In this work, we investigate the impact of electromagnetic effects on plasma turbulence self-organization at low magnetic shear using nonlinear gyrokinetics. Our previous electrostatic studies showed that turbulent eddies extend along magnetic field lines for hundreds of poloidal turns when the magnetic shear s is low or zero. Such ``ultra-long’’ eddies have significant consequences on turbulent transport due to parallel self-interaction. At low magnetic shear, parallel self-interaction induces strong corrugations in plasma profiles at low-order rational surfaces, including the formation of stationary current layers. When electromagnetic effects are considered, turbulence-generated currents lead to the development of stationary zonal magnetic potential, locally flattening the safety factor profile to form staircase structures or broaden the safety factor profile minimum. This represents a crucial feedback mechanism between turbulence and the imposed safety factor profile, resulting in a reduction in turbulent transport. We study the corrugated safety factor profiles using both the local flux tube code GENE and the global particle-in-cell code ORB5. To further explore this interaction, we employed a novel extension of the flux tube model, allowing simulations of non-uniform magnetic shear profiles, including minimum-q profiles relevant for Internal Transport Barrier (ITB) formation. Our findings indicate that turbulence-generated current layers can flatten the imposed non-uniformity across the entire domain or substantially widen rational surface regions, consistent with global simulation results. We believe these results are relevant for understanding ITB formation and inform long-standing experimental observations.
Schemes involving higher-beta fusion devices have led to intensified interest in electromagnetic turbulence in fusion plasmas. I will discuss this topic in two radically simplified settings: ITG-driven turbulence in a Z-pinch and interchange-driven turbulence in open-field-line systems. Both can be described by minimal models involving just a few “fluid” fields: density and temperature plus velocity and magnetic perturbations perpendicular to the mean magnetic field. Despite a drastic reduction in complexity that many seasoned fusion practitioners might consider excessive and indeed deeply immoral, these simple paradigms yield a wealth of interesting behaviour, some of which seems to capture key physical signatures of electromagnetic effects reported by more realistic numerical explorations — and might perhaps inform future such explorations (failing that, it is interesting enough to pique the curiosity of some theoreticians). Increasing beta in ITG-driven turbulence is shown to move the threshold for the low-to-high-transport (Dimits) transition towards lower temperature gradients. This is because in the struggle between the Reynolds and diamagnetic stresses that determines this transition in an electrostatic plasma [1], the Maxwell stress brought in by the Alfvenic activity associated with higher beta opposes the Reynolds stress and tips the system towards higher-transport states [2]. In the interchange-driven MHD turbulence, increasing beta (which in all such systems is equivalent also to increasing the parallel size of the system, increasing the temperature gradient or the field-line curvature) also pushes the system from an electrostatic low-transport state to an electromagnetic (very)-high-transport one — this is perhaps the simplest conceivable system exhibiting the Dimits transition [3] (simpler even than [1]). It is shown however that when 2D interchange motions are forbidden, the low-beta electrostatic state can be replaced by an electromagnetic one with very high transport and some rather strange properties (while it is interesting theoretically, as well as perhaps in some analogous astrophysical systems, the relevance of this last result to fusion devices is left to post-seminar discussion). Overall, the takeaway message is that electromagnetic turbulence in fusion plasmas is a physics-rich subject that, while challenging practical modellers, should also excite theoreticians.
[1] P. Ivanov et al. JPP 86, 855860502 (2020) & JPP 88, 905880506 (2022)
[2] Y. Zhang et al., in preparation (2024)
[3] W. Clarke et al., in preparation (2024)
A beam of runaway electrons (REs) within a cold, resistive background plasma can be unstable to resistive hose modes. Helical deflection of the current centroid of the beam away from its initial position is imperfectly canceled by eddy currents induced in the cold background plasma. The resulting macro-scale, over-stable, kink-like oscillations grow on a time scale determined by the resistivity of the cold plasma. Using a fluid model for the RE beam, we find that these high frequency modes are faster growing than resistive MHD instabilities when resistivity is large [Phys. Plasmas 31, 010701 (2024)]. Initial value calculations using the fluid RE model implemented in the extended MHD code NIMROD are compared to a 1D ODE eigenvalue code, and a semi-analytic dispersion relation for a uniform beam current equilibrium. Nonlinear simulations of the hose mode are presented, and the consequences for RE electron confinement in post-disruption tokamaks—where the background thermal plasma is cold and the resistivity is large—are discussed.
The Hasegawa-Mima equation is a local 2-D model for drift-wave turbulence in a magnetized plasma possessing a density gradient. Steady zonal flows with sinusoidal variation are known to be unstable to Kelvin-Helmholtz instabilities when u₀ / V* > λ² / Lₙ², where u₀ is the amplitude of the flow, V* is the electron diamagnetic drift velocity, λ is the wavelength of the flow, and Lₙ is the density gradient scale length [Zhu, et. al, Phys. Plasmas 25, 082121 (2018)]. With the addition of viscosity, we find that the parameter range for linear instability is modified, and that an additional unstable mode exists at very low kᵧ. The growth rate of the instability with viscosity is shown to go as kᵧ² at low kᵧ' and the frequency corresponds with the drift wave frequency ω*.
Accretion is a fundamental cosmic process involved in phenomena ranging from planet formation to X-ray binaries and the generation of jets from protostars and black holes. Evidence suggests magnetorotational instability (MRI) drives the turbulence that transports angular momentum outwards, allowing gas to spiral inwards in adisk. MRI-induced turbulence depends on coherent magnetic fields, which are susceptible to dissipation unless maintained by a dynamo. The fundamental processes sustaining MRI turbulence and dynamos remain elusive due to the historical separation of mean-field dynamo and angular momentum transport problems.
