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- Indico style - inline minutes
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- Indico Weeks View
The aim of the conference is to provide a discussion forum covering all areas of fusion-oriented theoretical activities.
Its main topics are:
D. Tskhakaya1, I. Borodkina1, O. Shyshkin1
1Institute of Plasma Physics of the CAS, Za Slovankou 3, 182 00 Prague 8, Czech Republic
Plasma transport modelling in the Scrape-Off Layer (SOL) represents one of the most complex numerical studies used in magnetic confinement fusion. These models require interdisciplinary approach to treat plasma, neutral and impurity particle dynamics and their nonlinear interaction with the plasma facing components of the fusion device [1]. Mentioned processes in the SOL influence plasma and power exhaust and hence, define plasma discharge performance and life-time of plasma facing components [2]. At present, the main numerical tools for SOL study represent fluid transport and kinetic codes; although new developments in edge turbulence fluid and gyrokinetic codes brought first impressive results (e.g. see [3, 4]). SOL fluid models are capable to treat realistic magnetic equilibria and wall geometry with relatively lower computing power requirements, but cannot self-consistently treat kinetic effect. The latter are usually provided via simplified full kinetic modelling of the SOL.
In the present work we describe the principles of kinetic and fluid modelling of the SOL and their limitations; discuss a number of long-standing problems, such as multi-dimensional boundary conditions, non-local heat transport, high density plasma edge effects; as well as consider number of new kinetic effects: finite source, kinetic drift and inverse temperature gradients at the wall. We estimate their influence on plasma transport in the SOL and discuss possible ways of their implementation into the fluid and gyro-kinetic SOL transport models.
References:
[1] P.S. Stangeby, ”Plasma Boundary of Magnetic Fusion Devices”, IOP Publishing, Bristol (2000).
[2] R.A. Pitts, et al, Nucl. Mater. Energy 20, 100696 (2019)
[3] A. Coroado and P. Ricci, Nucl. Fusion 62, 036015 (2022)
[4] D. Michels, et al., Physics of Plasmas 29, 032307 (2022)
[5] D. Tskhakaya, et al., Contrib. Plasma Phys. 48, 89–93 (2008)
Impurity seeding studies were performed for the first time in the slot divertor at DIII-D, showing that with suitable use of radiators, full detachment is possible without degradation of core confinement [1]. First ever multi species SOLPS-ITER simulations including full cross-field drifts and neutral-neutral collisions activated in DIII-D demonstrate the importance of target shaping and plasma drifts on divertor impurity leakage. The inclusion of the drifts in the
simulations enabled to study the behavior of these flows in a highly closed divertor showing the relevant role of convection on divertor asymmetry and divertor detachment in these conditions. The recycling source is affected by the superposition of the closure effect and plasma drifts. This results in a redistribution of plasma flow in the SOL and divertor plasma. Flow reversal [2] is found for both main ions and impurities affecting the SOL impurity transport and explaining the
dependence on strike point location of the detachment onset and impurity leakage found in the experiments. In addition to target shaping, the effect of different radiative species on power dissipation has been evaluated by replacing nitrogen with neon. The experimental results show
that Ne dissipates further upstream than N as confirmed by SOLPS-ITER modeling and analytic calculations using the 2-point model [3]. The two routes for dissipation identified here (using N through divertor radiation and with Ne radiating mantle upstream) lead to different pedestal responses. While Ne readily enters the pedestal, N remains compressed in the divertor without significantly affecting the profiles. This different leakage behavior is consistent with the higher
ionization potential for Ne compared to N. Neon injection leads to a reduced core ion transport as supported by TRANSP and GYRO simulations. The resulting increase in pedestal Ti improves pedestal stability through increased diamagnetic stabilization allowing higher pedestal pressure gradients. A self-enhancing mechanism of Ne build up has been identified as due to the increased pedestal stability and the radiative mantle. The findings of this work demonstrate that enhanced divertor dissipation and improved core-edge compatibility can be obtained by choosing appropriate radiative species for pedestal conditions, as well as by optimizing divertor geometry and tailoring drifts for particle entrainment.
Work supported by US DOES under DE-FC02-04ER54698 (DIII-D), DE-AC52-07NA27344 (LLNL), DE-AC02-09CH11466 (PPPL), and DE-AC05-00OR22725 (ORNL), DE-NA0003525 (SNL) and LDRD project 17-ERD-020.
[1] L. Casali et al. Phys. Plasmas 27, 062506 (2020)
[2] L. Casali et al. Nucl. Fusion 60 076011 (2020)
[3] L. Casali et al. Nucl. Fusion 62 026021 (2022)
On the road to fusion energy production, many efforts have been done in order
to address the well-known problem of anomalous transport in tokamak devices.
Understanding and predicting this phenomenon is a key issue towards the development of future fusion reactors. The understanding of turbulent transport has made tremendous
progresses in the last decade thanks to dedicated experimental campaigns, analytical modeling and huge efforts in numerical simulations. Numerical simulations allow us to analyse individually specific features and help us to understand and interpret experimental data. However, we have reached a modelling complexity that makes the simulations extremely challenging: somewhere this complexity has to be cut to obtain results with current computational resources. This does not come for free and each approximation hides some precious information that may be needed to describe experimental observations.
Here, we focus on global effects that can only be observed when the full torus is simulated. We show that in global simulations the Fick law is often an incorrect approximation and that the transport is frequently super-diffusive. This super-diffusivity is typically a result of the overlapping of local transport and avalanche-mediated transport triggered in another plasma region. \
Among the several features impacting transport non-locality, there are the well-known finite $\rho^*$ effects, the safety factor profile, and the dynamics of electrons. All of these factors will be discussed in the talk.
Boundary conditions as well, a critical issue of gyrokinetic simulations, turn out to be an important source of non-locality: the whole system is very sensitive to the way fluctuations are dissipated at the edge. Clearly it is possible to work in a simplified setup, like the gradient-driven simulations with just core turbulence, where the boundary conditions are less critical but this does not reflect the complexity of a real device.
Equilibrium geometry may also affects non-local properties and transport reduction of certain equilibria may be closely related to non-local effects; consider experiments with negative triangularity, where turbulent fluctuations and transport are reduced compared to the positive triangularity case, even well into the plasma core, where the difference in triangularity is negligible.\
In the talk, we mostly focus on these global effects comparing simulations performed on two TCV equilibria, corresponding to positive and negative triangularity configurations.
Turbulence-driven transport is still one of the main obstacles to overcome in order to obtain feasible thermonuclear reactors. For this reason, the microinstabilities that are found to drive turbulence have been extensively studied in the last decades, both analytically and numerically. In such studies, assumptions about plasma parameters and magnetic geometry are generally made, making difficult the discovery of any properties that might hold more generally.
Recently, it was shown by Helander and Plunk [1,2] that it is possible to obtain universal upper bounds on the growth rates of all local gyrokinetic instabilities via thermodynamic considerations. These bounds are valid for all possible microinstabilities that can be found both in tokamaks and stellarators. Some examples are ion- and electron-temperature-gradient-driven modes, trapped-electron modes, kinetic ballooning modes and microtearing modes. Moreover, these bounds are independent of the magnetic field configuration and some plasma parameters, such as the number of particle species, beta and collisions.
The validation of the upper bound for a hydrogen plasma with adiabatic electrons has already proved successful [3]. In particular, a comparison with results from linear, flux-tube gyrokinetic simulations has been carried out considering different magnetic field geometries, including various stellarator and tokamak configurations. The simulations have been performed with the gyrokinetic code stella [4]. The validation also included a comparison with analytical results. Here we extend the validation to the more general case in which electrons are treated kinetically [5].
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200— EUROfusion). Views and opinions expressed are however those of the author(s) only
and do not necessarily reflect those of the European Union or the European Commission. Neither the
European Union nor the European Commission can be held responsible for them.
References:
[1] P. Helander, and G. G. Plunk, Physical Review Letters 127, 155001 (2021)
[2] P. Helander, and G. G. Plunk, Journal of Plasma Physics 88, 905880207 (2022)
[3] L. Podavini, et al., “Validation of theoretical upper bounds on local gyrokinetic instabilities”, SmuK DPG Conference (Dresden, Germany), P 91 (P 11.43) (2023)
[4] M. Barnes et al., Journal of Computational Physics 391, 365 (2019)
[5] G. G. Plunk and P. Helander, Journal of Plasma Physics 88, 905880313 (2022)
The interaction between the plasma and the solid wall at the divertor/limiter target in tokamak devices affects turbulence in the plasma edge, thus impacting the overall confinement [1,2]. While the gyrokinetic framework allows one to describe turbulence and transport in the core of tokamak plasmas, most of present gyrokinetic codes still lack an adequate description of plasma-wall interaction [3,4]. In this perspective, plasma-wall interaction is studied with the VOICE code in a (1D-1V) kinetic approach along magnetic field lines, assumed to have a normal incidence [Bourne 2023]. Immersed boundary conditions are used to model the wall. Two different choices are made for the penalized wall region: either currents are allowed to flow within the material boundary or not [5].
The main properties of the Debye sheath physics are recovered, whatever the description adopted for the wall region. The formation of a positively charged layer in front of the plasma boundary defines the transition to the so-called Debye sheath. This non-neutral layer is accompanied by a drop of the electric potential that confines slow electrons and accelerates ions.
Most interestingly, discrepancies are found with respect to fluid predictions [6]. First, the non-vanishing conductive heat flux observed in kinetic simulations is usually neglected in fluid analyses. Its properties and impact on the overall dynamics reveal hardly tractable within the fluid framework. Second, we show that the expression of the plasma sound speed strongly depends on the chosen closure of the fluid hierarchy. The immediate consequence is that Bohm’s criterion defining the Debye sheath entrance in terms of the Mach number becomes non-operational. These departures from fluid predictions result in part from the single-species Fokker-Planck collision operator acting on the parallel transport.
Distribution functions at the sheath entrance depart from Maxwellians, leading to an ion heat transmission factor at the wall larger than usually predicted [7]. Parametric dependencies reveal that the observed kinetic sheath physics is robust: the sheath acts as a filter of high energy electrons to avoid any charge separation in the plasma on scales larger than a few Debye lengths.
Last, the lessons learnt from this reduced kinetic model are used to develop models of the sheath physics relevant to the gyrokinetic framework. We present a new approach, currently tested in the flux-driven framework of GYSELA [2,8], based on constraints on the averaged total current flowing to the limiter/divertor target.
References:
[1] S. Krasheninnikov et al., J. Plasma Phys. 74 (2008) 679. [2] G. Dif-Pradalier, et al. Commun Phys 5 (2022) 229 [3] S.E. Parker, et al., J. Comput. Phys. 104 (1993) 41 [4] E. L. Shi et al., Phys. Plasmas 22 (2015) 022504 [5] Y. Munschy et al., Kinetic plasma wall interaction using immersed boundary condictions, submitted to Nucl. Fusion (2023) [6] Y. Munschy et al., Kinetic plasma sheath self-organisation, Nucl. Fusion (2023) [7] P. C. Stangeby, The Plasma Boundary of Magnetic Fusion Devices (2000) [8] V. Grandgirard et al. Computer Phys. Comm. 207 (2016) 35
Turbulent transport has long been understood to be the dominant transport mechanism in tokamaks. Stellarators, such as W7X, that have been optimised to reduce collisional (neoclassical) transport are also expected to be limited by turbulent transport [1]. Combined theoretical, computational, and experimental progress has advanced our understanding of turbulence properties and the resultant transport, specifically pertaining to tokamaks. In stellarators, however, the more complicated magnetic geometry gives rise to differences in turbulent behaviour [2]. In particular, the magnetic geometry is no longer replicated along each field line, but instead varies between field lines within a given flux surface in a non-trivial way. This has the consequence that the standard approach of simulating a single flux tube may be insufficient to capture the mechanisms that influence transport; zonal flows that allow for communication across multiple field lines require consideration of the turbulent evolution across an annulus encompassing the entire flux surface.
In order to address this issue computationally, it is necessary to develop an approach to treat the entire flux annulus. We have thus developed a new algorithm to confront this problem and have implemented this into the $\delta f$-gyrokinetic code stella that employs a semi-implicit treatment of electron dynamics and retains spectral accuracy in the plane perpendicular to the mean magnetic field. We will describe the new algorithm and show results from its implementation and application to a given stellarator equilibrium. The explicit Full Flux Surface version of stella with adiabatic electrons has been benchmarked against the existing global code GENE, and scans in $\rho^*$ have been performed yielding good agreement with expectation when comparing to flux tube simulations performed with stella. To illustrate the efficacy of the new approach we will then compare the explicit version of stella with kinetic electrons with the equivalent results obtained using the semi-implicit time advance. We present the numerical results obtained thus far, with the aim that such a code can aid future discussions as to the effects of zonal modes in 3D geometries.
[1] Pedersen, T. S., Abramovic, I., Agostinetti, et. al., “Experimental confirmation of efficient island divertor operation and successful neoclassical transport optimization in Wendelstein 7-X. Nuclear Fusion”, 62(4), 042022, (2022).
[2] Proll, J. H. E., Mynick, H. E., Xanthopoulos, P., Lazerson, S. A., & Faber, B. J., “TEM turbulence optimisation in stellarators. Plasma Physics and Controlled Fusion”, 58(1), 014006, (2015).
see file below
JOREK [1] is one the most advanced non-linear simulation codes for studying MHD instabilities that can occur in magnetically confined fusion plasmas as well as their control. It leverages extensive parallel programming to obtain accurate results regarding the dynamics of the high-energy magnetically confined plasma inside the vacuum vessel of a tokamak or stellarator. In addition, a free-boundary and resistive wall extension was introduced via coupling to the STARWALL [2] and very recently also CARIDDI [3] codes, which both apply a Greens functions method to calculate densely populated matrices describing the electro magnetic interactions between plasma and conducting structures. MPI and OpenMP parallelization is exploited for the coupling [4].
To perform accurate simulations of the aforementioned interactions, the discretizations of the wall and of the plasma regions should be considered, and the Degrees Of Freedom determining the dimensions of the response matrices depend on these. Currently, due to the state-of-the-art limitations regarding computational resources and available memory, the reachable resolution in such simulations is not sufficient to describe all details of the wall structures of the ITER tokamak, which is under construction.
On the other hand, in the linear algebra literature, there exist factorization techniques that could lead to reducing memory consumption through compression techniques. The present work represents an effort of obtaining such matrix compression with the implementation of the Singular Value Decomposition through the adoption of routines from the ScaLAPACK library [5, 6]. The objective is leveraging matrix compression inside the JOREK code exploiting parallelism to allow handling more complex wall structures than presently possible by reducing memory consumption and computational costs.
After a brief overview of the activities currently carried on at the CIEMAT-BSC Advanced Computing Hub (ACH) in the EUROfusion Horizon Europe Program, this contribution will focus specifically on the effort in the JOREK code, describing the work that has been done so far, both from the theoretical and the implementation points of view. After that, results from preliminary tests will be shown and a glimpse of the next steps will be given.
References
[1] Hoelzl, Matthias, et al. Nuclear Fusion 61:6, 065001, (2021)
[2] Hoelzl, M., et al. Journal of Physics: Conference Series 401, 012010 (2012)
[3] Isernia, N., Schwarz N., et al. Physics of Plasmas (in preparation)
[4] Mochalskyy, S., M. Hoelzl, and R. Hatzky, arXiv preprint arXiv:1609.07441
[5] Blackford, L. Susan, et al. Society for Industrial and Applied Mathematics (1997)
[6] https://www.netlib.org/scalapack/
Electron cyclotron (EC) waves offer several advantages as a heating scheme in a tokamak fusion reactor, both from the technological (the launchers require small slots in the blanket; the ﬁrst tritium barrier can be incorporated into the vacuum vessel) and the physical (easy wave-plasma coupling; localized absorption) point of view. The most crucial applications of EC waves in a reactor are the sustainment of part of the plasma current and the stabilization of MHD instabilities like the neoclassical tearing mode (NTM). Here we focus on some recent theoretical advances and simulation results concerning the application of EC waves in DEMO plasmas.
When the electron temperature exceeds ca. 30 keV, the current drive efficiency is found to saturate [1], mainly due to parasitic absorption from the next cyclotron harmonic, in agreement with previous studies [2,3]. This problem can be mitigated by shifting the injection position towards the high-field side, e.g. to the top of the vacuum vessel. For temperatures below 30 keV, the maximum current drive is achieved as a balance between the need of heating suprathermal electrons and that of sufficient absorption [2]. A fast tool (HARE) to evaluate the maximum current drive has been developed on this basis [4]. The injection of extraordinary-mode waves at the fundamental harmonic for current drive in the outer plasma is discussed. For NTM stabilization, the situation is different, since the need for high current-drive efficiency, which is characterized by broad profiles, competes with the need for good localization. Simple criteria for NTM stabilization [5,6] provide a guidance for the determination of the optimum current drive profiles (i.e. those corresponding to NTM suppression at minimum injected power), as previously done for ITER [7]. The broadening of the absorption profiles due to beam scattering by density fluctuations becomes crucial in large machines [8] and needs to be assessed.
[1] E. Poli, Nucl. Fusion 53, 013011 (2013)
[2] G. R. Smith, R. H. Cohen, and T. K. Mau, Phys. Fluids 30, 3633 (1987)
[3] R. W. Harvey et al., Nucl. Fusion 37, 69 (1997)
[4] E. Poli et al., Phys. Plasmas 25, 122501 (2018)
[5] H. Zohm et al., Plasma Phys. Control. Fusion 49, B341 (2007)
[6] O. Sauter et al., Plasma Phys. Control. Fusion 52, 025002 (2010)
[7] E. Poli et al., Nucl. Fusion 55, 013023 (2015)
The numerical study of steady state solutions of equations describing the particle and energy balance rightfully gets ample attention since the ultimate goal of fusion research is to produce long-lasting quasi-stationary discharges in future fusion power stations. Transient states may, however, differ significantly from the steady state ultimately reached and will - in practice - impact on the actual fate of the discharge. In current-day machines, transient behavior is rule rather than exception. Using 2 models that cut away a maximum of effects while retaining crucial ingredients in order to bring out the specific impact of transient effects more clearly, the present EFTC contribution illustrates the different signature of radio frequency (RF) wave versus beam heating allowing to transiently trigger desirable effects that help steering the discharge. A simplified Fokker-Planck equation illustrates the differening temporal evolution of RF tail formation of minorities, majorities and beams, while also highlighting important distinctions between fundamental cyclotron and harmonic heating. An equally crude transport model allows to monitor the evolution of a discharge, e.g. showing the role of spatial localization of sources and its impact on transient values reached as well as on the steady state the discharge ultimately converges to.
Resonant mode-particle interactions crucially determine particle, energy and momentum transport and confinement performance in fusion devices. Non-axisymmetric perturbative modes exist either due to intrinsic instabilities, or due to intentionally applied magnetic fields and affect particles with specific kinetic characteristics by modifying specific locations of the particle’s phase space [1] and parts of their momentum distribution [2, 3]. Conditions for resonant interactions can be formulated on the basis of the Guiding Center (GC) approximation [4] and the utilization of the three Constants Of the Motion (COM), namely the energy, the canonical toroidal momentum, and the magnetic moment, uniquely labeling each GC orbit in an unperturbed axisymmetric equilibrium. The resonant conditions are determined by the mode numbers and the Orbital Spectrum of the GC motion that is the bounce/transit and bounce/transit-averaged toroidal precession frequencies of trapped and passing particles [5, 6]. In addition to the equilibrium magnetic field and the perturbative modes, a radial electric field, localized in the vicinity of the plasma edge, is known to accompany H-mode operation in fusion plasmas [7] with the consequent formation of an Edge Transport Barrier (ETB) suppressing the turbulence-driven transport due to E×B shear flow and improving plasma confinement [8-10].
In this work, we show that, although the radial electric field does not perturb the integrability of the GC motion in an axisymmetric equilibrium, it drastically modifies the topology of the phase space and the ion prompt losses, as well as the Orbital Spectrum and therefore the conditions for resonant mode-particle interactions under the presence of non-axisymmetric perturbations. Moreover, it enables the formation of Shearless Transport Barriers (STB) [11], persistently bounding particle orbits and reducing extended particle, energy and momentum transport. The calculation of a kinetic-q factor enables the accurate prediction of the location of the resonances and the STBs in the phase space of the system that is systematically confirmed by numerical particle tracing simulations, suggesting a valuable tool for investigating and controlling plasma transport under the presence of various non-axisymmetric perturbations.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] A. N. Kaufman, Phys. Fluids 15, 1063 (1972)
[2] W. W. Heidbrink and R. B. White, Phys. Plasmas 27, 030901 (2020)
[3] R. B. White and A. Bierwage, Phys. Plasmas 28, 032507 (2021)
[4] R. B. White and M. S. Chance, Phys. Fluids 27, 2455 (1984)
[5] P. A. Zestanakis, Y. Kominis, G. Anastassiou, and K. Hizanidis, Phys. Plasmas 23, 032507 (2016)
[6] Y. Antonenas, G. Anastassiou, and Y. Kominis, J. Plasma Phys. 87, 855870101 (2021)
[7] F. Wagner, et al., Phys. Rev. Lett. 49, 1408–1412 (1982)
[8] J. W. Connor and H. R. Wilson, Plasma Phys. Control. Fusion 42, R1-R74, (2000)
[9] E. Viezzer, et al., Nuclear Fusion 53, 053005 (2013)
[10] L. Sanchis, et al., Plasma Phys. Control. Fusion 61, 014038 (2019)
[11] I. L. Caldas, et al., Plasma Phys. Control. Fusion 54, 124035 (2012)
Inertial Fusion Energy production using lasers represents a key approach to nuclear fusion energy on earth. The concept of laser-driven Inertial Confinement thermonuclear Fusion (ICF) was proposed in 1972 in seminal papers by American and Russian scientists [Basov1972, Nuckolls1972], which initiated a worldwide effort to demonstrate inertial fusion in the laboratory. After five decades of continuous progress toward ignition last December at the Lawrence Livermore Laboratory the laser-driven inertial fusion principle has been demonstrated reaching ignition and burn, with a net Energy Gain (Fusion energy Versus Laser energy) of 1.5. The recent results in the US [Kritcher2022, Ralph2022, Shawareb2022, Zylstra2022,Wilson2022] have shown clearly that laser-fusion ignition is indeed possible, predictable, and repeatable. Such breakthrough has pushed forward the interest in Laser fusion of many countries, Universities, Research Centres and private companies. The approach pursued by the Livermore scientists is based on the “Indirect drive” scheme where the incoming laser radiation is first converted in soft X-rays in a gold cylinder cavity. Then, these X-rays symmetrically uniformly irradiate a spherical capsule filled with DT fuel and positioned in the center of the cylindrical cavity. The radiation ablates the outer layers of the capsule, compresses the fuel inside more than a thousand times and heats it to a temperature of hundred millions degrees. These are conditions where the fusion reactions take place and release a surplus of energy in a form of energetic neutrons, alpha-particles and radiation. The “direct drive” approach consists in the direct laser irradiation of a capsule with a DT fuel (thus bypassing the step of conversion in X-rays in the gold cylinder). It is more efficient and better suited for energy production, but implosion is less stable. Together with magnetic confinement fusion, direct drive ICF is a promising approach for construction of a fusion power plant: an abundant, clean, sustainable, and on-demand energy source for mankind. In this talk a general review of the laser and target optimization done at NIF to reach ignition will be presented together with an explanation of the “shock ignition”
[Betti2007] direct drive approach as a primary candidate for high gain laser fusion scheme.
