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A large number of magnetohydrodynamic (MHD) equilibrium calculations are often required for uncertainty quantification, optimization, and real-time diagnostic information, making MHD equilibrium codes vital to the field of plasma physics. In this paper, we explore a method for solving the Grad–Shafranov equation by using physics-informed neural networks (PINNs). For PINNs, we optimize neural networks by directly minimizing the residual of the partial differential equation as a loss function. We show that PINNs can accurately and effectively solve the Grad–Shafranov equation with several different boundary conditions, making it more flexible than traditional solvers. This method is flexible as it does not require any mesh and basis choice, thereby streamlining the computational process. We also explore the parameter space by varying the size of the model, the learning rate, and boundary conditions to map various tradeoffs such as between reconstruction error and computational speed. Additionally, we introduce a parameterized PINN framework, expanding the input space to include variables such as pressure, aspect ratio, elongation, and triangularity in order to handle a broader range of plasma scenarios within a single network. Parameterized PINNs could be used in future work to solve inverse problems such as shape optimization. [ABSTRACT FROM AUTHOR]

We demonstrate a fast adjoint-based method to optimise tokamak and stellarator equilibria against a pressure-driven instability known as the infinite- $n$ ideal ballooning mode. We present three finite- $\beta$ (the ratio of thermal to magnetic pressure) equilibria: one tokamak equilibrium and two stellarator equilibria that are unstable against the ballooning mode. Using the self-adjoint property of ideal magnetohydrodynamics, we construct a technique to rapidly calculate the change in the eigenvalue, a measure of ideal ballooning instability. Using the SIMSOPT optimisation framework, we then implement our fast adjoint gradient-based optimiser to minimise the eigenvalue and find stable equilibria for each of the three originally unstable equilibria. [ABSTRACT FROM AUTHOR]

This document is an erratum for an article titled "Grad-Shafranov equilibria via data-free physics informed neural networks" published in the journal Physics of Plasmas. The erratum states that an important reference was accidentally omitted from the original article. The missing reference is a prior work by D. A. Kaltsas and G. N. Throumoulopoulos titled "Neural network tokamak equilibria with incompressible flows." The authors of the erratum emphasize that this reference provides relevant background information for their own work on neural network surrogates for MHD equilibria. [Extracted from the article]

Local linear gyrokinetic simulations show that electron temperature gradient (ETG) instabilities are the fastest growing modes for in the steep gradient region for a JET pedestal discharge (92174) where the electron temperature gradient is steeper than the ion temperature gradient. Here, ky is the wavenumber in the direction perpendicular to both the magnetic field and the radial direction, and ρi is the ion gyroradius. At , the fastest growing mode is often a novel type of toroidal ETG instability. This toroidal ETG mode is driven at scales as large as and at a sufficiently large radial wavenumber that electron finite Larmor radius effects become important; that is, , where Kx is the effective radial wavenumber. Here, ρe is the electron gyroradius, R0 is the major radius of the last closed flux surface, and 1/LTe is an inverse length proportional to the logarithmic gradient of the equilibrium electron temperature. The fastest growing toroidal ETG modes are often driven far away from the outboard midplane. In this equilibrium, ion temperature gradient instability is subdominant at all scales and kinetic ballooning modes are shown to be suppressed by shear. ETG modes are very resilient to shear. Heuristic quasilinear arguments suggest that the novel toroidal ETG instability is important for transport. [ABSTRACT FROM AUTHOR]

Keywords: fusion plasma; plasma simulation; plasma instabilities EN fusion plasma plasma simulation plasma instabilities 1 1 1 07/24/23 20230801 NES 230801 The published article incorrectly contains a placeholder reference to an equation ("(??)") on page 2, paragraph 2, line 4: "The high-beta, I i ~ 1, regime has been explored previously in the context of asymptotic magnetohydrodynamic (MHD) equilibria by solving the Grad-Shafranov equation in the limit Graph (Hsu, Artun & Cowley 1996) where Graph is the aspect ratio of the tokamak and I q i is the safety factor defined in (??):." The placeholder ("(??)") should have been replaced by a pinpoint reference to equation 2.2 ("(2.2)"), as follows: "The high-beta, I i ~ 1, regime has been explored previously in the context of asymptotic magnetohydrodynamic (MHD) equilibria by solving the Grad-Shafranov equation in the limit Graph (Hsu, Artun & Cowley 1996) where Graph is the aspect ratio of the tokamak and I q i is the safety factor defined in (2.2)." Fusion plasma, plasma simulation, plasma instabilities. [Extracted from the article]