We investigate MRI turbulence and dynamo phenomena using direct statistical simulations in a zero net-flux, unstratified shearing box. Our approach begins with the development of a unified mean-field model that integrates the traditionally decoupled problems of large-scale dynamo action and angular momentum transport in accretion disks. The model includes a hierarchical set of equations, capturing up to second-order correlators, with a statistical closure approximation used for third-order correlators. We emphasize the complex web of interactions connecting various components of the stress tensors—Maxwell, Reynolds, and Faraday—through shear, rotation, correlators associated with mean fields, and nonlinear terms. By identifying the dominant interactions, we pinpoint the key mechanisms essential for the generation and maintenance of MRI turbulence. Our general mean-field model for MRI turbulence allows for a self-consistent formulation of the electromotive force, accounting for both inhomogeneities and anisotropies. In the context of large-scale magnetic field generation, we identify two critical mechanisms: the rotation-shear-current effect, which generates radial magnetic fields, and the rotation-shear-vorticity effect, responsible for vertical magnetic fields. We provide explicit, non-perturbative expressions for the transport coefficients associated with these dynamo effects. Importantly, both mechanisms rely on the inherent presence of large-scale vorticity dynamo within MRI turbulence. Lastly, I will discuss the associated magnetic helicity in such unstratified turbulent disks.
Microtearing instability is one of the major sources of turbulent transport in high-β tokamaks. These modes lead to very localized transport at low-order rational magnetic field lines, and we show that flattening of the local electron temperature gradient at these rational surfaces plays an important role in setting the saturated flux level in microtearing turbulence. This process depends crucially on the density of rational surfaces, and thus the system-size, and gives rise to a worse-than-gyro-Bohm transport scaling for system-sizes typical of existing tokamaks and simulations.
Entropy is a good fluid quantity for many quasi-equilibrium systems of physics, chemistry, and information. It has a good property of representing a thermodynamic state, a heat exchange, and a preferential direction of the irreversibility. Although the entropy concept can be useful to analyze the equilibrium (e.g. magnetic equilibrium), it has been hardly understood to be meaningful for the non-equilibrium, opened, and highly nonlinear system like fusion kinetic plasmas. In this presentation, we find the possibility of using entropy concept to interpret the nonlinear unsteady systems by two important examples of the fusion theory: (1) the zonal flow saturation for trapped electron mode (TEM) turbulence, and (2) the frequency chirping of Berk-Breizman (BB) model for energetic particle driven instability.
Thanks to the entropy property describing the energy exchange by ohmic heating and a triad transfer in the wave-particle interactions, the entropy balance analysis is useful to find the energy flow of a drift-wave turbulence between the driving component by external constraints (temperature and density gradients), the electromagnetic waves, the plasma kinetic free energy, and the collisional dissipation, which giving the free-energy cascade in terms of the spatial wavevector. In this study, we examine the contribution of the zonal flow on the TEM saturation quantitatively by implementing the entropy balance diagnostics in CGYRO simulations. For a TEM case of a high ratio of the electron density gradient to the electron temperature gradients, the effective zonal flow shearing rate (or the advection rate by the zonal flow) is much lower than the growth rate, so the perpendicular diffusion is important as much as the zonal flow saturation, as shown in the entropy transfer evaluation. It could be useful to quantify the role of the zonal flow and the perpendicular diffusion, and improve the existing saturation model of a reduced model (e.g. TGLF) for the TEM mode.
The BB model is widely used for interpreting the experimental observations of a periodic or a frequency chirping phenomena due to the non-disruptive MHD instability by energetic particles. If the bump-on tail instability is saturated by the external sink with a small collision, the free energy balance equation can be obtained by defining a new effective temperature and using the quasilinear part of the entropy production. The modified free energy balance is still useful to understand the nonlinear phenomena. In the chirping case, we found that the simulation results of the BB model regarding the chirping frequency and the kinetic distribution shape agree with the analysis maximizing the entropy of the background distribution function with the constraints of the external dissipation and Ampere’s law. It implies that the equipartition by entropy increase still sufficiently holds for the nonlinear unsteady process with a small Kubo number.
Stellarator magnetic fields must be optimized to achieve the confinement quality required for fusion reactors. Specifically, in order to exhibit radial neoclassical transport at low collisionality as small as tokamaks, stellarators need to be approximately omnigenous [1]. A magnetic field is omnigenous if, for all particles, the radial component of the drift velocity averages out to zero over the lowest-order orbits. Although radial neoclassical transport has typically been minimized indirectly by means of figures of merit such as the effective ripple [2], recent developments have enabled to carry out this minimization using accurate calculations of radial neoclassical transport at low collisionality [3]. However, not only neoclassical transport across flux-surfaces is important for stellarator optimization. In general, parallel transport produces a net electric current, known as bootstrap current, that can modify the magnetic field and, therefore, needs to be evaluated during the optimization process. This is particularly important if the goal is the design of approximately quasi-isodynamic fields (quasi-isodynamicity is the concept in which the magnetic configuration of Wendelstein 7-X is based on), a subclass of approximately omnigenous fields that have the additional property of giving small bootstrap current [4] and are compatible with island divertors. However, precise calculations of the bootstrap current were too slow to be included in a stellarator optimization loop so far.
In this seminar I will present MONKES (MONoenergetic Kinetic Equation Solver) [5], a new neoclassical code that solves the same monoenergetic drift-kinetic equation as DKES [6], the workhorse of neoclassical transport calculations in stellarators. MONKES was conceived, among other things, to satisfy the need for fast and accurate calculations of the bootstrap current. By exploiting the tridiagonal structure of the drift-kinetic equation in a Legendre basis, it is possible to obtain accurate results for all monoenergetic transport coefficients (i.e. those giving radial as well as parallel transport) at low collisionality using a single core in approximately one minute. These features make MONKES ideal for its inclusion in stellarator optimization suites for direct optimization of the bootstrap current. Apart from optimization, MONKES can be used for the analysis of experimental discharges and be integrated into predictive transport frameworks.
References
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[5] F. J. Escoto et al. Nuclear Fusion 64, 076030 (2024).
