Many plasmas of interest to the astrophysical and fusion communities are weakly collisional. In such plasmas, small scales can develop in the distribution of particle velocities, potentially affecting observable quantities such as turbulent fluxes. Consequently, it is necessary to monitor velocity space resolution in gyrokinetic simulations. In this paper, we present a set of computationally efficient diagnostics for measuring velocity space resolution in gyrokinetic simulations and apply them to a range of plasma physics phenomena using the continuum gyrokinetic code GS2. For the cases considered here, it is found that the use of a collisionality at or below experimental values allows for the resolution of plasma dynamics with relatively few velocity space grid points. Additionally, we describe implementation of an adaptive collision frequency which can be used to improve velocity space resolution in the collisionless regime, where results are expected to be independent of collision frequency.
The properties of the boundary plasma in a tokamak are now recognized to play a key role in determining the achievable fusion power and the lifetimes of plasma-facing components. Accurate quantitative modeling and improved qualitative understanding of the boundary plasma ultimately require five-dimensional gyrokinetic turbulence simulations, which have been successful in predicting turbulence and transport in the core. The additional challenges of boundary-plasma simulation necessitate the development of new gyrokinetic codes or major modifications to existing core gyrokinetic codes. In this t…
The first gyrokinetic simulations of plasma turbulence in the Texas Helimak device, a simple magnetized torus, are presented. The device has features similar to the scrape-off layer region of tokamaks, such as bad-curvature-driven instabilities and sheath boundary conditions on the end plates, which are included in these simulations. Comparisons between simulations and measurements from the experiment show similarities, including equilibrium profiles and fluctuation amplitudes that approach experimental values, but also some important quantitative differences. Both experimental and simulation …
3D2V continuum gyrokinetic simulations of electrostatic plasma turbulence in a straight, open-field-line geometry have been performed using the full-$f$ discontinuous-Galerkin code Gkeyll. These simulations include the basic elements of a fusion-device scrape-off layer: localized sources to model plasma outflow from the core, cross-field turbulent transport, parallel flow along magnetic field lines, and parallel losses at the limiter or divertor with sheath model boundary conditions. The set of sheath boundary conditions used in the model allows currents to flow through the walls. In addition …
We present the first-of-its-kind coupling of a continuum full-f gyrokinetic turbulence model with a 6D continuum model for kinetic neutrals, carried out using the Gkeyll code. Our objective is to improve the first-principles understanding of the role of neutrals in plasma fueling, detachment, and their interaction with edge plasma profiles and turbulence statistics. Our model includes only atomic hydrogen and incorporates electron-impact ionization, charge exchange, and wall recycling. These features have been successfully verified with analytical predictions and benchmarked with the DEGAS2 Mo…
Previous limiter-biasing experiments on the Texas Helimak, a simple magnetized torus, have been inconclusive on the effect of flow shear on turbulence levels. To investigate this, the first gyrokinetic simulations of limiter biasing in the Helimak using the plasma physics code Gkeyll have been carried out, and results are presented here. For the scenarios considered, turbulence is mostly driven by the interchange instability, which depends on gradients of equilibrium density profiles. An analysis of both experimental and simulation data demonstrates that shear rates are mostly less than than l…
The first linear global electromagnetic gyrokinetic particle simulation on the excitation of toroidicity induced Alfven eigenmode (TAE) by energetic particles is reported. With an increase in the energetic particle pressure, the TAE frequency moves down into the lower continuum.