[Atzeni2021] S Atzeni, D Batani, CN Danson, LA Gizzi, M Perlado, M Tatarakis, V Tikhonchuk, L. Volpe “An evaluation of sustainability and societal impact of high-power laser and fusion technologies: a case for a new European research infrastructure”, High Power Laser Science and Engineering, (2021), Vol. 9, e52, 4 pages.
[Atzeni2022] S Atzeni, D Batani, CN Danson, LA Gizzi, S Le Pape, JL Miquel, M Perlado, RHH Scott, M Tatarakis, V Tikhonchuk, L Volpe “Breakthrough at the NIF paves the way to inertial fusion energy”, Europhysics News 53 (1), 18-23
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[Kritcher2022] A. L. Kritcher, C. V. Young, H. F. Robey, C. R. Weber, A. B. Zylstra, O. A. Hurricane, D. A. Callahan,
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[Wilson2022] T. Wilson, Fusion energy breakthrough by US scientists boosts clean power hopes (2022), www.ft.com/content/4b6f0fab-66ef-4e33-adec-cfc345589dc7.
Global energy confinement studies based on empirical scaling expressions represent an important instrument for benchmarking experiments in tokamaks and stellarators, for specification of boundary conditions in modeling activities, for guiding the development of theoretical models of heat transport and for extrapolating plasma performance to new machines, such as ITER.
In this contribution, we report on a new H-mode confinement scaling with important implications regarding the dependence of confinement on plasma and machine parameters, including a remarkably weaker dependence on machine size. Referred to as ITPA20 and with a corresponding confinement enhancement factor H20, the new scaling has been derived from the recently updated International Tokamak Physics Activity (ITPA) global H-mode confinement database [1]. Compared to earlier versions of the database, new data have been added closer to ITER conditions, along with data from JET [2] and ASDEX Upgrade [3] with fully metallic walls. In addition, the range of several important parameters has been extended.
We first focus on the parameter estimates and predictions of the new scaling, obtained with innovative techniques grounded in Bayesian probability and information theory. This has revealed uncertainties that are significantly larger than those estimated with simple least squares, greatly influencing the significance of the various predictor variables in the scaling. These results are also relevant for other scaling laws derived from multi-machine databases that are routinely used within the fusion community.
We then touch upon the issue of the considerably weaker scaling with machine size (major radius) observed in the new scaling. In order to trace the origin of this reduced dependence, an optimization together with a clustering technique has been carried out to identify the subset of the database contributing to the reduction of the size scaling [4]. The clustering is interpreted in dimensionless space and shown to partly relate to the degree of multicollinearity between the predictor variables. Finally, the implications for ITER are discussed.
References:
[1] G. Verdoolaege, S. Kaye, C. Angioni, et al., Nucl. Fusion 61, 076006 (2021)
[2] M. Maslov, A. Boboc, J. Flanagan, et al., Nucl. Fusion 60, 036007 (2020)
[3] F. Ryter, C. Angioni, G. Tardini, et al., Nucl. Fusion 61, 046030 (2021)
[4] J. Hall, P. Zhang, and G. Verdoolaege, “Energy confinement scaling with machine size in the updated ITPA global H-mode confinement database”, IAEA Technical Meeting om Fusion Data Processing, Validation and Analysis (Vienna, Austria) (2021)
We present a new model to describe neoclassical transport in strong gradient regions in tokamaks such as internal transport barriers and the pedestal [1]. Previous work on neoclassical transport across transport barriers assumed large density and potential gradients but a small temperature gradient [2], or neglected the gradient of the mean parallel flow [3]. Using a large aspect ratio and low collisionality expansion, we relax these restrictive assumptions and keep gradient scale lengths that are of the size of the poloidal gyroradius. The poloidally varying part of the electric potential is retained because it is sufficiently large that it can detrap particles or trap them on the high-field side. We derive equations describing the neoclassical transport of particles, parallel momentum and energy by ions in the banana regime. Studying contributions from both passing and trapped particles, we show that the resulting transport is dominated by the trapped. Even so, passing particles cannot be neglected because they determine the poloidally varying piece of the electric potential which modifies the transport coefficients. We find that a non-zero neoclassical particle flux requires parallel momentum input which could be provided through interaction with turbulence or impurities. The energy flux across a transport barrier has upper and lower bounds in both temperature and density. Solutions to our transport equations are highly sensitive to the choice of sources and boundary conditions and do not always exist.
This work was supported by the U.S. Department of Energy (F.I.P., contract number DE-AC02-09CH11466) and (P.C., contract number DE-FG02-91ER-54109). S.T. was also supported by the German Academic Scholarship Foundation.
References:
[1] S. Trinczek, F. I. Parra, et al., J. Plasma Phys. 89, 905890304, (2023)
[2] P. J. Catto, F. I. Parra, et al., Plasma Phys. Control. Fusion 55, 045009 (2013)
[3] K. C. Shaing, C. T. Hsu, Physics of Plasmas 19, 022502 (2012)
Making use of a large experimental database of pedestals of H-mode ELMy JET-ILW pulses [1], we propose several approaches to systematic prediction of the height of the electron-temperature pedestal and of the electron temperature at the top of the density pedestal, with the engineering parameters and the density profiles as inputs. Simulations of ETG turbulence in steep-gradient regions of the pedestals of JET and other large tokamaks suggest that a simple scaling exists between the (gyroBohm-normalised) heat flux and the the local values of the electron density ($R/L_{n_e}$) and temperature ($R/L_{T_e}$) gradients [2]. This has previously been checked on a small subset of this database, and we now confirm that testing it against the entire database leads to consistent prediction of the electron temperature within 50% of the experimental values. The scaling proposed in [2] includes departures from the marginal stability, presumed to be achieved at $R/L_{T_e} = \eta_{e,NL} R/L_{n_e}$, where $\eta_{e,NL} $ is generally speaking a fitting parameter. Taking a simpler approach that assumes a definite local relationship between the gradients, we find that a range of power law scalings $R/L_{T_e} = A( R/L_{n_e})^\alpha$ with $\alpha$ between 0.33 and 1 correctly capture the behaviour of the electron temperature at the density-pedestal top. For $\alpha = 1$, $A\equiv \eta_{e,cr} $, which governs the turbulence saturation in the standard picture of slab-ETG modes. Measuring it halfway between the pedestal-density top and the separatrix yields a distribution of values (Figure 1) that lie considerably above the linear threshold $\eta_{e,lin} = 0.8$ [3]. This implies either that a nonlinear effect analogous to Dimits shift lifts $\eta_{e,cr}$ to an effective value just above 2 or that turbulent transport predominantly occurs in an order-unity-supercritical regime, characterised by this higher-than-marginal gradient ratio.
Figure 1: distribution calculated between top of electron density pedestal and separatrix for 1148 pulses.
Finally, we present a simple machine learning algorithm which, given the same experimental inputs as theory-based modelling, is able to predict the electron temperature within similar error bars for 20% of the database after being trained on the other 80%. This result confirms the conceptual possibility of accurate prediction and offers a baseline quality benchmark for improved models that rely on traditional theoretical understanding of turbulent transport.
References:
[1] L. Frassinetti et al. 2021 Nucl. Fusion 61, 016001.
[2] A.R. Field et al. 2023 Phil. Trans. R. Soc. A 381, 2021022.
[3] F. Jenko, W. Dorland, G. W. Hammett 2001 Phys. Plasmas 8, 4096.
Next-generation fusion reactors pose challenges for plasma-facing components (PFCs) made from solid materials due to the high heat and particle fluxes. To overcome these limitations, liquid-metal (LM) PFC concepts have been proposed recently, with electromagnetic restraint (Lorentz force) as a key mechanism to keep free-surface LM flows attached to reactor surfaces [1]. Therefore, the investigation of the potential of electromagnetic control for LM-PFCs is crucial.
The Stuart number, St = J0B0 δ/ρ*uτ2, represents the relative strength of the Lorentz force compared to the inertia force in electromagnetohydrodynamic (EMHD) flows, where J0 and B0, represents the current, and magnetic flux density at the wall, respectively. While δ, ρ, and uτ are the half channel height, density of the fluid and wall shear velocity. Regarding the fluid used, the St number is influenced by the properties of the fluid, i.e, its electrical conductivity, magnetic permeability, and density. Liquid metals, which typically have high electrical conductivity and magnetic permeability, tend to exhibit higher St numbers.
This study investigates the behaviour of the liquid under the Lorentz force effect to determine optimal St numbers for selecting liquid metal as the LM-PFC in tokamak reactors. To achieve this, spanwise oscillated Lorentz force is applied to a turbulent channel flow. The Lorentz force is obtained by solving Maxwell Equations for appropriate boundary conditions. A range of St numbers are used between St = 18 to 72 (Fig. 1 (a)). The applied Lorentz force reduces skin friction, resulting in lower viscous effects (Fig. 1(b)), therefore provides insights into obtaining suitable LM fluid that attaches to the reactor surface. Entropy generations are also investigated (Fig. 1(c)). Therefore, the findings of this study contribute to the understanding of the controllability of LM-PFCs through the application of Lorentz force, which aids in the selection of suitable LM-PFC design, and implementation in next-generation fusion reactors.
Future studies will focus on optimizing the Lorentz force parameters for LM-PFCs and conducting stability analyses that aim to enhance heat transfer efficiency and reliability of LM-PFC configurations in fusion reactors.
The presentation will introduce the principles of suppressing edge localized modes (ELMs) by external magnetic perturbations (RMPs). It will explain models that allow the description of the processes and highlight recent developments. Direct comparisons between simulations and experiments will be shown, and an outlook to the application for future machines like ITER will be given.
The suppression of ELMs by resonant magnetic perturbations RMPs in an ASDEX Upgrade plasma is modeled using the free boundary MHD code JOREK-STARWALL, which was recently extended to capture the plasma response up to the computational boundary allowing to reproduce experimentally measured flux surface corrugations precisely [1]. The simulations are performed with fully realistic plasma parameters and plasma flows. This realistic approach enables qualitative and quantitative comparisons of simulations with experimental observations, reveals important mechanisms, and forms a basis for more accurate predictive studies than previously possible.
Simulations show that, in the ELM suppressed state, there is a local structure in the radial displacement of the plasma around resonant surfaces that can be linked to the presence of magnetic islands. Together with recent experimental findings, this provides strong indications for the presence of a magnetic island chain at the pedestal top during ELM suppression in an ASDEX Upgrade discharge, contributing to resolving a long-standing open question.
Furthermore, the transition out of the ELM-suppressed phase into an ELM-unstable state is modeled through an increase of the pedestal density values. The simulations allow to disentangle the role for suppressing ELM instabilities of the edge pressure profile evolution on one hand and non-linear coupling between peeling-ballooning instabilities with the RMP-driven perturbations on the other hand.
References:
[1] V Mitterauer et al 2022 J. Phys.: Conf. Ser. 2397 012008
An accumulation of heavy impurities in the tokamak core is detrimental for its performance and can lead to disruption of the plasma. In smaller to medium size tokamaks the effective neoclassical transport in the pedestal is radially inwards [3]. In larger tokamaks---e.g. ITER--- where the temperature gradient is higher, the neoclassical transport is predicted to be outwards. The models are derived for axisymmetric quasi-steady-state plasmas. Applied 3D magnetic fields---i.e. Resonant Magnetic Perturbation (RMPs)--- can be used to suppress Edge Localized Modes (ELMs). Experimentally it has been observed in AUG, the use of RMPs enhances the outflow of heavy impurities in the pedestal. There is no model which can predict neoclassical heavy impurity transport in these ergodized 3D magnetic fields self-consistently.
In this contribution we present our tungsten transport simulation for an ASDEX Upgrade plasma with applied RMPs. Our model utilizes a full-orbit pusher, ionization, recombination, effective line and continuum radiation and neoclassical collisions with the background plasma. The effective collisional radiative rates are from the OpenADAS database, the neoclassical collision operator uses the framework of Homma [4].
We have compared the average radial transport between axisymmetric and 3D RMP scenarios in the pedestal region. RMPs clearly cause enhanced transport. However, it is not just enhanced axisymmetric diffusion, but perpendicular transport has a equivalently strong component in the n=2 toroidal harmonic. Additionally to the enhanced transport we found that part of the W gets trapped in 3D potential wells in the Scrape-off layer as shown in Fig.1. With the newly developed neutral model [5], we can combine the interaction in the divertor with the 3D RMPs to model the tungsten transport from the divertor towards the core of the plasma.
A possible triggering mechanism of Alfvén waves (AWs) in tokamak plasmas, based on localized perturbations induced by magnetic reconnection events, is discussed in the framework of nonlinear viscoresistive 3D MHD modeling.
Nicolas Lopez
University of Oxford
A wave near an isolated turning point is typically assumed to have an Airy function profile with respect to the separation distance. This description is incomplete, however, and is insufficient to describe the behavior of more realistic wavefields that are not simple plane waves [1]. Asymptotic matching to a prescribed incoming wavefield generically introduces a phasefront curvature term that changes the characteristic wave behavior from the Airy function to that of the hyperbolic umbilic function. This function, which is one of the seven classic ‘elementary’ functions from catastrophe theory along with the Airy function, can be understood intuitively as the solution for a linearly focused Gaussian beam propagating in a linearly varying density profile, as I shall discuss. The morphology of the caustic lines that govern the intensity maxima of the diffraction pattern as one alters the density lengthscale of the plasma, the focal length of the incident beam, and also the injection angle of the incident beam are presented in detail. This morphology includes a Goos-Hanchen shift and focal shift at oblique incidence that do not appear in a reduced ray-based description of the caustic. The enhancement of the intensity swelling factor for a focused wave compared to the typical Airy solution is highlighted, and the impact of finite lens aperture is discussed. Collisional damping and finite beam waist are included in the model and appear as complex components to the arguments of the hyperbolic umbilic function. The observations presented here on the behavior of waves near turning points should aid the development of improved reduced wave models to be used, for example, in designing modern nuclear fusion experiments.
[1] N. A. Lopez, E. Kur, and D. J. Strozzi, to appear in Phys. Rev. E, arXiv:2301.12788 (2023)
One of the most well-established codes for modeling non-linear Magnetohydrodynamics (MHD) for tokamak reactors is JOREK, which solves these equations with a Bézier surface based finite element method. This code produces a highly sparse but also very large linear system. The main solver behind the code uses either GMRES or Bi-CGSTAB with a physics-based preconditioner, but even with the preconditioner there are issues with memory and computation costs and the solver doesn’t always converge well. This work contains the first thorough study of the mathematical properties of the underlying linear system. It enables us to diagnose and pinpoint the cause of hampered convergence. In particular, analyzing the spectral properties of the matrix and the preconditioned system with numerical linear algebra techniques, will open the door to research and investigate more performant solver strategies, such as projection methods.
Tungsten divertors in tokamaks are designed to withstand and evacuate the excess heat coming from the hot plasma. Some of the tungsten divertor can melt, enter the plasma, and itself become a high-Z impurity plasma. If it enters the core, it can emit enough radiation to cause a loss of thermal plasma energy and degrade or terminate tokamak operation [1]. Thus, it is crucial to develop a theoretical and numerical framework to understand the transport processes for heavy impurities such that high performance scenarios can be developed. Present day tokamaks are often heated with unbalanced neutral beam injection, which in turn causes the plasma to rotate toroidally. The associated rotation of the tungsten plasma is typically super-sonic. Another common feature of tokamak plasmas is the existence of long-living non-axisymmetric magnetic fields – which can be due to saturated plasma instabilities – and/or symmetry breaking magnetic coils. The combination of a strongly rotating plasma in 3D magnetic fields is particular to tokamaks (heavy impurities in stellarators would rotate sub-sonically).
For heavy impurity simulations we use a δf PIC code called Venus-Levis [2]. To accurately trace the tungsten particles, one needs to consider the interaction with the background plasma. For this, a custom collision operator is used which relies on the calculation of neoclassical transport coefficients. These coefficients have presently been been calculated analytically [3-4]. To extend the work beyond ideal MHD, one needs to find these coefficients numerically. Current on-going work is being done to:
• develop a new numerical scheme based on Venus-Levis for these coefficients,
• port Venus-Levis to GPUs for a fast code which can take advantage of modern supercomputers.
References:
[1] T Pütterich, et al., the ASDEX Upgrade Team, and JET EFDA Contributors. Observations on the w-transport in the core plasma of jet and asdex upgrade. Plasma Physics and Controlled Fusion, 55(12):124036, nov 2013.
[2] D. Pfefferlé, W.A. Cooper, J.P. Graves, and C. Misev. Venus-levis and its spline-fourier interpolation of 3d toroidal magnetic field representation for guiding-centre and full-orbit simulations of charged energetic particles. Computer Physics Communications, 185(12):3127–3140, 2014.
[3] M Raghunathan, J P Graves, T Nicolas, W A Cooper, X Garbet, and D Pfefferlé. Heavy impurity confinement in hybrid operation scenario plasmas with a rotating 1/1 continuous mode. Plasma Physics and Controlled Fusion, 59(12):124002, oct 2017.
[4] E Lascas Neto, J P Graves, M Raghunathan, C Sommariva, D Pfefferlé, and JET Contributors. Heavy impurity transport in tokamaks subject to plasma rotation, NTV and the influence of saturated ideal MHD perturbations. Plasma Physics and Controlled Fusion, 64(1):014002, nov 2021.
The role of a runaway current in a post disruption plasma is investigated through numerical simulations in the single helicity limit. Linear results are verified against analytical theory and benchmarked against results already present in the literature. In particular, the presence of a microlayer below the resistive one is confirmed and the effect of the electron inertia on it is also investigated. Nonlinear results indicate that the distribution of runaways electron population undergoes significantly changes, being affected by the island growth and rotation. The combined effect of these mechanisms leads to the formation of a spiral-like structure which distribute the runaway electrons all over the island.
Evolving a two-fluid model based on the drift-reduced Braginskii equations [1], GBS [2] is a three-dimensional flux-driven turbulence code designed for simulating plasma turbulence and kinetic neutral dynamics in the tokamak boundary. The GBS simulation domain covers the entire tokamak volume, avoiding the need for an artificial boundary between the core and edge regions, thus preserving the core-edge-SOL interplay. The GBS code has been used to study turbulent transport regimes in the tokamak boundary, to delineate its operational limits and to describe their dependence on key physical parameters.
In the present work, we introduce a new formulation of the drift-reduced Braginskii equations that properly accounts for electromagnetic effects. In contrast to the implementation in typical turbulence codes, but similar to nonlinear resistive MHD codes such as JOREK [3], which have successfully simulated Edge-Localised-Modes (ELMs) dynamics and cycles, our model does not distinguish between equilibrium and fluctuating components of the time-evolving magnetic field and parallel current (i.e., it employs a full-f approach also for the electromagnetic fluctuations). Indeed, the model we developed allows for the simulation of MHD modes such as the peeling-ballooning instability, known to be the linear precursor to ELMs.
The newly developed electromagnetic model is implemented in the GBS code and we present the first simulation results in an X-point diverted geometry, focusing on the role that electromagnetic perturbations play in boundary transport and turbulence.
This work has been carried out within the framework of the EUROfusion Consortium, via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion) and funded by the Swiss State Secretariat for Education, Research and Innovation (SERI). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union, the European Commission, or SERI. Neither the European Union nor the European Commission nor SERI can be held responsible for them.
References:
[1] A. Zeiler et al., Physics of Plasmas 4, 872368 (1997)
[2] M. Giacomin et al., Journal of Computational Physics 463, 111294 (2022)
[3] A. Cathey et al., Nuclear Fusion 60, 124007 (2020)
To enhance the computational efficiency of particle codes in performing multi-n nonlinear simulations, piecewise finite elements have been developed in tokamak plasma, along with previous work [1,2,3]. Clebsch coordinates are constructed depending on the toroidal domain, which is consistent with the finite difference scheme [1]. In this work, the cubic spline finite element is adopted [3]. The grid is defined in $(r,\phi,\theta)$ coordinates as shown in Fig. 1, where $r$, $\phi$ and $\theta$ are radial-like, toroidal, and poloidal coordinates. However, the basis functions are aligned along the magnetic field in $(r,\phi,\eta)$, where $\eta=\theta-\int_{\phi_i}^\phi d\phi'/q(\theta')$, with $i$ denoting the index of the toroidal domain, $q$ representing the local safety factor, and the integration performed along the magnetic field. In addition to the benefit of avoiding grid deformation and reducing the grid number in one direction [1], the scheme can be extended to higher-order finite element methods. Furthermore, the $(r,\phi,\theta)$ grid is defined without a shift, allowing easy application of Fourier filters for linear benchmarking purposes.
The gyrokinetic electrostatic model and the electromagnetic model using the $p_\parallel$ formulation have been implemented. The matrix construction for equations is carried out in Clebsch coordinates, employing Monte-Carlo integration. Since the particles are represented in $(r,\phi,\theta)$ while the fields are defined in $(r,\phi,\eta)$, the field interpolation and density/current projection are performed using these two sets of coordinates. The linear benchmark using the Cyclone-like parameters shows reasonable agreement with previous work.
The capabilities of this scheme in multi-n nonlinear simulations are demonstrated, and its potential future applications in electromagnetic simulations in the MHD limit [3, 4], as well as employing unstructured meshes for whole volume simulations [5], are discussed. Furthermore, the connection to ongoing gyrokinetic studies is illustrated [6,7].
Figure 1: grids in $(x,y)$, i.e., $(\phi,\theta)$ (left); basis functions along B in $(x,y)$ (middle) and in torus (right).
References:
[1] B.D. Scott, Physics of Plasmas, 8, 447 (2001)
[2] Z.X. Lu, F. Zonca, A. Cardinali, Physics of Plasmas, 19, 042104 (2012)
[3] Z.X. Lu, G. Meng, R. Hatzky, M. Hoelzl et al, Plasma Phys. Controlled Fusion 65, 034004 (2023)
[4] R. Hatzky, R. Kleiber, A. Könies, A. Mishchenko et al, Journal of Plasma Physics. 85, 1 (2019)
[5] Z.X. Lu, Ph. Lauber, T. Hayward-Schneider, et al, Physics of Plasmas, 26, 122503 (2019)
[6] G.T.A. Huijsmans et al, Comparing linear stability of electrostatic kinetic and gyro-kinetic ITG modes, EPS conference (2023)
[7] A. Mishchenko et al, Plasma Phys. Controlled Fusion 65 (6), 064001 (2023)
This work deals with the relation/interaction between plasma flow and magnetic field during the process of reversed-field pinch (RFP) quasi-helical self-organization [1, 2], featuring improved confinement in the RFX-mod RFP [3].