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Understanding the impact of tungsten on tokamak performance and its mitigation by low-Z impurities is key to the success of ITER operations. A novel self-consistent gyrokinetic model, integrated into the XGC code [1,2], has been developed to comprehensively analyze tungsten transport and radiation. In this model, the tungsten ions are represented by a few bundles, with their fractional abundance determined by the atomic balance between ionization and recombination processes derived from ADAS rates [3]. The electron cooling by tungsten radiation is also derived from ADAS rates. This new model enables the study of tungsten radiation, transport dynamics, and interactions with low-Z species. Two complementary tungsten studies will be presented. First, we study the impact of nitrogen on the collisional and turbulent peaking factors of tungsten in a WEST plasma.Analysis of tungsten radiation with synthetic diagnostics will be discussed. Second, we explore the intricate interplay between collisional and turbulent transport in an H-mode plasma scenario of ASDEX Upgrade.
[1] Dominski et al. J. Plasm. Phys. (2019)
[2] Dominski et al. Phys Plasmas (2024)
[3] https://www.adas.ac.uk
The Columbia Stellarator eXperiment (CSX), currently in the design phase at Columbia University, is focused on investigating quasi-axisymmetric plasma with a small aspect ratio, and on validating recent developments in stellarator technology, theory, and optimization techniques. It is designed to test some of the theoretical predictions of quasi-axisymmetric plasmas, in particular plasma flow damping, MHD stability properties, and the study of trapped particle confinement. The magnetic field is generated by a set of two circular and planar poloidal field coils (PF coils) alongside two shaped interlinked coils (IL coils), with the potential consideration of additional coils to enhance shaping or experimental flexibility. The PF coils and vacuum vessel are repurposed from the former Columbia Non-Neutral Torus (CNT) experiment [1]. The two IL coils will be wound "in-house" at Columbia University, using non-insulated High-Temperature Superconducting (HTS) tapes. These coils undergo shape and strain optimization to produce the desired plasma configuration while adhering to numerous engineering constraints. Discovering a plasma shape that aligns with the physics objectives and can be produced by such a restricted number of coils poses a significant challenge. Indeed, the constrained coil set’s limited capacity to produce varied plasma shapes hinders the application of the traditional two-stage stellarator optimization approach. Instead, novel single-stage optimization techniques are employed, where plasma and coils are optimized concurrently. Despite an increased problem complexity due to the larger number of degrees of freedom, these methods find optimized plasma shapes that can be generated by coils that satisfy engineering constraints. We discuss two single-stage optimization methodologies[2, 3, 4]. We explore their application to the CSX experiment’s design, aiming to identify configurations that fulfill engineering constraints and generate a plasma within a desired regime for the experiment’s physics objectives.
[1] Pedersen, T. S. et. al. (2006). Construction and Initial Operation of the Columbia Nonneutral Torus. Fusion Science and Technology 50 (3), 372-381
[2] Jorge, R. et. al. (2023). Single-stage stellarator optimization: combining coils with fixed boundary equi libria. Plasma Physics and Controlled Fusion 67 (7), 074003
[3] Giuliani, A. et. al. (2022). Direct computation of magnetic surfaces in Boozer coordinates and coil opti mization for quasisymmetry. Journal of Plasma Physics 88 (4), 905880401
[4] Giuliani, A. et. al. (2023). Direct stellarator coil design using global optimization: application to a compre hensive exploration of quasi-axisymmetric devices. arXiv:2310.19097
New tridimensional plasma structures, that are oscillatory and classified as non-separable ballooning modes, can emerge in inhomogeneous plasmas and undergo resonant mode-particle interactions, e.g., with a minority population, that can lead them to modify their spatial profiles. Thus, unlike the case of previously known ballooning modes their amplitudes are not separable functions of space and time. The relevant resonance conditions are intrinsically different from those of the well-known Landau conditions for (ordinary) plasma waves: they involve the mode geometry and affect different regions of the distribution in momentum space at different positions in configuration space. The novel resonant mode-particle interactions constitute a direct (linear) process to exchange energy between different populations without the inefficiencies of nonlinear coupling processes. The new ballooning modes are relevant to circumbinary disks associated with pairs of black holes and to fusion burning plasmas that include an initially thermal population of fusion reacting nuclei and a population of high energy nuclei (reaction products). It is reasonable to expect that the distributions of the reacting nuclei in momentum space will not remain strictly Maxwellian and that the resulting reaction rates will be different from those evaluated for (conventional) thermalized plasmas.
In this talk we summarize some of our recent work of interest to PPPL researchers, in two parts:
Part I: First implementation of gyrokinetic exact linearized Landau collision operator and comparison with models. Previous gyrokinetic simulations have used model collision operators with approximate field-particle terms of unknown accuracy and/or have neglected collisional finite Larmor radius effects. This work demonstrates significant corrections using the first formulation [1, 2] and implementation [3, 4] of the gyrokinetic Fokker-Planck-Landau collision operator with the exact linearized field-particle terms. Realistic nonlinear gyrokinetic simulations of fusion plasma turbulence show significant corrections relative to the Sugama model collision operator for temperature-gradient-driven trapped electron mode turbulence and zonal flow damping, and for microtearing modes (the exact operator is now released in GENE-3.0). Future work will extend novel spectral methods implemented for the drift-kinetic operator [5] to the gyrokinetic operator.
Part II: Broadening of the Divertor Heat Flux Profile in DIII-D QH-Modes, Matched by XGC. Multi-machine empirical scaling predicts an extremely narrow heat exhaust layer in future high magnetic field tokamaks, producing high power densities that require mitigation. In the experiments presented [6], the width of this exhaust layer is nearly doubled using actuators to increase turbulent transport in the plasma edge. This is achieved in low collisionality, high confinement edge pedestals with their gradients limited by turbulent transport instead of ELMs or low-n MHD modes. The exhaust heat flux profile width and divertor leg diffusive spreading both double as a high frequency band of (TEM) turbulent fluctuations propagating in the electron diamagnetic direction doubles in amplitude. The results are quantitatively reproduced in electromagnetic XGC particle-in-cell simulations which show the heat flux carried by electrons emerges to broaden the heat flux profile, directly supported by Langmuir probe and infra-red imaging measurements.