The linear dynamics of Alfven modes in tokamaks is investigated here by means of the global gyrokinetic particle-in-cell code NEMORB. The model equations are shown and the local shear Alfven wave dispersion relation is derived, recovering the continuous spectrum in the incompressible ideal MHD limit. A verification and benchmark analysis is performed for continuum modes in a cylinder and for toroidicity-induced Alfven Eigenmodes. Modes in a reversed-shear equilibrium are also investigated, and the dependence of the spatial structure in the poloidal plane on the equilibrium parameters is descri…
Curvature-driven turbulence in a helical open-field-line plasma is investigated using electrostatic five-dimensional gyrokinetic continuum simulations in an all-bad-curvature helical-slab geometry. Parameters for a National Spherical Torus Experiment scrape-off-layer plasma are used in the model. The formation and convective radial transport of plasma blobs is observed, and it is shown that the radial particle-transport levels are several times higher than diffusive Bohm-transport estimates. By reducing the strength of the poloidal magnetic field, the profile of the heat flux to the divertor p…
We are developing a new continuum gyrokinetic code, Gkeyll, for use in edge plasma simulations, and here present initial simulations of turbulence on open field lines with model sheath boundary conditions. The code implements an energy conserving discontinuous Galerkin scheme, applicable to a general class of Hamiltonian equations. Several applications to test problems have been done, including a calculation of the parallel heat-flux on divertor plates resulting from an ELM crash in JET, for a 1x/1v SOL scenario explored previously, where the ELM is modeled as a time-dependent intense upstream…
A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a model collision operator satisfying these properties is described. This operator is based on the exact linearized test-particle collision operator, with approximations to the field-particle terms that preserve conservation laws and an H-Theorem. It includes energy diffusion, pitch-angle scattering, and finite Larmor radius effects corresponding to classical (real-space) diffusion. The numerical implementation in the continuum gyrokinetic code GS2 is fully implicit and guarant…
Understanding turbulent transport physics in the tokamak edge and scrape-off layer (SOL) is critical to developing a successful fusion reactor. The dynamics in these regions plays a key role in achieving high fusion performance by determining the edge pedestal that suppresses turbulence in H-mode. Additionally, the survivability of a reactor is set by the heat load to the vessel walls, making it important to understand turbulent spreading of heat as it flows along open magnetic field lines in the SOL. Large-amplitude fluctuations, magnetic X-point geometry, and plasma interactions with materia…
While in recent years, gyrokinetic simulations have become the workhorse for theoretical turbulence and transport studies in the plasma core, their application to the edge and scrape-off layer (SOL) region presents significant challenges. In particular, steep density and temperature gradients as well as large fluctuation amplitudes call for a "full-f" treatment. To specifically study problems in the SOL region, the gyrokinetic particle-in-cell (PIC) code PICLS has been developed. The code is based on an electrostatic full-f model with linearised field equations and uses kinetic electrons. Here…
The derivation and numerical implementation of a linearized version of the gyrokinetic (GK) Coulomb collision operator (Jorge R. et al., J. Plasma Phys. 85, 905850604 (2019)) and of the widely-used linearized GK Sugama collision operator (Sugama H. et al., Phys. Plasmas 16, 112503 (2009)) is reported. An approach based on a Hermite-Laguerre moment expansion of the perturbed gyrocenter distribution function is used, referred to as gyro-moment expansion. This approach allows considering arbitrary perpendicular wavenumber and expressing the two linearized GK operators as a linear combination of g…
This paper describes a conceptual framework for understanding kinetic plasma turbulence as a generalized form of energy cascade in phase space. It is emphasized that conversion of turbulent energy into thermodynamic heat is only achievable in the presence of some (however small) degree of collisionality. The smallness of the collision rate is compensated by the emergence of small-scale structure in the velocity space. For gyrokinetic turbulence, a nonlinear perpendicular phase mixing mechanism is identified and described as a turbulent cascade of entropy fluctuations simultaneously occurring a…
Two-fluid Braginskii codes have simulated open-field line turbulence for over a decade, and only recently has it become possible to study these systems with continuum gyrokinetic codes. This work presents a first-of-its-kind comparison between fluid and (long-wavelength) gyrokinetic models in open field-lines, using the GDB and Gkeyll codes to simulate interchange turbulence in the Helimak device at the University of Texas (T. N. Bernard, et. al., Phys. of Plasmas 26, 042301 (2019)). Partial agreement is attained in a number of diagnostic channels when the GDB sources and sheath boundary condi…