Experimental [3] and modelling [4] evidence shows that helical self-organization is characterized by quasi-periodical relaxation-reconnection events: partial conversion of magnetic into kinetic energy, current sheet formation, steepening of plasma current profiles, ion heating, locking of the angular phases between different Fourier components of the magnetic field. The latter is recognized as the three-dimensional trigger of the reconnection events [5].
In this work, the focus is on the behaviour and role of the plasma velocity field during the process. In fact, plasma velocity can interact with the magnetic field and with resonant magnetic perturbations in a rich variety of manner, classified in [6, 7].
Numerical results are shown, obtained by solving a three-dimensional nonlinear visco-resistive magnetofluid model (magnetohydrodynamics, MHD) that describe the hot current-carrying plasma.
We will discuss the interaction of a velocity field with the magnetic field, showing that the plasma flow halts the growth of magnetic instabilities. Then we will look at the velocity shear field that, as expected, curbs the amplitude of MHD modes [8] and at plasma vorticity.
The final part of the work will present a preliminary investigation on the role of a spatially localized peak of the shear velocity field. Such an increased shear appears to forerun the locking of the magnetic field modes that, in turn, creates the conditions for the relaxation event
References:
[1] S. Cappello, D. Biskamp, Nuclear Fusion, 36, 571 (1996)
[2] S. Cappello, D.F. Escande, Physical Review Letters, 85, 3838 (2000)
[3] R. Lorenzini et al, Nature Physics, 5, 570 (2009)
[4] D. Bonfiglio, M. Veranda et al., Physical Review Letters, 111 085002 (2013)
[5] M. Veranda et al., Rendiconti Lincei. Scienze Fisiche e Naturali, 31, 963, (2020)
[6] R. Fitzpatrick, Physics of Plasmas 5 , 3325 (1998)
[7] M. Bécoulet, G. Huysmans, X. Garbet et al., Nuclear Fusion 49, 085001 (2009)
[8] H. Biglari, P. H. Diamond, P. W. Terry, Physics of Fluids B 2 1 (1990)
ICRH is an attractive auxiliary heating system for future fusion reactors as it enables direct power deposition to the ions and does not suffer from high density cutoffs. However, ICRH launcher structure needs to be positioned close to the edge plasma to efficiently couple the launched power. This gives rise to enhanced plasma wall interactions near and far from the launching structure. One of these deleterious interactions is believed to be linked to the ponderomotive force, due to the strong electric field gradients in the plasma present near the antenna launcher.
Following the simple approach of the POND code described in [1], a first attempt to characterize the density perturbations expected in presence of ponderomotive force near an ICRH antenna launcher are investigated. A 1D and 3D version of the approach are implemented in the very simple case of a one-strap and two-strap shieldless antennas. Possible shortcomings and improvements in the model [1] are discussed.
The gyromoment (GM) approach was developed by B. J. Frei et al. [1] to address the challenges associated with the gyrokinetic (GK) modeling of turbulent dynamics in the boundary region of fusion devices. Based on expanding distribution functions onto a Hermite-Laguerre polynomial basis and evolving in space and time the expansion coefficients, the GM approach has the potential to efficiently simulate plasmas across a wide range of collisionalities. The GM approach was shown to be effective in delta-f simulations [2,3,4].
The present work is part of the broader effort to extend the GM method to full-f simulations. In particular, it addresses the challenge related to implementing numerically the nonlinear Coulomb collision operator, emphasizing the potential of the GM method for comprehensive and efficient simulations in fusion devices, especially in high collisionality scenarios.
The expansion of the full-f nonlinear Coulomb collision operator within the GM approach has been derived analytically [5,6] and has been numerically implemented in the delta-f regime [7]. However, the numerical evaluation of the full-f Coulomb collision operator requires the computation of traceless and symmetric parts of spherical harmonic basis tensor contractions at arbitrary ranks. Here, we introduce an innovative mathematical approach to evaluate spherical harmonic basis tensors and the trace removal of their contractions. This technique leverages the inherent symmetries of basis tensors to reduce the computational burden from factorial to polynomial complexity. This allows for the numerical evaluation of the GM expansion of the nonlinear Coulomb collision operator, thus enabling GM simulations with proper collisions.
[1] B. J. Frei, R. Jorge, P. Ricci, Journal of Plasma Physics 86 (2020).
[2] A. Hoffmann, B. J. Frei, P. Ricci, Journal of Plasma Physics 89 (2023).
[3] B. J. Frei, A. Hoffmann, P. Ricci, Journal of Plasma Physics 88 (2022).
[4] B. J. Frei, A. Hoffmann, P. Ricci, S. Brunner, Z. Tecchiolli, arXiv:2210.05799 (2022).
[5] R. Jorge, B. J. Frei, P. Ricci, Journal of Plasma Physics 85, 905850604 (2019).
[6] R. Jorge, P. Ricci, N. F. Loureiro, Journal of Plasma Physics 83 (2017).
[7] B. J. Frei, et al., Journal of Plasma Physics 87(5), 905870501 (2021).
Ion cyclotron resonance heating (ICRH) is known to create a population of fast ions in fusion plasmas. The non-Maxwellian distribution functions are obtained by solving the Fokker-Planck equation and are needed to describe the radio-frequency wave power deposition and transport phenomena.
This work uses wave fields and power deposition predicted by FEMIC [1], a finite element model for ICRH. When coupling a Fokker-Planck solver with FEMIC, it is beneficial to have a rapid modelling of distribution functions. Therefore, we propose a one-dimensional steady state model, which makes it possible to solve the Fokker-Planck equation by direct integration as done by Stix [2]. The benefits of this model are simplicity, transparency, and rapidity. The latter is especially important for self-consistent solvers.
In this work, a pitch angle averaged quasilinear operator for ions interacting with waves in the ion cyclotron frequency range is derived and tested. The operator is based on the work of Eriksson and Helander [3], although the finite drift orbit width is neglected, and the pitch angle dependence is averaged over. Consequently, the operator derived in this work depends only on the energy coordinate. The main advantage, compared to e.g. the operators in the PION [4] and FoPla [5] codes, is that it includes effects of the Doppler shift, such that ions at different energies are accelerated by different averaged electric fields. This is of particularly importance when the ion-ion hybrid layer is prominent, as in for example three-ion heating schemes.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] P. Vallejos, et al., Nuclear Fusion 59, 076022 (2019)
[2] T. H. Stix, Waves in Plasmas (New York: American Institute of Physics, 1992)
[3] L.-G. Eriksson and P. Helander, Physics of plasmas 1, 308–314 (1994) [4] L.-G. Eriksson et al., Nuclear Fusion 33, 1037 (1993) [5] D. Van Eester et al., Journal of Plasma Physics 87, 855870202 (2021)
We apply the Physics-informed Neural Networks (PINNs) to the magnetohydrodynamic (MHD) simulations. We build a neural network to find a solution of the MHD equations. We use the coordinate of space-time as model inputs and the outputs of the model are the magnetic field, the plasma bulk velocity, the plasma mass density, and the plasma thermal pressure. The MHD equations are combined into the loss function for a physics constrain. We show our training results and compare the computation time of the PINNs method and MHD numerical method.
Neoclassical tearing modes (NTM) are metastable magnetic islands in tokamaks; however, they appear frequently in experiments without any noticeable triggering event. In order to understand this, it has been numerically shown that turbulence can create a seed island by mode coupling [1,2,3], even remotely [4] ; such a seed island has been shown in 2D models to further grow from the NTM mechanism [5]. This amplification happens because of the island-induced pressure profile flattening. In turn, this flattening comes from the transport properties of the island, which are a consequence of the magnetic field perturbation. Therefore, characterizing magnetic transport both inside and outside a turbulence-driven magnetic island is crucial to understanding NTM triggering.
In this work, 3D reduced-MHD simulations of flux-driven ballooning turbulence are used to study the seed island creation in regimes where the classical tearing mode is linearly stable. A localized pressure source is used to control the radial position and strength of the turbulence. Several large-scale modes of island parity are generated in the nonlinear phase with different helicities, and stochasticity appears progressively in the region between those islands, although with significant stickiness of the field lines close to remaining KAM tori.
In this situation, even the definition of island size (or island boundary) becomes ambiguous. This is of particular importance since it is expected that flattening of the pressure profile appears only above a critical island size, depending on the ratio of parallel to perpendicular transport [6]. Several definitions of island size are compared, and a method for sorting magnetic field lines and calculating island width from Poincaré plots is presented.
We next study the transport properties of those magnetic fluctuations, first in an academic field featuring two large-scale modes. In particular, the role of Lagrangian Coherent Structures (LCS) are investigated from a statistical point of view. It is shown that field lines escape a tube over a finite length which is independent of tube size. However, this length is not uniform in the chaotic sea, and is minimum (indicating maximal transport) in the vicinity of LCS. Combined with the fact that LCS are not fixed but vary with time and velocity of particles, this could reduce their effectiveness as transport barriers when other processes exist.
Next, the robustness of those LCS with respect to small-scale turbulence is assessed. They are shown to persist even when the magnitude of small-scale fluctuations is enough to render the whole domain stochastic. Moreover, varying the wave number range of small scale fluctuations, we extract a critical magnetic spectrum determining whether LCS can be destroyed by small scale fluctuations.
Finally, analyzing MHD simulations in the light of the above results, we assess the role of the various scales in the onset of stochasticity and magnetic transport. In particular, we show that the main contribution to stochasticity comes from non-linearly generated large-scale modes.
References:
[1] A. Ishizawa et al, Phys. Plasmas 14, 040702 (2007)
[2] W. A. Hornsby et al, EPL 91 45001 (2010)
[3] M. Muraglia et al, Phys. Rev. Lett., 107, 095003 (2011)
[4] N. Dubuit et al, Phys. Plasmas 28, 022308 (2021)
[5] M. Muraglia et al, Nuclear Fusion 57 (7), 072010 (2017)
[6] R. Fitzpatrick et al, Phys. Plasmas 2 (3), 825 (1995)
Confinement quality in fusion plasma is significantly influenced by the presence of heavy impurities, e.g. Tungsten, which can lead to radiative heat loss and reduced confinement. This study explores impurity transport modeled by inertial particles in edge plasma, a previously unexamined aspect in plasma physics, using high-resolution direct numerical simulations of the Hasegawa-Wakatani equations, modeling electrostatic drift-waves in edge plasma. Our simulations employ one-way coupling of one million inertial point impurity particles.
We observe that with Stokes number, which characterizes the inertia of particles, being zero, the impurity particles behave as fluid particles. They follow the fluid streamlines, acting as "passive tracers." This results in a uniform distribution of particles, with no significant clustering. As the Stokes number increases, the inertia of the particles begins to dominate, causing them to deviate from the fluid streamlines. This results in the preferentially concentration of impurities, leading to distinct clustering and void formation. When the Stokes number is significantly larger, the impurity particles' inertia is so high that they tend to resist changes in their motion due to the fluid. As a result, they tend to maintain their trajectory, moving in a more random fashion, resulting in less clustering and more dispersion.
The study calculates impurity velocity divergence using modified Voronoi tessellation, which assigns specific volumes to impurity particles. By determining volume changes, the impurity velocity divergence can be calculated. Positive divergence indicates void formation, while negative signifies clustering. As the Stokes number increases, the probability density function (PDF) of divergence widens, reflecting more clustering. However, beyond a certain point, the PDF narrows indicating reduced clustering. The study also examines a modified H-W model with pronounced zonal flow and finds that it substantially decreases the divergence.
The dependence of L-H transition power threshold on plasma density is well documented and captured by the “ITPA 2008 scaling” [1] for high density D plasmas. In view of ITER operations, several studies with different H isotopes resulted in a 1/A mass dependence of the threshold [2]. Of particular interest for ITER H-mode access at low auxiliary power is the existence of a minimum L-H power threshold in density (n_{e,min}), observed also in JET with metallic wall (ILW) [3]. Below n_{e,min} (low-density branch), the power threshold is seen to increase again. The existence of n_{e,min} has been characterized in JET for different plasma species (H, D, T, He) [4]. In AUG and C-mod experiments with dominant electron heating, the appearance of n_{e,min} has been explained by the existence of an ion heat flux threshold, linear in density, to be exceeded for H-mode access. The inefficiency of electron-ion coupling in the low-density branch would then require increasing power to reach the ion-power threshold. Nevertheless, the linearity of the ion heat flux below n_{e,min} at the transition has not been observed in JET NBI-heated D plasmas [5]. In the recent JET D-T campaign, specific L-H transition experiments have been performed, including a plasma density scan, using NBI heating, at fixed magnetic field and plasma current. These experiments enables the characterization of density branches and n_{e,min} existence in D-T [6]. The data collected confirms the existence of n_{e,min} region in D-T plasmas. Interpretative transport simulations have been run to analyze D-T plasma behavior just before L-H transition. Thanks to transport modelling, a power balance analysis is carried out to evaluate the power terms contributing to the transition, separating the power coupled to ions and to electrons. The current contribution, after characterizing density branches in JET, NBI-heated D-T plasmas, presents the results of the power balance analysis and the dependence of the ion heat flux on D-T plasma density. The results are compared to JET D plasmas and to evidences from other fusion devices.
Acknowledgement: This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 - EUROfusion). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] Martin Y.R. et al 2008 J. Phys.: Conf. Ser. 123 012033
[2] E. Righi et al 1999 Nucl. Fusion 39 309
[3] Maggi C. et al 2014 Nucl. Fusion 54 023007
[4] E.R. Solano et al 2022 Nucl. Fusion 62 076026
[5] Vincenzi P et al 2022 Plasma Phys. Control. Fusion 64 124004
[6] E.R. Solano et al 2023 submitted to Nucl. Fusion
Parametric decay instabilities might play a significant role in various plasma physics phenomena and have garnered considerable interest in recent years [1]. In this study, we compare a model of parametric decay instabilities against the data observed during experiments conducted in the AUG (ASDEX Upgrade) fusion device while performing Electron Cyclotron Wall Conditioning (ECWC)[2].
During these experiments, a remarkable observation was made—clear spectral signals appeared at approximately half the main gyrotron frequency. Tomator-1D simulations [3] of the electron density and temperature evolution in the plasma demonstrated excellent
agreement with measurements obtained from AUG diagnostics.
The observed signals are generally associated to a parametric decay instability [4], which arises from a non-linear three-wave interaction where energy is transferred from a pump wave to daughter waves at shifted frequencies. The daughters, such as slow X-mode and Electron Bernstein waves (EBW), are naturally present in the plasma as thermal background. When these waves become trapped within a density bump formed between two reflective layers [5], an absolute decay is established, leading to exponential growth in the amplitude of the daughter waves. Subsequently, further decays into lower frequency waves, including EBW, Lower Hybrid (LH), or Ion Bernstein waves (IBW), might occur as the primary decay saturates.
In this work, we present a model that describes the necessary conditions for a primary decay to occur in conditions found during ECWC. By comparing the density profiles that create trapping conditions against Tomator-1D simulations and experimental profiles, we investigate the mechanisms that account for the observed features, such as half-frequency lines and localized light emission. This study contributes to the understanding of parametric decay instabilities in the context of electron cyclotron wall conditioning and its implications for the
future devices like ITER.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References
[1] S.K. Hansen, et al. Nucl. Fusion 60 106008 (2020)
[2] T. Wauters, et al. Nucl.Fusion 63 066018 (2023)
[3] T. Wauters, et al. Plasma Phys. Control. Fusion 62 3 (2020)
[4] A. Popov, et al. Plasma Phys. Control. Fusion 57 025022 (2015)
[5] M. G. Senstius, et al. Phys. Plasmas 27 062102 (2020)
Integrated modelling for magnetically confined tokamak plasmas is an indispensable tool in interpreting and in guiding the tokamak experiments. To evolve the plasma profiles (current, densities, temperatures), integrated modelling schemes solve a system of stiff diffusion-advection transport equations, constructed using a set of physical models for equilibrium, transport and sources. It is crucial for the solver of the transport equations to be accurate, numerically stable, conservative and fast converging to ensure robustness and to reduce the number of calls to numerically expensive physical models.
In this work we adapt the conservative Interpolated Differential Operation (IDO) scheme [1] to solve the Generalized Form of Transport Equations [2] (backwards-Euler time advance)
$ \qquad \qquad \qquad \qquad \qquad \qquad \frac{aY-bY_{M}}{\tau} + \frac{1}{c}\frac{\partial}{\partial x} \left( -d\frac{\partial Y}{\partial x} + eY \right) = f-gY, \qquad \qquad \qquad \qquad \qquad \qquad (1) $
which is used to formulate transport equations in ETS [2] for linearized iterations in the non-linear convergence loop. Here, $Y(x),Y_{M}(x)$ are the predicted profile on the present and previous time-steps respectively, $\tau$ is the time-step, and $a(x),b(x),..,g(x)$ are the transport coefficients obtained from physical models. To approximate $Y(x)$ we use 4th-order polynomials centred at the inner spatial grid points {$x_i$}$\phantom{}_{i=2}^{N-1}$ with grid size $N$. The coefficients of $i$-th polynomial are determined from {$Y_{i-1},Y_i,Y_{i+1},Y_{i-1/2}^x,Y_{i+1/2}^x$}, with the profile values $Y_i:=Y(x_i)$ and the cell-integrated values $Y_{i+1/2}^x:=\int_{x_i}^{x_{i+1}}Y(x)\mathrm{d}x$. The spatial derivatives $\frac{\partial Y}{\partial x},\frac{\partial^2 Y}{\partial x^2}$ are available via the approximating polynomials. A system of discretized linear equations is formed from: eq. (1) discretized at {$x_i$}$\phantom{}_{i=2}^{N-1}$; cell-integrals of eq. (1) at intervals {$[x_i,x_{i+1}]$}$\phantom{}_{i=1}^{N-1}$; and the boundary conditions. The free variables are {$Y _ {i+1},Y _ {i+1/2}^x$}$\phantom{}_{i=1}^{N-1}$. We evaluate integrals of the form $\int_{x_i}^{x_{i+1}}a(x)Y(x)\mathrm{d}x$ using Gauss-Legendre quadrature, where the off-grid values of $Y(x),a(x)$ are obtained from the approximating polynomials and a cubic splines representation respectively.
The proposed scheme has 4th order of convergence in space, which enables significant reduction of the spatial grid size while retaining high accuracy, thus minimizing calls to expensive physical models [3]. The proposed scheme is conservative by design, since it directly solves cell-integrated version of eq. (1). Additionally, as compared to IDO-based implementations [3], the scheme proposed here does not require the second-order derivative of diffusion and the first-order derivative of the source term, thus avoiding issues related to instabilities when taking numerical derivatives of transport coefficients computed by physical codes. The non-linear convergence using under-relaxation is similar to that reported in [3].
This work was supported by the Swedish Research Council - Vetenskapsrådet (2021-00182) and has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 - EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] Y. Imai, T. Aioki and K. Takizawa, Journal of Computational Physics 227, 4 (2008)
[2] D. P. Coster, et al., IEEE Transactions on Plasma Science 38, 9 (2010)
[3] J. M. Park, et al., Computer Physics Communications 214, (2017)
P. Costello1
, G. G. Plunk1
, and P. Helander1
1 Max-Planck-Institut für Plasmaphysik, Wendelsteinstraße 1, 17491 Greifswald, Germany
Since the development of gyrokinetic theory, a myriad of instabilities, which lead to
unwanted turbulent transport in tokamaks and stellarators, have been discovered. A recent
series of publications [1, 2, 3] have introduced a novel approach to computing rigorous upper
bounds for the growth rates of gyrokinetic instabilities in flux tube geometry. By maximising
the growth of a chosen gyrokinetic energy measure through optimal perturbations, known as
the optimal modes, these upper bounds represent the fastest-growing instabilities allowed by
the sources and sinks of this energy measure. One such choice for the energy is the so-called
generalised free energy, which is valid in the electrostatic limit. The generalised free energy
has the advantage of containing more information regarding the geometry of the magnetic
field than other energies, such as the Helmholtz free energy considered in [1, 2]. The
generalised free energy was recently considered in [3] for ion-temperature-gradient
instabilities (ITGs) with adiabatic electrons and was found to always give a tighter bound on
ITG growth than the Helmholtz energy. However, so far, the impact of trapped particles on
the optimal modes has not explicitly been dealt with, and thus, the optimal modes found were
largely agnostic to the variation of the magnetic field strength along the field line. In this
study we extend the theory of optimal modes to include a population of trapped particles. We
consider an electrostatic system with fully gyrokinetic ions and bounce-averaged, drift-kinetic
electrons, assuming a bounce time much shorter than the instability timescale. The central
result of this work is a system of integral equations that can be solved for a given flux tube.
The solutions to this system give the optimal modes of the generalised free energy. The
growth rates of the optimal modes in this setting depend on the magnetic field strength, fluxtube metric components, and the curvature of field lines as functions of the field-linefollowing coordinate. We analytically solve this system in a simple square magnetic well, and
numerically solve it for general magnetic field strengths. The resulting optimal modes provide
upper bounds on the growth rates of electrostatic instabilities, including ITGs with kinetic
electrons, trapped-electron modes, and ion-driven trapped-electron modes. Moreover, the
dependence of the upper bounds on magnetic geometry may be exploited in future stellarator
optimisation studies.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the
European Union via the Euratom Research and Training Programme (Grant Agreement No
101052200— EUROfusion). Views and opinions expressed are however those of the author(s) only
and do not necessarily reflect those of the European Union or the European Commission. Neither the
European Union nor the European Commission can be held responsible for them.
References:
[1] P. Helander and G. G. Plunk. Energetic bounds on gyrokinetic instabilities. Part 1. Fundamentals.
Journal of Plasma Physics, 88(2):905880207, April 2022. ISSN 0022-3778, 1469-7807.
doi:10.1017/S0022377822000277.
[2] G. G. Plunk and P. Helander. Energetic bounds on gyrokinetic instabilities. Part 2. Modes of optimal growth. Journal of Plasma Physics, 88(3):905880313, June 2022. ISSN 0022-3778, 1469-7807.
doi:10.1017/S0022377822000496.
[3] G. G. Plunk and P. Helander. Energetic bounds on gyrokinetic instabilities. Part III. Generalized
free energy, January 2023. doi:10.48550/arXiv.2301.00988
Advanced numerical tools play a determinant role in the understanding of plasma dynamics. The nonlinear three-dimensional magneto-hydrodynamic (3D MHD) code SPECYL [1] investigates magnetic self-organisation processes in fusion plasmas in the Reversed Field Pinch (RFP) and in the tokamak configurations. In the past, SPECYL has been used to investigate the Quasi Single Helicity states (QSH) in the RFP devices [2], where the dominance of a single component in the magnetic field spectrum sustains dynamo currents [3] and fosters the plasma confinement: QSH states in RFP plasmas are strongly influenced by the magnetic boundary [4]. Analogous self-organised helical structures have also been studied with SPECYL in the hybrid equilibria of tokamak plasmas [5,6].
