Cyclokinetic models and simulations for high-frequency turbulence in fusion plasmas

Gyrokinetics is widely applied in plasma physics. However, this framework is limited to weak turbulence levels and low drift-wave frequencies because high-frequency gyro-motion is reduced by the gyro-phase averaging. In order to test where gyrokinetics breaks down, Waltz and Zhao developed a new theory, called cyclokinetics [R. E. Waltz and Zhao Deng, Phys. Plasmas 20, 012507 (2013)]. Cyclokinetics dynamically follows the high-frequency ion gyro-motion which is nonlinearly coupled to the low-frequency drift-waves interrupting and suppressing gyro-averaging. Cyclokinetics is valid in the high-frequency (ion cyclotron frequency) regime or for high turbulence levels. The ratio of the cyclokinetic perturbed distribution function over equilibrium distribution function δf/F can approach 1.This work presents, for the first time, a numerical simulation of nonlinear cyclokinetic theory for ions, and describes the first attempt to completely solve the ion gyro-phase motion in a nonlinear turbulence system. Simulations are performed [Zhao Deng and R. E. Waltz, Phys. Plasmas 22(5), 056101 (2015)] in a local flux-tube geometry with the parallel motion and variation suppressed by using a newly developed code named rCYCLO, which is executed in parallel by using an implicit time-advanced Eulerian (or continuum) scheme [Zhao Deng and R. E. Waltz, Comp. Phys. Comm. 195, 23 (2015)]. A novel numerical treatment of the magnetic moment velocity space derivative operator guarantee saccurate conservation of incremental entropy.By comparing the more fundamental cyclokinetic simulations with the corresponding gyrokinetic simulations, the gyrokinetics breakdown condition is quantitatively tested. Gyrokinetic transport and turbulence level recover those of cyclokinetics at high relative ion cyclotron frequencies and low turbulence levels, as required. Cyclokinetic transport and turbulence level are found to be lower than those of gyrokinetics at high turbulence levels and low-Ω* values with stable ion cyclotron modes. The gyrokinetic approximation is found to break down when the density perturbation exceeds 20%, or when the ratio of nonlinear E×B frequency over ion cyclotron frequency exceeds 20%. This result indicates that the density perturbation of the Tokamak L-mode near-edge is not sufficiently large for breaking the gyro-phase averaging. For cyclokinetic simulations with sufficiently unstable ion cyclotron (IC) modes and sufficiently low Ω* ～10, the high-frequency component of the cyclokinetic transport can exceed that of the gyrokinetic transport. However, the low-frequency component of the cyclokinetic transport does not exceed that of the gyrokinetic transport. For higher and more physically relevant Ω* ?50 values and physically realistic IC driving rates, the low-frequency component of the cyclokinetic transport remains smaller than that of the gyrokinetic transport. In conclusion, the “L-mode near-edge short-fall” phenomenon, observed in some low-frequency gyrokinetic turbulence transport simulations, does not arise owing to the nonlinear coupling of high-frequency ion cyclotron motion to low-frequency drift motion.     

Entanglement dephasing of spatially separated multiparticle GHZ states coupled to a squeezed environment

In tokamak plasmas, it is recognized that ITG (ion temperature gradient instability) and trapped electron modes (TEM) are held responsible for turbulence giving rise to anomalous transport. The present work focuses on the building of a model including trapped kinetic ions and trapped kinetic electrons. For this purpose, the dimensionality is reduced by averaging the motion over the cyclotron motion and the “banana” orbits, according to the fact that the instabilities are characterized by frequencies of the order of the low trapped particle precession frequency. Moreover, a set of action-angle variables is used. The final model is 4D (two-dimensional phase space parametrized by the two first adiabatic invariants namely the particle energy and the trapping parameter). In this paper, the trapped ion and electron modes (TIM and TEM) are studied by using a linear analysis of the model. This work is currently performed in order to include trapped electrons in an existing semi lagrangian code for which TIM modes are already taken into account. This study can be considered as a first step in order to include kinetic trapped electrons in the 5D gyrokinetic code GYSELA [J. Abiteboul et al., ESAIM Proc. 32, 103 (2011)].     

Theoretical ROVibrational Energies (TROVE): A robust numerical approach to the calculation of rovibrational energies for polyatomic molecules

We present a new computational method with associated computer program TROVE (Theoretical ROVibrational Energies) to perform variational calculations of rovibrational energies for general polyatomic molecules of arbitrary structure in isolated electronic states. The (approximate) nuclear kinetic energy operator is represented as an expansion in terms of internal coordinates. The main feature of the computational scheme is a numerical construction of the kinetic energy operator, which is an integral part of the computation process. Thus the scheme is self-contained, i.e., it requires no analytical pre-derivation of the kinetic energy operator. It is also general, since it can be used in connection with any internal coordinates. The method represents an extension of our model for pyramidal XY₃ molecules reported previously [S.N. Yurchenko, M. Carvajal, P. Jensen, H. Lin, J.J. Zheng, W. Thiel, Mol. Phys. 103 (2005) 359]. Non-rigid molecules are treated in the Hougen-Bunker-Johns approach [J.T. Hougen, P.R. Bunker, J.W.C. Johns, J. Mol. Spectrosc. 34 (1970) 136]. In this case, the variational calculations employ a numerical finite basis representation for the large-amplitude motion using basis functions that are generated by Numerov-Cooley integration of the appropriate one-dimensional Schrödinger equation.     