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[3] Q. Pan, D. R. Ernst, and D. Hatch, Phys. Rev. E Lett. 103, L051202 (2021). https://doi.org/10.1103/PhysRevE.103.L051202
[4] Q. Pan, D. R. Ernst and P. Crandall, Physics of Plasmas 27, 042307 (2020). https://doi.org/10.1063/1.5143374
[5] M. Landreman and D. R. Ernst, J. Comput. Phys. 243, 130 (2013). https://doi.org/10.1016/j.jcp.2013.02.041
[6] D. R. Ernst, A. Bortolon, C. S. Chang, S. Ku et al., Phys. Rev. Lett. (2024) accepted for publication. https://arxiv.org/abs/2403.00185
Particle methods for kinetic simulations are numerically stable, easy to use, convenient for irregular geometries. The main challenge is the stochastic noise that could make the simulations unaffordable in low-speed problems (i.e., low signal-to-noise ratio) as well as transient problems where the time-averaging scheme is invalid. A lot of efforts have been made on the noise reduction by modifying the traditional direct simulation Monte Carlo (DSMC) method in solving the Boltzmann-like equations. At this presentation, I will introduce the direct simulation BGK (DSBGK) method, which can solve BGK-like equations as good approximations to the Boltzmann equation in many problems. As a duality of the DSMC method and the lattice Boltzmann method (LBM), the DSBGK method adopts a large number of simulated particles to represent the distribution function in the phase space, as used in the DSMC method but different from the LBM, while it updates the variables of each particle by integration of the kinetic equation along the corresponding trajectory, as modeled in the LBM but different from the DSMC method. The increments of particles’ variables inside each cell during each time step are obtained by the integration and the corresponding summations are used to regulate (not recompute) the macroscopic variables of the cell concerned, according to the mass, momentum and energy conservation laws. The previous values of cell’s variables are kept as anchors in the auto-regulation scheme to significantly reduce noise associated with the particles’ random movements into and out of each cell. Simulation results in several problems will be presented to show the noise reduction as well as the accuracy validation. Performance comparison with other particle and deterministic methods will be discussed.
Resonant interactions between runaway electrons (REs) and whistler waves in a tokamak may lead to pitch angle scattering of the REs. An increase in RE pitch angles may give rise to the energy dissipation of the runaways via synchrotron radiation. DIII-D experiments on whistler waves have indicated a possibility of intentionally launching whistler waves to mitigate the deleterious effects of REs on the plasma facing components via resonant interactions with whistlers [1,2]. In present work, we have use the coupled KORC-AORSA model to numerically analyze the complex nature of the interactions between whistler waves and runaway electrons in DIII-D. In this framework, we follow full orbit trajectories of large RE ensembles using the Kinetic Orbit Runaway Electron (KORC) code in the presence of whistler wave fields calculated by All Orders Spectral Algorithm (AORSA) code in a DIII-D experimental equilibrium. The nature of RE transport (diffusive/non-diffusive) [3] is analyzed in the presence of whistler fields and the impact of whistler field amplitudes and frequencies is observed on the pitch angle scattering of REs. Our findings indicate a significant increase in RE energy and scattering of the runaways to large pitch angles for whistler fields exceeding a threshold amplitude. The coupled KORC-AORSA simulation model can be further used to get physical insights into tokamak experiments on whistler waves- REs interactions.
[1] D. A. Spong et al., Phys. Rev. Lett., 120, 155002 (2018).
[2] W. W. Heidbrink at al., Plasma Phys. Control. Fusion, 61, 14007 (2019).
[3] D. del-Castillo-Negrete, Phys. Plasmas, 13, 082308 (2006).
A comprehensive understanding of electromagnetic effects on the microinstability properties of tokamak plasmas is becoming increasingly important as experimental values of the plasma beta and, therefore, electromagnetic fluctuations will be higher in reactor-relevant tokamak scenarios. Despite significant numerical progress in understanding the behaviour of instabilities such as the micro-tearing mode (MTM) or kinetic ballooning mode (KBM), there is still a lack of clarity about the fundamental physical processes that are responsible for them, owing to the complexity of full toroidal geometry. Constructing simplified models offers a path towards distilling the fundamental physical ingredients behind electromagnetic destabilisation. This talk focuses on electromagnetic instabilities driven by the electron-temperature gradient (ETG) in a local 'toy' model of a tokamak-like plasma. The model has constant equilibrium gradients (including magnetic drifts, but no magnetic shear) and is derived in a low-beta asymptotic limit of gyrokinetics. A new instability is shown to exist in the electromagnetic regime, the so-called 'thermo-Alfvénic instability' (TAI), whose physical mechanism hinges on a competition between diamagnetic drifts (due to the ETG) and rapid parallel streaming along perturbed field lines. Using linear gyrokinetic simulations, the TAI's presence is confirmed in slab geometry. The mapping of the TAI onto a more realistic tokamak equilibrium is considered, demonstrating that it survives aspects of the transition to toroidicity. A comparison is then drawn with the properties of the MTM and KBM, contextualising the TAI within the wider 'zoo' of electromagnetic instabilities commonly observed in tokamak simulations.
High energy particle resonances play an important role in particle confinement in toroidal fusion devices, both tokamaks and stellarators. In stellarators a resonance that matches the periodicity of the equilibrium field produces islands in particle orbits which increase in size with particle energy and can induce loss. As demonstrated in the Japanese stellarator LHD, the presence of a high frequency resonance invariably gives rise to a strong Alfven mode that causes particle loss. In a nonsymmetric stellarator a resonance does not produce a well defined island structure in the orbits, but typically scatters ten percent of orbits of all energies and pitch randomly, modifying mode growth and saturation properties. Avoiding the presence of high energy particle resonances should be a part of device design.