We present the implementation and verification against the independent 3D MHD nonlinear code PIXIE3D [7,8] of more realistic boundary conditions, featuring fully 3D boundary flow and including with increasing complexity a thin shell of variable resistivity in contact with the plasma, surrounded by an arbitrarily wide vacuum region that separates it from an outer ideal conductor [9]. This is a versatile formulation, capable of reproducing different experimental conditions: from an ideal wall attached to the plasma, to a free interface between plasma and vacuum, to a physical wall of finite resistivity at plasma boundary.
In the free plasma-vacuum interface regime, the 3D boundary flow is proven to have an essential role in reproducing free-boundary instabilities, such as the external kink modes in the tokamak configuration. Remarkably, the new implementation of SPECYL’s BCs is indeed shown to be capable of achieving robust and self-consistent agreement with the theoretical figures of merit of linear MHD [10], with great freedom of choice in the initial equilibrium and without the need of a pseudo-vacuum boundary region.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
[1] S. Cappello, D. Biskamp: “Reconnection processes and scaling laws in reversed field pinch magnetohydrodynamics” Nucl. Fus., 36, 571 (2008)
[2] D. Bonfiglio, M. Veranda, S. Cappello, D. F. Escande, L. Chacόn: “Experimental-like helical self-organization in reversed-field pinch modeling” PRL, 111, 085002 (2013)
[3] S. Cappello, D. Bonfiglio, D. F. Escande: “Magnetohydrodynamic dynamo in reversed field pinch plasmas: electrostatic drift nature of the dynamo velocity field” Phys. Plasmas, 13, 056102 (2006)
[4] M. Veranda, D. Bonfiglio, S. Cappello, et al.: “Magnetohydrodynamics modelling successfully predicts new helical states in reversed-field pinch fusion plasmas” Nucl. Fus., 57, 11, 116029 (2017)
[5] D. Bonfiglio, M. Veranda, S. Cappello, D. F. Escande, L. Chacόn: “Helical self-organization in 3D MHD modelling of fusion plasmas” Plasma Phys. Contr. Fusion, 57, 044001 (2015)
[6] P. Piovesan, D. Bonfiglio, et al.: “Role of a continuous MHD dynamo in the formation of 3D equilibria in fusion plasmas” Nucl. Fus., 57, 076014 (2017)
[7] L. Chacόn: “A non-staggered, conservative, ∇⋅b=0 finite volume scheme for 3d implicit extended magnetohydrodynamics in curvilinear geometries” Computer Phys. Communications, 163, 143-171 (2004)
[8] D. Bonfiglio, L. Chacόn, S. Cappello: “Nonlinear threedimensional verification of the SPECYL and PIXIE3D magnetohydrodynamics codes for fusion plasmas” Phys. Plasmas, 17, 082501 (2010)
[9] L. Spinicci: “3D Nonlinear MHD modelling studies: Plasma Flow and Realistic Magnetic Boundary Impact on Magnetic self-organisation in Fusion Plasmas”, PhD Thesis, chap. 7-8 (2023)
[10] J. P. Freidberg: “Ideal MHD”, chap. 8 and 11, Cambridge Univ. Press (2014)
The tokamak H-mode regime of confinement relies on the formation of an edge transport barrier, resulting in the appearance of the so-called “pedestal” in the plasma profiles. Understanding the mechanisms responsible for the pedestal evolution and features is crucial for optimizing fusion performance in the next-generation devices, such as ITER. However, a complete understanding of the role of turbulent transport in the pedestal region is lacking.
In this study, we focus on characterizing turbulent transport at the top of the pedestal in JET discharges in different regimes: a type-I edge-localized modes (ELM), and two baseline small-ELM (BSE) regimes [1]. To achieve this goal, we performed gyrokinetic stability analysis using the local versions of the GENE and GKW codes.
A detailed characterization of the most unstable modes around the pedestal top positions of the three discharges has been carried out. At the ion-scale, we found that kinetic-ballooning modes are dominant in the type-I ELM regime, while the BSE regimes are characterized by a hybrid TEM/ITG branch. At the electron-scale, both regimes are dominated by the toroidal and slab ETG branches [2]. Moreover, the sub-dominant modes spectra have been obtained by performing gyrokinetic eigenvalue computation, revealing that a rich variety of modes of different nature are destabilized. Starting from linear results, a quasi-linear (QL) model is applied to get an estimate of the turbulent fluxes to be compared with experimental values. Moreover, the impact of the electron scales modes in the QL estimates will be assessed.
Finally, to gain physical insights on magnetized plasma turbulence in regimes with sub-dominant modes, a mode decomposition technique is being developed. Firstly, we selected as reference case the Waltz standard case, linearly characterized by unstable ITG and TEM branches. Preliminary results on the decomposition of turbulent field structures will be shown. After its validation, the modal decomposition will be applied to non-linear gyrokinetic simulations with pedestal-like parameters.
The EC launcher conceptual design is moving towards an engineering design that will satisfy all physics requirements under the various system constraints, including maximum flexibility in case of further refinements of plasma scenarios and tasks [1]. In the current DEMO reference scenario, for the specific magnetic field and plasma density/temperature adopted, the gyrotron frequencies considered for fulfilling the EC-related tasks (as, e.g., plasma breakdown and ramp-up/down, central heating and MHD stabilization) are 136, 170 and 204 GHz.
During 2022, part of the investigation has been focused to the evaluation of the present design capabilities in case of a modification of the DEMO reference scenario (with either increase or decrease of the central magnetic field), regarding the tasks of EC-assisted breakdown and core plasma heating. For plasma breakdown, the primary goal is to direct EC beams to the region where the null point of the magnetic configuration is programmed to appear, whereas for core heating we also consider beams with frequency higher than 204 GHz (up to 240 GHz) for accommodating scenarios with elevated values of the central magnetic field.
This work presents and analyses beam tracing numerical simulations of the aforementioned scenarios, which have been performed in order to establish the parameter regions where the EC system performance is satisfactory with respect to the current DEMO requirements regarding the specific tasks [2]. The computations have been performed with the paraxial WKB beam tracing code TORBEAM [3], for magnetic equilibrium, plasma temperature and density profiles related to the DEMO design currently under consideration, and for selected EC wave frequencies, initial beam parameters (width, curvature radius and propagation direction) and launching positions, and provide the wave parameters related to DEMO heating and current drive (deposition radius, absorbed power and total driven current).
This work has been carried out within the framework of the EUROfusion Consortium, partially funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). The Swiss contribution to this work has been funded by the Swiss State Secretariat for Education, Research and Innovation (SERI). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union, the European Commission or SERI. Neither the European Union nor the European Commission nor SERI can be held responsible for them.
References:
[1] M. Q. T. Tran, et al., Fusion Engineering and Design 180, 113159 (2022)
[2] M. Siccinio, et al., Fusion Engineering and Design 179, 113123 (2022)
[3] E. Poli, et al., Computer Physics Communications 136, 90 (2001)
Università degli Studi Padova - Consorzio RFX
The strong uncertainty related to the estimate of the viscosity coefficient represents an important challenge in the application of magneto-hydrodynamics (MHD) simulation results to laboratory plasmas [1]. This is particularly relevant in the contest of the reversed-field pinch (RFP) configuration, where the viscosity together with the resistivity jointly rules the transition between different dynamical plasma regimes in non-linear MHD simulations, by means of the dimensionless Hartmann number [2] and the rise of helical self-organized states, exploiting non-ideal boundary conditions [3].
In the SpeCyl non-linear MHD code [4], the modelling of the viscosity is quite simple, considering a constant and scalar viscosity coefficient. In this work, we present a sensitivity study concerning the impact of different viscosity radial profiles. In particular, a profile inspired by the Braginskii perpendicular viscosity is implemented in SpeCyl. This choice is motivated by the fact that the MHD field instabilities relevant for the RFP configuration dynamics (resistive-kink / tearing modes) are active in the direction perpendicular to the magnetic field. Moreover, the analysis of a wide database of RFX-mod shots [5] has highlighted that the perpendicular Braginskii coefficient (with a suitably anomaly factor [6]) represents the contribution to the plasma viscosity which provides the best agreement, in the comparison of SpeCyl numerical simulations and RFX-mod data [7].
The non-monotonous viscosity displays the existence of local effects specifically due to the profile, in addition to the more consistent ones, due to the absolute value of the viscosity [8]. In particular, the Braginskii-like viscosity profile causes a localized damping of plasma flow in the regions where the viscosity is stronger, close to the plasma edge. This results in the reduction of the flow shear, in turn allowing the enhancement of ‘intermediate-resonant’ magnetic field modes amplitude, confirming a basic picture of interplay where the plasma flow counteracts the growth of magnetic instabilities [9]. Furthermore, a preliminary analysis on the magnetic topology is carried out, highlighting (as main result) the lowering of the connection length, interpreted as consequence of the higher MHD activity caused by the non-uniform profile.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion).
References:
[1] D. Montgomery, Plasma Physics and Controlled Fusion, 34, 1157 (1992)
[2] S. Cappello, D.F. Escande, Physical Review Letters, 85, 3838 (2000)
[3] D. Bonfiglio, M. Veranda et al., Physical Review Letters, 111 085002 (2013)
[4] S. Cappello, D. Biskamp, Nuclear Fusion, 36, 571 (1996)
[5] G. Spizzo et al., Nuclear Fusion, 55, 043007 (2015)
[6] R. Fridström et al., Physical Review Letters, 120, 225002 (2018)
[7] N. Vivenzi et al., Journal of Physics: Conference Series 2397, 012010 (2022)
[8] M. Veranda et al., Nuclear Fusion, 60, 016007, (2020)
[9] R. Fitzpatrick, Physics of Plasmas, 5, 3325, (1998)
United Kingdom Atomic Energy Authority, Culham Centre for Fusion Energy, Culham Science Centre
Simulations of transport parallel to open magnetic field lines have been a source of insight into the complex physics of the Scrape-Off Layer and divertor for a long time, ranging from 1D fluid model simulations of detachment [1] to various kinetic models [2,3,4] looking at modifications of parallel transport coefficients due to kinetic effects.
Most existing approaches focus on a specific set of equations when tackling the parallel transport problem. While this has produced a great number of important results, many applications, such as reduced modelling for machine learning and control, require frequent fundamental modifications to the equations. Thus a flexible framework easily accessible to a wide range of users is desirable.
This contribution will aim to present one such framework - ReMKiT1D(Reactive Multifluid and Kinetic Transport in 1D)[5], geared towards solving systems of 1D fluid equations with support for including electron kinetic effects (expanding on previous work[6]) and with in-built support for Collisional-Radiative Modelling. The software design behind the framework’s Modern Fortran back-end and Python interface enabling flexibility and rapid user-friendly modifications of models will be presented. Use cases and simple illustrative examples will also be shown, together with some widely used code benchmarking and verification tests. Finally, the ongoing and planned uses of the framework will be presented, with a special focus on the interplay between electron kinetic effects and various multi-fluid aspects common to the Scrape-Off Layer.
This work used the ARCHER2 UK National Supercomputing Service (https://www.archer2.ac.uk). This work has been part-funded by the EPSRC Energy Programme [grant number EP/W006839/1].
References:
[1] Dudson, B. D. et al. Plasma Phys. Control. Fusion, 61(6) (2019)
[2] Tskhakaya, D. et al. Contrib. Plasma Phys., 48(1–3), 89–93 (2008)
[3] Chankin, A. V. et al. Plasma Phys. Control. Fusion, 60 (2018)
[4] Mijin, S. et al. Plasma Phys. Control. Fusion, 62(9) (2020)
[5] Mijin, S. et al. ReMKiT1D - A framework for building reactive multi-fluid models of the tokamak Scrape-Off Layer with coupled electron kinetics in 1D – in preparation
[6] Mijin, S. et al. Computer Phys. Comm., 258, 107600 (2021)
Max Planck Institute for Plasma Physics
Energetic particles will play a central role in future burning plasma experiments, and their
confinement is an important aspect for a fusion reactor. Understanding the effects of energetic
particles (EPs) is essential, as they can strongly interact with the main plasma and drive
magnetohydrodynamic (MHD) instabilities. One notable example is the fishbone instability,
which arises from an internal kink mode with n=1 resonance to the precessional frequency of
EPs.
To investigate wave-particle interaction phenomena in tokamaks, numerical simulations serve
as a major research tool. This contribution focuses on applications and developments of the
nonlinear extended MHD code JOREK [1], which includes a kinetic module specifically for
energetic particles. In the present hybrid kinetic-MHD model implemented in JOREK, the
EPs are simulated using the particle-in-cell technique, while the bulk plasma is treated as a
fluid by solving the equations of MHD. A full-f formulation for the kinetic particles and an
anisotropic pressure coupling scheme to the fluid are used.
The kinetic particle module was applied to study toroidal Alfvén eigenmodes (TAE) and was
successfully validated in the linear regime [2]. The current setup is now being used to
investigate the fishbone instability both in the linear and in the nonlinear regime. Initial linear
simulations demonstrate good agreement with a benchmark case using the M3D-C1-K code
which is explained in [3].
Furthermore, a new hybrid kinetic-MHD model is currently under development for JOREK. It
aims at treating also the thermal ions kinetically and only describing the electrons as fluid.
The new model is based on the standard drift model. The ion density, parallel velocity and
pressure are determined through projections from the kinetic particles, while the electron
pressure and the MHD velocity are calculated using fluid equations. The new model is under
development in JOREK and we present the key concepts and the present state in this
contribution.
[1] M. Hoelzl et al. “The JOREK non-linear extended MHD code and applications to large-scale
instabilities and their control in magnetically confined fusion plasmas”. In: Nuclear Fusion
61.6 (May 2021), p. 065001. doi: 10.1088/1741-4326/abf99f. url: https://dx.doi.org/10.1088/1741-
4326/abf99f.
[2] T. J. Bogaarts et al. “Development and application of a hybrid MHD-kinetic model in JOREK”.
In: Physics of Plasmas 29.12 (2022), p. 122501. doi: 10.1063/5.0119435. eprint: https:
//doi.org/10.1063/5.0119435. url: https://doi.org/10.1063/5.0119435.
[3] C. Liu et al. “Hybrid simulation of energetic particles interacting with magnetohydrodynamics
using a slow manifold algorithm and GPU acceleration”. In: Computer Physics Communications, 275
(2022), p. 108313, url: https://doi.org/10.1016/j.cpc.2022.108313
University of Oxford
We investigate the saturation of turbulence in a three-field, fluid model of a magnetised plasma in a Z-pinch magnetic geometry. The model is derived by taking a long-wavelength limit of gyrokinetics and subsequently ordering the electron-temperature-gradient (ETG) to be much larger than all other equilibrium gradients, while still retaining the curvature and magnetic-field-gradient drifts. This system is linearly unstable to two-dimensional, curvature-mediated ETG modes, which themselves are known to undergo a secondary instability and generate zonal-flows, as seen in the Hasegawa-Mima (HM) system. By including the linear terms associated with the parallel physics in the secondary instability calculation, we find a three-dimensional branch to the HM secondary instability. The unstable secondary modes are Alfvénic, and their growth rate is comparable to that of the zonal-flows. We present numerical evidence that the level of heat transport in simulations is strongly tied to whether this Alfvénic secondary instability is adequately resolved. Further, we argue that this is the mechanism by which our model breaks up its two-dimensional, unstable structures and establishes a critically-balanced cascade of free energy.
The particle-in-cell code PICLS is a full-f finite element tool intended to simulate turbulence in the tokamak scrape-off layer using gyrokinetic ions and drift-kinetic electrons. Up until now however, PICLS has been a purely electrostatic code with a prescribed background magnetic field. This approach is not perfectly suited to represent unstable regimes occurring in the scrape-off layer, since although $\beta = 2\mu_0p/B^2$ can be small, turbulence there is still dominated by electromagnetic effects [1]. In order to capture those effects, an Ampère-solver is added to the code and the evolving magnetic field is taken into account in the particle pusher stage. In order to combat the Ampère-cancellation problem that arises from the Hamiltonian canonical Lagrangian formulation that PICLS is based on, we combine the newly added Ampère-solver with a pullback scheme akin to the one used in ORB5 [2]. This improved version of PICLS opens up possibilities in simulating $\beta$-dependent ITG-KBM transitions like illustrated in ref. 3 for the codes GENE, GKW, EUTERPE and ORB5, shear Alfvén waves, microtearing modes and more.
The Divertor Tokamak Test (DTT) [1],[2] is a superconducting device under construction in Frascati, Italy. DTT was proposed to assess the performance of a conventional ITER divertor and address the power exhaust issue that will affect future fusion devices as DEMO. DTT will be equipped with three auxiliary heating systems, including a Neutral Beam Injection (NBI) system. DTT NBI is a high-energy and high-power neutral beam system (ENBI = 250-510 keV, PNBI ≤ 10 MW) that generates a population of suprathermal particles, known as Energetic Particles (EPs). EP confinement is crucial to improve plasma performances and avoid EP losses to the machine first wall.
In this contribution, we characterize the beam-plasma interaction in axisymmetric magnetic field for different planned DTT plasma scenarios. The aim of the present investigation is to explore beam EP confinement in DTT plasmas, extending previous analyses [3] of the reference plasma scenario [4] to configurations at reduced toroidal magnetic field or current, taking into account the possibility of reducing NBI energy and power. The orbit-following Monte Carlo ASCOT suite of codes [5] is used. In particular, BBNBI [6] is used to evaluate the fraction of shine-through losses, i.e. beam particles lost to the first wall before ionization occurs. Through BBNBI we also obtain the information on newly-born fast ions required to populate the topological map built in the Constant of Motion (CoM) phase space [7]. This topological map is used to predict initial EP orbits and prompt losses, i.e. losses happening before collisions with background plasma occur. The ASCOT code is instead used to simulate the full slowing down collisional process and in order to gather information about EP distribution functions, beam contribution to the plasma in terms of power, current and momentum, and possible additional loss channels, e.g. orbit losses.
Predictive beam-plasma interaction modelling is essential to explore the use of the NBI system on DTT plasmas, defining its operability. This contribution presents a further step to optimize future DTT operations by contributing to the understanding of beam EP behavior in DTT.
References:
[1] DTT Interim Design Report, ENEA (2019).
https://www.dtt-dms.enea.it/share/s/avvglhVQT2aSkSgV9vuEtw
[2] R. Ambrosino, et al., Fusion Eng. Des. 167 (2021), 112330
[3] P. Vincenzi, et al., Fusion Eng. Des 189 (2023), 113436
[4] I. Casiraghi, et al., Plasma Phys. Control. Fusion 65 (2023), 035017
[5] E. Hirvijoki, et al., Comput. Phys. Commun. 185 (2014) 1310-1321
[6] O. Asunta, et al., Comput. Phys. Commun. 188 (2015) 33-46
[7] R. B. White. The theory of toroidally confined plasmas. Imperial college press (2014)
A linear theory of non local transport in relativistic unmagnetized plasmas is presented. The relativistic effects are due to high electron thermal energy. The relativistic Fokker-Planck equation is analytically solved for perturbed plasmas with respect to the global thermal equilibrium defined by the Maxwell–Boltzmann–Jüttner electron distribution function (EDF). The perturbed EDF is calculated for arbitrary collisionality range defined by the parameter λ_ei/L where λ_ei is the electron-ion mean free path, and L is the homogeneity scale length. and for arbitrary relativistic regime defined by the relativistic parameter z=m_e c^2/T, where m_e is the electron mass, T is the electron temperature (in energy units) and c is the speed of light. Using the Branginskii notations [1] we deduced in the Fourier space (x↔k), the nonlocal electron heat flux, δq and the transfer of momentum from ions to electrons, δR_ei:
δq=-K_T n_e c ik/k δT+α_u n_e Tδu+α_V n_e δV and δR_ei=-β_T n_e ikδT+β_u c/λ_ei n_e mδu where n_e is the electron density, δT is the perturbed temperature, δV is the perturbed fluid velocity, δu is the relative velocity of electron with respect to ions, K_T is the thermal conductivity, α_(u,V) are the relative and convective velocities respectively; β_T and β_u are the coefficients of the thermal force and the friction force respectively. Note here that these coefficients depend on both the Knudsen number kλ_ei and the relativistic parameter z. We illustrate our result in figure given below for the thermal conductivity K_T with respect of kλ_ei for two values of the relativistic parameter z:
We can see that the relativistic effects tend to increase the thermal conductivity and thus the dissipation of plasma modes. In the non relativistic limit and high collisonal regime kλ_ei<<1, the classical thermal conductivity [1], the frictional and thermal forces [1] were recovered. Furthermore, in the collisonless regime and arbitrary temperature values the result reported in Refs. 2 and 3 were also retrieved. New transport coefficients valid for arbitrary relativistic regime and collisionality were derived which can be used as reliable closure relations in fluid equations.
This work has been carried out within the framework of the PRFU 2023-2026 (Projet de Recherche et de Formation Universitaire) and under grant agreement No B00L02UN160420230002.
References:
[1] S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, NewYork, 1965),Vol. 1,p. 205.
[2] K. Bendib-Kalache, A. Bendib, and K. Mohammed El Hadj, Phys. Rev. E 82, 056401 (2010).
[3] B. Touil, A. Bendib and K. Bendib-Kalache Phys. Plasmas 24, 022111 (2017).
The presence of non-axisymmetric perturbations of an axisymmetric toroidal magnetic field results in the chaoticity of the magnetic field lines and strongly affects the charged particle motion and therefore the particle, energy and momentum transport in toroidal plasmas [1-2]. Particle chaoticity is determined by resonance conditions relating the unperturbed Orbital Frequencies of the particles with the toroidal and poloidal numbers of the perturbative modes [3]. The Guiding Center (GC) motion [4] of low-energy particles approximately follows the magnetic field lines so that magnetic and kinetic chaos have similar characteristics. However, higher-energy particles may undergo large drifts across the magnetic field lines and the chaoticity characteristics of their GC motion can be quite different from those of the underlying magnetic field. In fact, kinetic chaos may take place in different spatial locations from those of magnetic chaos, and the degree as well as the extent of the two chaoticities can be different depending on the particle orbit width. The systematic comparison of magnetic and kinetic chaos necessitates: (a) the efficient detection and quantification of chaos, and (b) the compact representation of particle orbits in a kinetic parameter space.