Parallel filtering in global gyrokinetic simulations

In this work, a Fourier solver [B.F. McMillan, S. Jolliet, A. Bottino, P. Angelino, T.M. Tran, L. Villard, Comp. Phys. Commun. 181 (2010) 715] is implemented in the global Eulerian gyrokinetic code GT5D [Y. Idomura, H. Urano, N. Aiba, S. Tokuda, Nucl. Fusion 49 (2009) 065029] and in the global Particle-In-Cell code ORB5 [S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T.M. Tran, B.F. McMillan, O. Sauter, K. Appert, Y. Idomura, L. Villard, Comp. Phys. Commun. 177 (2007) 409] in order to reduce the memory of the matrix associated with the field equation. This scheme is verified with linear and nonlinear simulations of turbulence. It is demonstrated that the straight-field-line angle is the coordinate that optimizes the Fourier solver, that both linear and nonlinear turbulent states are unaffected by the parallel filtering, and that the k_∥ spectrum is independent of plasma size at fixed normalized poloidal wave number.     

The global version of the gyrokinetic turbulence code GENE

The understanding and prediction of transport due to plasma microturbulence is a key open problem in modern plasma physics, and a grand challenge for fusion energy research. Ab initio simulations of such small-scale, low-frequency turbulence are to be based on the gyrokinetic equations, a set of nonlinear integro-differential equations in reduced (five-dimensional) phase space. In the present paper, the extension of the well-established and widely used gyrokinetic code GENE [F. Jenko, W. Dorland, M. Kotschenreuther, B.N. Rogers, Electron temperature gradient driven turbulence, Phys. Plasmas 7 (2000) 1904–1910] from a radially local to a radially global (nonlocal) version is described. The necessary modifications of both the basic equations and the employed numerical methods are detailed, including, e.g., the change from spectral methods to finite difference and interpolation techniques in the radial direction and the implementation of sources and sinks. In addition, code verification studies and benchmarks are presented.     

An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must either confront the challenge of controlling errors in the discrete divergence of the magnetic field, or else be faced with nonlinear numerical instabilities. One approach for controlling the discrete divergence is through a so-called constrained transport method, which is based on first predicting a magnetic field through a standard finite volume solver, and then correcting this field through the appropriate use of a magnetic vector potential. In this work we develop a constrained transport method for the 3D ideal MHD equations that is based on a high-resolution wave propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme developed by Rossmanith [J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (2006) 1766], and is based on the high-resolution wave propagation method of Langseth and LeVeque [J.O. Langseth, R.J. LeVeque, A wave propagation method for threedimensional hyperbolic conservation laws, J. Comput. Phys. 165 (2000) 126]. In particular, in our extension we take great care to maintain the three most important properties of the 2D scheme: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as cell-centered; (2) we develop a high-resolution wave propagation scheme for evolving the magnetic potential; and (3) we develop a wave limiting approach that is applied during the vector potential evolution, which controls unphysical oscillations in the magnetic field. One of the key numerical difficulties that is novel to 3D is that the transport equation that must be solved for the magnetic vector potential is only weakly hyperbolic. In presenting our numerical algorithm we describe how to numerically handle this problem of weak hyperbolicity, as well as how to choose an appropriate gauge condition. The resulting scheme is applied to several numerical test cases.     

AstroGK: Astrophysical gyrokinetics code

The gyrokinetic simulation code AstroGK is developed to study fundamental aspects of kinetic plasmas and for applications mainly to astrophysical problems. AstroGK is an Eulerian slab code that solves the electromagnetic gyrokinetic-Maxwell equations in five-dimensional phase space, and is derived from the existing gyrokinetics code GS2 by removing magnetic geometry effects. Algorithms used in the code are described. The code is benchmarked using linear and nonlinear problems. Serial and parallel performance scalings are also presented.     

Numerical Simulation of the 12 May 1997 CME Event

Our newly developed CESE MHD model is used to simulate sun-earth connection event with the well-studied 12 May 1997 CME event as an example. The main features and approximations of our numerical model are as follows： （1） The modifed conservation element and solution element （CESE） numerical scheme in spherical geometry is implemented in our code. （2） The background solar wind is derived from a 3D time-dependent numerical MHD model by input measured photospheric magnetic fields. （3） Transient disturbances are derived from solar surface by introducing a mass flow of hot plasma. The numerical simulation has enabled us to predict the arrival of the interplanetary shock and provided us with a relatively satisfactory comparison with the WIND spacecraft observations.     