Data from gyrokinetic turbulence codes are often difficult to visualize due their high dimensionality, the nontrivial geometry of the underlying grids, and the vast range of spatial scales. We present an interactive visualization framework that attempts to address these issues. Images from a nonlinear gyrokinetic simulation are presented. Index Terms--Gyrokinetics, plasma turbulence, visualization.

The velocity-space distribution of alpha particles born in fusion devices is subject to modification at moderate energies due to turbulent transport. Therefore, one must calculate the evolution of an equilibrium distribution whose functional form is not known a priori. Using a novel technique, applicable to any trace impurity, we have made this calculation for fully nonlinear gyrokinetic simulations not only possible but also particularly efficient. We demonstrate a microturbulence-induced departure from the local slowing-down distribution, an inversion of the energy distribution, and associated modifications to the alpha heating and pressure profiles in an ITER-like scenario. [ABSTRACT FROM AUTHOR]

The δf particle-in-cell algorithm has been a useful tool in studying the physics of plasmas, particularly turbulent magnetized plasmas in the context of gyrokinetics. The reduction in noise due to not having to resolve the full distribution function indicates an efficiency advantage over the standard ("full-f") particle-in-cell. Despite its successes, the algorithm behaves strangely in some circumstances. In this work, we document a fully resolved numerical instability that occurs in the simplest of multiple-species test cases: the electrostatic ΩH mode. There is also a poorly understood numerical instability that occurs when one is under-resolved in particle number, which may require a prohibitively large number of particles to stabilize. Both of these are independent of the timestepping scheme, and we conclude that they exist if the time advancement were exact. The exact analytic form of the algorithm is presented, and several schemes for mitigating these instabilities are also presented. [ABSTRACT FROM AUTHOR]

Linear gyrokinetic simulations covering the collisional-collisionless transitional regime of the tearing instability are performed. It is shown that the growth rate scaling with collisionality agrees well with that predicted by a two-fluid theory for a low plasma beta case in which ion kinetic dynamics are negligible. Electron wave-particle interactions (Landau damping), finite Larmor radius, and other kinetic effects invalidate the fluid theory in the collisionless regime, in which a general non-polytropic equation of state for pressure (temperature) perturbations should be considered. We also vary the ratio of the background ion to electron temperatures and show that the scalings expected from existing calculations can be recovered, but only in the limit of very low beta. [ABSTRACT FROM AUTHOR]

Collisions between counterpropagating Alfvén waves represent the fundamental building block of plasma turbulence, a phenomenon of great importance to a wide variety of fields, from space physics and astrophysics to controlled magnetic fusion. Proposed experiments to study Alfvén wave collisions on the Large Plasma Device (LAPD) [W. Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs, Rev. Sci. Instrum. 62, 2875 (1991)] at the University of California, Los Angeles, will benefit significantly from numerical modeling capable of reproducing not only the linear dispersive effects of kinetic and inertial Alfvén waves, but also the nonlinear evolution of the Alfvénic turbulence. This paper presents a comparison of linear simulation results using the astrophysical gyrokinetics code, AstroGK, to the measured linear properties of kinetic and inertial Alfvén waves in the LAPD plasma. Results demonstrate that: (1) finite frequency effects due to the ion cyclotron resonance do not prevent satisfactory modeling of the LAPD plasma using gyrokinetic theory; and (2) an advanced collision operator, recently implemented in AstroGK, enables the code to successfully reproduce the collisionally enhanced damping rates of linear waves measured in recent LAPD experiments. These tests justify the use of AstroGK in the modeling of LAPD Alfvén wave experiments and suggest that AstroGK will be a valuable tool in modeling the nonlinear evolution of proposed Alfvén wave collision experiments. [ABSTRACT FROM AUTHOR]

The article discusses techniques used to map direct numerical simulation (DNS) of turbulence through the use of modern graphics processing units (GPUs). Such units have high memory bandwidth and numerous single-program-multiple-data (SPMD) processors. Other topics include energy losses and catastrophic events caused by turbulence in plasma.