In this work, we introduce the Smaller Alignment Index (SALI) [5-6], utilized for the first time in the context of plasma physics, as an efficient measure for detecting and quantifying magnetic and kinetic chaos, and discuss its advantages in comparison with other standard chaos measures. Magnetic and kinetic chaos are visually compared in terms of appropriate Poincare surfaces of section for characteristic cases of low-energy, thermal, and higher-energy particles. Moreover, we construct detailed diagrams by assigning a SALI value to every point in the kinetic parameter space uniquely representing a particle orbit in terms of three Constants of the Motion, namely the energy, the canonical toroidal momentum and the magnetic moment. These diagrams contain aggregated information for the role of various non-axisymmetric perturbations on the chaoticity of specific particles characterized by being trapped or passing, and confined or lost, providing a valuable tool for understanding the interplay between perturbations and for transport prediction and control.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] A. Rechester and M. Rosenbluth, Phys. Rev. Lett 40, 38 (1978)
[2] K. Shinohara, et al., Nucl. Fusion 58, 082026 (2018)
[3] Y. Antonenas, G. Anastassiou and Y. Kominis, J. Plasma Phys. 87, 855870101 (2021)
[4] R. B. White and M. S. Chance, Phys. Fluids 27, 2455–2467 (1984)
[5] Ch. Skokos, J. Phys. A 34, 10029-10043 (2001)
[6] Ch. Skokos, Lect. Notes Phys. 790, 63-135 (2010)
In this talk, I will give an overview of the history and present state of stellarator optimisation. Wendelstein 7-X was optimised for good magnetic flux surfaces, small Pfirsch-Schlüter and bootstrap currents, high MHD beta limit, and good neoclassical confinement. Most of these properties have already been confirmed experimentally, giving confidence that theory-based optimisation can pay off handsomely in terms of plasma performance [1].
Thanks to the development of stellarator theory and computational power, it is becoming possible to optimise stellarators for additional properties. Experiments in LHD and W7-X have shown that most of the energy losses are due to plasma turbulence, which motivates recent efforts trying to reduce such losses. I will describe these efforts and discuss how they fit into the greater picture of neoclassical optimisation by means of quasisymmetry or quasi-isodynamicity.
[1] C.D. Beidler et al., Nature 596, 221 (2021).
The stellarator is an attractive concept for magnetic confinement fusion reactors, as it offers some advantages over the tokamak. Whereas in tokamaks part of the magnetic field is produced by driving a large inductive current in the plasma, the magnetic field of the stellarator is generated by external coils, avoiding the risk of current-driven instabilities and making steady-state operation easier. However, the lack of axisymmetry of the stellarator implies that good confinement is not guaranteed but requires careful tailoring (optimization) of the magnetic configuration. Wendelstein 7-X (W7-X) [1] is the first large stellarator whose magnetic configuration has been obtained using numerical optimization, and its reduced neoclassical transport of thermal species has been recently demonstrated [2]. This result has been a great success for the stellarator line of research and for the optimization effort in particular. However, the confinement of energetic particles, which is crucial for a reactor and improves with $\beta$ in W7-X, is barely acceptable, and only at high $\beta$, around 4%. Furthermore, the first experimental campaigns in W7-X have shown that turbulent transport limits its performance in most plasma regimes.
This work presents a new stellarator configuration [3] of the quasi-isodynamic (QI) type [4] (the same class to which W7-X belongs) that improves in fast ion confinement and turbulent transport with respect to W7-X while keeping all its other good properties. The new configuration, with four periods and aspect ratio A~10, has been obtained using the optimization suite of codes STELLOPT [5], in which the code KNOSOS [6] has been included and employed to compute novel orbit-averaged quantities that serve as proxies for the confinement of fast ions [7]. Monte Carlo calculations with ASCOT [8] confirmed that the confinement of fast ions is excellent not only for reactor-scale plasmas with $\beta$~4% but also for moderate values of $\beta$ (~1.5%), which can be transiently necessary for the operation of a reactor. The effective ripple is smaller than 0.5% in the plasma core and a preliminary evaluation allows us to foresee a small bootstrap current, as expected for QI devices. A significant magnetic well ensures the Mercier MHD stability in the whole plasma volume and the Ballooning stability up to $\beta$=5% has also been demonstrated using the code
COBRA [9]. The configuration satisfies the flat mirror property, recently introduced [10]. Partly as a consequence of its flat mirror, this configuration also possesses the maximum-J property already at low plasma $\beta$, exhibits reduced Trapped Electron Mode (TEM) turbulent transport, and its neoclassical properties are expected to be robust against error fields. Preliminary sets of filamentary coils that faithfully generate this configuration will also be presented.
[1] G. Grieger et al. Fus. Technol. 21, 1767 (1992).
[2] C. Beidler et al. Nature 596, 221 (2021).
[3] E. Sánchez et al. Nucl. Fusion 63 066037 (2023).
[4] L. S. Hall et al. Phys. Fluids 18, 552 (1975). P. Helander et al. Plasma Phys. Control. Fusion 51, 055004 (2009).
[5] S. A. Lazerson et al. (www.osti.gov/biblio/1617636-7pt).
[6] J. L. Velasco et al. J. Comput. Phys. 418, 109512 (2020).
[7] J. L. Velasco et al. Nucl. Fusion 61, 116059 (2021).
[8] E. Hirvijoki et al. Comput. Phys. Commun. 185, 1310 (2014).
[9] R. Sánchez et al. J. Comput. Physics 161, 576 (2000).
[10] Velasco et al. To be submitted.
See attached pdf file.
A relatively simple way to evaluate bootstrap current in stellarators involves the use of the long mean free path asymptotic formula by Shaing and Callen [1]. This formula contains all the information about device geometry in a geometrical factor, independent of plasma parameters. This method is particularly suited for stellarator optimization, where multiple quick estimates of bootstrap current are needed.
Interestingly, even though there have been a few different derivations of this formula [1,2,3], all leading to the same or almost the same result, these are never reproduced by the numerical drift kinetic equation (DKE) solvers in the $1/\nu$ regime. This regime is relevant for the electron component in fusion reactors [4]. Conversely, DKE solvers show the trend of bootstrap current to converge towards the Shaing-Callen limit in the presence of a radial electric field [4], which is not explicitly accounted for in the existing derivations of this formula.
Qualitatively, the reasons for such anomalous behavior have been recently identified in Ref.[5] using an adjoint approach, where Onsager symmetry between bootstrap coefficient and Ware-pinch coefficient has been applied. In the present report, the convergence of bootstrap current to the Shaing-Callen limit is studied both analytically and semi-analytically, as well as numerically with the aid of the NEO-2 code [6]. The analysis remains in agreement with the qualitative observations of Ref.[5]. In addition to this, special cases where bootstrap current converges to this limit in the $1/\nu$ regime are demonstrated. The analysis shows that in realistic toroidal stellarator geometry, bootstrap current diverges in the $1/\nu$ regime with decreasing collisionality as $ν^{-1/5}$, and, conversely, it converges to the Shaing-Callen limit in the presence of banana orbit precession (particularly due to the radial electric field) as $\nu^{3/5}$.
A method for evaluating bootstrap current in the presence of a radial electric field is also discussed, which entails the use of rapid computations of this current by NEO-2 in the $1/\nu$ regime.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 -- EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] K. C. Shaing and J.D.Callen, Phys. Fluids 26 (1983) 3315
[2] A. H. Boozer and H.J.Gardner, Phys. Fluids B 2 (1990) 2408
[3] P. Helander, J. Geiger, H.Maassberg, Phys. Plasmas 18 (2011) 092505
[4] C. D. Beidler et al, Nucl. Fusion 51 (2011) 076001)
[5] C. D. Beidler, "Final Report on Deliverable S2-WP19.1-T005-D005 Neoclassical and fast-ion transport" (2020) (MPG)
[6] W.Kernbichler et al, Plasma Phys. Control. Fusion 58 (2016) 104001
Recent advances in stellarator optimization have led to unprecedented improvement in neoclassical transport in Wendelstein 7-X [1] such that anomalous transport is contributing a significant portion of the transport, especially in the Scrape-Off-Layer (SOL) [2]. Here, we present isothermal fluid turbulence simulations using the BOUT++ framework [3,4] in the edge and scrape-off-layer (SOL) of an analytic stellarator configuration with an island divertor, thereby providing the first numerical insight into edge turbulence near islands in a stellarator. The overall transport follows a ballooning nature due to the dominant 1/R component of the curvature, but fluctuations are present throughout the island divertor region. It is determined that the island chain at the separatrix influences the amplitude and radial extent of the fluctuations. Fluxes not associated with ballooning transport can be observed near island X-points. The fluctuations exhibit a predominantly positive skewness, indicating blob-like perturbations for the transport into the outer SOL. It is determined that the separatrix is generally more correlated with the outer SOL, whereas the O-point regions
are decorrelated. The implications for W7-X, and the extension of a multi-fluid,global fluid turbulence framework to stellarator geometries will also be discussed.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 - EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
Prepared in part by LLNL under Contract DE-AC52-07NA27344. LLNL-JRNL-849204.
References:
[1] C. Beidler et al., Nature 596, 221 (2021)
[2] N. A. Pablant et al., Physics of Plasmas 25, 022508 (2018)
[3] B. D. Dudson et al., Phys. Computer Physics Communications 180, 1467 (2009)
[4] B. Shanahan, B Dudson and P Hill, hPlasma Physics and Controlled Fusion 61, 025007 (2019)
[5] B Dudson et al., arXiv:2303.12131 (2023)
Advanced transport models for energetic particles
Ph. Lauber 1, M. Falessi 2, A. Biancalani 3, A. Bottino 1, S. Briguglio 2, N. Carlevaro 2, V. Fusco 2,T. Hayward-Schneider 1, F. Holderied 1, A. Könies 4, Y. Li 2,6,Y.-Y. Li2 ,5,6, G. Meng 1, A. V. Milovanov 2, G. Montani 2, V.-A. Popa 1, S. Possanner 1, G. Vlad 2, X. Wang 1, M. Weiland 1, A. Zocco 4, F. Zonca 2
1 Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany
2 ENEA, Fusion and Nuclear Safety Department, C. R. Frascati, I-00044 Frascati (Rome), Italy
3 Léonard de Vinci Pôle Universitaire, Research Center, F-92916 Paris La Défense, France
4 Max-Planck-Institut für Plasmaphysik, D-17491 Greifswald, Germany
5 National Supercomputing Center of Tianjin, Tianjin, China
6 CREATE Consortium, Via Claudio 21, Napoli, Italy
In addition to increasingly realistic non-linear global simulations [1, 2, 3], a hierarchy of theory-based re- duced models is needed to complement the predictions concerning the performance of future burning plas- mas. Large parameter scans, sensitivity studies and multi-scale physics connecting energetic particle trans- port with neoclassical (transport) time scales require tools that go beyond what is presently feasible with first-principles numerical codes. In the view of this challenge we report in this work on the construction, validation and application of reduced energetic particle (EP) transport models pursued within the framework of the EUROFusion enabling research project ATEP (Advanced Transport models for EPs).
The general theoretical framework introduces the concept of long-lived toroidally symmetric structures in the particle phase space (phase space zonal structures, PSZS) that are separated from fast fluctuating contri- butions [4, 5, 6, 7]. Comprehensive transport equations have been derived that are designed to capture the evolution of PSZSs on collisional transport time scales while keeping the important non-linear interactions in a consistent multi-scale description. The model captures physics beyond simpler models (critical gradi- ent, kick model, quasi-linear) that, however, can be recovered in the appropriate limits. A generalisation of the theory to stellarator geometry has been started [8]. The DAEPS code [9] and the EP-stability workflow (EP-WF) [10] based on the code chain HELENA-LIGKA-HAGIS [11, 12, 13] deliver the necessary input for the PSZS transport equations, i.e. the orbit-and zonally-averaged response for a selected set of markers to a prescribed set of Alfvénic perturbations. In addition, neoclassical transport coefficients [14] for the same set of markers, and general EP distribution functions as calculated by various heating workflows[15] are pro- vided via standardised IMAS interfaces. The transport equation is then consistently evolved, or EP diffusion coefficients are evaluated for the use in standard transport codes. In addition, a 1d reduced model based on the beam-plasma bump-on-tail paradigm that is designed to go beyond the quasi-linear approximation and thus forecast possible EP transport transitions such as avalanching has been successfully compared to the LIGKA/HAGIS model. The formulation of the models allows one to carry out detailed analyses of transport scaling laws (diffusive/non-diffusive) for both Alfvénic gap and energetic particle modes using Lagrangian coherent structures (LCS) [16, 17]. The verification of the reduced models is carried out via comparison to numerical codes in the appropriate limits (HYMAGYC, (X)HMGC, STRUPHY, ORB5, HAGIS/LIGKA [18, 19, 20]). To that end, the implementation of PSZS diagnostics in the various codes [21, 22] provides a natural connection point for benchmarking with the reduced models. Several time-dependent scenarios from present-day and future experiments (in particular AUG [23], JT-60SA, TCV, DTT, JET,ITER) have been collected and are being analysed.
In summary, the PSZS transport theory and the LIGKA-HAGIS workflow ATEP provide a new and promis- ing approach to address the challenge of describing EP transport in fusion plasmas. With their ability to capture multi-scale physics, account for non-linear interactions, and forecast transport transitions, these re-
duced models have significant potential to enhance our understanding of EP transport.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 – EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] A. Mishchenko et al, Proc. 29th IAEA FEC, London UK (2023)
[2] T. Hayward-Schneider et al, Proc. 29th IAEA FEC, London UK (2023)
[3] A. Biancalani et al, Proc. 29th IAEA FEC, London UK (2023)
[4] F. Zonca et al, New Journal of Physics 17 013052 (2015)
[5] M V Falessi et al, Phys. Plasmas 26 022305 (2019)
[6] M. V. Falessi et al https://arxiv.org/pdf/2306.08642.pdf (2023)
[7] M V Falessi et al, Proc. 29th IAEA FEC, London UK (2023)
[8] A. Zocco et al, Journal of Plasma Physics 89, 905890307 (2023)
[9] Y. Li et al, PoP 27 062505, (2020).
[10] V.-A. Popa et al, submitted (2023)
[11] G.T.A. Huysmans et al, Proc. CP90 Conf. on Comp. Phys. Proc., 371, (1991)
[12] Ph. Lauber et al, J. Comp. Phys., 226/1 (2007)
[13] S.D. Pinches, Comp. Phys. Comm., 111 (1998)
[14] G Meng et al, Proc. 29th IAEA FEC, London UK (2023)
[15] M. Schneider et al, TH/6-1 Proc. 27th IAEA FEC, 2018
[16] N. Carlevaro et al, Proc. 48th EPS Conference on Plasma Physics, (2022)
[17] A. V. Milovanov et al, Phys. Rev. E 103 052218 (2021)
[18] G. Vlad et al, Nucl. Fusion 61 116026 (2021); Proc. 29th IAEA FEC, London UK (2023)
[19] X. Wang et al, PoP. 29 032512 (2022)
[20] F. Holderied et al, JCP 433 110143 (2021); 464 111329 (2022)
[21] S. Briguglio et al, PoP 21 112301 (2014)
[22] A. Bottino et al, J. Phys.: Conf. Ser. 2397 012019 (2022)
[23] Ph. Lauber et al, Proc. 27th IAEA FEC (2018)
In this contribution, we introduce the formulation of a self-consistent transport theory, based on the evolution of the renormalized plasma equilibrium in the presence of a finite level of electromagnetic fluctuations, which we call the zonal state (ZS). This formulation involves the derivation of the nonlinear equations for 1) the PSZS that are phase space structures undamped by collisionless processes and provides a proper definition of nonlinear equilibrium distribution function; 2) its counterpart, zonal fields (ZFs) that corresponds to the long-lived components of electromagnetic fields, and 3) toroidal symmetry breaking perturbations. The study employs a comprehensive gyrokinetic transport theory formulation to derive governing equations for the ZS. By employing the Chew Goldberger Low (CGL) description, the study effectively separates microscale structures from macro-/meso-scale components of the equilibrium. This research builds upon the Phase Space Zonal Structure (PSZS) concept [1-6] and provides a general gyrokinetic phase space transport theory that characterizes nonlinear dynamics of weakly collisional plasmas. The study addresses the limitations of current numerical frameworks used for simulations of EP-driven instabilities and transport in fusion devices. We note that the PSZS transport theory employed here offers a phase space perspective, which further extends the understanding of plasma transport processes. In this talk, we will focus on the derivation of the governing equation for the ZS, which encompasses the evolving renormalized nonlinear equilibrium evolving in transport time scale in as well as toroidal symmetry-breaking fluctuation spectrum and transport time scale ordering. As an illustrative example, we explore the physics of Geodesic Acoustic Modes (GAMs) and Energetic particle driven Geodesic Acoustic Modes (EGAMs), including the linear dielectric response, generation of zero frequency zonal fields, and the modulation of GAMs by zonal fields themselves. The nonlinear dynamics of EGAMs are also investigated, considering the action of sources, collisions, and the emission and re-absorption of GAM/EGAM fluctuations. We finally summarize the findings and discuss future directions for research.
References:
[1] M.V. Falessi, L. Chen, Z. Qiu, and F. Zonca, "Nonlinear equilibria and transport processes in burning plasmas," New J.Phys. (submitted)
[2] M.V. Falessi, F. Zonca, "Transport theory of phase space zonal structures," Phys.Plasmas 26, 022305 (2019).
[3] L. Chen and F. Zonca, "Physics of Alfvén waves and energetic particles in burning plasmas," Rev.Mod.Phys. 88, 015008 (2016).
[4] F. Zonca et al., "Energetic particles and multi-scale dynamics in fusion plasmas," Plasma Phys.Control.Fusion 57, 014024 (2015).
[5] F. Zonca, L. Chen, M.V. Falessi, Z. Qiu, "Nonlinear radial envelope evolution equations and energetic particle transport in tokamak plasmas," J.Phys.Conf.Ser. 1785, 012005 (2021)
[6] F. Zonca et al., "Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas," New J.Phys. 17, 013052 (2015).
Recent experiments using the 3-ion ICRH heating scheme [Kazakov NF 2015] have been successful at generating substantial populations of MeV range fast ions in the deep core of JET, mimicking the effect of fusion-born alpha particles in future burning plasmas. We analyze an ICRH heated L-mode in which fast ions destabilized a wide range of Alfvén eigenmodes (AEs) as observed using magnetics, reflectometer and Doppler backscattering (DBS) measurements. As ICRH heating power was increased and AEs were destabilized (DBS), we observed an increase in the electron thermal transport (dominant to the ion thermal transport inside rho = 0.4) and an increase in the deep core ion temperature. This is consistent with previous nonlinear turbulence simulations suggesting that AEs can stabilize ion-scale turbulence [DiSiena NF 2019, Mazzi Nat. Phys. 2022], however electron thermal transport remains a mystery. We report on the transport and gyrokinetic modelling using GS2 and CGYRO in conditions when Alfvén eigenmodes are both stable and unstable, as observed from magnetics and DBS measurements. We probe the origins of the anomalous electron thermal transport in the presence of MeV range fast ions and unstable Alfvén eigenmodes. The implications of these scenarios to burning plasmas will be discussed.
The success of magnetic confinement fusion as energy source relies crucially on reaching high temperatures for the fuel D and T ions. In a fusion reactor, plasma heating with waves in the ion cyclotron range of frequencies (ICRF) is the only system capable to provide a large fraction of bulk ion heating. Furthermore, in view of better understanding non-linear physics of alpha heating in ITER and future reactors, generation of MeV-range ions with ICRF and studying different mechanisms on how fast ions impact the plasma dynamics becomes progressively more important. Recent progress with the development of three-ion ICRF scenarios have expanded the use of these scenarios from dedicated ICRF studies [1] to a flexible tool with a broad range of applications [2].
In order to maximize bulk ion heating of D-T ≈ 50%-50% plasmas and increase Ti in the ramp-up phase with ICRF, ITER foresees the injection of a few percent of 3He ions [3]. Because 3He is a scarce gas, using the three-ion T-(IMP)-D ICRF scenario with a small amount of selected impurities (IMP) with 1/3 < (Z/A)imp < 1/2 as resonant absorbers was proposed in [4]. A fairly large number of low-Z and mid-Z impurities and their isotopes with (Z/A)imp ≈ 0.44-0.46 satisfy this criterion, including 7Li, 9Be, 11B, 22Ne, and Ar. Since these impurities have a higher atomic mass than 3He ions, they transfer an even larger fraction of the absorbed RF power to bulk D and T ions via Coulomb collisions.
The potential of 7Li and 9Be impurities (nimp/ne ≈ 1%) to absorb RF power efficiently in D-T plasmas was demonstrated at TFTR and recently at JET-ILW [5]. In particular, in recent JET-ILW experiments with the three-ion T-(9Be)-D ICRF scheme, a strong increase of Ti with ICRF was observed, in line with theoretical predictions. The experimental demonstration of this scenario at JET-ILW thus allows to extend the list of potential applications of three-ion ICRF scenarios in ITER and future fusion reactors.
At the moment, ITER is considering to switch from the Be/W to the full-W first wall, thus motivating the need to re-assess the potential of three-ion T-(IMP)-D ICRF scenarios in the absence of 9Be. In this contribution, we evaluate promising extrinsic impurities and the range of their concentrations for efficient bulk ion heating in D-T plasmas, with a focus on the plasma parameters in the ramp-up phase. We also discuss the potential of these novel ICRF scenarios for their applications in future high-magnetic field and high-beta spherical tokamaks [6, 7].
References:
[1] Ye.O. Kazakov et al., Nature Physics 13, 973 (2017)
[2] Ye.O. Kazakov et al., Phys. Plasmas 28, 020501 (2021)
[3] R. Dumont et al., Nucl. Fusion 53, 013002 (2013)
[4] Ye.O. Kazakov et al., Phys. Plasmas 22, 082511 (2015)
[5] Ye.O. Kazakov et al., “Progress With Applications of Three-Ion ICRF Scenarios for Fusion Research: A Review”, AIP Conf. Proc. (2023), accepted
[6] M. Gryaznevich et al., this conference
[7] M. Romanelli et al., this conference
In ideal magnetized plasmas, sheet-like field discontinuities, where current and vorticity peak, naturally form. According to the linear theory, these layers undergo fluid and magnetic instabilities whose strength depends on the amplitude of the local magnetic field and flow. In non-ideal plasmas, in presence of magnetic reconnection, the combined action of the sheared flow and the sheared magnetic field broadens the spectrum of the linearly unstable tearing modes and increases their growth rate compared to the static current layer case, both in collisional [1] and collisionless plasmas [2,3]. Recently [2], high resolution numerical simulations of a collisionless plasmas without electron temperature effects addressed the nonlinear dynamics of current and vorticity layers in a turbulent setup. They showed a complex situation in which, due to the presence of strong velocity shears, the typical plasmoid formation, observed to influence the energy cascade in the magnetohydrodynamic context, has to coexist with the Kelvin-Helmholtz instability. The competition among these instabilities affects not only the evolution of the current sheets, that may generate plasmoid chains or Kelvin-Helmholtz driven vortices, but also the energy cascade, that is different for the magnetic and kinetic spectra [2]. Here we present new results that extend the previous analysis by considering the the effect of the electron temperature which enters the plasma model equations via the ion sound Larmor radius.
