Massively Parallel Computing of Turbulent Transport in Tokamaks
Jeremy Kepner, Scott Parker, Viktor Decyk
备注：
A single encapsulated (Level 2) postscipt file (.eps) of our poster presented at the University of Minnesota Supercomputer Institute International Conference on Parallel Computing October 3-4, 1996. (This file is best examined in ghostview or other postscript previewer at high magnification)
With the advent of the gyrokinetic formalism, recent developments in low-noise nonlinear $δf$ methods, and enormous gains in computing power, large-scale gyrokinetic simulations have become an important tool for improved understanding of anomalous transport in tokamaks. Simulating the non-linear behaviour requires solving for the perturbations of distrbution function in five dimensions. We use a particle-in-cell approach to solve the equations via the non-linear characteristic method. The code has been parallelized for a variety of architecures (C90, CM-5, T3D) using a 1-D domain decomposition along the torroidal axis, for which the number of particles in each cell remains approximately constant. The quasi uniform distribution of particles, which minimizes load imbalance, coupled with the relatively small movement of particles across cells, which minimizes communications, makes this problem ideally suited to massively parallel architectures. We present the performance of the program as a function of number of processors and problem size, which demonstrates the near perfect scalability of the code. In addition, we discuss the scientific results obtained from the code and the types of problems that will be addressable as the next generation of super-computers become available.
A first-principles scaling law, based on turbulent transport considerations, and a multi-machine database of density limit discharges from the ASDEX Upgrade, JET and TCV tokamaks, show that the increase of the boundary turbulent transport with the plasma collisionality sets the maximum density achievable in tokamaks. This scaling law shows a strong dependence on the heating power, therefore predicting for ITER a significantly larger safety margin than the Greenwald empirical scaling (Greenwald et al, Nucl. Fusion, 28(12), 1988) in case of unintentional H-L transition.
Local nonlinear gyrokinetic simulations of tokamak plasmas demonstrate that turbulent eddies can extend along magnetic field lines for hundreds of poloidal turns when the magnetic shear is very small. By accurately modeling different field line topologies (e.g. low-order, almost rational, or irrational safety factor), we show that the parallel self-interaction of such "ultra long" eddies can dramatically reduce heat transport. This reveals novel strategies to improve confinement, constitutes experimentally testable predictions, and illuminates past observations of internal transport barriers.
The narrow power decay-length ($λ_q$), recently found in the scrape-off layer (SOL) of inner-wall limited (IWL) discharges in tokamaks, is studied using 3D, flux-driven, global two-fluid turbulence simulations. The formation of the steep plasma profiles measured is found to arise due to radially sheared $\vec{E}\times\vec{B}$ poloidal flows. A complex interaction between sheared flows and outflowing plasma currents regulates the turbulent saturation, determining the transport levels. We quantify the effects of sheared flows, obtaining theoretical estimates in agreement with our non-linear simu…
Several new results in the physics of edge poloidal flows, turbulent stresses and momentum transport are reported. These are based on experiments on the HL-2A tokamak. Significant deviation from neoclassical prediction for mean poloidal flow in Ohmic and L mode discharges is deduced from direct measurements of the turbulent Reynolds stress. The deviation increases with heating power. The turbulent poloidal viscosity is synthesized from fluctuation data, and is found to be comparable to the turbulent particle diffusivity. The intrinsic poloidal torque is deduced from synthesis, for the first ti…
A process specific to the dynamics of the heavy impurities in turbulent tokamak plasmas is found and analysed. We show that the parallel stochastic acceleration is strongly coupled to the perpendicular transport and generates a radial pinch velocity. The interaction is produced with the hidden drifts, a quasi-coherent component of the motion that consists of a pair of average radial velocities in opposite directions. The parallel acceleration breaks this symmetry and yields a radial average velocity that can be in the inward or outward direction. The transport of the tungsten ions in three-dim…
Turbulent transport in tokamak plasmas with rotational shear
利用非线性回旋动力学方法研究了托卡马克等离子体中存在旋转剪切时的湍流输运。在足够大的流动剪切，线性不稳定性被抑制，但瞬态增长模式驱动亚临界湍流的振幅增加与流动剪切。这导致热通量的局部最小值，表明等离子体约束的最佳E X B剪切值。局部最大的动量通量也观察到，允许在E × B剪切分叉的可能性。热流密度对温度梯度的敏感依赖性在大的流动剪切值下得到缓解，在较低的流动剪切值下，临界温度梯度增大。湍流普朗特数被发现在很大程度上是独立的温度和流量梯度，与一个接近统一的值。