This work reports on the development and numerical implementation of the linear electromagnetic gyrokinetic (GK) model in a tokamak flux-tube geometry using a moment approach based on the expansion of the perturbed distribution function on a velocity-space Hermite-Laguerre polynomials basis. A hierarchy of equations of the expansion coefficients, referred to as the gyro-moments (GM), is derived. We verify the numerical implementation of the GM hierarchy in the collisionless limit by performing a comparison with the continuum GK code GENE, recovering the linear properties of the ion-temperature…
A conservative scheme has been formulated and verified for gyrokinetic particle simulations of electromagnetic waves and instabilities in magnetized plasmas. An electron continuity equation derived from drift kinetic equation is used to time advance electron density perturbation by using the perturbed mechanical flow calculated from the parallel vector potential, and the parallel vector potential is solved by using the perturbed canonical flow from the perturbed distribution function. In gyrokinetic particle simulations using this new scheme, shear Alfvén wave dispersion relation in shearless …
Numerical techniques for discretization of velocity space in continuum kinetic calculations are described. An efficient spectral collocation method is developed for the speed coordinate - the radius in velocity space - employing a novel set of non-classical orthogonal polynomials. For problems in which Fokker-Planck collisions are included, a common situation in plasma physics, a procedure is detailed to accurately and efficiently treat the field term in the collision operator (in the absence of gyrokinetic corrections). When species with disparate masses are included simultaneously, a careful…
To faithfully simulate ITER and other modern fusion devices, one must resolve electron and ion fluctuation scales in a five-dimensional phase space and time. Simultaneously, one must account for the interaction of this turbulence with the slow evolution of the large-scale plasma profiles. Because of the enormous range of scales involved and the high dimensionality of the problem, resolved first-principles global simulations are very challenging using conventional (brute force) techniques. In this thesis, the problem of resolving turbulence is addressed by developing velocity space resolution d…
Conservative velocity mappings for discontinuous Galerkin kinetics
Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Helmholtz-Zentrum Dresden-Rossendorf (HZDR)（德累斯顿-罗森多夫亥姆霍兹中心）；Institute for Fusion Studies, University of Texas at Austin（德克萨斯大学奥斯汀分校聚变研究所）
The `Alfvén Paradox' is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfvén continuum. To resolve the paradox, the $ε$-pseudospectrum of the RMHD operator is considered. It is proven that for any $ε$, the $ε$- pseudospectrum contains the Alfvén continuum for sufficiently small resistivity. Formal $ε-pseudoeigenmodes$ are constructed using the formal Wentzel-Kramers-Brillouin-Jeffreys solutions, and it is shown that the entire stable half-annulus of complex frequencies with $ρ{|ω|^2}=|\bf{v} \cdot \bf{B}(x)|^2$ is resonant to …
The global total-$f$ gyrokinetic particle-in-cell code XGC, used to study transport in magnetic fusion plasmas, implements a continuum grid to perform the dissipative operations, such as plasma collisions. To transfer the distribution function between marker particles and a rectangular velocity-space grid, XGC employs a bilinear mapping. The conservation of particle density and momentum is accurate enough in this bilinear operation, but the error in the particle energy conservation can become undesirably large in special conditions. In the present work we update XGC to use a novel mapping tech…
A numerical method to implement a linearized Coulomb collision operator in the two-weight $δf$ Monte Carlo method for multi-ion-species neoclassical transport simulation is developed. The conservation properties and the adjointness property of the operator in the collisions between two particle species with different temperatures are verified. The linearized operator in a $δf$ Monte Carlo code is benchmarked with other two kinetic simulations, a $δf$ continuum gyrokinetic code with the same linearized collision operator and a full-f PIC code with Nanbu collision operator. The benchmark simulat…
Electrostatic gyrokinetic absolute equilibria with continuum velocity field are obtained through the partition function and through the Green function of the functional integral. The new results justify and explain the prescription for quantization/discretization or taking the continuum limit of velocity. The mistakes in the Appendix D of our earlier work [J.-Z. Zhu and G. W. Hammett, Phys. Plasmas {\bf 17}, 122307 (2010)] are explained and corrected. If the lattice spacing for discretizing velocity is big enough, all the invariants could concentrate at the lowest Fourier modes in a negative-t…
On nonlinear saturation of toroidal Alfvén eigenmode due to thermal plasma nonlinearities
Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Center for Nonlinear Plasma Science and C.R. ENEA Frascati（非线性等离子体科学中心及恩瑞亚弗拉斯卡蒂研究中心）；Institute for Fusion Theory and Simulation, School of Physics, Zhejiang University（浙江大学物理学院聚变理论与模拟研究所）；Key Laboratory of Frontier Physics in Controlled Nuclear Fusion and Institute of Plasma Physics, Chinese Academy of Sciences（中国科学院等离子体物理研究所及受控核聚变前沿物理重点实验室）；De Vinci Higher Education, De Vinci Research Center（德文奇高等教育机构及德文奇研究中心）
