokamak plasma experiments and physical research. Modeling of the linearized control response of plasma shape and position has been widely used for shape controller design in the last several years. But it usually contains much of the uncertainty, such as structured uncertainties and unmodeled dynamics. EAST tokamak plasma shape controller design is also based on a linear rigid plasma response model which integrated within a Matlab-based toolset known as TokSys. Meanwhile the PID control approach is currently used for EAST plasma shape control. Thi…
The magnetic field that supports tokamak and stellarator plasmas must be produced by coils well separated from the plasma. However the larger the separation, the more difficult it is to produce a given magnetic field in the plasma region, so plasma configurations should be chosen that can be supported as efficiently as possible by distant coils. The efficiency of an externally-generated magnetic field is a measure of the field's shaping component magnitude at the plasma compared to the magnitude near the coils; the efficiency of a plasma equilibrium can be measured using the efficiency of the …
The effect of magnetic turbulence in shaping the current density in axisymmetric magnetized plasma is analyzed using a turbulent extension of Ohm's law derived from the self-consistent action-angle transport theory. Besides the well-known hyper-resistive (helicity-conserving) contribution, the generalized Ohm's law contains an anomalous resistivity term, and a turbulent bootstrap-like term proportional to the current density derivative. The numerical solution of the equation for equilibrium and turbulence profiles characteristic of conventional and advanced scenarios shows that, trough "turbul…
It is believed that plasma rotation can affect the transitions to the advanced confinement regimes in tokamaks. In addition, in order to achieve fusion temperatures modern tokamaks rely on auxiliary heating methods. These methods generate pressure anisotropy in the plasma. For incompressible rotation with pressure anisotropy the equilibrium is governed by a Generalized Grad-Shafranov (GGS) equation and a decoupled Bernoulli-type equation for the effective pressure, $\bar{p}=(p_\parallel+p_\perp)/2$, where $p_\parallel$ ($p_\perp$) is the pressure tensor element parallel (perpendicular) to the …
Investigation of Toroidal Rotation Effects on Spherical Torus Equilibria using the Fast Spectral Solver VEQ-R
Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams of the Ministry of Education, School of Physics, Dalian University of Technology（大连理工大学物理学院教育部激光、离子和电子束材料改性重点实验室）；ENN Science and Technology Development Co., Ltd.（新奥科技发展有限公司）