Recently, a flurry of activities has been carried out on the isotopic effects in JT-60U [1], JET [2] and DIII-D [3], which has shown favorable confinement trend for heavier hydrogen isotopes. The consensus from these experimental observations was that this is an unsolved puzzle in tokamak plasmas. This is in fact not quite accurate. When the favorable effects was first observed on TFTR [4,5] for hydrogen, deuterium and tritium experiments, a theoretical attempt was indeed made to understand the results by Lee and Santoro [6]. Apparently, this paper has not attracted much attention in the community. Recently, a paper by Lee and White [7] on the H-mode physics has also described the isotope effects at the H-mode pedestal. In this talk, the theoretical interpretations on these isotopic effects based on 1) the resonance broadening theory [8] in the core as well as 2) the force balance equation for the pedestal from the gyrokinetic theory [7] will be described. The implementation of the related physics in an initial value code such as GTC [9] and/or GTS [10] will also be discussed.
[1] H. Urano and E. Narita , Plasma Phys. Control. Fusion 63, 084003 (2021)
[2] L. Horvath, C. F. Maggi, A. Chankin et al., Nuclear Fusion 61, 046015 (2021)
[3] L. Schmitz, Phil. Trans. R. Soc. A381: 20210237 (2022)
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[5] S. D. Scott, G. W. Hammett, C. K. Phillips et al., IAEA-CN-64/A6-6 (1997)
[6] W. W. Lee and R. A. Santoro, Phys. Plasmas 4, 169 (1997)
[7] W. W. Lee and R. B. White, Phys. Plasmas 26, 040701 (2019)
[8] T. H. Dupree, Phys. Fluids 11, 2680 (1968)
[9] Z. Lin, T. S. Hahm, W. W. Lee et al., Science 281, 1835 (1998).
[10] W. X. Wang, Z. Lin, W. M. Tang, W. W. Lee et al., 13, 092505 (2006)
The first principle gyrokinetic numerical experiments investigating the isotopic dependence of energy confinement achieve a quantitative agreement with experimental empirical scaling, particularly in Ohmic and L-mode tokamak plasmas. Mitigation of turbulence radial electric field intensity |δEr|2 and associated poloidal δE×B fluctuating velocity with the radial correlation length l_cr ∝ Mi^0.11 strongly deviating from the gyro-Bohm scaling is identified as the principal mechanism behind the isotope effects. Three primary contributors are classified, the deviation from gyro-Bohm scaling, zonal flow and trapped electron turbulence stabilization. Zonal flow enhances isotope effects primarily through reinforcing the inverse dependence of turbulence decorrelation rate on isotope mass with ω_c ∝ Mi^-0.76, which markedly differs from the characteristic linear frequency. The findings offer insights into isotope effects, providing critical implications for energy confinement optimization in tokamak plasmas.
The generation of small scale, mean or large scale magnetic fields in cosmos and astrophysical bodies is an important problem in astrophysical plasmas. A possible mechanism behind these multi scale magnetic energy growth is explained via dynamo action. Shear flows [1] often coexist in astrophysical conditions and the role of flow shear on the onset of dynamo is only beginning to be investigated. The paradigm of investigation of the exponential growth of magnetic field caused by the interaction of small-scale velocity fluctuations and a flow shear; is commonly referred to as the “shear dynamo problem” [2]. Various laboratory experiments [3], as well as numerical studies have been performed to understand these astrophysical scenarios in detail. According to conventional understanding, for a large scale or mean field dynamo, a lack of reflectional symmetry (e.g., non-zero fluid or kinetic helicity) is required, where as for small scale or fluctuation dynamo it is not. Obviously the role of fluid or kinetic helicity on the onset of dynamo action is a sensible question to ask.
In this present work we have analyzed kinematic dynamo model i.e, a case wherein (magnetic field does not back-react on velocity field) using a flow recently proposed by Yoshida and Morrison (YM) [4]. An interesting and useful aspect of this flow is that, it is possible to inject finite fluid helicity in the system, by systematically varying certain physically meaningful parameter. Using direct numerical simulation, we demonstrate that by systematically injecting finite fluid helicity, a systematic route emerges that connects “non-dynamo” to “dynamo” regime [5]. Time-averaged magnetic energy spectrum, for various magnitudes of injected fluid helicity is calculated and it is observed that, the spectra contain a visible maxima at a higher mode number, which is the distinguishing feature of small scale dynamo (SSD) [5]. However for a nonlinear dynamo or self-consistent dynamo model, the nonlinear effects start to change the flow (once the magnetic field is large enough) to stop further growth in magnetic field energy, i.e, the flow and magnetic field “back react” on each other. The influence of helical and non-helical drive in such a nonlinear or self-consistent dynamo model is shown to have some crucial dynamics [6]. Evidence of small-scale dynamo (SSD) activity is found for both helical and non-helical drives [6]. The spectrum analysis shows that the kinetic energy evolution adheres to Kolmogorov’s k^−5/3 law, while the magnetic energy evolution follows Kazantsev’s k^3/2 scaling. These scalings are observed to be valid for a range of magnetic Prandtl numbers (Pm) [6]. We have performed the above said studies using an in-house developed, multi-node, multi-card GPU based weakly compressible 3D Magnetohydrodynamic solver (GMHD3D) [7, 8]. Details of this study will be presented.
References:
[1] S. Biswas & R. Ganesh, Phys. Fluids 34, 065101 (2022).
[2] S. Biswas & R. Ganesh, Phys. Plasmas 30, 112902 (2023).
[3] R. Monchaux, M. Berhanu, et al., Phys. Rev. Lett. 98, 044502 (2007).
[4] Z. Yoshida & P. J. Morrison, Phys. Rev. Lett. 119, 244501 (2017).
[5] S. Biswas & R. Ganesh, Physica Scripta, Volume 98, Number 7.
[6] S. Biswas & R. Ganesh, Manuscript under Preparation (2024).