1) Biskamp, D. 2000, Magnetic Reconnection in Plasmas (Cambridge University Press)
2) Borgno, D. et al. 2022, ApJ 929, 62
3) Grasso, D. and Borgogno, D. 2022, Fluid models for collisionless magnetic reconnection (IOP Publishing)
The geodesic-acoustic-mode (GAM) is a plasma oscillation observed in fusion reactors with toroidal geometry (such as the Tokamak or Stellarator) and are recognized to be the non-stationary branch of the zonal flows (ZFs). Similarly to the ZFs, GAMs are understood to regulate cross-field turbulence and thus enhance energy confinement [1]. Still, their direct effect on turbulence is not yet fully understood [2], as GAMs are known to deplete the energy available to ZFs [3]. This complex contribution to the turbulence dynamics makes GAMs highly interesting in current fusion research.
The nonlinear Schrödinger equation (NLSE) model [4] for the isolated, undamped GAM predicts the susceptibility of GAM packets to the modulational instability (MI). The necessary conditions for this instability are analyzed analytically and numerically using the NLSE model. The predictions of the NLSE are compared to gyrokinetic simulations performed with the global particle-in-cell code ORB5, where the GAM packets are created from initial perturbations of the axisymmetric radial electric field $E_r$. An instability of the GAM packets with respect to modulations is observed, in both cases in which an initial perturbation is imposed and when the instability develops spontaneously. However, significant differences in the dynamics of the small scales are discerned between the NLSE and gyrokinetic simulations. These discrepancies are mainly due to the radial dependence of the strength of the nonlinear term and to the damping of higher spectral components, which we do not retain in the NLSE. The influence of the safety factor $q_s$, the ion Larmor radius $\rho_i$ as well as the perturbation wavenumber $k_\text{pert}$ on this effect is studied. The damping of the high-$k_r$ components can be understood in terms of Landau damping.
[1] G. D. Conway, A. I. Smolyakov, and T. Ido, Nuclear Fusion 62, 013001 (2021)
[2] A. I. Smolyakov, M. F. Bashir, A. G. Efimov, M. Yagi and N. Miyato, Plasma Physics Reports 42, 407 (2016)
[3] B. D. Scott, New Journal of Physics 7, 92 (2005)
[4] E. Poli, A. Bottino, O. Maj, F. Palermo, and H. Weber, Physics of Plasmas 28, 112505 (2021)
SOLPS-ITER modelling of plasma rotation with co-rotating atoms in the Magnum-PSI beam.
H.J. de Blank1 , J. Verstappen1, J. Gonzalez2, I. Classen1, E. Westerhof1
1 DIFFER - Dutch Institute for Fundamental Energy Research. De Zaale 20, 5612AJ, Eindhoven, The Netherlands
2 ARCNL, P.O.Box 93019, 1090BA Amsterdam, The Netherlands
In the ITER divertor heat loads of 10 MWm-2 are expected in steady state. The linear plasma device Magnum-PSI [1] can achieve plasma parameters close to those expected in the ITER divertor. In particular, Magnum-PSI experiments have been made with detached plasma state. [2], a state important for the ITER-divertor by reducing heat and particle fluxes. With easier access for some plasma diagnostics in Magnum-PSI than in tokamak divertors, it is attractive to have accurate simulations of both linear and divertor plasmas with the same numerical models. The goal is to learn from the linear plasma simulations more about the atomic and molecular processes, such as reactions and excitations, relevant in detachment.
SOLPS-ITER [3] simulations of the plasma and neutral particles have been shown to be possible for the relevant regimes in Magnum-PSI [4], making use of Thomson Scattering radial profiles of electron density and temperature near the plasma source and near the target plate. The Magnum-PSI plasma source can generate significant currents throughout the plasma beam which impact the discharge through Ohmic heating. This work presents improved calculations of these currents in SOLPS-ITER by including the rotation velocity of the neutral atoms as a consequence of friction with the plasma, which rotates mainly through the ExB drift. This component of the plasma velocity, being in the ignorable (symmetry) direction and perpendicular to the magnetic field, in the standard SOLPS-ITER code is not passed on from the plasma model (B2.5) to the neutrals model (EIRENE). Including this interaction, we find that, typically, the average rotation speed of the neutral atom population is about half of the plasma rotation velocity. This co-rotation affects the entire plasma beam in two ways: by somewhat reducing the neutral density in the plasma beam centrifugally, and by reducing the radial electric resistivity through friction between the ions and neutrals.
Each simulation uses one profile of the radial electric field near the plasma source as boundary condition. This profile is based on spectroscopic measurement of the H-atom rotation velocity on 30 parallel lines of sight, using some assumptions about the kinetic neutral particle velocity distribution. This paper will verify those assumptions using information about the neutral particle velocity distribution extracted from EIRENE in a limited number of plasma locations.
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was carried out on the EUROfusion High Performance Computer Marconi-Fusion hosted at Cineca (Bologna, Italy) and on the Dutch national e-infrastructure with the support of SURF Cooperative.
References:
[1] G. de Temmerman, et al., Fusion Engin. Design, 88, 483 (2013)
[2] R. Chandra et al., Plasma Phys. Control. Fusion, 63 (9):095006 (2021)
[3] S. Wiesen, et al., J. Nucl. Materials, 463, 480 (2015)
[4] J. Gonzalez et al., Plasma Phys. Control. Fusion , 65 055021 (2023)
A numerical tool modelling the excitation and evolution of electron avalanche ionization in the breakdown phase of start-up in tokamaks is presented. We estimate the energization efficiency of the nonlinear interaction between spatially localized Gaussian EC-fields propagating in vacuum with an ensemble of seed electrons. This process is coupled with the acceleration of electrons due to the induced loop voltage along the vacuum vessel, as well as the impact ionization and elastic collision events that lead to the abrupt increase of electron density during the avalanche process. Special care is taken to incorporate the effect of the toroidal magnetic field in the collision statistics [1]. Several numerical experiments are performed for configurations relevant to existing tokamaks [2] as well as ITER [3], dealing with all alternative start-up initiation procedures (ECRH pre-ionization [4], ECRH-assisted [5] and ohmic [6]). A simple analytical auxiliary tool based on the dynamics of avalanche evolution is developed in order to estimate the breakdown time as a function of the RF-field parameters, the loop voltage and the prefill pressure of the neutral gas.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] P. Ch. Papagiannis, etal. Proc. of the 48rd EPS Conf. on Plasma Physics, P5a.109 (2022)
[2] J. Stober etal., Nucl. Fusion, 51, 083031 (2011)
[3] P.C. de Vries and Y. Gribov, Nucl. Fusion, 59, 096043 (2019)
[4] J. Sinha etal. Nucl. Fusion 62, 6206601 (2022)
[5] D. Ricci, etal. Proc. of the 45th EPS Conf. on Plasma Physics, P4.1074 (2018)
[6] P.C. de Vries et al Nucl. Fusion 53, 053003 (2013)
The stellarator is a device designed to achieve controlled fusion by magnetic confinement.
Despite several advantages over tokamaks, its lack of axisymmetry creates some difficulties to confine the plasma: in an unoptimised stellarator reactor, both thermal ions and the fast ions produced by DT fusion would be lost faster than in a tokamak. On the other hand, stellarators are characterised by a wider set of parameters that may be varied in an optimisation procedure to shape the magnetic configuration appropriately towards improved confinement. For thermal ions, optimisation to reduce neoclassical transport has been shown to be effective for Wendelstein 7-X [1]; however, a stellarator with sufficiently reduced fast ion losses and low turbulent transport is yet to be built.
The object of this work is to obtain stellarator configurations that display good properties, particularly in terms of both thermal and fast ion confinement, while also satisfying the requirements of magnetohydrodynamic stability as determined by the Mercier criterion. To
this end, configurations are being pursued that follow the quasi-isodynamic (QI) concept and fulfil, already from intermediate ⟨β⟩, the maximum-J property. The closeness to QI serves to reduce neoclassical transport and maintain a low bootstrap current. We intend to allow for an island divertor design by having a monotonically increasing rotational transform profile which avoids low-order rationals and approaches 1 at the last closed flux surface, whereas the smallness of the bootstrap current is desirable to avoid unwanted changes to said profile. Additionally, the maximum-J property is expected to improve fast ion confinement and reduce turbulent transport. The design parameters chosen include an aspect ratio near 11-12 and a limited maximum elongation of the flux surfaces. The optimisation was carried out at an intermediate value of ⟨β⟩=2.5% as a strategy to maintain good properties for both intermediate and reactor relevant plasma pressures.
This optimisation work uses the code suite STELLOPT (CIEMAT branch), into which the neoclassical code KNOSOS [2] has been incorporated. The resulting configurations belong to the family of flat-mirror QI configurations [3], like the recently published design [4]. This contribution includes results for several periodicities. The main result is a particularly good 5 period configuration with small effective ripple (the figure of merit for neoclassical bulk energy transport) in the plasma core and no prompt losses of fast ions born at half radius, as verified by Monte Carlo guiding-centre simulations at reactor ⟨β⟩. This design also displays a low bootstrap current and fulfils the rotational transform requirements. The 3-period configuration has the advantage of being more compact than the 5 period design, although at the cost of some confinement capabilities. Finally, the 6 period configuration has similar confinement capabilities to the 5 period design at reactor ⟨β⟩.
In order to be candidates to fusion power plants, stellarators must be optimized, i.e. the magnetic field needs to be tailored to have sufficiently good confinement properties. When the optimization process is performed to minimize neoclassical losses, the goal is to obtain a magnetic field that is close to omnigeneity. A magnetic field is omnigenous [1] if the radial drift of collisionless particles averages out to zero, leading to levels of neoclassical transport similar to those in a tokamak. Quasi-isodynamic (QI) configurations are a subclass of omnigenous magnetic fields that have poloidally closed contours of the magnetic field strength, which grants them the additional property of having zero bootstrap current [2]. The bootstrap current can alter significantly the equilibrium magnetic field and, in particular, the geometry of a divertor relying on a specific structure of magnetic islands. In order to obtain a magnetic configuration with a viable island divertor, the bootstrap current must be kept sufficiently small, and this is typically ensured if the configuration is close enough to quasi-isodynamicity. Wendelstein 7-X is based on the concept of quasi-isodynamicity and has an island divertor. Another subclass of omnigenous magnetic fields consists of quasi-symmetric (QS) configurations. For QS stellarators, the bootstrap current is not small and therefore its effect must be explicitly considered during the optimization process [3].
Taking into account the bootstrap current in the optimization loop demands fast and accurate calculations at low collisionality, the relevant conditions in the core of reactor-relevant plasmas. However, accurate computations in these regimes are difficult (the exception are precisely QS stellarators, for which analytical formulas exist [3]). When the mean free path is long, boundary layers appear at the interfaces of different classes of trapped particles, and very high resolution in velocity space is required to solve correctly these boundary layers.
In this conference contribution we present MONKES (MONoenergetic Kinetic Equation Solver), a new neoclassical code for the evaluation of monoenergetic transport coefficients in large aspect ratio stellarators and tokamaks with broken symmetry. The code is spectral in the spatial and velocity coordinates, and employs a block tridiagonal algorithm for solving the resultant linear system of equations. MONKES is memory efficient and can run in a single core. It solves the same equation as the standard stellarator neoclassical code DKES [4], but it is more accurate and faster. This is demonstrated through a careful convergence study and a benchmark with DKES and SFINCS [5]. The above features make MONKES ideally suited for its integration into stellarator optimization suites. Apart from this, MONKES, complemented with KNOSOS [6,7], can be applied to a range of problems, such as the analysis of experimental discharges and predictive transport simulations.
Quasi-isodynamic (QI) stellarators are a uniquely attractive fusion reactor candidate due to their low neoclassical transport, excellent confinement of fusion-borne alpha particles, and vanishingly small bootstrap currents [1]. Due to the complexity of their geometries, QI stellarators must generally be designed through numerical optimization, which requires an objective metric that quantifies the degree to which a given design is QI. While once thought impossible, recent work has shown that nearly-perfectly QI geometries can be found using an appropriately-designed objective function [2]. In this work, we build upon this approach with the goal of finding QI geometries with reduced turbulence, improved MHD stability, and lower surface area-to-volume ratios. The configurations presented here were designed as potential candidates for future stellarator experiments and reactors, although work on this project is still preliminary. We find that, while these added objectives invariably worsen the QI quality of the results, excellent QI quality is still achievable, even when all other criteria are satisfied to the same (or greater) extent as other reactor-relevant designs [3] and Wendelstein 7-X.
This work has been carried out with in the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 - EUROfusion).
References:
[1] P. Helander and J. Nuhrenberg, Plasma Physics and Controlled Fusion 51(5) 055004, (2009)
[2] A. G. Goodman, et al., arXiv:2211.09829
[3] E. Sanchez et al., arXiv:2212.01143
In the last experimental campaign of Wendelstein 7-X (W7-X) the new ICRH antenna [1] was brought into operation. With only one of the two antenna straps being operable in the last campaign, the available phasing options as well as the amount of power that could be coupled into the core plasma were limited. Nevertheless, the antenna is expected to be fully functional for the next operational campaign. Hence, investigations have been started to evaluate how to best utilize all the heating systems of W7-X for the purpose of fast-ion generation.
Note that generating deeply trapped collisionless fast ions is an important prerequisite for proving that the confinement of fast ions improves in W7-X with increasing plasma beta [2]. So far, no heating scheme was able to generate such particles on its own in W7-X.
In this contribution we show novel simulation results, obtained with the SCENIC code package [3], regarding the possibility of generating fast particles in Wendelstein 7-X (W7-X) through combined radio-frequency (RF) heating and neutral beam injection (NBI) schemes. In contrast to previous work [4], the focus is on pure Hydrogen plasmas as well as Helium plasmas, respectively, as those will be the foreseen target plasmas for the next operation phase. All simulations are carried out with realistic W7-X profiles.
For both target plasmas, Hydrogen or Helium, various operating parameters are varied in order to find settings that maximise the generation of energetic particles and to guide the experimental planning in this way. Apart from changing the bulk-plasma composition itself, further actuators tried in this work include changing the antenna frequency, which affects the resonance with the Doppler-shifted NBI-ions, as well as varying the Hydrogen content (coming from NBI) in the plasma by changing the NBI heating power.
References:
[1] J. Ongena, et al., Physics of Plasmas 21, 061514 (2014)
[2] M. Drevlak, et al., Nuclear Fusion 54, 073002 (2014)
[3] M. Jucker, et al., Computer Physics Communications 182, 912-925 (2011)
[4] M. Machielsen et al., Journal of Plasma Physics 89, 955890202 (2023)
Concentrated exhaust power deposition must be avoided in a fusion power plant. A strategy to prevent this is to seed heavy impurities in the divertor, which radiate strongly at the local plasma temperature. This can help to maintain a uniform power deposition over the divertor structure. However, if the impurity migrates upstream, it can produce a number of detrimental effects, including fuel dilution and strong radiative power loss. Measures of the success with which the impurity is localised to the divertor are the impurity enrichment, which is the ratio of the impurity concentration in the divertor to that upstream, and the impurity compression, which is the ratio of the impurity density in the divertor to that upstream.
Using SOLPS-ITER simulations, we have investigated the localisation of seeded argon in a power plant-class connected double null diverted spherical tokamak geometry, with a well-baffled, extended outer divertor leg and fairly open, short, inner divertor leg. We find that the mean free path of the neutral argon is short compared to the size of the divertor legs for the range of plasma scenarios explored. The argon enrichment and compression can thus be increased past the detachment point of the outer leg, and the enrichment and compression of the inner divertor leg can be achieved with direct impurity seeding. By comparison, we find that neon, with a higher first ionization potential, has a longer mean free path [1,2] and weaker enrichment and compression, does not cool the plasma as effectively, and thus is less suitable for use at this scale.
[1] S. S. Henderson, et al., Nucl. Fusion 63 086024 (2023)
[2] I. Yu Senichenkov, et al., Plasma Phys. Control. Fusion 61 045013 (2019)
The neutral atoms in the plasma edge of nuclear fusion devices are typically modeled using a kinetic approach and more specifically the Monte Carlo (MC) code EIRENE [1]. Although EIRENE has been proved very reliable and effective, there are some drawbacks such as the statistical noise introduced by the MC techniques and the computational cost, which is significantly increased in high collisional regimes. Therefore, alternative approaches have been proposed including either the employment of more advanced kinetic stochastic codes (DSMC) [2] or of deterministic neutral fluid models [3] with considerable success. Always, the objective is the computationally robust and efficient coupling with the plasma edge model.
In the present work, instead of adopting a stochastic approach, an in-house deterministic solver, based on solid theoretical principles, utilizing a novel marching discrete velocity algorithm on unstructured grids [4] is developed and implemented to model the neutral particles in tokamak exhaust systems. The Boltzmann equation is accordingly replaced by suitable kinetic model equations, which are numerically solved, in a deterministic manner, by discretizing the particle velocity space via the method and the physical space via typical finite difference or volume schemes.
The bulk quantities are obtained by the moments of the particle distribution functions, while depending on the operating scenarios, various boundary conditions and coupling strategies with the plasma code may be applied. The validity of the proposed model is thoroughly assessed by modeling the neutral gas flow in a 2D-cut of the JET sub-divertor area and performing a systematic comparison with corresponding results in [2], where the DSMC method is introduced. Complimentary comparisons with the in-house gas distribution network code ARIADNE [5], which is also used to simulate the JET sub-divertor area are performed. The main advantages of the proposed approach compared to other available ones, include the detailed description of the flow at mesoscale without statistical noise with small computational effort.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 - EUROfusion).
References:
[1] D. Reiter et al, Fusion Sci. Technol., 47, 172-186 (2005).
[2] S. Varoutis et al, Fusion Eng. Des., 121, 13-21 (2017).
[3] Wim Van Uytven et al, Nucl. Fusion, 62, 086023 (2022).
[4] G. Tatsios, Advanced deterministic and stochastic kinetic modeling of gaseous microscale transport phenomena, Volos, Greece: Ph.D. Dissertation, University of Thessaly, 2019.
[5] N.Vasileiadis et al, Fusion Eng. Des., 103, pp. 125-135 (2016).
Supra-thermal Energetic Particles (EPs) can be found in a burning tokamak plasma due to external heating methods such as Neutral Beam Injection and fusion reactions. EPs travelling at velocities close to the Alfvén speed can interact at resonance with various discrete Alfvén Eigenmodes which appear in the frequency gaps of the shear Alfvén continuum. Toroidal Alfvén Eigenmodes (TAE) are created when two counter-propagating Alfvén waves are coupled due to the poloidally-varying magnetic field strength and may impact on plasma performance Through resonant wave-particle interactions, EPs can drive TAEs unstable in a tokamak plasma, leading to anomalous EP transport or even direct expulsion of EPs to the first wall. The stability of a TAE is dependent on the competition between EP drive and the various parameter-dependent damping mechanisms.
The global gyro-kinetic code ORB5 has been used to investigate the damping of TAEs. The instability has been simulated using a simple circular geometry and compared against (i) analytical theory (with good agreement found), and (ii) against the local version of the GENE code for one case. Additionally, a diagnostic developed for ORB5 has been used to determine the energy transfer for each species and to differentiate between damping due to drifts and parallel-streaming to allow each damping channel to be studied individually.
ORB5 has also been applied to study Alfvén Eigenmodes in spherical tokamak devices. Simulations have been run to match specific MAST-U shots using experimental data obtained in the most recent campaign, with the aim of developing a predictive capability for the excitation of these modes. Spherical tokamaks are associated with high beta and large inverse aspect ratio, two parameters which are believed to significantly modify the drive and damping of TAEs found in conventional tokamaks.
To accurately model the plasma dielectric properties in presence of rotational transform, most of the theoretical models and full-wave codes addressing radiofrequency wave propagation and absorption in tokamaks are based on toroidal and poloidal Fourier expansions of the RF fields (see for instance [1-4]). A significant drawback of this field representation is its lack of flexibility, in that it does not allow local refinements of numerical discretizations on a given magnetic surface.
As a first remedy to this, theoretical expressions have been obtained which are free from the poloidal mode expansion, but nevertheless preserve the description of wave dispersion along the curved inhomogeneous magnetic field [5]. These integral kernels, which describe the dielectric response of Maxwellian tokamak plasmas, were derived to lowest order in the Larmor radius and still made use of the Fourier expansion with respect to the toroidal angle.
The present communication generalizes these earlier results in two respects: (i) New theoretical expressions of the dielectric response have been obtained which are also free from the toroidal mode expansion. These mildly singular integral kernels depend on transcendental functions of a single variable (“kernel dispersion functions”) and incorporate the non-local nature of wave-particle interactions along the equilibrium magnetic field lines. They are independent of the RF field representation inside the plasma volume and therefore amenable to three-dimensional finite element discretizations. (ii) Moreover, our new results include finite Larmor radius effects to all orders in ρLT /λ⊥, i.e. (thermal Larmor radius) / (characteristic RF field lengthscale across the equilibrium magnetic field).
Once implemented in a finite element wave propagation code, this approach will provide full flexibility to implement local mesh refinements in the plasma, as required for instance near cyclotron resonance layers and in regions of rapid RF field variations. Moreover, it will easily interface with advanced antenna modelling codes based on the finite element method (e.g. [6]), and will hence enable the latter to accurately model plasma wave kinetic effects.
The paper will present the theoretical results and discuss their forthcoming application in ICRH modelling.
[1] D. J. Gambier and A. Samain, Nucl. Fusion 25, p. 283 (1985).
[2] M. Brambilla and T. Kruecken, Nucl. Fusion 28, p. 1813 (1988).
[3] P. U. Lamalle, Plasma Phys. Control. Fusion 39, 1409–1460 (1997).
[4] R. Dumont, Nucl. Fusion 49, 075033 (2009).
[5] P. U. Lamalle, AIP Conf. Proceedings 2254 100001 (2020), doi 10.1063/5.0014257.
[6] S. Shiraiwa et al, EJP Web of Conferences 157 (2017), doi 10.1051/epjconf/201715703048.
To understand plasma behaviour in the scrape-off layer (SOL), we need to know the boundary conditions for the plasma and electromagnetic fields near a divertor. At the plasma-wall boundary, in the direction perpendicular to the wall, there are four length scales of interest. These are the Debye length $\lambda_D$, the ion gyroradius $\rho_i$, the projection of the collisional mean free path in the direction normal to the wall $\lambda_\perp$ and the device size $L$. Assuming that the plasma near the divertor satisfies the scale separation $\lambda_D \ll \rho_i \ll \lambda_\perp \ll L$, we can split the plasma-wall boundary into three separate layers. The layer closest to the wall is the Debye sheath of width $\lambda_D$, then follows the magnetic presheath of width $\rho_i$ and then the collisional layer of width $\lambda_\perp$. Plasma dynamics in the first two layers are well understood [1-3]. In the SOL, at distances much greater than $\lambda_\perp$ from the wall, collisionality is high and Braginskii fluid equations are often used to model the plasma behaviour [4-6], here the ion distribution function is assumed to be approximately Maxwellian. The collisional layer, which we analyse in this work, connects this region of high collisionality with the collisionless magnetic presheath, where the ion distribution function is far from Maxwellian.