Electromagnetic gyrokinetic δf particle-in-cell turbulence simulation with realistic equilibrium profiles and geometry

The δf particle-in-cell method for gyrokinetic simulations with kinetic electrons and electromagnetic perturbations [Y. Chen, S. Parker, J. Comput. Phys. 189 (2003) 463] is extended to include arbitrary toroidal equilibrium profiles and flux-surface shapes. The domain is an arbitrarily sized toroidal slice with periodicity assumed in toroidal direction. It is global radially and poloidally along the magnetic field. The differential operators and Jacobians are represented numerically which is a quite general approach with wide applicability. Discretization of the field equations is described. The issue of domain decomposition and particle load balancing is addressed. A derivation of the split-weight scheme is given, and numerical observations are given as to what algorithmic change leads to stable algorithm. It is shown that in the final split-weight algorithm the equation for the rate of change of the electric potential is solved in a way that is incompatible with the quasi-neutrality condition on the grid scale. This incompatibility, while negligible on the scale of interest, leads to better numerical stability on the grid scale. Some examples of linear simulations are presented to show the effects of flux-surface shaping on the linear mode growth rates. The issue of long-term weight growth in δf simulation and the effect of discrete particle noise are briefly discussed.     

Collisional gyrokinetic full-f particle-in-cell simulations on open field lines with PICLS

Applying gyrokinetic simulations in theoretical turbulence and transport studies for the plasma edge and scrape-off layer (SOL) presents significant challenges. To particularly account for steep density and temperature gradients in the SOL, the “full-f” code PICLS was developed. PICLS is a gyrokinetic particle-in-cell (PIC) code, is based on an electrostatic model with a linearized field equation, and uses kinetic electrons. In previously published results, we applied PICLS to the well-studied 1D parallel transport problem during an edge-localized mode (ELM) in the SOL without collisions. As an extension to this collision-less case and in preparation for 3D simulations, in this work, a collisional model will be introduced. The implemented Lenard–Bernstein collision operator and its Langevin discretization will be shown. Conservation properties of the collision operator, as well as a comparison of the collisional and non-collisional case, will be discussed.     

SSS程序在电磁内爆中的数值模拟

 对SSS程序的边界条件进行修改,将修改后的SSS程序用于固体套筒内爆实验的数值模拟,并与现有的一维磁流体力学程序的计算结果进行比较,对比结果表明,对SSS程序的改造是成功的,也可以预见将SSS程序改造成一维磁流体力学计算程序是可行的。     

Efficient calculation of energy spectra using path integrals

A newly developed method for systematically improving the convergence of path integrals for transition amplitudes [A. Bogojević, A. Balaž, A. Belić, Phys. Rev. Lett. 94 (2005) 180403, A. Bogojević, A. Balaž, A. Belić, Phys. Rev. B 72 (2005) 064302, A. Bogojević, A. Balaž, A. Belić, Phys. Lett. A 344 (2005) 84] and expectation values [J. Grujić, A. Bogojević, A. Balaž, Phys. Lett. A 360 (2006) 217] is here applied to the efficient calculation of energy spectra. We show how the derived hierarchies of effective actions lead to substantial speedup of the standard path integral Monte Carlo evaluation of energy levels. The general results and the ensuing increase in efficiency of several orders of magnitude are shown using explicit Monte Carlo simulations of several distinct models.     

A lattice Boltzmann method for incompressible two-phase flows on partial wetting surface with large density ratio

This paper reports a new numerical scheme of the lattice Boltzmann method for calculating liquid droplet behaviour on particle wetting surfaces typically for the system of liquid–gas of a large density ratio. The method combines the existing models of Inamuro et al. [T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628–644] and Briant et al. [A.J. Briant, P. Papatzacos, J.M. Yeomans, Lattice Boltzmann simulations of contact line motion in a liquid–gas system, Philos. Trans. Roy. Soc. London A 360 (2002) 485–495; A.J. Briant, A.J. Wagner, J.M. Yeomans, Lattice Boltzmann simulations of contact line motion: I. Liquid–gas systems. Phys. Rev. E 69 (2004) 031602; A.J. Briant, J.M. Yeomans, Lattice Boltzmann simulations of contact line motion: II. Binary fluids, Phys. Rev. E 69 (2004) 031603] and has developed novel treatment for partial wetting boundaries which involve droplets spreading on a hydrophobic surface combined with the surface of relative low contact angles and strips of relative high contact angles. The interaction between the fluid–fluid interface and the partial wetting wall has been typically considered. Applying the current method, the dynamics of liquid drops on uniform and heterogeneous wetting walls are simulated numerically. The results of the simulation agree well with those of theoretical prediction and show that the present LBM can be used as a reliable way to study fluidic control on heterogeneous surfaces and other wetting related subjects.     