Two-fluid, three-dimensional, flux-driven, global, electromagnetic turbulence simulations carried out by using the GBS code are used to identify the main parameters controlling turbulent transport in the tokamak boundary and to delineate an electromagnetic phase space of edge turbulence. Four turbulent transport regimes are identified: (i) a regime of fully developed turbulence appearing at intermediate values of collisionality and $β$, with turbulence driven by resistive ballooning modes, related to the L-mode operation of tokamaks, (ii) a regime of reduced turbulent transport at low collisio…
The understanding of cross-field transport is crucial in order to optimize the properties of magnetic confinement in modern fusion devices. In this work, a two-dimensional, simplified model is used to study the turbulent dynamics in the region of the scrape-off layer. We show how the numerical model, based on the reduced Braginskii equations, is able to describe the formation and the evolution over time of blobs structures. We study these complex dynamics by using both classical Eulerian analysis and the Lagrangian approach, by varying the ambient conditions of the plasma. We have found that b…
We investigate the ExB diffusion of fusion born αparticles in tokamak plasmas. We determine the transport regimes for a realistic model that has the characteristics of the ion temperature gradient (ITG) or of the trapped electron modes (TEM) driven turbulence. It includes a spectrum of potential fluctuations that is modeled using the results of the numerical simulations, the drift of the potential with the effective diamagnetic velocity and the parallel motion. Our semi-analytical statistical approach is based on the decorrelation trajectory method (DTM), which is adapted to the gyrokinetic ap…
The results of flux-driven, two-fluid simulations in single-null configurations are used to investigate the processes determining the turbulent transport in the tokamak edge. Three turbulent transport regimes are identified: (i) a developed transport regime with turbulence driven by an interchange instability, which shares a number of features with the standard L-mode of tokamak operation, (ii) a suppressed transport regime, characterized by a higher value of the energy confinement time, low-amplitude relative fluctuations driven by a Kelvin-Helmholtz instability, a strong E x B sheared flow, …
The stochastic advection of low energy deuterium ions is studied in a three dimensional realistic turbulence model in conditions relevant for current tokamak fusion experiments. The diffusion coefficients are calculated starting from the test particles trajectories in the framework of the semi-analitical statistical model called the Decorrelation Trajectory Method. We show that trajectory trapping determined by the space correlation of the velocity field is the main cause for anomalous diffusion and we obtain the transport regimes corresponding to different values of the parameters of the turb…
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…
A pulsed oscillating power amplifier has been developed for high frequency biasing\cite{kn:deb1} and real time turbulent feedback experiment in STOR-M tokamak. It is capable to provide output peak to peak oscillating voltage of around $\pm60$V and current around 30A within frequency band 1kHz-50kHz without any distortion of any waveform signal. Overall output power is amplified by two stages power mosfet op-amp as well as nine identical push-pull amplifiers which are parallel connected in final stages. The power amplifier input signal, collected from plasma floating potential during plasma sho…
For the first time the pellet cycle of a multiple-isotope plasma is successfully reproduced with reduced turbulent transport modelling, within an integrated simulation framework. Future nuclear fusion reactors are likely to be fuelled by cryogenic pellet injection, due to higher penetration and faster response times. Accurate pellet cycle modelling is crucial to assess fuelling efficiency and burn control. In recent JET tokamak experiments, deuterium pellets with reactor-relevant deposition characteristics were injected into a pure hydrogen plasma. Measurements of the isotope ratio profile inf…
In previous work we provided the explicit form of the nonlinear PDEs, subjected to the appropriate boundary conditions, which have to be satisfied by transport coefficients for systems out of Onsager's region. Since the proposed PDEs are obtained without neglecting any term present in the balance equations (i.e., the mass, momentum, and energy balance equations), we propose them as a good candidate for describing also transport in thermodynamic systems in turbulent regimes. As a special case, we derive the nonlinear PDEs for transport coefficients when the thermodynamic system is subjected to …
The progress made in understanding spontaneous toroidal rotation reversals in tokamaks is reviewed and current ideas to solve this ten-year-old puzzle are explored. The paper includes a summarial synthesis of the experimental observations in AUG, C-Mod, KSTAR, MAST and TCV tokamaks, reasons why turbulent momentum transport is thought to be responsible for the reversals, a review of the theory of turbulent momentum transport and suggestions for future investigations.