For analysis of ion dynamics in the collisional layer, we solve the steady state drift kinetic equation in one spatial dimension with the full Fokker-Planck collision operator, together with the quasineutrality equation and the assumption of adiabatic electrons. We can neglect the asymptotically small magnetic presheath and apply the wall boundary conditions at the entrance of the magnetic presheath. For our boundaries we assume that all ions that reach the wall are absorbed (negatively charged wall) and we set the distribution function far away from the wall to be approximately a Maxwellian. From the magnetic presheath analysis [3], it is known that the distribution function must satisfy the Chodura condition at the entrance of the magnetic presheath. We show that this condition is also obtained from the collisional presheath analysis. We show that the scaling of the potential here is $\phi \sim \sqrt{x}$, where $x$ is distance from the wall, and the distribution function is exponentially small at small velocities parallel to the magnetic field. The latter is not true if neutral-ion collisions are included, however the potential scaling seems to remain valid. We also show that at the entrance of the collisional layer the flow of ions has to be supersonic. To analyse the collisional layer numerically we use the Galerkin method with quadratic finite element basis to solve the equations. To numerically determine the particle trajectories next to the wall we need to impose the derived potential scaling.
References:
[1] K -U Riemann, J. Phys. D: Appl. Phys. 24 493 (1991)
[2] A. Geraldini, F. I. Parra, and F. Militello, Plasma Phys. Control. Fusion, 60:125002 (2018)
[3] R. Chodura, The Physics of Fluids 25, 1628 (1982)
[4] P. Ricci, et al, Plasma Phys. Control. Fusion 54 124047 (2012)
[5] P. Tamain, et al, Journal of Computational Physics Volume 321, Pages 606-623 (2016)
[6] B. D. Dudson and J. Leddy, Plasma Phys. Control. Fusion 59 054010 (2017)
Y. Narbutt1
, A. Mishchenko1
, A. Zocco1
, K. Aleynikova1
and R. Kleiber1
1 Max Planck Institute for Plasma Physics, Greifswald, 17489, Germany
Magnetic confinement fusion requires high 𝛽 = 〈𝑝〉/(𝐵
2/2𝜇0), the ratio of plasma pressure to
magnetic pressure, to access high performances. Moderate 𝛽 can be beneficial for iontemperature-gradient (ITG) driven turbulence. However, as 𝛽 is increased above a certain
threshold, the so-called kinetic-ballooning-mode (KBM) [1] can be destabilized. This is a
plasma pressure gradient driven instability, which is inherently electromagnetic and can lead to
strong outwards-directed heat fluxes [2], degrading plasma confinement in the process. While,
linearly, KBMs have been successfully studied in the stellarator Wendelstein 7-X with fluxtube simulations [3, 4], it was also shown that the instability tends to be most unstable while
developing a global structure on the magnetic surface. This poster presents results of global
linear simulations of KBMs in W7-X geometry using the global gyrokinetic code Euterpe [5].
This work has been carried out within the framework of the EUROfusion Consortium, funded by the
European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200
— EUROfusion). Views and opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Commission. Neither the European
Union nor the European Commission can be held responsible for them.
References:
[1] W.M. Tang, J.W. Connor, and R.J. Hastie. “Kinetic-ballooning-mode theory in general geometry”.
In: Nuclear Fusion 20.11 (1980), p. 1439. doi: 10.1088/0029-5515/20/11/011.
[2] A. Mishchenko et al. “Gyrokinetic particle-in-cell simulations of electromagnetic turbulence in the
presence of fast particles and global modes”. In: Plasma Physics and Controlled Fusion 64.10 (2022),
p. 104009. doi: 10.1088/1361-6587/ac8dbc.
[3] K. Aleynikova, A. Zocco, and J. Geiger. “Influence of magnetic configuration properties on kinetic
ballooning modes in W7-X”. In: Journal of Plasma Physics 88.4 (2022), p. 905880411. doi:
10.1017/S0022377822000745.
[4] Ksenia Aleynikova and Alessandro Zocco. “Quantitative study of kinetic ballooning mode theory in
simple geometry”. In: Physics of Plasmas 24.9 (Aug. 2017). 092106. issn: 1070-664X. doi:
10.1063/1.5000052.
[5] R. Kleiber et. al. “EUTERPE: A global gyrokinetic code for stellarator geometry”. Submitted to
CPC. 2023
Integration of high fusion performance with sustainable power exhaust is one of the leading challenges of reactor-scale fusion devices. Current state-of-the-art simulation tools for scrape-off layer (SOL) plasmas, such as SOLPS-ITER, employ a finite volume plasma solver in 2D with either fluid or kinetic treatment of neutral transport [1]. However, with high-fidelity physics treatment, these are prone to relatively long computational times and convergence challenges. For computationally faster and dynamic simulations, 1D plasma solvers such as SD1D and DIV1D have been developed [2, 3]. In this study, the applicability of the Sparselizard library [4] for the simulation of fusion plasma physics is investigated, starting with the 1D SOL fluid models. Sparselizard is an open-source C++ finite-element (FE) library for the numerical implementation of multiphysics systems utilizing the domain decomposition methods for high-performance computing [4].
Firstly, the equations for the conservation of mass and momentum from the 1D isothermal fluid model were implemented and verified against an analytical model [5]. Then, a diffusive neutral model was introduced to determine the neutral distribution and particle source self-consistently. This was extended to the plasma temperature through the energy equation, assuming electron heat conduction as the sole energy transport mechanism. For verification, the results from the FE simulation were compared with the two-point model. Due to the strongly coupled and self-consistent interactions between various physical terms, a fully-coupled solver was implemented. Thus, all the conservation equations were solved simultaneously. In the FE formulation, stabilization techniques and Newton linearizations were employed. The resulting system of equations was solved using Newton iterations.
This research project, funded by Business Finland, is a collaboration with the EUROfusion Advanced Computing Hub at the University of Helsinki and Quanscient Oy.
References:
[1] S. Wiesen, et al. J. Nucl. Mat. 463 (2015) 480-484
[2] B.D. Dudson, et al. Plasma Phys. Control. Fusion 61 (2019) 065008
[3] G.L. Derks, et al. Plasma Phys. Control. Fusion 64 (2022) 125013
[4] A. Halbach, Sparselizard - the user friendly finite element c++ library. 2017
[5] P.C. Stangeby, The plasma boundary of magnetic fusion devices. Vol. 224. Philadelphia, Pennsylvania: Institute of Physics Pub., 2000.
Decades of research has demonstrated the necessity of using kinetic plasma models to accurately model the flux of heat and particles through the closed-field line region of tokamaks. In the much colder open-field-line region beyond the last closed flux surface (LCFS), fluid models are typically used to model the flux of heat and particles to the divertor. Recently, kinetic plasma models have been developed for the open-field-line region [1-4], using a variety of numerical methods with differing degrees of physics fidelity. These models will be important for assessing the importance of prompt losses of hot particles from the LCFS to the fluxes at the divertor plate, as well as determining whether there is a direct impact from the wall boundary on upstream physics within the LCFS.
In this work, we present a study of the role of the E×B drift in a drift kinetic model of the open-field-line region of a magnetic confinement device [5]. The model evolves drift-kinetic ions, and Boltzmann electrons, with the option to include a kinetic neutral species (please see the companion work [6]). The magnetic geometry is helical, with wall boundaries limiting the extent of the axial coordinate. We include a range of model collision operators, including Krook operators for ion-neutral collisions and a model pitch-angle scattering operator for ion-ion collisions. We include small spatial numerical viscosity as a proxy for finite-Larmor-radius physics. The numerical implementation is explicit in time using a strongly stability preserving Runge-Kutta algorithm, and a spectral-element discretisation for the spatial and velocity dimensions. The implementation is tested with manufactured solutions tests, demonstrating the potential for good performance.
With a source of particles injected into the centre of the domain and a plausible initial condition, the model allows us to study relaxation of the plasma and the formation of the steady-state sheath entrance at the wall boundary. In the absence of radial variation, we can verify that the steady-state solutions satisfy the kinetic Chodura condition [7]. We study the impact of a radially varying source and the consequent E x B drifts on the behaviour of the ion distribution function as it enters the sheath. In the absence of radial numerical dissipation, we find a wave-like instability that persists if fluctuations in the radial electric field are permitted to be non-zero. We carry out the linear stability analysis for the model drift kinetic system and we compare our findings with the instability observed in the numerical simulations.
References:
[1] C. S. Chang et al. Phys. Plasmas, 11:2649, (2004).
[2] E. L. Shi et al. J. Plasma Phys., 83:905830304, (2017).
[3] Q. Pan et al. Phys. Plasmas, 25:062303, (2018).
[4] M. Dorf et al. Phys. Plasmas, 23:56102, (2016).
[5] M. R. Hardman et al. Varenna-Lausanne Workshop P29 (2022)
[6] J. Omotani et al. European Fusion Theory Conference (2023)
[7] A. Geraldini et al. 2018. Plasma Phys. Control. Fusion 60, 125002.
Modelling the large amplitude fluctuations of the plasma edge, particularly across the separatrix into the hot scrape-off layers of future reactors, can require costly full-f kinetic simulations, with heavily restricted time-step due to uninteresting fast waves [1]. Here we demonstrate the first steps in the implementation of a method allowing consistent evolution of the system up to the transport timescale, at much lower numerical cost.
We formulate a coupled set of equations which describe the evolution of the low-order moments of the distribution function of each species, closed by the kinetic evolution of the species’ normalized distribution ‘shape’ function. Removal of the low-order moments allows accuracy to be retained when evaluating the closure across a domain where the plasma parameters vary strongly, typical of the plasma edge. It also ensures that the low-order moments remain consistent with the distribution function as the system evolves. Here we present the drift-kinetic limit of the system in a helical field geometry, with low magnetic field line pitch angle and assuming Boltzmann electrons [2].
We simplify to a 1D geometry, with a reduced description of essential ion-neutral interactions, and demonstrate agreement between numerical solutions obtained with the moment-kinetic method and those with a standard drift-kinetic solver. With periodic boundary conditions we identify a damped mode with charge-exchange reactions, and demonstrate numerical agreement with the analytically derived decay rates. With wall boundary conditions, we demonstrate successful formation of a steady-state solution with a sheath-edge boundary condition. Numerical implementation in the helical field geometry is ongoing.
This work has been funded by the EPSRC Energy programme [grant number EP/W006839/1]
References:
[1] W.W. Lee, “Gyrokinetic particle simulation model” J. Comput. Phys. 72 243–69 (1987)
[2] M. R. Hardman et al., “A kinetic model of ions and neutrals with wall boundary conditions in edge plasmas”, Varenna-Lausanne Workshop P29 (2022)
Stellarator magnetic configurations need to be optimized in order to meet all the required properties of a fusion reactor. The stellarator Wendelstein-X (W7-X) was optimized to be approximately quasi-isodynamic (QI). In an exactly QI field, trapped particles orbit, on average, in the poloidal direction, and therefore remain confined [1]. Neoclassical transport is thus expected to be low. Although the performance of W7-X has proven the success of neoclassical optimization [2], some issues remain open, such as fast ion losses and turbulent transport.
In this work, we present the concept of flat-mirror quasi-isodynamic stellarators [3]. We show that a nearly QI configuration with sufficiently small radial variation of the mirror term can achieve small radial transport of energy and good confinement of fast ions, even if it is not very close to exact quasi-isodynamicity, for a wide range of plasma scenarios (including low $\beta$ and small radial electric field). This opens the door to constructing better stellarator reactors, that would be easier to design (as they would be robust against error fields) and to operate (since, both during startup and steady-state operation, they would require less auxiliary power, and the damage to plasma-facing components caused by fast ion losses would be reduced).
The properties of a flat-mirror QI field can be understood in terms of the orbit-averaged drifts of trapped particles in the tangential and radial directions, which are connected to the radial and tangential derivatives of the second adiabatic invariant $J$, respectively. We have identified a strategy (small radial variation of the mirror term) to approach exactly QI magnetic fields that fulfill the maximum-$J$ property in vacuum. This is expected to have a beneficial impact on turbulence [4] and keeps neoclassical confinement good irrespective (to some extent) of the existence of a magnetic drift in the radial direction, of $\beta$ and of the radial electric field. We confront this strategy with the results obtained in the optimization campaign that led to the design of CIEMAT-QI [5], a stellarator magnetic configuration with excellent properties of bulk (neoclassical and turbulent) and fast particle transport at low $\beta$.
References:
[1] J.R. Cary and S.G. Shasharina, Phys. Plasmas 4, 3323 (1997)
[2] C. Beidler and the W7-X team, Nature 596, 221 (2021)
[3] J.L. Velasco, I. Calvo, E. Sánchez and F.I. Parra, arXiv:2306.17506
[4] P. Helander et al., Physics of Plasmas 20, 122505 (2013)
[5] E. Sánchez, J.L. Velasco, I. Calvo and S. Mulas, Nucl. Fusion 63, 066037 (2023)
This work is based on the variational principle for magnetic field lines introduced in 1983 by Cary and Littlejohn [1]. The action principles for magnetic field lines and for Hamiltonian mechanics are recalled to be analogous. It is shown that the first one can be rigorously proved from first principles without analytical calculations. Not only the action principles are analogous, but also a change of canonical coordinates is recalled to be equivalent to a change of gauge [2]. Furthermore, using the vector potential makes obvious the freedom in the choice of “time" for describing Hamiltonian dynamics. These features may be used for a new pedagogical and intuitive introduction to Hamiltonian mechanics. In the context of confined magnetic fields, the action principle for magnetic field lines makes practical calculations simpler and safer, with an intuitive background. In particular, with a new analytical result: the width of a magnetic island is proportional to the square root of an invariant flux related to this island, the magnetic flux through a ribbon whose edges are the field lines related to the O and X points of the island. This is the first expression of this width avoiding abstract Fourier components and obviously independent of the choice of coordinates. The same analytical calculation provides a simple way to compute numerically the width of a magnetic island. Also to apply Chirikov resonance overlap criterion. Moreover, a new formula provides explicitly the Boozer and Hamada magnetic coordinates from action-angle coordinates.
References:
[1] Cary JR, Littlejohn RG (1983), Annals of Physics 151(1):1-34
[2] Elsasser K (1986) 28(12A):1743
Since the early 90ties, 3D nonlinear MHD studies have been developing a fundamental framework for the understanding of the Reversed Field Pinch (RFP) self-organization. The simple visco-resistive MHD approximation clearly shows that 3D reconnection processes strongly characterize the dynamics in an ample range of the dimensionless Lundquist/Hartmann numbers, as well as in experimental discharges [1-9]. In fact, this is particularly evident during the nearly periodic relaxation-reconnection events (so-called RFP sawtoothing activity) observed in both Multiple and Quasi Helical regimes where the natural kinking of the current carrying plasma column triggers relaxations with localized shrinking/collapse of the global helical magnetic field perturbation. In this presentation we will provide a survey of the typical magnetic reconnection manifestation in nonlinear visco-resistive MHD of the RFP and some comparison with the similar behavior in the circular tokamak case. The main features are: magnetic into kinetic energy conversion (possibly providing ion heating), 3D current sheets formation and related flow patterns, mode phase locking, (toroidal collapse of the helix), excitation of Alfvén waves. The possibility to “tune” the sawtooth cycle by adopting suitable Resonant or Non resonant Magnetic Perturbation will be also presented. How much extended-MHD physics would be necessary to better capture the RFP experimental phenomenology still remains to be clarified, and we expect to address this aspect with the help of the RFX-mod2 device under renovation.
References:
[1] S. Cappello and D. Biskamp, Reconnection processes and scaling laws in reversed field pinch magnetohydro dynamics Nucl. Fusion 36 571(1996)
[2] S. Cappello, Bifurcation in the MHD behaviour of a self-organizing system: the reversed field pinch (RFP) Plasma Phys. Control. Fusion 46 B313 (2004)
[3] D. Bonfiglio, et al., Sawtooth mitigation in 3D MHD tokamak modelling with applied magnetic perturbations Plasma Phys. Control. Fusion 59 014032 (2017)
[4] M. Veranda, Helically self-organized pinches: dynamical regimes and magnetic chaos healing Nucl Fus 60 016007 (2020)
[5] M. Veranda et al., Magnetic reconnection in three‑dimensional quasi‑helical pinches, Rend. Fis. Acc. Lincei 31, 963–984 (2020),
[6] A. Kryzhanovskyy et al., Alfvén waves in reversed-field pinch and tokamak ohmic plasmas: nonlinear 3D MHD modeling and comparison with RFX-mod, Nucl. Fusion 62 086019 (2022)
[7] B. Momo et al The phenomenology of reconnection events in the reversed field pinch Nucl. Fus 60 056023 (2020)
[8] M. Gobbin et al., Ion heating and energy balance during magnetic reconnection events in the RFX-mod experiment
Nucl. Fusion 62 026030 (2022)
[9] L. Marrelli et al, The Reversed Field Pinch Nucl. Fusion 61 023001 (2021)
Importance of Parallel Dispersion in ICRF Modelling of Travelling Wave Antenna Concept in DEMO-Like Plasmas in 2D Axisymmetry
B. Zaar$^1$, T. Johnson$^1$, L. Bähner$^1$, R. Bilato$^2$, R. Ragona$^3$, and P. Vallejos$^4$
$^1$KTH Royal Institute of Technology, Stockholm, SE-114 28, Sweden
$^2$Max-Planck-Institut für Plasmaphysik, Garching, D-85748, Germany
$^3$Department of Physics, Technical University of Denmark, Kgs. Lyngby, DK-2800, Denmark
$^4$FOI Swedish Defence Research Agency, Norra Sorunda, SE-137 94, Sweden
Radio-frequency waves in the ion cyclotron range of frequencies (ICRF) in fusion plasmas are often modelled using the finite element method, spectral methods, or a combination thereof. Due to spatial dispersion, which is a non-local effect, the propagation and in particular dissipation of ICRF waves in fusion plasmas are best represented using spectral methods, since the wave-particle interaction is naturally described in Fourier space. However, spectral methods are inefficient when trying to accurately describe the complex geometries of the antenna and vessel wall, which are important for the power coupling to the plasma. Here, the finite element method is better suited. It has previously been shown that non-local effects can be added iteratively to a finite element solution [1], accounting for both parallel [2, 3] and perpendicular [4] spatial dispersion. Doing so, it is possible to use realistic geometry for antennas and vessel wall, while retaining the main plasma wave physics in the core plasma.
In this work, we follow the method described in [3] to determine the importance of parallel dispersion when modelling the travelling wave antenna (TWA) in a DEMO-like plasma [5] using a second harmonic tritium heating scheme. The proposed TWA for DEMO is an attractive antenna concept that can couple high power at lower voltages, allowing for longer distances between the antenna elements and the plasma boundary. ICRF waves launched from different positions will have different poloidal spectra and will be more or less susceptible to parallel spatial dispersion. To investigate this dependence, the antenna will be placed in a few different positions. An example of a top-launched ICRF wave is shown in Figure 1. Since iterative solutions may be computationally expensive, we also compare the iterative solution to a local plane wave approximation of the poloidal contribution to the parallel wavenumber, as an alterative way to estimate the effect of parallel dispersion.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] D. L. Green, and L. A. Berry, Computer Physics Communications 185, 736-743 (2014)
[2] O. Meneghini, et al., Physics of Plasmas 16, 090701 (2009)
[3] B. Zaar, et al., AIP Conference Proceedings (in press)
[4] P. Vallejos, et al., Plasma Physics and Controlled Fusion 62, 045022 (2020)
[5] R. Ragona and A. Messiaen, Nuclear Fusion 56, 076009 (2016)
M. F. F. Nave1, A. Mauriya1, M. Barnes2, E. Delabie3, J. Ferreira1, J. Garcia4 , A. Kirjasuo5, F.I. Parra6, M. Romanelli7 and JET Contributors*
EUROfusion Consortium, JET, Culham Science Centre, Abingdon, OX14 3DB, UK
1Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, P1049-001 Lisboa, Portugal
2Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK
3Oak Ridge National Laboratory, Oak Ridge, TN 37831-6169, USA
4CEA, IRFM, F-13108 Saint Paul Lez Durance, France
5VTT, Espoo, Finland
6Princeton Plasma Physics Laboratory, Princeton, NJ 08540, USA
7Tokamak Energy Ltd, 173 Brook Drive, Milton Park, Oxfordshire OX14 4SD, UK
*See the author list of J. Mailloux et al., (2022) https://doi.org/10.1088/1741-4326/ac47b4
Recent experiments in JET studied intrinsic rotation in Ohmic plasmas, which provided the first clear observation of rotation reversals in a large tokamak [1]. Main ion rotation measurements were made in H, D and T plasmas for a large density range that spanned over both the Linear Ohmic Confinement (LOC) and the Saturated Ohmic Confinement (SOC) phases. Two rotation reversals were clearly observed for each hydrogen isotope, with rotation profiles changing from peaked to hollow at a density close to the LOC-SOC transition, then to peaked again with increasing density. Most theories for intrinsic rotation attribute the observed rotation to a turbulent redistribution of momentum within the plasma core. For a preliminary analysis of the effect of the density on the core rotation observed at JET, we focus on one of the turbulence drives, namely the effect of neo-classical parallel velocity and heat flow on the turbulence [2-3]. Using a version of the GS2 code [4] that includes neoclassical flows, non-linear modeling of rotation profiles covering the whole density range has been performed for the H plasmas. GS2 simulations had previously shown that as the ion–ion collisionality increases, the momentum flux reverses direction in qualitative agreement with the low-density rotation reversal observed in many tokamaks [5]. In the GS2 simulations shown here, the signs (but not the magnitude) of the modelled velocity gradients agree with observations for both the rotation profiles measured during the low-density LOC phase and those measured during the higher-density SOC phase. In both cases the change in rotation shear seems to be driven by the change in the shape of the density and temperature profiles, not the change in ion–ion collision frequency.
“This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.”
References:
[1] MFF Nave et al. Nucl. Fusion 63 (2023) 044002
[2] F.I. Parra and P.J. Catto P.J. 2010 Plasma Phys. Control. Fusion 52 045004
[3] F.I. Parra, M. Barnes and P.J. Catto, Nucl. Fusion 51, 113001 (2011)
[4] W. Dorland et al, Phys. Rev. Lett. 85, 5579 (2000).