[7] S. Biswas, R. Ganesh et al. “GPU Technology Conference (GTC-2022)”,
https://www.nvidia.com/en-us/on-demand/session/gtcspring22-s41199/.[8] S. Biswas & R. Ganesh, Computers and Fluids 272 (2024) 106207.
Future devices like ITER will have limited capacity to drive toroidal rotation, increasing the risk of instabilities like resistive wall modes. Fortunately, many experiments have found that tokamak plasmas rotate “intrinsically”, that is, without applied torque. The modulated-transport model shows that such rotation may be caused by the interaction of ion drift-orbit excursions with the strong spatial variation of the turbulent momentum diffusivity [1]. The model predicts intriguing qualitative behavior, such as a strong dependence of edge intrinsic toroidal rotation on the major-radial position of the X-point, which was subsequently measured on TCV [2]. The model has also been experimentally validated through further dedicated tests [3, 4], as well as via application in the new European whole-device transport model IMEP [5]. However, certain applications will require a relaxation of the underlying assumptions. In particular, the original model required the turbulent momentum diffusivity to decay exponentially in the radial direction, while experiments often exhibit a more complicated variation. In this work, we generalize the modulated-transport model to allow the turbulent momentum diffusivity to depend on space in an axisymmetric but otherwise arbitrary way. To enable this generality, we assume that the normalized diffusivity is weak, roughly equivalent to assuming that the pedestal-top ion transit time is short compared to the transport time across the pedestal, a condition that is almost always met for experimental applications. Given the increased flexibility, along with a technically much easier calculation, the new approach may serve as a basis for future extensions, including shaped geometry and trapped particles as well as the retention of momentum transport by neutrals.
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[3] J. A. Boedo et al., Phys. Plasmas 23, 092506 (2016).
[4] A. Ashourvan, B. A. Grierson, D. J. Battaglia, S. R. Haskey, and T. Stoltzfus-Dueck, Phys. Plasmas 25, 056114 (2018).
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Axisymmetric modes in elongated plasmas are normally associated with a well-known ideal instability resulting in a vertical shift of the whole plasma column. This vertical instability is stabilized by means of passive feedback consisting of eddy currents induced by the plasma motion in a nearby wall and/or in plasma-facing components. When a thin resistive wall is considered, the n=0 mode dispersion relation can be studied analytically with reduced ideal MHD models and is cubic. Under relevant conditions, two roots are oscillatory and weakly damped. These oscillatory modes present Alfvénic frequency and are dependent on plasma elongation and on the relative position of the plasma boundary and of the wall. The third root is unstable and represents the so- called resistive wall mode (RWM) [1]. We focus on the two oscillatory modes, dubbed Vertical Displacement Oscillatory Modes (VDOM), that can be driven unstable due to their resonant interaction with energetic ions.
The fast ion drive, involving MeV ions in present days tokamak experiments such as JET, may overcome dissipative and resistive wall damping, setting an instability threshold, as described in Ref. [2]. The effects of energetic particles are added within the framework of the hybrid kinetic-MHD model. An energetic ion distribution function with ∂F/∂E > 0 is required to drive the instability, achievable with pitch angle anisotropy or with an isotropic distribution in velocity space with regions of positive slope as a function of energy. The latter situation can be achieved by considering losses of fast ions or due to fast ion source modulation [3-4]. The theory presented here is partly motivated by the observation of saturated n=0 fluctuations reported in [4,5], which were initially interpreted in terms of a saturated n=0 Global Alfvén Eigenmode (GAE). Modeling of recent JET discharges using the NIMROD [6] extended-MHD code will be presented, focusing on mode structure and frequency dependence. It is early for us to conclude whether the mode observed at JET is a VDOM rather than a GAE, nevertheless, we discuss the main points of distinction between GAE and VDOM that may facilitate their experimental identification.
References
[1] T. Barberis, et al. 2022, J.Plasma Phys. 88, 905880511
[2] T. Barberis, et al 2022 Nucl. Fusion 62 06400
[3] Ya.I. Kolesnichenko and V.V. Lutsenko 2019 Nucl. Fusion 59 126005
[4] V. G. Kiptily et al 2022 Plasma Phys. Control. Fusion 64 064001
[5] H. J. C. Oliver et al. 2017 Phys. Plasmas 24, 122505
[6] C. Sovinec et al. and the NIMROD Team 2004 J. Comp. Phys. 195 355
We discuss recent progress in understanding the role of transport physics in density limit phenomena. Our approach is one which combines theory and experiment. Contrary to the conventional wisdom that the density limit is enforced by MHD instability, findings indicate that the L−mode density limit is associated first with the degradation of the edge E × B shear layer. The latter occurs for k‖² vₜₕₑ² / ω νₑᵢ<1. Shear layer decay leads to strongly enhanced turbulence spreading and increased production of density 'blobs'. Interestingly, the spreading flux increases more rapidly with increasing n / n_G than does the particle flux. Shear layer decay is linked to a decline in zonal flow production.
A simple model for flow, fluctuation and density evolution reveals that the edge density will increase with edge heat flux (power). This favorable trend results from increased Reynolds stress flow drive at higher power. It provides physical insight into the power scaling of density limit, now observed in experiments. A scaling of n ∼ P^(1/3) is suggested for the case of ITG turbulence.
We briefly discuss recent density limit experiments in negative triangularity plasmas, as well as aspects of the H−mode density limit phenomenon. Implications for burning plasma are discussed.
Contributions from Ting Long, SWIP ; Rameswar Singh ,UCSD ; Rongjie Hong, UCLA and DIII−D ; Zheng Yan, Univ Wisc and DIII−D ; and George Tynan, UCSD are acknowledged.