Zonal flow helps reduce and regulate the turbulent transport level in tokamaks. Rosenbluth and Hinton have shown that zonal flow damps to a nonvanishing residual level in collisionless [M. Rosenbluth and F. Hinton, Phys. Rev. Lett. 80, 724 (1998)] and collisional [F. Hinton and M. Rosenbluth, Plasma Phys. Control. Fusion 41, A653 (1999)] banana regime plasmas. Recent zonal flow advances are summarized including the evaluation of the effects on the zonal flow residual by plasma cross-section shaping, shorter wavelengths including those less than an electron gyroradius, and arbitrary ion collisionality relative to the zonal low frequency. In addition to giving a brief summary of these new developments, the analytic results are compared with GS2 numerical simulations [M. Kotschenreuther, G. Rewoldt, and W. Tang, Comput. Phys. Commun. 88, 128 (1991)] to demonstrate their value as benchmarks for turbulence codes. [ABSTRACT FROM AUTHOR]

The direct quantitative correspondence between theoretical predictions and the measured plasma fluctuations and transport is tested by performing nonlinear gyrokinetic simulations with the GS2 code. This is a continuation of previous work with gyrofluid simulations [D. W. Ross et al., Phys. Plasmas 9, 177 (2002)], and the same L-mode reference discharge in the DIII-D tokamak [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)] is studied. The simulated turbulence is dominated by ion temperature gradient (ITG) modes, corrected by trapped-electron, passing-electron and impurity effects. The energy fluxes obtained in the gyrokinetic simulations are comparable to, even somewhat higher than, those of the earlier work, and the simulated ion thermal transport, corrected for E × B flow shear, exceeds the experimental value by more than a factor of 2. The simulation also overestimates the density fluctuation level. Varying the local temperature gradient shows a stiff response in the flux and an apparent up-shift from the linear mode threshold [A. M. Dimits et al., Phys. Plasmas 7, 969 (2000)]. This effect is insufficient, within the estimated error, to bring the results into conformity with the experiment. [ABSTRACT FROM AUTHOR]

In this work we numerically demonstrate both significant transient (i.e. non-modal) linear amplification and sustained nonlinear turbulence in a kinetic plasma system with no unstable eigenmodes. The particular system considered is an electrostatic slab with magnetic shear, kinetic electrons and ions, weak collisions and a density gradient, but with no temperature gradient. In contrast to hydrodynamic examples of non-modal growth and subcritical turbulence, here there is no sheared flow in the equilibrium. Significant transient linear amplification is found when the magnetic shear and collisionality are weak. It is also demonstrated that nonlinear turbulence can be sustained if initialized at sufficient amplitude. We prove that these two phenomena are related: when sustained turbulence occurs without unstable eigenmodes, states that are typical of the turbulence must yield transient linear amplification of the gyrokinetic free energy. [ABSTRACT FROM PUBLISHER]

Recent work has demonstrated that breaking the up–down symmetry of tokamak flux surfaces removes a constraint that limits intrinsic momentum transport, and hence toroidal rotation, to be small. We show, through MHD analysis, that ellipticity is most effective at introducing up–down asymmetry throughout the plasma. We detail an extension to GS2, a local δf gyrokinetic code that self-consistently calculates momentum transport, to permit up–down asymmetric configurations. Tokamaks with tilted elliptical poloidal cross-sections were simulated to determine nonlinear momentum transport. The results, which are consistent with the experiment in magnitude, suggest that a toroidal velocity gradient, (∂uζi/∂ρ)/vthi, of 5% of the temperature gradient, (∂Ti/∂ρ)/Ti, is sustainable. Here vthi is the ion thermal speed, uζi is the ion toroidal mean flow, ρ is the minor radial coordinate normalized to the tokamak minor radius, and Ti is the ion temperature. Though other known core intrinsic momentum transport mechanisms scale poorly to larger machines, these results indicate that up–down asymmetry may be a feasible method to generate the current experimentally measured rotation levels in reactor-sized devices. [ABSTRACT FROM AUTHOR]