[5] M. Barnes et al 2013 Phys. Rev. Lett. 111 055005
The dynamics of magnetic islands and the role they play in fusion plasmas are usually approached and predicted using extensions of the original theory by Rutherford [1, 2], on which estimates for their impact on the operation of present and future magnetic confinement devices are based. Likewise, diagnostics to detect their presence [3] and techniques to limit their impact are operated on the assumption that the fundamental physics of the phenomenon have been clarified. Still, there are experimentally reported examples of magnetic islands showing up in systems that are predicted to be stable against the formation of such structures [4], which indicates otherwise. The work presented here shows how turbulence that develops in the non-linear phase of a high-β system unstable to interchange modes is capable of generating magnetic islands [5, 6] in all conditions explored numerically, and how the dynamics of these turbulence driven magnetic islands (TDMIs) depend on the interaction with the zonal fields. In particular, competition is found to occur between the zonal flow and the magnetic island when it comes to the repartition the free energy of the system, and the growth of the magnetic island and of the zonal current are found to be tightly inter-dependent. Zonal flows and TDMIs can coexist in a stable manner throughout the simulation. The zonal flow is localized on the resonant surface, which is inside the separatrix of the TDMI, where it can maintain the pressure profile relatively stable, and the magnetic island grows on either side of the resonance, depending on geometry and other factors, where it flattens the pressure profile. From an experimental point of view, the effect of the zonal flow hides the presence of the magnetic island until it has reached a width as much as 4 times the critical width identified by Fitzpatrick [7]. These dynamics are studied by running non-linear simulations using a 6-field reduced electromagnetic fluid model [8] varying β and the magnetic shear, as well as the dissipations for the zonal flow. An analytical approach to the problem is also presented to highlight certain fundamental features of the interplay among the large scale structures and turbulence, in particular the fundamental difference in the magnetic and electric fields due to the former being divergence-free.
The project leading to this publication has received funding from the Excellence Initiative of Aix-Marseille University - A*Midex, a French “Investissements d’Avenir” program AMX-19-IET-013. The simulations in this article were run thanks to the support of EUROfusion and MARCONI-Fusion. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. Centre de Calcul Intensif d’Aix-Marseille is acknowledged for granting access to its high performance computing resources.
References:
[1] PH Rutherford. The Physics of Fluids, 16(11):1903–1908, 1973.
[2] H Reimerdes et al. Physical review letters, 88(10):105005, 2002.
[3] FL Waelbroeck. Nuclear Fusion, 49(10):104025, 2009.
[4] A Isayama et al. Plasma and Fusion Research, 8, 2013.
[5] M Yagi. Plasma and Fusion Research, 2:025–025, 2007.
[6] M Muraglia et al. Physical review letters, 107(9):095003, 2011.
[7] R Fitzpatrick. Physics of Plasmas, 2(3):825–838, 1995.
[8] D Villa et al. Journal of Plasma Physics, 88(6):905880613, 2022.
Implementation of an analytical Jacobian in the MEQ free-boundary tokamak equilibrium code suite
See attached PDF.
Recently, a general theoretical framework for the transport of Phase Space Zonal Structures (PSZS) has been developed \cite{zonca2015,falessi2019}. PSZS are the long-lived toroidal symmetric ($n=0$) structures that define the nonlinear equilibrium in the presence of fluctuations such as Alfv\'enic instabilities. In order to include sources and sinks and collisional slowing down processes, a new solver, ATEP-3D was implemented to describe the evolution of the EP distribution in the 3D Constants of Motion (COM) space. It is fully embedded in ITER IMAS framework and combined with the LIGKA/HAGIS codes \cite{LIGKA,HAGIS}. The new development is motivated by the need to use the COM (toroidal canonical momentum $P_\zeta$, energy $E$, and magnetic moment $\mu$) representation in the PSZS transport model.
The Fokker-Planck collision operator represented in the 3D COM space is derived and implemented in the HAGIS code giving orbit-averaged neoclassical transport coefficients.
For solving the PSZS equation including collisions, a finite volume method and the implicit scheme are adopted in the ATEP-3D code for optimized numerical properties. Open boundary conditions that allow the flux to pass through the boundaries without affecting the interior solution are implemented. ATEP-3D allows the analysis of the particle and power balance with sources and sinks in the presence of EP transport induced by Alfv\'enic fluctuations to evaluate the EP confinement properties. The first benchmarks and applications of this new reduced EP transport model in different parameter regimes are presented.
Building on previous work [1, 2, 3], we develop a new set of linear equations to determine the magnetic geometry coefficients needed for local gyrokinetic simulations on a flux surface of interest. The inputs required for the model are the shape of the flux surface, the radial derivative of that shape and four constants. One possible choice for these four constants is the pressure gradient, the gradient of the toroidal flux, and the rotational transform and its radial derivative at the flux surface of interest. When we apply our equations to rational flux surfaces, we find that, for flux surfaces to exist, two conditions must be satisfied. One of the conditions is the well-known Hamada condition [4], but the other has not been discussed in the literature to our knowledge.
References
[1] C.C. Hegna, Phys. Plasmas 7, 3921 (2000).
[2] A.H. Boozer, Phys. Plasmas 9, 3726 (2002).
[3] J. Candy and E.A. Belli, J. Plasma Phys. 81, 905810323 (2015).
[4] S. Hamada, Nucl. Fusion 2, 23 (1962).
Max-Planck-Institut für Plasmaphysik
Within the gyrokinetic formalism [1-2], we present the equations for an explicit treatment of the electromagnetic version of the collisionless Universal/Trapped-Electron, and Microtearing modes, in general geometry. The gradient of the plasma , the ratio of kinetic to magnetic pressure, is taken to be small enough to avoid perturbations of the magnetic field strength [3]. We highlight the role of trapped electrons in the resonant destabilization, or damping, via electromagnetic corrections to ideal Ohms's law, for electron-temperature-gradient driven modes whose frequency relates to the bounce-averaged electron curvature drift. We then investigate the stability properties of maximum-J devices [4] (where J is the second adiabatic invariant) at finite , that is, in the regime in which the maximum-J condition is more likely to be satisfied. Nonlinear energetic arguments will also be given.
References:
[1] W. M. Tang, J. W. Connor, and R. J. Hastie, Nucl. Fusion 20, 1493 (1980)
[2] E. A. Frieman , and L. Chen, Phys. Fluids 25, 502 (1982)
[3] A. Zocco, P. Helander, and J. W. Connor Plasma Phys. Control. Fusion 57 085003 (2015)
[4] P. Helander, Reports on Progress in Physics, 77, 08701 (2014)
Nonlinear inverse bremsstrahlung absorption (NLIBA) of intense electromagnetic waves in homogeneous plasmas may have significant impact on many physical phenomena through modifications of the electron distribution function (EDF). These modifications depend on the relevant parameter $ \alpha=\frac{v_0^2}{v_t^2\ } $, where $v_0 $ is the quiver velocity and $v_t $ is the electron thermal velocity. We address in this work the effects of the NLIBA on electron plasma waves (EPW).
The dispersion relation of the EPW is derived from the perturbed Vlasov-Poisson equations and we obtain for the real part, $\omega^2=\omega_p^2+\mathrm{\Gamma}k^2v_t^2 $ and for the imaginary part, the damping rate, $\gamma=\frac{\pi}{2\sqrt{2}}\frac{\omega^3}{\omega_p^2}\frac{\omega_p^2}{k^2v_t^2}\frac{d\hat{F}}{dx}\left(x=\frac{\omega}{\sqrt{2}k v_t}\right) $ where the different parameters have their usual meaning, $\hat{F}\left(\vec{v},\alpha\right)$ is the normalized reduced anisotropic EDF and $\mathrm{\Gamma}\left(\alpha\right)=12\sqrt{2}\displaystyle\int_{0}^{\infty}{x^2\hat{F}\left(x\right)}\mathrm{d}x $ is the polytropic index. For $\alpha \ll 1$, the plasma is Maxwellian and we recover the classical results of Bohm and Gross, and Landau, i. e., $\mathrm{\Gamma}_{Max}=3 $ and $ \frac{\gamma_{Max}}{\omega}=-\sqrt{\frac{\pi}{8}}\frac{\omega_p^3}{k^3v_t^3}\ \exp\left(-\frac{\omega^2}{2k^2v_t^2}\right)$. Solving numerically the Fokker-Planck equation for homogeneous plasmas in presence of strong laser field we calculated the EDF $ \hat{F}\left(\vec{v},\alpha\right)$ which presents strong temperature anisotropy induced by NLIBA. As a consequence, we found strong modification in the dispersion relation of the EPW. For $\alpha=1$ and $\alpha=2$ the polytropic index is $1.6$ and $2.6$ times greater than in the case of a Maxwellian plasma. The anisotropy effects affect also the damping of the EPW. The results for the damping rate are depicted in figure 1 where we give the damping rate normalized to the frequency of the EPW $\omega$ as a function of the dimensionless parameter $kv_t/\omega$. The dotted line corresponds to $\alpha\ll 1$ (Maxwellian plasma), the solid line corresponds to $\alpha=1$ and the dashed line corresponds to $\alpha=2$. We can note the modification of the damping rate. In particular, within the frequency range where these waves are weakly damped, i.e., $\frac{\omega}{\sqrt{2}kv_t}\gg 1$, we found that the damping is significantly greater for large $\alpha$. In particular for $ \frac{kv_t}{\omega}=0.15$ one obtains $\frac{\gamma}{\omega}=1.46\times {10}^{-5}$ instead of $\frac{\gamma\left(\alpha\ll 1\right)}{\omega}=4.6\times {10}^{-8}$ for Maxwellian plasmas. These changes in dispersion and damping of EPW, especially if $\alpha \sim 1$, should amend the thresholds and the growth rates of parametric instabilities involving the EPW in the coupling modes.
This work has been carried out within the framework of the PRFU 2023-2026 (Projet de Recherche et de Formation Universitaire) and under grant agreement No B00L02UN160420230002.
Magnetic reconnection consists in a modification of magnetic field topology leading to the formation of island-shaped magnetic structures. Magnetic reconnection is ubiquitous in magnetized plasmas. It is found in space plasmas (with the well-known example of sunspots of the solar flares[1]) as well as in fusion plasmas on earth[2].
The idea of the non-conservation of magnetic connectivity during the movement of a plasma emerged over the years[3]. Since then, many works based on theoretical and/or numerical models have given estimates of the growth rate of reconnected structures in disagreement with experimental observations (in space plasma in particular). In fusion plasmas, it is commonly accepted that the collisionality is too low to explain the existence of magnetic reconnection phenomena at large-scales[4] and at small-scales[5].
Thus, magnetic reconnection still raises many open questions. The work presented here falls within the context of hot fusion plasmas and aims to improve the fundamental knowledge about "the life of a magnetic island".
In the literature, studies mainly focus on how a reconnected structure (magnetic island) can grow, the phenomenon at the origin of magnetic reconnection being not distinguished from the phenomenon of growth. This leads generally to the disagreement between theory and experiences. However, there is no fundamental reason that the non-ideal mechanism at the origin of the reconnection is also the one that will allow the island to grow.
Here, in the light of the many works of the last 70 years, a new paradigm for understanding and studying the magnetic reconnection in fusion plasmas is proposed. The life of a magnetic island (whatever its scale) follows 3 phases : the origin, the growth and the saturation. The possible physical mechanisms at play in these 3 phases will be investigated from ionic Larmor radius scale to the large MHD scale. First, for the island origin, typical time scales in link with magnetic reconnection will be computed for 3 tokamaks of different sizes (TCV, WEST and JET) in order to check if magnetic reconnection is such an unexplained phenomenon in fusion plasmas. Second, for the island drive, the richness of possible mechanisms leading to "rapid" magnetic island growth will be presented from small[6] to large scales[7]. Third, comes the island saturation step. Results on the prediction of a large island size at saturation and its impact on transport will be presented.
References
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[3] J.W. Dungey, Phil. Mag. 44, 725 (1953).
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[6] M. Hamed et al., Phys. Plasmas 26, 092506 (2019).
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This talk introduces the application of a Hamiltonian gyrofluid approach to investigate magnetic reconnection and associated secondary instabilities, while considering finite Larmor radius corrections, parallel magnetic field fluctuations, and electron inertia. During the nonlinear development of spontaneous magnetic reconnection, non-collisional current sheets form, characterized by their small thickness similar to the electron skin depth. These sheets can become unstable, leading to the formation of plasmoids and facilitating high reconnection rates. We examine the marginal stability conditions for plasmoid formation in collisionless plasma, specifically focusing on the impact of finite but moderate βe, which has received limited attention previously. Our study shows that plasmoids can be obtained, in this context, from current sheets with an aspect ratio much smaller than in the collisional regime, and that the plasma flow channel of the marginally stable current layers maintains an inverse aspect ratio of 0.1. Our findings are validated through gyrokinetic simulations, demonstrating excellent agreement.
Moreover, we shall present preliminary results obtained in the regime of small βe and hot ions. we investigate the turbulent regime that arises from Kelvin-Helmholtz-like secondary instabilities outside magnetic islands, resulting in the creation of magnetic vortices.
In this work, we use local nonlinear gyrokinetic simulations of tokamaks to demonstrate that turbulent eddies can extend along magnetic field lines for hundreds of poloidal turns when the magnetic shear $\hat{s}$ is very weak or zero [1]. We find that as the magnetic shear is lowered, the parallel eddy length scales like 1/$\hat{s}$. At zero magnetic shear, their length is limited only by critical balance — the distance that electrons can travel along the field line within the lifetime of a turbulent eddy. Turbulent transport from these "ultra long" eddies is substantially impacted by parallel self-interaction, as individual ultra long eddies will often "bite their own tail" [2]. Thus the need for accurately treating field line topology—considering whether a flux surface has an integer, rational, near-rational, or irrational safety factor—becomes crucial. This is achieved by careful selection of the simulation domain length and the phase factor in the parallel boundary condition for $\hat{s}$ = 0 simulations. Consequently, we illustrate that field line topology can lead to transitions between different turbulent modes and fully stabilise Ion Temperature Gradient (ITG) turbulence, both linearly and nonlinearly. Additionally, we observe a novel physical effect termed “poloidal eddy squeezing” — when eddies become ultra long they can cover the full flux surface and, for specific values of the safety factor, strongly interact with themselves in the perpendicular direction. This can squeeze them, reducing their perpendicular size and ability to transport energy, thereby embodying an intriguing new strategy to improve confinement in tokamaks. Lastly, we explore the Internal Transport Barriers (ITBs) formation mechanism. Empirically, very weak or zero $\hat{s}$ has been identified as being one of the key conditions for facilitating ITBs [3]. We present low magnetic shear local gyrokinetic simulations that exhibit weak ITBs caused by the magnetic topology, which may inform a long-standing experimental observation that it is often easier to trigger ITBs where the safety factor has a low-order rational value [4]. Moreover, we use a novel extension of the flux tube model that enables the simulation of non-uniform magnetic shear profiles [5] to examine ITB formation at the safety factor minimum, including safety factor curvature effects. We observe a feedback mechanism of electromagnetic fluctuations on the imposed safety factor profile, occurring together with the formation of an ITB.
References:
[1] Volčokas A et al. 2023 Nuclear Fusion 63 014003
[2] Ball J et al. 2020 Journal of Plasma Physics 86 905860207
[3] Ida K et al. 2018 Plasma Physics and Controlled Fusion 60 033001
[4] Joffrin E et al. 2002 Plasma Physics and Controlled Fusion 44 1739–1752
[5] Ball J et al. 2023 Plasma Physics and Controlled Fusion 65 014004
Over the past three decades, extensive research has been conducted on halo currents to assess the thermal [1] and electromagnetic stresses [2] imposed on the ITER wall during plasma disruptions. However, the contribution of these currents to the generation of sideways forces on the tokamak vacuum vessel still remains poorly understood, which raises concerns [3, 4]. In order to address this critical priority for ITER, the ITPA community initiated a dedicated joint experiment in 2018. While significant progress has been made [4], it has become apparent that for reproducible studies of halo-kink-wall physics a specialized linear plasma device is needed.
Figure 1: The Halo Machine is designed for reproducible studies of kink-wall interaction and related currents.
We propose to study halo- and kink-related wall currents, plasma propulsion [5], and magnetic reconnection [6] in one machine to exploit the synergy between these closely interconnected research topics. Plasma guns generate two plasma current channels embedded in a background magnetic guide field Bz < 0.1 T, with electron temperature Te=5–20 eV, plasma density n=1–3x10^18m-3, column diameter a=5 cm and one-meter length. The diagnostics include fast cameras, electrostatic and magnetic sensors, and electron/ion energy analyzers. Moreover, the currents in hexagonal plasma facing components (PFCs) are measured with recently developed probes [7]. The design of PFC arrangement is supported by SPICE3 and BIT3 modelling.
References:
[1] M. Lehnen et al., Nucl. Fusion 53 093007 (2013)
[2] F. J. Artola et al., Nucl. Fusion 62 056023 (2022)
[3] R. Roccella et al., Nucl. Fusion 56 106010 (2016)
[4] M. Hron et al., Nucl. Fusion 62 042021 (2022)
[5] M. Zuin et al., Phys. Rev. Lett. 92 225003 (2004)
[6] T. P. Intrator et al., Nature Physics 5 521 (2009)
[7] F. Villone et al., “Design and experimental validation of an eddy currents probe”, 44th EPS Conference on Plasma Physics (Belfast, Northern Ireland), P1.106 (2017)
The performance of magnetic-confinement-fusion devices is often limited by the presence of turbulent fluctuations that lead to enhanced transport and energy losses. Both experimental and numerical investigations have shown that the turbulent fluctuations, and thus the transport properties of magnetised plasma, are influenced by the presence of sheared flows [1, 2, 3, 4]. Such flows can be either externally imposed or internally driven by the plasma itself. Here, we consider the effects of mean perpendicular flow shear on magnetised-plasma turbulence driven by microinstabilities. Assuming a saturated state with well-defined outer (energy-injection) scale and energy cascade [5], we derive scaling laws for the outer-scale wavenumbers and fluctuation amplitudes as functions of the imposed flow shear. In the case of anisotropic, 'streamer-dominated' turbulence, we find that flow shear can lead to significant suppression of turbulent transport at levels much lower than the rate of energy injection (i.e., than the growth rate of microinstabilities). Our theoretical predictions are confirmed by numerical results from both electrostatic gyrokinetic simulations and an electrostatic fluid model [6]. Finally, we discuss the consequences of our results for the cross-scale interactions between ion-scale flows and electron-scale turbulence.
References
[1] K. Burrell, Phys. Plasmas 4, 1499 (1997)
[2] R. Waltz et al., Phys. Plasmas 5, 1784 (1998)
[3] C. Roach et al., Plasma Phys. Control. Fusion 51, 124020 (2009)
[4] E. G. Highcock et al., Phys. Rev. Lett. 105, 215003 (2010)
[5] M. Barnes et al., Phys. Rev. Lett. 107, 115003 (2011)
[6] T. Adkins et al., J. Plasma Phys., accepted, preprint arXiv:2303.14834 (2023)
The ST40 tokamak [1], built and operated by Tokamak Energy, is a high field spherical tokamak (ST), B=2.2T at R0=0.4 – 0.5m, A=1.6 – 1.9 with a mission to extend the ST reactor physics basis [2]. Overall energy confinement in STs is largely determined by turbulent electron heat transport, with ion thermal conductivity close to neoclassical levels in higher collisionality H-modes. Electron confinement is found to increase with decreasing collisionality with evidence from Li wall conditioned plasmas in NSTX [3]. Experiments carried out in ST40, without Li and in future campaigns with Li, complement those of NSTX and allow to study electron confinement and turbulent heat transport scaling across a wide range of collisionalities accessible in the two devices. Dedicated parametric scans of both dimensional (B and Ip) and dimensionless (collisionality) parameters have been performed for both hot ion mode limited and diverted plasmas. The results of the scans have been used as the basis for gyrokinetic studies to understand the underlying microinstabilities driving transport, and for energetic particle transport studies. Similarity pulses with different impurity concentrations but the same line average electron density have been performed to provide information on the effect of impurities on energy confinement. Finally, a study to compare the ST40 operational regime with that of GLOBUS-M2 has been performed with the aim to extend the GLOBUS-M2 confinement scaling [4] to higher magnetic field and confirm the linear dependence of confinement on toroidal magnetic field which appears to characterize spherical tokamaks.
Until recently, theoretical studies in support of the commercial realization of fusion energy mostly focused on research in support of ITER and on the analysis of plasmas in existing tokamaks and stellarators, far from ignition conditions. Recent appearance of privately funded research has advanced alternative concepts, in particular the development of high-beta spherical configuration and high magnetic field tokamaks. Simultaneously with the development of technological and experimental research, the privately funded fusion industry is significantly contributing to the advancement of theoretical fusion physics.
At Tokamak Energy Ltd we are pursuing an alternative concept for the production of fusion power based on compact high field Spherical Tokamaks which offer a number of advantages discussed in this proposed presentation. We will present recent results from the high-field spherical tokamak ST40, where high ion temperatures Ti ~ 10 keV have been achieved at low plasma collisionality and plasma temperature required for burning conditions [1]. We will also present results of detailed transport analysis, carried out to benchmark simulations with the aim to extend studies to the burning and ignited plasma conditions [more details in 6].
We will review theoretical basis/grounds for a spherical tokamak-reactor, including studies of macro and micro instabilities, operational limits and disruptions, plasma confinement, neoclassical and turbulent plasma transport, optimization of magnetic confinement devices, divertor and edge physics. All these effects have significant differences in the ST geometry respect to conventional tokamaks, as detailed in this contribution.
Recent revisions of the Lawson criterium from necessary value of nTdttauE to that for nTdttauDT, where taudt is the DT burn time [2] results in narrowing the range of the needed plasma temperatures and avoids the need for a significant improvement of the energy confinement for the sustainment of a DT fusion burn. This restricts the choice of operational regimes and operation range of reactors and improved criterion equations of scientific breakeven, ignition, and engineering breakeven should be considered [3]. Both revisions are urgently needed for the ST path to Fusion that may have significant advantages [4] and we will discuss this in detail.
Three types of fluctuations can increase the particle transport and so affect this and as a result, ignition required conditions: fluctuations in the electric potential, in the locations of the magnetic field lines (for example due to ergodisation), and in the toroidal electric field, which may locally break the magnetic surfaces. These microinstabilities have significant aspects in the ST geometry and will be discussed. One of new regimes that require theoretical investigation of such instabilities is the hot ion mode (or FIRE mode on KSTAR [4] and a range of supershot, PEP and similar regimes). Ion temperatures required for reactor have been achieved on ST40 in the hot ion mode, as was predicted for ST40 [7].
References:
[1] S.A.M. McNamara et al Nucl. Fusion 63 054002 (2023) DOI 10.1088/1741-4326/acbec8
[2] A Boozer, Physics of Plasmas 30, 062503 (2023)
[3] S Entler et al, Nuclear Engineering and Technology 55 2687-2696 (2023)
[4] R Stambaugh Fusion Technol. 33 1 (1998)
[5] Yong-Su Na et al, Research Square, September 2021 DOI:10.21203/rs.3.rs-935325/v1
[6] M Romanelli, this conference
[7] A Yu Dnestrovskij, et al,. Plasma Phys. Control. Fusion 61 (2019) 055009