Starting from the assumption that saturation of plasma turbulence driven by temperature-gradient instabilities in fusion plasmas is achieved by a local energy cascade between a long-wavelength outer scale, where energy is injected into the fluctuations, and a small-wavelength dissipation scale, where fluctuation energy is thermalized by particle collisions, we formulate a detailed phenomenological theory for the influence of perpendicular flow shear on magnetized-plasma turbulence. Our theory introduces two distinct regimes, called the weakly and strongly sheared regimes, each with its own set of scaling laws for the scale and amplitude of the fluctuations and for the level of turbulent heat transport. We discover that the ratio of the typical radial and poloidal wavenumbers of the fluctuations, i.e., their aspect ratio, plays a central role in determining the dependence of the turbulent transport on the imposed flow shear. Our theoretical predictions are found to be in excellent agreement with numerical simulations of two models of magnetized plasma turbulence: (i) an electrostatic fluid model of slab electron-scale turbulence, and (ii) Cyclone-base-case gyrokinetic ion-scale turbulence.
We present the first local delta-f nonlinear gyrokinetic (GK) simulations based on a gyro-moment (GM) approach, which exploits the projection of the distribution functions onto a Hermite-Laguerre velocity- space basis. We first demonstrate that, in contrast to gyrofluid models, the GM approach reproduces the Dimits shift, notably, with a coarser velocity space resolution than the continuum GK GENE code. In addition, we reveal that the choice of collision operator model (Dougherty, Sugama, Lorentz and Landau) significantly impacts the level of turbulent transport through multi-species zonal flow damping.
In addition, we show for the first time that the GM approach is able to bridge the gap between GK and reduced fluid modelling by its exact equivalency to the model of Ivanov et al. 2020 when considering the same limits. Leveraging its efficiency and multi-fidelity capability, we finally use the GM approach to explore the impact of triangularity in realistic DIII-D edge conditions across a range of models, spanning from GK electron-ion multi-scale simulations to the reduced fluid limit.
Turbulence is one of the key ingredients in shaping H-mode pedestals. Identifying the relevant turbulent transport mechanisms in a pedestal, however, is a great scientific and numerical challenge. Here, we address this challenge by global, nonlinear gyrokinetic simulations of two pedestals: One from ASDEX Upgrade (Type-I ELMy H-mode) and one from JET (hybrid scenario H-mode). The global simulations permit to calculate heat fluxes due to ion-scale turbulence in the steep gradient region encompassing the full pedestal from top to foot. They are supported by detailed characterizations of gyrokinetic instabilities via local, linear simulations at pedestal top, center and foot as well as dedicated nonlinear electron-scale heat flux calculations. Simulations are performed with the gyrokinetic, Eulerian, delta-f code GENE (genecode.org) and employ a new code upgrade of its global, electromagnetic model that enables stable simulations at experimental plasma beta values.
In both investigated pedestals from AUG and JET, we find turbulent transport to have a complex radial structure that is multi-scale and multi-channel. Electron transport in the AUG pedestal is found to transition in scale. At the pedestal top ion-scale TEM/MTM instabilities fuel electron transport whereas in the pedestal center electron-scale ETG transport takes over. Turbulent ion heat flux is present at the pedestal top and strongly reduces towards the steep gradient region. Magnetic shear is found to locally contribute to the stabilization of microinstabilities and reduction of heat flux. In the JET pedestal, transport due to ITG is found to play a much more important role, particularly on the pedestal top/ outer core. In both pedestals, ExB shear is confirmed to strongly reduce heat fluxes in the global, nonlinear simulations. We discuss implications of our results for the applicability of quasi-linear transport models in the pedestal.
Understanding the formation of large-scale structures in weakly magnetized plasmas represents a crucial step towards developing predictive design capabilities for E×B devices dedicated to investigating fundamental plasma physics phenomena. MISTRAL is such a device based at PIIM laboratory to study plasmas in cross-field configuration (E⊥B). The formation of coherent rotating structures in MISTRAL is supposed to be due to an interplay between various instabilities and the E×B flow. However, a definitive understanding of which instabilities are accountable for their emergence and the specific triggers involved remains elusive. An experimental investigation of MISTRAL plasmas has been performed to lay the basis for the theoretical modeling. A two-fluid model has been developed to discuss the linear stability of rotating plasma columns. Prior works have demonstrated that rotating plasma columns are susceptible to centrifugal flute modes. However, most of the existing models rely on the low-frequency approximation (LFA), which holds true when the instability frequency and equilibrium flow frequency are considerably smaller than the ion-cyclotron frequency. This assumption is challenged in numerous laboratory plasma devices, including weakly magnetized plasma columns like MISTRAL. To address this limitation, a radially global dispersion relation describing the centrifugal instability without the LFA has been derived and linear stability analysis is performed. A comparison has been made between the results obtained using the dispersion relation with the radially local approximation and those obtained using the radially global dispersion relation. This comparison revealed the non-applicability of the local solution to MISTRAL-like plasma systems. Due to the high fraction of neutrals in the present plasma system, the model is further extended to include the effects due to ion-neutral collisions. In this first step, the ion-neutral collision frequency is assumed to be small as compared to the ion-cyclotron frequency. The dispersion relation is then solved with finite ion-neutral collisionality and the linear stability analysis is conducted.
In magnetic confinement fusion plasmas, many instabilities have a flute mode character. The field-aligned coordinates bring the benefit of efficient resolution of parallel mode structure along the magnetic field direction. However, the curvilinear coordinates make equations and codes more complex especially in high order PDE.
The Compile-time Symbolic Solver (CSS) is developed to solve PDEs and ODEs in finite difference method from vector equations directly. CSS is a general-purpose finite difference framework for generating finite difference codes easily and greatly reducing the risk of implementation mistakes.
For physics model, CSS supports arbitrary equations in arbitrary curvilinear coordinates and multiple boundaries for both PDEs and ODEs. For memory distribution, N-dimension distribution grids with hybrid TBB and MPI parallelization in arbitrary dimensions are implemented. For numerical method, CSS employs Method of Line in numerical difference with arbitrary grid points and offset. The N-dimension B-spline is implemented with arbitrary orders for pushing particles. CSS employs PARDISO to solve matrix problem and Runge–Kutta method for time advance. CSS is a C++20 template metaprogramming code which guarantee zero-overhead at runtime. Furthermore, the instruction optimization makes the codes generated by CSS much faster than usual codes.