We demonstrate that the universal mode driven by the density gradient in a plasma slab can be absolutely unstable even in the presence of reasonable magnetic shear. Previous studies from the 1970s that reached the opposite conclusion used an eigenmode equation limited to Lx ⪢ pi where Lx is the scale length of the mode in the radial direction, and pi is the ion Larmor radius. Here we instead use a gyrokinetic approach which does not have this same limitation. Instability is found for perpendicular wave numbers ky in the range 0.7 ≲ kypi, ≲ 100, and for sufficiently weak magnetic shear; Ls/Ln ≳ 17, where Ls and Ln are the scale lengths of magnetic shear and density. Thus, the plasma drift wave in a sheared magnetic field may be unstable even with no temperature gradients, no trapped particles, and no magnetic curvature. [ABSTRACT FROM AUTHOR]

We consider a nearly collisionless plasma consisting of a species of "test particles" in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field-a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space-known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C_{2}∝∫∫dxdv〈f^{2}〉-a quantity that here plays the role of (negative) entropy of the distribution function. We find that C_{2} is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C_{2} in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equation for the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C_{2} to high s (towards the collisional dissipation range) at every k, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C_{2} is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k-space (s-space) flux of C_{2} is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the (k,s) plane, with power-law asymptotics at large k and s. Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a "critical balance" in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

Gyrokinetic simulations of small-scale turbulent transport in a closed magnetic field-line plasma geometry are presented. The simulations are potentially applicable to dipolar systems such as the levitated dipole experiment (LDX) [J. Kesner, Plasma Phys. Rep. 23, 742 (1997).] and planetary magnetospheres, as well as simpler systems such as the Z pinch. We report here for the first time the existence of a robust particle (and weaker temperature) pinch regime, in which the particles are transported up the density gradient. The particle pinch is driven by non-MHD entropy-mode turbulence at k(⊥) ρ(i) ∼ 1 and particle pinch appears at larger η ≡ L(n)/L(T) ≳ 0.7, consistent with quasilinear theory. Our results suggest that entropy-mode transport will drive the LDX plasma profiles toward a state with η ∼ 0.7 and pressure gradients that are near marginal ideal MHD interchange-mode stability.

Gyrokinetic flux-tube simulations of turbulent transport due to small-scale entropy modes are presented in a ring-dipole magnetic geometry relevant to the Columbia-MIT levitated dipole experiment (LDX) [J. Kesner, Plasma Phys. J. 23, 742 (1997)]. Far from the current ring, the dipolar magnetic field leads to strong parallel variations, while close to the ring the system becomes nearly uniform along circular magnetic field lines. The transport in these two limits are found to be quantitatively similar given an appropriate normalization based on the local out-board parameters. The transport increases strongly with the density gradient, and for small eta=L(n)/L(T)<<1, T(i) approximately T(e), and typical LDX parameters, can reach large levels. Consistent with linear theory, temperature gradients are stabilizing, and for T(i) approximately T(e) can completely cut off the transport when eta greater or similar to 0.6.

Differential rotation occurs in conducting flows in accretion disks and planetary cores. In such systems, the magnetorotational instability can arise from coupling Lorentz and centrifugal forces to cause large radial angular momentum fluxes. We present the first experimental observation of the magnetorotational instability. Our system consists of liquid sodium between differentially rotating spheres, with an imposed coaxial magnetic field. We characterize the observed patterns, dynamics, and torque increases, and establish that this instability can occur from a hydrodynamic turbulent background.