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基于感应电磁场方程发展了低磁雷诺数条件下充分发展液态金属管道流的数值模拟程序。为了校正程序,分别计算了两种工况：液态金属在全绝缘管道和部分绝缘管道中的流动,数值结果与Hunt和Shercliff的解析解吻合的很好,表明该程序具有很高的精度。最后利用发展的程序对液态金属钠钾合金在管壁材料为304不锈钢的全导电管道中的流动进行了数值模拟,并对流速分布和MHD压降结果与实验结果进行了比较,结果表明数值与实验结果吻合较好。     

EAST位形磁流体平衡的模拟

对托卡马克平衡反演数值计算代码EFIT进行了改进,使之适用于EAST装置的磁流体平衡研究.用改进后的平衡反演代码EFIT模拟了EAST装置稳态阶段的偏滤器平衡位形.提出极向场线圈电流作为线性方程组主约束条件,期望位形作为从约束条件,求解非对称电流条件下自由边界平衡问题的方法.结果表明,该方法能较为容易地得到非对称电流条件产生的平衡位形,在非对称电流条件下自由边界平衡计算是收敛的.     

A LES/PDF simulator on block-structured meshes

A block-structured mesh large-eddy simulation (LES)/probability density function (PDF) simulator is developed within the OpenFOAM framework for computational modelling of complex turbulent reacting flows. The LES/PDF solver is a hybrid solution methodology consisting of (i) a finite-volume (FV) method for solving the filtered mass and momentum equations (LES solver), and (ii) a Lagrangian particle-based Monte Carlo algorithm (PDF solver) for solving the modelled transport equation of the filtered joint PDF of compositions. Both the LES and the PDF methods are developed and combined to form a hybrid LES/PDF simulator entirely within the OpenFOAM framework. The in situ adaptive tabulation method [S.B. Pope, Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combust. Theory Model. 1 (1997), pp. 41–63; L. Lu, S.R. Lantz, Z. Ren, and B.S. Pope, Computationally efficient implementation of combustion chemistry in parallel PDF calculations, J. Comput. Phys. 228 (2009), pp. 5490–5525] is incorporated into the new LES/PDF solver for efficient computations of combustion chemistry with detailed reaction kinetics. The method is designed to utilise a block-structured mesh and can readily be extended to unstructured grids. The three-stage velocity interpolation method of Zhang and Haworth [A general mass consistency algorithm for hybrid particle/finite-volume PDF methods, J. Comput. Phys. 194 (2004), pp. 156–193] is adapted to interpolate the LES velocity field onto particle locations accurately and to enforce the consistency between LES and PDF fields at the numerical solution level. The hybrid algorithm is fully parallelised using the conventional domain decomposition approach. A detailed examination of the effects of each stage and the overall performance of the velocity interpolation algorithm is performed. Accurate coupling of the LES and PDF solvers is demonstrated using the one-way coupling methodology. Then the fully two-way coupled LES/PDF solver is successfully applied to simulate the Sandia Flame-D, and a turbulent non-swirling premixed flame and a turbulent swirling stratified flame from the Cambridge turbulent stratified flame series [M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames I: Non-swirling flows, Combust. Flame 159 (2012), pp. 2896–2911; M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames II: Swirling flows, Combust. Flame 159 (2012), pp. 2912–2929]. It is found that the LES/PDF method is very robust and the results are in good agreement with the experimental data for both flames.     

空间傅里叶分析法识别MHD扰动模式

讨论了用空间傅里叶分析法确定托卡马克等离子体MHD磁扰动模式的方法。根据托卡马克实验的状况,对不同情况下的MHD磁扰动进行了详细的数值模拟,重点分析了该方法的系统误差,模拟结果与HL-1M装置上的实验结果进行了对比。比较结果表明,用空间傅里叶分析法识别托卡马克中的MHD扰动模式是一种可以行之有效的方法。     

Mean magnetic field and noise cross-correlation in magnetohydrodynamic turbulence: results from a one-dimensional model

We show that a recently proposed [J. Fleischer, P.H. Diamond, Phys. Rev. E 58, R2709 (1998)] one-dimensional Burgers-like model for magnetohydrodynamics (MHD) is in effect identical to existing models for drifting lines and sedimenting lattices. We use the model to demonstrate, contrary to claims in the literature, that the energy spectrum of MHD turbulence should be independent of mean magnetic field and that cross-correlations between the noise sources for the velocity and magnetic fields cannot change the structure of the equations under renormalisation. We comment on the scaling and the multiscaling properties of the stochastically forced version of the model. Received 29 October 1998 and Received in final form 8 December 1998     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.