We have used CSS to generate the Gyrokinetic-MHD Hybrid Code GMEC, 3D Field and the Particle calculation code FP3D and a fluid ITG code. For GMEC, we propose a new shifted metric method which is able to stabilize numerical instabilities and avoid the interpolation from MHD field-align grids to particle flux coordinate grids at the same time. The equilibriums can be analytical ones or numerical ones calculated by VMEC or DESC. We have used GMEC to simulate ballooning modes (IBM) with or without the diamagnetic drift term and tearing modes. The simulation results agree well with those of the eigenvalue code MAS. The n=20 IBM costs only 17 seconds using 448 cores. We have also used GMEC to simulate energetic particle-driven TAEs in a circular equilibrium and a CFETR equilibrium. The results of an n=3 TAE agree well with those of M3D-K code.
We have also used CSS to generate the test particle code FP3D for calculation of magnetic surfaces, rotation transform, particle orbits and neoclassical transport in both tokamaks and stellarators. We have used FP3D to simulate ripple losses in EAST tokamak and neoclassical transport coefficient in NCSX. The results are consistent with previous results. FP3D has been used in design and optimization of stellarators successfully.
[1] P. Y. Jiang, et al. CSS: Compile-time symbolic solver for finite difference method. To be submitted.
[2] P. Y. Jiang, et al. GMEC: Gyrokinetic-MHD Hybrid Code. To be submitted. [3] P. Y. Jiang, Z. C. Feng, G. D. Yu, and G. Y. Fu, FP3D: A code for calculating 3D magnetic field and particle motion. Submitted to POP.Accurately predicting lower hybrid current drive (LHCD) in the weak-damping regime is an outstanding challenge, which suggests important physics is missing in present-day ray-tracing/Fokker-Planck (RTFP) models. In this work, the impact of filamentary scrape-off layer (SOL) turbulence on LH waves is investigated using a new multi-scale scattering model. When coupled to an RTFP code, the resulting simulations of LHCD in Alcator C-Mod show RF power deposition profiles robustly peaked on-axis, leading to good agreement with experimental Motional Stark Effect and hard X-ray measurements. Therefore, it is shown that the rotation of the perpendicular wave-vector due to SOL turbulence is sufficient to bridge the discrepancy between simulation and experiment. Notably, this model predicts an asymmetric broadening of the transmitted wave-spectrum, which is attributed to full-wave scattering effects in the presence of spatially coherent turbulence. This asymmetry leads to rotation of incident power away from the plasma core when SOL densities are sufficiently high. RTFP modeling shows this effect plays a significant role in the anomalous drop in LHCD efficiency observed at high densities.
The multi-scale scattering model has two steps. (1) Single filament-wave interactions are solved in full-wave formalism using a Mie-scattering technique. (2) Multiple of these filament-wave interactions are modeled using the radiative transfer approximation, in which a photon’s scattering probability depends on the statistical properties of the filament population. The radiative transfer equation (RTE) is then solved using a Monte Carlo scattering term in a ray-tracing model, allowing for self-consistent coupling to RTFP codes. For verification and comparison against other models, the RTE is also solved in a simple slab geometry using a Markov chain. This model shows good agreement with ray-tracing in the Wentzel-Kramer-Brillouin (WKB) limit, and predicts greater, asymmetric scattering beyond the WKB limit. Good agreement is also found with numeric full-wave solutions at sufficiently low filament packing-fraction, which is consistent with the validity limit of the radiative transfer approximation.
It should be emphasized that this multi-scale scattering model retains many important full-wave effects while remaining computationally inexpensive, allowing fast parameter scans and inter-shot analysis. In addition, this model is highly applicable to the modeling of electron cyclotron wave scattering since the radiative transfer approximation is increasingly valid for waves at higher k.
B. Biswas et al., “Spectral broadening from turbulence in multiscale lower hybrid current drive simulations,” Nuclear Fusion, 63, 1 (2022).
B. Biswas et al., “A hybrid full-wave Markov chain approach to calculating radio-frequency wave scattering from scrape-off layer filaments,” Journal of Plasma Physics, 87, 5 (2021).
This is joint work with M. O'Neil, L. Greengard, and L.-M. Imbert-Gerard
Since the work of Sauter in 1931 it is known that quantum electrodynamics (QED) exhibits a so-called "critical" electromagnetic field scale, at which the quantum interaction between photons and macroscopic electromagnetic fields becomes nonlinear. One prominent example is the importance of light-light interactions in vacuum at this scale, which violates the superposition principle of classical electrodynamics. Furthermore, an electromagnetic field becomes unstable in this regime, as electron-positron pairs can be spontaneously created from the vacuum at the expenses of electromagnetic-field energy (Schwinger mechanism). Unfortunately, the QED critical field scale is so high that experimental investigations are challenging. One promising pathway to explore QED in the nonlinear domain with existing technology consists in the combination of modern (multi) petawatt optical laser systems with highly energetic particles. The suitability of this approach was first demonstrated in the mid-1990s at the seminal SLAC E-144 experiment. Since then, laser technology continuously developed, implying the dawn of a new era of strong-field QED experiments. For instance, the basic processes nonlinear Compton scattering and Breit-Wheeler pair production are expected to influence laser-matter interactions and in particular plasma physics at soon available laser intensities. Therefore, a considerable effort is being undertaken to include these processes into particle-in-cell (PIC) codes used for numerical simulations.
In the first part of the talk the most prominent nonlinear QED phenomena are presented and discussed on a qualitative level. Afterwards, the mathematical formalism needed for calculations with strong plane-wave background fields is introduced with an emphasize on fundamental concepts. Finally, the nonlinear Breit-Wheeler process is considered more in depth. In particular, the semiclassical approximation is elaborated, which serves as a starting point for the implementation of quantum processes into PIC codes.