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
The analysis of turbulence in plasmas is fundamental in fusion research. Despite extensive progress in theoretical modeling in the past 15 years, we still lack a complete and consistent understanding of turbulence in magnetic confinement devices, such as tokamaks. Experimental studies are challenging due to the diverse processes that drive the high-speed dynamics of turbulent phenomena. This work presents a novel application of motion tracking to identify and track turbulent filaments in fusion plasmas, called blobs, in a high-frequency video obtained from Gas Puff Imaging diagnostics. We comp…
We present an ultrafast neural network (NN) model, QLKNN, which predicts core tokamak transport heat and particle fluxes. QLKNN is a surrogate model based on a database of 300 million flux calculations of the quasilinear gyrokinetic transport model QuaLiKiz. The database covers a wide range of realistic tokamak core parameters. Physical features such as the existence of a critical gradient for the onset of turbulent transport were integrated into the neural network training methodology. We have coupled QLKNN to the tokamak modelling framework JINTRAC and rapid control-oriented tokamak transpor…
Transport barrier formation and its relation to sheared flows in fluids and plasmas are of fundamental interest in various natural and laboratory observations and of critical importance in achieving an economical energy production in a magnetic fusion device. Here we report the first observation of an edge transport barrier formation event in a gyrokinetic simulation carried out in a realistic tokamak edge geometry. The results show that turbulent Reynolds stress driven sheared ExB flows act in concert with neoclassical orbit loss to quench turbulent transport and form a transport barrier just…
The mean parallel current density evolution equation is presented using electromagnetic (EM) gyrokinetic equation. There exist two types of intrinsic current driving mechanisms resulted from EM electron temperature gradient (ETG) turbulence. The first type is the divergence of residual turbulent flux including a residual stress-like term and a kinetic stress-like term. The second type is named as residual turbulent source, which is driven by the correlation between density and parallel electric field fluctuations. The intrinsic current density driven by the residual turbulent source is negligi…
Edge shear flow and its effect on regulating turbulent transport have long been suspected to play an important role in plasmas operating near the Greenwald density limit $ n_G $. In this study, equilibrium profiles as well as the turbulent particle flux and Reynolds stress across the separatrix in the HL-2A tokamak are examined as $ n_G $ is approached in ohmic L-mode discharges. As the normalized line-averaged density $ \bar{n}_e/n_G $ is raised, the shearing rate of the mean poloidal flow $ ω_{\rm sh} $ drops, and the turbulent drive for the low-frequency zonal flow (the Reynolds power $ \ma…
Explicit structure-preserving geometric Particle-in-Cell (PIC) algorithm in curvilinear orthogonal coordinate systems is developed. The work reported represents a further development of the structure-preserving geometric PIC algorithm [1-12], achieving the goal of practical applications in magnetic fusion research. The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms, discrete exterior calculus, and explicit non-canonical symplectic integration. In addition to the truncated infinitely dimensional symplec…
The local intrinsic parallel current density driven by electron drift wave (DW) turbulence including electromagnetic (EM) effects is analytically studied. The scalings of the ratios of intrinsic current density driven by residual turbulent flux and by turbulent source to the bootstrap current density with electron density and temperature are predicted to be $T_e^{3/4}T_i/n_e$ and $T_eT_i/n_e$, respectively. Based on the typical parameters in DIII-D pedestal region, the local intrinsic current density driven by both the residual turbulent flux and the turbulent source is negligible. However, de…
Analysis of the time series obtained with the O-Mode reflectometer (Rhodes et al 1997 Plasma Phys. and Control. Fusion 40 (1998) 493-510) and the gas puff imaging (Cziegler, I. et al 2010 Phys. of Plasmas 17, No. 5 (2010) 056120) systems on the Alcator C-Mod tokamak reveals that the turbulent edge density fluctuations are chaotic. Supporting evidence for this conclusion includes: the observation of exponential power spectra (which is associated with Lorentzian-shaped pulses in the time series), the population of the corresponding Bandt-Pompe probability distributions (Bandt and Pompe 2002 Phys…
We construct a model for the turbulence near the X-point of a Tokamak device and, under suitable assumptions, we arrive to a closed equation for the electric field potential fluctuations. The analytical and numerical analysis is focused on a reduced two-dimensional formulation of the dynamics, which allows a direct mapping to the incompressible Navier-Stokes equation. The main merit of this study is to outline how the turbulence near the X-point, in correspondence to typical operation conditions of medium and large size Tokamaks, is dominated by the enstrophy cascade from large to smaller spat…
Turbulent dynamics in the scrape-off layer (SOL) of magnetic fusion devices is intermittent with large fluctuations in density and pressure. Therefore, a model is required that allows perturbations of similar or even larger magnitude to the time-averaged background value. The fluid-turbulence code GRILLIX is extended to such a global model, which consistently accounts for large variation in plasma parameters. Derived from the drift reduced Braginskii equations, the new GRILLIX model includes electromagnetic and electron-thermal dynamics, retains global parametric dependencies and the Boussines…
